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| How do you figure things out? | 1537 ways |
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| Choose from the domains above. Write to Andrius () to add more ways! |
Research & Culture & Business |
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Jesus
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Jesus . I'm studying the Gospel for ways that Jesus figured things out. He understands scriptural revelation. He makes sure to let people keep their freedom, by performing miracles and speaking in parables. He has expectations and emotional responses. He accepts people's hearts. Consider the decisions that Jesus made. What system do they form?  My diagram of Jesus' mind Here is my sketch of Jesus's mind based on the ways that he figured things out. See also the diagram above. He was very concerned not to cause others to stumble, and hoped he didn't cause John the Baptist to stumble, but ultimately, he caused his disciples to stumble. He spoke of two gates, the narrow to life and the wide to destruction. He urged people to repent and enter the Kingdom of Heaven, and he spoke to their belief, and some people did believe, and some were ready to die for him. He hoped that the Kingdom of Heaven could be thus established. Yet he accepted God's will to save all of the world, not just some, and so he accepted death and resurrection. So some people accepted him as he was, alive, and others dead, as they wished him, yet resurrected. And so the two branches are made one. Martha gave a great example of the practical work of the courts where the judge ignored the law to do what was right, where the judge gave a woman the right to parent a deaf child because she and her daughters had learned sign language and engaged him whereas his biological father had not. And so if there is nobody to dispute a ruling, then there is healing, there is justice in the most important sense. But if there is somebody to dispute it, then it can go up the system, and the laws can be worked out. Or as in the case of Solomon and the baby, it can become clear, through the process, what was not clear before it, such as who was the real mother. So the law supports a culture but ultimately it must be an internally based culture, and that is the "way", the truth and the life, the branch that Jesus offers, the "gate", the "light". Please crowdfund my philosophical portrait of Jesus! Thank you for contributing money and for spreading the word! |
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Math
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Paul Zeitz, I share with you my thoughts on the varieties of "deep structure" in mathematical "frames of mind". Your book has been profoundly helpful. I also share with Joanne Simpson Groaney ("Mathematics in Daily Life"), Alan Schoenfeld ("Learning to Think Mathematically..."), John Mason ("Thinking Mathematically"), Manuel Santos, and also Maria Druojkova and the where I am active. I have been looking for the "deep ideas" in mathematics. George Polya's book "Mathematical Discovery" documents four patterns (Two Loci, the Cartesian pattern, recursion, superposition) of the kind I'm looking for (and which bring to mind architect Christopher Alexander's pattern languages). Your book documents dozens more. I've found Joanne Groaney's book helpful and I think the other writings I mention will also be in this regard. You note in your "planet problem", pg.63, that "on the surface" it is a nasty geometrical problem but "at its core" it is an elegant logical problem. This distinction brings to mind linguist Noah Chomsky's distinction between the surface structure and the deep structure of a sentence. In general, what might that deep structure look like? George Polya ends his discussion of the pattern of "superposition" or "linear combination" to say that it imposes a vector space. In an example he gives, the problem of "finding a polynomial curve that interpolates N points in the plane" is solved by "discovering a set of particular solutions which are a basis for a vector space of linear combinations of them". The surface problem has a deep solution, and the deep solution is a mathematical structure! I list below 24 such deep structures which characterize the mathematical "frames of mind" by which we solve problems. I note in parentheses the related patterns, strategies, tactics, tools, ideas or problems. I have included every such that I have found in your book, as well as Polya's four patterns, "total order" and "weighted average" that I observed in Joanne Growney's book, and a few more that I know of. Please start with the illustrative example of constructing an equilateral triangle. |
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Gamestorming
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 Gamestorming |
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Andrius Kulikauskas's ways of figuring things out
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 Andrius Kulikauskas's ways of figuring things out How can we overview all of these ways of figuring things out with a general method? Here is a brief overview:
The goal of learning is to have the fullness of mind needed to truly obey. On the one hand, it starts with caring (being open to all outside us) and then recursively caring about caring (growing our mind), caring about caring about caring (recognizing our limits), caring about caring about caring about caring (acknowledging what transcends our limits). On the other hand, it starts with believing (referencing the absolute within us), believing in believing (allowing for others), believing in believing in believing (empathizing with their views), believing in believing in believing in believing (fostering the spirit amongst us). This growth in awareness is fueled by "relative learning" on specific matters through a cycle of taking a stand, following through and reflecting. That is essentially the scientific method, but we may apply it subjectively. We can realize that we are vulnerable to evil which may lead us astray in our conclusions. As our caring and believing unfold, we allow for a Person, a person-in-general or simply Jesus, who does what any good person would do, so that Caring about One is the same as Believing in One. Then we live that Person in four ways: Knowing Whether-What-How-Why One Knows. We thereby relate to ourselves in every way as Believers and Carers. Attention goes from whoever believes to whatever they care about in six ways, for example, Believer in Why caring about Whether. These pair ways of giving attention (why:concentrating, how:distinguishing, what:tracking) and what we give attention to (whether:whole, what:part, how:structure). These six ways have us think in terms of a conceptual language of absolute truth. The six ways are united in the perspective that believing is wider than caring, and this common outlook is God's perspective, which we take up by obeying. That's my summary of the big picture and I'm working to flesh it out. More simply, I have trouble making the most of life here and now. There is so much temptation to tune myself out and shut myself down. It's beyond my will to care to live forever. Yet there may be a God who loves me more than I love myself, wants me to be alive, sensitive, responsive more than I do. Such a God wants me to live forever, and grow forever, thus to learn forever. That is the will of God. How might I follow the will of God? I could simply obey. But I fear losing my freedom. Instead, I could believe one who does obey, such as Jesus, and follow his path and thus end up where he ends up. But there are many good reasons not to believe. If I have grown up believing something else, then I should be able to be true to my belief, whatever it is, for otherwise I'm not believing absolutely. Instead of obeying or believing, I claim that it is enough to care. If we care, then we will appreciate the success of the one who believes, and we will walk with them and end up in the same place. But our path will likely be much longer and harder. So be it. As people who care, we can share a culture of truth where, in each little domain, we can take our little leap of faith, and exercise our particular way of figuring things out. Through the integrity of those ways we can all vouch for the truth and participate in a culture of absolute knowledge as evident in our figuring things out so that all truth is available and tangible. As we practice this, and as we grow familiar with God who we engage in each particular way, then we can dare to try out many other ways, so that God and truth become absolutely real. Thus we appreciate the sense of believing, and even more, the sense in obeying. Yet we also see that forever we can learn by engaging others in the particular domains where they feel safe and learning along with them to grow beyond that. And so we all grow as one in the truth. As I document ways of figuring things out, I am considering how each way is structured internally.
Andrius's quest to know everything My ten minute video summary of my quest to know everything that I made in August, 2009. |
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Taxonomy
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Taxonomy I share below my response to a letter asking for how to create a taxonomy of anecdotes. I describe the House of Knowledge as a general framework for creating such taxonomies in any field. For example, I've written out almost 100 anecdotes about "my money mind", episodes in my life that informed my thoughts about money. I recently realized that each of them is informing a particular "way of figuring things out". And I've developed a system (a House of Knowledge) for categorizing those ways into 24 frames of mind (or rooms). Dave Gray found it quite readable, see here. In general, episodes are of the following types. You can think of them as dialogues with an abstract person or quality that becomes more and more concrete as this unfolds. Avoiding what would keep us from learning:
 Bob Lichtenbert, a Philosophical Portrait 2011.05 Andrius Kulikauskas: I'm creating a philosophical portrait of Bob Lichtenbert.  Andrius Kulikauskas, 2011.05.11: Pamela McLean of agreed that I draw her philosophical portrait by documenting and sharing her ways of figuring things out. She's a pioneer! Thank you, Pamela! |
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Morgan D. Jones of The Thinker's Toolkit
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Morgan D. Jones of The Thinker's Toolkit Morgan D. Jones is the author of He teaches ways of structuring the analysis of problems so that we don't jump to conclusions. He leads Thinker's Workshop, helped found Analytic Prowess, writes fiction, and taught analytic methods in Georgetown University's Master of Science Foreign Service graduate program. He worked in the C.I.A. for 36 years, including as an intelligence analyst of Soviet military affairs and chief of the Analytic Training Branch. I take his work to be representative of how the C.I.A. and other intelligence agencies analyze issues. Morgan D. Jones does a great job surveying the analytic techniques and providing intuition on how and where to use them. I'm especially interested in discovering how his analytic techniques relate to my ways of organizing, visualizing information. In his book, he considers applications in business and every day life. I think they also reflect on the analytical mindset at the C.I.A. where he learned, worked and taught. By his own logic, I think it's important to appreciate that bias. I suppose it also make sense to supplement them with espionage techniques, often clandestine, for gathering facts, as well as principles for finding moral bearings or not. |
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Making a Living
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Making a Living I apply the "ways of figuring things out" to my challenge of making a living. I draw on 60 anecdotes that I wrote up in 2006 about my "money mind". I thought this through in writing a letter to DaveGray, JayCross and others. We can start with: '''Working'''. Don't be idle, unproductive or destructive. Then there are two independent threads, Giving and Taking.
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Dee Guyton's ways of figuring things out
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 Dee Guyton's ways of figuring things out See my 40 minute interview of choir director Dee Guyton about ways she's figured things out in her life, faith and music. |
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Physics
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Physics I'm systematizing ways of figuring things out in physics. I have a B.A. in Physics from the University of Chicago. I'm grateful for conversations with Thomas Gajdosik and John Harland. Overall, in physics, we can divide the ways of figuring things out into experimental physics and theoretical physics. Experimental physics takes place "in the real world", which is to say, not divorced from the entirety of the universe. It is thus pre-systemic. We find ourselves thrust in an existing world that we are trying to make sense of. Theoretical physics models a subsystem of the world. It is systemic. It starts from scratch, as if we were God. Sources of examples include the Experiments, Science experiments, Physics experiments, List of experiments and Timeline of scientific experiments pages at Wikipedia. |
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Independent trials
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Independent trials |
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Avoiding evil
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Avoiding evil Avoiding what would keep us from learning. Avoiding evil, not shutting ourselves down |
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Measurement
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Measurement We can always start fresh with a new measurement. Each measurement assumes a partial view, an interest in some part of the system. We don't need a complete description, but rather we tease out whatever part of reality that we are interested, even though it is dubious in the big picture, yet our point of view (say, particle or wave) can be successful, even though incomplete. Yet therefore we need to keep working with independent measurements. Analogously, in math we can start fresh with a new piece of paper, or in life we can give a person a new chance. |
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Don't cause believers to stumble
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Don't cause believers to stumble Jesus warns us not to cause others to stumble. He is concerned that he himself not cause others (such as John the Baptist) to stumble. Ultimately, he causes his disciples to stumble. Paul says the law of righteousness is the stumbling stone. Whereas Peter says that Jesus is the stumbling stone. God is the one who causes Israel to stumble. For it wouldn't learn from being treated nicely, thus it must learn from being treated meanly. Thus God is a teacher and Israel may come to understand. But it is better for Israel, then it is for Assyria, which is God's instrument of stumbling. Woe, likewise, to Satan and all instruments of stumbling. Whereas those who God makes stumble may yet learn. And those who believe are those who accept some injustice because they know that justice will come, that the cornerstone is being laid. Don't cause the little ones who believe to stumble Mark 9:38 John said to him, "Teacher, we saw someone who doesn't follow us casting out demons in your name; and we forbade him, because he doesn't follow us." 9:39 But Jesus said, "Don't forbid him, for there is no one who will do a mighty work in my name, and be able quickly to speak evil of me. 9:40 For whoever is not against us is on our side. 9:41 For whoever will give you a cup of water to drink in my name, because you are Christ's, most certainly I tell you, he will in no way lose his reward. 9:42 Whoever will cause one of these little ones who believe in me to stumble, it would be better for him if he were thrown into the sea with a millstone hung around his neck. 9:43 If your hand causes you to stumble, cut it off. It is better for you to enter into life maimed, rather than having your two hands to go into Gehenna, into the unquenchable fire, 9:44 'where their worm doesn't die, and the fire is not quenched.' 9:45 If your foot causes you to stumble, cut it off. It is better for you to enter into life lame, rather than having your two feet to be cast into Gehenna, into the fire that will never be quenchedâ€" 9:46 'where their worm doesn't die, and the fire is not quenched.' 9:47 If your eye causes you to stumble, cast it out. It is better for you to enter into the Kingdom of God with one eye, rather than having two eyes to be cast into the Gehenna of fire, 9:48 'where their worm doesn't die, and the fire is not quenched.'* 9:49 For everyone will be salted with fire, and every sacrifice will be seasoned with salt. 9:50 Salt is good, but if the salt has lost its saltiness, with what will you season it? Have salt in yourselves, and be at peace with one another." Keep forgiving so they do not stumble Luke 17:1 He said to the disciples, "It is impossible that no occasions of stumbling should come, but woe to him through whom they come! 17:2 It would be better for him if a millstone were hung around his neck, and he were thrown into the sea, rather than that he should cause one of these little ones to stumble. 17:3 Be careful. If your brother sins against you, rebuke him. If he repents, forgive him. 17:4 If he sins against you seven times in the day, and seven times returns, saying, 'I repent,' you shall forgive him." Are you embracing what's right? or keeping others from embracing it? Matthew 23:14 "But woe to you, scribes and Pharisees, hypocrites! Because you shut up the Kingdom of Heaven against men; for you don't enter in yourselves, neither do you allow those who are entering in to enter. Be sure to not cause others to stumble Matthew 17:24 When they had come to Capernaum, those who collected the didrachma coins came to Peter, and said, "Doesn't your teacher pay the didrachma?" 17:25 He said, "Yes." When he came into the house, Jesus anticipated him, saying, "What do you think, Simon? From whom do the kings of the earth receive toll or tribute? From their children, or from strangers?" 17:26 Peter said to him, "From strangers." Jesus said to him, "Therefore the children are exempt. 17:27 But, lest we cause them to stumble, go to the sea, cast a hook, and take up the first fish that comes up. When you have opened its mouth, you will find a stater coin. Take that, and give it to them for me and you." Get behind me, Satan! Matthew 4:8 Again, the devil took him to an exceedingly high mountain, and showed him all the kingdoms of the world, and their glory. 4:9 He said to him, "I will give you all of these things, if you will fall down and worship me." 4:10 Then Jesus said to him, "Get behind me, Satan! For it is written, 'You shall worship the Lord your God, and you shall serve him only.'" Get behind me, Satan! Jesus rebukes Peter Matthew 16:21 From that time, Jesus began to show his disciples that he must go to Jerusalem and suffer many things from the elders, chief priests, and scribes, and be killed, and the third day be raised up. 16:22 Peter took him aside, and began to rebuke him, saying, "Far be it from you, Lord! This will never be done to you." 16:23 But he turned, and said to Peter, "Get behind me, Satan! You are a stumbling block to me, for you are not setting your mind on the things of God, but on the things of men." 16:24 Then Jesus said to his disciples, "If anyone desires to come after me, let him deny himself, and take up his cross, and follow me. 16:25 For whoever desires to save his life will lose it, and whoever will lose his life for my sake will find it. 16:26 For what will it profit a man, if he gains the whole world, and forfeits his life? Or what will a man give in exchange for his life? Is Jesus causing others to stumble? Jesus' is concerned that he himself not be the cause of others, such as John, to stumble. Matthew 11:2 Now when John heard in the prison the works of Christ, he sent two of his disciples 11:3 and said to him, "Are you he who comes, or should we look for another?" 11:4 Jesus answered them, "Go and tell John the things which you hear and see: 11:5 the blind receive their sight, the lame walk, the lepers are cleansed, the deaf hear, the dead are raised up, and the poor have good news preached to them. 11:6 Blessed is he who finds no occasion for stumbling in me." Jesus' prophesies can keep the disciples from stumbling John 16:1 These things have I spoken to you, so that you wouldn't be caused to stumble. Jesus' words cause them to have sin John 15:22 If I had not come and spoken to them, they would not have had sin; but now they have no excuse for their sin. Whoever causes others to stumble will be destroyed 18:7 "Woe to the world because of occasions of stumbling! For it must be that the occasions come, but woe to that person through whom the occasion comes! 18:8 If your hand or your foot causes you to stumble, cut it off, and cast it from you. It is better for you to enter into life maimed or crippled, rather than having two hands or two feet to be cast into the eternal fire. 18:9 If your eye causes you to stumble, pluck it out, and cast it from you. It is better for you to enter into life with one eye, rather than having two eyes to be cast into the Gehenna of fire. |
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Blank sheets
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Blank sheets Independent trials. We may think of our mind as blank sheets, as many as we might need for our work. We shouldn't get stuck, but keep trying something new, if necessary, keep getting out a blank sheet. We can work separately on different parts of a problem. This relates also to independent events (in probability), independent runs (in automata theory) and independent dimensions (in vector spaces). If something works well, then we should try it out in a different domain. Sarunas Raudys notes that we must add a bit of noise so that we don't overlearn. Avoid error-prone activity Simplify .... We could multiply out all the terms, but it would take a long time, and we'd probably make a mistake. We need a strategy. pg.166 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Get your hands dirty So we try another strategy, one of the best for beginning just about any problem: get your hands dirty. We try plugging in some numbers to experiment. If we are lucky, we may see a pattern. ... This is easy and fun to do. Stay loose and experiment. Plug in lots of numbers. Keep playing around until you see a pattern. Then play around some more, and try to figure out why the pattern you see is happening. It is a well-kept secret that much high-level mathematical research is the result of low-tech "plug and chug" methods. The great Carl Gauss ... was a big fan of this method. In one investigation, he painstakingly computed the number of integer solutions to x**2+y**2<=90,000. ... Don't skimp on experimentation! Keep messing around until you think you understand what is going on. Then mess around some more. pg.7, 30, 36 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Knowing when to give up Sometimes you just cannot solve a problem. You will have to give up, at least temporarily. All good problem solvers will occasionally admit defeat. An important part of the problem solver's art is knowing when to give up. pg.16, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Mental toughness, confidence and concentration But most beginners give up too soon, because they lack the mental toughness attributes of confidence and concentration. It is hard to work on a problem if you don't believe that you can solve it, and it is impossible to keep working past your "frustration threshold". ... You build upon your preexisting confidence by working at first on "easy" problems, where "easy" means that *you* can solve it after expending a modest effort. ... then work on harder and harder problems that continually challenge and stretch you to the limit ... Eventually, you will be able to work for hours single-mindedly on a problem, and keep other problems simmering on your mental backburner for days or weeks. ... developing mental toughness takes time, and maintaining it is a lifetime task. But what could be more fun than thinking about challenging problems as often as possible? pg.16, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Practice Practice by working on lots and lots and lots of problems. Solving them is not as important. It is very healthy to have several unsolved problems banging around your conscious and unconscious mind. pg.25, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Toughen up Toughen up by gradually increasing the amount and difficulty of your problem solving work. pg.24, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Vary the trials The mouse in the trap ... threw himself violently against the bars, now on this side and then on the other, and in the last moment he succeeded in squeezing himself through ... We must try and try again until eventually we recognize the slight difference between the various openings on which everything depends. We must vary our trials so that we may explore all sides of the problem. Indeed, we cannot know in advance on which side is the only practicable opening where we can squeeze through. The fundamental method of mice and men is the same; to try, try again, and to vary the trials so that we do not miss the few favorable possibilities ... a man can vary his trials more and learn more from the failure of his trials than the mouse. pg.16, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc., quoting "Mice and Men" by George Polya, Mathematical Discovery, Volume II, 1965. |
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Avoiding Evil
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Avoiding Evil Avoiding evil. Acknowledge what is evil or futile and stay away from it. Allow for real learning Appreciate futility What must it take for me to know everything? Conversing with: redirection Growing up, and seeking to know everything, I used to think that I could know human affairs from political science, which I could know from sociology, which I could know from psychology, which I could know from biology, which I could know from chemistry, which I could know from physics, which is based on mathematics. I was very interested to take high school physics and learn about the fundamental constituents of the universe. But the small bit that I surmised was that at the quantum level, reality itself seems to fade away, as if to say, you aren't meant to find anything here! And at that time, in the 1980's, it seemed that it would take billions of dollars of equipment and hundreds of scientists just to participate in the smallest way. So I concluded that if I am to know everything, God must make that possible. Consider what others avoid knowing What don't people want to know? I decided that if I am to possibly know everything, then God must make that knowledge easy to find and intuit. But others have not yet found it. So it must be a place where people could look, but choose not to. I surmised that it must be the wisdom of life, for there is no science of how to live our lives. Few people seem to wish that there be such a science. Who wants to listen to God? In 1995, just as I was preparing to engage others to make my philosophy practical, and so developing the good will exercises, I was inspired by Kangning Liou, who would pray for an hour every morning. I thought I should certainly pray more, but especially if I am to lead and even influence people, I need to be able to converse with God and listen to God. I thought that if this was possible for me, then it should be possible for others as well. Yet few people seem to hear God. Then I realized that few people want to hear God for they would then be accountable to obey him. Hang out with the disobedient What won't I know? Conversing with: conditional rightness I was a good kid, but would charge myself to hang out with disobedient or rebellious kids, to know about their life, to round out my knowledge. I learned what was meaningful about being "cool". Have rapid cycles of learning When I started my laboratory, Minciu Sodas, in 1998, I was living month-to-month, even week-to-week, not certain what I would try next. I think I learned a lot this way because I had very rapid and flexible cycles of learning. I had long term rhythms in my life for what I knew was important, but a snappy cycle for responding to opportunities. Live closer to real life What do people care about? After getting my Ph.D., I chose not to pursue an academic career because I wished to be closer to "real life". I moved to my grandmother's house and lived with her in a Chicago neighborhood, Marquette Park, which had "changed over" and gangs of Black-American youth were establishing themselves. Live precariously if that is more fruitful Conversing with: spiritual stagnation I grew up in a Southern Californian suburban utopia, comfortably, in a loving family. But I learned to "live on top of a question mark", as some of us say in Lithuanian. Live without assurance of income or in a neighborhood that others might not consider safe. Suppose that God will make things doable As a high school student I came to think that physics at the quantum level simply fades the way, and so is not a foundation for knowledge of life. So I concluded that if I am to know everything, God must make that possible. The truth must be at hand, easy for me to find, as if I was to search for a lost key at the foot of a lamp post, where the light happens to be best. Avoid real harm Avoid pointless debate Conversing with: true knowledge I have lost interest in debate because I have seen it driven primarily by people's ego and desire to "win", without an interest in appreciating, including and reconciling with other points of view, or discovering the essence or the truth, as if without a belief that it truly existed and was there to be found. At best, it is a waste of time, but it can also suck us into fruitless "hot button" topics. In watching debate, I note the clash of perspectives, as structured by divisions of everything, but I find it more productive to consider these clashes more calmly and starkly. Avoiding such debate, I have discovered ways to make progress, such as investigations, focusing on what we don't know but would like to know, rather than what we already think we know. Avoid whatever is unsafe Conversing with: possible misfortune As a child, I was always extra careful to stay away from anything that might injure me, such as saws and machinery. I still am. I don't play with danger. Hand evil over to Jesus Conversing with: lack of necessity I rarely have thoughts of the devil, but when I do, I simply say that Jesus has already defeated the devil, and I push them aside. Ignore the devil I ignore the devil because I want to live positively, I want to grow positive, and I don't think the devil is relevant. I thereby learn that the devil isn't relevant, at least not for me to live and grow and give fruit forever. Not to destroy what I treasured In 1992, I was very sad because the woman I had loved with all my heart had returned home to Lithuania and married a man she loved. I had so many feelings for her. Indeed, I stayed in love with her for about seventeen years. Early on, I thought, maybe I should kill my feelings for her. But I refused. They were my honest feelings that I felt for her. I wasn't going to dampen them. I wasn't going to look for her faults. I wasn't going to diminish myself, who I was and who I could be. I stayed true to them. Stay away from what can harm but can't be necessary I stay away from drugs and am not curious about them. I would tell myself that they can't be necessary for my quest and so they can only hurt me. I stay away from those who want to be degenerate, who want to descend into ruin, who want to involve me in their dysfunctionality. |
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Not shutting ourselves down
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Not shutting ourselves down Avoid depending on a single method Wikipedia: According to Badiou, philosophy is suspended from four conditions (art, love, politics, and science), each of them fully independent "truth procedures." (For Badiou's notion of truth procedures, see below.) Badiou consistently maintains throughout his work (but most systematically in Manifesto for Philosophy) that philosophy must avoid the temptation to suture itself (that is, to hand over its entire intellectual effort) to any of these independent truth procedures. When philosophy does suture itself to one of its conditions (and Badiou argues that the history of philosophy during the nineteenth and twentieth centuries is primarily a history of sutures), what results is a philosophical "disaster." Do not have contempt prior to investigation "There is a principle which is a bar against all information, which is proof against all arguments and which cannot fail to keep a man in everlasting ignorance - that principle is contempt prior to investigation." --Herbert Spencer Not rushing up from being knocked down Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: When you get knocked down, the first thing you do is try to compose yourself and not rush up. Don't jump up because of embarrassment because your equilibrium is still off. And if you jump up too fast, you're going to stumble. You're going to fall back. The key is to hopefully be near a rope which you can use as a brace to help you up. Stories of people who ended badly Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: "I had no intentions whatsoever to become a professional fighter because I had heard horror stories about former boxers who made money but, in the end, ended up with nothing. I didn't want to be one of those guys." Try different lines of questioning Her blood pressure was through the roof. I leafed through the chart and noted it had been controlled in the past. I began to dig to see if I could identify a cause. "How do you feel?" I probed. "All right," she mumbled with her Arkansas accent. "Are you having any chest pain or problems breathing?" "No suh." I tried another line of questioning before moving on. "Have you had any recent stress in your life?" Jackpot. Her eyes welled. Her voice remained emotionless. "My gran-chillin, got kilt. On my fron poich," she said. ... She cowered in panic on the other side of the door, inches from her grandchildren. "Ah was a-scared to open it. Ah din know it were them. Ah din know it were them," she repeated. ... Her blood pressure shot up. And remained high two weeks later. If I had not asked, she would not have told me. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.1934 |
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Don't be content with invalid explanations
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We feel the need to find explanations for everything, regardless of whether the explanations are accurate. ... Explanations, by making sense of an uncertain world, apparently render the circumstances of life more predictable and thus diminish our anxieties about what the future may hold. ... The compulsion to explain everything drives our curiosity and our thirst for knowledge of the world. ... Knowing - finding an explanation for an event - is one of the most satisfying of human experiences. There is great comfort in recognizing and making sense out of the world. Doing so creates order and coherence, and, where there is order, there is safety and contentment. We are instantly aware of the loss of this inner feeling of safety and contentment the moment we don't recognize a pattern in a situation that confronts us. ... Unfortunately, our compulsion to explain things can ... get us in trouble. When presented with an event that has no particular meaning, we find one anyway, and we subconsciously don't care whether the explanation is valid. ... the explanations we give for things don't have to be true to satisfy our compulsion to explain things. ... being satisfied, we are then blithely content to move on to something else without seriously questioning our explanation. This is one of the reasons we humans don't give sufficient consideration to alternatives ... the principal cause of faulty analysis. drawing from The Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich. Subconscious mental activity continuously influences our analysis and even determines our conclusions. ... Most of what goes on in the mind involves "mental shortcuts" that occur without our knowledge and beyond our conscious control. ... We cannot "teach" the mind how to work; it works as it works, and taking shortcuts is one of its ways. These shortcuts are beyond our conscious control. ... And where those shortcuts lead our thinking is anyone's guess. drawing from The Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich. Inflating or deflating probabilities to conform to our desires is the curse of wishful thinking: it won't happen to me because I don't want it to happen. Information is vivid because it was acquired either traumatically or recently and thus has made a strong impression on our memory. Information that is vivid is therefore more easily remembered than pallid, abstract information and, for that reason, has greater influence on our thinking. This information can far exceed what it actually deserves. drawing from The Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich. Another kind of patterning is the tendency of the human mind to look for cause-and-effect relationships. ... We seem to view the world in terms of cause and effect, and we somehow instinctively know the difference between the two concepts. ... So we by nature strive to know why something has happened, is happening, or will happen and what the result was, is, or will be. drawing from The Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich. We are driven to view the world around us in terms of patterns (especially cause and effect). The human mind instinctively views the world in terms of patterns, which it recognizes based on memories of past experiences. ... We see the patterns in situations ... in sequences of events ... our compulsion to see patterns can easily mislead us when we analyze problems. ... patterns can be captivating ... the mind can also easily miscontrue random events as nonrandom, perceiving a pattern where, in fact, none exists. ... when we want to see a particular pattern or expect to see it, or have become accustomed to seeing it, not only do we fill in missing information but our brain edits out features that don't fit the desired or familiar pattern. drawing from The Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich. Sometimes, when I offer to show people how to structure their analysis of problems, they say, "Sorry, but I'm awfully busy." "No, thank you. We don't use things like that here." "Very interesting, but it sounds a bit arcane." "I wouldn't think of using such Byzantine methods." The rejection is so spontaneous and emphatic that it could only be a mental knee-jerk reflex, not a reasoned response. What could be causing the unreasoning reflex? Because they have not learned and understood the techniques, they could not have performed any cost-benefit analysis on them. They therefore have not assimilated any data on which to base a reasoned judgment. The only conclusion one can draw is that their reactions are purely intuitive, meaning they come from the subconscious mind. And why would the subconscious mind reject these techniques out of hand? Because structuring one's analysis is fundamentally at odds with the way the human mind is accustomed to work. ... if we structure our analysis, the mind can't be free to mess around. That's why the subconscious rebels when it's asked to structure its thinking. We commonly begin our analysis of a problem by formulating our conclusions; we thus start at what should be the end of the analytic process. In keeping with our troublesome instinctive mental traits, we first select the scenario we intuitively believe is most likely correct ... Next, we look for evidence to support that scenario ... we become more and more convinced of our hypothesis and more and more resistant to, and contemptuous of, contradictory evidence and the alternative scenarios such evidence supports. Over the past several decades cognitive science has discovered that we humans are unknowingly victimized by instinctive mental traits that defeat creative, objective, comprehensive, and accurate analysis. As a result, we unwittingly, repeatedly, habitually commit a variety of analytical sins. He lists five analytic sins, which I think relate to the counterquestions, which serve to keep us from commiting them:
What else should I be doing? Our analysis usually focuses on the solution we intuitively favor; we therefore give inadequate attention to alternative solutions. Humans have a penchant to seek out and put store in evidence that supports their beliefs and judgments while eschewing and devaluing evidence that does not. ... "What am I supposed to be looking at? What's the point?" In other word's, what's the focus? ... focusing has immense downsides when we analyze problems. It tends to make us view problems one-dimensionally, meaning we tend to focus on (glom on to) the first solution that makes sense - that offers an explanation - as we interpret reality through the lens of our biases. Having focused on a solution, we are captivated by it and thus become uninterested in alternative solutions. Our fixation on our chosen solution therefore causes us to value evidence that supports that solution and to devalue, disbelieve, discredit, and discard evidence that does not. We therefore tend to accept at face value information that is consistent with our beliefs and to critically scrutinize and discount information that contradicts them. ... We tend to see in a body of evidence what, according to our mindset, we expect and want to see and tend not to see, again according to our mind-set, what we don't expect or don't want to see. drawing from The Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich. Stereotyping is a form of patterning ... I have to struggle against this impression to see this person as he or she really is; I have to struggle against my own mind... drawing from The Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich. "Satisficing" - picking the first solution that seems satisfactory, rather than examining all of the alternative hypotheses to identify not simply one that fits the evidence but the one that fits best. The problem is that most evidence is consistent with several hypotheses. ...we mess up in our efforts to analyze problems much more often than we are willing to admit. Indeed, the road to humankind's impressive achievements has invariably been paved with failures, many of which have set back our accomplishments by decades and longer. For every forward stride there have been telling missteps, because the problem-solving approach that has proven most practical and effective for the human species is the trial-and-error method. ... In all human affairs, from marriage to marketing to management, success is generally built upon failure. And while some failures are justly attributable to bad luck, most result from faulty decisions based on mistaken analysis. |
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Working
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Working Don't be idle! Don't be unproductive or destructive. After my freshman year my father arranged that I work the summer as an intern for his employer, Hughes Aircraft. There wasn't much for me to do there where I was assigned so I took the chance to teach myself programming, namely, Fortran. My supervisor wanted me to create a graphing program for him but I didn't have any real guidance and so I ended up focusing on what I thought was the coolest thing I could do, which was making a program for graphing derivatives of trigonometric functions, which was absolutely useless practically. So I made sure to gain from the experience, but it was a waste for my supervisor and they didn't have me back. When I was finishing my Ph.D., I wondered how I would make a living, if I was to devote myself to what I wanted to do. I thought that a university was a sheltered environment and so I would not learn about life. |
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Leaving
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Leaving Leaving Establishing a baseline Making an obervation many times under default, standard conditions provides the context for appreciating and measuring something exceptional, something radically different. An upper bound on the size of the cathode ray The small percentage of cathode rays deflected by air molecules allows for an estimate of their size.  Baseline: Geiger-Marsden Gold Foil Experiments A key part of this experiment was that the parameters (such as the foil) could be changed in many ways. Another key part was that the vast majority of the alpha particles were not affected by the foil at all, which made for a very prominent baseline. Between 1908 and 1913, Hans Geiger and Ernest Marsden under the direction of Ernest Rutherford determined that every atom contained a nucleus which consisted of most of its mass and its positive charge. They did this by emitting alpha particles (positively charged helium nuclei) from radium and other sources through thin foils (such as gold and aluminum) onto a flourescent screen. They observed that most of the particles passed through the foil but some particles were scattered by large angles, even bouncing back from the foil. This meant that most of the foil was empty space and that the mass was concentrated in positive nuclei that could knock the alpha particle off course. |
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Good Will: Building a culture of faith
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Good Will: Building a culture of faith This is the Kingdom of Heaven where Jesus is the Son of Man. Jesus urges us to go through the narrow gate that leads to life. Look for the ways to address Maslow's hierarchy of needs: Cling to what you have; get more than what you need; avoid extremes. Jesus believes in positivity, prefers it to negativity as taught by the Father, as given by his aloofness and prophets and warnings and having them resisted and humiliated and killed. But Jesus grows in reflection of the Father's way and ultimately does his will. Light as opposed to darkness. |
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Space
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Space A blank sheet is blank. We may or may not refer to that blankness. We may give it a name: identity, zero, one, empty set. The blankness is that origin point, that average, that center which is often unsaid but we may want to note as the natural, clever reference point, as in the case of the swimmer's hat that floated downstream (pg.64) Next, we can expand around the center by balancing positive and negative, numerator and denominator. We thereby introduce parity (Z2), odd or even, affirm or reject, where to reject rejection is to affirm. Next, we can expand terms as polynomials, as with "and" and "or", and thus create equations that construct and relate roots. Finally, we can consider a vector space in which any point can serve as the center for a basis. We thereby construct external "space". It is fully fledged in that it can define vectors, thus model time with arrows that have beginnings they come from and ends they go to. |
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Believing 1-2-3-4
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Believing 1-2-3-4 In real life, we discard the unessential to identify God which is deeper than our very depths, around such a core we allow for ourselves and others, we seek harmony of interests and we find a unity (Spirit) by which any person can serve as the center. These four frames are: believing; believing in believing; believing in believing in believing; believing in believing in believing in believing. |
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Our inner world
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Our inner world |
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Convergence
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Convergence means bringing together and moving toward one point. Whenever we take a narrower view of a problem, focusing on a single aspect of the puzzle or eliminating alternative solutions, we are in a convergent mode. |
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Giving
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Giving |
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Particle point of view
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Particle point of view Our measurement can take place within the frame of measurement. We have a natural frame of reference, for example, the center of mass. That center of mass can then be considered as balancing different masses, and integrating a system of masses, and ultimately defining a vector space. This is a static, spatial, nontemporal point of view. Every state has a location. We can speak of the state of a system. Analogously, in math we have a blank sheet with a natural frame, a center, a balance around that center, a polynomial algebra of constructions, and ultimately, a vector space where a basis makes explicit that every point can be the center. And in life, we can discard the unessential, presume only God, allow for both self and others, find harmony amongst our interests, and create a space for good Spirit. A particle is invariant even though it may have different sources |
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What you believe is what happens
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What you believe is what happens "What you believe is what happens" is the content of many of Jesus's parables, perhaps all of those that relate to the kingdom of heaven. Faith is the wish to understand, thus the irresistibility of coming to understand. Faith accepts God's love Ask in prayer Matthew 21:20 When the disciples saw it, they marveled, saying, "How did the fig tree immediately wither away?" 21:21 Jesus answered them, "Most certainly I tell you, if you have faith, and don't doubt, you will not only do what was done to the fig tree, but even if you told this mountain, 'Be taken up and cast into the sea,' it would be done. 21:22 All things, whatever you ask in prayer, believing, you will receive." Believe that God knows what you need even before you ask Matthew 6:8 Therefore don't be like them, for your Father knows what things you need, before you ask him. Believe that you have received Mark 11:22 Jesus answered them, "Have faith in God. 11:23 For most certainly I tell you, whoever may tell this mountain, 'Be taken up and cast into the sea,' and doesn't doubt in his heart, but believes that what he says is happening; he shall have whatever he says. 11:24 Therefore I tell you, all things whatever you pray and ask for, believe that you have received them, and you shall have them. People are saved by their faith Luke 7:47 Therefore I tell you, her sins, which are many, are forgiven, for she loved much. But to whom little is forgiven, the same loves little." 7:48 He said to her, "Your sins are forgiven." 7:49 Those who sat at the table with him began to say to themselves, "Who is this who even forgives sins?" 7:50 He said to the woman, "Your faith has saved you. Go in peace." Faith commands from God's certainty Assurance in his own path Matthew 8:22 But Jesus said to him, "Follow me, and leave the dead to bury their own dead." Certainty Mark 11:22 Jesus answered them, "Have faith in God. 11:23 For most certainly I tell you, whoever may tell this mountain, "Be taken up and cast into the sea" and doesn't doubt in his heart, but believes that what he says is happening; he shall have whatever he says. Command the forces of nature Matthew 8:26 He said to them, "Why are you fearful, O you of little faith?" Then he got up, rebuked the wind and the sea, and there was a great calm. 8:27 The men marveled, saying, "What kind of man is this, that even the wind and the sea obey him?" Faith evokes duty When we believe, we take ourselves to be believed in; and whatever obeys the belief, obeys out of duty. Thus faith evokes duty. Luke 17:6 The Lord said, "If you had faith like a grain of mustard seed, you would tell this sycamore tree, 'Be uprooted, and be planted in the sea,' and it would obey you. 17:7 But who is there among you, having a servant plowing or keeping sheep, that will say, when he comes in from the field, 'Come immediately and sit down at the table,' 17:8 and will not rather tell him, 'Prepare my supper, clothe yourself properly, and serve me, while I eat and drink. Afterward you shall eat and drink'? 17:9 Does he thank that servant because he did the things that were commanded? I think not. 17:10 Even so you also, when you have done all the things that are commanded you, say, 'We are unworthy servants. We have done our duty.'" God is glorified when Jesus' disciples bear fruit See video commentary by Dee Guyton! John 15:7 If you remain in me, and my words remain in you, you will ask whatever you desire, and it will be done for you. 15:8 "In this is my Father glorified, that you bear much fruit; and so you will be my disciples. Specific case follows from general principle Matthew 7:7 "Ask, and it will be given you. Seek, and you will find. Knock, and it will be opened for you. 7:8 For everyone who asks receives. He who seeks finds. To him who knocks it will be opened. Teach from authority Matthew 7:28 It happened, when Jesus had finished saying these things, that the multitudes were astonished at his teaching, 7:29 for he taught them with authority, and not like the scribes. Faith is for those who wish for change The Harvest, those who are attracted to his Good News The Good News is relevant to those who want healing. Matthew 9:35 Jesus went about all the cities and the villages, teaching in their synagogues, and preaching the Good News of the Kingdom, and healing every disease and every sickness among the people. 9:36 But when he saw the multitudes, he was moved with compassion for them, because they were harassed and scattered, like sheep without a shepherd. 9:37 Then he said to his disciples, "The harvest indeed is plentiful, but the laborers are few. 9:38 Pray therefore that the Lord of the harvest will send out laborers into his harvest." What does Jesus want? Matthew 8:2 Behold, a leper came to him and worshiped him, saying, "Lord, if you want to, you can make me clean." 8:3 Jesus stretched out his hand, and touched him, saying, "I want to. Be made clean." Immediately his leprosy was cleansed. 8:4 Jesus said to him, "See that you tell nobody, but go, show yourself to the priest, and offer the gift that Moses commanded, as a testimony to them." Who seeks a new outlook Jesus represents a new outlook and it is meaningful for those who want to repent and life fresh, live new. Those who are already attached to an outlook, such as the Pharisees or the disciples of John, don't get excited and attached to the new outlook, for they long for the old. Thus these passages come just after those where Jesus says he has come for the sick, not for the healthy. Luke 5:33 They said to him, "Why do John's disciples often fast and pray, likewise also the disciples of the Pharisees, but yours eat and drink?" 5:34 He said to them, "Can you make the friends of the bridegroom fast, while the bridegroom is with them? 5:35 But the days will come when the bridegroom will be taken away from them. Then they will fast in those days." 5:36 He also told a parable to them. "No one puts a piece from a new garment on an old garment, or else he will tear the new, and also the piece from the new will not match the old. 5:37 No one puts new wine into old wineskins, or else the new wine will burst the skins, and it will be spilled, and the skins will be destroyed. 5:38 But new wine must be put into fresh wineskins, and both are preserved. 5:39 No man having drunk old wine immediately desires new, for he says, 'The old is better.'" See also: Matthew 9:14, Mark 2:18 Faith is irresistible Note the relation between call and response Matthew 7:7 "Ask, and it will be given you. Seek, and you will find. Knock, and it will be opened for you. 7:8 For everyone who asks receives. He who seeks finds. To him who knocks it will be opened. What grows larger Matthew 13:31 He set another parable before them, saying, "The Kingdom of Heaven is like a grain of mustard seed, which a man took, and sowed in his field; 13:32 which indeed is smaller than all seeds. But when it is grown, it is greater than the herbs, and becomes a tree, so that the birds of the air come and lodge in its branches." What spreads through Matthew 13:33 He spoke another parable to them. "The Kingdom of Heaven is like yeast, which a woman took, and hid in three measures of meal, until it was all leavened." Why will it obey Luke 17:6 The Lord said, "If you had faith like a grain of mustard seed, you would tell this sycamore tree, 'Be uprooted, and be planted in the sea,' and it would obey you. Persistence of faith Having faith Matthew 17:20 He said to them, "Because of your unbelief. For most certainly I tell you, if you have faith as a grain of mustard seed, you will tell this mountain, 'Move from here to there,' and it will move; and nothing will be impossible for you. 17:21 But this kind doesn't go out except by prayer and fasting." Must keep praying and not give up Luke 18:1 He also spoke a parable to them that they must always pray, and not give up, 18:2 saying, "There was a judge in a certain city who didn't fear God, and didn't respect man. 18:3 A widow was in that city, and she often came to him, saying, 'Defend me from my adversary!' 18:4 He wouldn't for a while, but afterward he said to himself, 'Though I neither fear God, nor respect man, 18:5 yet because this widow bothers me, I will defend her, or else she will wear me out by her continual coming.'" 18:6 The Lord said, "Listen to what the unrighteous judge says. 18:7 Won't God avenge his chosen ones, who are crying out to him day and night, and yet he exercises patience with them? 18:8 I tell you that he will avenge them quickly. Nevertheless, when the Son of Man comes, will he find faith on the earth?" Not doubt Matthew 21:20 When the disciples saw it, they marveled, saying, "How did the fig tree immediately wither away?" 21:21 Jesus answered them, "Most certainly I tell you, if you have faith, and don't doubt, you will not only do what was done to the fig tree, but even if you told this mountain, 'Be taken up and cast into the sea,' it would be done. 21:22 All things, whatever you ask in prayer, believing, you will receive." Not doubt in your heart Mark 11:22 Jesus answered them, "Have faith in God. 11:23 For most certainly I tell you, whoever may tell this mountain, 'Be taken up and cast into the sea,' and doesn't doubt in his heart, but believes that what he says is happening; he shall have whatever he says. 11:24 Therefore I tell you, all things whatever you pray and ask for, believe that you have received them, and you shall have them. Requirement to participate in the Kingdom of Heaven The greatest commandment Matthew 22:35 One of them, a lawyer, asked him a question, testing him. 22:36 "Teacher, which is the greatest commandment in the law?" 22:37 Jesus said to him, "'You shall love the Lord your God with all your heart, with all your soul, and with all your mind.' 22:38 This is the first and great commandment. 22:39 A second likewise is this, 'You shall love your neighbor as yourself.' 22:40 The whole law and the prophets depend on these two commandments." What depends on what? 22:34 But the Pharisees, when they heard that he had silenced the Sadducees, gathered themselves together. 22:35 One of them, a lawyer, asked him a question, testing him. 22:36 "Teacher, which is the greatest commandment in the law?" 22:37 Jesus said to him, "'You shall love the Lord your God with all your heart, with all your soul, and with all your mind.' 22:38 This is the first and great commandment. 22:39 A second likewise is this, 'You shall love your neighbor as yourself.' 22:40 The whole law and the prophets depend on these two commandments." What is the greatest commandment? 22:34 But the Pharisees, when they heard that he had silenced the Sadducees, gathered themselves together. 22:35 One of them, a lawyer, asked him a question, testing him. 22:36 "Teacher, which is the greatest commandment in the law?" 22:37 Jesus said to him, "'You shall love the Lord your God with all your heart, with all your soul, and with all your mind.' 22:38 This is the first and great commandment. 22:39 A second likewise is this, 'You shall love your neighbor as yourself.' 22:40 The whole law and the prophets depend on these two commandments." What is the way to life? The gate of "belief" leads to life; the gate of "caring" leads to destruction. We can distinguish the gates by seeing who goes through them. Matthew 7:13 "Enter in by the narrow gate; for wide is the gate and broad is the way that leads to destruction, and many are those who enter in by it. 7:14 How narrow is the gate, and restricted is the way that leads to life! Few are those who find it. Who can enter the Kingdom of Heaven Luke 18:24 Jesus, seeing that he became very sad, said, "How hard it is for those who have riches to enter into the Kingdom of God! 18:25 For it is easier for a camel to enter in through a needle's eye, than for a rich man to enter into the Kingdom of God." Who can enter the Kingdom of Heaven John 3:5 ... "Most certainly I tell you, unless one is born of water and spirit, he can't enter into the Kingdom of God! 3:6 That which is born of the flesh is flesh. That which is born of the Spirit is spirit. 3:7 Don't marvel that I said to you, 'You must be born anew.' 3:8 The wind blows where it wants to, and you hear its sound, but don't know where it comes from and where it is going. So is everyone who is born of the Spirit." Who can or can't understand The word "overcome" is also translated "comprehended". God (who understands) and Jesus (who comes to understand) are the same God because they understand the same thing. Thus the darkness can't understand the light because it is not God. Only those who are of God can understand the light. John 1:5 The light shines in the darkness, and the darkness hasn't overcome it Speak to their faith Building on faith as an outlook for a culture Jesus looks to people's faith as the test of whether they have a shared culture. And given that faith, he speaks to that culture in them. Matthew 9:2 Behold, they brought to him a man who was paralyzed, lying on a bed. Jesus, seeing their faith, said to the paralytic, "Son, cheer up! Your sins are forgiven you." Jesus allows belief to be fulfilled Jesus' will allows for belief to be fulfilled. Matthew 8:13 Jesus said to the centurion, "Go your way. Let it be done for you as you have believed." His servant was healed in that hour. Jesus believes in the fruitfulness of their belief Matthew 9:27 As Jesus passed by from there, two blind men followed him, calling out and saying, "Have mercy on us, son of David!" 9:28 When he had come into the house, the blind men came to him. Jesus said to them, "Do you believe that I am able to do this?" They told him, "Yes, Lord." 9:29 Then he touched their eyes, saying, "According to your faith be it done to you." 9:30 Their eyes were opened. Jesus strictly commanded them, saying, "See that no one knows about this." 9:31 But they went out and spread abroad his fame in all that land. Jesus believes in them, just as they believe in him Jesus does as he is requested to do, and does not argue with them as to what to do, which shows that he believed in them. Matthew 9:18 While he told these things to them, behold, a ruler came and worshiped him, saying, "My daughter has just died, but come and lay your hand on her, and she will live." 9:19 Jesus got up and followed him, as did his disciples. ... 9:23 When Jesus came into the ruler's house, and saw the flute players, and the crowd in noisy disorder, 9:24 he said to them, "Make room, because the girl isn't dead, but sleeping." They were ridiculing him. 9:25 But when the crowd was put out, he entered in, took her by the hand, and the girl arose. 9:26 The report of this went out into all that land. Woman issuing blood Matthew 9:20 Behold, a woman who had an issue of blood for twelve years came behind him, and touched the fringe of his garment; 9:21 for she said within herself, "If I just touch his garment, I will be made well." 9:22 But Jesus, turning around and seeing her, said, "Daughter, cheer up! Your faith has made you well." And the woman was made well from that hour. Speaking to the good in others I did a study of the Gospel of Mark to understand Jesus' emotional responses, his cognitive expectations and how he responded to them to get things done. He spoke to the good in others. Why was Jesus born John 18:37 ... For this reason I have been born, and for this reason I have come into the world, that I should testify to the truth. Everyone who is of the truth listens to my voice." Their mindset traps them Incapacity is caused by a demon which can be cast out Matthew 9:32 As they went out, behold, a mute man who was demon possessed was brought to him. 9:33 When the demon was cast out, the mute man spoke. The multitudes marveled, saying, "Nothing like this has ever been seen in Israel!" 9:34 But the Pharisees said, "By the prince of the demons, he casts out demons." You *say* that Jesus frequently responds in a peculiar way, "You're the one who is saying that..." It seems that he infers that if somebody states a question, then their mind conceives the statement, so that given that all statements are true, then the person has already made it true simply by conceiving it, and choosing to say it, thus allow for it, make it real. Or how else to understand him? Where should the emphasis be? Perhaps: You *say* that... For then the statement is true as a statement, even if it is just part of a question. And for Jesus, it seems, if it is true as a statement, as a word, then it is true in reality. Judas asks if he will betray Jesus Matthew 26:25 Judas, who betrayed him, answered, "It isn't me, is it, Rabbi?" He said to him, "You said it." You say that I am a king. John 18:37 Pilate therefore said to him, "Are you a king then?" Jesus answered, "You say that I am a king. For this reason I have been born, and for this reason I have come into the world, that I should testify to the truth. Everyone who is of the truth listens to my voice." What you believe is what happens. "What you believe is what happens" is as much about our limitations - we are bound by the limits of our minds - as it is about our possibilities. For God is to reach us, and that must take place through our channels that we allow for. |
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Center
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Center A blank sheet is blank. We may or may not refer to that blankness. We may give it a name: identity, zero, one, empty set. The blankness is that origin point, that average, that center which is often unsaid but we may want to note as the natural, clever reference point, as in the case of the swimmer's hat that floated downstream (pg.64) (Blank sheet, what is so central that it is often left unsaid, origin of a coordinate system, natural or clever point of view, symmetrize an equation, average principle, choice of notation, convenient notation) Simple form Given any diophantine equation [an equation whose variables assume only integer values] ... Is the problem in "simple" form? Always make sure that you have divided out all common factors, or assume the variables share no common factors, etc. The purpose of math is to construct a simplification. pg.264 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Axiom of the empty set Wikipedia: There is a set such that no set is a member of it. The decimal point Dividing a number by 10, we are taught to move the decimal point to the left, but actually, it is the number that moves to the right. The decimal point is a fixed reference point. Similary, multiplying a number by 10, it may seem that the decimal point moves to the right, but actually, the number moves to the left. Zero is just a place holder Note that zero is just a place holder and so we don't have to multiply or add by zero in the algorithms. Zero is a fiction (as a numerical value outside of the system) and those operations are placeholder operations (meaningful only with regard to the particular system). Gospel Math. Change your point of view Changing your point of view is typically choosing the origin for a coordinate system. Changing the point of view is just another manifestation of peripheral vision. Sometimes a problem is hard only because we choose the "wrong" point of view. Spending a few minutes searching for the "natural" point of view can pay big dividends. pg. 64 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. The swimmer's hat A person dives from a bridge into a river and swims upstream through the water for 1 hour at constant speed. She then turns around and swims downstream through the water at the same rate of speed. As the swimmer passes under the bridge, a bystander tells her that her hat fell into the river as she originally dived. The swimmer continues downstream at the same rate of speed, catching up with the hat at another bridge exactly 1 mile downstream from the first one. What is the speed of the current in miles per hour? ... what if we look at things from the hat's point of view? pg. 64 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Considering the simplest point in an arbitrary coordinate system In the following problem, the "north pole" is the simplest point to consider, yet the coordinate system was arbitrary, and so the conclusion is valid for all points. We need some "notation". Let us assume that there is a universal coordinate system, such as longitude and latitude, so that we can refer to the "same" location on any planet. For example, if the planets were little balls floating in a room, the location "north pole" would mean the point on a planet which was closest to the ceiling. Given such a universal coordinate system, what can we say about a planet P which has a private point at location x? Without loss of generality, let x be at the "north pole". Clearly, the centers of all the other planets must lie on the south side of P's "equatorial" plane. But that renders the north poles of these planets public ... we have shown pretty easily that If location x is private on one planet, it is public on all the other planets. pg. 64 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Convenient notation Think about convenient notation. pg.29, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Elegant solution Algebra is commonly taught as a series of computational techniques. ... algebra is also an aesthetic subject. Sometimes one has to slog through messy thickets of algebraic expressions to solve a problem. But these unfortunate occasions are pretty rare. A good problem solver takes a more confident approach to algebraic problems. The wishful thinking strategy teaches her to look for an elegant solution. Cultivate this mindset: employ a light, almost delicate touch, keeping watch for opportunities that avoid ugly manipulations in favor of elegant, often symmetrical patterns. pg.162, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Invariant with respect to permutation of some of the roots of a polynomial The substitution u:=x + 1/x, which helped solve x**4+x**3+x**2+x+1=0 worked because u is invariant with respect to a permutation of some of the roots. This idea is the germ of ... Galois theory. pg. 103, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Invariants An invariant, as the name suggests, is merely some aspect of a problem - usually a numerical quantity - that does not change, even if many other properties do change. pg. 102, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Euler's formula Given a polyhedron without holes, the numbers of vertices, edges and faces satisfy V - e + f = 2, which is an invariant. See pg. 103, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Mean Value Theorem If f(x) is continuous on [a,b] and differentiable on (a,b), then there is a point u in (a,b) at which f'(u) = (f(b) - f(a))/(b-a). ... the proof is just one sentence: Tilt the picture for Rolle's theorem! The mean value theorem connects a "global" property of a function (its values at the endpoints a and b) with a "local" property (the value of its derivative at a specific point) and is thus a deeper and more useful fact than is apparent at first glance. ... Suppose f is differentiable on all real numbers and there is a constant k < 1 such that |f'(x)|<=k for all real x. Show that f has a fixed point. ... Since the derivative is at most k in absolute value, and since k < 1, the graph of y=f(x) to the right of the y-axis will be trapped within the dotted line "cone" and will eventually "catch up" with the graph of y=x. The mean value theorem lets us prove this is a satisfying way. Suppose that for all x>=0, we have f(x) does not equal to x. Then by the Intermediate Value Theorem we must have f(x)> x. Pick b>0 (think large). By the mean value theorem, there is a u in (0,b) such that f'(u) = (f(b)-f(0))/(b-0) ... Since f(b) > b, we have ... f'(u) > 1 - v/b Since b can be arbitrarily large, we can arrange things so that f'(u) becomes arbitrarily close to 1. But this contradicts |f'(u)| <=k < 1 ... The satisfying thing about this argument was the role that the mean value theorem played in guaranteeing exactly the right derivative values to get the desired contradiction. pg.297-298 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Motel room paradox Three women check into a motel room which advertises a rate of $27 per night. They each give $10 to the porter, and ask her to bring back 3 dollar bills. The porter returns to the desk, where she learns that the room is actually only $25 per night. She gives $25 to the motel desk clerk, returns to the room, and gives the guests back each one dollar, deciding not to tell them about the actual rate. Thus the porter has pocketed $2, while each guest spent 10-1 = $9, a total of 2 + 3 x 9 = $29. What happened to the other dollar? ... The actual "invariant" here is not $30, but $27, the amount that the guests spend, and this will always equal the amount that the porter took ($2) plus the amount that went to the desk ($25). pg. 22, 102, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Notation Adding notation is typically referencing everything with regard to an origin, a reference point. We need some "notation". Let us assume that there is a universal coordinate system, such as longitude and latitude, so that we can refer to the "same" location on any planet. For example, if the planets were little balls floating in a room, the location "north pole" would mean the point on a planet which was closest to the ceiling. Given such a universal coordinate system, what can we say about a planet P which has a private point at location x? pg. 64 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. |
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Believing
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Believing Referencing the absolute within us. Isolate what is most essential, what grounds everything else, what I truly want, what I love most intensely. What do I truly want? What can't be dismissed Conversing with: depths Accepting God's assertion I am in love with a woman who God told me I will marry. I don't pray that she marry me because God assures me that she will. I prayed with her that her aunt get well from cancer and God consistently assured me that she would. I asked if I should keep praying, if I should offer myself to him in some way, but he said no. She died, though. Yet they reported that just before she died she bore witness that God truly is, that he had greeted her, and she said that she loved them all very much, and she passed away. I didn't know what to make of this, and when I ask God, he doesn't say. Discard the unessential Conversing with: my depths We can ask ourselves, what is truly essential, and discard everything that is not, so that what remains is indeed essential. Descartes doubted everything, and was left with doubting. Kant imagined removing all objects and being left with time and space. Respect seriousness Conversing with: importance When I was little, our mother pulled us aside one day and briefly taught us about God and Jesus and how to make the sign of the cross. I appreciated that she was very serious and that there must be something real to it if she was so. I couldn't dismiss her utmost seriousness. God must be at least a possibility. What can or can't be removed? Conversing with: the unthinkable As a freshman at the University of Chicago, I was trying to explain how the concept of everything was an absolute. We seem to all have the concept, but perhaps others might not. One question that helped was to ask whether we could get rid of the concept of everything? My personal answer is no, I don't conceive of how I could let go of such a concept. Thus this supports the property of everything that it is a required concept. What must I believe? Conversing with: my ability to believe As a child, I pondered, what must I accept and believe so that I don't go astray in my thinking? and think evil things? I thought I should believe, as Jesus teaches, to believe God, and then also, to love my neighbor as myself, and for good measure, to believe that Jesus is God. What includes everything Conversing with: everything What does everything depend on? Conversing with: all that follows Jesus taught that the whole law and the prophets depends on "Love God with all of your heart, with all your soul, and with all your mind" and "Love your neighbor as yourself". What encompasses everything? Conversing with: everything As a child, I wondered how to apply myself. What was most important? I valued knowledge and I thought it was the key to all things. I thought it was important to know as broadly, deeply and generally as possible. I thus chose my quest to know everything and apply that knowledge usefully. Similarly, we can arrive at our deepest value in life by asking, what value includes all of our other values? |
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Rooting ourselves in our inner world
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Rooting ourselves in our inner world Having a complete desire to win Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: But there are a few, that have that intestinal fortitude, and they want to win at any cost. That's the guy who's listening to whatever his corner has to say. |
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Give everything away
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Give everything away I ultimately collected about two hundred classics of rock music but it was getting difficult to buy albums and CDs were very pricey. So I gave away my collection, mostly to some rock musicians from Lithuania. It was nice that they appreciated the albums. I won about thirty trophies in my chess playing days. And when I finished high school I made sure to give them all away to our high school chess club where they all ended up getting recycled. It was nice to clear that all away. One year, after I and my brother and sister had all finished our schooling, our parents gave us each a substantial sum of money. I used part to clear my loans, and then I decided that I should not save the rest, but give it away, taking JesusChrist seriously. I told my parents I didn't need it and could give it back, but they said no. So I gave more than half to my cousin, who had worked for a year to help his stepfather at his body shop without hardly any remuneration. I thought it could make clear that I cared. My parents were shocked and hurt and angry. They said that if they wanted to give charity (which they do) then they would give it themselves. My cousin was delighted and decided to spend it on three things that he had always wanted: back taxes on his land which his father had left him, a BMW motorcycle, and the Encyclopedia Britannica. All of which seemed a bit vain, but it was his choice, except I was not happy that he had spent thousands of dollars on the encyclopedia and I argued that it didn't make any sense given how much he would read it and how technology was evolving and that he should ask for his money back while he still had the chance. I felt relieved that he did indeed back out of the encyclopedia. So I learned that it was all right to give it all away, but it was good to care what happened to it. I didn't grow super close with my cousin except that I think we feel good about each other and I imagine that the money did have a good impact on his spirits because his life course seemed to pick up around then. |
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My actions
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My actions Doing what I love Doing what you want to do |
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Establishing a useful frame of reference
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Establishing a useful frame of reference A useful reference can be, for example, the center of mass or the center of momentum. |
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Replying to Scripture with Scripture
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Replying to Scripture with Scripture The greater context makes exceptions legitimate. Analogy of us to others Matthew 7:12 Therefore whatever you desire for men to do to you, you shall also do to them; for this is the law and the prophets. Appreciating the power of God Mark 12:24 Jesus answered them, "Isn't this because you are mistaken, not knowing the Scriptures, nor the power of God? 12:25 For when they will rise from the dead, they neither marry, nor are given in marriage, but are like angels in heaven. How much does Jesus demand of them? Matthew 11:29 Take my yoke upon you, and learn from me, for I am gentle and lowly in heart; and you will find rest for your souls. 11:30 For my yoke is easy, and my burden is light. Know Scripture Mark 12:24 Jesus answered them, "Isn't this because you are mistaken, not knowing the Scriptures, nor the power of God? ... 12:26 But about the dead, that they are raised; haven't you read in the book of Moses, about the Bush, how God spoke to him, saying, "I am the God of Abraham, the God of Isaac, and the God of Jacob" 12:27 He is not the God of the dead, but of the living. Ponder Scripture Matthew 22:41 Now while the Pharisees were gathered together, Jesus asked them a question, 22:42 saying, "What do you think of the Christ? Whose son is he?" They said to him, "Of David." 22:43 He said to them, "How then does David in the Spirit call him Lord, saying, 22:44 'The Lord said to my Lord, sit on my right hand, until I make your enemies a footstool for your feet?' 22:45 "If then David calls him Lord, how is he his son?" 22:46 No one was able to answer him a word, neither did any man dare ask him any more questions from that day forth. Replying to Scripture with Scripture In Matthew, the devil cites the Scripture in tempting Jesus, and Jesus responds with Scripture, and only Scripture, except for "Get behind me Satan!" with his last reply. Jesus seems to accord to Scripture greater assurance than anything he could say with his own mind. What do people need to hear? Matthew 10:7 As you go, preach, saying, 'The Kingdom of Heaven is at hand!' What God wishes for Matthew 9:12 When Jesus heard it, he said to them, "Those who are healthy have no need for a physician, but those who are sick do. 9:13 But you go and learn what this means: 'I desire mercy, and not sacrifice,' for I came not to call the righteous, but sinners to repentance." What is the law and the prophets? Matthew 7:12 Therefore whatever you desire for men to do to you, you shall also do to them; for this is the law and the prophets. What lets one read Scripture with fresh eyes Matthew 13:51 Jesus said to them, "Have you understood all these things?" They answered him, "Yes, Lord." 13:52 He said to them, "Therefore every scribe who has been made a disciple in the Kingdom of Heaven is like a man who is a householder, who brings out of his treasure new and old things." What might have changed the course of history Matthew 11:20 Then he began to denounce the cities in which most of his mighty works had been done, because they didn't repent. 11:21 "Woe to you, Chorazin! Woe to you, Bethsaida! For if the mighty works had been done in Tyre and Sidon which were done in you, they would have repented long ago in sackcloth and ashes. 11:22 But I tell you, it will be more tolerable for Tyre and Sidon on the day of judgment than for you. 11:23 You, Capernaum, who are exalted to heaven, you will go down to Hades. For if the mighty works had been done in Sodom which were done in you, it would have remained until this day. 11:24 But I tell you that it will be more tolerable for the land of Sodom, on the day of judgment, than for you." Who declares that he is the Son of God Jesus wants to act as man, not as God. Matthew 8:29 Behold, they cried out, saying, "What do we have to do with you, Jesus, Son of God? Have you come here to torment us before the time?" Who is worthy of Jesus We must love God beyond individuals in order to love God through an individual, Jesus. Matthew 10:37 He who loves father or mother more than me is not worthy of me; and he who loves son or daughter more than me isn't worthy of me. 10:38 He who doesn't take his cross and follow after me, isn't worthy of me. What makes exceptions legitimate Jesus notes that in Scripture there are cases of exceptions. And those exceptions are made legitimate, for example, by the greater context, such as the temple, or even greater than that, the Son of Man. Matthew 12:1 At that time, Jesus went on the Sabbath day through the grain fields. His disciples were hungry and began to pluck heads of grain and to eat. 12:2 But the Pharisees, when they saw it, said to him, "Behold, your disciples do what is not lawful to do on the Sabbath." 12:3 But he said to them, "Haven't you read what David did, when he was hungry, and those who were with him; 12:4 how he entered into God's house, and ate the show bread, which was not lawful for him to eat, neither for those who were with him, but only for the priests? 12:5 Or have you not read in the law, that on the Sabbath day, the priests in the temple profane the Sabbath, and are guiltless? 12:6 But I tell you that one greater than the temple is here. |
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Balance
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Balance We can expand around the center by balancing positive and negative, numerator and denominator. We thereby introduce parity (Z2), odd or even, affirm or reject, where to reject rejection is to affirm. (Parity, Z2: affirm-reject, multiplication by one, addition of zero, union with empty set, expansion around center) Add zero creatively Many [factoring] problems involve combinations of these formulas, along with basic strategies (for example, wishful thinking), awareness of symmetry, and the value add zero creatively tool. Zeitz gives the example of factoring x**4 + 4, thinking wishfully that it was the difference of two squares, and making more perfect squares appear by adding 0 = 4x**2 - 4x**2. pg. 163, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Completing the square by adding zero x**2 + a*x = x**2 + a*x + a**2/4 - a**2/4 = (x + a/2)**2 - (a/2)**2 One way to discover this completing-the-square formula is to add zero creatively, as above. pg. 163, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Appeal to Physical Intuition Let x1, x2, ... xn be positive real numbers with product P and sum S. Prove that the largest value of P is attained when all the xi are equal. ... Imagine the n positive numbers as "physical" points on the number line, each with unit weight. The balancing point (center of mass) of these weights is located at the arithmetic mean value A = S/n. Notice that if we move the points around in such a way that they continue to balance at A, that is equivalent to saying that their sum stays constant. Our strategy, inspired by the symmetry-product principle, is to consider situation where the xi are not all equal and show that we can make them "more equal" and increase their product without changing their sum. If the points are not all clustering at A, then at least one will be to the left of A (call it L) and another will be to the right of A. Of these two points, move the one which is closest to A right up to A, and move the other so that the balancing point of the two points hasn't changed. ... This proof is called "algorithmic" ... pg.195-196 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Easy invariants Be on the lookout for "easy" invariants. Check to see if you can rearrange your problem to get simple numbers such as zero or one. pg.106, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Homogeneous coloration Parity is invoked by a coloration which is homogeneous or not. Is it possible to tile a 66 x 62 rectangle with 12 x 1 rectangles? ... Color the squares of the 66 x 62 rectangle with 12 colors in a cyclic "diagonal" pattern... This coloring has the nice property that any 12 x 1 rectangle in the tiling consists of 12 differently colored squares ... each color occurs in the same number of squares. We will call such a coloration "homogeneous". ... We can break it up into 4 sub-rectangles ... the entire large rectangle is not homogeneous... pg.111, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Make an expression uglier Sometimes you may want to make an expression uglier because it then yields more information. ... Which is bigger 1998/1999 or 1999/2000? ... here is an argument that uses the define a function tool: Let f(x) = x/(x+1) ... How does this function grow? We have f(x) = x/(x+1) = 1/(1 + 1/x) and now it is easy to check that as x>0 increases, the 1/x term decreases, causing f(x) to increase ... "If the denominator increases, the fraction decreases, and vice versa." ... f(x) is monotonically increasing for positive x. ... Notice how we actually made an expression uglier... it is much easier to analyze the behavior of the function. pg.165, 191-192 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Modular arithmetic Modular arithmetic arguments are typically parity arguments (divisible by N - not divisible by N). Let N be a 4-digit number with decimal representation abcd. Then n = 10**3 a + 10**2 b + 10 c + d. ... 10**k = 1**k = 1 (mod 9) ... n = 10**3 a + 10**2 b + 10 c + d = 1 a + 1 b + 1 c + d (mod 9) ... The important thing is to be aware of the possibility that an invariant may be a quantity modulo m for a properly chosen m. pg.110, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Multiply cleverly by one The sister to the add zero creatively tool is the multiply cleverly by one. pg. 163, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Overcount and rectify In explaining combinations and thus the coefficients of the binomial theorem... To count the number of ways a joint event occurs, multiply together the number of choices for each sub-event. To rectify uniform overcounting, divide by the overcounting factor. Such overcounting occurs, for example, when some of the objects are indistinguishable but we label them to make them easier to count, then take off the labels. In general, the number of ways you can select a subset of r distinct elements from a set of n distinct elements, where the order of selection doesn't matter, is P(n,r)/r! = (n r). This is called a combination. If the order does matter, then the number of ways is P(n,r) and it is called a permutation. Compare overcounting with the use of "negative numbers". pg.207-208 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Parity Integers are divided into two parity classes: even and odd. The even integers are divisible by 2, the odds are not. Note that zero is even. ... The parity of a sum of a set of integers is odd if and only if the number of odd elements is odd. The parity of a product of a set of integers is odd if and only if there are no even elements in the set. ... knowledge of parity is sometimes all that is needed, especially if parity is involved in the statement of the problem. ... Whenever a problem involves integers, ask yourself if there are any parity restrictions. Experiment with different values than the given if necessary. ... Parity works amazingly well, but it is rather crude. After all, we are reducing the infinite universe of integers into a tiny world inhabited by just two entries, "even" and "odd". pg. 104, 106, 110 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. 127 people in a tennis tournament If 127 people play in a singles tennis tournament, prove that at the end of the tournament, the number of people who have played an odd number of games is even. ... each game has exactly two people playing it... the sum counts every game that has been played exactly twice! ... the sum above is even, and is a sum of an odd number (127) of elements. If an odd number of them were odd, the sum would not be even... pg. 102, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Parity problem: Dominos on a chessboard Remove the two diagonally opposite corner squares of a chessboard. Is it possible to tile this shape with thirty-one 2 x 1 "dominos"? ... At first, it seems like a geometric/combinatorial problem with many cases and subcases. But it is really just a question about counting colors. The two corners that were removed wre both (without loss of generality) white, so the shape we are interested in contains 32 black and 30 white squares. Yet any domino, once it is placed, will occupy exactly one black and one white square. The 31 dominos thus require 31 black and 31 white squares, so tiling is impossible. pg. 60 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Parity of a sum or product Let a1, a2, ..., aN represent an arbitrary arrangement of the numbers 1, 2, 3,...N. Prove that, if N is odd, the product (a1-1)(a2-2)...(aN-N) is an even number. ... The crux move: consider the sum (a1-1) + (a2-2) + ... + (aN-N) pg. 105 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Simplicity You have been taught that "simplification" is to combine things in "like terms". This sometimes simplifies an expression, but the good problem solver has a more focused, task-oriented approach, motivated by the wishful thinking strategy. Avoid mindless combinations unless this makes your expressions simpler. Always move in the direction of greater simplicity and/or symmetry and/or beauty (the three are often synonymous). pg.165, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. |
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Respect - Implicitly Care
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Respect - Implicitly Care Five-Fingered Consensus gauge extent of consensus Help Me Understand know what participants want to know Red:Green Cards honor feedback |
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Believing in believing
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Believing in believing Allowing for self and others. Listen to others and God, appreciate what they care about as given by their choices, words and actions. Take that to heart. Disposing myself with regard to another Conversing with: goodness Empathize with abstract person Conversing with: person's fate I may think things through without experience but by empathizing with an abstract person. This helps me to be sensitive to human weaknesses, to respond seriously to any talk of suicide, to think compassionately about the education system, health care system, to expect meaning in physical intimacy. I can consider, how might I help a person-in-general in particular circumstances?  Existential conversation with God How should I apply myself? Conversing with: God I think of an "existential conversation" with God as one where I don't presume to hear directly from God, but rather, link up my existential situation with God's existential situation, and walk through conclusions based on that. Good wins out over bad Conversing with: goodness In presuming that good must be more relevant than bad, I understood to give priority to good will exercises that grew out of positive feelings (and so expressed right thinking we could share) instead of those that addressed negative feelings. Increase God's possibilities Conversing with: God's will In 2002, I had reached my limits financially. I had failed to generate interest in an import/export standard for tools for organizing thoughts. I was in Chicago, and I had to think of finding any job. Indeed, my plan had been that, having a mathematics Ph.D. and some programming skills, I could always find a job. However, the US was in a recession. And I realized that my resume did not reflect a standard career and, with hundreds of people applying for each job, I would be weeded out early on for almost all jobs. Yet my own rationalizations could keep me from finding work, too. How could I know what to do? So I decided to increase God's possibilities by taking a fourfold approach. I applied for some full time positions. I applied for work through temp agencies (but the little work they had went to their earlier workers). I wrote some ambitious proposals for my lab (including writing a paper which won a travel award and took me to India and then home to Lithuania). I looked for part-time work through my network and was hired by Shannon Clark for a few months of corporate work that paid well. By taking all approaches I felt that I kept myself open to God's plans for me. What would be fair Conversing with: God As a six-year-old child, I engaged God in an existential conversation. I wished to apply myself to know everything, partly for its own sake, and partly to make good use of that knowledge. I was concerned that to seek such knowledge I would need to think freely, which could be very dangerous and lead me astray. I thus offered that, if he let me think freely, even think that God doesn't exist, then on my part, I would always believe in him. What would be reasonable As a child, I had made an offer to God, which seemed reasonable, and I felt God's presence, and I wondered if I should have a sign to confirm this understanding. But then I thought it would be wrong to ask for a sign, to look for a sign, to demand a sign, as I might invent a sign, or depend on a sign, and be lead astray. So I told God that my sign would be that I have no sign, and that if he ever meant otherwise, he would give me a sign. Take another to heart Conversing with: other Acknowledging people's natural inclination Conversing with: people's inclinations In developing good will exercises, I learned that people who are riled about some surface subject typically aren't too interested in the deeper issue that fuels it. People, in general, aren't interested in whatever would make them more responsible, such as truly knowing everything or hearing from God. Be interrogated Conversing with: my virtue I try to accept provocations from Jehovah's Witnesses and evangelicals regarding my life. I feel disturbed when they don't show interest in me, but treat me presumptuously as an infidel or sinner. Yet this is how I stay open, how I learn about myself and the weakness of their doctrine. Correspond openly Conversing with: extensions By corresponding openly, through mailing groups with public archives, I am able to learn to what extent I shape a new culture or clash with an existing one. I learn how others maintain or influence a culture. I saw how Janet Feldman felt offended by my righteousness when I openly criticized Peter Ongele for asking me for funds rather than explaining when he would repay me the money I loaned him. Dialogue with another Conversing with: others as they understand themselves In speaking with others, I developed 12 questions that I thought were especially useful for understanding, engaging and supporting them. Heartful interaction Conversing with: others as equals As I developed good will exercises, through many conversations with others, I learned that I could open up to new angles by alternatively stepping in (looking subjectively) and stepping out (stepping back, looking objectively). I learned how to get past the surface issue, the "hot button" issue, and get to the deep issue underlying it. Imagine another's mindset Conversing with: person In studying Jesus's emotions as documented in the Gospel of Mark, I imagined Jesus's mindset, what were his expectations that led to his emotions, and how did he respond? In talking with a person, I can imagine what keeps them from saying things they might say. Note what is interesting to others Conversing with: their attention I learn more with and from others by being open to what they are interested. In looking for participants for good will exercises, I would ask what issues riled them. In looking for work or clients, I look for who I might serve, and how. In organizing online groups, I center them around the deepest value of a leader. That way I hope that at least two of us, they and I, will be active. In general, I learn that there are people who are fascinated by things that I am not, but could be, and so they help me stretch and grow. Share what God says Conversing with: people's relationship with God Almost every morning I link up with God, listen to him and write down what he says. Sometimes I share that with other people. That helps me appreciate that a relationship with God is, it seems, most relevant on a personal level. Show good will Conversing with: another's wishes My father taught me to always show good will. I saw him practice that many times in many ways. Showing good will helps me imagine but also notice others' wishes, purpose and good will. Take a remark to heart Conversing with: another's perspective I take to heart remarks by my parents and my loved ones. My father ever taught me to stand erect and to walk with my feet straight. I take to heart remarks by my students and their parents. Some of them would worry about grades and exams. I may not have the same concerns, and I may not agree with their conclusions, but I can't deny their feelings. I want to make sure that I am doing what's right and so I have to be mindful of their concerns and address them by changing my behavior, by making it understandable or at least, by showing respect for them. Take counsel to heart Conversing with: loving care As a boy, I was in love with a popular girl, Lacie Diaz, who said but a few words to me, but which I took to heart: Not to curse like the other kids, for it did not suit me. To wear wire-framed glasses, which I only could many years later. Another young woman taught me to say compliments. And a young man, Johnny Gonzales, after many years of saving me, told me to fight my own battles. Take to heart Jesus's ideas Conversing with: spirit free of this world Love your enemy, give everything away, be true to your wife, pray in twos and threes, engage God. Take up another's perspective Conversing with: other's wisdom I imagined God's perspective to think that "days of creation" might mean "divisions of everything". I embraced Jesus' perspective such as "love your enemy" and "give everything away". I committed myself to my parents' perspective of living our Lithuanian identity and culture. I respected my childhood peers' perspective of the meaningfulness of being cool. Take up my enemy's point of view Conversing with: excluded person When I take up the point of view of a person antagonistic to me, I can learn what they want from me and others. I can focus on that. Taking to heart people's perspective Conversing with: people's inclinations People weren't very interested to do good will exercises with me. They aren't very interested in my laboratory's services. I keep their interests in mind to know that my activities aren't central or basic in most people's lives and that I may be fundamentally on the wrong track. |
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Presuming others likewise
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Presuming others likewise Pamela McLean, June 10, 2011. Perhaps all we ever really do in dig deeper into our own understanding, and develop that, in response to someone else sharing their own explorations of understanding. Having someone who sounds urgent but not desperate Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: The key is having someone who's composed, someone who has experience. Someone who wants you to get out there and go at the guy but who doesn't sound so desperate or frustrated. And Angelo Dundee was that guy, who was in my corner, who said the right things at the right time." Make eye contact that you are OK Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: While you're doing that, make eye contact with the referee to give him [the] sense that you're OK. Trust the person you are improvising with Fresh Air interview of Steve Coogan, June 9, 2011: Coogan tells Fresh Air's Dave Davies that playing a version of himself was harder than playing fictional characters, in part because of the way he had to think about himself in front of a camera. "My biggest fear, really, was that it would be self-indulgent," he says. "Actors play my parents and actors play various people I meet, but the only real people are Rob and I. So it's a very difficult thing to try and be creative in that type of environment. But if you trust the person you're with, then you know when you're improvising you'll be able to create stuff and come up with ideas." |
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Major factors, major issues
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First, nearly all situations, even the more complex and dynamic, are driven by only a few major factors. Factors are things, circumstances or conditions that cause something to happen. Factors, in turn, beget issues, which are points or questions to be disputed or decided. ... Major factors and major issues are the navigational aids of analysis; they tell us where our analysis should be headed. And they normally change as we become aware of new information and gain a deeper understanding of the problem. If we lose sight of them, we lose our way in the analytic process. We should concentrate our analytic efforts on the major factors and issues. Studying subtleties (lesser factors and issues), incorporating them into our analysis, and weighting their impact on the situation and its possible outcomes are usually a waste of time because subtleties never play a significant role. ... The first insight, therefore, is to create at the outset and maintain throughout the problem-solving process a list of major factors and issues, adding and deleting items as necessary. |
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Appreciate others
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Appreciate others As we give, we appreciate what others can and do give, for we may have them give on our behalf. When I go traveling and staying with people it is good to have some kind of present to express my appreciation. So I try to think through and carry gifts that would be special but for people who I might not even know yet. So in Lithuania I sometimes buy small pieces of art or blank leather bound books. My father taught me and my brother how to play chess. He would not play easy but rather he would handicap himself by removing several of his pieces from the board, enough to even out our chances. As we got better and would beat him, he would reduce the handicap. First it was a queen and two rooks, later a queen, then a rook, then a bishop, then a knight, until one day we could beat him. Because you can't learn from somebody who is not playing to win! I was always impressed by how smart and kind he was as a teacher. My parents have a principle never to give me a loan, even a short term loan that might be very helpful. However, they are very steadfast in giving me gifts if I need help, even if they are unhappy about it. When I was in Lithuania, during the independence movement, I saw that the newly forming youth organizations could work together and be stronger. I thought that they could share things like video equipment. But it turned out that nobody ever wanted to share, each deeply preferred to have their own. After the war in Soviet-occupied Lithuania there was not much food and our relatives ate from one bowl and they say the children who ate slower ended up smaller. I worked a summer for a Lithuanian organization on a special project making cassettes for the blind. One weekend they wanted me to come in to help stuff envelopes. I refused because I said it wasn't part of my job description. They were unhappy. But I let them down. |
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Hear
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Hear Listen Asking our group Consider what others say Get feedback from people Identify with the words of a song Listening to elders Listening to others Listening to what you have to say Notice what a person does Taking music and words to heart Others' awareness Considering who knows Considering who needs to know what Knowing if someone is paid Others' intentions Asking about intentions Noting choices people could have made Noting choices people make Serving others Think of others as equals |
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Accepting God's justice
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Accepting God's justice We may have real reason to note that what God does in particular cases is unfair. Yet by accepting God's justice, even God's unfairness to us, we allow him more possibilities and we participate with him to create an outstanding justice in the big picture. We thus heal ourselves, and next we can heal others. Adding our slack, which is mercy. Dependence on circumstances Matthew 9:14 Then John's disciples came to him, saying, "Why do we and the Pharisees fast often, but your disciples don't fast?" 9:15 Jesus said to them, "Can the friends of the bridegroom mourn, as long as the bridegroom is with them? But the days will come when the bridegroom will be taken away from them, and then they will fast. Add his faith where faith is lacking Matthew 8:23 When he got into a boat, his disciples followed him. 8:24 Behold, a violent storm came up on the sea, so much that the boat was covered with the waves, but he was asleep. 8:25 They came to him, and woke him up, saying, "Save us, Lord! We are dying!" 8:26 He said to them, "Why are you fearful, O you of little faith?" Then he got up, rebuked the wind and the sea, and there was a great calm. Be thankful to God Matthew 11:25 At that time, Jesus answered, "I thank you, Father, Lord of heaven and earth, that you hid these things from the wise and understanding, and revealed them to infants. 11:26 Yes, Father, for so it was well-pleasing in your sight. God's justice rewards those who have not yet been rewarded Matthew 6:1 "Be careful that you don't do your charitable giving before men, to be seen by them, or else you have no reward from your Father who is in heaven. 6:2 Therefore when you do merciful deeds, don't sound a trumpet before yourself, as the hypocrites do in the synagogues and in the streets, that they may get glory from men. Most certainly I tell you, they have received their reward. 6:3 But when you do merciful deeds, don't let your left hand know what your right hand does, 6:4 so that your merciful deeds may be in secret, then your Father who sees in secret will reward you openly. How others value followers Matthew 10:24 "A disciple is not above his teacher, nor a servant above his lord. 10:25 It is enough for the disciple that he be like his teacher, and the servant like his lord. If they have called the master of the house Beelzebul, how much more those of his household! Know the opposing culture, thus know what they are thinking Jesus is familiar with a culture based on faith (the Kingdom of Heaven) and also with a culture that is not (the Kingdom of Man). If people don't show faith, then he knows they are of the opposing culture, he knows how they are interpreting things and he knows what they are thinking. Matthew 9:2 Behold, they brought to him a man who was paralyzed, lying on a bed. Jesus, seeing their faith, said to the paralytic, "Son, cheer up! Your sins are forgiven you." 9:3 Behold, some of the scribes said to themselves, "This man blasphemes." 9:4 Jesus, knowing their thoughts, said, "Why do you think evil in your hearts? Men reward each other in this world Matthew 6:1 "Be careful that you don't do your charitable giving before men, to be seen by them, or else you have no reward from your Father who is in heaven. 6:2 Therefore when you do merciful deeds, don't sound a trumpet before yourself, as the hypocrites do in the synagogues and in the streets, that they may get glory from men. Most certainly I tell you, they have received their reward. Showing good will and allowing others to condemn you Jesus, John the Baptist and Paul show good will to others by satisfying their norms, at least to an extent, and yet allow others to condemn them even so. Matthew 11:16 "But to what shall I compare this generation? It is like children sitting in the marketplaces, who call to their companions 11:17 and say, 'We played the flute for you, and you didn't dance. We mourned for you, and you didn't lament.' 11:18 For John came neither eating nor drinking, and they say, 'He has a demon.' 11:19 The Son of Man came eating and drinking, and they say, 'Behold, a gluttonous man and a drunkard, a friend of tax collectors and sinners!' But wisdom is justified by her children." The Son of Man is nowhere acceptable Jesus is the Son of Man in that Man teaches him by making an example out of him. Thus the Son of Man is never acceptable. This is an injustice that Jesus bears for the sake of healing. Matthew 8:19 A scribe came, and said to him, "Teacher, I will follow you wherever you go." 8:20 Jesus said to him, "The foxes have holes, and the birds of the sky have nests, but the Son of Man has nowhere to lay his head." Unfairness towards the good son Luke 15:25 "Now his elder son was in the field. As he came near to the house, he heard music and dancing. 15:26 He called one of the servants to him, and asked what was going on. 15:27 He said to him, 'Your brother has come, and your father has killed the fattened calf, because he has received him back safe and healthy.' 15:28 But he was angry, and would not go in. Therefore his father came out, and begged him. 15:29 But he answered his father, 'Behold, these many years I have served you, and I never disobeyed a commandment of yours, but you never gave me a goat, that I might celebrate with my friends. 15:30 But when this, your son, came, who has devoured your living with prostitutes, you killed the fattened calf for him.' 15:31 "He said to him, 'Son, you are always with me, and all that is mine is yours. 15:32 But it was appropriate to celebrate and be glad, for this, your brother, was dead, and is alive again. He was lost, and is found.'" What a disciple should expect Matthew 10:24 "A disciple is not above his teacher, nor a servant above his lord. 10:25 It is enough for the disciple that he be like his teacher, and the servant like his lord. If they have called the master of the house Beelzebul, how much more those of his household! What does it take for people to repent? People may repent if they wrong the righteous. Matthew 23:33 You serpents, you offspring of vipers, how will you escape the judgment of Gehenna? 23:34 Therefore behold, I send to you prophets, wise men, and scribes. Some of them you will kill and crucify; and some of them you will scourge in your synagogues, and persecute from city to city; 23:35 that on you may come all the righteous blood shed on the earth, from the blood of righteous Abel to the blood of Zachariah son of Barachiah, whom you killed between the sanctuary and the altar. What does mercy require? Matthew 9:11 When the Pharisees saw it, they said to his disciples, "Why does your teacher eat with tax collectors and sinners?" 9:12 When Jesus heard it, he said to them, "Those who are healthy have no need for a physician, but those who are sick do. 9:13 But you go and learn what this means: 'I desire mercy, and not sacrifice,' for I came not to call the righteous, but sinners to repentance." What is one's cross Matthew 10:38 He who doesn't take his cross and follow after me, isn't worthy of me. What to expect of a disciple Matthew 10:24 "A disciple is not above his teacher, nor a servant above his lord. 10:25 It is enough for the disciple that he be like his teacher, and the servant like his lord. If they have called the master of the house Beelzebul, how much more those of his household! What will Jesus give to those who are burdened Matthew 11:28 "Come to me, all you who labor and are heavily burdened, and I will give you rest. Whether to be afraid Matthew 10:29 "Aren't two sparrows sold for an assarion coin? Not one of them falls on the ground apart from your Father's will, 10:30 but the very hairs of your head are all numbered. 10:31 Therefore don't be afraid. You are of more value than many sparrows. Who bears the brunt of hate Matthew 10:22 You will be hated by all men for my name's sake, but he who endures to the end will be saved. Who should come to Jesus Matthew 11:28 "Come to me, all you who labor and are heavily burdened, and I will give you rest. Why did he come? Matthew 20:24 When the ten heard it, they were indignant with the two brothers. 20:25 But Jesus summoned them, and said, "You know that the rulers of the nations lord it over them, and their great ones exercise authority over them. 20:26 It shall not be so among you, but whoever desires to become great among you shall be your servant. 20:27 Whoever desires to be first among you shall be your bondservant, 20:28 even as the Son of Man came not to be served, but to serve, and to give his life as a ransom for many." What makes exceptions possible The Sabbath is a day of exceptions, and the Son of Man is, by his slack, his mercy, the one who makes exceptions possible. Sacrifice is just a sign of slack. Thus there is no guilt for those who the Son of Man gives slack to. Matthew 12:7 But if you had known what this means, 'I desire mercy, and not sacrifice,' you would not have condemned the guiltless. 12:8 For the Son of Man is Lord of the Sabbath." |
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Polynomials
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Polynomials We can expand terms as polynomials, as with "and" and "or", and thus create equations that construct and relate roots. (Or, And, method of undetermined coefficients, expansion, construction) Addition means combine like units, list different units We can practice this principle by considering "units", but also numbers used as units ("tens", "thousands", "millions", "sevenths", "percentages"), imaginary units ("zillions"), any numbers ("twos", (-1)), unknown numbers ("X"), squares of unknown numbers ("X**2"). We can mix units such as 2x + 3y -5x + 4y and likewise. Gospel Math. Designing algorithms with index cards I have the student write on an index card, "I'm thinking of a number". Then they create subsequent index cards, each with their own instruction: "Double the number", "Add 5 to the number", "Halve the number", "Square the number" and so on. Then I have them input a number and get the output and write down all of the intermediate answers. They do this for several numbers. Then we can consider an abstract input X. We can discuss if the algorithm can be simplified, and can it be reversed. Basic Math. And Simple multiplication. If there are a varieties of soup and b varieties of salad, then there are ab possible ways to order a meal of soup and salad. pg.205 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Combinatorial tactics But mostly, combinatorics is a tactical game. You have already learned the fundamental tactics of multiplication, division, addition, permutations, and combinations. pg.212 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Complement Each selection of 10 winners from a group of 17 is simultaneously a selection of 7 losers from this group. ... The Symmetry Identity ... (n r) = (n n-r) ... The combinatorial argument shows why it is true, while algebra merely shows us how it is true. pg.209 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Complete the square Counting the Complement A particular application of [changing your point of view] in counting problems is If the thing you wish to count is confusing, try looking at its complement instead. How many n-bit strings contain at least 1 zero? ... an easier approach is to first note that there are 2**n possible n-bit strings, and then count how many of them contain no zeros. A complement is possible when there is a closed system, which is what polynomials define. pg.225 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Division Algorithm Let f(x) and g(x) be polynomials in K[x], where K is one of Z, Q, R, C or Zn. Then f(x) = Q(x)g(x) + R(x) where Q(x), R(x) are elements of K[x] and the degree of R(x) is less than the degree of g(x). ... For example ... by doing "long division" we get ... The important thing is that the quotient Q(x) is also in Z[x], i.e., also has integer coefficents. pg.180 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Eliminate radicals Assume that you can make good progress. What will the result look like? If this expression is to simplify, we will probably be able to eliminate radicals. If we multiply any two terms, we can use the difference of two squares formula and get expressions which contain only one radical. pg.166 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Extracting squares The tactic of extracting squares includes many tools in addition to completing the square. ... It is easy enough to see how this works, but why is another matter. For now, hindsight will work: remember that many useful squares lurk about and come to light when you manipulate "cross-terms" appropriately (making them cancel out or making them survive as you see fit). pg.164-165, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Factor Theorem A reinterpretation of this theorem from a problem solver's perspective is To know the zeros of a polynomial is to know the polynomial. In other words, if you don't know the zeros of the polynomial under consideration, either expend some effort to find them, or shift your focus to a new polynomial whose zeros are apparent. pg.182 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Infinitely many primes There are infinitely many primes ... We start by assuming that there are only finitely many primes p1, p2, ... , pN. Now (the ingenious crux move!) consider the number Q = (p1p2...pN) + 1. Either it is prime, or it is composite. ... We construct a value p1p2...pN that satisfies all conditions exactly and then adjust it ever so slightly +1 so that it satisfies none of them. In other words, we create a model p1p2...pN that satisfies all and then we force that model to break down. We design this value with regards to the closed system as a whole. pg.244 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Modulo n filtering Another essential tactic [in analyzing diophantine equations] is to "filter" the problem modulo n for a suitably chosen n. This tactic often helps to show that no solutions are possible, or that all solutions must satisfy a certain form. (Use of the division algorithm is closely related to the factor tactic.) ... the complete theory of Pythagorean triples: Find all solutions to x**2 + y**2 = z**2 ... we can assume that our solution is not just primitive, but relatively prime in pairs ... Next, a little parity analysis; ie., let's look at things modulo 2. Always begin with parity. You never know what you will discover. ... exactly one of x and y is even... pg.265-267 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Number theoretic functions Of the infinitely many functions with domain N, we will single out a few that are especially interesting. Most of these functions are multiplicative ... f(ab) = f(a)f(b) whenever a and b are relatively prime ... Define sigma_r(n)= sum over divisors d of n of d**r ... Let F(n) = sum over divisors d of n of f(d). If f is multiplicative, then F will be multiplicative as well. Define phi(n) to be the number of positive integers less than or equal to n which are relatively prime to n. ... We can use the principle of inclusion-exclusion to evaluate phi(n) ... We define [the Mobius function] mu(n) = 1 if n=1, =0 if p**2 divides n for some prime p, = (-1)**r if n = p1p2...pr, each p a distinct prime. This is a rather bizarre definition, but it turns out that the Mobius function very conveniently "encodes" PIE. ... show that sum over divisors d of n of mu(d) = 1 if n=1; =0 if n>1. The values of mu(n) alternate sign depending on the parity of the number of prime factors of n. This is what makes the Mobius function related to PIE. ... phi(n) = sum over divisors d of n of mu(d)(n/d). pg.257-261 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Or Simple addition. If there are a varieties of soup and b varieties of salad, then there are a+b possible ways to order a meal of soup or salad (but not both soup and salad). pg.205 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Substitution Let a, b, c be positive real numbers such that abc=1. Prove that 1/(a**3(b+c)) + 1/(b**3(c+a)) + 1/(c**3(a+b)) >= 3/2. ... What is the worst thing about this problem? It is an inequality involving fairly ugly fractions. Wishful thinking tells us that it would be nicer if the fractions either were less ugly or did not exist at all. ... There is a pretty obvious substitution - but only obvious if you have the idea of substitution in the forefront of your consciousness. The substitution is x=1/a, y=1/b, z=1/c, which transforms the original inequality (use the fact that xyz=1) into x**2/(y+x) + y**2/(z+x) + z**2/(x+y) >= 3/2. This inequality is still not that easy to deal with, but the denominators are much less complicated, and the problem has been reduced in complexity. pg.170 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Symmetric functions of zeroes we can get a series of expressions for the coefficients of a polynomial in terms of its zeros x**4 + a3x**3 + a2x**2 + a1x + a0 = (x-p)(x-q)(x-r)(x-s) ... Equating like terms, we have a3 = - (sum of all zeros), a2 = +(sum of all products of two different zeros), a1 = -(sum of all products of three different zeros), a0 = +(product of the zeros), where it is understood that "different" here has a purely symbolic meaning; i.e. we multiply only zeros with different labels, such as p and q, even if their numerical values are the same. ... we see, the pattern, and can write the formulas in general ... These formulas are very important, and should be committed to memory ... note the role that the power of -1 plays. pg.184-185 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. The Fundamental Theorem of Algebra The Factor Theorem above tells us that x-a is a factor of polynomial P(x) if a is a zero. But how do we know if a polynomial even has a zero? The Fundamental Theorem of Algebra guarantees this: Every polonomial in C[x} has at least one complex zero. pg.182 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Viewing a problem modulo m Viewing a problem modulo m for a suitably chosen m is a wonderful simplification tactic because it reduces the infinite universe of integers to the finite world of Zm. You have encountered this idea before with parity ... which is just the case m=2 ... Often (but not always) we turn to prime values of m, since primes are simpler, more "fundamental" objects which are generally easier to understand. In general, When beginning a number theory investigation, assume that the variables are prime or at least relatively prime. Often the general case follows from the prime case, with just a few "technical" details. ... Fermat's Last Theorem. Let n>=3. Prove that the equation x**n + y**n = z**n has no non-zero integer solutions. ... we shall point out two simplifications ... Without loss of generality, n is prime ... we may assume that ... x, y, z have no common factor (other than 1). ... One reason that primes are so convenient is that unique "multiplicative inverses" exist. Viewing modulo m emphasizes the closed nature of a system. pg.252-253 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc x**2 + y**2 = 13 This is a diophantine equation, so we should try factoring. "But it doesn't factor", you say. Sure it does! We can write (x+yi)(x-yi) = 13, where i is of course equal to the square root of -1, The only problem, and it is a huge one, is that i is not an integer. But let's stay loose, bend the rules a bit, and make the problem easier. It is true that the square root of -1 is not an integer, but what if we looked at the problem in Z13? Notice that 5**2 = 25 == -1 (mod 13). In other words, i "makes sense" modulo 13. pg.274-275 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc |
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Plan - Explicitly Commit
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Plan - Explicitly Commit Graphic Gameplan agree on executable action plan RACI Matrix clarify responsibility, improve morale Scenario Slider identify and prepare for key scenarios Who/What/When Matrix make clear commitments WhoDo brainstorm, plan and prioritize actions |
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Believing in believing in believing
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Believing in believing in believing Empathizing with their views. Appreciate people's commitments, their investigations, endeavors and positions, and integrate them. Allowing us to let go of our positions Conversing with: conviction Create public space Conversing with: principles By organizing Minciu Sodas, an online laboratory, entirely public, I had the chance to watch people grow, mature, learn as they acted on their beliefs and reflected on the consequences. Disagreement of views Conversing with: contradiction In organizing the good will exercises, I learned early on that, for a given issue, we wouldn't agree what truth came from the world. Often I felt that it was clear that people couldn't see past their own principle, couldn't acknowledge the dead end that they were in. People kept focusing on whatever got their buttons. They would disagree about an issue, such as gang violence, but actually, when you heard them out, in their minds they were approaching that issue from their own very personal angle, often only tentatively linked to the "hot button" words. So I developed a way to ask about their feelings, to balance negative feelings with positive feelings, and derive their real issue issue from the subject matter of their feelings. Empathy for disputants Conversing with: conviction By having empathy for people, I was aware how intensely they clung to their truths, often unknowingly, the truth of the world. Internal dialogue with someone dear or critical Conversing with: convictions I often have dialogues in my mind with people I know. I suppose they are sparked by my feelings and my conscience. In reflect about events in my life, I will feel a wish to say to somebody the truth. But am I being fair to them? What would they say? My mind provides their reply, what they might say. And then I think and reply. In this way, my mind rehearses conversations that may be several minutes long. I recall doing this in grade school as I rode home on the bus, thinking to myself. I have had many conversations with my parents, with good friends, with girls and women I have been in love with, and even certain peripheral individuals who are critical of me. I have a crush on a woman, but then imagine, what if she had an abortion, would I still love her, and what would I say? I have had many conversations in my mind with my parents about the things I do that irk them, such as not getting a hair cut or my difficulty in making a living or my adventures in life or my philosophical ideas, or their aspirations for me, such as being Lithuanian or being a good person, kind and of good will. I learned that people in my mind were more real, vibrant, honest, direct, intense, distilled, than they were in real life. True, in real life, when they said something unexpected, I had to adjust my understanding of them, yet in mind, there could also be something similar, when I managed to see them in a new light, and indeed, I would converse with them in my mind until I could resolve all of my feelings. I noticed that my conversations with God are quite similar, and I could explain to others that, in that sense, God is very real to me, just as the people in my mind are more real than they are in real life. In 2011, in speaking about this with my father, I learned that he actually doesn't have any such conversations. I suppose I have them because I was never able to talk with my parents and others as much as I liked about what I cared about, and because when I did have a chance, I didn't want the conversations to go on unhelpful or hurtful tracks. Reconsider Conversing with: stubbornness Sometimes I take up a position and defend a principle with great passion in my mind, especially when I feel attacked. I may find many good reasons to stay firm. But then, when I feel that I've weathered the storm, a mild voice grows in me that I might now relent, that I don't have to look at things so starkly, that the issue isn't the only one or the most important, that time brings forth other issues. Ultimately, I always do relent, when I myself feel ready. This also helps me to avoid getting into arguments, to not be afraid to see them through honestly and vigorously, but to be prompt and resolute in letting go of them. Choo-choo train One day, perhaps for my birthday or for Christmas, my parents got me a black plastic choo-choo train that I could sit on and pretend to ride around the apartment. I must have been three years old because we were still living at the Page Street apartments in Buena Park. It was the first impressive toy that I had and I treasured it because it was "mine", because it was given to "me", and it was an acknowledgment that I should "have" things. My parents invited over Robbie's family. He was a little boy who used to live at the apartments, with whom I had played sometimes, but didn't really know, as I basically played by myself. My mother and father asked that I show him my choo-choo train and let him ride it. I said No. I felt it was mine and I wanted to keep it that way. They insisted. They said I was bad. This disturbed me greatly. My parents said later that I refused to talk to them for three days. But I remember some of what I was thinking: that it was deceitful of them to give me things and later to say what I must do with them; it was the same as to take them from me; that thus no notion of property could ever be secure; and thus there could be no ground for civilization; and this made progress untenable; yet progress was our highest ideal. I think finally they allowed that I had not behaved badly. And then I relented that my parents were loving people and that I should not make too much of this. But I told myself that I would never get attached to things. Relent Conversing with: injustice When I was fourteen or so, I had a chess coach, Robert Snyder, who one day locked the room, sat next to me and told me I was "special". I told him to get away from me and then I told my mother it was improper and we should call the police. She resisted, explaining that it could destroy his life, and so I didn't call the police and even stayed on as his student, with nothing like that recurring. Decades later he was convicted of molesting minors. I learned that I could be not so harsh. In college, I had a roommate who sold pot, said he wouldn't do it anymore, but took on a new shipment anyways. I was appalled that he lied, and I wanted to call the police, but my other roommate, who had made the discovery, had us talk to housing instead. He was kicked out of housing, but stayed in the university and never faced the police. I realized what a gross injustice there was compared to how Black American youth were treated. Writing letters in my mind I remember writing many letters in my mind, notably while I would ride my bike through the orchards of California's Central Valley and towards the Sierra mountains, when I worked in Visalia in 2001 as a software developer. Often, I never had a chance to write these letters, so they kept evolving. Relating our positions Conversing with: capability Collect investigatory questions Conversing with: person's vitality What is a question that you don't know the answer to, but wish to answer? I have collected such investigatory questions from about 200 people. They are deep, beautiful, profound, inspiring questions. They have enormous variety. They are related to people's deepest values. They show how a person is growing. They can be studied in terms of their structure. Collecting endeavors Conversing with: aims In 2007-2009, in leading the Minciu Sodas laboratory, I collected endeavors from more than 100 participants. This helped me know who actually wanted support and how we might organized around them. What endeavors were key, strategically? I noticed how narrower endeavors supported broader endeavors. Identify strategic endeavors Conversing with: productiveness I drew a map of endeavors for the Minciu Sodas laboratory and our participants to show how they can all be related. Then, as Direktorius, I chose strategic endeavors which I should invest my efforts in because they would best leverage our efforts on behalf of all of our endeavors, including my quest to make Minciu Sodas viable so that I could make a living.  Relate endeavors Conversing with: might In 2007, I asked participants of Minciu Sodas, my online laboratory, what did they want to achieve? Then I organized the endeavors with a diagram, a map, where broader endeavors led to narrower endeavors. The broadest endeavor I took to be God's endeavor, to reach the hard to reach. |
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Integrating some of us
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Integrating some of us Honoring a request She expressed no bitterness about her illness or about my inability to cure her. During her last hospitalization, she made one request. She lay in the hospital bed, her chest heaving to get air. We held hands. "Doctor Ansen?" Her eyes sparkled. "Ah want ta see yo chile. Please, can ah see yo chile?" We hatched a plan. On the next Saturday morning, I brought my son to County Hospital for her to see. She came to the sixth floor window of the Medical A building. I stood outside and held my infant son up over my head like a gift offering to the gods. She smiled and waved at us from the sixth floor window, oxygen tubes dangling from her nose. We waved back. I cried when she died a few days later. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.1934 Let each other offend each other Fresh Air interview of Steve Coogan, June 9, 2011: So the two comedians decided to give each other the artistic license to offend each other while filming. "What makes it interesting is that there's an edge to it and a discomfort to it that makes it engaging," Coogan says. "It's not just a couple of actors saying, 'Get a load of me. I'm laughing at myself.' There are a couple of moments where I find Rob irritating — genuinely — and I respond naturally, but not the way that I would in reality." Revealing that you have been abused Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: I saw an episode of Oprah and [former child TV star] Todd Bridges finally came forth and said that he was sexually abused. I hear people always say that when you surrender and admit these things, it's a sense of freedom too. What I miss Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: "Of all the things I miss in boxing, I miss the preparation of a bout, I miss choreographing tactics and moves and things like that," he says. "I miss all of my guys, my entourage being around me and working out with me, getting in better shape. |
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Consider the analysis underlying numbers
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We must resist at all costs being captivated by the numbers (like weights and utility values) we use in applying some of these techniques. We must constantly focus on and fully understand the analysis underlying these numbers, for they inevitably tend to take on a life of their own and to drive, not reflect, our analysis. According to Laplace, if we're trying to determine which of two or more outcomes will occur, but we don't have reliable evidence to judge which is more likely, we should assume the probability is equal for all outcomes. ... when we have all the data, as in a deterministic problem, calculate probability by arithmetic computation. Conditionally dependent events are those in which the occurrence of one event depends upon the occurrence of another. Multiply probabilities of the "and" type. Mutually exclusive events preclude one another. Add probabilities of the "either... or..." type. If they are completely exhaustive, then they should add up to 100%. Whether we are analyzing a problem alone or collaborating with others, we should make it a rule to highlight all probability expressions and translate them into percentiles. ... but only in the analytic phase. Never use percentiles in final written products unless ... the numbers are based on definitive evidence and precise calculations. |
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Learn what is fair
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Learn what is fair When I started my business I wanted my colleagues in Lithuania to earn the same as me. But then I realized that I am the one investing, sticking my neck out, running the risk. And they themselves stated that first I need to make money, and then they can. In general, different people helped or not in different ways, as they were able and felt comfortable. I was running a social enterprise, I support a community center, I didn't owe the government any money, but it was going to destroy me through this paperwork. |
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Value others
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Value others Appreciating people Appreciating people's leadership Appreciating people's skills Realizing what others can do Recognizing what others could do Recognizing who should be involved Proportionate scaling Proportionality lets us conduct an experiment on one scale and have it yield results on another scale, multiplicatively.  Proportionality: Erastothenes measured the Earth's circumference One key step in Erasthotenes's estimate is the measurement, in different places, of the extreme of a periodic phenomenon. Another key step is the application of proportionality to this extreme. Erastothenes estimated the Earth's circumference. He knew that in the Ancient Egyptian city of Swenet, on the summer solstice, at noon, the sun would reflect on the water at the bottom of the well. Whereas in his city of Alexandria, 5000 stadia away to the North, on the summer solstice, at noon, the shortest shadow of a pole would form a ratio of 1 to 50. Thus he could estimate by geometry that Alexandria and Swenet were separated by 1/50 of the Earth's circumference. Erastothenes thus depended on a noteworthy fact about Swenet (that it lies on the Tropic of Cancer, as we would say) that makes it different from Alexandria. He was able to entertain an idea of the world as a sphere. He was able to rely on a regularity that held in two different locations, that the shortest shadow is on the summer solstice. He was able to express that distinction as a mathematical ratio. And he was able to project that distinction of local results as a consequence of a global context. The idea is that the same measurement will yield different results in a different context. And he focused that difference on a particular parameter, the angle. The small angle in the measurement is equal to the earth's angle is equal to the earth's sector. Note that Erastothenes's measurement depends on the Sun's rays being essentially parallel to the Earth. This indicates that the diameter of the Sun and the distance between the Sun and the Earth are extremely large compared to the diameter of the Earth. This last concept appears in the sad last line of this Chinese poem: http://www.chinese-poems.com/bo3.html Feelings on Watching the Moon Bai Juyi The times are hard: a year of famine has emptied the fields, My brothers live abroad- scattered west and east. Now fields and gardens are scarcely seen after the fighting, Family members wander, scattered on the road. Attached to shadows, like geese ten thousand li apart, Or roots uplifted into September's autumn air. We look together at the bright moon, and then the tears should fall, This night, our wish for home can make five places one. |
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Healing
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Healing Jesus heals many people and they are made whole and do not stumble because of him. Yet others are upset that he heals on the Sabbath and thus stumble because of him. Jesus keeps believers from stumbling. God gives the reward, and the reward is Jesus. The good does not receive the reward, because they had the good from the beginning, for the reward is sharing in all things, and so Jesus appreciates the logic, for Jesus had appreciated the wealth of God from the beginning, and received it likewise, and so could have no reward, except through those who were given to him by God. Am I causing others to stumble? Jesus doesn't want others to stumble because of him, yet it may happen, and yet he suggests that there are those who he doesn't cause to stumble, and thus more fundamentally, people need not stumble. Luke 7:19 John, calling to himself two of his disciples, sent them to Jesus, saying, "Are you the one who is coming, or should we look for another?" ... Go and tell John the things which you have seen and heard: that the blind receive their sight, the lame walk, the lepers are cleansed, the deaf hear, the dead are raised up, and the poor have good news preached to them. 7:23 Blessed is he who finds no occasion for stumbling in me. Anticipating Matthew 17:24 When they had come to Capernaum, those who collected the didrachma coins came to Peter, and said, "Doesn't your teacher pay the didrachma?" 17:25 He said, "Yes." When he came into the house, Jesus anticipated him, saying, "What do you think, Simon? From whom do the kings of the earth receive toll or tribute? From their children, or from strangers?" Distinguish between people and their demons Matthew 8:28 When he came to the other side, into the country of the Gergesenes, two people possessed by demons met him there, coming out of the tombs, exceedingly fierce, so that nobody could pass that way. 8:29 Behold, they cried out, saying, "What do we have to do with you, Jesus, Son of God? Have you come here to torment us before the time?" 8:30 Now there was a herd of many pigs feeding far away from them. 8:31 The demons begged him, saying, "If you cast us out, permit us to go away into the herd of pigs." 8:32 He said to them, "Go!" They came out, and went into the herd of pigs: and behold, the whole herd of pigs rushed down the cliff into the sea, and died in the water. Encourage others to pray for intercession Matthew 9:37 Then he said to his disciples, "The harvest indeed is plentiful, but the laborers are few. 9:38 Pray therefore that the Lord of the harvest will send out laborers into his harvest." Give as was given to you By giving freely as was freely given to us, we extend the giving of the giver, support it, confirm it and include ourselves in it. Matthew 10:8 Heal the sick, cleanse the lepers, and cast out demons. Freely you received, so freely give. How Jesus treats us Jesus treats us as equals, as we treat him. Matthew 10:32 Everyone therefore who confesses me before men, him I will also confess before my Father who is in heaven. 10:33 But whoever denies me before men, him I will also deny before my Father who is in heaven. How will one be saved Fleeing is key to being saved, and the variety of places to go, and the fact that there are more places than needed to "teach by making an example of". Matthew 10:22 You will be hated by all men for my name's sake, but he who endures to the end will be saved. 10:23 But when they persecute you in this city, flee into the next, for most certainly I tell you, you will not have gone through the cities of Israel, until the Son of Man has come. Pray to God for vocation Matthew 9:37 Then he said to his disciples, "The harvest indeed is plentiful, but the laborers are few. 9:38 Pray therefore that the Lord of the harvest will send out laborers into his harvest." The down-to-earth nature of the good Luke 7:19 John, calling to himself two of his disciples, sent them to Jesus, saying, "Are you the one who is coming, or should we look for another?" ... Go and tell John the things which you have seen and heard: that the blind receive their sight, the lame walk, the lepers are cleansed, the deaf hear, the dead are raised up, and the poor have good news preached to them. 7:23 Blessed is he who finds no occasion for stumbling in me. What are the consequences of believing? Matthew 7:24 "Everyone therefore who hears these words of mine, and does them, I will liken him to a wise man, who built his house on a rock. 7:25 The rain came down, the floods came, and the winds blew, and beat on that house; and it didn't fall, for it was founded on the rock. What confirms the Kingdom of Heaven? Matthew 10:8 Heal the sick, cleanse the lepers, and cast out demons. Freely you received, so freely give. What does God control Matthew 10:29 "Aren't two sparrows sold for an assarion coin? Not one of them falls on the ground apart from your Father's will, 10:30 but the very hairs of your head are all numbered. 10:31 Therefore don't be afraid. You are of more value than many sparrows. What happens to one who seeks or loses one's life Matthew 10:39 He who seeks his life will lose it; and he who loses his life for my sake will find it. What is more relevant? Spiritual health is more relevant than physical health. Jesus naturally is alert to a person's spiritual well being and focuses on that rather on their physical well-being. Thus he takes liberty to interpret the person's sadness or happiness as regarding his sins, not his illness. Matthew 9:2 Behold, they brought to him a man who was paralyzed, lying on a bed. Jesus, seeing their faith, said to the paralytic, "Son, cheer up! Your sins are forgiven you." What is pleasing to God The Kingdom of Heaven is hidden from the wise and understanding as part of the union of all through God's impartiality. It is not for those who understand (as God the Father) but who are coming to understand (as the Son of Man who is ultimately, upon understanding, the Son of God). Matthew 11:25 At that time, Jesus answered, "I thank you, Father, Lord of heaven and earth, that you hid these things from the wise and understanding, and revealed them to infants. 11:26 Yes, Father, for so it was well-pleasing in your sight. What is the reward for leaving everything to follow Jesus Matthew 19:27 Then Peter answered, "Behold, we have left everything, and followed you. What then will we have?" 19:28 Jesus said to them, "Most certainly I tell you that you who have followed me, in the regeneration when the Son of Man will sit on the throne of his glory, you also will sit on twelve thrones, judging the twelve tribes of Israel. 19:29 Everyone who has left houses, or brothers, or sisters, or father, or mother, or wife, or children, or lands, for my name's sake, will receive one hundred times, and will inherit eternal life. 19:30 But many will be last who are first; and first who are last. Who is blessed Jesus thinks about who is blessed. He feels that those who aren't disappointed in him will be blessed. Matthew 11:6 Blessed is he who finds no occasion for stumbling in me. Who needs a physician? Matthew 9:11 When the Pharisees saw it, they said to his disciples, "Why does your teacher eat with tax collectors and sinners?" 9:12 When Jesus heard it, he said to them, "Those who are healthy have no need for a physician, but those who are sick do. 9:13 But you go and learn what this means: 'I desire mercy, and not sacrifice,' for I came not to call the righteous, but sinners to repentance." Who will be saved Matthew 10:22 You will be hated by all men for my name's sake, but he who endures to the end will be saved. |
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Vector space
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Vector space We can consider a vector space in which any point can serve as the center for a basis. (Superposition, linear combination, duality) Indicator function We shall now present a proof of the complement function of PIE, using the "binary" language of indicator functions. Recall that the indicator function of A is denoted by 1A and is a function with domain U (where U is a "universal set" containing A) and range {0,1} defined by 1A(x) = 0 if x not in A, =1 if x in A, for each x in U. ... 1A(x)1B(x) = 1 A-intersect-B(x). 1 - 1A(x) = 1 complement-of-A(x). ... the product of two indicator functions is the indicator function of the intersection of two sets and the indicator function of a set's complement is just one subtracted from the indicator function of that set. ... Define the function g(x) = (1 - 1A1(x))(1 - 1A2(x)) (1 - 1A3(x)) ... N0 = sum over x in U of g(x). pg.230-31 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Linear combination ...the result was attained in two steps. First, we were lucky enough to spot a particularly accessible case, a special situation, and gave a solution well adapted, but restricted, to this special situation ... Then, by combining particular cases to which the restricted solution is applicable, we obtained the full, unrestricted solution, applicable to the general case... The first step deals with a particular case which is not only especially accessible, but also especially useful; we can appropriately call it a leading particular case: it leads the way to the general solution. The second step combines particular cases by a specific algebraic operation. ... n particular solutions, after being multiplied by given constants, are added to form the general solution. ... we add and subtract equations dealing with the special situation to obtain the general proof. Let us call the algebraic operation employed ... linear combination or superposition. We may use the terms introduced to outline our pattern: Starting from a leading special situation we attain the general solution by superposition of particular cases. "Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving" by George Polya, 1962, John Wiley & Sons. |
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Include - Care
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Include - Care Air Time Mastermind attract solutions to pressing problems Brainwriting include everybody in evolving an idea Button pay attention to everyone Open Space network around active participants and their purposes Quaker Conversation invite everybody and let conversation flow Talking Chips include every voice equally |
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Believing in believing in believing in believing
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Believing in believing in believing in believing Fostering the spirit amongst us. Create space to include everybody as they are, both actively immersed and passively reflected, alternatively, so God's spirit may speak through them. Creating an environment to generate an idea Create conditions Conversing with: inspiration In 1995, encouraged by Joe Damal, I wanted to start applying my philosophy practically. I decided to consider situations where I wanted to follow the truth of the heart, as opposed to the world. But where to start? I gave myself a year to develop good will exercises. Every two weeks I invited friends and all who might be interested to help me think this through. In between, I would write up and mail out and email out a newsletter reflecting on the last meeting and preparing for the next meeting. I conducted the meetings so that they always yielded some new idea, some step forward. Dedicating my best hour Starting around 1994, when I started to work from home as a software developer, I made sure to start my day by working on my philosophy for an hour or two. Even later, as I struggled and failed to make a living from my lab, Minciu Sodas, I always dedicated my best hour or two to my philosophy. My best hour is in the morning, when my mind is fresh and uncluttered with concerns. I pray to God, do some calisthenics, eat breakfast, check my emails to keep them off my mind, and then apply my mind to my philosophy, preferably to the deepest question that I can. My goal is to get a new idea every day. Then I feel that my day has gone well and it doesn't matter what else happens. I typically continue by reviewing, writing and sharing my strategy for applying myself and making a living. As the day wears on, I make some effort to make a living. But I don't let that have my best energies. I believe that we all have a right and duty to spend one or two hours each day of our best time to apply ourselves and do what we were created and inspired to do. Invite all people Conversing with: God's will In my activities, I've tried to be and stay open to everybody and not be exclusive. I have tried not to worry about people's intelligence, competence, reliability, wealth or niceness. This has helped me to be more flexible with regard to what might happened, how things might develop, and be more appreciative as to what other people might contribute. It has also gotten me to develop relevant filters, such as expecting certain behavior. I've focused on "independent thinkers", expected people to be accountable to their own deepest value in life, or to engage me based on their own question that they wish to answer. Step in, step out *** How to generate a profound, new and relevant idea? Conversing with: spirit In 1996, in developing the good will exercises, I wanted to make sure that each meeting yielded some insight that helped me develop them further. It was not easy because people didn't quite believe that something could come from such ideas and they kept drifting off in tangents. Yet those tangents were like gradients that showed what directions our minds could move in. I was able to note those gradients as others talked. But as they talked, they drifted away from the main point I cared about. So I had to talk to keep bringing them back to the main point. And as I talked I would immerse myself, pay attention to what I was feeling and thinking, and myself perhaps go to far afield. I started explaining this dynamic as "step in, step out". We should try to develop a rhythm where as one person "steps in" and brings forth from their personal experience, others should step back and watch the dynamics of the conversation. This yields a flutter, the Holy Spirit, as happens at the Chinese bible study that I'd go to sometimes with my friend Shu-Hong. I think it allows us to feel how different mental paths are possible from a particular point, and thus makes it real in its freedom. One question that came up thereby was raised by Bob, which was, how does he know that I won't brainwash him? This led me to think, how do I keep from being brainwashed, and so I recalled science fiction questions that I pondered in junior high school such as, "How do I know that I'm not a robot?" and the answers that I developed, counterquestions such as "Would it make any difference?" So I developed a whole theory and structure of counterquestions. Drawing insight from others Accept everyone the same, as if God Conversing with: simplicity In accepting a person as myself, if they are God, I give them my full attention, and realize how surprisingly intelligent they are, or troubled or proud or dignified or harmonious. It makes whole the many things to learn. Listen to faithless sermons Conversing with: spiritual patience Many priests speak without inner faith and so I note how they miss Jesus's meaning. The good son, the brother of the prodigal son, reminds me of Jesus, what he must have felt.. Observe teachers teach Conversing with: successful communication As a child, I often already knew what the teacher was teaching, or felt it was taught slowly, so I watched them teach, and considered how I would teach differently, what is the deep meaning? Tune myself to my sweetheart Conversing with: mood My sweetheart is a great influence on me. I have every desire to be one with her. So I open myself to her values that I perceive. She loves to worship God and she is a patriot of Lithuania. I can be skeptical of all of that, but not when I think of her, and I am glad that it is what I truly wish to be a part of, too. She is beautiful and she makes me feel chaste, too, for I devote myself to her. |
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Fostering an environment for integration
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Fostering an environment for integration I-Thou dialogue Bob Lichtenbert writes up a dialogue where one person is the Seeker and the other is the Tipper (typical person). Then volunteers read that dialogue out loud. Then they pause discuss. This yields a more interdisciplinary approach to questions because the audience is from different fields and aren't philosophers. |
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The Analytic Power of a Group
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Finally, experiments in group process have shown that, in most circumstances, the analytic power of a group of analysts is greater than that of any of its single members. For that reason, the group's consensus judgments are likely to be more accurate than the judgments of any individual member. Yet when a group of people sits around a table and analyzes a problem, rare is the group member who believes that the other members collectively know more about the problem, understand it better, can come up with a better solution than that member can, particularly when that member's opinion is at odds with the group's. Structuring group analysis of problems facilitates the exchange of ideas and the examination of alternatives that are necessary for building a consensus. ... A host of things - such as individual mind-sets, conflicts over who is in authority, domination by a clique, lack of group focus - can decimate the effectiveness of a group ... These interactions within the group tend to divide and confuse its members and to defeat their common purpose. ... By organizing in a sensible, informative way the problem being analyzed, structuring greatly facilitates a group's work. Because group analysis tends to jump erratically from one topic to another as members press for acceptance of competing ideas, a principal beneficial effect of structuring is to help the group perceive the problem's full dimensions, to focus its attention on individual aspects of the problem, and to keep track of where the group is in the analytic process. |
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Invest in giving
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Invest in giving Invest in a culture of giving In the town in China where I was staying I wanted to buy a bicycle. However, the family was very anxious for my safety and refused to let me. It was as if I was a small child. But as my Chinese improved I was able to ask in ways that gave them confidence in me. So they arranged that I could buy a bicycle. I went to school in Chicago in part to be closer to my relatives, all of whom lived in the area. Some of them would say how family was much more important than friends because in the end you always counted on family. But that never bore out. Partly, my mother didn't want me to ever have or discuss any money issues with anybody in our extended family. And simply in practice, whenever I discussed my business needs or for loans or leads or mentors, there was no help. Whereas I have good friends who backed me up with significant long term or short term loans. In general, different people helped or not in different ways, as they were able and felt comfortable. |
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Others' limits
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Others' limits Accepting a person the way they are Keeping a person in mind |
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Decomposing the contributions of independent forces along independent dimensions
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Decomposing the contributions of independent forces along independent dimensions This requires not just experimental measurement but also a theoretical system of coordinates. It also typically builds on a theory of forces in terms of dimensional parameters such as distance.  Schiehallion experiment An example of decomposing the contributions of two independent forces along two independent dimensions. In 1774, a pendulum's plumb-bob was placed near an isolated mountain, Schiehallion. The discrepancy from the true zenith, as measured with regard to the stars, allows the size of the force to be calculated as given by Newton's gravitational force law. This makes it possible to compare the density of the mountain with the density of the Earth. The Earth turned out to be about twice as dense as the mountain. This suggested that the core of the Earth was metallic. The same particle is yielded by different sources If rays with the same properties are yielded by different sources, then this suggests that they consist of the same particles. Deducing that cathode rays consists of particles Varying the atomic source of cathode rays and getting the same estimate for their mass suggests that they are composed of a universal particle. |
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Good heart: Saving those who stumble
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Good heart: Saving those who stumble The theology of grace, of saving the sinner. Here Jesus is the Son of God. This is the wide gate that leads to destruction. Jesus apparently has in mind a progression: great (master), greater (teacher), greatest (lord) which corresponds to the math ideas of extreme point, greatest lower bound and limit, and also brings to mind the ways to address Maslow's hierarchy of needs: self-esteem, opportunity, self-fulfillment. Darkness as opposed to light. |
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Time
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Time C1) Sequence C2) Poset with maximal or minimal elements C3) Least upper bounds, greatest lower bounds C4) Limits The act of ever getting a new sheet (blank or otherwise) makes for a countably infinite list. That is what we need for mathematical induction. Next, we may prefer some sheets as more noteworthy than others, which we ignore, so that some are most valuable. Such extremes are assumed by the extreme principle. An example is the square as the rectangle of a given perimeter that yields the most area. Next, we construct monovariants which say, in effect, that the only results which count are those that beat the record-to-beat, which yields sequences of increasing minimums, thus a greatest lower bound, or alternatively, a least upper bound. Finally, we allow such a boxing-in or boxing-out process to continue indefinitely, yielding (or not) a limit that may very well transcend the existing system (as the reals transcend the rationals). We thereby construct internal "time" which is fully fledged in that the continuum without gaps may be used to model space. |
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Caring 1-2-3-4
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Caring 1-2-3-4 In real life, we can open our mind to all thoughts, we can collect and sort them by way of values, we can push ourselves to our personal limitations, and we can allow for an ideal person (such as Jesus) who transcends our limitations. These four frames are: caring; caring about caring; caring about caring about caring; caring about caring about caring about caring. |
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Our outer world
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Our outer world |
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Divergence
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Divergence... means to branch out, to go in different directions, from a single point. Whenever we take a broader view of a problem, whether by examining evidence more thoroughly, gathering new evidence, or entertaining alternative solutions, we are in a divergent mode. |
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Taking
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Taking |
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Wave point of view
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Wave point of view Our measurement can take place outside of our frames of measurement and thus link several such frames. This is a dynamic point of view where there is no distinction between the future and the past so that all is reversible. We can think of the wave in terms of where it starts and where it ends. It includes all paths between these two points. This is a deterministic, nonspatial point of view, which establishes time, an ideal continuum that is beyond the frames but thus relevant for us. Analogously, in math we may have a sequence of sheets, as with mathematical induction, some of which may be of ultimate importance, as with the extreme principle, thus allowing for boxing in with greatest lower bounds and least upper bounds, leading to limits that may transcend, go beyond what we can account for. Or in life, we can be open to care about everything, then care about our minds by which we care, then come up against our personal limits, then allow for an ideal (such as Jesus) that transcends our limits. |
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What you find is what you love
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What you find is what you love "What you find is what you love" is the content of several of Jesus's parables, such as the coin that the widow lost but found, or the lost sheep that the shepherd went after, leaving ninety-nine behind. It is, I think, the most important message of his parables, even more important than "what you believe is what happens", for it gives God's point of view, as such. Favor the lost sheep of the House of Israel Matthew 10:5 Jesus sent these twelve out, and commanded them, saying, "Don't go among the Gentiles, and don't enter into any city of the Samaritans. 10:6 Rather, go to the lost sheep of the house of Israel. Jesus has compassion for those who suffer Matthew 8:5 When he came into Capernaum, a centurion came to him, asking him, 8:6 and saying, "Lord, my servant lies in the house paralyzed, grievously tormented." 8:7 Jesus said to him, "I will come and heal him." Jesus healed Peter's mother-in-law Matthew 8:14 When Jesus came into Peter's house, he saw his wife's mother lying sick with a fever. 8:15 He touched her hand, and the fever left her. She got up and served him. People love as much as they are forgiven Luke 7:41 ... A certain lender had two debtors. The one owed five hundred denarii, and the other fifty. 7:42 When they couldn't pay, he forgave them both. Which of them therefore will love him most?" 7:43 Simon answered, "He, I suppose, to whom he forgave the most." He said to him, "You have judged correctly." 7:44 Turning to the woman, he said to Simon, "Do you see this woman? I entered into your house, and you gave me no water for my feet, but she has wet my feet with her tears, and wiped them with the hair of her head. 7:45 You gave me no kiss, but she, since the time I came in, has not ceased to kiss my feet. 7:46 You didn't anoint my head with oil, but she has anointed my feet with ointment. 7:47 Therefore I tell you, her sins, which are many, are forgiven, for she loved much. But to whom little is forgiven, the same loves little. Somebody suggests to Jesus what Jesus might want Matthew 8:2 Behold, a leper came to him and worshiped him, saying, "Lord, if you want to, you can make me clean." 8:3 Jesus stretched out his hand, and touched him, saying, "I want to. Be made clean." Immediately his leprosy was cleansed. 8:4 Jesus said to him, "See that you tell nobody, but go, show yourself to the priest, and offer the gift that Moses commanded, as a testimony to them." What is the way to destruction? The gate of "belief" leads to life; the gate of "caring" leads to destruction. We can distinguish the gates by seeing who goes through them. Matthew 7:13 "Enter in by the narrow gate; for wide is the gate and broad is the way that leads to destruction, and many are those who enter in by it. 7:14 How narrow is the gate, and restricted is the way that leads to life! Few are those who find it. What you find is what you love. |
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Sequence
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Sequence The act of ever getting a new sheet (blank or otherwise) makes for a countably infinite list. That is what we need for mathematical induction. Axiom of Infinity Wikipedia: Let S(x) abbreviate x U {x}, where x is some set. Then there exists a set X such that the empty set {} is a member of X and, whenever a set y is a member of X, then S(y) is also a member of X. More colloquially, there exists a set X having infinitely many members. The minimal set X satisfying the axiom of infinity is the von Neumann ordinal ω, which can also be thought of as the natural numbers N. Mathematical induction Edward Cherlin: The defining characteristic of the standard natural numbers is induction.
Standard induction This is a very powerful method for proving assertions that are "indexed" by integers... Each assertion can be put in the form, P(n) is true for all integers n >= n0, where P(n) is a statement involving the integer n, and n0 is the "starting point". In standard induction: 1. Establish the truth of P(n0). This is called the "base case" and is usually an easy exercise. 2. Assume that P(n) is true for some arbitrary integer n. This is called the inductive hypothesis. Then show that the inductive hypothesis implies that P(n+1) is also true. This is sufficient to prove P(n) for all integers n>=n0, since P(n0) is true by (1) ... pg.46, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Prove n! > 2**n where integer n >= 3 Weitz, pg.51 Prove that the sum of the interior angles of any n-gon is 180(n-2) degrees Weitz, pg.50 Transfinite induction Edward Cherlin: Proofs by transfinite induction are entirely analogous to proofs by finite induction. Edward Cherlin: A similar construction [to the natural numbers] and a similar inductive condition define the transfinite ordinals, but with some differences. If S is a set of ordinals with no largest member, then the union of its members is a larger ordinal. This allows us to define the first transfinite ordinal as the order type of the set of natural numbers. This ordinal is called omega. Then we get omega + 1, and so on. Taking all countable ordinals together gives us the first uncountable ordinal, and so on again. |
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Relax - Innovate
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Relax - Innovate 6-8-5 generate many rough ideas quickly Break challenge assumptions that constrain imagination Image-ination get unstuck with new ideas Mission Impossible let ideas settle, open room for new thoughts Object Brainstorm suggest associations The Anti-Problem get unstuck, note the obvious |
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Caring
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Caring Being open to all outside us. Be open to any and all thoughts and associations. Note what captures my attention. Am I doing anything about this? Being open to everything Conversing with: infinity Observing what is going on around me As a nearsighted child, I noticed "floaters" that moved across my eye, chains of blocks. I learned about cells in my biology, and then, one day, sitting in a dentist's chair, I thought, the floaters must be chains of cells which I can see because they are close up, right on my eyeball. My mother said no. But later an optometrist confirmed that, yes, indeed they were! I thought it was wonderful that I could see human cells with my own naked eyes! Take in all thoughts Conversing with: unlimited Once in a while, I marvel at the amazing variety of input that engages me and enters my mind. I could easily believe that there is an objective world around me, which I can't help but accept, so as to get by. It is a world that can easily overturn any model which I might suggest for it. Thus, I don't look to it for assurance, but rather I stay open to it. I feel obliged to take seriously whatever I note, whatever I am told, so that my character might be completely informed and thus ever perfected. John Muir has celebrated in detail, with awe and joy, the wonders of the Sierra mountains. Wait Conversing with: associations I learned to wait sometimes, to give a chance for solutions to arise, for my unconscious to percolate or for the situation to develop. I learned to give God a chance. For example, I was living with David Ellison-Bey and his home was confirmed as foreclosed and then sold, and by law, I had to leave. I decided not to rush to leave, but waited to find a suitable place. Noting what we find no analogue for What can't we have learned We can't have learned the concept of everything because everything we know in the real world is bounded, whereas everything is unbounded. |
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Being completely open to the outer world
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Being completely open to the outer world Unexpected recording of unconscious Keith Richards of the Rolling Stones came up with the riff for the song "Satisfaction" in his sleep, recorded it unexpectedly, and woke up to find it on the tape. Fresh Air: In an interview on Fresh Air, Richards recounts how he woke up just long enough to record the famous opening riff of "Satisfaction" on a cassette player he'd placed next to his bed. "I go to bed as usual with my guitar, and I wake up the next morning, and I see that the tape is run to the very end," Richards tells Terry Gross. "And I think, 'Well, I didn't do anything. Maybe I hit a button when I was asleep.' So I put it back to the beginning and pushed play and there, in some sort of ghostly version, is [the opening lines to 'Satisfaction']. It was a whole verse of it. And after that, there's 40 minutes of me snoring. But there's the song in its embryo, and I actually dreamt the damned thing." |
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Expose ourselves to new information
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The mother of all biases is the "mind-set" ... the distillation of our accumulated knowledge about a subject into a single, coherent framework or lens through which we view it. ... the summation or consolidation of all of our biases about a particular subject. ... feelings come upon us in a flash; we don't think about them before we sense them. Such is the power of mind-set. ... When we read a novel ... the author furnishes us bits of information ... At some point, our absorption and interpretation of these small details solidifies into a particular mind-set. ... mind-sets enable us to put events and information immediately into context without having to reconstruct from memory everthing that previously happened ... provide us instant insight into complex problems ... Being an aggregate of countless biases and beliefs, a mind-set represents a giant shortcut of the mind ... The influence a mind-set has on our thinking is thus magnified by many orders of magnitude over that of a simple bias ... There is only one way to change undesirable biases and mind-sets, and that is by exposing ourselves (our minds) to new information and letting the mind do the rest. Fortunately, the mind is, to a large degree, a self-changing system: give it new information, and it will change the bias. ... Most biases and mind-sets, however, are highly resistant to alteration and are changed only gradually, eroded away by repeated exposure to new information. drawing from The Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich. |
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Accept what I am given
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Accept what I am given When we first got allowance, it was five cents a week. Then we realized that we couldn't buy anything for that! So we let our parents know and it went up to ten cents per week. Jesus was supported by women's possessions Luke 8:1 It happened soon afterwards, that he went about through cities and villages, preaching and bringing the good news of the Kingdom of God. With him were the twelve, 8:2 and certain women who had been healed of evil spirits and infirmities: Mary who was called Magdalene, from whom seven demons had gone out; 8:3 and Joanna, the wife of Chuzas, Herod's steward; Susanna; and many others; who served them from their possessions. Andrius: "I'm always wondering how I could make a living from documenting and sharing "ways of figuring things out". Perhaps I should do that for business and economics." Bradford Hansen-Smith, 2011.04.25: What works in my experience is to start where you are. This lessens the load of trying to figure things out before hand. |
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My awareness
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My awareness Drifting attention Keep us attentive Noting whether I pay attention |
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Wait for the master, share in the treasure.
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Wait for the master, share in the treasure. Jesus appeals to people: Choose what to invest in. Distinguish direct and indirect importance Matthew 6:25 Therefore I tell you, don't be anxious for your life: what you will eat, or what you will drink; nor yet for your body, what you will wear. Isn't life more than food, and the body more than clothing? 6:26 See the birds of the sky, that they don't sow, neither do they reap, nor gather into barns. Your heavenly Father feeds them. Aren't you of much more value than they? Figure things out soundly The "eye" is how the body figures things out. We should choose to figure things out soundly. Matthew 6:22 "The lamp of the body is the eye. If therefore your eye is sound, your whole body will be full of light. 6:23 But if your eye is evil, your whole body will be full of darkness. If therefore the light that is in you is darkness, how great is the darkness! Incompatibility Matthew 6:24 "No one can serve two masters, for either he will hate the one and love the other; or else he will be devoted to one and despise the other. You can't serve both God and Mammon. Invest in what is eternal, not temporary Matthew 6:19 "Don't lay up treasures for yourselves on the earth, where moth and rust consume, and where thieves break through and steal; 6:20 but lay up for yourselves treasures in heaven, where neither moth nor rust consume, and where thieves don't break through and steal; 6:21 for where your treasure is, there your heart will be also. Jesus took our infirmities Jesus dealt with the human condition, including illness, whether or not his own. Matthew 8:16 When evening came, they brought to him many possessed with demons. He cast out the spirits with a word, and healed all who were sick; 8:17 that it might be fulfilled which was spoken through Isaiah the prophet, saying: "He took our infirmities, and bore our diseases." Jesus wishes to see faith Matthew 8:8 The centurion answered, "Lord, I'm not worthy for you to come under my roof. Just say the word, and my servant will be healed. 8:9 For I am also a man under authority, having under myself soldiers. I tell this one, 'Go,' and he goes; and tell another, 'Come,' and he comes; and tell my servant, 'Do this,' and he does it." 8:10 When Jesus heard it, he marveled, and said to those who followed, "Most certainly I tell you, I haven't found so great a faith, not even in Israel. Relative comparison Mark 12:43 ... "Most certainly I tell you, this poor widow gave more than all those who are giving into the treasury, 12:44 for they all gave out of their abundance, but she, out of her poverty, gave all that she had to live on." The relationship between what we let go and receive What we are able to let go of is more than matched by what we are able to receive. Luke 18:28 Peter said, "Look, we have left everything, and followed you." 18:29 He said to them, "Most certainly I tell you, there is no one who has left house, or wife, or brothers, or parents, or children, for the Kingdom of God's sake, 18:30 who will not receive many times more in this time, and in the world to come, eternal life." Thinking while working I think of two people working together, but one is thinking and the other is not. The one who thinks a few new thoughts each day will have thousands of thoughts in their mind, and the other will not. One is "taken away" and the other is not. Luke 17:35 There will be two grinding grain together. One will be taken, and the other will be left. We speak out of the abundance of our hearts Luke 6:45 The good man out of the good treasure of his heart brings out that which is good, and the evil man out of the evil treasure of his heart brings out that which is evil, for out of the abundance of the heart, his mouth speaks. What creates difficulty Matthew 19:20 The young man said to him, "All these things I have observed from my youth. What do I still lack?" 19:21 Jesus said to him, "If you want to be perfect, go, sell what you have, and give to the poor, and you will have treasure in heaven; and come, follow me." 19:22 But when the young man heard the saying, he went away sad, for he was one who had great possessions. 19:23 Jesus said to his disciples, "Most certainly I say to you, a rich man will enter into the Kingdom of Heaven with difficulty. 19:24 Again I tell you, it is easier for a camel to go through a needle's eye, than for a rich man to enter into the Kingdom of God." What do they seek? They devour widows' houses. Matthew 23:13 "Woe to you, scribes and Pharisees, hypocrites! For you devour widows' houses, and as a pretense you make long prayers. Therefore you will receive greater condemnation. What is preferred Matthew 13:44 "Again, the Kingdom of Heaven is like a treasure hidden in the field, which a man found, and hid. In his joy, he goes and sells all that he has, and buys that field. What is the right scope for anxiety? Matthew 6:34 Therefore don't be anxious for tomorrow, for tomorrow will be anxious for itself. Each day's own evil is sufficient. What keeps us from thinking correctly? Matthew 7:3 Why do you see the speck that is in your brother's eye, but don't consider the beam that is in your own eye? 7:4 Or how will you tell your brother, 'Let me remove the speck from your eye;' and behold, the beam is in your own eye? 7:5 You hypocrite! First remove the beam out of your own eye, and then you can see clearly to remove the speck out of your brother's eye. What must one give up Luke 18:24 Jesus, seeing that he became very sad, said, "How hard it is for those who have riches to enter into the Kingdom of God! 18:25 For it is easier for a camel to enter in through a needle's eye, than for a rich man to enter into the Kingdom of God." What one is willing to give up for it Matthew 13:45 "Again, the Kingdom of Heaven is like a man who is a merchant seeking fine pearls, 13:46 who having found one pearl of great price, he went and sold all that he had, and bought it. What shall be in your heart Deuteronomy 6:4 Hear, Israel: Yahweh is our God; Yahweh is one: 6:5 and you shall love Yahweh your God with all your heart, and with all your soul, and with all your might. 6:6 These words, which I command you this day, shall be on your heart; 6:7 and you shall teach them diligently to your children, and shall talk of them when you sit in your house, and when you walk by the way, and when you lie down, and when you rise up. Where to have one's treasure Luke 18:21 He said, "I have observed all these things from my youth up." 18:22 When Jesus heard these things, he said to him, "You still lack one thing. Sell all that you have, and distribute it to the poor. You will have treasure in heaven. Come, follow me." 18:23 But when he heard these things, he became very sad, for he was very rich. Who is our master? Matthew 23:10 Neither be called masters, for one is your master, the Christ. Who is the greatest in the Kingdom of Heaven The greatest is the one who is most humble, thus most healing. 18:1 In that hour the disciples came to Jesus, saying, "Who then is greatest in the Kingdom of Heaven?" 18:2 Jesus called a little child to himself, and set him in their midst, 18:3 and said, "Most certainly I tell you, unless you turn, and become as little children, you will in no way enter into the Kingdom of Heaven. 18:4 Whoever therefore humbles himself as this little child, the same is the greatest in the Kingdom of Heaven. 18:5 Whoever receives one such little child in my name receives me, 18:6 but whoever causes one of these little ones who believe in me to stumble, it would be better for him that a huge millstone should be hung around his neck, and that he should be sunk in the depths of the sea. Words are things brought of one's treasure Matthew 12:33 "Either make the tree good, and its fruit good, or make the tree corrupt, and its fruit corrupt; for the tree is known by its fruit. 12:34 You offspring of vipers, how can you, being evil, speak good things? For out of the abundance of the heart, the mouth speaks. 12:35 The good man out of his good treasure brings out good things, and the evil man out of his evil treasure brings out evil things. 12:36 I tell you that every idle word that men speak, they will give account of it in the day of judgment. 12:37 For by your words you will be justified, and by your words you will be condemned." Your heart is where your treasure is Matthew 6:19 "Don't lay up treasures for yourselves on the earth, where moth and rust consume, and where thieves break through and steal; 6:20 but lay up for yourselves treasures in heaven, where neither moth nor rust consume, and where thieves don't break through and steal; 6:21 for where your treasure is, there your heart will be also. Behavior depends on emotion Matthew 9:14 Then John's disciples came to him, saying, "Why do we and the Pharisees fast often, but your disciples don't fast?" 9:15 Jesus said to them, "Can the friends of the bridegroom mourn, as long as the bridegroom is with them? But the days will come when the bridegroom will be taken away from them, and then they will fast. What attracts people Matthew 11:7 As these went their way, Jesus began to say to the multitudes concerning John, "What did you go out into the wilderness to see? A reed shaken by the wind? 11:8 But what did you go out to see? A man in soft clothing? Behold, those who wear soft clothing are in king's houses. 11:9 But why did you go out? To see a prophet? Yes, I tell you, and much more than a prophet. |
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Poset with maximal or minimal elements
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Poset with maximal or minimal elements We may prefer some sheets as more noteworthy than others, which we ignore, so that some are most valuable. Such extremes are assumed by the extreme principle. An example is the square as the rectangle of a given perimeter that yields the most area. Four times a right triangle is the difference of two squares A right triangle rotated four times brings us back to where we were. This yields a difference of two squares from which the Pythagorean theorem follows. Note also that sin(x) and cos(x) are equal to their fourth derivatives. Gospel Math. Right triangles are more basic than circles Consider a line segment of length AB and consider all of the right triangles with hypotenuse AB. Drop the altimeter Z so that the hypotenuse is X + Y. The radius is the average, the arithmetic mean (X+Y)/2 and the altimeter is the geometric mean of X and Y. This makes clear that the arithmetic mean is greater than the geometric mean because the radius is greater than the altimeter. It also shows that the square has greater area than other rectangles of the same perimeter. Then the right triangles form a semicircle without the two endpoints. Circles depend on distance and thus presume right triangles. Gospel Math. Arithmetic-Geometric Mean Inequality 1998 x 2000 ? 1999 x 1999 Your intuition probably tells you that the question mark should be replaced with "<", for the left-hand side is the area of a rectangle that is not quite a square, while the right hand side is the area of a square with the same perimeter as the rectangle (namely 4x1999). It makes sense that given a fixed amount of fencing, the rectangle of maximum area is a square. ... (x+y)/2 >= square-root of xy. ... The arithmetic mean of two positive real numbers is greater than or equal to the geometric mean, with equality only if the numbers are equal. ... Here is a nice geometric proof of AM-GM. Let AC be the diameter of a circle, and let B be any point on the circle. Recall that ABC will be a right triangle. Now locate point D so that BD is perpendicular to AC. Then triangles ABD and BCD are similar; hence x/g = g/y. Thus g = square root of xy, the geometric mean of x and y. Indeed, that's why it is called a geometric mean! pg.193-194 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Calculus of variations Wikipedia: Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown functions and their derivatives. The interest is in extremal functions that make the functional attain a maximum or minimum value - or stationary functions - those where the rate of change of the functional is precisely zero. Perhaps the simplest example of such a problem is to find the curve of shortest length, or geodesic, connecting two points. If there are no constraints, the solution is obviously a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Extremal arguments involving infinite sets In more complicated problems, it is not always obvious what entities should be monotonized, and the Well-Ordering Principle is not always true for infinite sets. In situations involving infinite sets, sometimes extremal arguments work, but you need to be careful. .... Let f(x) be a polynomial with real coefficients of degree n such that f(x)>=0 for all x in R. Define g(x):= f(x) + f'(x) + f''(x) ... + f(n)(x). Show that g(x)>=0 for all x in R. pg.88, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Extreme principle If possible, assume that the elements of your problem are "in order". Focus on the "largest" and "smallest" elements, as they may be constrained in interesting ways. ... Work by contradiction; assume that whatever you wish to prove is not true. Then look at the minimal (or maximal) element and develop an argument that creates a smaller (or larger) element, which is the desired contradiction. As long as a set of real numbers is finite, it will have a minimum and a maximum element. pg.82, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. When black and white points are interspersed... Let B and W be finite sets of black and white points, respectively, in the plane, with the property that every line segment which joins two points of the same color contains a point of the other color. Prove that both sets must lie on a single line segment. ... The extreme principle comes to the rescue: Assume that the points do not all lie on a line. Then they form at least one triangle. Consider the triangle of smallest area. Two of its vertices are the same color, so between them is a point of the other color, but this forms a smaller triangle - a contradiction! pg.82, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Least squares Wikipedia: The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e. sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every single equation. Maximum We conceive or find the maximum. Laffer Curve Wikipedia: In economics, the Laffer curve is a theoretical representation of the relationship between government revenue raised by taxation and all possible rates of taxation. It is used to illustrate the concept of taxable income elasticity (that taxable income will change in response to changes in the rate of taxation). The curve is constructed by thought experiment. First, the amount of tax revenue raised at the extreme tax rates of 0% and 100% is considered. It is clear that a 0% tax rate raises no revenue, but the Laffer curve hypothesis is that a 100% tax rate will also generate no revenue because at such a rate there is no longer any incentive for a rational taxpayer to earn any income, thus the revenue raised will be 100% of nothing. If both a 0% rate and 100% rate of taxation generate no revenue, it follows that there must exist at least one rate in between where tax revenue would be a maximum. Monotonizing Always be aware of order and maximum/minimum in a problem, and always assume, if possible, that the elements are arranged in order ... Think of this as "free information". pg.83, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Sign of the derivative You certainly know that the derivative f'(x) of a function f(x) has two interpretations: a dynamic definition as rate of change [of f(x) with respect to x], and a geometric definition as slope of the tangent line to the graph y=f(x) at the point (x,f(x)). The rate-of-change definition is especially useful for understanding how functions grow. More elaborate information comes from the second derivative f''(x), which of course measures how fast the derivative is changing. Sometimes just simple analysis of the signs of f' and f'' is enough to solve fairly complicated problems. pg.294 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Sum of Squares The Cauchy-Schwarz inequality states that [(the sum of aibi)**2 <= (sum of ai**2)(sum of bi**2)] with equality holding if and only if a1/b1 = a2/b2 = ... = an/bn. ... another way to prove [this] uses the simple but important tool that A sum of squares of real numbers is non-negative, and equal to zero if and only if all the numbers are zero. ... 0 <=(ay-bx)**2 + (az-cx)**2 + (bz-cy)**2. pg.198-199 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Symmetry-Product Principle We can extract more information from our algebraic proof of AM-GM. ... S**2 - 4P = D**2, where S, P, D are respectively the sum, product and difference of x and y. ... As the distance between two positive numbers decreases, their product increases, provided that their sum stays constant. This agrees with our intuition: As a rectangle becomes more "squarish", i.e., more symmetrical, it encloses area more "efficiently". pg.193-194 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Tangent-line approximation The tangent-line definition of the derivative stems from its formal definition as a limit. ... This suggests a useful, but less well-known, application of the derivative, the tangent-line approximation to the function. ... In general, analyzing f(a+h) with its tangent-line approximation f(a) + hf'(a) is very useful, especially when combined with other geometric information, such as convexity. ... Prove Bernoulli's Inequality (1 + x)**alpha >= 1 + alpha x for x>= -1 and alpha >= 1, with equality when x = 0. ... For integer alpha, this can be proven by induction ... But induction won't work for arbitrary real alpha. Instead, define f(u) = u**alpha, and note that f'(u) = alpha u**(alpha - 1) and f''(u) = alpha(alpha-1)u**(alpha-2). ... Thus the graph y=f(u) is concave up...Therefore, the graph of y=f(u) lies above the tangent line for all u>=0. ... f(1+x) is always strictly greater than its linear approximation 1+alpha x, except when x=0, in which case we have equality. pg.296-297 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Use of extreme principle with induction Often, problems involve the extreme principle plus a particular argument style, such as contradiction or induction. A common tactic is to use the extremes to "strip down" a problem into a smaller version ... The formal argument then uses induction. ... Several positive integers are written on a blackboard. One can erase any two distinct integers and write their greatest common divisor and least common multiple instead. Prove that eventually the numbers will stop changing. pg.85, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Well-Ordering Principle Every non-empty set of positive integers has a least element. pg.83, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Division algorithm The extreme principle in number theory ... Let a and b be positive integers, b>=a. Then there exist integers q,r satisfying q>=1 and 0<=r pg.92, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. |
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Recombine - Explicitly Innovate
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Recombine - Explicitly Innovate Forced Analogy introduce new angles Heuristic Ideation Technique generate ideas by recombining possibilities |
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Caring about caring
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Caring about caring Growing our mind. Accumulate, select, organize and reorganize thoughts. Recognize influences, inject feeling, direct my thinking and create my own mind, my own personal language. Opening up possibilities Accept a position of responsibility Conversing with: opportunities I have learned by accepting invitations to participate. I complemented Audrey Barnes on the gospel choir at St. Benedict the African and she invited me to join. I wrote a Lithuanian rock song for Loreta Grikaviciute and she invited me to sing it. Then in Lithuania we were invited to perform in the rock festival Roko marsas per Lietuva. As a youth, I was invited to lead a scout group. In Uzhupis, I was invited to show my drawings, do big works and paint new works. Generally, this expanded my possibilities and opened me up to my talents. I appreciated others' talents and I learned a bit about the logic and sensibility of each domain. Bus station lottery When I came to Vilnius, Lithuania in 1997, I enjoyed the Old Town very much, but every weekend I felt an impulse to get out of the city, have an adventure, get some exercise, and also, open up some time to be with God. I also wanted to get to know the country better. But I didn't like the pressure of planning ahead and sticking to a plan. So on Sundays I would go to the central bus station and look for which buses were duly leaving and choose from among them. I thought of this as a "bus station lottery". Bus tickets at the time were very cheap for me, so that for $2 or $4 I could travel for an hour or two or more. I could get off wherever I wanted to along the way. Meanwhile, I would read the cultural newspapers I brought, work on my philosophical notes and engage God a bit. I would get out somewhere, note the schedule of the returning buses, and go for a walk. It was a fun way of mixing things up. Keep my mind as open as possible As a child, perhaps five-years-old, appreciating the concept of God, and of believing in God or not, I wondered if such a concept might interfere with my thinking. But I asked myself, which would close more doors in my thinking, to accept God or to reject God? I realized that to not reject God was to accept God as a possibility, and thus keep that more or less open either way, whether God was real. Although to accept God as a possibility was, I thought, to accept God as a reality, in that God is real, first and foremost, as a concept. But to reject God was to close that door completely. And so I chose to accept the possibility of God. Live independently Conversing with: life When I went to elementary school it was understood that I was best to address any troubles at school by myself, without involving my parents or teachers. I felt this way especially because I was younger than the other children as I had skipped kindergarten. During recess, I would venture out far into the yard, away from the teachers. Every so often, a troubled child would come and threaten me. Then I would ask God to save me. And always, some stronger child would come and protect me. When I was deciding what university to go to, I thought that it would be best to live away from home because I was very good and obedient and respected my parents' wisdom and experience, and I thought that I would best develop my own decision making if I lived independently. Looking back, I suppose that I learned the most from the many conversations that I had with my roommates. Reading a book that somebody recommends I've learned quite a bit from several books that others recommended that I read. An IrDA member from Oregon introduced me to Christopher Alexander's "The Timeless Way of Building". It's a wonderfully poetic book and I rank Alexander along with Plato and Kant for his insightful theory and practice of pattern languages. Malcolm Duerod recommended that I read "The Shack", a book depicting God's relationship with himself as the Holy Trinity, but it got me thinking that, in my imagination, God is alone. June Terry recommended "A Purpose Driven Life" and I've just read the first chapter, but it made me realize that, as a child, I appreciated that my happy life did not come from my own merits, yet it was I who appreciated that, and I who decided to apply myself, and I who engaged God regarding that, as I myself thought best. Sometimes I set aside a day, often Sunday, for an excursion by bus and/or a long walk. I may bring along a philosophical problem that I work on, along with relevant diagrams or notes which I add to along the way. On such trips I often try to spend time with God, too. The trip helps me spend longer time on a problem, wrap my mind around it, clears my mind a bit, perhaps includes some random inputs or insights, helps me feel peaceful and reach deeper into my feelings, balance them out. What would make my life easier? In China, one night in bed, I was thinking to myself, how hard it is for me to be Lithuanian. I didn't have anybody to speak to in Lithuanian, I didn't have time to read in Lithuanian, I was learning Chinese and so my Lithuanian was atrophying. I was raised to be Lithuanian, yet it would take an hour a day to truly be so. So I thought that my life would be easier if I lived in Lithuania. A second reason was that it would make it more likely that I could fall in love with a Lithuanian woman, as may unconscious was not allowing me otherwise, for I had grown up in a Lithuanian family and was committed to being Lithuanian. What would make things more likely? In China, one night in bed, I was thinking to myself ... that if I lived in Lithuania, it would make it more likely that I could fall in love with a Lithuanian woman, as my unconscious was not allowing me otherwise, for I had grown up in a Lithuanian family and was committed to being Lithuanian. Shaping my interests Accumulate thoughts Conversing with: my mind I have kept notes of my philosophical thoughts. In high school, I jotted down scribbles of my thoughts on physical reality, energy and matter and light. In college and graduate school, I would jot notes on paper, but my thoughts were so murky, and headed off in so many directions, I found it impossible to write them up. Then I purchased an Amstad word processor with memory and I started writing up my ideas. I continued with my father's old Macintosh which he gave me. I also kept sketch books with my notes and ideas. Then I put the papers in presentation books with transparent pockets. I programmed a FoxPro database for collecting examples of structures. Later I created Microsoft Access databases for recording data from my good will exercises, my conversations with God, and my structures and diagrams. I used Windows Paint to create diagrams. Later I used Dia diagram editor for Linux. I used Stephen Danic's Lucid Fried Eggs. Ultimately, the best tool for accumulating and organizing my notes was a wiki. I used PmWiki and ProWiki. Now I have created a database for collecting ways of figuring things out. Since I graduated in 1993, I work almost every morning on my philosophy. I try to come up with at least one new idea every day. I remember my ideas by connecting them with my overall system and also by phrasing relevant questions which I can keep in mind. Writing down my thoughts and organizing them helps me to see where they fit within the overall system. I would forget them if I didn't write them down because there are so many. Accumulating first hand experiences *** What is the variety of experience? Conversing with: others' experiences I've asked hundreds of people, What is their deepest value in life which includes all of their other values? I learned that some people have a deepest value and have formulated it, others feel it but haven't expressed it in words, and others don't have one that they know of. I learned that whether or not they have one seems to relate to their maturity as independent thinkers; that the deepest value seems to arise from the challenge of integrating competing values; that two people may give it the same name, but when asked further, mean different things; thus that each person seems to have a unique deepest value. Similarly, I've interviewed people about their questions that they don't know the answer to, but wish to answer, as well as their endeavors and dreams. I've also collected episodes in my life, such as ways that I addressed doubts with counterquestions. 2006.09.03 Earlier this year I thought through with John Rogers an online learning environment for community currency design that was based on first hand knowledge of the money mind. In particular, we expected personal stories to play a key role. I had made for myself a list of my own personal experiences with money. Today I wrote up more than 60 of these vignettes and am sorting through them and analyzing them. I wanted to see how the data compared to the theory. These stories are little episodes in life that I recall and that I learn from. It seems to me that each of them focuses on one of three emotional dimensions: Calm vs. Riled; Positive vs. Negative; Sensitive vs. Insensitive. Each episode seems to help distinguish between the opposites in some way. Atune my mind In 2000 or so, I noted eight areas in my personal life where I wanted to improve myself: Be with God, foster my conscience, foster my willpower, foster my stewardship, be curious, serve others, support others' endeavors, be successful. Each morning, after I prayed, I would run through each of these and imagine how I might do that during the day. I didn't try to plan to actually do them because it would be too contrived to carve out so much time. Rather, I would imagine how I might do that, and then during the day I would be open to similar opportunities and seize them. Over several years this helped me change in these ways and I felt better about myself. Afterwards, I gave up this practice because it had achieved its purpose and it took up time, perhaps twenty minutes. Center conversation around philosophical questions Conversing with: comprehension of the essence As a student and a young man, in conversation with others, I would bring up my philosophical questions as often as I could, because they were meaningful to me, and I thought they might influence me and other in a positive way. They also helped me learn if people were serious, what they cared about, whether it could be fruitful to be with them, and after a year or so of such dialogue I would be interested to be friends in a broaders sense. Create my own personal language Conversing with: my insightfulness As I noticed conceptual structures such as divisions of everything, I had to give names to them, both in English and in Lithuanian. Sometimes I adapted existing words and sometimes I made up new words or expressions. Thus I created a private language of a few hundred concepts. This helped me see that I could think nonverbally, because I was able to imagine and think about and relate these concepts without words, although the words were very helpful, too. Dedicate my mind Is God good? On Christmas morning once, in Lithuania, I was living in my parents' office, and I thought about God. I realized that there was God outside us and God inside us and they must be the same God, but how? I considered that God inside us is God in system, the Good. I put this together with what I had been considering from the Gospel of John, and realized that Life is the fact that God is good, that they are the same, but eternal life is understanding that fact, keeping them separate, so that God need not be good. So I dedicated my mind to what my heart believed was important. I matched my personal intuition with a statement that I felt must be meaningful. This let me observe and engage a paradox which in a way I resolved. Direct my thinking Conversing with: capability By directing my thinking to thoughts or objects, important or otherwise, I note what I can be sensitive to, the variety of what can engage my attention. I can temporarily focus my attention on any thought, or away from it by focusing on another, or even be still in my mind, yet ultimately my mind is always drawn by peripheral thoughts or events. I like to think. It is not something I want to turn off. Rather, I want to make the most of my thinking. I see that by ever thoughtfully directing my thinking I have in large part shaped the mind that I live in. Influencing myself Conversing with: my own essence I exert my will power to influence myself to be good in all ways. As a young person, I didn't care much for fruits or vegetables, but as I got older, I would at times encourage or even force myself to eat them, so that now I eat many more of them. I was weak and didn't have friends, so after school I would go outside and play sports on our block, and then later, I went out for the water polo team in high school. At home we spoke Lithuanian, but it took great effort to think in Lithuanian, and yet I chose to do so and persisted, successfully. I think this especially taught me to value "caring about thinking", which became my motto for Minciu Sodas, my lab of independent thinkers. But even more importantly, I was vigilant in my mind to weed out mean thoughts and to think good thoughts. Muddle through Conversing with: future In 1992, I was elated that the woman from Lithuania who I loved with all my heart had agreed to visit me in California. She stayed for three months, which is long enough that I couldn't quite remember not being with her. When she left, I felt devastated. Within a few months she married a man who she loved, but who hadn't been in love with her. I couldn't help but think about her every fifteen minutes. I knew that it would be a long time before I felt better. I made a plan to combat my depression with regular activities to pull me through. Throughout Lent, I made sure to go to church an extra day each week, on Wednesdays, to be closer to God. I read the Bible every night, hopping around from book to book, reading it all of it for the first time. I read a Chinese classic, "The Three Kingdoms", to entertain myself. And I did sports every day, mostly jogging. The year coincided with my writing my Ph.D. thesis, which was good, because that was a bit of drudgery, too. Three months had gone by when I realized that I was still thinking about her every fifteen minutes, but that my thoughts had changed. I was no longer thinking about our time together, but rather of the feelings that I had thinking about her. And so I realized that my emotional window encompassed the last three months, and although my feelings seemed just as strong, but they were feelings about reflections. And so each three months my reflections were once removed, like copies of copies. Prime my mind In 2000 or so, I made a list of eight areas in my personal life where I wanted to improve myself: Be with God; foster my conscience; foster my willpower; foster my stewardship; be curious; serve others; support others' endeavors; be successful. Then every morning after I prayed I would spend about twenty minutes going through the list and just imagining examples of how I might work on each goal that day. In this way, I primed my mind so that I was ready to make time for such activities as they came up naturally. My life was very flexible and this mental exercise was much better for me than trying to schedule activities. After several years, I felt that these matters had become quite natural, my feelings had subsided concerning them, and I didn't have to do this routine, but they were part of my life. Recognizing influences on my perspective Conversing with: my own thinking I notice how television engrosses my thinking and how I am a freer person by not having one. Radio takes up mental bandwidth, too. The cultures that I have lived in have shaped my thinking of what's important. My efforts to link up with God have fostered my appetite for being with God. Selecting thoughts Conversing with: my own attention There are so many things to think about, to occupy myself with or be distracted by. I try to focus on those thoughts that build towards a greater purpose. For this reason, I very much enjoy my quest "to know everything and apply that usefully". I take note of any thought that might contribute towards that, but especially, deep ideas, profound ideas, foundational ideas. I thus imprint them in my mind. I often write them down. I look for them whenever I find my mind drifting towards shallow concerns. This helps me conclude what I truly care about. Sorting thoughts Conversing with: limits of my mind I've written my thoughts as notes on paper, in sketchbooks, and on index cards and in presentation books. I've stored them in various databases, including FoxPro, Access and OpenOffice Base. But I've found it especially helpful to write them down on HTML pages, especially wiki pages. I've used Lucid Fried Eggs, PmWiki, ProWiki and my own interface. Sometimes I work on them on one humungous page, sorting and resorting my thoughts and notes, making hundreds and hundreds of edits. Other times I work with several hundred different wiki pages. Working with the master page helps me look for and develop the overall view. Take care of myself Conversing with: my welfare In 1992, I was deeply distraught when the woman I loved with all my heart returned to Lithuania and married the man she loved, who hadn't loved her. I made a plan to combat my loss with regular activities to pull me through. Throughout Lent, I made sure to go to church an extra day each week, on Wednesdays, to be closer to God. I read the Bible every night, hopping around from book to book, reading it all of it for the first time. I read a Chinese classic, "The Three Kingdoms", to entertain myself. And I did sports every day, mostly jogging. The year coincided with my writing my Ph.D. thesis, which was good, because that was a bit of drudgery, too. |
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Focusing and maximizing our openness
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Focusing and maximizing our openness Ask for something more substantial from participants Fresh Air interview of Steve Coogan, June 9, 2011: The idea for The Trip was first proposed by Michael Winterbottom, who directed Coogan and Brydon in A Cock and Bull Story. Winterbottom wanted something "more substantial" from the two actors, Coogan says. Being invited by the right person Fresh Air interview of Stephen Colbert, June 15, 2001: That's when Sondheim wrote Colbert a personal note. "[He said that] against his instincts, he had a good time on my show and would I consider playing Harry in Company?" he says. "And he ended the letter with the sentence 'You have a perfect voice for musical theater.' And I read it to my wife and she said, 'Boy, you have to do this. No one, let alone Stephen Sondheim is going to ask you to do Sondheim.' And I said, 'You're right, I have to do it.' " Changing the level of protein intake Can the amount of protein that we eat change the activity of an enzyme? How does protein intake affect cancer initiation? Our first test was to see whether protein intake affected the enzyme principally responsible for aflatoxin metabolism, the mixed function oxidase (MFO). This enzyme is very complex because it also metabolizes pharmaceuticals and other chemicals, friend or foe to the body. Paradoxically, this enzyme both detoxifies and activates aflatoxin. It is an extraordinary transformation substance. At the time we started our research, we hypothesized that the protein we consume alters tumor growth by changing how aflatoxin is detoxified by the enzymes present in the liver. ... After a series of experiments, the answer was clear. Enzyme activity could be easily modified simply by changing the level of protein intake. pg.51, T.Colin Campbell, Thomas M.Campbell, The China Study, 2006. Health. Enlisting a team Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: After deciding to go pro, Leonard enlisted Morton to bring others onto his team. Morton asked several men, including lawyer Mike Trainer and Angelo Dundee, Muhammad Ali's trainer, to work with Leonard. Hear stories from real life I heard similar stories from my other patients .. He was killed when he pushed a girl out of the way of the bullets. My patient, the mother of the dead boy, climbed into bed with her mother, also my patient, and they held each other and cried together. Her two surviving children struggled at school. She developed diabetes and hypertension and some heart abnormalities. The grandmother's health deteriorated as well. How can these experiences not affect health and accelerate death in our patients? Each story left me, mouth agape, in shock and dismay. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.1934 Pamela McLean, June 10, 2011. I don't pretend to read all the group posts that land in my inbox - but I do try to dip in to yours now and again to sample your thoughts and activities. What keeps people interested Fresh Air interview of J.J. Abrams, June 13, 2011: "Whether it's commercial, whether it's online, whether it's any number of magazines, by the time a movie comes out — and all it is is entertainment — people are force-fed information they didn't even want. ... So by the time you go to the movie, it doesn't have that power anymore." |
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Control what our mind focuses on
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Third, structuring helps us to focus our analysis. The mind instinctively focuses. That's how the mind works, so it's going to focus whether we want it to or not. Therefore, we're better off to work with the mind than against it and, in so doing, control what it focuses on. If we don't, it will do its own focusing, and its shortcuts can lead us down the wrong path. [Edumnd Bolles, A Second Way of Knowing] We shift attention from sensation to sensation, watching only part of the visual field, listening to only some sounds around us, savoring certain flavors among many. Our capacity to select and "impart intensity" to sensations prevents us from being slaves to the physical world around us. Reality is a jumble of sensations and details. Attention enables us to combine separate sensations into unified objects and lets us examine objects closely to be sure of their identity. ... drawing from The Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich. |
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Maximize what I can take
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Maximize what I can take When I was at the end of my resources in 2000, I asked, who would be the best person for me to work for, for my personal growth. So I turned to Cass McNutt who was active at our lab, interested in tools for thinking, and experienced as an entrepreneur. I was especially impressed by his visceral love of God. Working for him was trickier than we both had thought. I'm glad I had the chance and that I was able to turn to him and he was able to take me. At their suggestion, I agreed to cancel my request, which I did, but then they said they would still have to audit me. I said that it would destroy me and that I would have to consider instead devoting that time to publicly making my case. In graduate school I started collecting record albums because I worked as a teaching assistant and grading math exams was numbingly mindless. I also wanted to expand my horizons. When I was a child, I got just enough allowance each week to buy a chocolate bar. I liked chocolate very much and I would buy a Nestle's crunch bar because I had figured out that it was thin and so I could spend the longest time eating it. I think I timed myself once and I spent a whole half hour nibbling through one bar. That was like heaven, but a bit ridiculous. |
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Be present
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Be present Appreciating attentiveness Being totally present Focusing on my role Imagining the proper mindset Put in feeling Putting to memory Tune in to the season |
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Variational principle
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Variational principle Wikipedia: A variational principle is a scientific principle used within the calculus of variations, which develops general methods for finding functions which minimize or maximize the value of quantities that depends upon those functions. For example, to answer this question: "What is the shape of a chain suspended at both ends?" we can use the variational principle that the shape must minimize the gravitational potential energy. According to Cornelius Lanczos, any physical law which can be expressed as a variational principle describes an expression which is self-adjoint. These expressions are also called Hermitian. Such an expression describes an invariant under a Hermitian transformation. Felix Klein's Erlangen program attempted to identify such invariants under a group of transformations. In what is referred to in physics as Noether's theorem, the Poincaré group of transformations (what is now called a gauge group) for general relativity defines symmetries under a group of transformations which depend on a variational principle, or action principle. See: History of variational principles in physics and Brachistochrone, the curve of fastest descent Fermat's principle Wikipedia: In optics, Fermat's principle or the principle of least time is the principle that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light. However, this version of the principle is not general; a more modern statement of the principle is that rays of light traverse the path of stationary optical length. Fermat's principle can be used to describe the properties of light rays reflected off mirrors, refracted through different media, or undergoing total internal reflection. It follows mathematically from Huygens' principle (at the limit of small wavelength), and can be used to derive Snell's law of refraction and the law of reflection.  Gauss' principle of least constraint Wikipedia: The principle of least constraint is a least squares principle stating that the true motion of a mechanical system of N masses is the minimum of the quantity above for all trajectories satisfying any imposed constraints, where m-k, r-k and F-k represent the mass, position and applied forces of the kth mass. Gauss' principle is equivalent to D'Alembert's principle. The principle of least constraint is qualitatively similar to Hamilton's principle, which states that the true path taken by a mechanical system is an extremum of the action. However, Gauss' principle is a true (local) minimal principle, whereas the other is an extremal principle. Hertz's principle of least curvature Wikipedia: Hertz's principle of least curvature is a special case of Gauss' principle, restricted by the two conditions that there be no applied forces and that all masses are identical. Principle of least action Wikipedia: In physics, the principle of least action - or, more accurately, the principle of stationary action - is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system. The principle led to the development of the Lagrangian and Hamiltonian formulations of classical mechanics. ... Maupertuis felt that "Nature is thrifty in all its actions" ... the action principle is not localized to a point; rather, it involves integrals over an interval of time and (for fields) an extended region of space. Moreover, in the usual formulation of classical action principles, the initial and final states of the system are fixed, e.g., Given that the particle begins at position x1 at time t1 and ends at position x2 at time t2, the physical trajectory that connects these two endpoints is an extremum of the action integral. In particular, the fixing of the final state appears to give the action principle a teleological character which has been controversial historically. |
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Follow the teacher, share in the virtue.
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Follow the teacher, share in the virtue. Jesus asks in different ways, What is greater? The teacher is greater than the student. At best, the student is the equal of the teacher. Jesus came to set people at odds. Distinguishing, who can hurt the body, and who the soul. A disciple is not above his teacher Luke 6:39 ... "Can the blind guide the blind? Won't they both fall into a pit? 6:40 A disciple is not above his teacher, but everyone when he is fully trained will be like his teacher. Are you doing bad? and even worse? Matthew 23:15 Woe to you, scribes and Pharisees, hypocrites! For you travel around by sea and land to make one proselyte; and when he becomes one, you make him twice as much of a son of Gehenna as yourselves. Are you making any converts? Matthew 23:15 Woe to you, scribes and Pharisees, hypocrites! For you travel around by sea and land to make one proselyte; and when he becomes one, you make him twice as much of a son of Gehenna as yourselves. Are you making people better? Matthew 23:15 Woe to you, scribes and Pharisees, hypocrites! For you travel around by sea and land to make one proselyte; and when he becomes one, you make him twice as much of a son of Gehenna as yourselves. Being a blind guide Matthew 23:23 "Woe to you, scribes and Pharisees, hypocrites! For you tithe mint, dill, and cumin, and have left undone the weightier matters of the law: justice, mercy, and faith. But you ought to have done these, and not to have left the other undone. 23:24 You blind guides, who strain out a gnat, and swallow a camel! Can different values coexist Matthew 10:29 "Aren't two sparrows sold for an assarion coin? Not one of them falls on the ground apart from your Father's will, 10:30 but the very hairs of your head are all numbered. 10:31 Therefore don't be afraid. You are of more value than many sparrows. Distinguish between individuals and multitudes Matthew 8:18 Now when Jesus saw great multitudes around him, he gave the order to depart to the other side. Give authority over unclean spirits Matthew 10:1 He called to himself his twelve disciples, and gave them authority over unclean spirits, to cast them out, and to heal every disease and every sickness. Having them answer Mark 12:15 He said to them, "Whose is this image and inscription?" They said to him, "Caesar's." How people think in their hearts Matthew 5:22 But Jesus, perceiving their thoughts, answered them, "Why are you reasoning so in your hearts? Hypocrisy as cause for greater condemnation The condemnation is greater because of the pretense. Matthew 23:13 "Woe to you, scribes and Pharisees, hypocrites! For you devour widows' houses, and as a pretense you make long prayers. Therefore you will receive greater condemnation. Listening to teachers and asking them questions Luke 2:46 It happened after three days they found him in the temple, sitting in the midst of the teachers, both listening to them, and asking them questions. 2:47 All who heard him were amazed at his understanding and his answers. Looking for exceptions Matthew 7:9 Or who is there among you, who, if his son asks him for bread, will give him a stone? 7:10 Or if he asks for a fish, who will give him a serpent? What is Jesus' name to believe in? "Jesus' name" by which to believe is the "one and only Son of God". It seems to be enough to believe in his name (his ideal, his possibility) rather than the historical Jesus. John 3:17 For God didn't send his Son into the world to judge the world, but that the world should be saved through him. 3:18 He who believes in him is not judged. He who doesn't believe has been judged already, because he has not believed in the name of the one and only Son of God. 3:19 This is the judgment, that the light has come into the world, and men loved the darkness rather than the light; for their works were evil. 3:20 For everyone who does evil hates the light, and doesn't come to the light, lest his works would be exposed. 3:21 But he who does the truth comes to the light, that his works may be revealed, that they have been done in God." What is the nature of wisdom? Wisdom approaches people where they are at, or where they claim to be at. Matthew 11:18 For John came neither eating nor drinking, and they say, 'He has a demon.' 11:19 The Son of Man came eating and drinking, and they say, 'Behold, a gluttonous man and a drunkard, a friend of tax collectors and sinners!' But wisdom is justified by her children." What is the source of hate Matthew 10:22 You will be hated by all men for my name's sake, but he who endures to the end will be saved. What should we be anxious about? Matthew 6:34 Therefore don't be anxious for tomorrow, for tomorrow will be anxious for itself. Each day's own evil is sufficient. Whether people are aware of what they are asking Matthew 20:20 Then the mother of the sons of Zebedee came to him with her sons, kneeling and asking a certain thing of him. 20:21 He said to her, "What do you want?" She said to him, "Command that these, my two sons, may sit, one on your right hand, and one on your left hand, in your Kingdom." 20:22 But Jesus answered, "You don't know what you are asking. Whether they think Jesus is of God Jesus wants to know how they think of him, whether or not he is from God, because that determines how he interacts with them. Luke 18:18 A certain ruler asked him, saying, "Good Teacher, what shall I do to inherit eternal life?" 18:19 Jesus asked him, "Why do you call me good? No one is good, except one—God. 18:20 You know the commandments: 'Don't commit adultery,' 'Don't murder,' 'Don't steal,' 'Don't give false testimony,' 'Honor your father and your mother.'" Who finds or loses life Matthew 10:39 He who seeks his life will lose it; and he who loses his life for my sake will find it. Who is greater Matthew 11:11 Most certainly I tell you, among those who are born of women there has not arisen anyone greater than John the Baptizer; yet he who is least in the Kingdom of Heaven is greater than he. Who is judged or not John 3:17 For God didn't send his Son into the world to judge the world, but that the world should be saved through him. 3:18 He who believes in him is not judged. He who doesn't believe has been judged already, because he has not believed in the name of the one and only Son of God. 3:19 This is the judgment, that the light has come into the world, and men loved the darkness rather than the light; for their works were evil. 3:20 For everyone who does evil hates the light, and doesn't come to the light, lest his works would be exposed. 3:21 But he who does the truth comes to the light, that his works may be revealed, that they have been done in God." Who is the teacher? Matthew 23:8 But don't you be called 'Rabbi,' for one is your teacher, the Christ, and all of you are brothers. Who only kills the body The body can be resurrected. Matthew 10:26 Therefore don't be afraid of them, for there is nothing covered that will not be revealed; and hidden that will not be known. 10:27 What I tell you in the darkness, speak in the light; and what you hear whispered in the ear, proclaim on the housetops. 10:28 Don't be afraid of those who kill the body, but are not able to kill the soul. Rather, fear him who is able to destroy both soul and body in Gehenna. Why Jesus came Jesus brings out every person on their own. Matthew 10:34 "Don't think that I came to send peace on the earth. I didn't come to send peace, but a sword. 10:35 For I came to set a man at odds against his father, and a daughter against her mother, and a daughter-in-law against her mother-in-law. 10:36 A man's foes will be those of his own household. Why should they learn from Jesus Matthew 11:29 Take my yoke upon you, and learn from me, for I am gentle and lowly in heart; and you will find rest for your souls. 11:30 For my yoke is easy, and my burden is light. Will this be relevant to people? Matthew 23:36 Most certainly I tell you, all these things will come upon this generation. Wisdom is expressed through many styles Wisdom caters to all styles, yet those who wish will reject each style. Luke 7:31 ... To what then will I liken the people of this generation? What are they like? 7:32 They are like children who sit in the marketplace, and call one to another, saying, 'We piped to you, and you didn't dance. We mourned, and you didn't weep.' 7:33 For John the Baptizer came neither eating bread nor drinking wine, and you say, 'He has a demon.' 7:34 The Son of Man has come eating and drinking, and you say, 'Behold, a gluttonous man, and a drunkard; a friend of tax collectors and sinners!' 7:35 Wisdom is justified by all her children. Working independently of his disciples Matthew 11:1 It happened that when Jesus had finished directing his twelve disciples, he departed from there to teach and preach in their cities. |
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Least upper bounds, greatest lower bounds
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Least upper bounds, greatest lower bounds We construct monovariants which say, in effect, that the only results which count are those that beat the record-to-beat, which yields sequences of increasing minimums, thus a greatest lower bound, or alternatively, a least upper bound. Least-upper-bound assumes second-order logic Wikipedia Second-order logic is more expressive than first-order logic. For example, if the domain is the set of all real numbers, one can assert in first-order logic the existence of an additive inverse of each real number by writing ∀x ∃y (x + y = 0) but one needs second-order logic to assert the least-upper-bound property for sets of real numbers, which states that every bounded, nonempty set of real numbers has a supremum. If the domain is the set of all real numbers, the following second-order sentence expresses the least upper bound property [...] In second-order logic, it is possible to write formal sentences which say "the domain is finite" or "the domain is of countable cardinality." To say that the domain is finite, use the sentence that says that every surjective function from the domain to itself is injective. To say that the domain has countable cardinality, use the sentence that says that there is a bijection between every two infinite subsets of the domain. It follows from the upward Löwenheim-Skolem theorem that it is not possible to characterize finiteness or countability in first-order logic. Algorithmic proof AM-GM reformulated...We have managed to change two of the n numbers in such a way that
Bounded monotonic sequence In practice, there are several possible methods of showing that a given sequence converges to a limit. ... Show that the sequence is bounded and monotonic. A sequence (an) is bounded if there is a finite number B such that |an|<=B for all n. The sequence is monotonic if it is either non-increasing or non-decreasing. ... Bounded monotonic sequences are good, because they always converge. To see this, argue by contradiction: if the sequence did not converge, it would not have the Cauchy property, etc. ... pg.285 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Convergence of upper and lower bounds In practice, there are several possible methods of showing that a given sequence converges to a limit. ... Show that the terms of the sequence are bounded above and below by the terms of two convergent sequences that converge to the same limit. For example, suppose that for all n, we have 0 < xn < (0.9)**n. This forces the limit as n goes to infinity of xn to be zero. Conversely, if the terms of a sequence are greater in absolute value than the corresponding terms of a sequence that diverges (has infinite limit), then the sequence in question also diverges. pg.285 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Muchnik [1956] and Friedberg [1957] invented a computable approximation to this forcing argument, called the finite injury method , where they allow a requirement to finitely often be injured (i.e, previous progress to be destroyed), and then they begin again to satisfy the given requirement, until after it is finally satisfied and never injured thereafter. We present finite injury priority arguments and a solution to Post's problem of c.e. sets A and B of incomparable Turing degree. Robert Soare's upcoming book, "Computability Theory and Applications: The Art of Classical Computability". Growth rates of functions It is important to understand the hierarchy of growth rates for the most common functions. The best way to learn about this is to draw lots of graphs. Any quadratic function of x will dominate any linear function of x, provided that x is "large enough". ... x**a will eventually dominate x**b provided that a > b > 0. ...if a is any positive number, and b > 1, then b**x eventually dominates x**a ... exponential functions grow faster than polynomial functions ... if a is any positive number, and b > 1, then x**a eventually dominates log(b)x. In summary, the hierarchy of growth rates, from slowest to highest, is logarithms, powers, exponents. pg.190-191 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Limits and Colimits in Category Theory In category theory, the limit of a diagram is the "least upper bound" for the diagram, in the sense that it is a cone (generalization) of the diagram such that it is more specific than any other such cone. Similarly, the colimit of a diagram is the "greatest lower bound" for the diagram, in the sense that it is a co-cone that is more general than any other such co-cone. Wikipedia: Limits are also referred to as universal cones, since they are characterized by a universal property (see below for more information). As with every universal property, the above definition describes a balanced state of generality: The limit object L has to be general enough to allow any other cone to factor through it; on the other hand, L has to be sufficiently specific, so that only one such factorization is possible for every cone. Limits may also be characterized as terminal objects in the category of cones to F. It is possible that a diagram does not have a limit at all. However, if a diagram does have a limit then this limit is essentially unique: it is unique up to a unique isomorphism. For this reason one often speaks of the limit of F. Massage 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... Show that the harmonic series diverges ... 1/3 + 1/4 >= 1/2, 1/5+1/6+1/7+1/8 >= 1/2, etc. ... Therefore the entire harmonic series is greater than or equal to 1 + 1/2 + 1/2 + 1/2 + ... which diverges. The key idea used above combines the obvious fact that a>=b => 1/a <= 1/b with the nice trick of replacing a "complicated" denominator with a "simpler" one. This is an example of the many-faceted massage tool - the technique of fiddling around with an expression, with whatever method works (adding zero, multipying by one, adding or subtracting a bit, etc.) in order to make it more manageable. ... Yet another idea inspired by the Wishful Thinking strategy. ... The philosophy of massage is to "loosen up" an expression in such a way that it eventually becomes easier to deal with. ... Sometimes the first stage of massage seemingly worsens the difficulty. But that is temporary, much like physical massage, which can be rather painful until your muscles magically relax. Here is an instructive example which combines massage with its frequent partner, telescoping. ... pg.176-177, 197 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Monovariant A monovariant is a quantity which may or may not change at each step of a problem, but when it does change, it does so monotonically (in only one direction). Another term used for monovariant is semi-invariant. Monovariants are often used in the analysis of evolving systems, to show that certain final configurations must occur, and/or determine the duration of the system. Many monovariant arguments also make use of the extreme principle (at least the well-ordering principle). pg.112, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. John Conway's Checker Problem Put a checker on every lattice point ... in the plane with y-coordinates less than or equal to zero. The only legal moves are horizontal or vertical "jumping". By this we mean that a checker can leap over a neighbor, ending up 2 units up, down, right, or left of its original position, provided that the destination point is unoccupied. After the jump is complete, the checker that was jumped over is removed from the board. ... Is it possible to make a finite number of legal moves and get a checker to reach the line y=5? pg.114, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Number of games in an elimination-style tournament Find a formula for the number of games that must be played in an elimination-style tournament starting with n contestants. ... The number of people who are left in the tournament is clearly a monovariant over time. This number decreases by one each time a game is concluded. Hence if we start with n people, the tournament must end after exactly n-1 games! pg.112, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Reversing cards in a deck The n cards of a deck ... are labeled 1,...,n. Starting with the deck in any order, repeat the following operation: if the card on top is labeled k, reverse the order of the first k cards. Prove that eventually the first card will be 1 (so no further changes occur). pg.112, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Constraint optimization Wikipedia: Constraint optimization (also called constrained optimization) seeks for a solution maximizing or minimizing a cost function. A constraint optimization problem can be defined as a regular constraint satisfaction problem in which constraints are weighted and the goal is to find a solution maximizing the weight of satisfied constraints. Alternatively, a constraint optimization problem can be defined as a regular constraint satisfaction problem augmented with a number of "local" cost functions. The aim of constraint optimization is to find a solution to the problem whose cost, evaluated as the sum of the cost functions, is maximized or minimized. The regular constraints are called hard constraints, while the cost functions are called soft constraints. Note that solutions of constraint optimization generally involve a series of maximums or minimums. Tom Wayburn, 2011.04.28: Richard Tapia of Rice University math department suggests that Noah must have been one of the earliest practitioners of constrained optimization, as he had to house all the animals and in such a way that none got eaten. Branch and bound Wikipedia: Constraint optimization can be solved by branch and bound algorithms. These are backtracking algorithms storing the cost of the best solution found during execution and use it for avoiding part of the search. More precisely, whenever the algorithm encounters a partial solution that cannot be extended to form a solution of better cost than the stored best cost, the algorithm backtracks, instead of trying to extend this solution. First-choice bounding functions Wikipedia: In the branch and bound method, one way for evaluating this upper bound for a partial solution is to consider each soft constraint separately. For each soft constraint, the maximal possible value for any assignment to the unassigned variables is assumed. The sum of these values is an upper bound because the soft constraints cannot assume a higher value. It is exact because the maximal values of soft constraints may derive from different evaluations: a soft constraint may be maximal for x = a while another constraint is maximal for x = b. Russian doll search Wikipedia: This method runs a branch-and-bound algorithm on n problems, where n is the number of variables. Each such problem is the subproblem obtained by dropping a sequence of variables x_1, ..., x_i from the original problem, along with the constraints containing them. After the problem on variables x_i+1, ..., x_n is solved, its optimal cost can be used as an upper bound while solving the other problems. In particular, the cost estimate of a solution having x_i+1, ..., x_n as unassigned variables is added to the cost that derives from the evaluated variables. Virtually, this corresponds on ignoring the evaluated variables and solving the problem on the unassigned ones, except that the latter problem has already been solved. |
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Appeal - Implicitly Innovate
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Appeal - Implicitly Innovate Ethos, Logos, Pathos identify weakness in presentation Heart, Hand, Mind appeal to a person holistically |
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Caring about caring about caring
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Caring about caring about caring Acknowledging what transcends our limits. Let go of my personal limits and live the vision of an unlimited God who can live through all. My personal limitations Conversing with: impossibility Appreciate my vulnerabilities I was in first or second grade when I got glasses. It turned out that my vision was very bad. I can hardly make out the features of people's faces. Although I resisted needing glasses, I appreciated that they were a great gift. I wondered, how would I have survived before glasses were invented? I might not be able to work at all, I might have to be a beggar. And so I appreciated that I was not master of my own circumstances, and thus not free to do as I pleased. Battle with myself Conversing with: my own fate Many times, I've comforted myself by touching myself, and despite that pleasure, I know well that afterwards I feel dissipated and depressed. Yet my mind is so creative in coming up with justifications at the time. But it does seem to make a difference that I battle my mind, even if I lose. And from this battle I have learned many things, but especially that my mind and my will are different. I am my will and not my mind. Be myself Conversing with: my destiny When I have fallen in love, it has generally been wholehearted, and without much chance that my feelings would be returned. Yet I have been true to myself and thereby, I think, grown in integrity and learned what it is that I find lovely in another person. Care for mind, body and spirit Conversing with: harmonious life In eighth grade we were taught the Greek view of harmonious growth in mind, body and spirit. This encouraged me to take up sports, even though I was very weak. In high school, I joined the water polo and swimming teams. We swam from 6:30 to 8:00 in the morning and from 2:30 to 5:00 in the afternoon. I was always the worst, but I grew to be quite well built and of at least average strength. Come to see or learn Conversing with: my understanding I was in love with a wonderful woman, but she married another man. I didn't stay in touch with her, but I did stay in love with her for almost twenty years. Once she wrote me with condolences on the death of my niece. I wrote her back that she was the dearest person in my life, and for her husband's sake, I didn't think it good that I be in touch with her, but I truly wished that she apply her creative talents somehow, as she was a beautiful soul, and Lithuania needed that very much. A year or so passed and my love finally faded, in part because I thought she wasn't living her talents, and in part because she was working as a cog in the European Union, apparently, not for the short term. Then I fell in love with another woman. It happened, finally, that I had reason to call her, and she was delighted. She told me all of her troubles. Then I raised my concerns that she was not applying herself and that she had sold out. She took it very badly and berated me. I saw how sensitive she was to herself, and not to me, and I wondered, why did I find her attractive all of those years? I realized that the world had changed. During the Soviet occupation of Lithuania, when people did what they could to avoid an inner life, she stood out to me as a person with an amazingly rich one, in which I believed. But now I thought it apparent that it revolved around her own feelings. Desire to live as an example Conversing with: my destiny I wanted to put my philosophy into practice and so I started developing good will exercises to address situations where we are riled because we believe one thing in our hearts, but in the world it is otherwise. In pursuing this, I wanted to live as an example. After two years I stopped because I realized that being riled meant that my mind was thinking wrongly, and so why start from that and encourage myself and others to focus on that? Later, after I completed my video summary "I wish to know", I wanted to start a culture, but more and more I realized that it was not by living as an example to emulate, but by playing a role that God put me in a position to play. End a relationship Conversing with: inability I had a roommate who was very bright but, as time went on, I wasn't able to engage with him in a constructive way. I decided that we were injuring each other as people and so I told him that I was ending our relationship. I learned my limitations, that I was unable to have a healthy relationship with him. Exercise a secondary principle Conversing with: respect My Mother taught me that when I go to a dance, I must invite all of the girls to dance and not just the ones I thought were pretty. And although this was not most important to me, but even so, many times when I did not know whom to ask, I remembered her words and obeyed them. She made me care more about other people. Stay true to God Conversing with: fate As a child in grade school, I was weak and young, but during recess, I didn't want to be protected by the teachers, so I would walk far off into the field. And quite often a disturbed child would come up to bully me. I was a very good child and my response was to be as good as possible and to appeal in my heart to God to save me. Always some stronger kid would come and save me. Try out all possible solutions Conversing with: impossiblity In avoiding bankruptcy, I'm trying to take every reasonable step along the way. In trying to make a living from my business, Minciu Sodas, I kept looking for the most promising and sensible opportunity. In pursuing my sweetheart, I'm trying to show that I care about her in every way. In trying to save David's home, and especially, in encouraging him to take action to save it, I've tried to show that I'm ever interested. These are all areas where I can't assure my own success yet must try to do my part. Note the limits of thinking Conversing with: wholeness Applying Scripture Conversing with: heritage I wanted to have a way to hear from God, so I relied on the prayer "Our Father", which I had prayed many times, was part of my personal heritage, my family's heritage and my religious upbringing. Justification Conversing with: given Certain ideas I have thought are dangerous, and justification helped me to realize, in what sense I think they are. Also, I was born in California but raised Lithuanian, and justification made me embrace my Lithuanian heritage in a very conscious way. Limits of the mind Conversing with: limitlessness I have thought about Everything all of my life, and have found it to have only four properties. Participate in discussion Conversing with: variety of perspectives What is happiness? I've seen in discussion led by Michael Gillespie that such a question leads to only a few types of answers, namely: joy, contentment, assurance. Which is to say, it is a trigger for a representation of the threesome in terms of doing, thinking and being. Recognize that I do not know how else to think Conversing with: unavoidability I learned of the divisions of everything by noting the limits of my mind and my inability to think in more than just a few ways. |
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Recognizing the limits of our openness
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Recognizing the limits of our openness Be immersed in the lives of those who suffer The hand of institutional racism was invisible to most white people, including my friends, who tended to avoid institutions or neighborhoods that catered to black people out of fear for their own safety or discomfort. My weekly session with my patients in the General Medicine Clinic heightened my sensitivity to the issues of race in America. In 1906, W.E.B. DuBois said, "The problem of the twentieth century is the problem of the color line." I was a middle-class white man from a small city in upstate New York. I had never been in a position to understand the meaning of these words until I was immersed in the lives of my patients that revealed their truth so powerfully and so tragically. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.1934 Being introduced to an activity Fresh Air interview of boxer Sugar Ray Leonard, June 10, 2011: It was Leonard's shyness as a child, he says, that prompted his brother Roger to first take him to a boxing gym as a young teenager. Have hang ups and neuroses Fresh Air interview of Steve Coogan, June 9, 2011: "There is a certain insecurity and a certain kind of malcontentedness that [I] just channel into [my] work," he says. "Being creative means some of the things that bother you stop bothering you, because you exploit them creatively. So it's a kind of a strange process, but you need to have hangups and neuroses to be creative. If you're just in a state of nirvana, you're not going to be very interesting or funny." Let people fight it out The oral surgery clinic had a perverse policy. They would treat only fifty patients daily. No appointments. Fantus' doors opened at 7:00 a.m. Patients with toothaches, loose teeth, oral tumors and mouth abscesses lined up in painful silence during the dark hours of the early morning. When the doors to Fantus were opened, it was like the starting gate at Arlington race track. They're off! The crowd scrambled through the open Fantus gates. Patients, some in wheelchairs, others with canes and crutches, raced to get to the Oral Surgery clinic to win one of the fifty prized slots that guaranteed a dentist would see them. This system had persisted through the years despite its inhumanity. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.1934 Not wanting to listen to your trainer Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: "When you're losing, you really don't want to hear what your trainer has to say. He may tell you the right thing, but you're so exhausted, you're so beat up, that your lungs are burning, your legs are tired. And you've given up Things getting out of control Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: I shared the wealth. And I tried to give [my family] a home. I bought homes for most of my family members. And I bought cars. I bought 10 cars, they were Pintos. They were free [for them.] I just tried to help them because they're my brothers and sisters. But things just got out of control. Things became too frequent. What a typical person goes through Just about every man had a scar from a knife or bullet wound. Almost every woman had lost a close family member to violence. The names of lost loves and relatives were tattooed onto the arms and in the memories of my patients. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.1934 |
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Become aware of bias
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We instinctively rely on, and are susceptible to, biases and assumptions. Bias - an unconscious belief that conditions, governs, and compels our behavior ... Immanuel Kant theorized that the mind is not designed to give us uninterpreted knowledge of the world but must always approach it from a special point of view ... with a certain bias. ... Acquiring a bias is not a conscious mental process ... we are stuck with biases whether we want them or not ... It is bias that enables us to repeat an action we have taken before without going through all of the mental steps that led to the original act. ... assumptions beget biases, biases beget assumptions ... much more often than not, our biases lead us to correct conclusions and reactions, and they do it exceedingly fast. ... the rapidity of this process and the fact that it is unconscious - and thus uncontrolled - have the unfortunate effect of strengthening and validating our biases at the expense of truth. The reason ... is that we tend to give high value to new information that is consistent with our biases, thus reinforcing them, while giving low value to, and even rejecting, new information that is inconsistent. New information that is ambiguous either is construed as consistent with our biases or is dismissed as irrelevant. ... because most biases are hidden from our consciousness, we aren't aware of their existence or of their effects ... The trouble with biases is that they impose artificial constraints and boundaries on what we think. The insidious part is that we aren't even aware that our thinking is restricted. drawing from The Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich. |
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Learn my limits
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Learn my limits Learn my limits, typically by overstepping them. During the independence movement, the dollar was worth a lot and I could hire an organizer. I paid him for one or two months but then he needed his labor book to be signed by somebody official. So I suggested that we go the reform movement's office, that they could do that formality, and I was surprised that they refused. In high school we used to raise money for our swim team by having a swimathon. We would go door to door and ask for a penny or more per lap that we swam. We ended up swimming 200 laps. It was good practice asking people for support and quite surprising how supportive people were. When I was in high school, some of my classmates started getting part time jobs. I wondered whether I should get one, whether that would help me be a more balanced person. I once wanted a telescope for Christmas and my parents hesitated but I was surprised to find that they gave me one. I found it early in the morning and tried to put it together but apparently I broke it. And to my sadness that was the end of the telescope, my parents did not replace it. As a graduate student I arranged to spend a year at the University of Vilnius in Soviet-occupied Lithuania. However, I didn't hear back from them. The vice president of the Academy of Sciences was a friend of a family friend and he offered to take care of the matter. It didn't seem right to have to go through such a private channel. But it made all the difference. I received my invitation as planned. It was very strange to see the system in action and the amount of authority required to handle such a small matter. When I was in college, I used to work in the summer, but would save my money so that I wouldn't have to work during the year. I wanted to study and learn as much as I possibly could, and I was glad that my parents were supportive. When I was finishing my Ph.D., I wondered how I would make a living, if I was to devote myself to what I wanted to do. I thought that a university was a sheltered environment and so I would not learn about life. Also, if I made money from my philosophical pursuits, then I would corrupt myself and my thinking. I decided that I should work part-time and be free to spend the rest of my time on my own projects. When I was a student I went to Soviet-occupied Lithuania and so my grandfather gave me some money to buy items for some old anti-communist guerilla fighters at the dollar store. And he said be sure to bring back the receipts. So I met the people and I bought what they wanted but they wanted to keep the receipts and not thinking I let them. I didn't understand why they were so important, so long as the money was spent. And my grandfather was very unhappy because he said that he will not have receipts to give to the emigre organization and they won't be giving any more money because they can't account for it. My father wanted us to make the most of the opportunity that we had not having to work. I felt encouraged by his view because it was his first hand account, having had worked, and I accepted it, and I felt I did not need to relive it. It's interesting that in my life I have never had a full time job. I had a close friend who I stayed with in the United States when he invited me for his wedding. I was just starting my business and I stayed at his home with his parents when he left for his honeymoon. And they started to wonder what I was doing there. And then I realized that although I believed in giving and receiving and my friend had been quite supportive but that was not normal and it was possible to quickly run up against limits. I was anxious and I told God I was too anxious to work part-time on my philosophy and on the business too at that point in my life. I offered a deal with him to let me focus on the business and I would make the business model work for him, too. Then I felt more at ease. When I completed my graduate courses, it was time to write a Ph.D. thesis. However, it wasn't my goal to be a mathematician, I had simply wanted to get an education. So I didn't know for what reason to continue, although it felt a bit awkward to stop. My mother supplied the reason, she said that I wanted to be a person who finishes things, and I wouldn't have to explain for the rest of my life why I hadn't finished. That was the reason I continued. |
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Applying myself
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Applying myself My helpfulness Knowing what belongs to me Knowing what I will help with Recognizing how I can help others Recognizing the kind of person I am My limits Being limited Not finding alternatives Realizing my limitations  Millikan-Fletcher Oil Drop Experiment The Millikan-Fletcher Oil Drop Experiment in 1909 measured the size of the quantum of electrical charge. A tiny charged oil drop is suspended in air by an eletrical force working against gravity. It's terminal velocity (in the face of air resistance) is measured by adjusting the electric field so that the drop falls down and then falls up or down again. The charge of the drop can be measured from the terminal velocities. Many such charges are measured. It can then be showed by considering differences that they are multiples of a quantum value of charge. There is a lower bound on significant differences. |
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Belong to the Lord, share in the favor of the Lord.
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Belong to the Lord, share in the favor of the Lord. We receive each other, we receive Jesus, we receive a reward from God. Jesus is our reward. Cast out with a word Matthew 8:16 When evening came, they brought to him many possessed with demons. He cast out the spirits with a word, and healed all who were sick; 8:17 that it might be fulfilled which was spoken through Isaiah the prophet, saying: "He took our infirmities, and bore our diseases." Do people and Jesus truly know each other? Matthew 7:22 Many will tell me in that day, 'Lord, Lord, didn't we prophesy in your name, in your name cast out demons, and in your name do many mighty works?' 7:23 Then I will tell them, 'I never knew you. Depart from me, you who work iniquity.' God through Jesus and angles saves those who stumble God, Jesus, angels look to protect and save those who stumble. Matthew 18:10 See that you don't despise one of these little ones, for I tell you that in heaven their angels always see the face of my Father who is in heaven. 18:11 For the Son of Man came to save that which was lost. How are we one with Jesus and God Matthew 10:40 He who receives you receives me, and he who receives me receives him who sent me. 10:41 He who receives a prophet in the name of a prophet will receive a prophet's reward. He who receives a righteous man in the name of a righteous man will receive a righteous man's reward. 10:42 Whoever gives one of these little ones just a cup of cold water to drink in the name of a disciple, most certainly I tell you he will in no way lose his reward." How can judgment be escaped? Matthew 23:33 You serpents, you offspring of vipers, how will you escape the judgment of Gehenna? How does one receive We receive by receiving on behalf of another, of God in them, by giving through them. Matthew 10:40 He who receives you receives me, and he who receives me receives him who sent me. 10:41 He who receives a prophet in the name of a prophet will receive a prophet's reward. He who receives a righteous man in the name of a righteous man will receive a righteous man's reward. 10:42 Whoever gives one of these little ones just a cup of cold water to drink in the name of a disciple, most certainly I tell you he will in no way lose his reward." Note from whom God hides and to whom reveals The significance of Jesus' miracles are hidden from John the Baptist, from Bethsaida and Capernaum, from the wise and understanding, but "infants" are delighted and uplifted, those who are fresh in faith, who don't want to justify the world, but want it bettered, by those who suffer and labor, who want healing. Matthew 11:25 At that time, Jesus answered, "I thank you, Father, Lord of heaven and earth, that you hid these things from the wise and understanding, and revealed them to infants. 11:26 Yes, Father, for so it was well-pleasing in your sight. What are the consequences of not believing? Matthew 7:26 Everyone who hears these words of mine, and doesn't do them will be like a foolish man, who built his house on the sand. 7:27 The rain came down, the floods came, and the winds blew, and beat on that house; and it fell—and great was its fall." What happens to the bad tree? Matthew 7:19 Every tree that doesn't grow good fruit is cut down, and thrown into the fire. What is beyond this world will manifest in this world Matthew 6:6 But you, when you pray, enter into your inner room, and having shut your door, pray to your Father who is in secret, and your Father who sees in secret will reward you openly. What is needed for judgment Jesus is the judge because he is a "son of man" and knows what it means to be human. Yet he seeks not his own will, but that of his Father, who sent him and gave him life. John 5:24 "Most certainly I tell you, he who hears my word, and believes him who sent me, has eternal life, and doesn't come into judgment, but has passed out of death into life. 5:25 Most certainly, I tell you, the hour comes, and now is, when the dead will hear the Son of God's voice; and those who hear will live. 5:26 For as the Father has life in himself, even so he gave to the Son also to have life in himself. 5:27 He also gave him authority to execute judgment, because he is a son of man. 5:28 Don't marvel at this, for the hour comes, in which all that are in the tombs will hear his voice, 5:29 and will come out; those who have done good, to the resurrection of life; and those who have done evil, to the resurrection of judgment. 5:30 I can of myself do nothing. As I hear, I judge, and my judgment is righteous; because I don't seek my own will, but the will of my Father who sent me. What is possible for man and God God can change one's heart. Luke 18:26 Those who heard it said, "Then who can be saved?" 18:27 But he said, "The things which are impossible with men are possible with God." What is the ultimate outcome? Matthew 24:1 Jesus went out from the temple, and was going on his way. His disciples came to him to show him the buildings of the temple. 24:2 But he answered them, "You see all of these things, don't you? Most certainly I tell you, there will not be left here one stone on another, that will not be thrown down." What makes people believe God the Father changes people's hearts. John 4:47 When he heard that Jesus had come out of Judea into Galilee, he went to him, and begged him that he would come down and heal his son, for he was at the point of death. 4:48 Jesus therefore said to him, "Unless you see signs and wonders, you will in no way believe." 4:49 The nobleman said to him, "Sir, come down before my child dies." 4:50 Jesus said to him, "Go your way. Your son lives." The man believed the word that Jesus spoke to him, and he went his way. What sanctifies what? 23:16 "Woe to you, you blind guides, who say, 'Whoever swears by the temple, it is nothing; but whoever swears by the gold of the temple, he is obligated.' 23:17 You blind fools! For which is greater, the gold, or the temple that sanctifies the gold? 23:18 'Whoever swears by the altar, it is nothing; but whoever swears by the gift that is on it, he is obligated?' 23:19 You blind fools! For which is greater, the gift, or the altar that sanctifies the gift? 23:20 He therefore who swears by the altar, swears by it, and by everything on it. 23:21 He who swears by the temple, swears by it, and by him who was living in it. 23:22 He who swears by heaven, swears by the throne of God, and by him who sits on it. Where did the good and the bad come from Matthew 13:37 He answered them, "He who sows the good seed is the Son of Man, 13:38 the field is the world; and the good seed, these are the children of the Kingdom; and the darnel weeds are the children of the evil one. 13:39 The enemy who sowed them is the devil. The harvest is the end of the age, and the reapers are angels. 13:40 As therefore the darnel weeds are gathered up and burned with fire; so will it be at the end of this age. 13:41 The Son of Man will send out his angels, and they will gather out of his Kingdom all things that cause stumbling, and those who do iniquity, 13:42 and will cast them into the furnace of fire. There will be weeping and the gnashing of teeth. 13:43 Then the righteous will shine forth like the sun in the Kingdom of their Father. He who has ears to hear, let him hear. Who comes to the light John 3:17 For God didn't send his Son into the world to judge the world, but that the world should be saved through him. 3:18 He who believes in him is not judged. He who doesn't believe has been judged already, because he has not believed in the name of the one and only Son of God. 3:19 This is the judgment, that the light has come into the world, and men loved the darkness rather than the light; for their works were evil. 3:20 For everyone who does evil hates the light, and doesn't come to the light, lest his works would be exposed. 3:21 But he who does the truth comes to the light, that his works may be revealed, that they have been done in God." Who is our Father? Matthew 23:9 Call no man on the earth your father, for one is your Father, he who is in heaven. Who loves the world It is the Father who loved the world and gave up his Son for it. John 3:14 As Moses lifted up the serpent in the wilderness, even so must the Son of Man be lifted up, 3:15 that whoever believes in him should not perish, but have eternal life. 3:16 For God so loved the world, that he gave his one and only Son, that whoever believes in him should not perish, but have eternal life. 3:17 For God didn't send his Son into the world to judge the world, but that the world should be saved through him. Whom to fear and why Fear God. God is the only one who can destroy completely. Matthew 10:26 Therefore don't be afraid of them, for there is nothing covered that will not be revealed; and hidden that will not be known. 10:27 What I tell you in the darkness, speak in the light; and what you hear whispered in the ear, proclaim on the housetops. 10:28 Don't be afraid of those who kill the body, but are not able to kill the soul. Rather, fear him who is able to destroy both soul and body in Gehenna. Why is a reward received Matthew 10:40 He who receives you receives me, and he who receives me receives him who sent me. 10:41 He who receives a prophet in the name of a prophet will receive a prophet's reward. He who receives a righteous man in the name of a righteous man will receive a righteous man's reward. 10:42 Whoever gives one of these little ones just a cup of cold water to drink in the name of a disciple, most certainly I tell you he will in no way lose his reward." What does Jesus offer, as Lord? The Sabbath is a day of exceptions, and the Son of Man is, by his slack, his mercy, the one who makes exceptions possible. Sacrifice is just a sign of slack. Thus there is no guilt for those who the Son of Man gives slack to. Matthew 12:7 But if you had known what this means, 'I desire mercy, and not sacrifice,' you would not have condemned the guiltless. 12:8 For the Son of Man is Lord of the Sabbath." |
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Limits
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Limits We allow a boxing-in or boxing-out process to continue indefinitely, yielding (or not) a limit that may very well transcend the existing system (as the reals transcend the rationals). Continuum hypothesis Wikipedia: There is no set whose cardinality is strictly between that of the integers and that of the real numbers. ... The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that this continuum hypothesis can neither be disproved nor be proved using the axioms of Zermelo-Fraenkel set theory, the standard foundation of modern mathematics, provided ZF set theory is consistent. Cauchy property In practice, there are several possible methods of showing that a given sequence converges to a limit. ... Show that the ai eventually get arbitrarily close to one another. More precisely, a sequence (an) possesses the Cauchy property if for any (very tiny) epsilon > 0 there is a (huge) N such that |am - an| < epsilon for all m, n >= N. If a sequence of real numbers has the Cauchy property, it converges. The Cauchy property is often fairly easy to verify, but the disadvantage is that one doesn't get any information about the actual limiting value of the sequence. pg.285 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Convergence We say that the real-valued sequence (an) converges to the limit L if ... That if we pick an arbitrary distance epsilon > 0, eventually, and forever after, the ai will get within epsilon of L. More specifically, for any epsilon > 0 (think of epsilon as a really tiny number), there is an integer N (think of it as a really huge number, one that depends on epsilon) such that all of the numbers aN, aN+1, aN+2, ... lie within epsilon of L. In other words, for all n >= N, |an - L| < epsilon. pg.285 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Shoenfield [1961] and Sacks independently [1963a,b,c] invented the infinite injury method for computable constructions in which a given requirement may be injured infinitely often, but only computably so, and still somehow succeed. Yates, Martin, and Lachlan extended and refined this method and added others such as the minimal pair construction of a pair of noncomputable c.e. sets A and B whose degree have infimum degree 0 the degree of the computable sets. By the end of the initial period of computability 1930--1970 the constructions and proofs had become so complicated that it had become very difficult to obtain any intuition about them. Fortunately, Lachlan introduced: (1) the method of games in computability Lachlan [1970a] (see Chapter on Lachlan Games); (2) the topology of priority arguments Lachlan [1973], which gave rise to the true stages method for infinite injury (refined in Chapter 8); and (3) the use trees in Lachlan [1975] to divide a strategy for a priority argument into a collection of strategies, one assigned to each node of the tree, and each equipped with a guess about the outcome of higher priority strategies. Lachlan's original use of trees [1975] for a 0''' argument was very complicated, but the tree method with simplifications was presented in Soare [1985] and is further refined in this book in Chapters 9 and 10. It has been used very extensively in computability theory ever since. These three improvements and their refinements allow the reader of this book to help keep in mind what Leo Harrington calls the mountain top view of the proof: a small number of key elements, their interaction, how we resolve conflicts between conflicting requirements, and the underlying beauty of the proof. Robert Soare's upcoming book, "Computability Theory and Applications: The Art of Classical Computability". Integrals The fundamental theorem of calculus gives us a method for computing definite integrals. ... There is a lot of interplay between summation, integration, and inequalities; many problems exploit this. Compute the limit as n approaches infinity of the sum from k =1 to n of n/(k**2 + n**2). The problem is impenetrable until we realize that we are not faced with a sum, but the limit of a sum; and that is exactly what definite integrals are. So let us try to work backwards and construct a definite integral whose value is the given limit. ... Even finite sums can be analyzed with integrals. If the functions involved are monotonic, it is possible to relate integrals and sums with inequalities... pg.301-302 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc |
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Dream - Explicitly Care
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Dream - Explicitly Care Cover Story thinking big Customer, Employee, Shareholder imagine futures from several perspectives Make a World sketch a vision of a future state Storyboard imagine and create possibilities The Comeback Album of The Decade Game imagining a new life |
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Caring about caring about caring about caring
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Caring about caring about caring about caring Acknowledging what transcends our limits. Let go of my personal limits and live the vision of an unlimited God who can live through all. Appreciate that God's will and my will are independent Conversing with: divineness Conversing with: divineness Acknowledge that God behaves inconsistently in Scripture Conversing with: God's truth I find it very freeing to note and consider inconsistencies in God's behavior in Scripture. I note that the priest Eli's sons were wicked, and God had them killed in battle, and the Philistines took the ark of Yahweh, and upon hearing that, Eli fell backward and died. (1 Sam 4) Yet the priest Samuel's sons perverted justice, and the people refused them, and wanted a king, and God granted them a king. (1 Sam 8) Or when Zacharias asks the angel Gabriel, "How can I be sure of this?", that his barren wife will give birth, then he is made mute until the child is born, because he did not believe. But when Mary asks the angel Gabriel, "How can this be?", that she will give birth to a child, being a virgin, then the angel explains how. (Luke 1) People draw conclusions from the Scriptures. They may not notice such discrepancies; they may not choose to notice them; and if they do notice them, then they may explain them away in many ways, so as to defend an idea that God is consistent. But given such discrepancies, I don't see how I can draw any conclusions, except that God's reasoning is hidden, or more constructively, that God is inconsistent, practically speaking, from our point of view. Ultimately, God does as God pleases, and God is free, and such a thought is freeing, whereas people are consistent, just as machines are consistent in their outcomes, and alcoholics are consistent in their goals. I've been taught that there are four Gospels so as to have different witnesses tell the same story and corroborate each other. But once I checked their versions of the Resurrection and was astonished to see that they differed in absolutely every fact: who saw Jesus first; how many people were there; where did that occur and so on. They disagree on absolutely every fact and are completely incompatible! Which typically would not be the case if they were lying or inventing. Which suggests that the Resurrection involved a total breakdown of time and space, whether real or imaginary. Noticing such difficulties supports my hope that the Scripture is a perfect text in that it transcends the particular wording or translation, but says something constructive to anybody who reads it in good faith. Acknowledging Scripture Conversing with: self-imposed assumptions I acknowledge that Scripture, such as the Gospels of Mark, Matthew, Luke and John, is the message of God intended for us to take to heart. I am attracted to what Jesus says, and I accept his logic. As a youth I read the Gospels, including the Sermon on the Mount, where Jesus says that if your eye causes you to sin, then you should pluck it out, for it is better to go to heaven without an eye, then to go to hell in one piece. Jesus' logic seemed clear and sensible. If I were not to follow it, then I would be denying him and denying Scripture. Ask God to intercede Conversing with: hope By asking God to intercede, I figure out, what do I truly want? I and other prayed for Lithuania to become free of Soviet occupation and it came true! I loved a woman with all my heart, but I didn't think I should interfere with her free will, so I asked God to fulfill at least two of three wishes of mine, that she visit me, that she be happy and that she fall in love with me. And she did visit me and she was happy, but she didn't fall in love with me. Do not go along with God Conversing with: baselessness I don't always do as God has me do. In China, as I was writing up how to do the good will exercises, and I was engaging God for his help to work together, God told me to sit on the window sill. I did not want to offend God, but I thought that it was too much to ask of my faith. God was a bit miffed, and his reply was that this put our relationship in perspective. In Chicago, I prayed God that my friend David, who I was living with, not lose his home. God told me that he would stay there, and indeed I would live there with my sweetheart. Even so, I looked for another place to live, because I did not want to live there after it was confirmed sold. God was not mad at me. First hand experience Conversing with: my life experiences First hand experience makes me think twice about statements that may otherwise seem questionable, such as Jesus' exhortations that what we believe is what happens. I and others prayed that Lithuania be free of the Soviet Union, and amazingly, this did come to pass. I dedicated myself to know everything and apply that knowledge usefully, and I enjoyed many discoveries which led in 2009 to my overview of knowledge, achieving my goal, and yet continuing onward. I wished to fall in love with a woman whom I could love with all my heart, more than any other, and God blessed me just so. Give in to myself Conversing with: tolerance By succumbing to myself, my vices or my weaknesses, I appreciate God's forgiveness, God's slack for others. I am not so stark or harsh. Hold God's behavior to at least my own standard I loved a woman with all of my heart, but she chose to marry another man. I told God that I still loved her, but that for me to be true to pursuing her, I would have to kill this man. I told God that I wasn't going to do that. And so I told God that he owed me, for I had loved her so completely, and that I wouldn't love a woman unless I loved her more than I had loved this woman. Seventeen years later, unexpectedly, God brought me to such a woman! And I love her unreservedly, and God encourages me. Imagine that God is responsible Conversing with: God's wishes When my computer crashes, if I lose a letter or file, then I often stop and wonder, what was the point of losing that? what would God have me do otherwise? And so I try to make good of the loss, often thinking and writing more kindly, or focusing on the key point. In Chicago, when my boss told me he wouldn't give me more hours because I was unwilling to change my approach, then I considered, maybe God doesn't want me to have more hours there, which surprised him, as he was a devout Christian. Jesus focused on what is natural Sometimes, as my debts grew, I would wonder if I should have foresworn from ever going into debt. However, I realized the positive aspects of my debts. And I recalled that Jesus discouraged saving, but had much to say about debtors, which he seemed to take as the natural human condition, and which makes sense, given that we're created by God and raised by our parents. Make sense of Scripture Conversing with: God's thinking I learn a different way of looking at things by trying to make sense of concepts from Scripture. Perceive a structure in Scripture *** What does structure mean to God? Conversing with: divineness I am encouraged when I observe in Scripture a structure that I am aware of from elsewhere. I noted that Jesus' antitheses in the Sermon on the Mount are six of the counterquestions. I identified Jesus' condemnations, "Woe to you, Pharisees", with the six expressions of the will. I related his parable of the sower and Satan's temptations with the levels of the foursome. This helps me think of these structures from a fresh perspective. Stay in touch Conversing with: good will Sometimes my values seem to diverge so sharply from people I know that I don't see the point of maintaining a relationship. But then I realize that, even so, I can have good will and maintain it at some most basic level. I allow and recognize that there may be some purpose greater than my own perspective. Accentuate the core of my ideal Conversing with: my purpose Conversing with: my purpose Analyzing the ideal person Conversing with: perfection I analyzed Jesus's emotional responses as found in the Gospel of Mark. From those episodes I deduced Jesus's expectations. Choose what is most comprehensive Conversing with: my point of departure I was finding it useful to ask Minciu Sodas leaders, staring with Franz Nahrada, about their "key concept", and better said, their "deepest value" in life. I thought I should formulate my own as well, and finally did, as I was engaging a young man in Lithuania about that. I found it difficult to do. As a child, I had greatly valued honesty, of saying what you truly think and feel, of never lying under any circumstances, not accepting any excuse for that. I also valued knowledge and learning. I thought that the truth was key for all other values to be true. In surveying my values, I realized that it was not knowing the truth or having the truth but living by truth that expressed what I felt important. Life was more than truth, but truth could and must be my key to live my life with integrity. Exhaustive review Conversing with: significance In 2009, to my surprise, I completed my quest "to know everything" with a 10 minute video summary, "I Wish to Know". I needed to find my bearings afterwards. I considered, in the broadest sense, what did I learn from my quest? And what came next? I learned two things. I need not be cold or distant to God as I had been, I need not keep God at arm's length, for God's point of view had made sense of absolutely every structure that I had uncovered in my search. I realized also that the useful application of this knowledge would be to foster a culture of truth, of self-learning, of figuring things out, which from my experience at Minciu Sodas was very much Jesus' vision of the Kingdom of Heaven. Make meaningful Conversing with: meaningfulness In Vilnius, I babysat my god daughter Deboryte and helped her with her homework. That ended badly because I was interested to teach her more broadly, but she was focused on doing what was assigned, and so I made her upset. Even so, I tried to draw meaning from the evening, and I remembered that I was a good tutor. I had worked at Ivy League Tutoring in Chicago, and since I had exhausted my hopes to make a living from Minciu Sodas, perhaps I could go back and work there. I called the founder, Adrian Hunter, and indeed, he agreed to hire me, and I left for Chicago within a month. More generally, as a youth, I tried to draw meaning from every episode in my life, every talent I developed, and build on them, although certain skills, such as my Spanish, always seemed underutilized, and I spent time on others, such as playing guitar, that I never sufficiently developed to get past the hump so I could build on them. Seek the most straightforward meaning In reading the Scripture, especially, passages where the meaning is not obvious, I try to consider the most straightforward interpretation, even the most childlike one. It may not be the traditional interpretation because people may want to take away a different meaning than is intended. At the Last Supper, Jesus equated his bread with his body and his wine with his blood, and gave it to his disciples to eat and to drink. Taking him literally, and trying to understand what he might mean, most straightforwardly, I thought, "Given that this is his bread, he will eat it, and it will become his body. In that sense, it is his body already, by its destiny, its purpose. And so, if he gives what is truly his to them instead, he changes its course, and they are eating his body. In the parable of the Good Samaritan, by which Jesus explains the meaning of "neighbor" in "Love your neighbor", he asks, "Who was the neighbor to the Samaritan?" People don't pay attention to the fact that he defines the neighbor as the one who helped, not the one who was hurt. Sort out my own feelings Conversing with: my knowledge Sorting through my feelings, I recognized various distinctions. I disavowed feelings of shame because they came from others and not myself, and so I felt it was wrong to shame and to feel shame. In contrast, embraced feelings of guilt as healthy because they came from inside my own self, and I could resolve them internally. I felt pleasure or comfort in fondling myself, but afterwards it inevitably made me feel depressed and spent, and so I concluded from my own feelings that it was wrong for me to do and I should ever resist that, even though I constantly failed. I thought it was my own personal addiction, sin that helped me identify with other people's addictions, failings of character, sins. Yet I felt that, in any event, it was not central to my life, it was not what was key to understanding me. |
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Allowing for an ideal that transcends our limits
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Allowing for an ideal that transcends our limits Being drawn to an activity Fresh Air interview of boxer Sugar Ray Leonard, June 10, 2011: "For some reason, I was drawn towards boxing," he says. "Or maybe boxing drew me towards it — because once I put those gloves on, after about six months, boxing was my life." Finding the value that lies between those things we know Bradford Hansen-Smith, 2011.04.25: Meaning is not in explaining what we know, rather in finding the value that lies between those things we know. Alignment with that which is out of our range extends our understanding revealing greater symmetries and ways of solving problems, like ways of making a living without having to get into someone else's box. What we're not taught in school If I had not asked, she would not have told me. I might have just adjusted her medicine and had her return in three months. I did not learn how to treat this in medical school. There was no medicine for grief, for the inevitability of urban violence. I felt powerless. I mumbled my sympathy and asked her to return in a month to recheck her blood pressure. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.1934 Surprising interaction Bob Lichtenbert's aphorisms: "I'm so glad when other people respond to me." Somebody responded to me and I was surprised that they didn't change the subject. |
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Establish rational, systematic frameworks
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Fifth, by establishing rational, systematic frameworks within which to analyze problems, analytic structuring techniques enable us to impose our analytic will on our subconscious mind, preventing it from following the instinctive mental traits that lead to faulty analysis. |
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Steward what I am given
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Steward what I am given Our vocation. And given that I need to pay taxes in both Lithuania and the United States I think that I should do all the paperwork myself because I need to be on top of it myself and there is nobody here who knows both systems anyways. At one point our leader Algis Razminas of the Folk Creativity Club Atzalynas won enough projects to make a living, but they were cancelled by the government when the minister changed and the budget was redone. When my parents used to travel in Soviet-occupied Lithuania they would carry two suitcases full of presents, household items such as calculators and bottles of aspirin. We had hundreds of relatives because our grandparents were active in keeping in touch with their brothers and sisters and cousins. A lot of care went into thinking what people might need, and once there, deciding who to give what. In high school there was a student who worked long hours in the cafeteria. His work was often his explanation for what he was doing with his time. When he got paid he would ask for his wage to be given to him in rolls of quarters. Then he would throw his quarters down the sidewalk with the hope, it seemed, that people would run after them. I concluded that work could very well be a spiritual dead end. |
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Making sense of fate
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Making sense of fate Making sense of fate's intent |
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Millstone hung around the neck
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Millstone hung around the neck In the gospels of Mark, Matthew and Luke, Jesus refers to a "huge millstone" hung around the neck, which I think is the "relative learning" of taking a stand, following through and reflecting, for it is very slow learning, yet it is better than evil. In Greek it is "mulos onikos", which is a donkey-powered millstone, and I take him to compare such a ponderous "relative learner" (or scientist) with an ass. Millstone in Luke Luke 17:1 He said to the disciples, "It is impossible that no occasions of stumbling should come, but woe to him through whom they come! 17:2 It would be better for him if a millstone were hung around his neck, and he were thrown into the sea, rather than that he should cause one of these little ones to stumble. 17:3 Be careful. If your brother sins against you, rebuke him. If he repents, forgive him. 17:4 If he sins against you seven times in the day, and seven times returns, saying, 'I repent,' you shall forgive him." Millstone in Mark Mark 9:38 John said to him, "Teacher, we saw someone who doesn't follow us casting out demons in your name; and we forbade him, because he doesn't follow us." 9:39 But Jesus said, "Don't forbid him, for there is no one who will do a mighty work in my name, and be able quickly to speak evil of me. 9:40 For whoever is not against us is on our side. 9:41 For whoever will give you a cup of water to drink in my name, because you are Christ's, most certainly I tell you, he will in no way lose his reward. 9:42 Whoever will cause one of these little ones who believe in me to stumble, it would be better for him if he were thrown into the sea with a millstone hung around his neck. 9:43 If your hand causes you to stumble, cut it off. It is better for you to enter into life maimed, rather than having your two hands to go into Gehenna, into the unquenchable fire, 9:44 'where their worm doesn't die, and the fire is not quenched.' Millstone in Matthew Matthew 18:1 In that hour the disciples came to Jesus, saying, "Who then is greatest in the Kingdom of Heaven?" 18:2 Jesus called a little child to himself, and set him in their midst, 18:3 and said, "Most certainly I tell you, unless you turn, and become as little children, you will in no way enter into the Kingdom of Heaven. 18:4 Whoever therefore humbles himself as this little child, the same is the greatest in the Kingdom of Heaven. 18:5 Whoever receives one such little child in my name receives me, 18:6 but whoever causes one of these little ones who believe in me to stumble, it would be better for him that a huge millstone should be hung around his neck, and that he should be sunk in the depths of the sea. What is lowly 12:14 Jesus, having found a young donkey, sat on it. As it is written, 12:15 "Don't be afraid, daughter of Zion. Behold, your King comes, sitting on a donkey's colt." 12:16 His disciples didn't understand these things at first, but when Jesus was glorified, then they remembered that these things were written about him, and that they had done these things to him. Zechariah 9:9 Rejoice greatly, daughter of Zion! Shout, daughter of Jerusalem! Behold, your King comes to you! He is righteous, and having salvation; lowly, and riding on a donkey, even on a colt, the foal of a donkey. |
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Thread
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Thread T) Extend the domain F) Continuity R) Self-superimposed sequence These three frames are the cycle of the scientific method: take a stand (hypothesize), follow through (experiment), reflect (conclude). I imagine that they link B1, B2, B3, B4 with C1, C2, C3, C4 to weave all manner of mathematical ideas, notions, problems, objects. |
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Relative Learning
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Relative Learning In real life, this is taking a stand, following through and reflecting, but it is important to avoid evil, keep varying and not fall into a rut of self-fulfillment. |
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Relative Learning
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Relative Learning Changing one's attitudes, beliefs and values that reinforce one's criminal behavior. Effective strategies should also focus on changing the attitudes, beliefs, and values that reinforce offenders' criminal behavior. This approach teaches the offenders to (1) pay attention to their thoughts and feelings; (2) recognize how those thoughts and feelings lead to negative behavior; (3) use new thinking to reduce risk; and (4) practice, practice, and practice those new thinking and positive behavioral skills. This is not something that can be done in a classroom setting. Offenders must be given daily opportunities to practice on their own, especially since intensive repetition is a key to intervention's success. Without it, the approach is nothing more than a lecture, and will never effectively reduce delinquency. Dispatches from Juvenile Hall: Fixing a Failing System, by John Aarons, Lisa Smith, Linda Wagner, Penguin Books, 2009. Page 180. Astronomy experiments At Wikipedia, the experiment page gives examples of how the scientific method is applied in astronomy. There is also a page listing key astronomy experiments. Biology experiments Wikipedia documents more than 80 biology experiments. The experiment page gives examples of how the scientific method is applied in biology. There is also a page listing key biology experiments. Chemistry experiments Wikipedia documents 7 chemistry experiments. The experiment page gives examples of how the scientific method is applied in chemistry. There is also a page listing key chemistry experiments. Design of experiments Wikipedia has more than 150 pages on experimental design. Field experiment Wikipedia: A field experiment applies the scientific method to experimentally examine an intervention in the real world (or as many experimentalists like to say, naturally-occurring environments) rather than in the laboratory. Field experiments, like lab experiments, generally randomize subjects (or other sampling units) into treatment and control groups and compare outcomes between these groups. Examples include:
Professor Shakhashiri (?) of the University of Wisconsin chemistry department: Add joy and share the fun of scientific experimentation with friends and family members. Safety comes first. Be sure to follow each instruction carefully. HAVE FUN! Psychology experiments Wikipedia documents 55 psychology experiments. There is also a page listing key psychology experiments. Schiehallion experiment Wikipedia: The Schiehallion experiment was an 18th-century experiment to determine the mean density of the Earth. ... The experiment involved measuring the tiny deflection of a pendulum due to the gravitational attraction of a nearby mountain. Space science experiments Wikipedia documents 20 space science experiments. Tim Hunkin's experiments Cool cartoons that will have you experimenting with food, light, sound, clothes, and a whole lot more!! Hundreds of cartoon experiments from cartoonist, broadcaster and engineer Tim Hunkin. |
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Work-in-parallel
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Work-in-parallel Full-fledged giving and taking feed each other. In my experience, they are only loosely coupled, however, by Working-in-Parallel based on a three-cycle of Aligning our principles, Pursuing them, and Taking Stock. ("Common knowledge" sometimes assumes that giving and taking are tightly coupled, but that goes counter to some of my personal experience.) |
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Scientific method
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Scientific method We design experiments that link together, tangle together the two incomplete outlooks of space and time, single frame and multiple frames, particle and wave, static and dynamic, free and deterministic. This is because each experiment presumes an experimenter and thus takes place both within a frame of measurement and beyond it. Each experiment includes a hypothesis, an experimental test, and an appraisal of the results. Analogously, in math, given a constraint, we extend its domain to include a new domain, we stitch them together by presuming continuity, and we relate the two applications by superimposing them, yielding a more general constraint. In life, we take a stand, follow through and reflect. Physics experiments Wikipedia documents more than 200 physics experiments. The experiment page gives examples of how the scientific method is applied. There is also a page listing key physics experiments. What is easier? It is hard to forgive somebody, harder than to tell somebody, get up and walk!Yet it is easier to forgive somebody than to let somebody walk who is unable. Jesus plays on this ambiguity so they could believe or not, as they choose. Matthew 9:5 For which is easier, to say, 'Your sins are forgiven;' or to say, 'Get up, and walk?' 9:6 But that you may know that the Son of Man has authority on earth to forgive sins..." (then he said to the paralytic), "Get up, and take up your mat, and go up to your house." |
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As you value the little, so you value the big.
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As you value the little, so you value the big. The Pharisees do the little matters of justice, but not the big ones. Jesus experiments by sending out his disciples. They are the little and he is the big. As with little, so with much Mark 16:10 He who is faithful in a very little is faithful also in much. He who is dishonest in a very little is also dishonest in much. 16:11 If therefore you have not been faithful in the unrighteous mammon, who will commit to your trust the true riches? 16:12 If you have not been faithful in that which is another's, who will give you that which is your own? Doing the little things, but not the big ones Matthew 23:23 "Woe to you, scribes and Pharisees, hypocrites! For you tithe mint, dill, and cumin, and have left undone the weightier matters of the law: justice, mercy, and faith. But you ought to have done these, and not to have left the other undone. 23:24 You blind guides, who strain out a gnat, and swallow a camel! Stay with those who are worthy Matthew 10:11 Into whatever city or village you enter, find out who in it is worthy; and stay there until you go on. The sender experiments by sending Matthew 10:16 "Behold, I send you out as sheep in the midst of wolves. What is more and less important Matthew 23:23 "Woe to you, scribes and Pharisees, hypocrites! For you tithe mint, dill, and cumin, and have left undone the weightier matters of the law: justice, mercy, and faith. But you ought to have done these, and not to have left the other undone. 23:24 You blind guides, who strain out a gnat, and swallow a camel! |
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Extend the Domain
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Extend the Domain Consider a constraint such as (2**X)(2**Y) = 2**(X+Y). It may make sense in one domain, such as integers X,Y > 2. If we hold true to the constraint, then we can extend the domain to see what it implies as to how 2**X must be defined for X=1,0,-1,... We can then think of the constraint (2**X)(2**Y) = 2**(X+Y) as stitching together unrelated domains. Such stitching I think allows us, in differential geometry, to stitch together open neighborhoods and thus define continuity for shapes like the torus. Apply calculus ideas to a discrete problem For any sequence of real numbers A=(a1,a2,...) define delta-A to be the sequence (a2-a1, a3-a2, a4-a3,...) whose nth terms is a_n+1 - a_n. Suppose that all of the terms of the sequence delta-(delta-A) are 1, and that a19=a94=0. Find a1. Even though this is not a calculus problem - the variables are discrete, so notions of limit make no sense - we can apply calculus-style ideas. Think of A as a function on the subscript n. The delta operation is reminiscent of differentiation; thus the equation delta-(delta-A) = (1,1,1...) suggests the differential equation d2A/dn2 = 1. Solving this (pretending that it makes sense) yields a quadratic function for n. None of this was "correct", yet it inspires us to try guessing that an is a quadratic function of n. And this guess turns out to be correct! pg.313 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Appropriate new ideas ...cultivating a confident attitude, so that when you see a beautiful solution, you no longer think, "I could never have done that", but instead think, "Nice idea! It's similar to ones I've had. Let's put it to work!" Learn to shamelessly appropriate new ideas and make them your own. There's nothing wrong with that; the ideas are not patented. If they are beautiful ideas, you should excitedly master them and use them as often as you can, and try to stretch them to the limit by applying them in novel ways. pg.20, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Complete set of solutions Given any diophantine equation [an equation whose variables assume only integer values] ... Can we find all solutions? Once one solution is found, we try to understand how we can generate more solutions. It is sometimes quite tricky to prove that the solutions found are the complete set. pg.264 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Define a Function Show, without multipying out, that (b-c)/a + (c-a)/b + (a-b)/c = (a-b)(b-c)(a-c)/abc. Even though it is easy to multiply out, let us try to find a more elegant approach. Notice how the right-hand side factors. We can deduce this factorization by defining f(x) = (b-c)/x + (c-x)/b + (x-b)/c. Notice that f(b) = f(c) = 0. By the factor theorem, if we write f(x) as a quotient of polynomials f(x) = P(x)/xbc, then P(x) must have x-b and x-c as factors. ... By symmetry, we could also define the function ... pg.167 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Eulerian mathematics In the last few pages, we have been deliberately cavalier about rigor, partly because the technical issues involved are quite difficult, but mostly because we feel that too much attention to rigor and technical issues can inhibit creative thinking, especially at two times: The early stages of any investigation.; The early stages of any person's mathematical education. We certainly don't mean that rigor is evil, but we do wish to stress that lack of rigor is not the same as nonsense. A fuzzy, yet inspired idea may eventually produce a rigorous proof; and sometimes a rigorous proof completely obscures the essence of an argument. There is, of course, a fine line between a brilliant, non-rigorous argument and poorly thought-out silliness. To make our point, we will give a few examples of "Eulerian mathematics", which we define as non-rigorous reasoning which may even (in some sense) be incorrect, yet which leads to an interesting mathematical truth. We name it in honor of the 18th-century Swiss mathematician Leonard Euler, who was a pioneer of graph theory and generatingfunctionology, among other things. Euler's arguments were not always rigorous or correct by modern standards, but many of his ideas were incredibly fertile and illuminating. Most of Euler's "Eulerian" proofs are notable for their clever algebraric manipulations... Sometimes a very simple yet "wrong" idea can help solve a problem. ... They are excellent illustrations of the "bend the rules" strategy pg.312 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Existence of solutions Given any diophantine equation [an equation whose variables assume only integer values] ... Do there exist solutions? Sometimes you cannot actually solve the equation, but you can show that at least one solution exists. pg.264 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Experimentation When it comes to strategy, combinatorial problems are no different from other mathematical problems. The basic principles of wishful thinking, penultimate step, make it easier, etc. are all helpful investigative aids. In particular, careful experimentation with small numbers is often a crucial step. For example, many problems succumb to a three-step attack: experimentation, conjecture, proof by induction. pg.212 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Generalizing the scope of a problem The define a function tool ... is part of a larger idea, the strategy of generalizing the scope of a problem before attacking it. pg. 98, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Guess the limit Somehow guess the limit L, and then show that the ai get arbitrarily close to L. pg.285 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Inequalities I think inequalities become relevant when we patch together different solution spaces. Inequalities are important because many mathematical investigations involve estimation, optimizations, best-case and worst-case scenarios, limits, etc. Equalities are nice, but are really quite rare in the "real world" of mathematics. ... Of the many ways of proving inequalities, the simplest is to perform operations that create logically equivalent but simpler inequalities. More sophisticated variants include a little massage, as well. pg.189-191 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Nonexistence of solutions Given any diophantine equation [an equation whose variables assume only integer values] ... Are there no solutions? Quite frequently, this is the first question to ask. As with argument by contradiction, it is sometimes rather easy to prove that an equation has no solutions. It is always worth spending some time on this question when you begin your investigation. pg.264 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Substitute Convenient Values If the polynomial P(x) is divided by x-a the remainder will be P(a). ... To see why the Remainder Theorem is true in general, divide the polynomial P(x) by x-a, getting Q(x) with remainder r. Using the division algorithm, we write P(x) = Q(x)(x-a) + r. The above equation is an identity; i.e., it is true for all values of x. Therefore we are free to substitute in the most convenient value of x, namely x=a. This yields P(a) = r, as desired. pg.181 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. |
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Circumscribe - Consent
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Circumscribe - Consent Pie Chart Agenda agree quickly on agenda The 7Ps Framework plan out a meeting Visual Agenda increase interest in a meeting |
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Take a stand
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Take a stand Taking a stand. Recognize what is important in the given scope. Building on what I believe Acknowledge my assumptions Conversing with: unknown I wish to assume as little as possible. Thus I note the assumptions that I do make. I assume the possibility of God rather than reject that possibility. I assume the possibility of others. By recognizing these as assumptions, I do not take them for granted, but allow that they are simply assumptions, which may be wrong and may be questioned. Appreciate my circumstances Conversing with: good fortune I was born to a loving family. All of our needs were met. Furthermore, we lived in Southern California, where the people around us for miles and miles were also doing fine, it seems. I was instinctively good. I had a fantastic mind. My future seemed bright and unlimited. I appreciated that this was not my doing, that others did not have that, and that I could have easily been born as a different person in different circumstances. Thus, early on, I considered that I must apply myself, my talents and my opportunities on behalf of all. Ask for extraordinary assistance Conversing with: commonality Sometimes I have asked for extraordinary help from people I know with power: to go to university in Soviet-occupied Lithuania; to get a visa for my love; to get help from Lithuania's foreign ministry for Kenyans; to have a place to stay for the night from the Church in Warsaw. I learned how help was not forthcoming, or how it was much more or much less than what I merited. Believing in my superiority My mother thought that I might need glasses, but I resisted that idea, I suppose because I didn't wish to allow that I was imperfect in any way. One day we were driving home from Lithuanian Saturday school on Highway 5. I would read out loud the exit signs as they came up. My sister Rima managed to read our exit "Carmenita" before I did! How could that be? I realized that she saw it before me. That's when I admitted that I needed glasses. Endow an existing term with new meaning Conversing with: possibilities of language In my philosophy, I made use of the terms "representation" and "topology" from mathematics. I used them in my own way, but also drew from their mathematical meaning. The term "representation" allowed me to distinguish between what is denoted (a matrix representation) and what it denotes (an abstract group). The term "topology" allowed me to consider us imagining not things (as suggested by "categories"), but worlds, backdrops, canvases, which might be simple and fundamental, yet not necessarily expressible in terms of a single statement, rule, principle or quality. Make a back up plan Conversing with: support In embarking on my business, Minciu Sodas, to organize independent thinkers, I hedged myself and my debts with the thought that I would find work if my efforts failed. But then later I came to learn that, having taken my path, it was not so likely that I could get a job or that it could be honest work. Making explicit my circumstances Conversing with: complete stranger As I appreciate that I can assume practically nothing about others, and recognize the difficulty of communicating with anybody, yet I can leverage my limitations. I can suppose that they may likewise have such difficulties; that I know primarily about myself and primarily my own limitations; that I can hope that they do likewise; and that any attempt to communicate will leverage those limitations; and the structure of our effort, the message C from A to B, will be both the content and the form of our initial communication; and that this is a "constructive hypothesis" which we can apply, and to the extent it is by some miracle fruitful, we can build upon. Recognizing a solution Conversing with: making meaningful As a freshman at the University of Chicago, I would regularly come up against the question, Is it possible to know anything at all? I had been discovering "divisions of everything". I was learning how to define what I meant by "everything" and formalizing its four properties. I realized that everything functioned as a first absolute, as an anchor that I and others could know, could communicate, and could refer to. Remembering a solution *** What life experience is relevant? Conversing with: memory In 1996, I was organizing meetings to think through "good will exercises" for living by our hearts. One of our participants, Bob, asked how we might be sure that I wasn't going to brainwash them? I thought about this and remembered how in junior high school I had struggled with questions such as, "How do I know that I'm not a robot?" Back then, I had come up with a reply, a counterquestion, namely, "Would it make any difference?" I thus considered several dozens such situations and developed a theory of doubts and counterquestions. Reuse existing solutions Conversing with: solution As a computer programmer, I learned to solve many challenges by reusing and rewriting existing code that I or others had developed for similar problems. Generalize from examples Conversing with: validity In developing the good will exercises, and considering situations that riled me, I saw that, in the case of a homeless person, I myself was not following the truth of my heart. From this one example, I looked for this principle in general. Similarly, when I noticed, in discussing the question, What is happiness? that there was a division of three at play, I started considering, what would it mean to divide everything into two parts? or four parts? and so on. Being myself Believe in God Conversing with: inexperience I wondered, why were people so hesitant to do "good will exercises" with me? Why did they not want to let go of their experience in the sense that I was? Why was I able to live on the edge? I realized that I do believe in God. I do believe that God watches over me. I can therefore do risks to do the right thing. I then appreciated that people might not be able to do that if they don't believe in God. I drew this conclusion even though I had been very careful to think that people might not need to explicitly believe in God, but might have some comparable concept. Yet I saw that, to my dismay, they generally didn't. Comprehending my role Conversing with: everybody After I concluded my quest to know everything with my video summary, "I Wish to Know", and after I failed to make Minciu Sodas a viable business because our culture was too weak, I thought about my own role. I had worked to include others as equals in a culture, although of evidently different levels of maturity. But now I thought that, given my goal to apply my findings and foster a culture, I should appreciate my own role in that. I myself might be more central to such a culture than I had supposed. Get involved Conversing with: my own limits I have exposed my failings and strengths by getting involved and not always holding back. At Lithuanian scout camp Rakas, as a nineteen year old "Brother of the Forest", I spied on the teenage girl scouts from out of the forest. They saw me, but I didn't retreat. I'm sure I looked pretty stupid. In 2009, I went to a philosophy workshop in Lithuania, which had rejected my wish to talk about my findings in my quest to know everything, including my video. At mid-day, one of the speakers gave a long screed against America, women, homosexuals, minorities. At question time, I asked, "Maybe Hitler was right?" and he replied, "As soon as I talk about such things, I am inevitably compared to Hitler." As nobody kept him from speaking so irresponsibly, I got up and punched him, saying that the Nazis had executed my grandfather. (It seems that subsequently he stopped pursuing his line of thinking, at least publicly.) I have many such incidents, stupid or noble, because I am willing to step out of norms. Stick to what is right Conversing with: world I've tried, even if ever so slightly, to stay true to what is right. For example, I force myself to at least skim the terms of service when I am asked to click that I have read and understood them. At work, in public, and socially, I note and stand up for what is right. Sometimes I lose out as a consequence, and sometimes I gain in different ways. This helps me appreciate and identify with the righteous, the downtrodden and the marginalized. |
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Extend application
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Extend application Embracing a role I was at a crossroads. Ready or not, here I was, "Presenting Dr. Ansell." A "real" doctor. And while I felt like a poseur, a fraud, I decided that despite my insecurity and inexperience, I needed to act as if I knew what the hell I was doing. One month into my internship, on the West Side of Chicago, in a steamy corner of the fourth floor of Fantus clinic, at the County Hospital. An epiphany. I suppressed a wave of panic and shoved my doubts aside. Oh. I got it. I was a "real" doctor now. The patients expected no less. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.1934 Making little adjustments Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: It's just little pointers. It's like golf — making those little adjustments. The same thing occurs in boxing. You make adjustments." |
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Frequency-and-experience
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When we don't have all the facts, we estimate probability based on frequency and experience. Frequency is how often an event has occurred in the past; experience is what happened during each event. ... the more we know about the circumstances of an event whose probability we are estimating, the more accurate our estimate will be. |
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Align
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Align Working for Cass McNutt I learned about visibility. I had always been inclined to work from foundations. But I observed from him that it was the client relationship that was most important. Therefore it was key that the client continuously feel real progress and benefit, even if it was in simple things. And that these benefits were actually quite real and significant, whereas deeply pursued software initiatives might very often not bear any fruit at all and be completely counterproductive. At the Atzalynas club I have had one or more computers which I have let the leaders use. But I have never allowed them to add pirated software. This would frustrate the leader of the computer club. The club got a few computers, too. I agreed that they have pirated programs on the club's computers, but not on mine. |
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My duties
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My duties Appreciating my freedom Considering what I have to do Considering what I have to do Knowing what I can do at least Noting what I do wrong What rules do I not follow? |
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Hypothesis
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Hypothesis Wikipedia: A hypothesis is a proposed explanation for a phenomenon. The term derives from the Greek, hypotithenai meaning "to put under" or "to suppose." For a hypothesis to be put forward as a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous observations that cannot satisfactorily be explained with the available scientific theories. Edward Cherlin, 2011.04.24: I like your cycle of scientific method: take a stand (hypothesize), follow through (experiment), reflect (conclude), although I find that there is more to it. It has been pointed out that a hypothesis must include a model (usually mathematical) and a mapping between parts of the model (observables) and observations, including experiments. |
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As you value others, so are you valued.
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As you value others, so are you valued. Continuity between ourselves and others... Each test is a test of the tested, the tester and the test. This is the cross, that man is taught by making an example out of. Don't judge so you won't be judged - let the experiment take place. The experiment should be stark and self-sustaining, not have extra resources. Inspecting a coin Mark 12:15 ... But he, knowing their hypocrisy, said to them, ´Why do you test me? Bring me a denarius, that I may see it. 12:16 They brought it. Inspecting a fig tree Mark 11:13 Seeing a fig tree afar off having leaves, he came to see if perhaps he might find anything on it. When he came to it, he found nothing but leaves, for it was not the season for figs. Are you a hypocrite? Matthew 7:3 Why do you see the speck that is in your brother's eye, but don't consider the beam that is in your own eye? 7:4 Or how will you tell your brother, 'Let me remove the speck from your eye;' and behold, the beam is in your own eye? 7:5 You hypocrite! First remove the beam out of your own eye, and then you can see clearly to remove the speck out of your brother's eye. As you do, so God does to you Mark 11:25 Whenever you stand praying, forgive, if you have anything against anyone; so that your Father, who is in heaven, may also forgive you your transgressions. 11:26 But if you do not forgive, neither will your Father in heaven forgive your transgressions." Calling out hypocrisy Each test is a test of the tested, but also a test of the tester and their test. Mark 12:14 ... "...Is it lawful to pay taxes to Caesar, or not? 12:15 Shall we give, or shall we not give?" But he, knowing their hypocrisy, said to them, "Why do you test me? Bring me a denarius, that I may see it." Cultures experiment with each other Matthew 10:17 But beware of men: for they will deliver you up to councils, and in their synagogues they will scourge you. Don't judge and you won't be judged If you don't judge, then you are perfect, as God is perfect, and you do as God, and so Jesus won't judge you, for God does not judge, but Jesus judges, yet he does not and need not judge those who do as God does. Luke 6:36 Therefore be merciful, even as your Father is also merciful. 6:37 Don't judge, and you won't be judged. Don't condemn, and you won't be condemned. Set free, and you will be set free. Having the least bit Matthew 17:20 He said to them, "Because of your unbelief. For most certainly I tell you, if you have faith as a grain of mustard seed, you will tell this mountain, 'Move from here to there,' and it will move; and nothing will be impossible for you. 17:21 But this kind doesn't go out except by prayer and fasting." How we should treat Jesus Jesus treats us as equals, as we treat him. Matthew 10:32 Everyone therefore who confesses me before men, him I will also confess before my Father who is in heaven. 10:33 But whoever denies me before men, him I will also deny before my Father who is in heaven. How will you be judged? Matthew 7:1 "Don't judge, so that you won't be judged. 7:2 For with whatever judgment you judge, you will be judged; and with whatever measure you measure, it will be measured to you. If you forgive, then God forgives you, and if you don't, then God won't forgive you Matthew 6:14 "For if you forgive men their trespasses, your heavenly Father will also forgive you. 6:15 But if you don't forgive men their trespasses, neither will your Father forgive your trespasses. Let a social experiment be stark Matthew 10:16 "Behold, I send you out as sheep in the midst of wolves. Therefore be wise as serpents, and harmless as doves. Let the bad and the good grow distinct Matthew 13:24 He set another parable before them, saying, "The Kingdom of Heaven is like a man who sowed good seed in his field, 13:25 but while people slept, his enemy came and sowed darnel weeds also among the wheat, and went away. 13:26 But when the blade sprang up and brought forth fruit, then the darnel weeds appeared also. 13:27 The servants of the householder came and said to him, 'Sir, didn't you sow good seed in your field? Where did this darnel come from?' 13:28 "He said to them, 'An enemy has done this.' "The servants asked him, 'Do you want us to go and gather them up?' 13:29 "But he said, 'No, lest perhaps while you gather up the darnel weeds, you root up the wheat with them. 13:30 Let both grow together until the harvest, and in the harvest time I will tell the reapers, "First, gather up the darnel weeds, and bind them in bundles to burn them; but gather the wheat into my barn."'" Let the tester be tested by his testing Every test is also a test of the tester, of their values and their intent. Jesus lets John be tested by his testing. Matthew 11:2 Now when John heard in the prison the works of Christ, he sent two of his disciples 11:3 and said to him, "Are you he who comes, or should we look for another?" 11:4 Jesus answered them, "Go and tell John the things which you hear and see: 11:5 the blind receive their sight, the lame walk, the lepers are cleansed, the deaf hear, the dead are raised up, and the poor have good news preached to them. 11:6 Blessed is he who finds no occasion for stumbling in me." Relative comparison "Twice as much" Matthew 23:15 Woe to you, scribes and Pharisees, hypocrites! For you travel around by sea and land to make one proselyte; and when he becomes one, you make him twice as much of a son of Gehenna as yourselves. Remove the beam from your own eye Luke 6:41 Why do you see the speck of chaff that is in your brother's eye, but don't consider the beam that is in your own eye? 6:42 Or how can you tell your brother, 'Brother, let me remove the speck of chaff that is in your eye,' when you yourself don't see the beam that is in your own eye? You hypocrite! First remove the beam from your own eye, and then you can see clearly to remove the speck of chaff that is in your brother's eye. What is one worthy of? If we are laborers, then we should not have to depend on anything but our labor. Matthew 10:9 Don't take any gold, nor silver, nor brass in your money belts. 10:10 Take no bag for your journey, neither two coats, nor shoes, nor staff: for the laborer is worthy of his food. What should we depend on? If we are laborers, then we should not have to depend on anything but our labor. Matthew 10:9 Don't take any gold, nor silver, nor brass in your money belts. 10:10 Take no bag for your journey, neither two coats, nor shoes, nor staff: for the laborer is worthy of his food. Who empathizes with whom? Matthew 23:29 "Woe to you, scribes and Pharisees, hypocrites! For you build the tombs of the prophets, and decorate the tombs of the righteous, 23:30 and say, 'If we had lived in the days of our fathers, we wouldn't have been partakers with them in the blood of the prophets.' 23:31 Therefore you testify to yourselves that you are children of those who killed the prophets. 23:32 Fill up, then, the measure of your fathers. Who is more valuable Matthew 6:25 Therefore I tell you, don't be anxious for your life: what you will eat, or what you will drink; nor yet for your body, what you will wear. Isn't life more than food, and the body more than clothing? 6:26 See the birds of the sky, that they don't sow, neither do they reap, nor gather into barns. Your heavenly Father feeds them. Aren't you of much more value than they? Who is our enemy Matthew 10:21 "Brother will deliver up brother to death, and the father his child. Children will rise up against parents, and cause them to be put to death. Who is the greatest among us? Matthew 23:11 But he who is greatest among you will be your servant. Your measure will be used to measure for you Luke 6:38 "Give, and it will be given to you: good measure, pressed down, shaken together, and running over, will be given to you. For with the same measure you measure it will be measured back to you." |
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Continuity
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Continuity As in Polya's discussion of Descartes' universal method, we can apply continuity to consider the implications of a constraint or an equation. Polya asks about an iron ball floating in mercury, if we pour water on it, will the ball sink down or float up or stay the same? He answers this by first imagining that the water has no specific gravity (like a vacuum) and then increasing it continuously until it approaches and surpasses that of iron. Varying the variable is putting the constraint to the test, presuming that there is a solution point, just as we do and can in physical reality. At what points will the model break or hold? Continuity is the thread that we sew. Continuity Informally, a function is continuous if it is possible to draw its graph without lifting the pencil. Of the many equivalent formal definitions, the following one is the easiest to use. Let f: D -> R and let a be an element of D. We say that f is continuous at a if the limit as n approaches infinity of f(x_n) = f(a) for all sequences (x_n) in D with limit a. ... Continuity is a condition that you probably take for granted. This is because virtually every function you have encountered (certainly most that can be written with a simple formula) are continuous. ... Consequently, we will concentrate on the many good properties that continuous functions possess. Here are two extremely useful ones. Intermediate-Value Theorem. If f is continuous on the closed interval [a,b], then f assumes all values between f(a) and f(b). ... Extreme-Value Theorem. If f is continuous on the closed interval [a,b], then f attains minimum and maximum values on this interval. ... The extreme-value theorem seems almost without content, but examine the hypothesis carefully. If the domain is not a closed interval, it may not be true. pg.288-289 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Intermediate-Value Theorem If f is continuous on the closed interval [a,b], then f assumes all values between f(a) and f(b). ... the IVT, while "obvious"... has many practical applications. ... Let f:[0,1]->[0,1] be continuous. Prove that f has a fixed point; i.e., there exists x in [0,1] such that f(x) = x. ... Let g(x) = f(x)-x. Note that g is continuous and that g(0) = f(0) >=0 and g(1) = f(1)-1<=0. By the IVT, there exists u in [0,1] such that g(u) = 0. But this implies f(u)=u. pg.288-289 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Interpret dynamically To understand what a moving curtain is, we shall explore, in some detail, the most important idea of elementary calculus. ... What is the fundamental theorem of calculus (FTC), what does it mean, and why is it true? ... We start with the very useful define a function tool .. Let g(t) = integral from a to to of f(x) with regard to x. We choose the variable t on purpose, to make it easy to visualize g(t) as a function of time. As t increases from a, the function g(t) is computing the area of a "moving curtain" ... Notice that g(a) = 0. ... Differentiation is not just about tangent lines - it has a dynamic interpretation as instantaneous rate of change. Thus g'(t) is equal to the rate of change of the area of the curtain at time t. ... The area grows fast when the leading edge of the curtain is tall, and it grows slowly when the leading edge is short. It makes intuitive sense that g'(t) = f(t) since in a small interval of time delta-t, the curtain's area will grow by approximately f(t)delta-t. ... The crux move was to interpret the definite integral dynamically, and then observe the intuitive relationship between the speed that the area changes and the height of the curtain. This classic argument illustrates the critical importance of knowing as many possible alternate interpretations of both differentiation and integration. Note that the variable "t" is understood here as time and that is part of the necessary implicit context for understanding. pg.282-284 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Stretching things Three women check into a motel room which advertises a rate of $27 per night. They each give $10 to the porter, and ask her to bring back 3 dollar bills. The porter returns to the desk, where she learns that the room is actually only $25 per night. She gives $25 to the motel desk clerk, returns to the room, and gives the guests back each one dollar, deciding not to tell them about the actual rate. Thus the porter has pocketed $2, while each guest spent 10-1 = $9, a total of 2 + 3 x 9 = $29. What happened to the other dollar? ... try stretching things a bit: what if the actual room rate had been $0? Then the porter would pocket $27 and the guests would spend $27, which adds up to $54! pg. 22, 102, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Topologically equivalent ... the original problem was almost immediately equivalent to the modified easier version. That happened for a mathematical reason: the problem was a "topological" one. This "trick" of mutating a diagram into a "topologically equivalent" one is well worth remembering. pg.19, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. |
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Perfect - Explicitly Validate
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Perfect - Explicitly Validate Do, Redo & Undo make mistakeproof Wizard of Oz test the completeness of a design |
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Follow through
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Follow through Following through. Live out fully the consequences of my principles. Live out my ideas Conversing with: self-check Apply my discoveries Conversing with: significance of knowledge I have not simply wanted to know everything, but also to apply that knowledge usefully. This has shaped the questions that I've chosen to take up. In 1995, encouraged by Joe Damal, I set upon applying my philosophy practically. I addressed situations where we believe one thing in our heart, and the world teaches us differently, and we feel riled. As I meant to lead and influence people, I sought for a way to pray to God that I might listen to him. With the good will exercises, I found ways to capture and express people's intution. In Lithuania, I needed a way to make a living, so I started up Minciu Sodas, a laboratory for independent thinkers, where I tried to make use of conceptual structures to structure our online space and activity. In fostering a culture of truth, I am sharing, documenting and structuring ways of figuring things out. My practical impulse has thus focused me on questions that engage what's at the heart of my personal life. Assume self-consistency My study of narration, and Lithuanian folk tales in particular, yielded an interesting result, that the tone of voice (forcing, commanding, explaining or caring) at the the beginning of a story was constant, as well as at the end of the story, but the two tones of voice were different, which meant that something noteworthy happened in between. This theory was very fruitful because in each unit of narrative (where tension was created and relaxed) I could consider the tone of voice. Sometimes it was clear, and in the cases where it was note, I could infer it from the constancy. For example, I could deduce that crying was a form of commanding. Thus I built up a catalog of ways the tones of voice manifested themselves and then test that catalog to see if it indeed held true. I could build up such catalogs for various cultures and genres and even write computer programs to make such inferences, tests and deductions. Be naive Conversing with: my assumptions As I studied, I supposed that the more I learned in school, the more kinds of jobs I could do and find when the time came. I thought that math was practical and that I would find many ways to apply a Ph.D. Later, as I took loans and risks to start my lab, Minciu Sodas, I thought that I could always find a regular job if I failed. I thought so, accepting tenants of my environment. Now, I see differently, but my wholehearted naivete then, my lack of doubt then helps me as I now reflect and conclude. Object to the existing system ever so slightly Conversing with: justice In the course of my usual activities, I address injustices that I notice, even though it may get me into trouble. I reported election violations in Chicago. When I won a Knight News Challenge award, I spoke up about the need to address "ethnic cleansing" in Chicago. I smuggled books as I traveled. I poured out alcohol at events where I felt people were pressured to drink. I would walk across the Black American neighborhood to show myself that I could do it. Real life exercise Conversing with: response I developed "good will exercises" by which I and others assimilated "truths of the heart" through real life exercises, not just hypothetically. Allow for the unexpected Conversing with: goodness Find out for myself Conversing with: limits set by constraints Sometimes I'm able to observe and learn first hand rather than through books and schools. In Lithuania, in 1988 and 1989, I was able to observe up close Lithuania's reform movement which achieved independence from the Soviet Union. In 1980, I visited Lithuania as a 15 year old, and was able to talk to relatives of various backgrounds. I walked the streets of Vilnius and saw how different buildings that I recognized from Saturday school all fit together spatially with regard to each other. I traveled through Europe in the summer of 1989 and saw with my own eyes the plaza of Venice which Marco Polo departed from. From 1998 to 2010, I sought clients for Minciu Sodas and gained first hand experience of sorts in business. Get out of my comfort zone Conversing with: self-check At times, I made an effort to go outside my comfort zone, especially to reach out to those who are marginalized. I chose to live with my grandmother in her neighborhood, Marquette Park, which had become primarily Black American. I would practice engaging the youth hanging out on the corner. I ventured into the Black American community to join a Catholic church there, St. Benedict the African. In 2006, I stayed for three weeks in Nablus, Israeli-occupied Palestine, and taught nonviolence there. Keep checking Conversing with: goodness Over the years, when I have a chance, I ask priests, how can I participate in the Catholic Church to pursue Jesus's vision of the Kingdom of Heaven? I keep checking and so I feel I can tentatively conclude that nobody else is interested. Test out assumptions Conversing with: understanding Is it true that everyone can be a genius? As a tutor for mentally challenged adults I realized that there are some objective challenges to learning yet everybody may have special gifts, some extra scraps of cortex for individual talents. |
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Drive to logical conclusion
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Drive to logical conclusion Destroying a prototype Fresh Air interview of Michael Hiltzik about the Hoover Dam, June 8, 2010: "The federal government created two labs for the concrete to be tested; they enlisted the University of California [and] the University of Colorado to test other formulations. And they even built a test dam in a California river valley — they built it so it could be destroyed, so they could see what kinds of stresses and strains this dam would undergo." Black Hole as a Natural Laboratory "Seeing a star get ripped apart by a black hole from almost 4 billion light-years away, that's a remarkable thing," says astronomer Dave Goldberg, co-author of A User's Guide to the Universe: Surviving the Perils of Black Holes, Time Paradoxes, and Quantum Uncertainty, who was not part of the studies. "We want to study black holes because they are tremendous natural laboratories for what happens to matter at very high energies." "Rare Cosmic Blast Traced to Black Hole" by Dan Vergano, USA TODAY, June 16, 2011 Practice with sparring partners similar to one's opponent Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: Dundee, Morton and Trainer were responsible for managing Leonard's fights and for selecting sparring partners who were similar in height, speed and power to the opponents Leonard was about to face. These sparring partners worked with Leonard for weeks to prepare him for a match. ... To prepare, Leonard chose a sparring partner who idolized Duran. "He fought like Roberto Duran — he used his head and dirty tactics and what-have-you," he says. "And it made me more aware, from a defensive standpoint, so when I faced Duran, I was prepared." Practicing as a child Fresh Air interview: When J.J. Abrams was 13, his grandfather gave him a Super 8 camera to shoot homemade movies. He started off making horror films — and killing off all of his relatives. "I would take anyone who was available — my sister, my mother, any friends — and I would kill them in crazy ways," he tells Fresh Air's Terry Gross. "We would do makeup effects. ... I would make blood and ask my mom if I could borrow her makeup — which didn't trouble her because she knew I was going to basically just kill someone with it. It was all ridiculous." Abrams went onto a successful film and television career, with credits including Alias, Lost, Fringe, Mission: Impossible III, Star Trek (2009) and Cloverfield. Natural experiment Wikipedia: A natural experiment is an observational study in which the assignment of treatments to subjects has been haphazard: That is, the assignment of treatments has been made "by nature", but not by experimenters. Thus, a natural experiment is not a controlled experiment. Natural experiments are most useful when there has been a clearly defined and large change in the treatment (or exposure) to a clearly defined subpopulation, so that changes in responses may be plausibly attributed to the change in treatments (or exposure). Natural experiments are considered for study designs whenever controlled experimentation is difficult, such as in many problems in epidemiology and economics. |
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Hypothesis Testing
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 Hypothesis testing is primarily about analyzing individual hypotheses and so I list it here. Secondarily, it is also related to ranking. ...a hypothesis helps to narrow the scope of our thinking. It thus provides a framework - a mind-set - within which to analyze and interpret information ... it can have dramatically negative effects if it leads to satisficing - to focusing on one hypothesis to the exclusion of alternatives. ... [Hypothesis testing ranks] competing hypotheses by the degree to which relevant evidence is inconsistent. The favored hypothesis has the least inconsistent evidence, not the one with the most consistent evidence. ... Consistent evidence proves nothing, because evidence can, and usually does, support more than one hypothesis. ... Only inconsistent evidence has any real value in determining the credibility of a hypothesis and in ranking hypotheses by the degree of their credibility.
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Pursue
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Pursue I was very much in love with a young lady from Lithuania, a wonderful person. She finally agreed to visit me in the United States after years of hesitation. I didn't have much savings but I decided that I would borrow everything that I might need from credit cards so that I could take care of her during her stay and she would have a good time. Ultimately, she married another man who she had loved for many years, but I was very glad that I did not think twice about doing all I could for her and to be with her and that money was not an issue. I am due a rather large refund in Lithuania for value added tax which I have paid as a business, for my income comes from outside of the country, but I have accumulated expenses within the country. I had to request permission for a refund. Then they informed me that they would need to do a complete audit of all of the books for my business. Which is to say, not just the documents relevant to the value added tax, but absolutely everything. I told them that it would take me hundreds of hours to prepare. They said that they would give me plenty of time, six weeks. |
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Practice
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Practice Considering a principle's consequences Living with integrity Practicing our principles Trying |
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Experimental design
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Experimental design Wikipedia: In general usage, design of experiments (DOE) or experimental design is the design of any information-gathering exercises where variation is present, whether under the full control of the experimenter or not. Edward Cherlin, 2011.04.24: But that is not enough. We must also think of other possible models, and design experiments to rule them in or out, and we must think of every possible experiment that could invalidate our model. This is the great service that Einstein performed for Quantum Mechanics, because he disliked it so much. Every time he thought he had found a contradiction or something nonsensical in the math, the lab boys verified that it really worked that way in experiments. |
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Good word: As you value the fruit, so you value the tree.
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Good word: As you value the fruit, so you value the tree. To deny this is the "sin against the Holy spirit". Jesus denounces those who do not receive him and his word, who do not repent. Repentance is the fruit, the person is the tree. How can I show my authority? Jesus' miracles manifest his authority: Judge the tree by the fruit. Matthew 9:5 For which is easier, to say, 'Your sins are forgiven;' or to say, 'Get up, and walk?' 9:6 But that you may know that the Son of Man has authority on earth to forgive sins..." (then he said to the paralytic), "Get up, and take up your mat, and go up to your house." Judge the tree by the fruit Mark 11:14 Jesus told it, "May no one ever eat fruit from you again!" and his disciples heard it. Acknowledging faith Matthew 15:25 But she came and worshiped him, saying, "Lord, help me." 15:26 But he answered, "It is not appropriate to take the children's bread and throw it to the dogs." 15:27 But she said, "Yes, Lord, but even the dogs eat the crumbs which fall from their masters' table." 15:28 Then Jesus answered her, "Woman, great is your faith! Be it done to you even as you desire." And her daughter was healed from that hour. Are works truly good? Matthew 7:22 Many will tell me in that day, 'Lord, Lord, didn't we prophesy in your name, in your name cast out demons, and in your name do many mighty works?' 7:23 Then I will tell them, 'I never knew you. Depart from me, you who work iniquity.' Are you making people better? Matthew 23:15 Woe to you, scribes and Pharisees, hypocrites! For you travel around by sea and land to make one proselyte; and when he becomes one, you make him twice as much of a son of Gehenna as yourselves. Childhood can't be presumed, but must be demonstrated Jesus distinguishes between the ethnic children and the spiritual children of the Jewish patriarchs. Spiritual childhood is perhaps the point of being "born again". Thus all people are inherently equal. Matthew 8:11 I tell you that many will come from the east and the west, and will sit down with Abraham, Isaac, and Jacob in the Kingdom of Heaven, 8:12 but the children of the Kingdom will be thrown out into the outer darkness. There will be weeping and gnashing of teeth. Could it be better not to be born? Matthew 26:24 The Son of Man goes, even as it is written of him, but woe to that man through whom the Son of Man is betrayed! It would be better for that man if he had not been born. Denouncing Why does Jesus denounce the cities? Perhaps they are free to repent, perhaps others will learn from them, and perhaps he is confirming God's will. Matthew 11:20 Then he began to denounce the cities in which most of his mighty works had been done, because they didn't repent. 11:21 "Woe to you, Chorazin! Woe to you, Bethsaida! For if the mighty works had been done in Tyre and Sidon which were done in you, they would have repented long ago in sackcloth and ashes. 11:22 But I tell you, it will be more tolerable for Tyre and Sidon on the day of judgment than for you. 11:23 You, Capernaum, who are exalted to heaven, you will go down to Hades. For if the mighty works had been done in Sodom which were done in you, it would have remained until this day. 11:24 But I tell you that it will be more tolerable for the land of Sodom, on the day of judgment, than for you." Distinguish between those he torments and those he heals Matthew 8:28 When he came to the other side, into the country of the Gergesenes, two people possessed by demons met him there, coming out of the tombs, exceedingly fierce, so that nobody could pass that way. 8:29 Behold, they cried out, saying, "What do we have to do with you, Jesus, Son of God? Have you come here to torment us before the time?" Do they make life easier or harder? Matthew 23:4 For they bind heavy burdens that are grievous to be borne, and lay them on men's shoulders; but they themselves will not lift a finger to help them. Does one take up their cross and follow Jesus Matthew 10:38 He who doesn't take his cross and follow after me, isn't worthy of me. Judge by the outcome that you hear and see Matthew 11:2 Now when John heard in the prison the works of Christ, he sent two of his disciples 11:3 and said to him, "Are you he who comes, or should we look for another?" 11:4 Jesus answered them, "Go and tell John the things which you hear and see: 11:5 the blind receive their sight, the lame walk, the lepers are cleansed, the deaf hear, the dead are raised up, and the poor have good news preached to them. 11:6 Blessed is he who finds no occasion for stumbling in me." No good tree brings forth rotten fruit Luke 6:43 For there is no good tree that brings forth rotten fruit; nor again a rotten tree that brings forth good fruit. 6:44 For each tree is known by its own fruit. For people don't gather figs from thorns, nor do they gather grapes from a bramble bush. The tree is known by the fruit. Matthew 7:15 "Beware of false prophets, who come to you in sheep's clothing, but inwardly are ravening wolves. 7:16 By their fruits you will know them. Do you gather grapes from thorns, or figs from thistles? 7:17 Even so, every good tree produces good fruit; but the corrupt tree produces evil fruit. 7:18 A good tree can't produce evil fruit, neither can a corrupt tree produce good fruit. 7:19 Every tree that doesn't grow good fruit is cut down, and thrown into the fire. 7:20 Therefore by their fruits you will know them. What will happen to the good and the bad Matthew 13:37 He answered them, "He who sows the good seed is the Son of Man, 13:38 the field is the world; and the good seed, these are the children of the Kingdom; and the darnel weeds are the children of the evil one. 13:39 The enemy who sowed them is the devil. The harvest is the end of the age, and the reapers are angels. 13:40 As therefore the darnel weeds are gathered up and burned with fire; so will it be at the end of this age. 13:41 The Son of Man will send out his angels, and they will gather out of his Kingdom all things that cause stumbling, and those who do iniquity, 13:42 and will cast them into the furnace of fire. There will be weeping and the gnashing of teeth. 13:43 Then the righteous will shine forth like the sun in the Kingdom of their Father. He who has ears to hear, let him hear. What will happen to the good and the bad Matthew 13:47 "Again, the Kingdom of Heaven is like a dragnet, that was cast into the sea, and gathered some fish of every kind, 13:48 which, when it was filled, they drew up on the beach. They sat down, and gathered the good into containers, but the bad they threw away. 13:49 So will it be in the end of the world. The angels will come forth, and separate the wicked from among the righteous, 13:50 and will cast them into the furnace of fire. There will be the weeping and the gnashing of teeth." Whether they succeed Matthew 17:19 Then the disciples came to Jesus privately, and said, "Why weren't we able to cast it out?" 17:20 He said to them, "Because of your unbelief. For most certainly I tell you, if you have faith as a grain of mustard seed, you will tell this mountain, 'Move from here to there,' and it will move; and nothing will be impossible for you. 17:21 But this kind doesn't go out except by prayer and fasting." Whether to prepare what to say Matthew 10:19 But when they deliver you up, don't be anxious how or what you will say, for it will be given you in that hour what you will say. Who is the cause of the result? Matthew 6:28 Why are you anxious about clothing? Consider the lilies of the field, how they grow. They don't toil, neither do they spin, 6:29 yet I tell you that even Solomon in all his glory was not dressed like one of these. 6:30 But if God so clothes the grass of the field, which today exists, and tomorrow is thrown into the oven, won't he much more clothe you, you of little faith? Who is worthy? Matthew 10:11 Into whatever city or village you enter, find out who in it is worthy; and stay there until you go on. Who speaks through whom Matthew 10:19 But when they deliver you up, don't be anxious how or what you will say, for it will be given you in that hour what you will say. 10:20 For it is not you who speak, but the Spirit of your Father who speaks in you. Whose judgment brings out whose testimony Matthew 10:18 Yes, and you will be brought before governors and kings for my sake, for a testimony to them and to the nations. |
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Self-superimposed sequence
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Self-superimposed sequence We can formulate what we have learned in general. We do this by considering a local constraint on values as a recurrence relation (on values a1, a2, ..., aN) and then superimposing the resulting sequence upon itself, as with a generating function, yielding a global relationship of the function with itself. If the model holds, then it can be tested further. This automata is the hand that makes the stitch. (Recurrence relation as an automata, auto-associative memory of neurons as in Jeff Hawkins' "On Intelligence", generating function, telescoping tool, shift operator) Arithmetic Mean An arithmetic sequence is a sequence of consecutive terms with a constant difference ... a, a+d, a+2d, ... An arithmetic series is a sum of an arithmetic sequence. The sum of an arithmetic sequence is a simple application of the Gaussian pairing tool ... Upon adding, we immediately deduce that S = n((A+L)/2), the intuitively reasonable fact that the sum is equal to the average value of the terms multiplied by the number of terms ... another term for "average" is arithmetic mean. pg.172-173 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Auto associative memory of neurons Jeff Hawkins discusses auto-associative memory of neurons in his book "On Intelligence", where cortical columns use time-delay to relate patterns to themselves. Be on the lookout for new ideas Always be on the lookout for new ideas. Each new problem that you encounter should be analyzed for its "novel idea" content. pg.20, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Be wary interchanging a “limit of a limit" ...something disturbing happened. Each an(x) is continuous (in fact, differentiable), yet the infinite sum of these functions is discontinuous. This example warns us that infinite series of functions cannot be treated like finite series. There are plenty of other "pathologies", for example, a function f(x) defined to be the infinite sum of fi(x), yet f'(x) is not equal to the sum of the fi'(x). The basic reason behind these troubles is the fact that properties like continuity, differentiation, etc. involve taking limits, as does finding the sum of a series. It is not always the case that a "limit of a limit" is unchanged when you interchange the order. Luckily, there is one key property that prevents most of these pathologies: uniform convergence, which is defined in the same spirit as uniform continuity ... We say that the sequence of functions (fn(x)) converge uniformly to f(x) if the "N response" to the "epsilon challenge" is independent of x. pg.309 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Catalyst Sometimes telescoping won't work with what you start with, but the introduction of a single new term will instantly transform the problem. We call this the catalyst tool. ... Simplify the product (1 + 1/a)(1 + 1/a**2)(1 + 1/a**4) ... (1 + 1/a**(2**100)). Call the product P and consider what happens when we multiply P by 1 - 1/a. The "catalyst" is the simple difference of two squares formula (x-y)(x+y) = x**2 - y**2. (1 - 1/a)P = (1 - 1/a)(1 + 1/a)(1 + 1/a**2)(1 + 1/a**4) ... (1 + 1/a**(2**100)) = (1 - 1/a**2)(1 + 1/a**2)(1 + 1/a**4) ... (1 + 1/a**(2**100)) etc. = (1 - 1/a**201). pg.175 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Define a function The define a function tool ... is part of a larger idea, the strategy of generalizing the scope of a problem before attacking it. pg. 98, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Generating functions The crossover tactic of generating functions owes its power to two simple facts.
Geometric series tool Let 1=a0=a1=a2=... Then the corresponding generating function is just 1 + x + x**2 + x**3 + ... This is an infinite geometric series which converges to 1/(1-x), provided that |x|<1 pg.144, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Power series A power series is a special case of a series of functions, namely one where each term has the form an(x-c)**n. ... What makes power series so useful is that they converge uniformly so long as you contract the radius of convergence a bit. ... Thus, once you are in possession of a uniformly convergent power series, you can abuse it quite a bit without fear of mathematical repercussions. You can differentiate or integrate term by term, multiply it by other well-behaving power series, etc. and be sure that what you get will behave as you think it should. ... Not only are they easy to manipulate, but they provide "ideal" information about the way the function grows. pg.311-312 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Recurrence Many problems involving the natural numbers require finding a formula or algorithm that is true for all natural numbers n. If we are lucky, a little experimenting suggests the general formula, and we then try to prove our conjecture. But sometimes the problem can be so complicated that at first it is difficult to "globally" comprehend it. The general formula may not be at all apparent. In this case it is still possible to gain insight by focusing on the "local" solution, the transition from n=1 to n=2, and then, more generally, the transition from n to n+1. ... In how many ways can an nx2 rectangle be tiled by 1x2 dominos? ... we have the recurrence formula t_n+1 = t_n + t_n-1 for n=2, 3,... Have we solved the problem? Yes and no. [The formula], plus the boundary values t_1 = 1, t_2 = 2, completely determine t_n for any value of n, and we have a simple algorithm for computing the values: just start at the beginning and apply the recurrence formula! ... These values are precisely the Fibonacci numbers... So the problem is "solved", in that we have recognized that the tiling numbers are just Fibonacci numbers. Of course you may argue that the problem is not completely solved, as we do not have a "simple" formula for t_n (or f_n). ... While it is nice to have a "simple" formula that generates the Fibonacci sequence, knowing the recurrence formula is almost as good, and sometimes it is impossible or extremely difficult to get a "closed-form" solution to a recurrence. ... Compute the number of different triangulations of a convex n-gon ... t_n = sum over u+v = n+1 of t_u t_v ... known as the Catalan numbers ... Recurrence formulas ... may seem rather complicated, but they are really straightforward applications of standard computing ideas (partitioning and simple encoding). Algebraically, the sum should remind you of the rule for multiplying polynomials ... which in turn should remind you of generating functions ... Cn = (1/(n+1))(2n n) The purpose of math is to create models that simplify (which is, however, why they hold only tentatively). When there is no closed-form solution, then the recurrence relation may feel unsatisfactory because it has not led to the desired simplification, but has simply reproduced, redenoted the original problem. pg.233-239 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Restate It pays to reread a problem several times. As you rethink classification, hypothesis and conclusion, ask yourself if you can restate what you have already formulated. For example, it may seem that the hypothesis is really trivial, and you just have to repeat it verbatim from the statement of the problem. But if you try to restate it, you may discover new information. Sometimes just reformulating hypothesis and conclusion with new notation helps ... Normally, one reads a problem silently. But for many people, reciting the sequence out loud is enough of a restatement to inspire the correct solution (as long as a number such as "1211" is read "one-two-one-one"...) pg.30, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Shifted sequence Let f(x) = a0 + a1 x + a2 x**2 + ... be the generating function corresponding to the sequence (aN). Now look at x f(x) = a0x + a1 x**2 + a2 x**3 + ... This is the generating function of the original sequence, but shifted.... Now we make use of the relationship between aN and a(N-1)... [Then compare and subtract and rephrase the resulting infinite geometric series.] ... This method was technically messy, since it involved using the geometric-series tool repeatedly as well as partial fractions. But don't get overwhelmed by the technical details - it worked because multiplying a generating function by x produced the generating function for the "shifted" sequence. Likewise, dividing by x will shift the sequence in the other direction. These techniques can certainly be used for many kinds of recurrences. pg.146, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Solve for the limit There is a simple but very productive tool that often works when a sequence is defined recursively. Let us apply it to the previous example: x_n+1 = 1/2(x_n + alpha/x_n) If xn approaches L, then for really large n, both xn and x_n+1 approach L. Thus, as n approaches infinity, the equation x_n+1 = (x_n + alpha/x_n)/2 becomes L = (1/2)(L + alpha/L), and a tiny bit of algebra yields L = square root of alpha. This solve for the limit tool does not prove that the limit exists, but it does show us what the limit must equal if it exists. pg.288 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Telescope A geometric sequence is exactly like an arithmetic sequence except that now the consecutive terms have a common ratio ... a, ar, ar**2, ar**3... The Gaussian pairing tool is no help for summing geometric series, because the terms are not additively symmetric. However, the wonderful telescope tool comes to the rescue .... S = a + ar + ar**2 + ... + ar**(n-1) and rS = ar + ar**2 + ar**3 + ... + ar**n. Observe that S and rS are nearly identical, and hence subtracting the two quantities will produce something really simple ... all terms cancel except for the first and the last. (That's why it's called "telescoping", because the expression "contracts" the way some telescopes do.) We have S - rS = a - ar**n and solving for S yields S = (a - ar**n)/(1-r). ... The important thing is to be aware of the possibility for telescoping, which is really just an application of the adding zero creatively tool. And quite often, a telescoping attempt won't work perfectly, but will reduce the complexity of a problem. pg.173-174 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Telescoping tool Zeta Function The zeta function z(s) is defined by the infinite series z(s) = 1/1**s + 1/2**s + 1/3**s + ... pg.177 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. |
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Reevaluate - Implicitly Validate
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Reevaluate - Implicitly Validate How-Now-Wow Matrix identify what is both new and achievable NUF Test reality check for ideas Pre-Mortem be upfront about risks Start, Stop, Continue reevaluate activities SWOT Analysis gauge and shape possibilities of success |
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Reflect
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Reflect Reflecting. Allow for mistakes and solutions, recognize them and make good of them. Learning from experience Conversing with: life Conversing with: life I was working to promote an import export standard through the Infrared Data Association. After a difficult start there was a meeting where we had a good chance of making it happen. However, as Rob Lockhart explained to me afterwards, I oversold, I tried too hard, I was too vocal and aggressive, and members backed away, and then afterwards the chance was gone for the whole endeavor. So that is where I learned what it means to oversell. Yet in my mind I felt that if that's what it all amounted to, then it was not a very important matter, and I would not fret that. Check myself Conversing with: view from the side When people criticize me, I check myself, but often their criticism doesn't seem fair. Over time, I realized that the criticism is often more applicable to the criticizer, and I concluded that I am a mirror for other people's weaknesses. Good deed Conversing with: justice Lord Baden-Powell taught scouts to do a good deed every day. As a child, I took this seriously. Later, I appreciated the opportunity to do good. This concept helped me to see the dynamics of good, the way that it worked, its significance and advantages, not yielding return directly, and yet allowing for good things to happen to me as well as others, and importantly, fostering my character. Learning from my failures Conversing with: other possibilities I learned not to brag about my mind because that can hurt others; to be ready to lead because others may not be ready;to use the simplest vocabulary so that I would be understood by everybody; to make clear my main point so that people could help me; to not be modest so that people would be aware of what I can do Learning from real life activity Conversing with: life After spending a year developing good will exercises, and another year conducting them, I came to understand that the person who is riled is wrong and confuses the truth of the world for the truth of the heart. I realized that people weren't keen to do such exercises. By doing them, I was focusing on what riled people, whereas I should be focusing on peace, as Jesus greeted in peace. In organizing Minciu Sodas, I learned that people don't want to dedicate themselves to a culture of independent thinkers. Take to heart an exception to a rule I had noted four tests to distinguish truths of the heart from truths of the world. The most important test was that the person who is riled regarding a subject confuses the truth of the heart and of the world. I tried to apply these tests as carefully and openmindedly as possible to make sure that they held true and I was not biasing the results. In developing a good will exercise with Annette Robinson about an issue that she cared about, it happened that what she thought and felt was the truth of the heart, what she kept coming back to, was indeed the truth of the heart as determined by the other three tests. This gave me pause to think. Later, I came across a few other such cases. After developing some forty exercises, I realized that I should be focused primarily on feelings of peace rather than what made people riled, and so my good will exercises were not grounded on the right foundation. Subsequently, I realized that this particular test compared with the other three tests may simply indicate whether a person was confused or not regarding the truth of the heart. If they were confused, then their frustration would continue to build. If they were no longer confused, then their frustration was dissipating. So it was actually more useful as an indication of their state of mind rather than whether the truth itself was of the heart or of the world. The other three tests could account for that. What is fruitful is positive Sometimes, as my debts grew, I would wonder if I should have foresworn from ever going into debt. But time and again I realized that the pressure of my debts kept me from sitting still, from stagnating, and forced me to be enormously creative in ways that I probably would not have been otherwise. I realized that it was good, just as negative numbers are good, because they open up more possibilities. Appreciating my circumstances Appreciating that I am happy As a child I realized that I was very happy, I was very blessed, and as far as I could see, I would always be happy, for certainly I had the means to be. I had a mother and a father, wise and loving. We lived in a wonderful time and place, the utopia of the Californian suburbs of the 1970s, where for miles and miles around all of the people lived more or less the same, with their needs met and each family having a little something more, and people living ever more comfortably in a society that was ever more just. I had a brilliant mind that, as far as I could tell, was limitless in its potential. I was a very good child, able to keep myself safe and busy, keen to do the right thing, as if I had a good seed and it could not be otherwise. I also had enormous will, an exacting conscience, and a fierce ambition to apply myself, especially my mind. I would always have people who cared about me. I appreciated the many aspects of my happiness and, consequently, I could consider, were they my doing? Appreciating that I could have been born anybody As a child, I realized that I was profoundly happy, but that it was not of my own merits. I could have been born in Cambodia, defenseless and left to die or be killed. Therefore I thought that my happiness, gifts and talents were not meant simply to serve me, to keep me happy, but rather, to serve absolutely everybody, for I could have been anybody. How could I serve us all? What would others want me to do? Critically reviewing my life In 2000 or so, I critically reviewed my life, all my weaknesses which I perceived, much like Ben Franklin did, as he notes in autobiography, which my brother alerted me to. I found eight areas in my personal life where I wanted to improve myself: Be with God, foster my conscience, foster my willpower, foster my stewardship, be curious, serve others, support others' endeavors, be successful. |
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Accept outcome
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Accept outcome Comparing effort and outcome Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: When they announced the decision, I felt like I had given 100 percent, just for the wrong fight. Early victories Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: He started competing in amateur matches, eventually winning the 1973 National Golden Gloves Lightweight Championship. The following year, he won the Golden Gloves title again, along with the National AAU Lightweight Championship. Those early victories, he says, helped him break out of his shell. Look at yourself in the mirror Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: It's a gradual progression of getting better and, as weeks go by, you look into the mirror and you see a different person. It evolves. One minute, you look kind of soft and then, within 6-8 weeks, your muscle and all that definition appears. The mirror doesn't lie. It tells you exactly what you are." Natural experiment Conversing with: creativity of nature We can recognize that nature itself may be "conducting experiments" that we would like to conduct, but can't. Nature has created the conditions for the experiment and we simply have to note and analyze the results. There are natural nuclear reactions in uranium, bacteria growing in high levels of arsenic, and many types of remarkable configurations in microbiology and astronomy. Such searches have the spirit of Stephen Wolfram's "A New Kind of Science". Suffering from the actions of others Fresh Air interview of J.J. Abrams: His justification for such secrecy has to do with a script for a Superman film he wrote 15 years ago, he explains. "It was leaked out of the studio, and someone reviewed the script online in great detail, and it became this big thing and ultimately the film didn't get made, A hypothesis is a declarative statement that has not been established as true. ... We assert the truth of a hypothesis by offering supporting evidence. But, as the late philosopher Karl Popper established in his writings in the 1930s, we can never really prove a hypothesis true. We can and do, however, for countless reasons, accept hypotheses as true until they are proven to be false. ... We disprove a hypothesis with evidence. "Information" becomes "evidence" only when we connect it with a hypothesis. ... When we find "evidence", we should try to establish its validity by answering four questions:
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Plausible reasoning
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"How high can a giraffe fly?" Instantly our mind tells us the question is "illogical". ... how do we know? An educated guess is that the mind compares the new information about the giraffe with the old information the mind has stored away about the animal. ... it clashes violently, and the mind tells us it doesn't match, meaning it doesn't make sense. This is a far cry from reasoning or rational thinking, and it's certainly light-years away from formal logic. ... "Plausible" means "seemingly true at first glance." ... As Morton Hunt put it, "Natural [plausible] reasoning often succeeds even when it violates laws of logic. What laws then does natural reasoning obey?" He cited two: plausibility and probability. "In contrast to logical reasoning," said Hunt, "natural reasoning proceeds by steps that are credible [plausible] but not rigorous and arrives at conclusions that are likely [probable] but not certain. ... Because logic, to be effective, requires total consistency and total certainty, logical thinking is unsuited for dealing with the real world in which the only certainty (other than taxes and TV commercials) is ultimate death. But plausible reasoning requires neither consistency nor certainty. Plausibility strives for the gap, just as truth does. drawing from The Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich. |
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Take stock
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Take stock When I was seven years old my parents gave me seventy five cents and I decided to spend it to go up the Space Needle. It was something I really wanted to do. But then I cried and they asked why, wasn't I happy? And I said, yes, it was good but it wasn't worth seventy five cents. I had thought that it would be a life time experience and then felt disappointed that it wasn't and it was over and I could have gotten some army soldiers instead. I have always hoped to attract programmers who could help for free on our projects, but that never worked that way. Instead, I had to learn how to program. And I have helped self-educate some programmers, but they have proved not quite reliable. And yet every so often I am positively surprised, they help in significant ways. I let my brother borrow money on my credit cards. It was my suggestion, as he was out of school and trying to establish himself as a commercial artist and our parents weren't especially supportive. Overall, it worked out for him, and he found work. Although he suffered from the loans but he paid me back. I think that due to him I got quite a large credit limit which I drew on later when I started my business. I never liked to haggle and I felt that it was the best policy to pay full price, partly because I believed that everybody should pay the same price, and that haggling was an inefficient system. And when I moved to Lithuania I felt that I was even so part of a different economic system and shouldn't pretend otherwise. But when I started traveling to attend conferences and how expensive they were, it was suggested that I ask for a discount because I am from Lithuania. And then I decided that I needed the help and if I could get it that way then I would take it. |
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Let things unfold
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Let things unfold Allowing failure Be ready to improvise Giving things time Making good of mistakes Waiting patiently |
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Observe
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Observe Edward Cherlin, 2011.04.24: We know that our models are reasonably complete and accurate at best in the areas we have been able to observe, and that every new addition to our senses in improved scientific instruments, going back to Galileo's first telescope, reveals surprises like the mountains of the moon, the constancy of the speed of light (interferometers) or neutrino oscillation (simple but quite large neutrino detectors). |
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Caring for = Believing in
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Caring for = Believing in This matches the two branches, Believing and Caring, completely and quite likely, inversely. For Believing leads to Caring, and Caring leads to Believing. And so matching the two branches as wholes leads to this perspective. |
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Sets of objects exist
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Sets of objects exist We can fuse the particle and wave points of view to work with a partial reality. For example, we can talk about a banana, an apple or an orange as well defined objects that mean something more than a random assemblage of half of the atoms in a banana with half of the atoms in the apple. We are then no longer talking about the symmetry of the universe. John says: A symmetry group commutes with the underlying symmetry of a particular phenomenon, its spacial symmetry, as the set of possible transformations, possible futures. Previously, we worked with the entire universe, and if we translated it abstractly in a symmetry group and then ran things forward in the translated frame, it was exactly isomorphic to what it would have been if we had not translated it. But now, as we want to compartmentalize the universe, then the price to pay is that we are not translating by the full symmetry group, but only by some part of it. This is analogous to having a tensor product and considering only one component, so that we have partial symmetry. We are going to treat one part of the universe as a compartment. This gives the reality to the symmetry group because otherwise it couldn't be measured. Andrius: This compartmentalization is also what allows us to define entropy and the one-way direction of time, which says that states drift away from deliberateness, which is expressed by the compartmentalization. Compartmentalization also indicates the philosophical gaps or boundaries that allow for measurement to take place, allow for objectivity, the separation of the observer and the observed. Analogously, in math we have symmetry groups, and in life we have meanings, the essence of what we wish to say, which me take to be absolute, in cases where we have fundamental agreement. |
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Linking heaven to earth
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Linking heaven to earth God fights by retreating. Thus the Kingdom of Heaven is taken violently by those who want to enter into it by killing Jesus. Jesus connects earth and heaven, light and darkness, identity and perfection, metaphysically. He is the light of the world. He is the substitute for each of us, by whom we can substitute for each of us. The last day, when there is no more night, thus no distinction between night and day, the two branches. Jesus's expectations I did a study of the Gospel of Mark of Jesus's emotional responses to see what he was responding to, what his expectations were. His expectations were that we are all one. Allegiance to Jesus' name What is Jesus' name? It is a substitute for him that shows who is accepting his culture. It is perhaps a unifier for the two branches. A name allows for ambiguity and resolves it to unity. 18:1 In that hour the disciples came to Jesus, saying, "Who then is greatest in the Kingdom of Heaven?" 18:2 Jesus called a little child to himself, and set him in their midst, 18:3 and said, "Most certainly I tell you, unless you turn, and become as little children, you will in no way enter into the Kingdom of Heaven. 18:4 Whoever therefore humbles himself as this little child, the same is the greatest in the Kingdom of Heaven. 18:5 Whoever receives one such little child in my name receives me, 18:6 but whoever causes one of these little ones who believe in me to stumble, it would be better for him that a huge millstone should be hung around his neck, and that he should be sunk in the depths of the sea. Analogy of God to us as parent to children Matthew 7:11 If you then, being evil, know how to give good gifts to your children, how much more will your Father who is in heaven give good things to those who ask him! Are we the same or different? Matthew 23:8 But don't you be called 'Rabbi,' for one is your teacher, the Christ, and all of you are brothers. Asking them to make evident their belief Jesus has them say if they are believers or not so that he can know how to speak to them. Mark 11:29 Jesus said to them, "I will ask you one question. Answer me, and I will tell you by what authority I do these things. 11:30 The baptism of John - was it from heaven, or from men? Answer me." Call his disciples Matthew 10:1 He called to himself his twelve disciples, and gave them authority over unclean spirits, to cast them out, and to heal every disease and every sickness. Children do as the parent does Children of Wisdom Wisdom approaches people where they are at, or where they claim to be at. Matthew 11:19 ... But wisdom is justified by her children." Creating a tangible reference to an abstract question Jesus calls a child to make his point. 18:1 In that hour the disciples came to Jesus, saying, "Who then is greatest in the Kingdom of Heaven?" 18:2 Jesus called a little child to himself, and set him in their midst, 18:3 and said, "Most certainly I tell you, unless you turn, and become as little children, you will in no way enter into the Kingdom of Heaven. 18:4 Whoever therefore humbles himself as this little child, the same is the greatest in the Kingdom of Heaven. 18:5 Whoever receives one such little child in my name receives me, 18:6 but whoever causes one of these little ones who believe in me to stumble, it would be better for him that a huge millstone should be hung around his neck, and that he should be sunk in the depths of the sea. Eightfold structure of Lord's prayer Matthew 6:9 Pray like this: 'Our Father in heaven, may your name be kept holy. 6:10 Let your Kingdom come. Let your will be done, as in heaven, so on earth. 6:11 Give us today our daily bread. 6:12 Forgive us our debts, as we also forgive our debtors. 6:13 Bring us not into temptation, but deliver us from the evil one. For yours is the Kingdom, the power, and the glory forever. Amen.' Essential concepts Jesus cares about law, justice, mercy faith. Matthew 23:23 "Woe to you, scribes and Pharisees, hypocrites! For you tithe mint, dill, and cumin, and have left undone the weightier matters of the law: justice, mercy, and faith. But you ought to have done these, and not to have left the other undone. 23:24 You blind guides, who strain out a gnat, and swallow a camel! God's will: the bread of life Bread of life - DIFFERENCE/UNITY IN BELIEVING - guarantee eternity - to do the Father's will that all he draws to Jesus do indeed believe in him unwaveringly, thus have eternal life, as Jesus' gift is unwavering How are distinctions reconciled? The last day means that there is no more alternation of night and day and so the two branches are reconciled. John 6:39 This is the will of my Father who sent me, that of all he has given to me I should lose nothing, but should raise him up at the last day. 6:40 This is the will of the one who sent me, that everyone who sees the Son, and believes in him, should have eternal life; and I will raise him up at the last day." How is good related to God Matthew 19:16 Behold, one came to him and said, "Good teacher, what good thing shall I do, that I may have eternal life?" 19:17 He said to him, "Why do you call me good? No one is good but one, that is, God. How to unify Jesus unifies the two branches by laying down his life. John 10:14 I am the good shepherd. I know my own, and I'm known by my own; 10:15 even as the Father knows me, and I know the Father. I lay down my life for the sheep. 10:16 I have other sheep, which are not of this fold. I must bring them also, and they will hear my voice. They will become one flock with one shepherd. 10:17 Therefore the Father loves me, because I lay down my life, that I may take it again. 10:18 No one takes it away from me, but I lay it down by myself. I have power to lay it down, and I have power to take it again. I received this commandment from my Father." Identify Peter as the Stone Matthew 16:15 He said to them, "But who do you say that I am?" 16:16 Simon Peter answered, "You are the Christ, the Son of the living God." 16:17 Jesus answered him, "Blessed are you, Simon Bar Jonah, for flesh and blood has not revealed this to you, but my Father who is in heaven. 16:18 I also tell you that you are Peter, and on this rock I will build my assembly, and the gates of Hades will not prevail against it. 16:19 I will give to you the keys of the Kingdom of Heaven, and whatever you bind on earth will have been bound in heaven; and whatever you release on earth will have been released in heaven." Identifying himself as born of spirit, not born of woman The Kingdom of Heaven is rejected violently by those who prefer Jesus to be dead. Thus the violent enter into the Kingdom of Heaven, by killing Jesus so that he could save them. 11:12 From the days of John the Baptizer until now, the Kingdom of Heaven suffers violence, and the violent take it by force. Is a name or word appropriate? Matthew 23:8 But don't you be called 'Rabbi,' for one is your teacher, the Christ, and all of you are brothers. 23:9 Call no man on the earth your father, for one is your Father, he who is in heaven. 23:10 Neither be called masters, for one is your master, the Christ. Jesus knows himself ("most certainly") Matthew 8:8 The centurion answered, "Lord, I'm not worthy for you to come under my roof. Just say the word, and my servant will be healed. 8:9 For I am also a man under authority, having under myself soldiers. I tell this one, 'Go,' and he goes; and tell another, 'Come,' and he comes; and tell my servant, 'Do this,' and he does it." 8:10 When Jesus heard it, he marveled, and said to those who followed, "Most certainly I tell you, I haven't found so great a faith, not even in Israel. Love God "Love God" is the positive commandment that unites four of the commandments. It is what unites us as people with God through love for him. Naming Matthew 16:17 Jesus answered him, "Blessed are you, Simon Bar Jonah, for flesh and blood has not revealed this to you, but my Father who is in heaven. 16:18 I also tell you that you are Peter, and on this rock I will build my assembly, and the gates of Hades will not prevail against it. Note from whom God hides and to whom reveals The significance of Jesus' miracles are hidden from John the Baptist, from Bethsaida and Capernaum, from the wise and understanding, but "infants" are delighted and uplifted, those who are fresh in faith, who don't want to justify the world, but want it bettered, by those who suffer and labor, who want healing. Matthew 11:25 At that time, Jesus answered, "I thank you, Father, Lord of heaven and earth, that you hid these things from the wise and understanding, and revealed them to infants. 11:26 Yes, Father, for so it was well-pleasing in your sight. Planning Matthew 16:18 I also tell you that you are Peter, and on this rock I will build my assembly, and the gates of Hades will not prevail against it. 16:19 I will give to you the keys of the Kingdom of Heaven, and whatever you bind on earth will have been bound in heaven; and whatever you release on earth will have been released in heaven." Seeking the greatest commandment Mark 12:28 One of the scribes came, and heard them questioning together. Knowing that he had answered them well, asked him, "Which commandment is the greatest of all?" 12:29 Jesus answered, "The greatest is, "Hear, Israel, the Lord our God, the Lord is one: 12:30 you shall love the Lord your God with all your heart, and with all your soul, and with all your mind, and with all your strength." This is the first commandment. 12:31 The second is like this, "You shall love your neighbor as yourself." There is no other commandment greater than these." Speak from what they can know John 3:10 Jesus answered him, "Are you the teacher of Israel, and don't understand these things? 3:11 Most certainly I tell you, we speak that which we know, and testify of that which we have seen, and you don't receive our witness. 3:12 If I told you earthly things and you don't believe, how will you believe if I tell you heavenly things? The Jewish prophets are the Kingdom of Heaven Jesus calls the Jewish people "the children of the Kingdom of Heaven" and says that Abraham, Isaac and Jacob will be in the Kingdom of Heaven. But this means that Abraham, Isaac and Jacob are already in the Kingdom of Heaven and even define the Kingdom of Heaven. Therefore the "children" ethnically are not necessarily the children spiritually. Jesus is, it seems, discovering this distinction. Note also that Isaac is the child of Abraham, and Jacob is the child of Isaac. Matthew 8:11 I tell you that many will come from the east and the west, and will sit down with Abraham, Isaac, and Jacob in the Kingdom of Heaven, 8:12 but the children of the Kingdom will be thrown out into the outer darkness. There will be weeping and gnashing of teeth. To whom will I give the keys of the Kingdom of Heaven Matthew 16:15 He said to them, "But who do you say that I am?" 16:16 Simon Peter answered, "You are the Christ, the Son of the living God." 16:17 Jesus answered him, "Blessed are you, Simon Bar Jonah, for flesh and blood has not revealed this to you, but my Father who is in heaven. 16:18 I also tell you that you are Peter, and on this rock I will build my assembly, and the gates of Hades will not prevail against it. 16:19 I will give to you the keys of the Kingdom of Heaven, and whatever you bind on earth will have been bound in heaven; and whatever you release on earth will have been released in heaven." What do they already know Jesus is sensitive to what they already know, even as they ask. Luke 18:18 A certain ruler asked him, saying, "Good Teacher, what shall I do to inherit eternal life?" 18:19 Jesus asked him, "Why do you call me good? No one is good, except one—God. 18:20 You know the commandments: 'Don't commit adultery,' 'Don't murder,' 'Don't steal,' 'Don't give false testimony,' 'Honor your father and your mother.'" What does God deliver to Jesus God the Father (understanding) delivers all things to God the Son (who is coming to understand) for that is how all things are understood. Matthew 11:27 All things have been delivered to me by my Father. What example does he give? Matthew 20:24 When the ten heard it, they were indignant with the two brothers. 20:25 But Jesus summoned them, and said, "You know that the rulers of the nations lord it over them, and their great ones exercise authority over them. 20:26 It shall not be so among you, but whoever desires to become great among you shall be your servant. 20:27 Whoever desires to be first among you shall be your bondservant, 20:28 even as the Son of Man came not to be served, but to serve, and to give his life as a ransom for many." What is done for the sake of his name Matthew 19:27 Then Peter answered, "Behold, we have left everything, and followed you. What then will we have?" 19:28 Jesus said to them, "Most certainly I tell you that you who have followed me, in the regeneration when the Son of Man will sit on the throne of his glory, you also will sit on twelve thrones, judging the twelve tribes of Israel. 19:29 Everyone who has left houses, or brothers, or sisters, or father, or mother, or wife, or children, or lands, for my name's sake, will receive one hundred times, and will inherit eternal life. 19:30 But many will be last who are first; and first who are last. What is good Jesus thinks that what is good, what is important, is that people open up to spiritual life, embrace the Kingdom of Heaven as possible, and thus live a faith in eternal life. I think that he felt this should be in tandem, physically and spiritually. Matthew 11:2 Now when John heard in the prison the works of Christ, he sent two of his disciples 11:3 and said to him, "Are you he who comes, or should we look for another?" 11:4 Jesus answered them, "Go and tell John the things which you hear and see: 11:5 the blind receive their sight, the lame walk, the lepers are cleansed, the deaf hear, the dead are raised up, and the poor have good news preached to them. 11:6 Blessed is he who finds no occasion for stumbling in me." What is good What is good is what we believe in (what you believe is what happens) and/or what we care about (what you find is what you love). Luke 18:18 A certain ruler asked him, saying, "Good Teacher, what shall I do to inherit eternal life?" 18:19 Jesus asked him, "Why do you call me good? No one is good, except one—God. 18:20 You know the commandments: 'Don't commit adultery,' 'Don't murder,' 'Don't steal,' 'Don't give false testimony,' 'Honor your father and your mother.'" Where is Jesus from John 3:13 No one has ascended into heaven, but he who descended out of heaven, the Son of Man, who is in heaven. Whether Jesus or Satan has them Jesus prayed that Peter's faith not fail. Luke 22:31 The Lord said, "Simon, Simon, behold, Satan asked to have you, that he might sift you as wheat, 22:32 but I prayed for you, that your faith wouldn't fail. You, when once you have turned again, establish your brothers." Whether their faith will fail Jesus prayed that Peter's faith not fail. Luke 22:31 The Lord said, "Simon, Simon, behold, Satan asked to have you, that he might sift you as wheat, 22:32 but I prayed for you, that your faith wouldn't fail. You, when once you have turned again, establish your brothers." Whether they think he is of God or not Matthew 19:16 Behold, one came to him and said, "Good teacher, what good thing shall I do, that I may have eternal life?" 19:17 He said to him, "Why do you call me good? No one is good but one, that is, God. Who can know God God is understanding. To know God is to come to understanding, thus is by and through Jesus, for Jesus is coming to understand. Jesus reveals God to those who believe, those of the Kingdom of Heaven. God is impartial, and gives judgment to Jesus, whom he teaches to be impartial. Jesus is God's impartial judgment, for Jesus is distinct and independent of God, thus impartial with regard to God, for Jesus came to understanding. To know God is to know his impartiality, aand to know his Son is likewise to know his impartiality. The impartiality of God manifests in his Son who manifests it, Matthew 11:27 All things have been delivered to me by my Father. No one knows the Son, except the Father; neither does anyone know the Father, except the Son, and he to whom the Son desires to reveal him. Who can know the Son of God Only God the Father (understanding) knows the Son of God, for God is of beyond this world and only so can be apart from it and understand, be distinct, from one who is firmly in it. Matthew 11:27 All things have been delivered to me by my Father. No one knows the Son, except the Father; neither does anyone know the Father, except the Son, and he to whom the Son desires to reveal him. Who does God listen to John 9:30 The man answered them, "How amazing! You don't know where he comes from, yet he opened my eyes. 9:31 We know that God doesn't listen to sinners, but if anyone is a worshipper of God, and does his will, he listens to him. Who is a substitute for Jesus A child is a substitute for Jesus. Matthew 18:1 In that hour the disciples came to Jesus, saying, "Who then is greatest in the Kingdom of Heaven?" 18:2 Jesus called a little child to himself, and set him in their midst, 18:3 and said, "Most certainly I tell you, unless you turn, and become as little children, you will in no way enter into the Kingdom of Heaven. 18:4 Whoever therefore humbles himself as this little child, the same is the greatest in the Kingdom of Heaven. 18:5 Whoever receives one such little child in my name receives me, 18:6 but whoever causes one of these little ones who believe in me to stumble, it would be better for him that a huge millstone should be hung around his neck, and that he should be sunk in the depths of the sea. Who revealed this Matthew 16:15 He said to them, "But who do you say that I am?" 16:16 Simon Peter answered, "You are the Christ, the Son of the living God." 16:17 Jesus answered him, "Blessed are you, Simon Bar Jonah, for flesh and blood has not revealed this to you, but my Father who is in heaven. 16:18 I also tell you that you are Peter, and on this rock I will build my assembly, and the gates of Hades will not prevail against it. 16:19 I will give to you the keys of the Kingdom of Heaven, and whatever you bind on earth will have been bound in heaven; and whatever you release on earth will have been released in heaven." |
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Symmetry
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Symmetry Symmetry group We unify internal and external points of view, link time and space, by considering a group of actions in time acting on space. Some aspects of the space are invariant, some aspects change. Actions can make the space more or less convoluted. At this point, we have arrived at a self-standing system, one that can be defined as if it was independent of our mental processes. Our problem has become "a math problem". Analogously, in real life, after projecting more and more what we mean in general by people, including ourselves and others, we finally take us for granted as entirely one and the same and instead make presumptions towards a universal language by which we might agree absolutely. Axiom schema of specification Wikipedia: If z is a set, and P is any property which may characterize the elements x of z, then there is a subset y of z containing those x in z which satisfy the property. The "restriction" to z is necessary to avoid Russell's paradox and its variants. I think this relates to the idea that we can focus on the relevant symmetry and the relation between the locations affected or not by the symmetry group and the actions of that group. A bank of useful derivatives of "functions of a function" We conclude our discussion of differentiation with two examples that illustrate a useful idea inspired by logarithmic differentiation. ... Logarithmic differentiation is not just a tool for computing derivatives. It is part of a larger idea: developing a bank of useful derivatives of "functions of a function" that you can recognize to analyze the original function. If a problem contains or can be made to contain the quantity f'(x)/f(x), then antidifferentiation will yield the logarithm of f(x), which in turn sheds light on f(x). pg.300 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Algebraic symmetry Sequences can have symmetry, like this row of Pascal's Triangle: 1, 6, 15, 20, 20, 15, 6, 1 ... In just about any situation where you can imagine "pairing" things up, you can think about symmetry. pg.74, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Combination of techniques We shall end the chapter with an exploration of the diophantine equation x**2 + y**2 = n ... where n is a prime p. Our exploration will use several old strategic and tactical ideas, including the pigeonhole principle, Gaussian pairing, and drawing pictures. The narrative will meander a bit, but please read it slowly and carefully, because it is a model of how many different problem-solving techniques come together in the solution of a hard problem. pg.274 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Complex Numbers Complex numbers are the crossover artist's dream: like light, which exists simultaneously as wave and particle, complex numbers are both algebraic and geometric. You will not realize their full power until you become comfortable with their geometric, physical nature. This in turn will help you to become fluent at translating between the algebraic and the geometric in a wide variety of problems. ... We strongly urge you to read at least the first few chapters of our chief inspiration for this section, Tristan Needham's Visual Complex Analysis. This trail-blazing book is fun to read, beautifully illustrated, and contains dozens of geometric insights that you will find nowhere else. ... If z=a+bi, we define the conjugate of z to be z-bar = a-bi. Geometrically, z-bar is just the reflection of z about the real axis. Complex numbers add "componentwise" ... Geometrically, complex number addition obeys the "parallelogram rule" of vector addition ... Multiplication by the complex number rCisTheta is a counterclockwise rotation by Theta followed by stretching by the factor r. So we have a third way to think about complex numbers. Every complex number is simultaneously a point, a vector, and a geometric transformation, namely the rotation and stretching above! pg.131-134, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Exploit underlying symmetry in polynomials Algebra problems with many variables or of high degree are often intractable unless there is some underlying symmetry to exploit. ... Solve x**4 + x**3 + x**2 + x**1 + 1 = 0 ... we will use the symmetry of the coefficients as a starting point to impose yet more symmetry, on the degrees of the terms. Simply divide by x**2 yielding x**2 + x + 1 + 1/x + 1/x**2 then make the substitution u := x + 1/x. pg. 75, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Fixed objects When pondering a symmetrical situation, you should always focus briefly on the "fixed" objects which are unchanged by the symmetries. For example, if something is symmetric with respect to reflection about an axis, that axis is fixed and worthy of study (the stream in the previous problem played that role). pg. 72 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Four bugs chasing each other a classic problem which exploits rotational symmetry along with a crucial fixed point ... Four bugs are situated at each vertex of a unit square. Suddenly, each bug begins to chase its counterclockwise neighbor. If the bugs travel at 1 unit per minute, how long will it take for the four bugs to crash into one another? pg.71 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Geometric symmetry The simplest geometric symmetries are rotational and reflectional. pg. 71 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Fetching water for Grandma Your cabin is 2 miles due north of a stream which runs east-west. Your grandmother's cabin is located 12 miles west and 1 mile north of your cabin. Every day, you go from your cabin to Grandma's, but first visit the stream (to get fresh water for Grandma). What is the length of the route with minimum distance? pg.71 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Square inscribed in circle inscribed in square A square is inscribed in a circle which is inscribed in a square. Find the ratio of the areas of the two squares. pg.70 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Harmony An informal alternate definition of symmetry is "harmony". ... If you can do something that makes things more harmonious or more beautiful, even if you have no idea how to define these two terms, then you are often on the right track. pg. 70 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Invariant with respect to transformations this topic [symmetry] is logically contained within the concept of invariants. If a particular object (geometrical or otherwise) contains symmetry, that is just another way of saying that the object itself is an invariant with respect to some transformation or set of transformations. For example, a square is invariant with respect to rotations about its center of 0, 90, 180 and 270 degrees. pg. 103, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Not quite symmetrical The strategic principles of peripheral vision and rule-breaking tell us to look for symmetry in unlikely places, and not to worry if something is almost, but not quite symmetrical. In these cases, it is wise to proceed as if symmetry is present, since we will probably learn something useful. pg. 70 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Completing the square by trying to symmetrize x**2 + a*x = x*(x + a) = (x + a/2 -a/2)*(x + a/2 + a/2) = (x + a/2)**2 - (a/2)**2 Above is a way to discover the completing-the-square formula by trying to symmetrize the terms, then adding zero creatively. pg. 163, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Roots of Unity The zeros of the equation x**n = 1 are the nth roots of unity. These numbers have many beautiful properties that interconnect algebra, geometry and number theory. One reason for the ubiquity of roots of unity in mathematics is symmetry: roots of unity, in some sense, epitomize symmetry... pg.131-134, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Search for order Many fundamental problem-solving tactics involve the search for order. Often problems are hard because they seem "chaotic" or disorderly; they appear to be missing parts (facts, variables, patterns) or the parts do not seem connected. ... we will begin by studying problem-solving tactics that help us find or impose order where there seemingly is none. pg. 69 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Symmetrize the coefficients Solve the system of equations ... The standard procedure for solving systems of equations by hand is to substitute for and/or eliminate variables in a systematic (and tedious) way. But notice that each equation is almost symmetric, and that the system is symmetric as a whole. Just add together all five equations; this will serve to symmetrize all the coefficients ... Now we can subtract this quantity from each of the original equations to immediately get ... pg.166-167 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Symmetry Symmetry involves finding or imposing order in a concrete way, for example, by reflection. ... We call an object symmetric if there are one or more non-trivial "actions" which leave the object unchanged. We call the actions that do this the symmetries of the object (Footnote: We are deliberately avoiding the language of transformations and automorphisms that would be demanded by a mathematically precise definition.) ... Why is symmetry important? Because it gives you "free" information. If you know that something is, say, symmetric with respect to 90-degree rotation about some point, then you only need to look at one-quarter of the object. And you also know that the center of rotation is a "special" point, worthy of close investigation. pg. 69-70 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. The Gaussian pairing tool Gauss, as a child, added up the sum 1 + 2 + 3 + ... + 100, presumably by pairing up the number 1 and 100, 2 and 99, 3 and 98, ... 50 and 51, yielding 50 pairs of 101 for a total of 5,050. Paul Zeitz notes this as an example of symmetry and calls it the Gaussian pairing tool. pg. 75, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Tilt the picture we will present, with a "hand-waving" proof, one important theoretical tool which will allow you to begin to think more rigorously about many problems involving differentiable functions. We begin with Rolle's theorem, which certainly falls into the "intuitively obvious" category. If f(x) is continuous on [a,b] and differentiable on (a,b), and f(a) = f(b), then there is a point u in (a,b) at which f'(u) = 0. The "proof" is a matter of drawing a picture. There will be a local minimum or maximum between a and b, at which the derivative will equal zero. Rolle's theorem has an important generalization, the mean value theorem. If f(x) is continuous on [a,b] and differentiable on (a,b), then there is a point u in (a,b) at which f'(u) = (f(b) - f(a))/(b-a). ... the proof is just one sentence: Tilt the picture for Rolle's theorem! pg.297-298 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Transformation The pattern of superposition points out a path from a leading special case (or from a few such cases) to the general case. There is a very different connecting path between the same endpoints with which the ambitious problem-solver should be equally acquainted: it is often possible to reduce the general case to a leading special case by an appropriate transformation. ... For a suggestive discussion of this topic see J. Hadamard, Lecons de geometrie elementaire. Geometrie plane, 1898; Methodes de transformation, pp. 272-278. "Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving" by George Polya, 1962, John Wiley & Sons. |
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Distill - Implicitly Transform
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Distill - Implicitly Transform Back of the Napkin evoke informal, focused answers Draw the Problem describe problems so that we care Elevator Pitch sum up what is remarkable about your solution Pecha Kucha / Ignite share, present concisely Poster Session communicate ideas passionately with simple images The Pitch focus on feasibility and viability |
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Caring for = Believing in
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Caring for = Believing in Distilling the essence. Distill the key idea and communicate it simply and vividly. Capture with a personal expression Artistic process Conversing with: subconscious Making my statue "Troskimai (Wishes)" and my video summary "I Wish to Know" spurred the breakthroughs by which I pulled together my philosophy, namely that the evolving structures show that God is Not necessary (rather than is). My painting of the muses of the days of creation showed me how their smiles grew ever more serious. Focusing on the differences between me and God reminded me of my childhood experiences. Consider how a word is used in other passages Is God good? Jesus in the Gospels seems to speak in a private language, a personal code, much as I think in my philosophy. I find that about a third of his sayings are unclear as to their meaning. I look for other passages where he uses the same word or image or idea and that helps me decode what I think he means. In the Gospel of John, I traced down that the "Son of Man" means one who is taught by man, and man teaches by making an example out of him. I did a comprehensive review of Jesus's words in the Gospel of John and how he uses and explains his words, I chased them down and found that they centered on doing the will of God, which is that we have eternal life. With further contemplation, I concluded that "life is the fact that God is good, but eternal life is understanding that fact, that God need not be good." Similarly, I've tried to decode Kant's Critique of Pure Reason and the beginning of Heidegger's Being and Time. As I do that, I look for approaches and structures that I myself have uncovered in my own thinking. I think a similar approach led to the decoding of the Rosetta stone and of the Egyptian hieroglyphics as well as many others. Equate two statements Conversing with: issue In developing the good will exercises, I would ask people, if they were riled about a surface topic, what made them feel bad about it? What would make them feel worse? What would make them feel good? And what would make them feel even better? I would express their answers as starkly as possible. Then I would look for what their statements had in common, ignoring the negative and positive dimensions, and focusing only on the semantic content. This would yield the deep issue. Find something comparable in my own experience Conversing with: insight What does Jesus mean by the "poor-in-spirit"? From my efforts towards a culture of independent thinkers, I deduced that the "rich-in-spirit" are the martyrs, whereas the "poor-in-spirit" want to take many small, sensible leaps of faith, not one big one. Illustrative examples Conversing with: fundamental knowledge Certain examples from real life make vivid for me the reality of the sciences. In physics, I was most impressed when our professor Isaac Abella showed us that if we place our forefinger just above our thumb and look through that gap in the light, then we will see diffraction effects, ripples of light and darkness, which is evidence that light is a wave, which interferes with itself constructively (brightening the light) and destructively (canceling out, leaving darkness). I was deeply impressed that such a physical fact could be observed and deduced from our bare hands. It was available to all humankind, yet who cared to make sense of it? Watching human cells floating on my eyes As a nearsighted child, I noticed "floaters" that moved across my eye, chains of blocks. I learned about cells in my biology, and then, one day, sitting in a dentist's chair, I thought, the floaters must be chains of cells which I can see because they are close up, right on my eyeball. My mother said no. But later an optometrist confirmed that, yes, indeed they were! I thought it was wonderful that I could see human cells with my own naked eyes! Visualizing my feelings In 1992, I was devastated because the woman I loved with all my heart, but hadn't been in love with me, returned to her home in Lithuania and married a man she loved, but who hadn't been in love with her. Sometimes I would visualize how I felt. I felt that I was in a deep abyss, so far down that I couldn't even see or imagine the light above, and the abyss went down, down, down. But over the months, as I chugged along, I started to feel the light; I started to imagine the edge of the cliff; I was still down below, but I knew there was a pasture above that edge; that was about six months later. And then one day I felt that I was over the edge. Express in a universal language Considering inseparability of perspectives I am hypothesizing, based on my results with just a few rooms, that for each room of the House of Knowledge there are two groups of ways, one that relates to the heart and another to the world. I'm wondering if those two groups are inseparable, if they somehow must go together and thus fit together, for otherwise there could be 48 rooms and not 24 rooms, but there are many structural reasons why there should be only and exactly 24 rooms, but especially because the omniscope has 24 perspectives, and also because it matches well with the number of divisions 24/3=8, topologies 24/2=12 and representations 24/4=6. Furthermore, inseperability would be an intriguing notion. It would also relate to what I call anti-structure or the division of everything into -1 perspectives, which is given by an opposite, as in the perspective of free will, where good and bad coexist and are inseparable, unable to stand alone. Distill the essence *** What are the constituent elements? Conversing with: ideal interlocutor Good will exercises. Andrius's and God's answers to the 12 questions. Deep ideas in math, algebra. Doubts and counterquestions. The truth of the world proceeds from the truth of the heart. Express in Lithuanian Conversing with: meaning of an expression Lithuanian words are transparent in meaning. So I try to express concepts in Lithuanian to understand them. I thus expressed "culture" as "commonality". Notice inherent ambiguity I noticed the ambiguity in Jesus' explanation of why he speaks plainly to his believers, but in parables to others, that "they might not understand, they might be damned", which is to say, that he not impinge on their freedom, but that they might be free to accept him or not. The word "might" or "can" can be understood in two ways, by fate or by free will. Similarly, the word "meaningful" in English and in Lithuanian, and I suppose in every language, tends towards two different meanings, one where a word in a system has meaning within it, and another where a life in a world has meaning beyond it. Recognize a structure from its parts Conversing with: my own knowledge In the good will exercises, I recognized the relevance of the threesome because, with regard to the homeless, I was not following through on my stand. I therefore considered that likewise there could be exercises which help me reflect upon following through, or take a stand upon reflecting. Reexpress in a different domain Conversing with: answer Restating structures Conversing with: science I was inspired by the works of Christopher Alexander and Stephen Toulmin, but I rethought and accepted their ideas from my own point of view. Early on, I organized Minciu Sodas around a system of ten categories (four for supporting caring, six for supporting thinking) much as in the absolute learning of the House of Knowledge. I was hoping that what we learn from our activity would yield insights to the metaphysical structures and vice versa. Ultimately, I abandoned this to organize around individuals and their deepest values. |
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Match inner and outer worlds through person-in-general
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Match inner and outer worlds through person-in-general Maximum Margin Classifier Starting with the Minimum Empirical Error Classifier, the number of dimensions exceeds the number of training samples, and proper training leads to zero empirical error. Zero empirical error can also be obtained when n = N1 + N2 exceeds the number of dimensions p, if the distance between the pattern classes is sufficiently large. Let the empirical error be zero, and consider the Euclidean distance D between the discriminant hyperplane and the learning-set vector closest to it. Consider likewise the distance between the discriminant hyperplane and a second learning-set vector. Then as the weights grow, a related ratio diminishes to zero, which implies that the relative contribution of the second learning-set vector becomes insignificant. There is a tendency for the learning algorithm to put the decision hyperplane further from the closest learning-set vector. At the end of the learning process, several vectors are at the same distance from the discriminant hyperplane. Only the ones closest to the discriminant hyperplane contribute to the cost function and the final hyperplane location. This is the maximum margin classifier. (See: Sarunas Raudys, Evolution and generalization of a single neurone: I. Single-layer perceptron as seven statistical classifiers", Neural Networks 11 (1998) 283-296) Address the needs of loved ones Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: When he got back to D.C.'s Maryland suburbs, where his family lived, he realized that his father was extremely sick. His mother had her own health issues and the family needed money — right away. "Janks Morton, who eventually became my trainer and mentor, told me: 'You turn pro, you make money. You can pay your father's hospital bills.' ... I turned professional for that reason." Overloading terms Jerry Michalski of The REXpedition overloaded the terms of Yin and Yang. Taoism traditionally understands Yin as receptive, feminine, dark Earth energy and Yang as active, male, outward, bright energy. Jerry extends these terms to our times by attributing to Yang what is rational, logical, Cartesian, hierarchical, command and control, analytic energy, and to Yin what is emotional, spiritual, ecological, systemic, biological, emergent, abundant, creative energy. Taoism recommends creative tension. Jerry thinks that we're in a great rebalancing, we've been suffering too much Yang, and Yin is showing up on the Internet, interpersonal relationships. Jerry brings attention to it, collaborating with people who are making it happen and creating things that accelerate the process. Relationships with people who face hardship I learned a lot from the patients. I was discovering the tools of medicine from them. Many of my patients and I grew up together. They had seen me become a father for the first time, and they consoled me when my father died. I had seen their children grow up, having children themselves. I had helped them through family crises, tragedies, diseases, and deaths. I had no idea in those first weeks and months of General Medicine Clinic how much I would grow from these relationships. I am on a first-name basis with many of my original patients. From them I gained insight into illness and the dignity with which people can face hardship that has helped me through difficult times in my life. I have taken care of three generations of some families, and have seen the destruction that poverty, poor diet, obesity, diabetes, and hypertension can unleash on a family's tree. I learned that sometimes giving hope or an embrace is as therapeutic as a drug. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago. Why are we using the word we are using? Jerry Michalski of The REXpedition is turned off by the word "consumer" because of what the dictionary says it actually means (to destroy, often wastefully) and because of the violent metaphors that it encourages (military campaigns, branding cattle, driving traffic). Defining basic terms Bob Lichtenbert, Seekers' Dialogue on Motivating For More Meaning Let's start at the same place that we always ought to and do begin, namely, with a definition of boundary setting on basic terms. Now by "motivating" I shall primarily mean "moving another person to care or be interested or do in anything". I am mostly interested in personal motivation in the psychological and philosophical (or value) senses. So "motivation" as I mean it here in general is "to arouse others toward goal-directed behavior". Applying the letter of the law and watching for the spirit of the law Perry Recker notes that in this episode the two mothers are like the two branches, one of harshness and destruction and the other of forgiveness and healing. I note that the Talmud has a tradition that if two conflict over a coat, and it is not clear who owns it, then it should be divided in half. Solomon may have simply applied the same logic here. His application revealed who the true owner was. If so, he was wise by being true to the law, yet alert to what the law then reveals of our hearts. The law is a process of clarifying, not a decision. 1st Kings 3:23 Then the king said, "The one says, 'This is my son who lives, and your son is the dead;' and the other says, 'No; but your son is the dead one, and my son is the living one.'" 3:24 The king said, "Get me a sword." They brought a sword before the king. 3:25 The king said, "Divide the living child in two, and give half to the one, and half to the other." 3:26 Then the woman whose the living child was spoke to the king, for her heart yearned over her son, and she said, "Oh, my lord, give her the living child, and in no way kill it!" But the other said, "It shall be neither mine nor yours. Divide it." 3:27 Then the king answered, "Give her the living child, and in no way kill it. She is its mother." God's law is down-to-earth and known Deuteronomy 30:11 For this commandment which I command you this day, it is not too hard for you, neither is it far off. 30:12 It is not in heaven, that you should say, "Who shall go up for us to heaven, and bring it to us, and make us to hear it, that we may do it?" 30:13 Neither is it beyond the sea, that you should say, "Who shall go over the sea for us, and bring it to us, and make us to hear it, that we may do it?" 30:14 But the word is very near to you, in your mouth, and in your heart, that you may do it. |
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Focus on one element at a time
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Fourth, structuring focuses on one element at a time, which, compared to our instinctive scattershot approach of tackling all elements simultaneously, is more systematic, more thorough, and more productive of relevant ideas. At any point in the analytic process, from the very beginning to the very end, we are in one of two modes: convergent or divergent. ... Both divergence and convergence are necessary for effective problem solving. Divergence opens the mind to creative alternatives; convergence winnows out the weak alternatives and focuses on, and chooses among, the strong. Without divergence, we could not analyze a problem creatively or objectively; without convergence, we would just keep on analyzing, never coming to closure. It is therefore vital to effective problem solving that the analyst be prepared and able to shift back and forth between divergent and convergent approaches easily and at will, using each mode to its best effect as the problem-solving process dictates. What is more, our conscious awareness of (1) the diametrically opposed roles of convergence and divergence and (2) which mode we are in at any given moment in the analytic process will, by itself, greatly enhance our ability to solve problems. Unfortunately, it is extremely difficult for humans to shift back and forth between these two ever-opposite, ever-warring approaches. Most of us are not inherently good divergers; divergence is not one of our instinctive processes. Indeed, most of us habitually resist divergence - sometimes passionately, even angrily. |
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Live as if we are all one person
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Live as if we are all one person When I completed my graduate courses, it was time to write a Ph.D. thesis. However, it wasn't my goal to be a mathematician, I had simply wanted to get an education. So I didn't know for what reason to continue, although it felt a bit awkward to stop. My mother supplied the reason, she said that I wanted to be a person who finishes things, and I wouldn't have to explain for the rest of my life why I hadn't finished. That was the reason I continued. We had a fund raiser for our travel to the national chess championships. We went around to different businesses to ask for donations. I thought that the money should go to the kids who needed it the most. But our teacher said that the kids should all share it equally but especially because we had made such an extra effort to collect it. When we were children, our father decided that he would supplement our allowance by giving us one penny for each page of Lithuanian books that we read for our own enjoyment. He wouldn't pay for English books, only Lithuanian. It was a really great support because the Lithuanian books were extra effort but I really wanted to get good at reading them. It made it extra rewarding. My father taught me and my brother how to play chess. He would not play easy but rather he would handicap himself by removing several of his pieces from the board, enough to even out our chances. As we got better and would beat him, he would reduce the handicap. First it was a queen and two rooks, later a queen, then a rook, then a bishop, then a knight, until one day we could beat him. Because you can't learn from somebody who is not playing to win! I was always impressed by how smart and kind he was as a teacher. It never seemed right to make a living in a way that was special regarding me, but rather to find a way that anybody could make a living. Because making a living is a big problem in life and what good is it if I find a solution that only works for me? I used to collect postage stamps. I kept a very large and orderly collection with thousands stamps from my father and uncle and grandfather and quite a few that I had purchased at a shop. I used the catalogs to organize my collection and was aware of how much the various stamps were worth. It was worth hundreds of dollars, the albums alone! When I was leaving high school for college I was quite happy to give my collection away, and saddened to realize that there wasn't any person, any child to give it away to. I sent it to a nun who was collecting stamps and selling them in bulk for pennies to raise money, earning perhaps less than the shipping costs. It was sad, but even sadder that throughout my life I never met a child who collected stamps and would care about all the complicated world history and geography that they represented. This, even though I worked as a tutor with dozens of disadvantaged children. I would lend money to my friends and have them pay off my own loans, thus hurting my credit rating. I felt that I was helping by letting them make good use of my credit. And that I shouldn't have to be after them as they should be on top of things. But none of them were perfect as to that. I suppose it's important to keep on top of the use of credit if you're going to share it, you have to manage that. When I was in high school, some of my classmates started getting part time jobs. I wondered whether I should get one, whether that would help me be a more balanced person. Or perhaps my brother wanted to get one. But my father said that he had worked in his youth because he had had to. And we would be working all our lives. So he did not want us to work until we had to. He wanted us to make the most of the opportunity that we had not having to work. I felt encouraged by his view because it was his first hand account, having had worked, and I accepted it, and I felt I did not need to relive it. It's interesting that in my life I have never had a full time job. |
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Our message
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Our message Building on a message Realizing what we're communicating What is our message? What is the meaning of an activity? |
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Conservation
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Conservation Noether's Theorem Wikipedia: Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. ... The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action. Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. ... For illustration, if a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally symmetric; from this symmetry, Noether's theorem shows the angular momentum of the system must be conserved. ... Noether's theorem is important, both because of the insight it gives into conservation laws, and also as a practical calculational tool. It allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it allows researchers to consider whole classes of hypothetical Lagrangians to describe a physical system. For illustration, suppose that a new field is discovered that conserves a quantity X. Using Noether's theorem, the types of Lagrangians that conserve X because of a continuous symmetry can be determined, and then their fitness judged by other criteria. Conservation law Wikipedia: Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. ... The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action. Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. ... For illustration, if a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally symmetric; from this symmetry, Noether's theorem shows the angular momentum of the system must be conserved. ... Noether's theorem is important, both because of the insight it gives into conservation laws, and also as a practical calculational tool. It allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it allows researchers to consider whole classes of hypothetical Lagrangians to describe a physical system. For illustration, suppose that a new field is discovered that conserves a quantity X. Using Noether's theorem, the types of Lagrangians that conserve X because of a continuous symmetry can be determined, and then their fitness judged by other criteria. Conservation of angular momentum Wikipedia: In a closed system angular momentum is constant. This conservation law mathematically follows from continuous directional symmetry of space (no direction in space is any different from any other direction). If a planet is found to rotate slower than expected, then astronomers suspect that the planet is accompanied by a satellite, because the total angular momentum is shared between the planet and its satellite in order to be conserved. The conservation of angular momentum is used extensively in analyzing what is called central force motion. If the net force on some body is directed always toward some fixed point, the center, then there is no torque on the body with respect to the center, and so the angular momentum of the body about the center is constant. Conservation of color charge Wikipedia: In particle physics, color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD). ... Color charge is conserved, but the book-keeping involved in this is more complicated than just adding up the charges, as is done in quantum electrodynamics. Conservation of electric charge Wikipedia: In physics, charge conservation is the principle that electric charge can neither be created nor destroyed. The quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is always conserved. Charge conservation is a physical law that states that the change in the amount of electric charge in any volume of space is exactly equal to the amount of charge flowing into the volume minus the amount of charge flowing out of the volume. In essence, charge conservation is an accounting relationship between the amount of charge in a region and the flow of charge into and out of that region. Conservation of linear momentum Wikipedia: The law of conservation of linear momentum is a fundamental law of nature, and it states that if no external force acts on a closed system of objects the momentum of the closed system remains constant. One of the consequences of this is that the center of mass of any system of objects will always continue with the same velocity unless acted on by a force from outside the system. Conservation of momentum is a mathematical consequence of the homogeneity (shift symmetry) of space (position in space is the canonical conjugate quantity to momentum). So, momentum conservation can be philosophically stated as "nothing depends on location per se". Conservation of mass-energy Wikipedia: In physics, mass-energy equivalence is the concept that the mass of a body is a measure of its energy content. In this concept the total internal energy E of a body at rest is equal to the product of its rest mass m and a suitable conversion factor to transform from units of mass to units of energy. If the body is not stationary relative to the observer then account must be made for relativistic effects where m is given by the relativistic mass and E the relativistic energy of the body. Conservation of probability density Wikipedia: In quantum mechanics, the probability current (sometimes called probability flux) is a concept describing the flow of probability density. In particular, if one pictures the probability density as an inhomogeneous fluid, then the probability current is the rate of flow of this fluid (the density times the velocity). ... This is the conservation law for probability in quantum mechanics.... Conservation of weak isospin Wikipedia: In particle physics, weak isospin is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction. ... The weak isospin conservation law relates the conservation of T3; all weak interactions must preserve T3. It is also conserved by the other interactions and is therefore a conserved quantity in general. CPT symmetry Wikipedia: CPT symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity, and time simultaneously. ... The CPT theorem requires the preservation of CPT symmetry by all physical phenomena. It assumes the correctness of quantum laws and Lorentz invariance. Specifically, the CPT theorem states that any Lorentz invariant local quantum field theory with a Hermitian Hamiltonian must have CPT symmetry. Lorentz symmetry Wikipedia: In standard physics, Lorentz symmetry is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space". |
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The prophets
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The prophets Jesus speaks of "the prophets and the law". |
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Truth
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Truth 0 Truth, 1 Model, 2 Implication, 3 Variable We now think of the problem as relating two sheets, one of which has a wider point of view because it includes what may vary, not just what is fixed. There are four ways to relate two such sheets. They are given by the questions Whether it is true? What is true? How is it true? Why is it true? Truth is what is evident, what can't be hidden, what must be observed, unlike a cup shut up in a cupboard. The fixed sheet is the level of our problem and the varying sheet is our metalevel from which we study it. |
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Knowing
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Knowing In real life, I can say from my work on "good will exercises" that on any subject (such as "helping the homeless") there are two truths (of the heart and of the world) that pull in different directions. For example, "my help can make things worse" and "I should help those who need help". There are four tests that agree as to which truth is of the heart (the metalevel, the solution space) and which is of the world (the level, the problem space). |
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Game maker vs. Game player
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Game maker vs. Game player  Models for Systems Analysis I presented our paper "Organizing Thoughts into Sequences, Hierarchies and Networks" at a conference in Heidelberg in 1999. At that conference I think there was a talk about the different kinds of modeling systems being used, and how they might be used together eclectically, heterogeneously. I created this diagram to show how they might be organized by the six kinds of representations. Four of them correspond to Whether-What-How-Why. The other two are representations in terms of Observer and Thing, and may correspond to the branches of Believe and Care, respectively. |
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Valuation
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Valuation Giving and Taking are in sync when we Live As If We Were All One Person. We can then substitute for ourselves with Valuation. Such a substitution (much like a model substituting for reality) takes place on four levels:
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Experiments and Theory
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Experiments and Theory Experiments (specific instances) and theory (general laws) are related as level and metalevel. There is a dualism. But, actually, they are not qualitatively different. For an experiment is never a single instance, but always a set of instances, for it must be reproducible. In that sense, every experiment has a generality, just as a theory does. These two levels can be conflated, which is how we view Reality, where the facts and the laws coincide. Or the levels can be distinct to various degrees, and completely distinct when the facts are considered to be applications of the rules. Andrius: There are four possible levels (Whether, What, How, Why) for relating facts and rules, and there are six pairs of possible levels, with the wider level reserved for the rules (the imagined observer) and the narrower level reserved for the facts (the imagined observed). Analogously, in Math we have the mathematical structures that describe (on paper) our problem, and we have the mathematical structures that describe how our minds are solving the problem. The two are conflated as Truth. They are distinguished as Model, Implication and Variable. There are six kinds of variables. In life, we have four ways of distinguishing the truths of the heart and the world, given by Whether, What, How, Why we know what we know, and there are six ways that the two truths may be related. |
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Feel moved
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Feel moved Emotions are the most intense way that we are attached to the world and engaged in it. They are our and Jesus' complete engagement. Especially compassion for others and their plight in this world. Jesus felt moved Be profoundly troubled Matthew 26:36 Then Jesus came with them to a place called Gethsemane, and said to his disciples, "Sit here, while I go there and pray." 26:37 He took with him Peter and the two sons of Zebedee, and began to be sorrowful and severely troubled. 26:38 Then he said to them, "My soul is exceedingly sorrowful, even to death. Stay here, and watch with me." Compassion 9:36 But when he saw the multitudes, he was moved with compassion for them, because they were harassed and scattered, like sheep without a shepherd. Compassion for what they may suffer Matthew 15:32 Jesus summoned his disciples and said, "I have compassion on the multitude, because they continue with me now three days and have nothing to eat. I don't want to send them away fasting, or they might faint on the way." Note their feelings Luke 24:36 As they said these things, Jesus himself stood among them, and said to them, "Peace be to you." 24:37 But they were terrified and filled with fear, and supposed that they had seen a spirit. 24:38 He said to them, "Why are you troubled? Why do doubts arise in your hearts? What is hidden will be revealed All will be revealed We should strive to understand of our own free will, for it will be revealed regardless, even after we have lost it. Luke 8:16 "No one, when he has lit a lamp, covers it with a container, or puts it under a bed; but puts it on a stand, that those who enter in may see the light. 8:17 For nothing is hidden, that will not be revealed; nor anything secret, that will not be known and come to light. 8:18 Be careful therefore how you hear. For whoever has, to him will be given; and whoever doesn't have, from him will be taken away even that which he thinks he has." How are Jesus' disciples different from him Jesus' disciples may speak plainly what he spoke in parables. As disciples, they add freedom to what Jesus says, for they took freely what he offered, and so they can give freely. Unlike the Son, who was taught by the Father? Matthew 10:26 Therefore don't be afraid of them, for there is nothing covered that will not be revealed; and hidden that will not be known. 10:27 What I tell you in the darkness, speak in the light; and what you hear whispered in the ear, proclaim on the housetops. What can be contained or not Matthew 10:26 Therefore don't be afraid of them, for there is nothing covered that will not be revealed; and hidden that will not be known. 10:27 What I tell you in the darkness, speak in the light; and what you hear whispered in the ear, proclaim on the housetops. What is hidden Matthew 13:34 Jesus spoke all these things in parables to the multitudes; and without a parable, he didn't speak to them, 13:35 that it might be fulfilled which was spoken through the prophet, saying, "I will open my mouth in parables; I will utter things hidden from the foundation of the world." Psalm 78:2 78:1 Hear my teaching, my people. Turn your ears to the words of my mouth. 78:2 I will open my mouth in a parable. I will utter dark sayings of old, 78:3 Which we have heard and known, and our fathers have told us. 78:4 We will not hide them from their children, telling to the generation to come the praises of Yahweh, his strength, and his wondrous works that he has done. |
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Truth: Whether it is true?
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Truth: Whether it is true? Truth: Whether it is true? The two sheets may be conflated in which case we may interpret the problem as statements that we ourselves are making which may be true or false and potentially self-referential. Together they allow for proofs-by-contradiction where true and false are kept distinct in the level, whereas the metalevel is in a state of contradiction where all statements are both true and false. In my thinking, contradiction is the norm (the Godly all-things-are-true) and non-contradiction is a very special case that takes great effort, like segregating matter and anti-matter. Deep structure "solution spaces" allow us, as with Euclid's equilateral triangle, to step away from the "solution" and consider the candidate solutions, indeed, the failed solutions. Argument by contradiction Instead of directly trying to prove something, we start by assuming that it is false, and show that this assumption leads us to an absurd conclusion. A contradiction argument is usually helpful for proving directly that something cannot happen. ... When you begin thinking about a problem, it is always worth asking, What happens if we negate the conclusion? Will we have something that is easier to work with? If the answer is "yes", then try arguing by contradiction. pg.46, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Square root of 2 is not rational A classic example of proof by contradiction. pg.46, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Edward Cherlin, 2011.04.05: Yale Professor Fred B. Fitch's book, Symbolic Logic presents a system of logic that can be proven consistent. Dropping the law of Excluded Middle was essential to the construction. Gödel's theorem depends on Excluded Middle, so it doesn't apply to this proof of consistency. If R is the set of all sets that are not members of themselves (with further precision required that does not concern us here), then R is a member of R if and only if R is not a member of R. In the presence of Excluded Middle, this results in contradiction. In its absence, it is merely undecidable both in terms of provability and of truth. |
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Feel - Implicitly Innovate
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Feel - Implicitly Innovate Bodystorming sense what might work in the real world Mood Board capture the feel of an idea Synesthesia appreciate visceral, memorable aspects |
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Knowing whether
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Knowing whether Knowing Whether one knows, through our feelings. Caring about Carer. Listen to my feelings, whether I am riled or at peace, and why I feel that way. The person who is riled is wrong! I used to be very bothered when I engaged the homeless. It was because I focused on the truth "my help can make things worse" as if that were the truth of the heart, the truth that I should be thinking. Tuning in to my feelings Conversing with: I Clarify assumptions *** Why am I thinking that? Conversing with: question In replying to a doubt with a counterquestion, I am opening myself to consider and clarify my assumptions. Feeling good before God When I wrestle with how to make a living, and I don't know how to think that through, so that even my blood pressure feels high, then I listen especially, what God wants me to do. And when I hear from God, as I do, and take up in my heart what he would have me do, then I feel a great energy of goodness and freedom that overcomes my anxiety. I hold on to that good feeling and it makes me feel that everything will be fine if I am true to God. As I do related work, I feel elated. How I feel Conversing with: myself Sometimes, when I feel pressured by life, I ask myself, What do I truly want? and that helps guide me. Riled is wrong Conversing with: my own peace In doing the good will exercises, I concluded that I and others who feel riled are actually thinking wrongly. We have been living from the truth of the world instead of the truth of the heart. We confused the two. I suppose that is why people are suspicious of others who speak agitatedly, but receptive of those who speak calmly. Confronting my own outlook Doubt myself Conversing with: self-understanding Sometime I learn by being open to doubt and responding to it. In eighth grade or so, at Lithuanian Saturday school, the priest who taught us religion asked me and others if we might become priests. I didn't feel such a desire, but I asked myself, why shouldn't I become a priest? This had me think through my reasons: I knew what I wanted to do with my life, and I didn't want to hand over my freedom; I didn't want to do what priests mainly do, which is to distribute the sacraments; I didn't think that Jesus would be a priest, but rather an ordinary person so as to make clear where true authority resides; I wanted to be open to having a sweetheart, marrying her and raising a family. As a grad student, I wondered, how do I know that I'm not gay? Especially if I find some men handsome? I took up this question and explained to myself that I myself had no interest in being gay, and no interest in dwelling on men, handsome or otherwise. Pondering my own legends Conversing with: freedom As a child, certain mental events became reference points. The most significant was engaging God to let me think freely that I might pursue my quest to know everything. Another was in third grade, the day we came back to school from summer. A girl, Rachael Baca, was running around the field in new boots. She was kicking me, as if it was a way to show that she liked me. I wasn't interested. I told God, so this is what girls are all about? I don't want to be any part of this. God said, really? I said, yes, it's not sensible. But then, I thought, I was too harsh, too hurried. Maybe some day I will want to fall in love and have a family? So I told God, not for the next ten years, until I'm seventeen. But I wasn't sure if God heard that, if he and I hadn't already sealed my fate. As it turned out, for at least ten years, and more than that, I was completely incapable of talking to girls, but would regularly fall in love. I thought I was cursed. Perhaps in seventh grade, I told God that there was one reward I would ask for figuring everything out, and that was to have a sweetheart, the most wonderful, beautiful, good and true woman in the world. He asked, do you want her to be your companion in your work? And I said, no, I can do that myself. I just wanted to enjoy her. Truly, when I finished my quest to know everything, at the age of 44, I found her right away! It was if I could look at life and people differently. I am blessed. It was a long wait, but I'm glad. Turn a question on its head Conversing with: circumstances Given a question, such as, Why is there evil? I may invert it, assume "there is evil" and ask, What does that say about God? Rather than pretend that there is no evil, or that it only appears to be evil, or presume that God is good, I allow myself to think more simply that God isn't primarily concerned with evil, that God wants absolutely all of the good, and is willing to allow for evil if that's what it takes to include every least bit of good. In this way, I can say that there is some good that comes with evil, yet there is other good that need not, and so evil is not necessary in general. Vouch for my beliefs with my own life Conversing with: my rightness As I developed the good will exercises, I wanted to learn from other people's experiences, and I wanted to foster a shared culture. But I thought it would be wrong to lead people to do what I wouldn't do myself. That would be to abuse them. So I've always looked to experiment on my own life, and especially in spiritual matters, to vouch for my beliefs with my own life. That helps me be in touch with my own beliefs. |
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Gut feeling
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Gut feeling Emotional devastation of loved ones Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: When they announced the decision, I felt like I had given 100 percent, just for the wrong fight. But the devastation — the emotional devastation that went across the board to my family and friends — was unbelievable. I saw them crying. Everyone was crying but me." Feeling release Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: Yet he was also hiding something: He was the victim of sexual abuse. "It's like a paradox," said Leonard. "I'm a fighter but yet, I'm so fearful. I don't fight back and I don't tell anyone. I don't confront it. So I lived with those periods for 30-some years but I remember too, that when I drank heavily, when my emotions were not as stable, I would cry, sob and the pain — it felt good. I felt embarrassed but it felt good because I released some of those memories or that poison that was in my stomach." Following what makes me feel uncomfortable Jerry Michalski of The REXpedition: The genesis of my Relationship Economy thesis was a realization, back around 1994 when I was writing Esther Dyson's monthly tech newsletter Release 1.0, that the word "consumer" made me really uncomfortable. I followed that energy, and it proved invaluable. Ideas kept unfolding from that initial premise. I began to notice the consumerization of so many spheres of human activity, from how we educate our children to how we elect our governments and how we pray to our Gods. I paid attention to the language of marketing to consumers, to the metaphors and business models that had spun out as a result. Sweetest feeling Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: "There is no sweeter feeling than when you throw the perfect punch. You get a signal — you get this little tingling sensation that shoots up and down your arm to let you know that you've hit the jackpot. And you know. You know right away that guy's gone." |
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Argue the other side
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Ask your friend to identify an issue about which he or she has strong feelings. Ask your friend to state his or her position on the issue and to defend tat position with argumentation. ... ask him or her to make the opposite case. ... Most people can't do it effectively; some can't do it at all and become quite upset when asked to. They are so focused on their interpretation of the issue that they cannot adequately articulate the other side (or sides). drawing from The Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich. Learned churchmen took the devil's position simply for argument's sake to challenge the rationale presented in the nomination for sainthood, the idea being that through this process the truth, perforce, would out. ... challenge the proposition not because they disbelieve it but simply to test its validity. Devil's advocacy does this by seeking, with either the same or other evidence, to prove the opposite of whatever the challenged view holds. ... Devil's advocacy is analytically useful because it by design focuses on a contrary or opposite viewpoint and in doing so activates all of the instinctive behaviors associated with focusing: viewing the problem one-dimensionally through the lens of biases connected with that viewpoint; eschewing alternative solutions; valuing evidence that supports that viewpoint; devaluing and discarding evidence that does not. Devil's advocacy thus engages the very same mental tactics employed by the primary advocate but applies them in opposition, thereby promoting objectivity. What's more, the devil's advocate goes a step further by seeking out and obtaining new evidence, evidence about which the prime advocate has no knowledge or curiosity or that was disbelieved and discredited. This further step is the secret weapon of devil's advocacy, the extra dimension that makes it a formidable analytic technique. ... Devil's advocacy will ... open your mind to new dimensions and perceptions of the problem, poking holes in fallacious self-serving arguments and stripping away thinly reasoned and thinly supported analysis. The hallmark of focusing is the revered human institution of advocacy - taking a position on an issure, marshaling supporting evidence, and defending that position against the arguments of those holding (focusing on) opposing views. ... advocacy works. It is ... the foundation of our judicial system, in which the prosecution prosecutes, the defense defends, and neither side does both. ... So advocacy does work, when someone other than the advocator makes the decision - such as a judge or a jury, the electorate, the board of directors, or the boss, whoever that might be. But ... advocacy feeds and perpetuates our mind-sets, biases, beliefs, and prejudices. It thus nurtures our tendency to focus and, in doing so, defeats objectivity ... our full understanding of the problem. drawing from The Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich. We let go of untrue beliefs by seeking the truth. We tend to cling to untrue beliefs in the face of contradictory evidence. ... many of our most cherished individual beliefs, upon careful examination, are simply untrue ... we tend to hold fast to these beliefs in the very face of incontrovertible contradictory evidence ... We simply rationalize away the disparity. ... Robert Abelson offers compelling insight. He contends that we humans treat belief like material possessions. We acquire and retain material possessions because of the functions they serve and the value they offer. ... our possessions make us feel good ... he notes the similarity between beliefs and possessions is revealed in our language. We have beliefs. We adopt, inherit, acquire, hold, and cling to them. drawing from The Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich. |
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Peace of mind
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Peace of mind A decision substitutes for the decision maker. They are conflated. Whether there is value. I was anxious and I told God I was too anxious to work part-time on my philosophy and on the business too at that point in my life. I offered a deal with him to let me focus on the business and I would make the business model work for him, too. Then I felt more at ease. We took in a lady I knew from the streets, helped her quit drinking, and took her in for two years and she worked as a cleaning lady. But then I drove her out because I thought that she should move on, as she was not active in the club's activity with the youth, but just wanted them to stay away so the club would stay clean, and didn't want to do anything more with her life. I walked to the river and I stumbled across a statue of the Japanese consul who had written 6,000 visas by hand to Jews so they could escape the country and the Nazi threat. I sat down and cried. I thought there should be a national exam that all the bureaucrats should be required to pass regarding their ability and responsibility to be creative and proactive to do the right thing to the extent that they can. |
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My state of mind
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My state of mind How I feel My state of mind Pray with my music Recognizing how I was moved Reflecting on my mindset Something feels right or not Something was missing What I like What is touching me |
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Self destruction of the experimental apparatus
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Self destruction of the experimental apparatus The experimental apparatus may be destroyed by the experiment itself. This is the case with an atomic bomb. Another example would be trying to cut a grape with a saw, destroying the grape in the process. Of course, the experimenter may be destroyed along with the experiment. And yet there may be someone who manages to observe it nonetheless, as with a supernova. Thus the experiment and the experimenter may be in the same plane. The question is then to what extent they can be segregated, to what extent the experiment can tolerate the experimenter. |
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Do not take away their freedom
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Do not take away their freedom Jesus does not speak plainly to those who don't believe so as not to take away their freedom to not believe. Others accept Jesus' word and testify to it and so can declare it to others for they believe of their own. Whereas perhaps Jesus believes because he was sent by the Father and he obeyed the Father. Jesus believes the Father because he has seen the Father; blessed are those who believe without having seen. Distinguish what they say and do Distinguish between what people say and do Matthew 23:1 Then Jesus spoke to the multitudes and to his disciples, 23:2 saying, "The scribes and the Pharisees sat on Moses' seat. 23:3 All things therefore whatever they tell you to observe, observe and do, but don't do their works; for they say, and don't do. Does the outside match the inside? Matthew 7:15 "Beware of false prophets, who come to you in sheep's clothing, but inwardly are ravening wolves. How things appear on the inside and the outside Matthew 23:27 "Woe to you, scribes and Pharisees, hypocrites! For you are like whitened tombs, which outwardly appear beautiful, but inwardly are full of dead men's bones, and of all uncleanness. 23:28 Even so you also outwardly appear righteous to men, but inwardly you are full of hypocrisy and iniquity. Whose seat they sat on? What does it mean, to sit in Moses' seat? Matthew 23:1 Then Jesus spoke to the multitudes and to his disciples, 23:2 saying, "The scribes and the Pharisees sat on Moses' seat. 23:3 All things therefore whatever they tell you to observe, observe and do, but don't do their works; for they say, and don't do. Jesus speaks so they are free to understand or not How can I show my authority? without forcing belief? By having the force of character to heal the paralytic, Jesus suggests that he has the strength of character to forgive him as well. People who believe his character can appreciate his personal ability to personally forgive. (Why a personal Jesus is relevant). People who don't believe can think that he's referring to a divine mystical authority and not acknowledge his person. Matthew 9:5 For which is easier, to say, 'Your sins are forgiven;' or to say, 'Get up, and walk?' 9:6 But that you may know that the Son of Man has authority on earth to forgive sins..." (then he said to the paralytic), "Get up, and take up your mat, and go up to your house." Jesus wanted people to understand him of their own Matthew 11:15 He who has ears to hear, let him hear. Luke 8:8 As he said these things, he called out, "He who has ears to hear, let him hear!" People are free to not believe Luke 18:1 He also spoke a parable to them that they must always pray, and not give up, 18:2 saying, "There was a judge in a certain city who didn't fear God, and didn't respect man. 18:3 A widow was in that city, and she often came to him, saying, 'Defend me from my adversary!' 18:4 He wouldn't for a while, but afterward he said to himself, 'Though I neither fear God, nor respect man, 18:5 yet because this widow bothers me, I will defend her, or else she will wear me out by her continual coming.'" 18:6 The Lord said, "Listen to what the unrighteous judge says. 18:7 Won't God avenge his chosen ones, who are crying out to him day and night, and yet he exercises patience with them? 18:8 I tell you that he will avenge them quickly. Nevertheless, when the Son of Man comes, will he find faith on the earth?" Speaking in parables Jesus spoke in parables so that people would be free to understand him or not, as they themselves chose. He does not force them to understand, and thus to believe, which would be the case if he spoke clearly. Thus he speaks plainly to those who already believe. Matthew 13:10 The disciples came, and said to him, "Why do you speak to them in parables?" 13:11 He answered them, "To you it is given to know the mysteries of the Kingdom of Heaven, but it is not given to them. 13:12 For whoever has, to him will be given, and he will have abundance, but whoever doesn't have, from him will be taken away even that which he has. 13:13 Therefore I speak to them in parables, because seeing they don't see, and hearing, they don't hear, neither do they understand. 13:14 In them the prophecy of Isaiah is fulfilled, which says, 'By hearing you will hear, and will in no way understand; Seeing you will see, and will in no way perceive: 13:15 for this people's heart has grown callous, their ears are dull of hearing, they have closed their eyes; or else perhaps they might perceive with their eyes, hear with their ears, understand with their heart, and should turn again; and I would heal them.' 13:16 "But blessed are your eyes, for they see; and your ears, for they hear. 13:17 For most certainly I tell you that many prophets and righteous men desired to see the things which you see, and didn't see them; and to hear the things which you hear, and didn't hear them. Who to be related to Allow people to be doomed completely. If they aren't interested, let them go. Matthew 10:14 Whoever doesn't receive you, nor hear your words, as you go out of that house or that city, shake off the dust from your feet. 10:15 Most certainly I tell you, it will be more tolerable for the land of Sodom and Gomorrah in the day of judgment than for that city. Who to give peace to Peace is invasive, apparently. It shouldn't be forced upon others. Matthew 10:13 If the household is worthy, let your peace come on it, but if it isn't worthy, let your peace return to you. Jesus thinks by analogy Analogy and ambiguity of physical and spiritual health I think that Jesus' miracles of healing allowed him to refer ambiguously to physical and spiritual health, as with "seeing" and "hearing", "walking" and "raising of the dead", which can be taken physically or spiritually. Matthew 11:2 Now when John heard in the prison the works of Christ, he sent two of his disciples 11:3 and said to him, "Are you he who comes, or should we look for another?" 11:4 Jesus answered them, "Go and tell John the things which you hear and see: 11:5 the blind receive their sight, the lame walk, the lepers are cleansed, the deaf hear, the dead are raised up, and the poor have good news preached to them. 11:6 Blessed is he who finds no occasion for stumbling in me." Analogy from human nature Mark 12:9 What therefore will the lord of the vineyard do? He will come and destroy the farmers, and will give the vineyard to others. Analogy with children's ways Matthew 11:16 "But to what shall I compare this generation? It is like children sitting in the marketplaces, who call to their companions 11:17 and say, 'We played the flute for you, and you didn't dance. We mourned for you, and you didn't lament.' Analogy with society Luke 18:1 He also spoke a parable to them that they must always pray, and not give up, 18:2 saying, "There was a judge in a certain city who didn't fear God, and didn't respect man. 18:3 A widow was in that city, and she often came to him, saying, 'Defend me from my adversary!' 18:4 He wouldn't for a while, but afterward he said to himself, 'Though I neither fear God, nor respect man, 18:5 yet because this widow bothers me, I will defend her, or else she will wear me out by her continual coming.'" 18:6 The Lord said, "Listen to what the unrighteous judge says. 18:7 Won't God avenge his chosen ones, who are crying out to him day and night, and yet he exercises patience with them? 18:8 I tell you that he will avenge them quickly. Nevertheless, when the Son of Man comes, will he find faith on the earth?" Analogy with the law Matthew 12:1 At that time, Jesus went on the Sabbath day through the grain fields. His disciples were hungry and began to pluck heads of grain and to eat. 12:2 But the Pharisees, when they saw it, said to him, "Behold, your disciples do what is not lawful to do on the Sabbath." 12:3 But he said to them, "Haven't you read what David did, when he was hungry, and those who were with him; 12:4 how he entered into God's house, and ate the show bread, which was not lawful for him to eat, neither for those who were with him, but only for the priests? 12:5 Or have you not read in the law, that on the Sabbath day, the priests in the temple profane the Sabbath, and are guiltless? 12:6 But I tell you that one greater than the temple is here. Analogy with the natural world Matthew 6:25 Therefore I tell you, don't be anxious for your life: what you will eat, or what you will drink; nor yet for your body, what you will wear. Isn't life more than food, and the body more than clothing? 6:26 See the birds of the sky, that they don't sow, neither do they reap, nor gather into barns. Your heavenly Father feeds them. Aren't you of much more value than they? Foxes have holes and birds have nests 8:20 Jesus said to him, "The foxes have holes, and the birds of the sky have nests, but the Son of Man has nowhere to lay his head." Lilies of the field Matthew 6:28 Why are you anxious about clothing? Consider the lilies of the field, how they grow. They don't toil, neither do they spin, 6:29 yet I tell you that even Solomon in all his glory was not dressed like one of these. 6:30 But if God so clothes the grass of the field, which today exists, and tomorrow is thrown into the oven, won't he much more clothe you, you of little faith? Analogy with the unrighteous world Luke 18:1 He also spoke a parable to them that they must always pray, and not give up, 18:2 saying, "There was a judge in a certain city who didn't fear God, and didn't respect man. 18:3 A widow was in that city, and she often came to him, saying, 'Defend me from my adversary!' 18:4 He wouldn't for a while, but afterward he said to himself, 'Though I neither fear God, nor respect man, 18:5 yet because this widow bothers me, I will defend her, or else she will wear me out by her continual coming.'" 18:6 The Lord said, "Listen to what the unrighteous judge says. 18:7 Won't God avenge his chosen ones, who are crying out to him day and night, and yet he exercises patience with them? 18:8 I tell you that he will avenge them quickly. Nevertheless, when the Son of Man comes, will he find faith on the earth?" Looking for comparisons Matthew 11:16 "But to what shall I compare this generation? Social rules as models for general rules Matthew 9:14 Then John's disciples came to him, saying, "Why do we and the Pharisees fast often, but your disciples don't fast?" 9:15 Jesus said to them, "Can the friends of the bridegroom mourn, as long as the bridegroom is with them? But the days will come when the bridegroom will be taken away from them, and then they will fast. Miracles allow for belief and unbelief How they figure things out The sayings seem arbitrary as the same word is used for red with different results. Matthew 16:1 The Pharisees and Sadducees came, and testing him, asked him to show them a sign from heaven. 16:2 But he answered them, "When it is evening, you say, 'It will be fair weather, for the sky is red.' 16:3 In the morning, 'It will be foul weather today, for the sky is red and threatening.' Hypocrites! You know how to discern the appearance of the sky, but you can't discern the signs of the times! 16:4 An evil and adulterous generation seeks after a sign, and there will be no sign given to it, except the sign of the prophet Jonah." Miracles It seems that Jesus did miracles so that people would be able to not understand his goodness, would be attracted to hear him but need not understand him. I and others might think, "I would do miracles, too, if I had such powers," and not appreciate that if we had such powers, then we might do very nasty things, or selfish things, or not apply ourselves. We don't appreciate the care with which Jesus did his miracles, that his goodness was not what he did, but how and why he did it. What sign will he give Matthew 16:1 The Pharisees and Sadducees came, and testing him, asked him to show them a sign from heaven. 16:2 But he answered them, "When it is evening, you say, 'It will be fair weather, for the sky is red.' 16:3 In the morning, 'It will be foul weather today, for the sky is red and threatening.' Hypocrites! You know how to discern the appearance of the sky, but you can't discern the signs of the times! 16:4 An evil and adulterous generation seeks after a sign, and there will be no sign given to it, except the sign of the prophet Jonah." What distinguishes the cultures Born of woman vs. born of spirit Matthew 11:11 Most certainly I tell you, among those who are born of women there has not arisen anyone greater than John the Baptizer; yet he who is least in the Kingdom of Heaven is greater than he. Distinguish foretelling and establishing Jesus came to establish, to put into place, to fate, not to foretell or warn or heighten free will. Matthew 11:13 For all the prophets and the law prophesied until John. Do people and Jesus truly know each other? Matthew 7:22 Many will tell me in that day, 'Lord, Lord, didn't we prophesy in your name, in your name cast out demons, and in your name do many mighty works?' 7:23 Then I will tell them, 'I never knew you. Depart from me, you who work iniquity.' Do they want their works to be seen by men or by God? Matthew 23:5 But all their works they do to be seen by men. They make their phylacteries broad, enlarge the fringes of their garments, 23:6 and love the place of honor at feasts, the best seats in the synagogues, 23:7 the salutations in the marketplaces, and to be called 'Rabbi, Rabbi' by men. Do what I say and you are unshakeable Luke 6:46 "Why do you call me, 'Lord, Lord,' and don't do the things which I say? 6:47 Everyone who comes to me, and hears my words, and does them, I will show you who he is like. 6:48 He is like a man building a house, who dug and went deep, and laid a foundation on the rock. When a flood arose, the stream broke against that house, and could not shake it, because it was founded on the rock. 6:49 But he who hears, and doesn't do, is like a man who built a house on the earth without a foundation, against which the stream broke, and immediately it fell, and the ruin of that house was great." How to receive the Kingdom of God Children and fools don't live by experience. Luke 18:15 They were also bringing their babies to him, that he might touch them. But when the disciples saw it, they rebuked them. 18:16 Jesus summoned them, saying, "Allow the little children to come to me, and don't hinder them, for the Kingdom of God belongs to such as these. 18:17 Most certainly, I tell you, whoever doesn't receive the Kingdom of God like a little child, he will in no way enter into it." Identifying himself as born of spirit, not born of woman Jesus is important as a person born of spirit, not of woman. John the Baptizer is more impressive than Jesus as a person born of woman. 11:11 Most certainly I tell you, among those who are born of women there has not arisen anyone greater than John the Baptizer; yet he who is least in the Kingdom of Heaven is greater than he. What distinguishes the ways to life and to destruction? The gate of "belief" leads to life; the gate of "caring" leads to destruction. We can distinguish the gates by seeing who goes through them. Matthew 7:13 "Enter in by the narrow gate; for wide is the gate and broad is the way that leads to destruction, and many are those who enter in by it. 7:14 How narrow is the gate, and restricted is the way that leads to life! Few are those who find it. What happens to those who listen to Jesus? Matthew 7:24 "Everyone therefore who hears these words of mine, and does them, I will liken him to a wise man, who built his house on a rock. 7:25 The rain came down, the floods came, and the winds blew, and beat on that house; and it didn't fall, for it was founded on the rock. 7:26 Everyone who hears these words of mine, and doesn't do them will be like a foolish man, who built his house on the sand. 7:27 The rain came down, the floods came, and the winds blew, and beat on that house; and it fell—and great was its fall." What is the difference between cultures? Matthew 10:16 "Behold, I send you out as sheep in the midst of wolves. Who do disciples say that I am? Matthew 16:15 He said to them, "But who do you say that I am?" 16:16 Simon Peter answered, "You are the Christ, the Son of the living God." 16:17 Jesus answered him, "Blessed are you, Simon Bar Jonah, for flesh and blood has not revealed this to you, but my Father who is in heaven. Who do people say that I am? Matthew 16:13 Now when Jesus came into the parts of Caesarea Philippi, he asked his disciples, saying, "Who do men say that I, the Son of Man, am?" 16:14 They said, "Some say John the Baptizer, some, Elijah, and others, Jeremiah, or one of the prophets." What his disciples should not yet tell others Confiding How do we know of the devil's temptation of Jesus? There were no witnesses except for Jesus himself. The Holy Spirit may have informed the evangelists or Jesus may have told his disciples himself. If they were to know, then it seems that he might as well have told them himself. Why would he then tell about the temptations? Perhaps to say that they were the "down-to-earth" salt of the earth that he had chosen, that it was he who needed them. Jesus also seems to have confided much to John, his best friend, who seems to present Jesus' point of view as Jesus understood himself. Telling only his disciples Matthew 16:20 Then he commanded the disciples that they should tell no one that he was Jesus the Christ. Who should know Matthew 16:20 Then he commanded the disciples that they should tell no one that he was Jesus the Christ. 16:21 From that time, Jesus began to show his disciples that he must go to Jerusalem and suffer many things from the elders, chief priests, and scribes, and be killed, and the third day be raised up. |
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Model: What is true?
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Model: What is true? The metalevel may simplify the problem at the level. Such a relationship may develop over stages of "wishful thinking" so that the metalevel illustrates the core of the problem. Ultimately, the metalevel gives the solution's deep structure and the level gives the problem's surface structure. A right triangle is half a rectangle This morphism is the basis for the area of a right triangle, but also for all of trigonometry, and shows that a function need not be a formula, and shows how two domains - angles and ratios - can be linked, as by shapes. Gospel Math. Recasting geometry/combinatorics as parity Remove the two diagonally opposite corner squares of a chessboard. Is it possible to tile this shape with thirty-one 2 x 1 "dominos"? ... At first, it seems like a geometric/combinatorial problem with many cases and subcases. But it is really just a question about counting colors. The two corners that were removed wre both (without loss of generality) white, so the shape we are interested in contains 32 black and 30 white squares. Yet any domino, once it is placed, will occupy exactly one black and one white square. The 31 dominos thus require 31 black and 31 white squares, so tiling is impossible. pg. 60 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Apply algebra ideas to a calculus problem Our final example is also due to Euler. Here the tables are turned: ideas from polynomial algebra are inappropriately applied to a calculus problem, resulting in a wonderful and correct evaluation of an infinite series (although in this case, complete rigorization is much more complicated). ... Is there a simple expression for zeta(2) = 1 + 1/2**2 + 1/3**2 + ... ? Euler's wonderful, crazy idea was inspired by the relationship between zeros and coefficients which says that the sum of the zeros of the monic polynomial x**n + a_n-1 x**n-1 + ... + a1 x + a0 is equal to - a_n-1; this follows from an easy argument that examines the factorization of the polynomial into terms of the form (x-ri), where each ri is a zero. Why not try this with functions that have infinitely many zeros? A natural candidate to start with is sin x ... pg.315 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Crossover A crossover ... is an idea that connects two or more different branches of math, usually in a surprising way. ... perhaps the three most productive crossover topics: graph theory, complex numbers, and generating functions. pg.119, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Deliberately misleading presentation Three women check into a motel room which advertises a rate of $27 per night. They each give $10 to the porter, and ask her to bring back 3 dollar bills. The porter returns to the desk, where she learns that the room is actually only $25 per night. She gives $25 to the motel desk clerk, returns to the room, and gives the guests back each one dollar, deciding not to tell them about the actual rate. Thus the porter has pocketed $2, while each guest spent 10-1 = $9, a total of 2 + 3 x 9 = $29. What happened to the other dollar? ... This problem is deliberately trying to mislead the reader into thinking that the profit that the porter makes plus the amount that the guests spend *should* add up to $30. pg. 22, 102, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. e Determine, with proof, the largest number which is the product of positive integers whose sum is 1976. ... Once again, we shall inappropriately apply calculus to a discrete problem. It makes intuitive sense for the numbers whose sum is 1976 to be equal (see the discussion of the AM-GM inequality...) But how large should these parts be? Consider the optimization question of finding the maximum value of f(x) = (S/x)**x, where S is a positive constant. An exercise in logarithmic differentiation (do it!) shows that S/x = e. Thus, if the sum is S each part should equal e and there should be S/e parts. Now this really makes no sense if S=1976 and the parts must be integers, and having a non-integral number of parts makes even less sense. But at least it focuses our attention on parts whose size is close to e=2.71828... Once we start looking at parts of size 2 and 3, the problem is close to a solution... pg.313 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Encoding In contrast [to partitioning], the encoding tactic attempts to count something in one step, by first producing a bijection (a fancy term for a 1-1 correspondence) between each thing we want to count and the individual "words" in a simple "code". ... Instead of partitioning the collection of subsets into many classes, look at this collection as a whole and encode each of its elements (which are subsets) as a string of symbols. Imagine storing information in a computer. How can you indicate a particular subset of S = {a,b,c}? There are many possibilities, but what we want is a uniform coding method that is simple to describe and works essentially the same for all cases. That way it will be easy to count. For example, any subset of S is uniquely determined by the answers to the following yes/no questions. Does the subset include a? Does the subset include b? Does the subset include c? We can encode the answers to these questions by a three-letter string which uses only the letters y and n. For example, the string yyn would indicate the subset {a,b}. Likewise, the string nnn indicates the empty set and yyy indicates the entire set S. Thus There is a bijection between strings and subsets. ... And it is easy to count the number of strings; two choices for each letter and three letters per string mean 2**3 different strings in all. ... Proper encoding demands precise information management. ... try to think carefully about "freedom of choice": ask yourself what has already been completely determined from previous choices ... Beginners are often seduced by the quick answers provided by encoding and attempt to convert just about any counting problem into a simple multiplication or binomial coefficient Note that strings have an additional structure which makes the counting easy: the strings presume a total order of positions, from left to right, whereas the elements of a set need not be ordered. This ordering comes for free and makes the bijection work. pg.213-214 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Fantasize an answer When looking at the conclusion of the problem, especially for a "to find" problem, sometimes it helps to "fantasize" an answer. Just make something up, and then reread the problem. Your fantasy answer is most likely false, and rereading the problem with this answer in mind may help you to see why the answer is wrong, which may point out some of the more important constraints of the problem. pg.30, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Interpreting algebraic variables as coordinates Whenever a problem involves several algebraic variables, it is worth pondering whether some of them can be interpreted as coordinates. pg. 59 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Recast a problem from one domain into another domain The powerful idea of converting a problem from words to pictures is just one aspect of the fundamental peripheral vision strategy. Open your mind to other ways of reinterpreting problems. ... what appeared to be a sequence of numbers was actually a sequence of descriptions of numbers ... Another example was the locker problem in which a combinatorial problem metamorphosed into a number theory lemma. "Combinatorics <=> Number Theory" is one of the most popular and productive such "crossovers", but there are many other possibilities. Some of the most spectacular advances in mathematics occur when someone discovers a new reformulation for the first time. pg. 60 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Recast an inequality as an optimization problem AM-GM reformulated ... we altered our point of view and recast an inequality as an optimization problem. pg.195-196 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Recasting [In combinatorics,] the strategy of recasting is especially fruitful: to counteract the inherent dryness of counting, it helps to creatively visualize problems (for example, devise interesting "combinatorial arguments") and look for hidden symmetries. Many interesting counting problems involve very imaginative multiple viewpoints ... to see if a combinatorial identity is true, examine how each side of the equation counts a representative element pg.212, 228 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Recasting geometry as algebra Descartes' idea of recasting geometric questions in a numeric/algebraic form led to the development of analytic geometry, which then led to calculus. pg. 60 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Edward Cherlin: Proving that two seemingly unrelated, even apparently incompatible objects are equivalent in some way, or can model each other, is one of the deepest ideas in math. Edward Cherlin: Mathematicians lost interest in Brouwer's Intuitionistic logic and set theory when it was shown that it and the more usual non-constructive logics and set theories can model each other. Edward Cherlin: The fact that each of elliptic/Riemannian, Euclidean, and hyperbolic/Lobachevskian geometries contain models of each other shows that all three are equally valid. For example, a Clifford's surface in Riemannian space and a horosphere in Lobachevskian space both have locally Euclidean geometry. Galois theory Galois theory maps the roots of a given polynomial equation (in field theory) to the Galois group of permutations of the roots. Edward Cherlin: At one time it was thought that there was a vast space of possible String Theories in physics. It turns out that all of them are maps of each other. Edward Cherlin: The proof of Fermat's Last Theorem depends on the Taniyama-Shimura Theorem that all elliptic functions are modular, that is, that there is a structure-preserving mapping between elliptic functions and modular forms. Two different ways Keeping a flexible point of view is a powerful strategy. This is especially true with counting problems where often the crux move is to count the same thing in two different ways. To help develop this flexibility, you should practice creating "combinatorial arguments". This is just fancy language for a story that rigorously describes in English how you count something. ... Pay attention to the building blocks of "algebra to English" translation, and in particular, make sure you understand when and why multiplication rather than addition happens, and vice versa. Examples include addition (or), multiplication (and), exponentiation, combination, permutation, distinct members, products of choices, sums of choices, complements of combinations. pg.208 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Make it easier The easier problem may actually be the more informative, relevant, natural, instructive problem. If the given problem is too hard, solve an easier one. ... For example, if the problems involves big, ugly numbers, make them small and pretty. If a problem involves complicated algebraic fractions or radicals, try looking at a similar problem without such terms. At best, pretending that the difficulty isn't there will lead to a bold solution... At worst, you will be forced to focus on the key difficulty of your problem, and possibly formulate an intermediate question, whose answer will help you with the problem at hand. And eliminating the hard part of a problem, even temporarily, will allow you to have some fun and raise your confidence. pg.18, 31 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.  Wishful thinking It is helfpul to try to loosen up, and not worry about rules or constraints. Wishful thinking is always fun, and often useful. For example, in this problem, the main difficulty is that the top boxes labeled A and C are in the "wrong" places. So why not move them around to make the problem trivially easy? ... Ask yourself, "What is it about the problem that makes it hard?" Then, make the difficulty disappear! You may not be able to do this legally, but who cares? Temporarily avoiding the hard part of a problem will allow you to make progress and may shed light on the difficulties. pg.18, 31 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. |
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Agree - Implicitly Understand
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Agree - Implicitly Understand Campfire learn from stories, share culture Graphic Jam visualize abstract concepts, absorb complexity History Map appreciate relevance of organization's culture Memory Wall appreciate people, foster team Visual Glossary create a shared language Welcome to My World share our internal map of reality |
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Knowing what
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Knowing what Knowing What one knows, its origins. Caring about Believer. Appeal to shared inner implicit inherent culture despite explicit experience of examples that suggest differently. The truth of the world is easy to point to, can be shown by examples, whereas the truth of the heart must already be in you, is evoked by analogy. It is easy to show examples that "my help can make things worse". But how can I show that I "should" help? I can't observe that, but rather, the notion must already be in me. Likewise, I can point to the surface structure of a problem, but as for the deep structure, I have to appeal to you that you are already familiar with it. Sense the spirit of a discipline Attend lectures Conversing with: science I learned much physics and math in college through lectures, especially physics lectures where professors such as Isaac Abella imparted their intuition. He would say things that the books weren't saying or weren't able to make prominent, but got across the idea at hand and the spirit of the subject. Center conversation within their expertise Conversing with: people's wisdom People aren't generally interested in my philosophical thinking or can't engage me regarding it. So I've learned it to speak with people about their subjects of expertise or interest, especially with my friends, such as psychology with Shu-Hong Zhu, math and physics with John Harland, cognitive science and neuroscience with Giedrius Buracas, and subsequently, with thinkers at up to twenty working groups at my online lab, Minciu Sodas. Through these fields I'm able to express and develop my philosophical ideas and find where they are relevant. Get to know the classics Conversing with: content I like to read the classics in literature, philosophy, religion and other fields because they help me understand what is pertinent, fundamental, creative, fruitful and influential, what is a real contribution. I enjoyed learning about and listening to classics of rock, jazz and classical music. This also helps me find my own way, what I myself am attracted to, makes me feel safe to do that and not feel manipulated by the spirit of our times. I appreciate the classics as an efficient way to learn. Making sense of a statement through personal intuition Is God good? I was contemplating God and the idea that he was both inside us and outside of us, yet the same God. I remembered an idea of Jesus from the Gospel of John and, in contemplating both, I made sense of its meaning: that life is the fact that God is good, uniting them, but understanding that fact, distinguishing them, is eternal life, by which God need not be good. I believed there was meaning in a statement, and by leveraging my personal intuition, I was able to get that meaning. I think that Father Dave Martin prepared his sermons similarly, relating his personal intuition to a passage from the Gospel, noting how it differed, and contemplating the difference. Note the nature of the origin of a statement Conversing with: sources In developing the good will exercises, I noted the following test to distinguish the truths of the heart and of the world: We can point to the truth of the world with examples, but we can't show or learn the truth of the heart and it must already be in us. I can show many examples that "My help could make things worse", whereas I can in no way show that "I should help those who need or want help", but it must be innate. I can appeal to one's knowledge of the latter with illustrations and analogies, but I can't show it in the world. Learn the issues of a discipline Ask for advice Conversing with: know how I learn many things by asking people who are knowledgeable in particular fields such as computers, Linux, Lithuanian, languages, websites. Dialogue with those responsible Conversing with: concern In speaking with leaders of the Chicago Archdiocese's Office of Catechesis, I realized that in the mainstream churches I might best connect with those who love to worship God. Interrogate Conversing with: advisor When I go shopping for something I don't know much about, such as shoes, bicycles, hiking equipment, computers or gifts for women, I try to ask the salesperson questions, both to learn about what to look for, and to discern if they are genuinely helping me and I should follow their advice. Survey existing knowledge Conversing with: what is known As a child, when I started my quest to know everything, I was keen to learn all that what was already known, starting with history, starting with the beginning, the Mesopotamians. Unfortunately, I couldn't find any books on them. So then I read about the Ancient Egyptians. In college, I was excited to be able to learn on a deeper, more serious level, but I had to give up the idea or reading everything and knowing it as it was presented. Yet even when I worked independently, I would try to survey relevant fields. My friend Shu-Hong Zhu made me a list of books relevant for decision making and emotion, and in one book I learned of a study of the emotions that mothers interpreted in faces of babies, namely: content, sad, excited, surprised, frightened, disgusted. I then thought through a model of these as emotional responses based on cognitive expectations. Try out advice received Conversing with: instruction I've tried out advice from people in nutrition, health, hiking, music, culture, literature... Sometimes these are knowledgeable people and sometimes it just seems that they know something they found useful and are glad to share. Such knowledge tends to be useful, handling some issue, and may be fruitful conceptually, in that it makes personal sense and develops sensitivity in the relevant field. |
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Origin
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Origin Apply constructs of source domain to target domain George Lakoff and Mark Johnson describe how a metaphor allows us to apply the constructs of a source domain (such as WAR) to a new, target domain (such as ARGUMENT). ...This is an example of what it means for a metaphorical concept, namely, ARGUMENT IS WAR, to structure (at least in part) what we do and how we understand what we are doing when we argue. The essence of metaphor is understanding and experiencing one kind of thing in terms of another. It is not that arguments are a subspecies of war. Arguments and wars are different kinds of things - verbal discourse and armed conflict - and the actions performed are different kinds of actions. But ARGUMENT is partially structured, understood, performed and talked about in terms of WAR. The concept is metaphorically structured, the activity is metaphorically structured, and, consequently, the language is metaphorically structured. Moreover, this is the ordinary way of having an argument and talking about one. From Chapter 1, Concepts We Live By, of Metaphors We Live By, by George Lakoff and Mark Johnson, The University of Chicago, 1980 Tikslinti #803 Ask people why Why did County patients tolerate these waits and abusive conditions? Our patients declared that they came because County had "the best doctors." This was not true. There is no way we were the best. We were young, uninitiated, and worse, unsupervised. But many of our patients had been turned away from other institutions or had family or friends with the same experience "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.1934 Pamela McLean, June 10, 2011. Maybe that is why we need story telling (and pattern languages) in our post-web world - so that communication is tied firmly into context through anecdotes. The beauty of a role model Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: "I watched Muhammad Ali, how when he would speak, how it was such a thing of beauty. It sounded so wonderful. And I wanted to be like him." Pamela McLean, June 10, 2011. Maybe generalisations only work effectively when people are living in the same cultural groups or working in the same silos. Identifying with a larger equivalent Socrates explores the human soul by identifying it with a city. |
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Fact and Judgment: Taxonomy of Problem Types
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While problems come in all varieties, shapes, and sizes, each can be categorized in terms of the roles that fact and judgment play in analysis of the problem. ... there is an inverse relationship between the number of facts and the amount of judgment required to solve a problem. ... the fewer facts we have, the more judgment is required ... the more facts - the less judgment. From this relationship we can define four basic types of problems:
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Motivation (internal and external)
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Motivation (internal and external) External motivation substitutes for internal motivation. What is valued. We talked a long time about the difference between the two systems, Lithuania and US, that the IRS gives priority to the facts at hand rather than the documents, that they give people refunds based on what the people say they are due (they could not believe this), and that people are audited typically once in their lifetime. In junior high school and high school I and my brother played actively in adult tournaments. The stronger tournaments were for money. The money seemed to suggest a very different way of playing that was foreign to me and so I put it out of my mind. It seemed to interfere with the idea of trying to go up against the best players rather than win a prize amongst the weaker players. I had believed that it was a bad idea to work for a friend or relative as it could destroy our relationship which is much more precious than any work. But then I had the chance to work for my friend John Harland. And indeed there were times of stress but it really deepened our friendship and I learned that much more what a wonderful person he is. |
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Study
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Study Collect recordings and lyrics Expand my interests Listen to examples Looking at different aspects of a subject Study videos, concerts, services |
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Monkeying around
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Monkeying around If the experimental apparatus is stable, then the experimenter can make all kinds of changes to it and see what happens. They can see how changing an input variable affects an output variable. Aborption of cathode rays is proportional to density of material they pass through Wikipedia: Philipp Lenard observed that the absorption of cathode rays was, to first order, proportional to the density of the material they were made to pass through. This appeared to contradict the idea that they were some sort of electromagnetic radiation. Alterations: Geiger-Marsden Gold Foil Experiments A key part of this experiment was that the parameters (such as the foil) could be changed in many ways. The same experiment could be done many times with just a slight alteration. Another key part was that the vast majority of the alpha particles were not affected by the foil at all, which made for a very prominent baseline. Between 1908 and 1913, Hans Geiger and Ernest Marsden under the direction of Ernest Rutherford determined that every atom contained a nucleus which consisted of most of its mass and its positive charge. They did this by emitting alpha particles (positively charged helium nuclei) from radium and other sources through thin foils (such as gold and aluminum) onto a flourescent screen. They observed that most of the particles passed through the foil but some particles were scattered by large angles, even bouncing back from the foil. This meant that most of the foil was empty space and that the mass was concentrated in positive nuclei that could knock the alpha particle off course. |
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The first will be last and the last will be first
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The first will be last and the last will be first In this world, the truth of the world is prior to the truth of the heart; but in peace, the truth of the heart is prior to the truth of the world. Jesus may be relating the principle of reflectivity, which changes the direction of the twosome, so that reflection switches what is first and what is last. Thus we can never be certain, which will be first and which will be last, but must stay on our toes. How God cares and judges The land of Galilee shall be exalted Isaiah 9:1 But there shall be no more gloom for her who was in anguish. In the former time, he brought into contempt the land of Zebulun and the land of Naphtali; but in the latter time he has made it glorious, by the way of the sea, beyond the Jordan, Galilee of the nations. 9:2 The people who walked in darkness have seen a great light. Those who lived in the land of the shadow of death, on them the light has shined. 9:3 You have multiplied the nation. You have increased their joy. Believe that God hears you when you speak once Matthew 6:7 In praying, don't use vain repetitions, as the Gentiles do; for they think that they will be heard for their much speaking. Children by heredity and by spirit will be distinguished Jesus distinguishes between the ethnic children and the spiritual children of the Jewish patriarchs. Spiritual childhood is perhaps the point of being "born again". Matthew 8:11 I tell you that many will come from the east and the west, and will sit down with Abraham, Isaac, and Jacob in the Kingdom of Heaven, 8:12 but the children of the Kingdom will be thrown out into the outer darkness. There will be weeping and gnashing of teeth. Consequence of general principle If we can serve only one master, and that master is God, then God will provide for us, and we need not be anxious. Matthew 6:24 "No one can serve two masters, for either he will hate the one and love the other; or else he will be devoted to one and despise the other. You can't serve both God and Mammon. 6:25 Therefore I tell you, don't be anxious for your life: what you will eat, or what you will drink; nor yet for your body, what you will wear. Isn't life more than food, and the body more than clothing? 6:26 See the birds of the sky, that they don't sow, neither do they reap, nor gather into barns. Your heavenly Father feeds them. Aren't you of much more value than they? Correction of our self-conception Matthew 23:12 Whoever exalts himself will be humbled, and whoever humbles himself will be exalted. How would the Gentiles have reacted Matthew 11:20 Then he began to denounce the cities in which most of his mighty works had been done, because they didn't repent. 11:21 "Woe to you, Chorazin! Woe to you, Bethsaida! For if the mighty works had been done in Tyre and Sidon which were done in you, they would have repented long ago in sackcloth and ashes. 11:22 But I tell you, it will be more tolerable for Tyre and Sidon on the day of judgment than for you. 11:23 You, Capernaum, who are exalted to heaven, you will go down to Hades. For if the mighty works had been done in Sodom which were done in you, it would have remained until this day. 11:24 But I tell you that it will be more tolerable for the land of Sodom, on the day of judgment, than for you." Who is convinced of their own righteousness What is the difference between faith in oneself and being convinced of one's own righteousness? What is the difference between praying to oneself and praying to God? Luke 18:9 He spoke also this parable to certain people who were convinced of their own righteousness, and who despised all others. 18:10 "Two men went up into the temple to pray; one was a Pharisee, and the other was a tax collector. 18:11 The Pharisee stood and prayed to himself like this: 'God, I thank you, that I am not like the rest of men, extortioners, unrighteous, adulterers, or even like this tax collector. 18:12 I fast twice a week. I give tithes of all that I get.' 18:13 But the tax collector, standing far away, wouldn't even lift up his eyes to heaven, but beat his breast, saying, 'God, be merciful to me, a sinner!' 18:14 I tell you, this man went down to his house justified rather than the other; for everyone who exalts himself will be humbled, but he who humbles himself will be exalted." Why are you anxious? Matthew 6:28 Why are you anxious about clothing? Consider the lilies of the field, how they grow. They don't toil, neither do they spin, 6:29 yet I tell you that even Solomon in all his glory was not dressed like one of these. 6:30 But if God so clothes the grass of the field, which today exists, and tomorrow is thrown into the oven, won't he much more clothe you, you of little faith? How will God reward Distinguishing those who will lose and those who will gain In the passage below from Luke, which can be compared to the Sermon on the Mount in Matthew, Jesus refers to the "poor" rather than the "poor-in-spirit", and the "Kingdom of God" rather than the "kingdom of heaven". It reminds me of "the last will be first and the first will be list" as with pop-up automata or climbing into and out of a bus. Luke 6:20 ... Blessed are you who are poor, for yours is the Kingdom of God. 6:21 Blessed are you who hunger now, for you will be filled. Blessed are you who weep now, for you will laugh. 6:22 Blessed are you when men shall hate you, and when they shall exclude and mock you, and throw out your name as evil, for the Son of Man's sake. 6:23 Rejoice in that day, and leap for joy, for behold, your reward is great in heaven, for their fathers did the same thing to the prophets. 6:24 "But woe to you who are rich! For you have received your consolation. 6:25 Woe to you, you who are full now, for you will be hungry. Woe to you who laugh now, for you will mourn and weep. 6:26 Woe, when men speak well of you, for their fathers did the same thing to the false prophets. Jesus wants to be rewarded by God Jesus often forbids his works from being known. I imagine it is because he wants to be rewarded by God, not by this world. Matthew 9:30 ... Jesus strictly commanded them, saying, "See that no one knows about this." 9:31 But they went out and spread abroad his fame in all that land. Reward is from the one who sees Matthew 6:17 But you, when you fast, anoint your head, and wash your face; 6:18 so that you are not seen by men to be fasting, but by your Father who is in secret, and your Father, who sees in secret, will reward you. Reward is that what is secret is made open Matthew 6:5 "When you pray, you shall not be as the hypocrites, for they love to stand and pray in the synagogues and in the corners of the streets, that they may be seen by men. Most certainly, I tell you, they have received their reward. 6:6 But you, when you pray, enter into your inner room, and having shut your door, pray to your Father who is in secret, and your Father who sees in secret will reward you openly. Who decides what is fair Matthew 19:29 Everyone who has left houses, or brothers, or sisters, or father, or mother, or wife, or children, or lands, for my name's sake, will receive one hundred times, and will inherit eternal life. 19:30 But many will be last who are first; and first who are last. 20:1 "For the Kingdom of Heaven is like a man who was the master of a household, who went out early in the morning to hire laborers for his vineyard. 20:2 When he had agreed with the laborers for a denarius a day, he sent them into his vineyard. 20:3 He went out about the third hour, and saw others standing idle in the marketplace. 20:4 To them he said, 'You also go into the vineyard, and whatever is right I will give you.' So they went their way. 20:5 Again he went out about the sixth and the ninth hour, and did likewise. 20:6 About the eleventh hour he went out, and found others standing idle. He said to them, 'Why do you stand here all day idle?' 20:7 "They said to him, 'Because no one has hired us.' "He said to them, 'You also go into the vineyard, and you will receive whatever is right.' 20:8 When evening had come, the lord of the vineyard said to his manager, 'Call the laborers and pay them their wages, beginning from the last to the first.' 20:9 "When those who were hired at about the eleventh hour came, they each received a denarius. 20:10 When the first came, they supposed that they would receive more; and they likewise each received a denarius. 20:11 When they received it, they murmured against the master of the household, 20:12 saying, 'These last have spent one hour, and you have made them equal to us, who have borne the burden of the day and the scorching heat!' 20:13 "But he answered one of them, 'Friend, I am doing you no wrong. Didn't you agree with me for a denarius? 20:14 Take that which is yours, and go your way. It is my desire to give to this last just as much as to you. 20:15 Isn't it lawful for me to do what I want to with what I own? Or is your eye evil, because I am good?' 20:16 So the last will be first, and the first last. For many are called, but few are chosen." Who do people seek a reward from? Matthew 6:16 "Moreover when you fast, don't be like the hypocrites, with sad faces. For they disfigure their faces, that they may be seen by men to be fasting. Most certainly I tell you, they have received their reward. What is first from which the rest follows What follows from what? Matthew 6:31 "Therefore don't be anxious, saying, 'What will we eat?', 'What will we drink?' or, 'With what will we be clothed?' 6:32 For the Gentiles seek after all these things; for your heavenly Father knows that you need all these things. 6:33 But seek first God's Kingdom, and his righteousness; and all these things will be given to you as well. What should be dealt with first Matthew 23:25 "Woe to you, scribes and Pharisees, hypocrites! For you clean the outside of the cup and of the platter, but within they are full of extortion and unrighteousness. 23:26 You blind Pharisee, first clean the inside of the cup and of the platter, that its outside may become clean also. Who is first among others? Matthew 20:24 When the ten heard it, they were indignant with the two brothers. 20:25 But Jesus summoned them, and said, "You know that the rulers of the nations lord it over them, and their great ones exercise authority over them. 20:26 It shall not be so among you, but whoever desires to become great among you shall be your servant. 20:27 Whoever desires to be first among you shall be your bondservant, 20:28 even as the Son of Man came not to be served, but to serve, and to give his life as a ransom for many." What sanctifies The circumstances are greater than what is in them Heaven is greater than God, for heaven sanctifies God. God is Godly because of his throne, because of heaven. 23:16 "Woe to you, you blind guides, who say, 'Whoever swears by the temple, it is nothing; but whoever swears by the gold of the temple, he is obligated.' 23:17 You blind fools! For which is greater, the gold, or the temple that sanctifies the gold? 23:18 'Whoever swears by the altar, it is nothing; but whoever swears by the gift that is on it, he is obligated?' 23:19 You blind fools! For which is greater, the gift, or the altar that sanctifies the gift? 23:20 He therefore who swears by the altar, swears by it, and by everything on it. 23:21 He who swears by the temple, swears by it, and by him who was living in it. 23:22 He who swears by heaven, swears by the throne of God, and by him who sits on it. Who regards what is holy or valuable? Matthew 7:6 "Don't give that which is holy to the dogs, neither throw your pearls before the pigs, lest perhaps they trample them under their feet, and turn and tear you to pieces. |
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Implication: How is it true?
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Implication: How is it true? Implication: How is it true? The metalevel may relate to the level as cause and effect by way of a flow of implications. The metalevel has us solve the problem, typically by working backwards. The level presents the solution, arguing forwards. Deduction Also known as "direct proof", deduction is merely the simplest form of argument in terms of logic. A deductive argument takes the form "If P, then Q" or "P=>Q" or "P implies Q". Sometimes the overall structure of an argument is deductive, but the smaller parts use other styles. pg.46, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Penultimate step Once you know what the desired conclusion is, ask yourself, "What will yield the conclusion in a single step?" Sometimes a penultimate step is "obvious", once you start looking for one. And the more experienced you are, the more obvious the steps are. For example, suppose that A and B are weird, ugly expressions that seem to have no connection, yet you must show that A = B. One penultimate step would be to separately argue that A ≥ B AND B ≥ A. Perhaps you want to show instead that A ≠ B. A penultimate step would be to show that A is always even, while B is always odd. pg. 30, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Recast geometry as logic ...a problem that is geometric on the surface, but not at its core ... We are given n planets in space, where n is a positive integer. Each planet is a perfect sphere and all planets have the same radius R. Call a point on the surface of a planet private if it cannot be seen from any other planet. ... We conjecture that the total private area is always exactly equal to the area of one planet, no matter how the planets are situated. It appears to be a nasty problem in solid geometry, but must it be? The notions of "private" and "public" seem to be linked with a sort of duality; perhaps the problem is really not geometric, but logical. ... If location x is private on one planet, it is public on all other planets. After this nice discovery, the penultimate step is clear: to prove that Given any location x, it must be private on some planet. ... pg. 63 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. |
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Improve - Explicitly Innovate
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Improve - Explicitly Innovate 3-12-3 Brainstorm fresh, incremental improvements Challenge Cards identify, address challenges, improve product Plus/Delta constructive feedback |
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Knowing how
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Knowing how Knowing How one knows, contrasts. Believing in Carer. Improve results by taking a more fundamental point of view. The truth of the world follows from the truth of the heart, but not the other way around. If "I should help those who need help", then I won't want my help to make things worse. But if I simply don't want to make things worse, I will never help anybody. Heightening the pull in opposite directions Contrast two statements, positive and negative Conversing with: comparability In developing a good will exercise, if I am riled by something negatively, then I ask myself, what would make me feel even worse? and this opens up space for me to ask, what would make feel good? and even better than that? Then I contrast the resulting statements so that their parts align and contrast sharply. Counterbalance one's personal feelings Conversing with: fairness In developing good will exercises with people, they were generally riled by a surface topic at which they could only look at one way. If the topic made them feel bad, then I would ask them, why? and what would make them feel even worse? That would open up possibility inside of them for me to ask, what would make them feel good? and even better than that? I would contrast the four answers so they were as intense and comparable as possible, and then compare them, what did they all have in common, yielding the deep issue. Heighten the contrast of statements pulling us in opposite directions On any abstract topic, contrasting truths of the heart and of the world arise, so that they pull in opposite directions, they suggest opposite actions. I heighten that contrast. I thought "life is will" and I asked, what is the difference? At first, I supposed that life is the "unity" of the "representations" of "anything", whereas will is the "unity" of the "representations of anything". Subsequently, I concluded that anything is the structure of life, choosing is the representations of anything, and will is the unity of choosing, thus the will is the unity of the representations of the structure of life. Distinguishing primary and secondary perspectives Compare outcomes Conversing with: causality The truth of the world follows from the truth of the heart, but not the other way around. I have observed this logical direction in 40+ good will exercises that I developed. It serves as one of four tests for distinguishing the truths of the heart from the truths of the world. For example, if "I should help those who need or want help", then soon it will be relevant that "My help could make things worse". But if I start with "My help could make things worse", then I will never help anyone! So "I should help those who need or want help" is the truth of the heart, whereas "My help could make things worse" is the truth of the world. Identifying the issue Conversing with: conditionality I may solve an issue by appealing to the heart of it. For example, I may wonder whether God would let me hear him, whether God would allow for that. Yet of all the things that I could ask for, isn't that the most ordinary one for God to grant? If Jesus encourages us to ask God for things, then isn't this the one that he can't credibly deny me? |
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Implication
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Implication Compare total expenses Fresh Air interview of Michael Hiltzik about the Hoover Dam, June 8, 2010: Dams are very expensive. And the water that they provide for users is very expensive water because of the capital expense of building a dam. It's wiser today to look for other sources of water supply, including conservation and reclamation, and this is what we try to do now because it's much cheaper, more efficient and ecologically friendly. Decide to be different from a leading influence Fresh Air interview with Keith Richards of the Rolling Stones: "If you're talking image-wise, we probably did make a decision to not be The Fab Four. They were basically differences between the bands. The Beatles were basically a vocal band. They all sang and one song, John would take the lead. Another, Paul [would] or George and sometimes Ringo. Our band set up totally differently — with one frontman, one lead singer, and what I loved about it is that there's an incredible difference in it between The Beatles and ourselves, but at the same time, we were there at the same time, and you're dealing with each other. And it was a very, very fruitful and great relationship between the Stones and The Beatles. It was very, very friendly. The competition thing didn't come into it as far as we were concerned." The path I want to take Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: "I wanted to win the goal medal and then go home and further my education in college," he says. |
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Weigh one element against another
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Second, structuring allows us to compare and weigh one element against another. Instead of looking at a whole bowl of vegetable soup, we look at the soup's ingredients, one ingredient at a time. This identifies which factors and relationships are critical not only to our analysis but also to the concerns of those who will make use of our findings. |
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Money (compensation of difference)
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Money (compensation of difference) Compensation substitutes for disparity in contributions. How there is value. When we were children, our father decided that he would supplement our allowance by giving us one penny for each page of Lithuanian books that we read for our own enjoyment. He wouldn't pay for English books, only Lithuanian. It was a really great support because the Lithuanian books were extra effort but I really wanted to get good at reading them. It made it extra rewarding. I remember as a child our mother would give us 5 cents for every caterpillar that we would remove from the tomato vines. The caterpillars were large, light green, gross to pick up, and scary because their alien body ended in what looked like a stinger. It didn't seem worth it. But then my mother said that it wasn't really about the money, she needed my help. So her need got me to do it, and the reward encouraged me to work efficiently. Mielas Andriau, Buvo smagu skaityti apie tavo pinigine odiseja. I really made you work for that nickel. Su meile, Mama RutaKK, 2011.05.05 |
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God's significance
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God's significance Championing faith What brings people to God Whether my prayers were answered |
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Establishing the reliability of the apparatus
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Establishing the reliability of the apparatus Once the apparatus is working, then there arises a question as to whether the interesting results it gives may be explained by some anomalies in the apparatus itself. Thus it becomes important to investigate the apparatus itself and establish its reliability. The apparatus becomes the subject of experiments. This is the case with many modern instruments, for example, neutrino detectors. |
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See oneself in Scripture
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See oneself in Scripture Jesus distinguishes between four levels of receiving the word. By receiving each other, as we receive Jesus, we are receiving the word. Thus we can take the word to heart, not just other people. Concepts from Scripture Instruments of stumbling will be destroyed Instruments of stumbling (such as Assyria) will be destroyed, not simply punished Those who are punished are not destroyed Psalm 118:17 I will not die, but live, and declare Yah's works. Jesus cites this Psalm. There is a distinction between being punished and being destroyed. We are punished by those who will ultimately be destroyed. The stone they reject will become the head of the corner. 118:18 Yah has punished me severely, but he has not given me over to death. 118:19 Open to me the gates of righteousness. I will enter into them. I will give thanks to Yah. 118:20 This is the gate of Yahweh; the righteous will enter into it. 118:21 I will give thanks to you, for you have answered me, and have become my salvation. 118:22 The stone which the builders rejected has become the head of the corner. 118:23 This is Yahweh's doing. It is marvelous in our eyes. 118:24 This is the day that Yahweh has made. We will rejoice and be glad in it! 118:25 Save us now, we beg you, Yahweh! Yahweh, we beg you, send prosperity now. 118:26 Blessed is he who comes in the name of Yahweh! We have blessed you out of the house of Yahweh. 118:27 Yahweh is God, and he has given us light. Bind the sacrifice with cords, even to the horns of the altar. Identify in Scripture others who you personally know Matthew 11:9 But why did you go out? To see a prophet? Yes, I tell you, and much more than a prophet. 11:10 For this is he, of whom it is written, 'Behold, I send my messenger before your face, who will prepare your way before you.' Jesus' role An action that others will understand later 12:14 Jesus, having found a young donkey, sat on it. As it is written, 12:15 "Don't be afraid, daughter of Zion. Behold, your King comes, sitting on a donkey's colt." 12:16 His disciples didn't understand these things at first, but when Jesus was glorified, then they remembered that these things were written about him, and that they had done these things to him. Zechariah 9:9 Rejoice greatly, daughter of Zion! Shout, daughter of Jerusalem! Behold, your King comes to you! He is righteous, and having salvation; lowly, and riding on a donkey, even on a colt, the foal of a donkey. Appreciating the power of God Mark 11:1 When they drew near to Jerusalem, to Bethsphage and Bethany, at the Mount of Olives, he sent two of his disciples, 11:2 and said to them, "Go your way into the village that is opposite you. Immediately as you enter into it, you will find a young donkey tied, on which no one has sat. Untie him, and bring him. 11:3 If anyone asks you, "Why are you doing this?" say, "The Lord needs him and immediately he will send him back here." Contemplating Scripture Mark 12:35 Jesus responded, as he taught in the temple, "How is it that the scribes say that the Christ is the son of David? 12:36 For David himself said in the Holy Spirit, "The Lord said to my Lord, "Sit at my right hand, until I make your enemies the footstool of your feet."" 12:37 Therefore David himself calls him Lord, so how can he be his son?" Finding confirmation in Scripture Mark 12:10 Haven't you even read this Scripture: "The stone which the builders rejected, the same was made the head of the corner. 12:11 This was from the Lord, it is marvelous in our eyes" Foretelling Jesus foretells several events that came to pass, and several that may still come to pass. How did he know? Did he gather that from the Scripture? Or his knowledge of God? Foretelling based on Scripture Jesus foretells several events, and of these, some seem to be based on his interpreting Scripture. Matthew 26:21 As they were eating, he said, "Most certainly I tell you that one of you will betray me." 26:22 They were exceedingly sorrowful, and each began to ask him, "It isn't me, is it, Lord?" 26:23 He answered, "He who dipped his hand with me in the dish, the same will betray me. 26:24 The Son of Man goes, even as it is written of him, but woe to that man through whom the Son of Man is betrayed! It would be better for that man if he had not been born." Fulfilling Scripture Jesus enters Jerusalem with great fanfare on Palm Sunday, takes a look around the temple, and simply leaves. Mark 11:11 Jesus entered into the temple in Jerusalem. When he had looked around at everything, it being now evening, he went out to Bethany with the twelve. Jesus took our infirmities Jesus dealt with the human condition, including illness. Matthew 8:16 When evening came, they brought to him many possessed with demons. He cast out the spirits with a word, and healed all who were sick; 8:17 that it might be fulfilled which was spoken through Isaiah the prophet, saying: "He took our infirmities, and bore our diseases." How he chose to speak to the multitudes Matthew 13:34 Jesus spoke all these things in parables to the multitudes; and without a parable, he didn't speak to them, 13:35 that it might be fulfilled which was spoken through the prophet, saying, "I will open my mouth in parables; I will utter things hidden from the foundation of the world." Psalm 78:2 78:1 Hear my teaching, my people. Turn your ears to the words of my mouth. 78:2 I will open my mouth in a parable. I will utter dark sayings of old, 78:3 Which we have heard and known, and our fathers have told us. 78:4 We will not hide them from their children, telling to the generation to come the praises of Yahweh, his strength, and his wondrous works that he has done. Interpreting Scripture metaphorically Jesus sees himself as the bridegroom, and the apostles as his friends. I presume this is an image that he has taken from Scripture. Matthew 9:15 Jesus said to them, "Can the friends of the bridegroom mourn, as long as the bridegroom is with them? But the days will come when the bridegroom will be taken away from them, and then they will fast. Knowing from Scripture Matthew 20:17 As Jesus was going up to Jerusalem, he took the twelve disciples aside, and on the way he said to them, 20:18 "Behold, we are going up to Jerusalem, and the Son of Man will be delivered to the chief priests and scribes, and they will condemn him to death, 20:19 and will hand him over to the Gentiles to mock, to scourge, and to crucify; and the third day he will be raised up." Reference point in Scripture The coming of the Son of Man is for Jesus a reference point in Scripture. Matthew 10:22 You will be hated by all men for my name's sake, but he who endures to the end will be saved. 10:23 But when they persecute you in this city, flee into the next, for most certainly I tell you, you will not have gone through the cities of Israel, until the Son of Man has come. Remind of what he said beforehand Luke 24:44 He said to them, "This is what I told you, while I was still with you, that all things which are written in the law of Moses, the prophets, and the psalms, concerning me must be fulfilled." 24:45 Then he opened their minds, that they might understand the Scriptures. 24:46 He said to them, "Thus it is written, and thus it was necessary for the Christ to suffer and to rise from the dead the third day, 24:47 and that repentance and remission of sins should be preached in his name to all the nations, beginning at Jerusalem. 24:48 You are witnesses of these things. Requesting of Father, if it is possible Jesus sees the distinction between his desires and God's desires. Matthew 26:39 He went forward a little, fell on his face, and prayed, saying, "My Father, if it is possible, let this cup pass away from me; nevertheless, not what I desire, but what you desire." Role as a Shepherd Matthew 9:36 But when he saw the multitudes, he was moved with compassion for them, because they were harassed and scattered, like sheep without a shepherd. Statements In the gospel of John, Jesus makes eight statements of the form "I am...", namely: I am the good shepherd; I am the light of the world; I am the way, the truth and the life; I am the gate for the sheep; I am the bread of life; I am the true vine; I am the resurrection and the life; I am. Thank you to Adrian Hunter for alerting me to these statements. I have related them to the lines of the Beatitudes, Maslow's hierarchy of need, and operating principles to address those needs. The "I am..." statements apply the operating principles more broadly to transcend each of the needs. How did the "I am..." statements arise in Jesus' mind? Perhaps as he read Scripture he kept thinking, What is that? What does it mean? and sometimes he would answer, I am that! This refers to me. Telling them what they may understand later Luke 18:31 He took the twelve aside, and said to them, "Behold, we are going up to Jerusalem, and all the things that are written through the prophets concerning the Son of Man will be completed. 18:32 For he will be delivered up to the Gentiles, will be mocked, treated shamefully, and spit on. 18:33 They will scourge and kill him. On the third day, he will rise again." 18:34 They understood none of these things. This saying was hidden from them, and they didn't understand the things that were said. What is Jesus lord of? The Sabbath is a day of exceptions, and the Son of Man is, by his slack, his mercy, the one who makes exceptions possible. Sacrifice is just a sign of slack. Thus there is no guilt for those who the Son of Man gives slack to. Matthew 12:7 But if you had known what this means, 'I desire mercy, and not sacrifice,' you would not have condemned the guiltless. 12:8 For the Son of Man is Lord of the Sabbath." What is written in the prophets There will be a culture of the righteous. Isaiah 54:13 All your children shall be taught of Yahweh; and great shall be the peace of your children. 54:14 In righteousness you shall be established: you shall be far from oppression, for you shall not be afraid; and from terror, for it shall not come near you. John 6:45 It is written in the prophets, 'They will all be taught by God.' Therefore everyone who hears from the Father, and has learned, comes to me. 6:46 Not that anyone has seen the Father, except he who is from God. He has seen the Father. Who was he sent to Matthew 15:22 Behold, a Canaanite woman came out from those borders, and cried, saying, "Have mercy on me, Lord, you son of David! My daughter is severely demonized!" 15:23 But he answered her not a word.His disciples came and begged him, saying, "Send her away; for she cries after us." 15:24 But he answered, "I wasn't sent to anyone but the lost sheep of the house of Israel." Why Jesus came Matthew 9:12 When Jesus heard it, he said to them, "Those who are healthy have no need for a physician, but those who are sick do. 9:13 But you go and learn what this means: 'I desire mercy, and not sacrifice,' for I came not to call the righteous, but sinners to repentance." Led by the Spirit Fast and be hungry Matthew 4:1 Then Jesus was led up by the Spirit into the wilderness to be tempted by the devil. 4:2 When he had fasted forty days and forty nights, he was hungry afterward. Go to the wilderness Matthew 4:1 Then Jesus was led up by the Spirit into the wilderness to be tempted by the devil. 4:2 When he had fasted forty days and forty nights, he was hungry afterward. Led by the Spirit into temptation by the devil Matthew 4:1 Then Jesus was led up by the Spirit into the wilderness to be tempted by the devil. The word and the foursome Foursome: The parable of the sower Luke 8:5 "The farmer went out to sow his seed. As he sowed, some fell along the road, and it was trampled under foot, and the birds of the sky devoured it. 8:6 Other seed fell on the rock, and as soon as it grew, it withered away, because it had no moisture. 8:7 Other fell amid the thorns, and the thorns grew with it, and choked it. 8:8 Other fell into the good ground, and grew, and brought forth fruit one hundred times." ... 8:11 Now the parable is this: The seed is the word of God. 8:12 Those along the road are those who hear, then the devil comes, and takes away the word from their heart, that they may not believe and be saved. 8:13 Those on the rock are they who, when they hear, receive the word with joy; but these have no root, who believe for a while, then fall away in time of temptation. 8:14 That which fell among the thorns, these are those who have heard, and as they go on their way they are choked with cares, riches, and pleasures of life, and bring no fruit to maturity. 8:15 That in the good ground, these are such as in an honest and good heart, having heard the word, hold it tightly, and bring forth fruit with patience. Representations of the foursome Matthew and Luke describe three ways that Satan tempts Jesus. They bring to mind for me the levels of the foursome: Whether, What, How, Why. Satan places Jesus at the pinnacle of the temple and encourages him to jump down on the rocks below, on their sensory image, What. Satan leads him up to a high mountain and shows him the kingdoms of the world, all that he will give him; he's using the rock as a How. Satan tempts him to turn the stone into food, which plays with Why and Whether it is a stone. After these temptations, Jesus calls forth his disciples, and he tells Simon that he will be called Peter, which is rock or stone on which he will build his church. One reading is that turning the stone into food was Why, whereas Peter is stone as Whether. Satan tempts Jesus with idealism that favors Why, How, What over Whether, but Jesus rejects it as empty unless it is grounded in Whether. The kingdom-of-heaven is down-to-earth (the salt of the earth) and Peter is, too. When Peter later gets caught up in Jesus as king, then Jesus rebukes him, "Get behind me, Satan!" just as he did to Satan in the desert. Speak and hear by way of the foursome The foursome is how Jesus speaks to several audiences, and how the word reaches people, and their hearts: the level at which the words is received: Whether, what, how or why. Also, the word is what they "find", and these are the four levels of the "finding" in the House of Knowledge. Mark 4:13 He said to them, "Don't you understand this parable? How will you understand all of the parables? 4:14 The farmer sows the word. 4:15 The ones by the road are the ones where the word is sown; and when they have heard, immediately Satan comes, and takes away the word which has been sown in them. 4:16 These in the same way are those who are sown on the rocky places, who, when they have heard the word, immediately receive it with joy. 4:17 They have no root in themselves, but are short-lived. When oppression or persecution arises because of the word, immediately they stumble. 4:18 Others are those who are sown among the thorns. These are those who have heard the word, 4:19 and the cares of this age, and the deceitfulness of riches, and the lusts of other things entering in choke the word, and it becomes unfruitful. 4:20 Those which were sown on the good ground are those who hear the word, and accept it, and bear fruit, some thirty times, some sixty times, and some one hundred times." Tempted by conditionals Barbara Pecker noticed that in Matthew, the devil's temptations of Jesus are all based on conditionals ("If..."): "If you are the Son of God, command that these stones become bread." "If you are the Son of God, throw yourself down, for it is written, 'He will put his angels in charge of you.' and, 'On their hands they will bear you up, so that you don't dash your foot against a stone.'" 4:8 Again, the devil took him to an exceedingly high mountain, and showed him all the kingdoms of the world, and their glory. 4:9 He said to him, "I will give you all of these things, if you will fall down and worship me." |
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Variable: Why is it true?
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Variable: Why is it true? The metalevel and the level may be distinct in the mind, as complements. Given the four levels (why, how, what, whether), the metalevel is associated with the wider point of view (why being the widest) and the level with a narrower point of view. We may think of them concretely in terms of the types of signs: symbol, index, icon, thing. The pairs of four levels are six ways to characterize the relationship. I believe that each way manifests itself through the relationship that we suppose for our variables: dependent vs. independent, known vs. unknown, given vs. arbitrary, fixed vs. varying, concrete vs. abstract, defined vs. undefined, evaluated vs. unevaluated, specialized vs. generalized, domain given or not, determined vs. undetermined (as in the problem of measuring the shortest distance to the river and grandmother's) and so on. I need to study the variety that variables can express. I suppose that, mentally, the varying variables are active in both levels, whereas the fixed variables are taken to be in the level. The levels become apparent when, for example, we draw a picture because that distinguishes the aspects of our problem that our iconic or indexical or symbolic. Likewise, our mental peripheral vision picks up on aspects specific to a particular level. Free variables and Bound variables Wikipedia: In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation that specifies places in an expression where substitution may take place. The idea is related to a placeholder (a symbol that will later be replaced by some literal string), or a wildcard character that stands for an unspecified symbol. The variable x becomes a bound variable, for example, when we write 'For all x, (x + 1)2 = x2 + 2x + 1.' or 'There exists x such that x2 = 2.' In either of these propositions, it does not matter logically whether we use x or some other letter. However, it could be confusing to use the same letter again elsewhere in some compound proposition. That is, free variables become bound, and then in a sense retire from being available as stand-in values for other values in the creation of formulae. Bend the rules Don't let self-imposed, unnecessary restrictions limit your thinking. Whenever you encounter a problem, it is worth spending a minute (or more) asking the question, "Am I imposing rules that I don't need to? Can I change or bend the rules to my advantage?" pg.23, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Draw a picture I imagine that drawing a picture brings out its inner logic at the level of "icon" or "what". Central to the open-minded attitude of a "creative" problem solver is an awareness that problems can and should be reformulated in different ways. Often, just translating something into pictorial form does wonders. pg.59, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Draw pictures In practice, there are several possible methods of showing that a given sequence converges to a limit. ... Draw pictures whenever possible. Pictures rarely supply rigor, but often furnish the key ideas that make an argument both lucid and correct. ... consider the sequence (xn) defined by x0=alpha and x_n+1 = 1/2(x_n + alpha/x_n) ... In the picture below... Notice that the y-coordinate of the midpoint of the line segment AB is the average of these two numbers, which is equal to x_1 ... To show convergence with this picture, we would need to carefully argue why we will never "bounce" away from the convergence point. .... The picture suggests two things: that the sequence decreases monotonically, and that it decreases to square root of alpha. ... The trickiest part in the example above was guessing that the limit was alpha. What if we hadn't been lucky enough to have a nice picture? pg.285-288 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Drawing the monk problem A monk climbs a mountain. He starts at 8 am and reaches the summit at noon. He spends the night on the summit. The next morning, he leaves the summit at 8am and descends by the same route he used the day before, reaching the bottom at noon. Prove that there is a time between 8 am and noon at which the monk was at exactly the same spot on the mountain on both days One solution is to draw the paths on a distance-time graph, which makes it clear that the paths must cross and so they must meet. The pictures brings out the two conditions and shows how they come together. pg.19, 59 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Invent a font The next example combines "Complement PIE" with other ideas, including the useful encoding tool, invent a font, whereby we temporarily "freeze" several symbols together to define a single new symbol. ... Four young couples are sitting in a row. In how many ways can we seat them so that no person sits next to his or her "significant other?" Define Ai to be the set of all seatings for which bi and gi sit together. To compute |Ai|, we have two cases: either bi is sitting to the left of gi or vice versa. For each case, there will be 7! possibilities, since we are permuting 7 symbols: the single symbol bigi (or gibi), plus the 6 other people... Note that alphabetical order in a Spanish language dictionary treats "ch" and "ll" as letters so that "ch" comes after "cz" and "ll" comes after "lz". pg.230 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Loosen up Loosen up by deliberately breaking rules and consciously opening yourself to new ideas (including shamelessly appropriating them!) Don't be afraid to play around, and try not to let failure inhibit you. pg.24, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Peripheral vision One way to heighten your receptiveness to new ideas is to stay "loose", to cultivate a sort of mental peripheral vision. ... Likewise, when you begin a problem solving investigation, you are "in the dark". Gazing directly at things won't help. You need to relax your vision and get ideas from the periphery. Like Polya's mouse, constantly be on the lookout for twists and turns and tricks. Don't get locked into one method. Try to consciously break or bend the rules. pg.20, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Without loss of generality Note the use of the phrase "without loss of generality" in the following problem. The color "white" is chosen arbitrarily, yet its value is fixed. This is one way that variables can be employed. Remove the two diagonally opposite corner squares of a chessboard. Is it possible to tile this shape with thirty-one 2 x 1 "dominos"? ... At first, it seems like a geometric/combinatorial problem with many cases and subcases. But it is really just a question about counting colors. The two corners that were removed wre both (without loss of generality) white, so the shape we are interested in contains 32 black and 30 white squares. Yet any domino, once it is placed, will occupy exactly one black and one white square. The 31 dominos thus require 31 black and 31 white squares, so tiling is impossible. pg. 60 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Create notation You can make progress on a math problem simply by creating a relevant notation for it, which allows you to think about it in a new way, in a new level. Hard and soft constraints Wikipedia: The aim of constraint optimization is to find a solution to the problem whose cost, evaluated as the sum of the cost functions, is maximized or minimized. The regular constraints are called hard constraints, while the cost functions are called soft constraints. |
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Invert - Explicitly Transform
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Invert - Explicitly Transform Design the Box translate features into benefits Flip It open up opportunities Product Pinnochio improve the end user experience The Virtuous Cycle shift focus from process to recurring value Understanding Chain unfold your knowledge from your audience's point of view |
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Knowing why
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Knowing why Knowing Why one knows, scopes. Believing in Believer. Invert my point of view by taking up another's wider point of view. Given a subject such as "helping the homeless", and the four questions Why? How? What? Whether?, then the heart considers a broader question than the world. The world asks, What is helpful? (what makes things better, not worse) but the heart asks Why are we helpful? (because we should). This makes for six types of issues. Define a concept in terms of more fundamental concepts Define a concept in terms of other concepts I wanted to be sure to include and define all of the basic concepts in life. I wanted to do that in terms of the most basic concepts, and ultimately, ground them in the structures that I was discovering. I defined life as "the fact that God is good" and love as "support for life" and also as "the unity of the representations of the structure of God", thus "the unity of wishing", "the unity of the representations of everything". I organized these definitions using TheBrain and then later exported that to an HTML hierarchy. I knew that this kind of definition was, by itself, problematic and so I looked for other ways of defining as well, such as by way of "mind games" as with the topologies. 
Diagram what follows from what Conversing with: assumptions In studying argumentation, I drew diagrams to track which principle builds on which principle. In studying ways of figuring things out, I likewise considered, which way depended on which. Expressing the essence as a relation of concepts *** What are the elements of an experience? Conversing with: concepts I formulated the seven counterquestions as perspectives placed in situations. I recognized the qualities of signs as pairs of levels from the foursome. Similarly, I recognized that I could express the secondary structures as injections of God of one level into a primary structure of another level. In studying verbalization, I realized that the deep issue of a good will exercise could be considered as built up from two concepts which are put together differently by the truth of the heart and the truth of the world. Greimas's semiotic square The Lithuanian semiotician Algirdas Julius Greimas developed the semiotic square, for example: White (all light), Black (no light), Not-Black (=grey), Not-White (=colored). Wikipedia: The Semiotic Square, also known as the Greimas Square, is a tool used in the structural analysis of the relationships between semiotic signs. Greimas considered the Semiotic Square to be the elementary structure of meaning. It is associated with the Aristotelian Square of Opposition, Boole's syllogistic, "the logical hexagon of R. Blanche ... as well as to the structures called, in mathematics, the Klein four-group, and, in psychology, the Piaget group. Identifying regularities *** What are independent factors? Conversing with: regularity In studying narration and Lithuanian folk tales, I noticed that the tone of voice creating tension (forcing, commanding, explaining, caring) was constant in the beginning of the story and constant at the end. The voice receiving the tension was unfolded to involve ever more levels of Maslow's hierarchy, until the climax of the story was reached when the need for self-fulfillment was attacked, and then the character was folded back up again. This was all true in each story. In studying the counterquestions, I expressed each as a pair of a perspective and a situation. 
Inverting conclusions Conversing with: conclusions As I considered the truths of the heart and of the world from some forty goodwill exercises, I noticed that both were related to their topic, but in different ways. Given a deep issue such as "helping", there was a truth of the heart, "I should help those who ask for help", and a truth of the world, "My help should not make things worse", each teaching something about "helping". I thought of the former as "accomodating of helping" and the latter as "helping of reprioritizing", and in general, I noticed that each truth was such a "double gerund". In the truth of the heart, the issue "helping" was qualified by "accomodating", whereas in the truth of the world, "helping" was a qualifier for "reprioritizing". I realized that "helping" could be defined as "accomodating of reprioritizing" and then realized that I could turn this process around. I could work backwards to ask, what are the truths of the heart and of the world about "accomodating"? how does it break down as a pair of gerunds? and work my way backwards to the most basic issue. And I knew the six most basic issues in that the truths of the heart and of the world took up different levels of the foursome, making for six pairs in all. So I had a method for working backwards from real life examples "in the field" to a theoretical foundation for them. Phenomena coincide in that they appear together Conversing with: identity In phonology, semiotics and other structural disciplines, there is a principle that two phenomena are the same if they appear together, so that no change in meaning is ever signified should one stay the same, but the other change. Phenomena coincide in that they never appear together Conversing with: continuity In phonology, semiotics and other structural disciplines, there is a principle that two phenomena are the same if they never appear together. For example, Superman and Clark Kent must be the same person because they are never seen together. Relate scopes Comparing levels of the foursome: Why? How? What? Whether? Conversing with: scope of a statement On a given subject, one way to distinguish the truths of the heart and of the world is that, given four questions - whether, what, how, why - with why being the broadest - the heart asks the broader question. This also leads to six different pairs, thus categories into which the related topic and good will exercise fall into.  Cross check Conversing with: truth I found four different ways to distinguish between the truths of the heart and of the world. Three of the ways corresponded with each other all of the time in the more than 40 good will exercises that I developed. These tests thereby affirmed each other: We can point to the truth of the world with examples from experience, but not the truth of the heart, which must be in us already; the truth of the world follows from the truth of the heart, but not the other way around; and given four questions, whether-what-how-why, why being the broadest, the truth of the heart asks a broader question than the truth of the world. The fourth test is that the person who is riled is wrong, and confuses the truths of the heart and the world, but in some cases they didn't and I had to ammend my conclusions regarding that test. |
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Levels
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Levels Edward Cherlin, 2011.04.05: This idea [of dropping the excluded middle] can be followed into a realm of multiple-valued logics. Buddhist logic considers the possibilities
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Problem Restatement
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How we define a problem usually determines how we analyze it. ... We frequently discover, based on information and perceptions gained midway through the analysis, that the initial problem statement was far off the mark. ... narrow definition of a problem caused their analysis to be shortsighted, overlooking alternative and possibly more beneficial solutions. ... Every problem ... can be viewed from multiple conflicting perspectives. And what drives these differing perspectives? Biases, those unseen killers of objective truth, determine our perspective of any problem. That perspective in turn drives our analysis, our conclusions, and ultimately our recommendations. ... it makes good sense at the outset ... to deliberately strive to identify and examine our biases as they relate to the problem at hand. ... the human mind by design works to conceal the biases that drive our thinking ... introspection is impracticable ... I recommend an indirect approach, which is to restate (redefine) the problem in as many different ways as we can think of. We simply shift our mental gears into a divergent mode ... and start pumping out restatements without evaluating them. The key here ... is letting ideas flow freely, without attempting to justify them. Sometimes restating the problem is difficult because the original statement was poorly articulated. ... One can generally gain most of the benefits of restating a problem in five or ten minutes. ... A problem restatement session will rather quickly, almost magically, focus on the crux of a problem - the core issues - and reveal what the problem is really all about ... if, as often is the case, we are analyzing a problem for someone else's benefit, it is best to generate the problem restatements in that person's presence. Doing so in an open discussion will reveal our consumer's prime concerns and what he or she considers to be the key issues. This will facilitate reaching agreement at the start concerning what the problem is and what our analysis will aim to find out. Two basic approaches: 1) Paraphrase: Reword the problem without shifting its primary focus. 2) Make a 180-Degree Turn: Shift the problem's focus by viewing it from the opposite direction. ... Most important of all, restatements should, whenever possible, be put into writing so we - and our consumer ... can study them. ... the goal of problem restatement is to expand our thinking about the problem, not to solve it. ... A valuable tip when restating problems is to make them simple, positive, and in active voice. The mind works more easily and quickly with simple, positive, active-voice sentences than with complex, negative, passive-voice sentences. Finally, all of the structuring methods presented in this book are visual processes that involve writing or depicting elements of a problem or on a display board or computer screen, where we can see them. Why is seeing them important? By enabling the brain actually to see the words or numbers or depictions of the problem, we engage more brainpower in analyzing and solving the problem and so gain added insights. Indeed, when elements are seen visually, we often discover correlations we missed when we simply thought about them. The old adage - "A picture is worth a thousand words" - speaks to the power of engaging the brain's visual capabilities. Georgia O'Keeffe separated levels in her mind, emotion and perception. ...in [Edmund] Bolles's book A Second Way of Knowing: "Georgia [O'Keeffe] was suddenly struck by the realization that her feelings governed the way she saw the scene. It was a moment of transformation: the entire visual world, she realized, was dependent on the emotional world." Said Bolles: "That day she learned the artist's secret; what you perceive depends on who you are. Analytical thinkers have generally assumed that we perceive reality as it is; they then use a process of abstract reasoning to interpret that perception. O'Keefe realized that the perception is the interpretation. It rests on an internal reality that governs the meaning we find in our sensations." The internal reality Bolles speaks of is, in fact, controlled by the mental traits I've been talking about. ... We view the world through a dense veil of burdensome, thought-warping biases and mind-sets. drawing from The Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich. A matrix is typically used to compare and make explicit the internal structure of records, thus to uncover correlations, rather than to organize or visualize externally. I think the matrix is a way to restate a problem so that we can think of it on two different levels, the whole and the parts. This internal structuring yields a systemic boundary that opens the way for external restructuring. First, we think of as many different explanations as we can... We call that diverging. We then sort the explanations (cluster them) into definable groups, which we'll call categories, for want of a better term. ... We also discern basic explanations ... which I have listed below by category ... Let's examine each category ... [matrix analysis begins] that's what structuring analysis does: it enables us to analyze each element of a problem separately, systematically, and sufficiently. ... We can find out by determining whether there was a correlation ... To analyze this question, I have constructed chronologies using a matrix. ... a matrix is nothing more than a grid with as many cells as needed for whatever problem is being analyzed. ... a matrix enables us ... to separate elements of a problem; categorize information by type; compare one type of information with another; compare pieces of information of the same type; see correlations (patterns) among the information. ... putting the numerical data into matrix form has the effect of isolating the data so they can be analyzed more easily, both separately and in combination. ... is there a correlation ...? [...mapping out relationships...] ... which one of the following circumstances would be best ... possible outcomes ... facilitate discussion of the six possible scenarios (alternative outcomes) to ensure that the pros and cons of each were analyzed separately, systematically, and sufficiently. ... the first thing I do, when confronted with a problem, is to ask myself how I can represent the problem in a matrix. ... when I can portray it in a matrix, the problem immediately opens itself to analysis, like the petals of a flower opening up to reveal its inner parts. Moreover, displaying a matrix like this one on a screen to guide discussion at a meeting can be extremely helpful. ... By providing a visual means of focusing our mind on each option, one at a time, the matrix enables us to easily compare and rank... |
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Humility
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Humility Autonomy substitutes for morality. Why there is value. I remember as a child that our parents used to give us allowance. They made it very clear that the allowance was not for any work that we did. The purpose of the allowance was for us to learn how to spend thoughtfully and manage our money. We also had to help with some basic chores. And we were to do them because we had to, it was independent of our allowance. I took from them that our responsibility to do our work and our need to be thoughtful about money were two separate issues. After my freshman year my father arranged that I work the summer as an intern for his employer, Hughes Aircraft. There wasn't much for me to do there where I was assigned so I took the chance to teach myself programming, namely, Fortran. My supervisor wanted me to create a graphing program for him but I didn't have any real guidance and so I ended up focusing on what I thought was the coolest thing I could do, which was making a program for graphing derivatives of trigonometric functions, which was absolutely useless practically. So I made sure to gain from the experience, but it was a waste for my supervisor and they didn't have me back. My godson in Lithuania became quite interested in alternative music. I was very glad that he could educate me because I always felt doubtful about my own tastes and was glad to have somebody thoughtful to discuss music with. He would buy pirated music as it was the only way to get much of the music, and later, the only way to afford it. I would gladly listen to him play, but I could never make a copy for myself as that seemed a boundary I felt I shouldn't cross. Once in a while I would borrow his disks but I was repulsed by the idea of having pirated music. |
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Other's actions
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Other's actions Noting how others do things Noting the standard Preferring the standard |
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Environmental variables
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Environmental variables Environmental variables are what we change and what we measure, the inputs and outputs. |
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Symmetry breaking
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Symmetry breaking Explicit symmetry breaking Wikipedia: Explicit symmetry breaking indicates a situation where the dynamical equations are not manifestly invariant under the symmetry group considered. This means, in the Lagrangian (Hamiltonian) formulation, that the Lagrangian (Hamiltonian) of the system contains one or more terms explicitly breaking the symmetry. Such terms can have different origins:
Spontaneous symmetry breaking Wikipedia: Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state. For spontaneous symmetry breaking to occur, there must be a system in which there are several equally likely outcomes. The system as a whole is therefore symmetric with respect to these outcomes (if we consider any two outcomes, the probability is the same). However, if the system is sampled (i.e. if the system is actually used or interacted with in any way), a specific outcome must occur. Though we know the system as a whole is symmetric, we also see that it is never encountered with this symmetry, only in one specific state. Because one of the outcomes is always found with probability 1, and the others with probability 0, they are no longer symmetric. Hence, the symmetry is said to be spontaneously broken in that theory. Nevertheless, the fact that each outcome is equally likely is a reflection of the underlying symmetry, which is thus often dubbed "hidden symmetry", and has crucial formal consequences, such as the presence of Nambu-Goldstone bosons. When a theory is symmetric with respect to a symmetry group, but requires that one element of the group is distinct, then spontaneous symmetry breaking has occurred. The theory must not dictate which member is distinct, only that one is. A ball on top of a hill Wikipedia: A common example to help explain this phenomenon is a ball sitting on top of a hill. This ball is in a completely symmetric state. However, its state is unstable: the slightest perturbing force will cause the ball to roll down the hill in some particular direction. At that point, symmetry has been broken, because the direction in which the ball rolled has a visible feature that distinguishes it from all other directions. The "choice" of direction is immaterial, however, as any other direction would do, i.e. the system is still bearing traces of the symmetry of the hill, albeit now somewhat less apparent. Symmetry breaking Wikipedia: Symmetry breaking in physics describes a phenomenon where (infinitesimally) small fluctuations acting on a system which is crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observer unaware of the fluctuations (or "noise"), the choice will appear arbitrary. This process is called symmetry "breaking", because such transitions usually bring the system from a disorderly state into one of two definite states. Since disorder is more symmetric, in the sense that small variations to it don't change its overall appearance, the symmetry gets "broken". Symmetry breaking is supposed to play a major role in pattern formation. Phase transition Wikipedia: A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another. A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium certain properties of the medium change, often discontinuously, as a result of some external condition, such as temperature, pressure, and others. For example, a liquid may become gas upon heating to the boiling point, resulting in an abrupt change in volume. The measurement of the external conditions at which the transformation occurs is termed the phase transition point. The term is most commonly used to describe transitions between solid, liquid and gaseous states of matter, in rare cases including plasma. |
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The Law: You have heard it said..., but I say to you...
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The Law: You have heard it said..., but I say to you... The relationship between life and eternal life is that of "believing" and "not having doubt". The positive commandments are ways that we believe, and the negative commandments, as expressed by the counterquestions, are ways of overcoming doubt, thus having certainty. Life (in this world) is the positive commandments and eternal life (beyond this world) is the negative commandments and so they are connected. These are ways of receiving the word, just as we receive others or Jesus or our reward. In the Sermon on the Mount, Jesus speaks six antitheses regarding God's commandments from the Old Testament. They match up with six counterquestions that I have found for addressing doubts. I expect he counters the truths of the world with the truths of the heart, the narrower perspective with the broader perspective. These are six of the commandments, and along with the four other commandments, this may be thought of as "the prophets and the law". The six are pairs that express the gap between the ways that evil grows clear, the world as opposed to the heart (continuously, feeling, as the little, so the big; distinguishing inside and outside, as the work, so the worker; distinguishing the first and the last, staying within our domain, as others, so us) and the ways of fixing bad relationships, setting the heart right (with ourselves, with others, with God). The gap means that we can fix bad relationships before they are bad, and thus, we can foster good relationships, even without any bad, by being obedient proactively. It also means that we must be ever adaptive to what it means to be good, ever alert to our doubts and ever responsive to change and grow. It is the way in which God creates, but making room, allowing for a gap, giving slack, being flexible, alive, sensitive, responsive.  Jesus's antitheses I'm trying to understand the structure of Jesus's antitheses, "You have heard it said... but I say unto you..."  Ten Commandments I think that the ten commandments give the 6+4 rooms of the House of Knowledge and that Jesus expressed this connection. But I'm not sure exactly what it is. I don't remember, where did I come up with the structure: engage, wait, believe, rely, love, suffer? The will of God is expressed as the commandments, and Jesus' interest not to force understanding, but rather to encourage it.
Receiving Ways of uniting the two branches. One branch is receiving, unity with God, and the other branch is not receiving, difference from God and others. |
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Restructuring
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 10 Tree of variations, 20 Adjacency graph, 21 Total order, 32 Powerset lattice, 31 Decomposition, 30 Directed graph The structures above are graph-like geometries. They are six ways that we visualize structure. We visualize by restructuring a sequence, hierarchy or network. We don't and can't visualize such structures in isolation, but rather, we visualize the restructuring of, for example, a network which becomes too robust so that we may restructure it with a hierarchy of local and global views, which we visualize as an Atlas, or we may restructure it with a sequence, which we visualize as a Tour that walks about the network. Here are the six visualizations, accordingly: ("Hierarchy => Sequence" means "Hierarchy restructured as Sequence", etc.)
Graph The concept of a graph is very simple: merely a finite collection of vertices and edges. ... Just about any situation involving "relationships" between "objects" can be recast as a graph, where the vertices are the "objects" and we join vertices with edges if the corresponding objects are "related". pg.120, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.  Six ways of thinking of multiplication: Fractal, Proportion, Tally, Box, Label, Divide out. (Andrius thinking out loud) I think that the six pairs of levels, six kinds of variables, six Zermelo-Frankel axioms of set theory can be illustrated by models of multiplication as Maria Droujkova has been studying. Question: What does it mean to cancel out units as physicists do?
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Absolute Learning
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Absolute Learning In real life, we address our doubts (surface problems) with counterquestions (deep solutions). I may doubt, How do I know I'm not a robot? and because that has me question all of my experiential knowledge, I can't resolve that by staying in the same level as my problem. Instead, I ask a counterquestion that takes me to my metalevel: Would it make any difference? If there's a difference, then I can check if I'm a robot. If there's not a difference, then it's just semantic and I'm fine with being a robot (by analogy, #3 and #4 may actually be equivalent in some total order). My counterquestion in this case forced you to pin down your variable, like forcing an "arbitrary" epsilon to be fixed so that I could choose my delta accordingly. There are six doubts answered by six counterquestions:
Investigation of 6+4 I'm investigating the conceptual structure of 6+4. I'll be keeping some notes here as I find convenient. I'm wondering if error may play a role in pairing two levels. Directed, cyclic, causal graphs may allow for the study of whether there are errors and how they may be corrected.  Caring about thinking I organized the Minciu Sodas laboratory in terms of ten online spaces to support ten different goals, four of re-caring and six of re-thinking, towards our overall goal of "caring about thinking".  Primary and Secondary Structures I came to realize that the six secondary structures (divisions, representations, topologies, argumentation, verbalization, narration) were given by pairs of levels of the foursome that injected God into primary structures. Symmetric functions I wrote my Ph.D. on the combinatorics of the symmetric functions of the eigenvalues of a matrix. I think it may well be relevant here because it gives the foundations of all matrix combinatorics including walks, cycles, words, Lyndon words, and surely all manner of trees and priorities. The usual symmetric functions may also be relevant. |
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Experiencing
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Experiencing Single Layer Perceptron Sarunas Raudys considers the non-linear Single Layer Perceptron as a process in which the weights of the perceptron are increasing, and the cost function of the sum of squares is changing gradually. He shows that when this single layer perceptron is trained by adaptive optimization techniques, then it is not any single kind of classifier, but rather, in the course of its evolution, it exhibits the behavior of a very rich family of linear classifiers. During the backpropagation training, the decision boundary of a single-layer perceptron become close to or identical to that of seven statistical classifiers:
Andrius: In my understanding, the basic intuition is that in learning we must watch out for over-learning as this keeps us from learning new approaches. For this reason it is important that training always include some noise. My hypothesis is that this yields a hierarchy of learning approaches that accord with the statistical classifiers. Each of them I expect is of increasing sophistication and able to solve the problems of the preceding approaches, although perhaps not as quickly. I imagine that the sophistication is characterized by the number of point of views entertained by the approach so that it functions as a Division Of Everything into one, two, three, four, five, six or seven parts. Basically, I think that a classifier is, in its world, a division of everything. I wish to understand intuitively the statistical approaches so that I might clarify or reject my hypothesis. (See: Sarunas Raudys, Evolution and generalization of a single neurone: I. Single-layer perceptron as seven statistical classifiers", Neural Networks 11 (1998) 283-296)  An example of the 6+4 model is in my paper , see especially the diagram which I got to present to Joseph Goguen's class because I related it to his algebraic semiotics of user interface design. (Yes, he was brilliant! and a great shame that he passed away.) |
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Structuring analysis of problems
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Structuring analysis has us tend to our mind's weaknesses and counter them before they become evident. Each counterquestion thus addresses a different weakness. Perhaps they complement each other in pairs. "What else should I be doing?" is divergent and "Am I able to consider the question?" is convergent. In general, perhaps the competition between convergence and divergence is given by the pairs of levels. Convergence takes us from questions to answers, whereas divergence takes us from answers to questions. Which is the broader view, the view of the heart? I also note that each of the six analytic tools seems to have us analyze the complementary levels. For example, a decision-probability-utility tree is defined by whether-what but has us analyze why and how, whereas a causal flow diagram is based on whether-why but has us analyze what and how, and so on. All of us regularly make mistakes of judgment based on faulty analysis ... While some of these errors can be blamed on a lack of information or education, most occur because of the way our minds work. Our minds frequently mislead us, giving us a false understanding of events and circumstances and causing our analysis of events and circumstances to be flawed ... By learning about the mental barriers and pitfalls that impede effective analysis and acquiring the skills and techniques to overcome them, our batting average can be improved, and significantly so. ... The skills and techniques I speak of are what this book is about: ways of organizing, or structuring, our analysis of problems. This book explains what it means to structure analysis; identifies and describes the mental traits that tend to lead us astray; explains how structuring our analysis of problems defeats the ill effects of these traits; describes fourteen easily understood structuring techniques; and provides exercises through which the reader can begin to master them. ... The word analysis means separating a problem into its constitutent elements. Doing so reduces complex issues to their simplest terms. ... But other than showing, as Bertrand Russel said, whether the elements of a problem are sensible or nonsensical, what does structuring buy us? ...
Bertrand Russell ... in 1901 wrote in his Principles of Mathematics that, with regard to interpreting the language of philosophic theories, the solution is the analytic method - the breaking down of language until a theory shows itself to be either a set of sensible substatements or just nonsense. In this way, he said, many philosophical "problems" just disappear. Russell's book changed the way English philosophy was conducted by establishing the "analytic approach" as the only reputable method of studying philosophical questions. |
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Service
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Service Humility is concrete as six forms of Service that pairs the four levels of Servant and Served. These forms consider what can be done "within the system" (the market, the firm) (whereas Giving and Taking take place "outside the system", among family, friends, neighbors). The mindsets are given by the following questions:
 The Business Equation by John Caswell The diagram above presents my structuring of The Business Equation taught by John Caswell of GroupPartners.net It does seem to me to correlate with the methods from my "money mind" stories. |
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Forces
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Forces Forces are expressions of causality, of the relationship between before and after. They allow for a system to break down into subsystems. Rules are applied in six different ways to link states before and after an event. In math, these are the kinds of subsystems that Implication forms. In life, these are the ways that we visualize. |
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We sin in our hearts
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We sin in our hearts Matthew 5:27 "You have heard that it was said, 'You shall not commit adultery;' 5:28 but I tell you that everyone who gazes at a woman to lust after her has committed adultery with her already in his heart. 5:29 If your right eye causes you to stumble, pluck it out and throw it away from you. For it is more profitable for you that one of your members should perish, than for your whole body to be cast into Gehenna. 5:30 If your right hand causes you to stumble, cut it off, and throw it away from you. For it is more profitable for you that one of your members should perish, than for your whole body to be cast into Gehenna. Empathize with the fearful Matthew 8:25 They came to him, and woke him up, saying, "Save us, Lord! We are dying!" 8:26 He said to them, "Why are you fearful, O you of little faith?" The glory of God: the resurrection and the life The resurrection and the life - DIFFERENCE/UNITY IN HEARING - the transcendence of death through belief in, acceptance of God's perspective through us - all will hear the voice of the Son of God; and those who believe in him will not be judged, but will live and never die; but judgment is given to the Son of Man; and those who do evil will be judged |
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Tree of variations
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Tree of variations Model truth (can distinguish possibilities). Weighted averages, moves in games. 10 Evolution: Hierarchy => Sequence (for determining weights). Examination of cases. Axiom of pairing Wikipedia: If x and y are sets, then there exists a set which contains x and y as elements. This relates to evolution perhaps as a notion of "counting up" or "sorting out". Multiplication, division by 10 moves the number One example of fractal multiplication that comes to mind is multiplication by 10. I noticed this year that the decimal point is not logically positioned. It should be at (under or over) the "ones" place. Then the system would be symmetric. The space to the left would be the "tens" and the space to the right would be the "tenths". A little (video) essay could explain that when we multiply or divide by 10, it is the number that actually moves, not the decimal point. It's like the fact that the Sun is at (or near) the center of the solar system and the Earth revolves around it, not the other way around, as it may seem. The way that the decimal point is placed is very destructive pedagogically, it makes people (like me) think that there is something between the digit; that there is some mysterious difference between the whole numbers and the fractional numbers; and most sadly, that (jaded) teachers don't see that there's something wrong with the system, as I remember thinking as a child; and that there's no reason to care. I suspected that the decimal point is where it is for typographical reasons; maybe because it arose along with printing? or derived from accounting notation? My point being that an adult who appreciates what I've just written (and more along these lines) would appreciate some thing very deep about math (including how to calculate 10% discounts by shifting numbers to the right). Gospel Math. Algorithmic Proof ... the argument below can easily be rewritten as an induction proof. But it is much more instructive to present a new type of argument, an algorithmic proof where we give a general recipe for the construction of an Eulerian path. Consider first a graph with exactly two odd-degree vertices, which we will call s and f. Let us try to naively draw an Eulerian path, starting from s. We will travel randomly ... But this doesn't quite work ... We would be stuck at vertex f, with no way to "backtrack" and traverse the other edges. In this case, let us temporarily remove the edges that we have traveled. We are left with the subgraph ... Since the original graph was connected, the subgraph "intersected" some of the edges that we removed ... Now let us apply the "naive" algorithm to the subgraph ... So now we can perform "reconstructive surgery" on our original path and get an Eulerian path for the entire graph. ... This method will work in general. We may have to repeat the "remove the traveled edges and travel the subgraph" step several times ... but since the graph is finite, eventually we will finish. pg.124-125, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Dividing into cases Sometimes you can reduce the number of pigeonholes by dividing into cases. pg.96, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Examination of cases One of the methods of proof is examination of cases, for example, considering odd and even separately, as in a proof of 1+2+3+...+ n = n(n+1)/2.  Fractal multiplication: Recopying the whole A whole can be recopied (however many copies), then again, then again. This is like fractal multiplication, as with your five-legged starfish whose each leg holds another five-legged starfish. It is like multiplying by powers of 10. The addition rule is at work, adding exponents as in (10**2)(10**3) = 10**5. Multiplying by 10 or dividing by 10 shifts the number with regard to the decimal point, although it looks like the decimal point is moving. We may think of this as simply changing the units, the base unit, which may be unknown. Repeatedly folding paper
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Empathize - Implicitly Understand
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Empathize - Implicitly Understand Empathy Map develop a customer profile Fishbowl observe conversational dynamics Give-and-Take Matrix map out actors' motivations, interactions Pain-Gain Map understand a person's motivations and decisions Show and Tell elicit stakeholders' views Show Me Your Values understand employee's perceptions of what values are involved Stakeholder Analysis identify stakeholders and how to engage them |
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How does it seem to me?
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How does it seem to me? Tracking the whole, introspecting. Imagine how people see things based on their circumstances. Empathetically modeling a particular mind's thinking Contemplating revelation Conversing with: greater than human perspective I studied the Gospel of John to try to decode what he was saying, specifically in his "I am..." statements, but also more generally, for in that gospel he speaks as if in an algebraic code. He keeps defining abstract words in terms of other abstract words, on and on, and I chased them as if they were equations. At the heart of that seemed to be the will of God that we have eternal life. And that perspective helped me appreciate the tension between presuming God to be good or not. And thus I realized that life is the fact that God is good, which conflates God and good as if they were the same, but eternal life is the understanding that God does not have to be good, so that God and good are separate, and there is an eternal life in reconciling God beyond the system and good within the system. Imagine another person's mind Conversing with: mind In tutoring, I would often understand my students by trying to model their mind, and so imagine the obstacles that were keeping them from solving a problem. Imagining a mind's circumstances Conversing with: presumptiveness I was studying emotions and read of a set of six that were observed crossculturally. Mothers would see in their newborn babies six emotions: content, sad, excited, surprised, frightened, disgusted. I realized that I could generate these, in my mind, by imagining myself as a child surrounded by alphabet blocks and trying to guess what letter was underneath. Learn from other's mistakes Conversing with: menaces Noting other people's failings and failures, I took them as confirmation not to use drugs, not to manipulate other people or think of us as different. I appreciated the privileges of my life, my opportunities and my need to apply myself and my virtues. Feeling out my own mind as a mind-in-general Conversing with: mind Conversing with: mind Feel through a specific case Conversing with: characteristicness The truth of the world follows from the truth of the heart, but not the other way around. How one thought extends another thought In studying argumentation, I drew a diagram of how my principles unfolded in organizing my Minciu Sodas laboratory. I then considered the ways in which one thought extends another thought, especially on that part Z given by God. I related them to the twelve topologies.
Introspection Conversing with: limits of my mind By introspection I sensed the wholeness of various divisions of everything and how a division's perspectives fit together. Structural aesthetics Conversing with: internal imagination I am sometimes informed by my own personal sense of what is attractive structurally. I sense that the foursome, the division of everything into four perspectives, is structured to favor idealism over materialism, and the human over the divine, so that How precedes What, Why precedes Whether, and the former shift precedes the latter shift. I've never quite confirmed that, but it just seems to my moral sensitivity the way those outlooks should fit together. Similarly, I understand that good and bad are opposites, but I generally don't think of them as equals, for example, thinking of good as refering to God beyond the system. Good may not be able to stand on its own, but there is a sense in which it doesn't need bad. Tracking my thinking Conversing with: introspection In watching my own mind and noting how I look at different cases I see that the direction of the twosome changes upon reflection. Our mind shifts readily from "opposites coexist" to "all things are the same" and not the other way around. We shift from same to different because "same" implies opposites (for things to be the same they must also be different) whereas "different" doesn't (different things are just different). But if we think about the words "different" and "same", and thereby distance ourselves from what they mean to our minds, as we so often do, then it seems that different involves opposites ("opposites coexist"), and same does not ("all things are the same"). |
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Onesome: What-Whether
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Onesome: What-Whether Euclidean Distance Classifier Suppose that at the beginning of training the weights are small and the activation function acts as the linear function. (See: Sarunas Raudys, Evolution and generalization of a single neurone: I. Single-layer perceptron as seven statistical classifiers", Neural Networks 11 (1998) 283-296) Serving people of a different background My patients' lives were a window into a slice of American life I had never known — sharecroppers, wooden shacks on dusty backroads, back-breaking cotton picking for pennies a pound. Towns whose names littered civil rights history — Philadelphia, McComb, Indianola, Yahoo City, Little Rock, Montgomery, Birmingham. Life under Jim Crow. "Yes, suh. No, suh." The humiliation of survival in places where being black meant no chance for justice. The Illinois Central ride to Chicago. The promise of jobs. The disappointment of segregation and the urban violence that greeted them. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.1934 What is most enjoyable for viewers Fresh Air interview of J.J. Abrams: it's literally wanting people to have a good time and to have a little bit of a surprising time. So whenever I'm trying to keep things quiet, it is 100 percent an effort to make the experience of actually seeing the movie or TV show more enjoyable for the viewers." Who had an impact on my growth? I sometimes measure my life progress by thinking of the people who have had an impact on my growth as a human being. My parents, my wife, my children, my friends and colleagues. I number my patients among them. Health. Chicago. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. |
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The Decision/Event Tree
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 A decision/event tree is a diagram that graphically shows choices and their outcomes at different junctures in alternative sequences or chains of events. Each sequence or chain of events is a separate scenario.
the most difficult analytic problems are of the random and indeterminate type, where, because facts are scarce, our analytic product depends heavily on subjective judgments. For that reason, there can be no certainty when dealing with random or indeterminate problems. ... Estimating is what we do when we run out of data. ... The two types of probability we encounter most often in analysis are mutually exclusive (the "or" type, where we add probabilities) and conditionally dependent (the "and" type, where we multiply them). ... There are three inviolable rules for constructing probability trees:
 utility is the benefit that someone has received, is receiving, or expects to receive from some situation. ... It is the reason why that person has taken, is taking, or will take a certain action. ... humans, when confronted with a choice between alternative courses of action (alternative options), choose the course that offers them the greatest utility, and the person making the choice defines what that utility is. ... we call it self-interest. ... The purpose of utility analysis is to rank any number of options according to how they serve the decision maker's self-interest.
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Appraise
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Appraise How does it seem to me? Appraise. The calm Servant provides the motivated Served with an appraisal, judgment, price. This is a microeconomic perspective. When I was in junior high school I used to spend my allowance on Mad magazine. It seemed at first like very intelligent, fresh, stimulating humor and I very much looked forward to it. But over time the formula became apparent. And after a few years I became disappointed and finally lost interest and stopped buying it. I went through the same cycle with Newsweek and other publications, where what originally seemed fresh turned out to be an editorial formula that grew stale with familiarity. I have a godson in Lithuania and I would give him coins I collected in my travels, but his father would take them away and say that he would keep them until he got older, so he wouldn't lose them. In time I caught on that he was too keen on the coins himself to ever give them to his son so I resigned myself, but I always bring him some from my travels. I always knew that the main aim of the collection was to educate myself, but it was a bit sad that this education was not relevant to anybody else, at least not directly. Internalize another person's reflections Conversing with: inner contradictions In China, I was visiting a family, and a young doctor told me how he grappled with the moral challenges of bribes and gifts given to doctors, explaining that he would only take gifts after he had treated the patients, and yet they would think ill of him because he would accept their bribes and so they thought he wasn't giving them the best service. As I listened to his story, I thought about my own work as a software developer, how it most likely wasn't creating any good in society, but was simply the result of a business opportunity that arose through arbitrary quirks of government regulation. I realized that I couldn't let myself make a living from work that did not add positively to society. I sold my car. It was a practical car, well kept and I tried to get the blue book price. Somebody wanted to buy it but they were hoping I would go down and I refused to. I thought those prices were accurate. So instead I ultimately sold it for quite a bit lower than that, to a friend. |
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Empathize
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Empathize Assuming intersubjectivity Caring what loved ones believe Considering how others may see things Imagine a listener Imagining what others may think What others may like Your state of mind |
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Reconcile quickly
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Reconcile quickly Matthew 5:21 "You have heard that it was said to the ancient ones, 'You shall not murder;' and 'Whoever shall murder shall be in danger of the judgment.' 5:22 But I tell you, that everyone who is angry with his brother without a cause shall be in danger of the judgment; and whoever shall say to his brother, 'Raca!' shall be in danger of the council; and whoever shall say, 'You fool!' shall be in danger of the fire of Gehenna. 5:23 "If therefore you are offering your gift at the altar, and there remember that your brother has anything against you, 5:24 leave your gift there before the altar, and go your way. First be reconciled to your brother, and then come and offer your gift. 5:25 Agree with your adversary quickly, while you are with him in the way; lest perhaps the prosecutor deliver you to the judge, and the judge deliver you to the officer, and you be cast into prison. 5:26 Most certainly I tell you, you shall by no means get out of there, until you have paid the last penny. Distinguish between what they say and what they do Matthew 23:2 ... "The scribes and the Pharisees sat on Moses' seat. 23:3 All things therefore whatever they tell you to observe, observe and do, but don't do their works; for they say, and don't do. God's intent: The gate for the sheep The gate for the sheep - DIFFERENCE/UNITY IN LISTENING - Jesus is the entry to the culture of those who live in belief, by which they can go in and out of their culture What should we seek first? Matthew 6:31 "Therefore don't be anxious, saying, 'What will we eat?', 'What will we drink?' or, 'With what will we be clothed?' 6:32 For the Gentiles seek after all these things; for your heavenly Father knows that you need all these things. 6:33 But seek first God's Kingdom, and his righteousness; and all these things will be given to you as well. What should you do first? Matthew 7:3 Why do you see the speck that is in your brother's eye, but don't consider the beam that is in your own eye? 7:4 Or how will you tell your brother, 'Let me remove the speck from your eye;' and behold, the beam is in your own eye? 7:5 You hypocrite! First remove the beam out of your own eye, and then you can see clearly to remove the speck out of your brother's eye. |
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Adjacency graph
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Adjacency graph Imply truth (can determine connectedness) (Connectedness, coloring, triangulation of polygon) 20 Atlas: Network => Hierarchy (for determining connections) All parabolas have the same shape I've found in teaching the quadratic equations (and searching for a use for them) that the parabola is arguably an ideal curve for learning to graph. That's because all parabolas have one and the same shape, if you discount zooming in and out. Each parabola, if you zoom in, will look flat, and if you zoom out, will look narrow, in exactly the same proportions. You can see this if you substitute x -> ax and y->ay and thereby you can transform x = y**2 to xa = y**2 a**2 so x = a y**2 effectively where a can be as you like. This means that all parabolas look the same and their graphs differ only in how you move them around, up and down, left or right, negative or positive, zoom in or zoom out. Also, I teach my students to draw too graphs because most never realize how the parabola completely flattens out at the bottom where -1 < x < 1. It's a bit like filming a movie where you have to combine full length shots of people with head shots. One shot won't do it. Gospel Math. Axiom of extensionality Wikipedia: Two sets are equal (are the same set) if they have the same elements. Note that there are thus two levels of equality. Equality is a bidirectional relationship. And the two levels are like levels of an atlas. An atlas defines, at different levels, what can be consider the "same" point or location from far away, though may be different from close up. The "equivalence" and equivalence classes may be "turned on or off" selectively in the atlas. Coloring The use of coloring is related to parity and modular arithmetic, except that we are not constrained by the algebraic properties of integers. pg.111, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Divisibility Given two natural numbers a, b, their greatest common factor (a,b) ... is defined to be the largest integer which divides both a and b. ... If the GCD of two numbers is 1, we say that the two numbers are relatively prime. ... If g divides a and g divides b, then g divides ax + by, where x and y can be any integers ... An important consequence of the division algorithm, plus [the last fact], is that The greatest common divisor of a and b is the smallest positive linear combination of a and b. ... a great showcase for the use of the extreme principle plus argument by contradiction. Define u to be the smallest positive linear combination and let g=(a,b) ... certainly g divides u ... suppose that u does not divide a. Then by the division algorithm, there exists a quotient k>=1 and positive remainder r < u such that a = ku+r. But then r = a-ku is positive and less than u. This is a contradiction, because r is also a linear combination of a and b ... Consequently, u divides a, and likewise u divides b. So u is a common divisor; thus u=g. ... This linear combination characterization of the GCD is really quite remarkable, for at first one would think that PPF's are needed to compute the GCD of two numbers. But in fact, computing the GCD does not rely on PPF's ... but we can use [instead the fact that if there exists x, y such that ax + by = 1, then a and b are relatively prime]. ... we do not need to assume the truth of the FTA in order to compute the GCD. The GCD is the grid size that results on a number line from taking steps back and forth of size a and b. We are thus thinking not in terms of primes as components, but of grids (the numbers they divide) that can be included or not in each other, and are thus organized by a lattice of conditions (of which points can be reached or not), where satisfying all conditions means including all points and being relatively prime and writing a ratio in reduced form. pg.245 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Graph Just about any situation involving "relationships" and "objects" can be recast as a graph, where the vertices are the "objects" and we join vertices with edges if the corresponding objects are "related". pg.120, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Handshake Lemma In any graph, the sum of the degrees of all the vertices is equal to twice the number of edges. pg.121, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Sleeping mathematicians During a certain lecture, each of five mathematicians fell asleep exactly twice. For each pair of these mathematicians, there was some moment when both were sleeping simultaneously. Prove that, at some moment, some three of them were sleeping simultaneously. pg.120, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Least Common Multiple ...we define the least common multiple, or LCM, of a and b to be the least positive integer which is a multiple of both a and b. pg.245 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc  Proportion multiplication: Rescaling the whole A whole can be rescaled. This is proportion, as with your teddy bear projected on a screen. The rescalings are actions that can be composed, magnifying and shrinking. They are all multiplying against some unknown, acting upon it, yielding actions x actions x (object with units). They can be reorganized and canceled away. I sometimes talk to my students about "magnifying drops" (each drop multiplying by 10) and "shrinking drops" (each drop dividing by 10) and ask what happens when we add one drop after another drop. Also, actions can be decomposed into component actions, into primes. I relate this to the adjacency graph and the Atlas view because it consists of a hierarchy of global and local views upon a network, thus the same relation can appear at different scales. Bicycle gearing Bicycle gears are given by the formula: (number of front teeth / number of back teeth) x circumference of wheel = distance traveled per revolution. That's an example of the proportional scaling ("rescaling the whole", like with projecting a teddy bear ). You could imagine starting out with two gears (the gear fixed to the wheel, and the gear you are pedaling) each the size of the back wheel. Replacing those giant gears with smaller gears is an action of magnifying or shrinking the distance traveled. You could add more gears to further magnify or shrink that ratio. Adding teeth to the gears is changing the units (and that aspect could be considered "rescaling the multiple", as with skip counting, but that is the correspondence of the number of teeth to the circumference of the gear, NOT the relationship between the two gears). The fact that multiplication is taking place at two different levels makes it challenging to think about because the gear ratio is "abstract" and not concretely, directly related to the size of the wheel. The Two Men of Tibet Two men are located at opposite ends of a mountain range, at the same elevation. If the mountain range never drops below this starting elevation, is it possible for the two men to walk along the mountain range and reach each other's starting place while always staying at the same elevation? ... As long as it is legal to walk backward, it is pretty easy... But why does it work? ... define a graph G whose vertices are all ordered pairs (x,y) where x,y [are the "interesting" places] and x and y are at the same elevation. ... the vertices of G consist of all possible legal configurations of where the two dots could be ...we shall join two vertices by an edge if it is possible to travel between the two configurations in one "step" ... if we can show that there is a path from (a,s) to (s,a) we'd be done. ... by the handshake lemma, the sum of the degrees of the vertices of this subgraph must be even. Since the only two vertices with odd degree are (a,s) and (s,a), this subraph must contain (s,a) as well. ... we solved this hard problem with a very simple parity analysis. Of course, we first needed the insight of constructing a graph, and the crux move of defining the vertices and edges in a very clever way. pg.126-128, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Triangulating, then coloring The walls of a museum gallery form a polygon with n sides, not necessarily regular or even convex. Guards are placed at fixed locations inside the gallery. Assuming that guards can turn their heads, but do not walk around, what is the minimum number of guards needed to assure that every inch of wall can be observed? ... A coloring reformulation comes to the rescue: Triangulate the gallery polygon. pg. 43, 61 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. |
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Get to Know - Implicitly Care
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Get to Know - Implicitly Care Ice Breaker introduce people as creative spirits Low-Tech Social Network get to know each other Trading Cards embrace each other as self-defined |
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What else should I be doing?
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What else should I be doing? Discerning the whole, choosing a data set. Engage a dataset with a full variety of roles. Deriving a structure from a complete variety of examples Conversing with: phenomenon  Allowing for inconsistency I considered the variety of prayer. I noticed that they have us think of God inconsistently, as one who has fated everything in advance, one who fixes and manages the situation, and one who can guide us spontaneously. Building my own collection of examples I reflect on my own activity and study it. I developed and conducted about 40 good will exercises. As I collected them, I studied how they fell into groups, and analyzed the nature of the system as a whole. Similarly, I am studying the ways of figuring things out and organizing them in a House of Knowledge which turned out to have 24 rooms. Choose a fruitful data set Conversing with: circumstances for a phenomenon In developing a theory, I often choose a dataset of examples that makes vivid the phenomenon I am looking for and represents the complete variety of possibility. I analyzed Jesus' feelings and expectations by studying episodes from the Gospel of Mark because it was the most emotional of the Gospel. I studied the content of Jesus' parables in the Gospel of Luke. I studied what is "good" in the Gospel of Matthew. I tried to chase down Jesus' algebra of expressions to understand his "I am..." statements in the Gospel of John. I studied Lithuanian folk tales for my narrative theory because they are old, engaging, consistent and pure. I studied how God is imagined in the first 40 psalms because I was considering topologies as the variety of ways that we can imagine and I thought of God as the most generic yet intimate of concepts. I studied Lakoff and Johnson's target spaces from Metaphors We Live By because they had noted an important phenomenon and their many examples seemed to capture the variety of possibility. Collect ways of looking at something Conversing with: variety inherent in a phenomenon I developed and conducted good will exercises and noted different tests for distinguishing the truth of the heart and the truth of the world. In general, three of the tests agreed with each other. I think the fourth and most important test, whether the person confuses the truths of the heart and the world, indicates whether the person has resolved the conflict or not. Similarly, I noted the different properties of everything, and more generally, the divisions of everything along with their various representations. Compare structural qualities I noticed that the foursome, fivesome, sixsome each involve structural shifts but they are of a different quality. This was helpful in considering how the primary structures vary.  Survey all of the structures that I know I made a survey of all of the conceptual structures that I had found and was familiar with. I was able to put them together in a system 1 + 6 + 4 + 1 that is much like half of the House of Knowledge. Recognize that I have nothing better to do Conversing with: causes Why do I want to know everything? Why am I organizing the kingdom of heaven? The most relevant answer that I have found is simply that I have nothing better to do. Manifesting the complete variety that a structure allows for Conversing with: wholeness Leveraging hypotheses of structural symmetry I often make large and fruitful leaps by considering the structures that operate at the highest meta level, assuming that they work as efficiently, elegantly, effectively as possible, thus supposing that they make profound use of symmetry. In 1988, in noting the role of the threesome in the eightfold way, I looked for and found three variants (Lord's prayer, St.Peter's Keys to Heaven, Beatitudes). I expected that each served as a framework for a language and that there were, accordingly, three languages: argumentation, verbalization and narration. I saw that Maslow's hierarchy was a backbone for the Beatitudes and I expected it to play a key role in narration, which I found to be the case. I discovered that there were eight divisions, six representations and twelve topologies, and that narration moved us from the divisions to the representations. I also saw that these structures along with argumentation and verbalization could be considered as negations of the representations of the onesome (no internal structure, simplest algorithm, no external context) and negations of the representations of the nullsome (significant, constant, direct) and so these six structures were themselves a spine for the eightfold way. This allowed me to deduce by symmetry that argumentation moves us from topologies to representations, and verbalization moves us from topologies to divisions. Such thinking helps me be alert as to what to look for. Applying structural possibilities *** What can structure mean? Conversing with: unfolding I studied the statements and truths that arose in developing good will exercises. I noted that given the four questions Why? How? What? Whether?, the heart would address a broader question than the world, with Why being the broadest question. This allowed me to categorize the exercises in terms of the pairs of levels, and indeed, they related to the counterquestions. This left in each case a pair of levels not involved in the counterquestion. I realized that they were relevant to the doubts which the counterquestions addressed. And the order in which they were taken up determined whether they led to positive or negative feelings. Completing a structure by comparing it to a known template The divisions of everything help me to recognize and complete many structures. For example, Peirce's three kinds of signs (symbol, index, icon) match with three of the four levels of the foursome. I have noticed that very often, philosophers give only three of the four levels because if they are working from a materialist representation, then Why is irrelevant, and if from an idealist representation, then Whether is irrelevant. Thus Why and Whether are null perspectives for the relevant representations. They are the baselines which the others are thought in terms of.  Completing a structure by permuting it Conversing with: missing I can complete a structure by noticing the available permutations. I noticed that the virtues in Plato's republic (beauty, bravery, self-control, justice) were the four levels of the foursome colored by the threesome (universal, individual, communal, communal). The virtues in St.Paul's hymn to love (love, hope, faith, loyalty) were likewise the four levels of the foursome colored by the threesome (individual, communal, universal, universal). They had a general form (emotion, virtue, internal perspective, external perspective) and a logic (internalizing the external perspective as an internal perspective fixes the emotion permanently as a virtue). So I deduced a third set of virtues (communal, universal, individual, individual). I searched them out in my understanding of life and found what I was familiar with, honesty. I permuted the emotions and virtues which I was familiar with from my understanding of the division of everything into six perspectives. I realized that was one representation and that the other representation of the sixsome was in terms of internal and external perspectives. Permuting I realized that feelings of closeness or intimacy where immortalized as honesty when we internalize duty as caring. Thus by permuting I noted three expressions of the "meaning of life". Similarly, in studying Buddha's eightfold way and Jesus' prayer "Our Father", I noticed that they were permutations of the same eightfold structure, and a third permutation was the Beatitudes of the Sermon on the Mount. Completing a structure with an eighth perspective *** What is the bigger picture? I am familiar with many structures that have seven-eight perspectives. The seven perspectives form a complete system, yet there is an eighth perspective that looks from beyond the system. Thus I conjectured that there is an eight narrative that takes us from Caring to Commanding, and thus is a story of Creation. Likewise, I conjectured that there must be an eighth counterquestion that stands aside from any doubt, namely, What do I truly want? Look for what's missing Conversing with: incompleteness In childhood I often chose to study whatever I knew the least about. In sketching out a system for self-education, as in mathematics, I keep thinking, what to add? |
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Twosome: How-Whether
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Twosome: How-Whether Regularized Linear Discriminant Analysis Starting with the Euclidean Distance Classifier, assume that tj(1) = 1 and tj(2) = -1. Analyze a change in the weight vector after the second and following iterations. This yields the weight vector resulting from regularized linear discriminant analysis. (See: Sarunas Raudys, Evolution and generalization of a single neurone: I. Single-layer perceptron as seven statistical classifiers", Neural Networks 11 (1998) 283-296) Imagining a different conceptual geometry Jerry Michalski of The REXpedition thought of the growth of knowledge as expanding beyond what a single person (like Leibniz) could know, expanding as if in a radial plane where various disciplines (math, history,...) carved out and walled off their own portions. Then one day he reimagined that plane as a sphere where each person was a point on that sphere with tangents heading off in different directions, but the points on that sphere may be starting to converge, so that people of different jargons are saying similar things.  Dave Gray reimagines Jerry Michalski's geometry of knowledge. He draws it as a basket. The idea here is about the advance of knowledge. The spines are the various disciplines which must necessarily diverge as we attempt to understand a universe where the unknown always exceeds the known. ... The horizontal weaving strands are the threads we weave as we attempt to connect the various disciplines. ... Without the weaving the spines just lay flat on the ground and radiate out. And without the spines the basket is just a coiled garden hose on the ground. ... Over time it holds more and more meaning, life, knowledge and goodness. It's a cornucopia! :) |
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Divergent Thinking
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Most of us were not endowed with an inherent ability to think divergently. ... Why is opening up such a struggle? ... we humans are the fastest criticizers and ridiculers on the planet ... Negative thoughts flood our minds and overwhelm those tiny rivulets of positive notions. The foremost challenge in analyzing problems is to think divergently, not just at crucial points in our analysis but at every stage of the process. ... I regard as unnecessary overkill the wild, frenzied brainstorming methods that some people advocate ... Nevertheless, if blitzkrieg methods suit your purpose, use them. ... I have witnessed many a brainstorming session in which intelligent people go through the motions of generating ideas but treat the entire process as kind of a game. ... To analyze a problem effectively, we need to be open to ideas, at any time and from any source, from the very outset to the very end of the analytic process. ... 1) The more ideas, the better. ... 2) Build one idea upon another. The power of brainstorming lies in the freedom and spontaneity of the process: one idea spawns another and another and another in rapid succession. Participants are encouraged to say aloud that the idea one is suggesting "builds on" someone else's idea. This statement, which rewards and flatters that someone else and reaffirms the connectivity of ideas, adds momentum to the process. ... 3) Wacky ideas are okay. ... open the way to new, practical ideas ... incite humor, which relaxes our paranoid grip on the mind's generative capabilities and liberates more ideas. This commandment, however, bothers most people ... What troubles people is not that an idea is silly or foolish but that they will feel and look silly and foolish suggesting it. ... 4) Don't evaluate ideas. Neither yours nor especially someone else's. This rule liberates people from their self-imposed restraints in generating ideas. ... In brainstorming, the practicality of ideas is irrelevant. ... When we've finished diverging (exhausted our reservoir of ideas), the next step is to focus by winnowing out the impractical stuff and clustering ideas that are similar. We are now in a convergent mode. [In his example, he came up with six clusters and a seventh cluster of impractical ideas. I think I could structure his six clusters.] ... In the third step, we select those ideas that are intuitively practical and promising. ... a further useful step one can take is to perform PROs-CONs-and-FIXes on each of the selected ideas to identify their strengths and weakness and to evaluate their feasibility. This step also provides an opportunity to revise the ideas, enhancing and fine-tuning them. ... Frequently, when I'm discussing a problem with someone, I'll switch to a divergent mode. But I know that if I don't tell the person I'm switching, he or she will get upset, thinking I'm either off on some wild tangent or that I'm really serious about some idea I've raised for discussion when, in fact, I'm only testing the analytic waters. So I always tell people when I'm brainstorming ... |
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Audit
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Audit What else should I be doing? Audit. The calm Servant provides the rich Served with a budget, alternatives, substitutions. When I was fifteen I visited Lithuania for a summer and I would go to the dollar store and the lady there would exchange my US money for coins from all sorts of countries, I was very excited about that. I even brought her a flower. Later it seemed kind of ridiculous how I had cared so much about those coins. |
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Play our part
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Play our part Learn roles Considering who is responsible Emulating Learning by teaching Learning each role Learning every role Mastering a language Our Duties Considering our duties Considering what we need worry about Take charge Taking up a responsibility |
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Be perfect.
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Be perfect. Matthew 5:43 "You have heard that it was said, 'You shall love your neighbor, and hate your enemy.' 5:44 But I tell you, love your enemies, bless those who curse you, do good to those who hate you, and pray for those who mistreat you and persecute you, 5:45 that you may be children of your Father who is in heaven. For he makes his sun to rise on the evil and the good, and sends rain on the just and the unjust. 5:46 For if you love those who love you, what reward do you have? Don't even the tax collectors do the same? 5:47 If you only greet your friends, what more do you do than others? Don't even the tax collectors do the same? 5:48 Therefore you shall be perfect, just as your Father in heaven is perfect. Be merciful as your Father is merciful Luke 6:36 Therefore be merciful, even as your Father is also merciful. 6:37 Don't judge, and you won't be judged. Don't condemn, and you won't be condemned. Set free, and you will be set free. Distinguish the living and the dead based on their priorities Matthew 8:21 Another of his disciples said to him, "Lord, allow me first to go and bury my father." 8:22 But Jesus said to him, "Follow me, and leave the dead to bury their own dead." God is kind toward the unthankful and evil Luke 6:27 "But I tell you who hear: love your enemies, do good to those who hate you, 6:28 bless those who curse you, and pray for those who mistreat you. 6:29 To him who strikes you on the cheek, offer also the other; and from him who takes away your cloak, don't withhold your coat also. 6:30 Give to everyone who asks you, and don't ask him who takes away your goods to give them back again. 6:31 "As you would like people to do to you, do exactly so to them. 6:32 If you love those who love you, what credit is that to you? For even sinners love those who love them. 6:33 If you do good to those who do good to you, what credit is that to you? For even sinners do the same. 6:34 If you lend to those from whom you hope to receive, what credit is that to you? Even sinners lend to sinners, to receive back as much. 6:35 But love your enemies, and do good, and lend, expecting nothing back; and your reward will be great, and you will be children of the Most High; for he is kind toward the unthankful and evil. God's example: the way, the truth and the life The way, the truth and the life -- DIFFERENCE/UNITY IN DOING -- Jesus is the way that links the life (heaven to earth) and the truth (from earth to heaven), thus people know the Father by knowing the Son, and do their works (of life) by asking the Son and keeping his commandments, and see and know and receive as counselor the Spirit of Truth, which the world doesn't see or know or receive How to be perfect Matthew 19:20 The young man said to him, "All these things I have observed from my youth. What do I still lack?" 19:21 Jesus said to him, "If you want to be perfect, go, sell what you have, and give to the poor, and you will have treasure in heaven; and come, follow me." 19:22 But when the young man heard the saying, he went away sad, for he was one who had great possessions. 19:23 Jesus said to his disciples, "Most certainly I say to you, a rich man will enter into the Kingdom of Heaven with difficulty. 19:24 Again I tell you, it is easier for a camel to go through a needle's eye, than for a rich man to enter into the Kingdom of God." Should you judge? Matthew 7:1 "Don't judge, so that you won't be judged. 7:2 For with whatever judgment you judge, you will be judged; and with whatever measure you measure, it will be measured to you. Touch and feel Luke 24:38 He said to them, "Why are you troubled? Why do doubts arise in your hearts? 24:39 See my hands and my feet, that it is truly me. Touch me and see, for a spirit doesn't have flesh and bones, as you see that I have." 24:40 When he had said this, he showed them his hands and his feet. 24:41 While they still didn't believe for joy, and wondered, he said to them, "Do you have anything here to eat?" 24:42 They gave him a piece of a broiled fish and some honeycomb. 24:43 He took them, and ate in front of them. What distinguishes the Gentiles and God? Matthew 6:31 "Therefore don't be anxious, saying, 'What will we eat?', 'What will we drink?' or, 'With what will we be clothed?' 6:32 For the Gentiles seek after all these things; for your heavenly Father knows that you need all these things. What is your reward? Matthew 5:46 For if you love those who love you, what reward do you have? Don't even the tax collectors do the same? 5:47 If you only greet your friends, what more do you do than others? Don't even the tax collectors do the same? 5:48 Therefore you shall be perfect, just as your Father in heaven is perfect. Which is the way that leads to life? The gate of "belief" leads to life; the gate of "caring" leads to destruction. We can distinguish the gates by seeing who goes through them. Matthew 7:13 "Enter in by the narrow gate; for wide is the gate and broad is the way that leads to destruction, and many are those who enter in by it. 7:14 How narrow is the gate, and restricted is the way that leads to life! Few are those who find it. Who does one love more Matthew 10:37 He who loves father or mother more than me is not worthy of me; and he who loves son or daughter more than me isn't worthy of me. 10:38 He who doesn't take his cross and follow after me, isn't worthy of me. Who will find it more tolerable Matthew 11:20 Then he began to denounce the cities in which most of his mighty works had been done, because they didn't repent. 11:21 "Woe to you, Chorazin! Woe to you, Bethsaida! For if the mighty works had been done in Tyre and Sidon which were done in you, they would have repented long ago in sackcloth and ashes. 11:22 But I tell you, it will be more tolerable for Tyre and Sidon on the day of judgment than for you. 11:23 You, Capernaum, who are exalted to heaven, you will go down to Hades. For if the mighty works had been done in Sodom which were done in you, it would have remained until this day. 11:24 But I tell you that it will be more tolerable for the land of Sodom, on the day of judgment, than for you." Why should we be anxious or not? Matthew 6:34 Therefore don't be anxious for tomorrow, for tomorrow will be anxious for itself. Each day's own evil is sufficient. |
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Total order
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Total order Imply model (can order procedures) (Strong induction, decision making, total ranking, integers) 21 Canon: Sequence => Network (for determining priorities) Well-ordering theorem Wikipedia: Axiom 9: For any set X, there is a binary relation R which well-orders X. This means R is a linear order on X such that every nonempty subset of X has a member which is minimal under R. ... Given Zermelo-Fraenkel axioms 1-8, there are many statements provably equivalent to axiom 9, the best known of which is the axiom of choice (AC), which goes as follows. Let X be a set whose members are all non-empty. Then there exists a function f from X to the union of the members of X, called a "choice function", such that for all Y element of X one has f(Y) element of Y. Since the existence of a choice function when X is a finite set is easily proved from axioms 1-8, AC only matters for certain infinite sets. AC is characterized as nonconstructive because it asserts the existence of a choice set but says nothing about how the choice set is to be "constructed." The Well-ordering theorem seems tightly related to the concept of a total order. It's interesting that it's related to the axiom of choice. If you can count it, it's an integer Show that the product of k consecutive integers is divisible by k! ... simply observe that m(m+1)...(m+k-1)/k! = (m+k-1 k) and binomial coefficients are integers! The moral of the story: Keep your point of view flexible. Anything involving integers is fair game for combinatorial reasoning. pg.273 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Permutations Permutations are the ways of reordering a collection of objects, all distinct. A permutation labels a total order. Strong induction Strong induction gets its name because we use a "stronger" inductive hypothesis. After establishing the base case, we assume that for some n, all of the following are true: P(n0), P(n0+1), P(n0+2), ... , P(n), and we use this assumption to prove that P(n+1) is true. Sometimes strong induction will work where standard induction doesn't. ... Behind the idea of strong induction is the notion that one should stay flexible when it comes to defining hypotheses and conclusions. pg.52, 54, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.  Tally multiplication: Rescaling the multiple A multiple can be rescaled. This is like skip counting or repeated addition. Note that here the numbers added are cardinals, which is to say, we don't care in each subgroup what order they had, it's not relevant, we're simply adding up the sums. This is a recounting, a shift from larger units to smaller units. 3 x (23 x dollars) becomes (3 x 23) x dollars. Amount x (large unit) becomes action x (amount x small unit) becomes (action x amount) x small unit becomes amount x small unit. We can count coins by grouping together the pennies, nickels, dimes, quarter, placing them in rows or groups of 4 or 5 or 10. Number x (Value x Unit) => (Number x Value) x Unit. Cutting a stack of cheese slices If I have a stack of 10 slices of cheese and I slice them all in two, then:
Bucket elimination Wikipedia: Bucket elimination is a satisfiability algorithm. It can be defined as a reformulation of adaptive consistency. Its definitions uses buckets, which are containers for constraint, each variable having an associated bucket. A constraint always belongs to the bucket of its highest variable. The bucket elimination algorithm proceeds from the highest to the lowest variable in turn. At each step, the constraints in the buckets of this variable x_i are considered. By definition, these constraints only involve variables that are lower than x_i. The algorithm modifies the constraint between these lower variables (if any, otherwise it creates a new one). In particular, it enforces their values to be extendible to x_i consistently with the constraints in the bucket of x_i. This new constraint, if any, is then placed in the appropriate bucket. Since this constraint only involves variables that are lower than x_i, it is added to a bucket of a variable that is lower than x_i. This algorithm is equivalent to enforcing adaptive consistency. Since they both enforce consistency of a variable with all its parents, and since no new constraint is added after a variable is considered, what results is an instance that can be solved without backtracking. Since the graph of the instance they produce is a subgraph of the induced graph, if the induced width is bounded by a constant the generated instance is of size polynomial in the size of the original instance. As a result, if the induced width of an instance is bounded by a constant, solving it can be done in polynomial time by the two algorithms. |
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Prioritize - Implicitly Commit
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Prioritize - Implicitly Commit $100 Test assign relative value 20/20 Vision agree how to rank initiatives Build the Checklist order tasks or rank tasks Dot Voting converge on ideas which capture most interest Forced Ranking agree on priorities Impact and Effort Matrix ground commitments Pro/Con List identify key pros and cons to weight for decision Value Mapping get feedback on what features are important |
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Would it make any difference?
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Would it make any difference? Discerning the parts, contrasting dimensions. Compare and contrast what aspects are most important. Tease out contradictions. Comparing instances of a structure Conversing with: example Compare perspectives Conversing with: dialogue I compared my answers to the 12 questions with what I imagine God answers to be. I realized that I live in circumstances but God does not; and that I myself wish for God to be, but God need not wish to be. My perspective sometimes differs from my parents' and other people's.  Comparing structures *** What is this an example of? Conversing with: generality I note and compare different versions or variants of the same structure, such as St.Peter's Keys to Heaven (his string of virtues from faith to love, Peter 2 1:5-7) and Buddha's eightfold way and other examples of what I call the "eightfold way". The counterquestions, kinds of prayer, ways of engaging the violent, ways we change our mind, all exhibit the same underlying structure. This helps me look at the same structure from different perspectives. The variations may fall into different groups, as with the various representations of the divisions into everything, or the four families of primary structures. The different examples help to confirm the shared structure and its basic features.  Properties of living systems I compared various structures I thought were related to properties of life, of living systems, expressing them from different points of view. Compatibility with structures I know In Scripture I sometimes find references to images and numbers that bring to mind the conceptual structures that I have been documenting. I noticed how the seven days of creation could mean events for God, thus the seven divisions of everything, which they match in number. And so I think of creation as an operation +1 of reflection. And I notice with interest that the creations of the first three days are governed by the creations of the next three days, as noted by bishop Skvireckas in his notes to his translation of the Bible into Lithuanian. Similarly, I notice that Ezekiel's chariot of God is carried by four creatures, like the four representations of the nullsome, or the four representations of everything. I notice that there are 24 elders in Revelations. Such coincidences spark my mind and encourage me to think that I may be on track, overall. As I studied Jesus' emotional responses, I looked for correlations between his emotional responses, why he had the emotional response, ways he got things done, ways he showed good will, the avenues opened up for the good, and the directions of the good. Noting a familiar structure In developing the good will exercises, I was alert to the relevance of structures that I was familiar with, such as the foursome (which arose as the four tests of the truth of the heart and the truth of the world) and the threesome (which gave the structure of the exercises). Similarly, in organizing the ways of figuring thing out with the House of Knowledge, I recalled the structure 4 + 6 where there are 6 pairs of 4 levels. Selecting canonical formulations Conversing with: exceptionality Structures refer to perspectives that are deeper than words, deeper than any particular example. How can we think about them and talk about them? I try to choose words and instances that are canonical. Where possible, I look for the simplest words from every day life that have basically that meaning. I used to call the threesome "being, doing, thinking" or "existing, acting, reflecting" or "communal, individual, universal". Over time, I realized that these were better thought of as representations of the threesome. I now talk about the deeper structure as "taking a stand, following through, reflecting". Contrasting possibilities within a system Conversing with: opposing view Appreciate a person's weakness Conversing with: obstacles I once asked a colleague what their deepest value was and, when they told me, I was taken aback, because I thought it personally happened to be their weak point. As I thought about myself and others, I decided that a person's deepest value is both their strong point and their weak point. Contrast dimensions *** What are the components? Conversing with: tension in possibilities In analyzing emotional responses, I noted three dimensions: positive vs. negative, calm vs. riled, sensitive vs. insensitive. I later saw how these were relevant in the good will exercises. In analyzing the counterquestions, I saw how they related a perspective with a situation. Finding the obstacle Conversing with: weaknesses Recognize in the three-cycle (take a stand, follow through, reflect) where am I stuck and thus feel riled. Searching for a self-grounded structure Conversing with: order In high school, I was attracted to structures that I tried to make sense of, namely the Holy Trinity, and also the seperation of powers. I ran for student body president, successfully, on a platform of strengthening the separate branches of student body government. I was especially interested in figuring out the internal structure, how and why the various parts related to each other, and how that made for the overall structure. My contemplations led the way to my documenting "divisions of everything". Spanning the obstacle Conversing with: strengths In designing good will exercises, I expected them to consist of three parts - taking a stand, following through, and reflecting - that we might assimilate the truth of the heart. I realized, from a particular example, and from contemplating the general situation, that we must be having difficulty with one of those three parts. Therefore, we should place that one in the middle of the exercise, so that the exercise might help us overcome that obstacle by spanning it from and to what we knew we could do. |
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Threesome: How-What
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Threesome: How-What Standard Fisher Linear Discriminant Function Given the Regularized Linear Discriminant Analysis, when t goes to infinity, then we get the Fisher discriminant function. The standard Fisher linear discriminant function was proposed by Fisher in 1936. It is the first known classification rule. In 1951, Anderson showed that it can be obtained from optimal statistical decision function theory. We assume multivariate Gaussian classes with a common covariance matrix (GCCM). Into the model we insert maximum likelihood sample estimates instead of unknown parameters. The Fisher rule requires the inversion of a sample-based covariance matrix. This is problematic when we have a small learning set and large dimensionality. The adaline linear classifier of Widrow and Hoff is identical to the Fisher rule when we have the same number of learning vectors from both competing classes. (See: Sarunas Raudys, Evolution and generalization of a single neurone: I. Single-layer perceptron as seven statistical classifiers", Neural Networks 11 (1998) 283-296) How may society be changing for the better? Jerry Michalski of The REXpedition notes that businesses are turning away from "consumer" culture and towards person-to-person relationships with customers, clients, guests, citizens, members, participants, households... How one concept misrepresents another Fresh Air interview of doctor David Ansell, June 15, 2001: Working at County, Ansell says, made him realize just how much the current payment system drives health care inequalities. "There's a misunderstanding that if you just go to the [emergency room], that's health care," he says. "It's not. ... And I don't think the public or politicians really understand that. Health. Chicago. Surveys of different populations Many years later, colleagues of mine conducted door-to-door health surveys in Chicago's poorest neighborhoods. More than a third of those surveyed had higher rates of depression, asthma, hypertension and smoking than those in white communities. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.1934 |
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Pairwise Ranking
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The paramount feature of weighted ranking is its method of ranking each item against each other. ... We thus systematically compare each item with every other item. ... If, by chance, two items end up with the same number of marks because our analysis was inconsistent, we simply rank these two items head to head to break the tie. ... It doesn't permit us to take analytic shortcuts that shortchange our analysis. We must analyze and make a decision on each member of every pair. ... Ranking things in pairs is something the mind can do easily, quickly, and effectively. The mind does love to focus. ... Pair ranking also provides reliability. ... What happens is that the process delivers the rankings. ... As you practice pair ranking, you'll discover that sometimes you can quickly determine with certainty which item is going to be first or which last. When you do, make a note of it and don't include that item anymore as you rank the others. Now rank these fifteen movies from best to worst according to your personal likes and dislikes. ... The main defect in our instinctive method lies in our tendency ... to view problems one-dimensionally ... to focus on (glom on to) the first solution that makes sense. Thus, the moment we think of a reason - any sound, persuasive reason - to like or dislike one of the movies on our list, we tend to latch on to that reason, make our ranking decision on that basis, and move on without considering our other likes and dislikes, which may actually be more important to us. ... It is well and good to use stringent criteria in making decisions, but we should be sure we have considered all relevant criteria and have applied all of the criteria equitably to all of the decisions. ... By applying different criteria to different ranking decisions, we distort our analysis, which can lead to gravely disappointing outcomes that later make us wonder what went wrong. ... There are two inherent weaknesses in our instinctive method. First, we tend to apply different criteria to the different items being ranked. Second, we tend to regard all of our criteria as having equal importance to us ... For that reason, rankings generated with our instinctive method tend to be inconsistent and are thus unreliable. ... your mind doesn't distinguish between vital and nonvital matters when it comes to ranking.  In most problems there is a bevy of outcome classes that we consider when choosing among alternative courses of action. ... each class equates to a different perspective.
 It is frequently advisable, even necessary, ..., to assess utilities from the vantage of more than one perspective. ... Utility analysis handles multiple perspectives quite easily by analyzing each perspective separately (constructing a utility matrix for each) and then merging the results to rank the options. Because the options being weighted are the same for each, the matrices are identical in structure. Ensuring this uniformity is critical to the process. Each matrix contains the same options and outcomes, and the probabilities of the outcomes are identical. Only the utility values will vary because each matrix ... views the problem from a distinctly different perspective ... with different self-interests in mind. ... This technique for utility analysis of multiple perspectives has wide application in problem-solving situations where conflicting interests render a choice among alternative courses of action difficult. Multiple-perspective utility analysis produces decisions that take equitable account of each party's interests, especially when these parties actively participate in the analysis.
...humans are compulsively negative... negative thoughts defeat creative object thinking... Negative thoughts can quickly overwhelm and preempt positive consideration. A powerful but simple technique for dealing with this problem is called PROs-CONs-and-FIXes. This rigorous technique compensates for negative thinking by forcing us to identify the positives first. Only then are we allowed to indulge joyously in the negatives. But the technique goes a step further by examining the negatives and trying to think of actions that could be taken to "fix" them, either converting them into positives or, if that isn't feasible, eliminating them altogether. Those negatives (cons) that can't be fixed represent the price one must pay, the burden one must bear, if the thing being evaluated is to be adopted or accepted.
A matrix offers two important advantages over a tree for performing utility analysis. First, the relative differences in utility values of outcomes are more easily perceived in a matrix than in a tree. Second, arithmetic calculations are easier to perform. This is due, in part, to the different configurations of the two structuring devices: a tree is busy, sort of sprawls out, and is sometimes unsymmetrical, while a matrix is a compact, tightly organized, symmetrical unit. Also, the focus of a utility matrix is squarely on alternative outcomes, while a tree portrays whole scenarios in which outcomes are only a part. Depicting scenarios - the strong suit of trees - tends to overshadow and divert attention from the outcomes. ... Because it facilitates arithmetic operations ... a matrix is more suitable than a tree for analyzing a problem from different perspectives and with different classes of outcomes.
 Weighted ranking is essentially pair ranking, but it includes the use of weights (setting the contribution of factors, as in B3) and a table (the factors yielding the internal structure of a record).
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Negotiate
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Negotiate Would it make any difference? Negotiate. The motivated Servant provides the rich Served with choices, priorities, contracts. When I came to study in Lithuania the university administration placed me in the dormitory for foreigners. This was in the Soviet days and that dormitory was practically empty and isolated from the local students. I told the housing authority that they give me a place in the regular dormitory or I would find somebody to trade places with me. That shook them up and so grudgingly they found a room for me in the regular dormitory with a student of Soviet history. My priorities are to make sure that the government has my money, to turn in my declarations, to keep a record of my receipts, and only then to go through the artifice of keeping formal books. I said also that my income comes primarily from the United States and since I had to abide by both systems, I gave priority to the United States. |
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Compare
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Compare Comparing how things have been Comparing us Comparing what I give and get Considering what is the best we can do Disagreeing Knowing what is valued more than money |
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Let your 'Yes' be Yes and your 'No' be No
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Let your 'Yes' be Yes and your 'No' be No What is more than 'yes' and 'no' are conditionals 'if', counterfactuals, as by which Satan tempted Jesus. Matthew 5:33 "Again you have heard that it was said to them of old time, 'You shall not make false vows, but shall perform to the Lord your vows,' 5:34 but I tell you, don't swear at all: neither by heaven, for it is the throne of God; 5:35 nor by the earth, for it is the footstool of his feet; nor by Jerusalem, for it is the city of the great King. 5:36 Neither shall you swear by your head, for you can't make one hair white or black. 5:37 But let your 'Yes' be 'Yes' and your 'No' be 'No.' Whatever is more than these is of the evil one. Do I have authority? Jesus is the Son of Man in that Man ever teaches him by making an example out of him. So Jesus suffers much abuse. And, like any abused person, Jesus has the authority to forgive his abusers. To the extent that I and others of us maintain and justify this abusive culture, I abuse Jesus, too. And he can thus forgive me and forgive all as the person upon whom all of our sins and all of the abusive culture ultimately bear down. Matthew 9:5 For which is easier, to say, 'Your sins are forgiven;' or to say, 'Get up, and walk?' 9:6 But that you may know that the Son of Man has authority on earth to forgive sins..." (then he said to the paralytic), "Get up, and take up your mat, and go up to your house." Can we prolong our lives? Matthew 6:27 "Which of you, by being anxious, can add one moment to his lifespan? God's love: the good shepherd The good shepherd - DIFFERENCE/UNITY IN RECOGNIZING - Jesus lays down his life for his sheep, thus they know each other, just as he and God know each other, and he will bring in another flock, and unify them What is anxious for itself? Matthew 6:34 Therefore don't be anxious for tomorrow, for tomorrow will be anxious for itself. Each day's own evil is sufficient. |
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Powerset lattice
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Powerset lattice Vary implication (can satisfy various conditions) 32 Chronicle: Sequence => Hierarchy (for determining solutions) Axiom of power set Wikipedia: In mathematics, the axiom of power set is one of the Zermelo-Fraenkel axioms of axiomatic set theory. Given any set A, there is a set P(A) such that, given any set B, B is a member of P(A) if and only if B is a subset of A. Distributing with box mathematics We can distribute 23 x 15 using grid paper as (20 + 3) x (10 + 5). We can change the tens to hundreds or thousands or millions. Likewise we can distribute (2X + 3) x (X + 5). Compare with two digit multiplication. Gospel Math.  Box multiplication: Redistributing the set A set can be redistributed. A set is anything which can be thought as multiple units, where each element is distinct. A set can be the rows of a chessboard or an array. It can be the breakdown of a length, for example, 2 feet and 1/2 foot. We multiply the set by applying the distributive law and multiplying each element of the set separately. And that multiplication can be noncommutative, which is to say, we can make sure that the element is followed by the action. Such multiplication typically looks like a set matched with another set, yielding "box multiplication", as when we multiply (3 feet + 1/4 foot)(2 feet + 1/2 foot) and get 4 products which we then sum up. This is also the standard computation of multiplication, for example, multiplying 2 digit numbers together. Multiplication thus gives the ways of matching units, multiple units times multiple units, as in box multiplication, accounting for all possibilities. Units times units means that conditions are satisfied, thus generating all of the solutions. Here the units are all known, they form sets, thus there are distinct terms in expressions with multiple units. Multiply in a particular order Simplify .... We could multiply out all the terms, but it would take a long time, and we'd probably make a mistake. We need a strategy. If this expression is to simplify, we will probably be able to eliminate radicals. If we multiply any two terms, we can use the difference of two squares formula and get expressions which contain only one radical. pg.166 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Polya's pattern of two loci  Euclid's first problem in his Elements is: In drawing an equilateral triangle, given the first side AB, how do we draw the other two? The solution is: to draw a circle c(A) around A of length AB and to draw a circle c(B) around B of length AB. The third point C of the equilateral triangle will be at a point where the two circles intersect. (There are two such points, above and below the line segment.) Polya notes that this solution is a particular example of a general pattern of "two locii", which is to say, we can often find a desired point by imagining it as the intersection of two curves. I note further that each curve may be thought of as a condition (X="points within a distance AB of A", Y="points within a distance AB of B"). The solution created four regions:
Reimagining the monk problem A monk climbs a mountain. He starts at 8 am and reaches the summit at noon. He spends the night on the summit. The next morning, he leaves the summit at 8am and descends by the same route he used the day before, reaching the bottom at noon. Prove that there is a time between 8 am and noon at which the monk was at exactly the same spot on the mountain on both days One solution is to allow for two monks traveling, one up and the other down, so that it is clear they must meet. In this way the solution is where the two conditions are both satisfied. pg.19, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Constraint satisfaction Wikipedia: In artificial intelligence and operations research, constraint satisfaction is the process of finding a solution to a set of constraints that impose conditions that the variables must satisfy. A solution is therefore a vector of variables that satisfies all constraints. Creative rethinking Many creative methods have us rethink, reorganize, untangle, recombine the conditions that define a problem. The two ropes Paul Zeitz gives the following problem that can be solved (the two ropes can be retrieved) by teasing out the conditions that the solution must satisfy and organizing them thoughtfully. You are locked in a 50 x 50 x 50-foot room which sits on 100-foot stilts. There is an open window at the corner of the room, near the floor, with a strong hook cemented into the floor by the window. So if you had a 100-foot rope, you could tie one end to the hook, and climb down the rope to freedom. (The stilts are not accessible from the window.) There are two 50-foot lengths of rope, each cemented into the ceiling, about 1 foot apart, near the center of the ceiling. You are a strong, agile rope climber, good at tying knots, and you have a sharp knife. You have no other tools (not even clothes). The rope is strong enough to hold your weight, but not if it is cut lengthwise. You can survive a fall of no more than 10 feet. How do you get out alive? pg. 27, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. |
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Ground - Explicitly Care
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Ground - Explicitly Care Context Map show effects of context The 5 Whys discover the root cause The Blind Side acknowledge unknowns |
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What do I have control over?
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What do I have control over? Focusing on structure, appreciating its purpose. Relate a structure with its context by making its purpose explicit. The purpose of my circumstances Conversing with: possibility Address the causes of my bad mood Conversing with: responsibility I wondered about the causes of my moods and concluded that sometimes they have no reason, and perhaps that is their purpose, to kick us out of our usual tracks. Coinciding of endeavor and circumstances Conversing with: possibility I wanted to document my accumulated intuition and I realized that I should be able to do this with a good will exercise that focused on what got going my positive feelings. Distinguish what I have control over Conversing with: irresponsibility As a child, I was a bit afraid of plants. My mother had a beautiful yard with many plants, including a section with dense ivy and large throated jack-of-pulpits. I was suspicious of plants because they seemed very patient, which could be a ruse. I didn't want to allay my suspicions simply based on what other people did or thought. Yet I realized that, if indeed plants might be malevolent, it was beyond my capabilities to establish that, and so I should accept what people thought. Entertain a doubt Conversing with: possibility As I child, I found plants to be suspicious, because of their possibly great patience, thus unpredictable and dangerous. Moral imagination Conversing with: how things should be I imagined what the end of the world must be like, the making tangible of the Holy Spirit, the return of Christ as the identity of all. As a child, I imagined what death must be like, a slowing down of time to allow for a review and untangling of all of our life. What would others want me to do for us all? As a child, I appreciated that I could have been born anybody. How could I serve us all? What would others want me to do? Even though I could not ask them, yet I understood that I could imagine what they would have me do, but especially, if they understood my situation. Ultimately, I decided that the best thing I could do would be to know everything and apply that knowledge usefully. As I did that, I kept mindful of others around the world, thinking how my thoughts might be fruitful for them, imagining them rooting for me, and even allowing me to get some rest and relaxation. As an adult, I have even met such people and we've actually helped each other. A structure's purpose Conversing with: answer Applying a structure Conversing with: applicability I knew that my mind could encompass six perspectives, but not seven or eight, as in the Lord's prayer, and this helped me figure out how to listen to God. I realized that if I tried to contemplate each line of the Lord's prayer, then this would overload my mind, flatten me out, and indeed I would find myself before God, as if in my world there was a rift that opened up above me. Augmenting a structure with its purpose *** What is the significance of a structure? Conversing with: meaningfulness I had noticed that the good will exercises could be organized into six groups based on the pairs of questions that they asked from whether? what? how? why?, the heart asking the broader question, with why? being the broadest. Then I noticed, starting with a few examples, that the questions could be answered yes or no, and that the heart would answer one way, and the world another way. Given a doubt, Is this truly wrong?, and the counterquestion, Is this the ways things should be?, the world says, "It is wrong, it should Not be this way", but the heart says, "It is wrong, it Should be this way", which is to say, that wrong behavior like the crucifixion of Jesus happens for a purpose. I realized that this is indeed part of the very purpose of the counterquestions, to distinguish between the heart and the world, and thus this observation adds new meaning to the counterquestions, aside from their ability to wrench us out of our doubts and our experience.  Conversing with: fruitfulness As I collected more than 100 "ways of figuring things out", I wanted to analyze them in more detail, so I needed to focus on a sample. I decided to choose the most important, the most characteristic ones. First, I considered which ones were important "nodes" in my diagram of how they unfolded in my thinking, from start to finish. Later, I looked at ways that belonged to more than one thread in my life. Finally, I went through all of them and noted for each, what did they achieve? I grouped them into categories accordingly. Then I rated the importance of each way on a scale from 2 to 5 and got the average for the group. I put the most important groups in the center of a diagram. It seemed that the most important ways were the ones that I lived subjectively, from the inside, and the less important ways I lived objectively, from the outside.  Considering the structure of a null structure In considering representations of the foursome, I noticed the role of "null perspectives" as reference points for the other perspectives. Later, in analyzing the primary structures, and considering how God is injected in them, I noticed that I could think of these structures each defined by "null structures" having their own increasingly complex structure. I think I was much inspired by the idea in modern algebra of a "kernel" to a group homomorphism, a structure which is mapped to the identity action. Evoking structure with a question Conversing with: answer As a freshman in college, I had an excellent teacher, Michael Gillespie, for Political Order and Change. He would start each class with a relevant question and discussion, and only then proceed to the text we had read, thus showing how the issues arose organically in conversation. Early on, he asked, What is happiness? (The text was the Declaration of Independence.) It was the kind of question to which I sought an absolute answer. I noticed amongst the answers that some people emphasized joy, as in the joy of eating ice cream, but others emphasized contentment, as in reflecting on that joy, and others emphasized assurance, as in knowing that the joy was right and good and real and not a dream. Thus, in the conversation, I saw a threefold structure of doing (joy) and thinking (contentment) and being (assurance). I later realized that the question, What is happiness?, triggered this structure, evoked it.  Pregnant structure A structure may be able to say more than I know it to say, especially along with other structures. When I realized that I could think of the secondary structures could be expressed as God of one level of the foursome injected into a primary structure from another level, I noticed in particular in 2003 what happened to the seventh perspective of the primary structures. They showed an interesting variety of behavior, which I thought of as six expressions of the will, six representations of anything:
Recognize the significance of a concept Conversing with: combination The importance of faith in every aspect of the "ways of figuring things out". *** Which structures are distinct? Conversing with: purpose A very powerful technique is to consider the purpose of a structure. Over several years I had collected a dozen or more examples of structural frameworks consisting of seven or eight perspectives, depending on how you looked at them. I had tried to conceive of them as a single structure, yet that seemed less and less tenable. Finally, I made a list of what seemed to be the purpose of each structure. I noticed four purposes:
Recognizing and reexamining an assumption Conversing with: purpose In 1997, after a year developing the good will exercises and a year of doing about forty of them, and after much theoretical reflection on them, I realized that it was better to be peaceful than to be riled. The person who is riled is thinking incorrectly, confusing the truths of the heart and the world, whereas the person at peace is thinking correctly. But my whole purpose of the good will exercises was to deal with situations where we are riled, where we believe one thing in our hearts, and we see another thing in the world, and we struggle to follow our hearts. It turns out that we are riled because we are confused. I also saw that, aside from myself, people didn't seek to participate. The exercises helped me with several issues that I had in my life, and they made me more at peace and appreciate that. So I realized that the exercises were problematic in that they had us start with and focus on our being riled instead of our being at peace. And so I stopped doing them.  Conversing with: goal of evolution As I analyzed the "ways of figuring things out", I noticed that several of them seemed to be dialogues with God or myself or others. Indeed, I saw that I could think of each of them as a dialogue with some quality. I went through my list of ways and wrote down the quality that I imagined it had me be in dialogue with. Then I grouped those qualities. I had a chart of the first 55 ways that I had noted, especially from my philosophical work, where I had organized them by how they had become relevant as my inquiry unfolded. After studying the ways in terms of their various aspects, I finally tuned into how the "conversant" evolved, from very abstract "inner depths" or "infinity beyond" to a full-fledged "human-in-general", as we presumed ever more aspects. The "human-in-general" conflates us with our conversant. I noticed then that at that point the conversant was no longer imagined, but became presumed, and so instead, the ways were dialogues with a shared conceptual language, which seemed to require us to presume God as well. (I realize now that this also marks the distinction between defining ourselves beyond any system and then defining the system that we are in.) We start to dialogue with our conversants' circumstances, consider them from God's point of view. Subsequently, I saw how the four tests for the heart and the world were bridges between us and our conversant, and how the ways related to structural questions were perhaps six groups of pairs of these four tests, and that taking up God's point of view was the ultimate way. Taking up a structure Conversing with: structure By taking up a structure, and appreciating its internal relationships, we can make sense of it. I did this with the "meaning of life", deducing a third set of values to correspond with Plato's and St.Paul's. I did this with Steve Bonzak's principles of Kung Fu. What a structure must do, at a minimum In 1988, in Vilnius, I was studying Lithuanian folk tales in order to develop a theory of narration. I realized that I should break down a tale in terms of atomic units. I wondered, what is the nature of such a unit? What is its purpose? And I realized that, at a minimum, it must be able to keep our attention. I noticed that it must create tension, but soon enough, it must relax that tension. Only so can it keep our attention, by switching back and forth. This made it possible to define the unit. And I could it define it directly with regard to the listener or reader (!) and only indirectly, slightly arbitrarily, with regard to the teller or writer. |
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Foursome: Why-How
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Foursome: Why-How Fisher Linear Discriminant Function With Pseudoinverse Covariance Matrix Starting with the Standard Fisher Linear Discriminant Function, we may then allow that the covariance matrix is not invertible. This is always the case when the number of learning samples is 2N < p + 2 If we write the weight vector in a certain way, then the weights approach the Fisher linear discriminant function with pseudoinverse covariance matrix. This assumes that the inputs of the activation function vary in the linear interval. We need to have non-limit target values abs(tj)<1. (See: Sarunas Raudys, Evolution and generalization of a single neurone: I. Single-layer perceptron as seven statistical classifiers", Neural Networks 11 (1998) 283-296) Pamela McLean, June 10, 2011. Maybe that is why we need story telling (and pattern languages) in our post-web world - so that communication is tied firmly into context through anecdotes. |
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Chronologies and time lines
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 ... a chronology, or what Harvard professors Richard Neustadt and Ernest May call a time line in their thoroughly engaging, highly informative book Thinking in Time. ... I stumble over information that I expect to be chronological but isn't ... This compulsion to view events chronologically is related to our instinct to view the world in terms of cause-and-effect relationships ... the mind is programmed to see cause first and effect second. ... the mind, in order to interpret [otherwise], silently reverses the events, just as it reverses the order of object and actor in passive voice. ... A chronology (time line) allows us to understand and appreciate the context in which events occurred, are occurring, or will occur. We are then in a far better position to interpret the significance of each event with respect to the problem. Sometimes putting events in chronological order points to a solution. ... [Neustadt and May] "To see the story behind the issue [the problem], it can help merely to mark on a piece of paper the dates one first associates with [the problem's] history. ... Implicity: vicarious experience acquired from the past, even the remote past, gives such guidance to the present that [the history of the problem] becomes more than its own reward. .  A special form of time line is what Neustadt and May call "placement", whcih means arraying chronologically the events in public history and details in private life that may affect the outlook of a decision maker whether the latter be a person or an organization. From this history and details we draw inferences about the decision maker's thinking - motives, prejudices, and so on. |
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Sell
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Sell What do I have control over? Sell. The rich Servant provides the humble Served with possibilities, favors, exceptions. I am due a rather large refund in Lithuania... At their suggestion, I agreed to cancel my request, which I did, but then they said they would still have to audit me. I said that it would destroy me and that I would have to consider instead devoting that time to publicly making my case. My priorities are to make sure that the government has my money, to turn in my declarations, to keep a record of my receipts, and only then to go through the artifice of keeping formal books. I said also that my income comes primarily from the United States and since I had to abide by both systems, I gave priority to the United States. We talked a long time about the difference between the two systems, that the IRS gives priority to the facts at hand rather than the documents, that they give people refunds based on what the people say they are due (they could not believe this), and that people are audited typically once in their lifetime. I was running a social enterprise, I support a community center, I didn't owe the government any money, but it was going to destroy me through this paperwork. Finally, the inspector said, but what could she possibly do, she had to keep to the law. I said, simply do whatever you can within the bounds of your powers. So she spoke to her supervisor and he spoke to his and they made an exception for me and I finally escaped. Finally, the inspector said, but what could she possibly do, she had to keep to the law. I said, simply do whatever you can within the bounds of your powers. So she spoke to her supervisor and he spoke to his and they made an exception for me and I finally escaped. Algis Razminas, the leader of the Atzalynas Club, tried to make a living from it by winning money from projects. But when we analyzed the projects, the money was tied to particular expenses, with lots of money for consumables, some for wages, but none for essentials like heating or electricity. |
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Declaring my purpose
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Declaring my purpose Recognizing our purpose My intentions Being offered opportunities Declaring my intent How would I want to be treated? Keep setting goals Knowing what I don't want to do Knowing what I love Knowing what I truly want Knowing what I want to devote myself to Recognizing goals What do I try to do? What do I want from others? Our purpose Being true to our purpose Considering our purpose Take charge Being held responsible for your example to others Being responsible Living as a leader |
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Don't resist one who is evil
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Don't resist one who is evil Matthew 5:38 "You have heard that it was said, 'An eye for an eye, and a tooth for a tooth.' 5:39 But I tell you, don't resist him who is evil; but whoever strikes you on your right cheek, turn to him the other also. 5:40 If anyone sues you to take away your coat, let him have your cloak also. 5:41 Whoever compels you to go one mile, go with him two. 5:42 Give to him who asks you, and don't turn away him who desires to borrow from you. Allowing demons to do as they wish Matthew 8:31 The demons begged him, saying, "If you cast us out, permit us to go away into the herd of pigs." 8:32 He said to them, "Go!" They came out, and went into the herd of pigs: and behold, the whole herd of pigs rushed down the cliff into the sea, and died in the water. Do as people wish him to Matthew 8:33 Those who fed them fled, and went away into the city, and told everything, including what happened to those who were possessed with demons. 8:34 Behold, all the city came out to meet Jesus. When they saw him, they begged that he would depart from their borders. God's work: the light of the world Light of the world -- DIFFERENCE/UNITY IN SEEING - give people the chance to escape darkness What questions are productive? Matthew 6:31 "Therefore don't be anxious, saying, 'What will we eat?', 'What will we drink?' or, 'With what will we be clothed?' Why don't people believe? To believe is to accept a way of figuring things out. Matthew 6:28 Why are you anxious about clothing? Consider the lilies of the field, how they grow. They don't toil, neither do they spin, 6:29 yet I tell you that even Solomon in all his glory was not dressed like one of these. 6:30 But if God so clothes the grass of the field, which today exists, and tomorrow is thrown into the oven, won't he much more clothe you, you of little faith? |
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Decomposition
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Decomposition Vary model (can variously combine factors) (Pigeonhole principle, partitions, factorizations, encoding, full range of outputs, principle of inclusion-exclusion) 31 Catalog: Hierarchy => Network (for determining redundancies) Axiom of union Wikipedia: For any set F there is a set A containing every set that is a member of some member of F Reorganizing addition to do it in your head Adding 256 + 256 in your head is easier to do from left to right than from right to left because the intermediate response (500) is simpler because there is nothing to carry. Gospel Math. Reorganizing multiplication to do it in your head Given a problem 5,000 x 50,000, we can break it down in different ways such as 5 x 50 x 1,000 x 1,000 so that we can do it in our head. Gospel Math. Binomial theorem Pascal's Triangle contains all of the binomial coefficients. ... We derived the binomial theorem above by observing that the coefficients in the multiplication of the polynomial (x+y)**k by (x+y) obeyed the summation formula. Here is a more direct "combinatorial" approach, one where we think about how multiplication takes place in order to understand why the coefficients are what they are. Consider the expansion... notice how we can read off the "history" of each term...Now let us think about combining like terms... pg.211 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Complement PIE The combination of PIE with counting the complement is so common, it is worth noting (and verifying) the following alternative formulation of PIE [principle of inclusion-exclusion] ... Given N items, and k sets A1, A2, ..., Ak, the number of these items which lie in none of the Aj is equal to N-S1+S2-S3+...+-Sk, where the Si is the sum of the cardinalities of the intersections of the Aj's taken i at a time, and the final sign is plus or minus, depending on whether k is even or odd. pg.230 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Conspicuous ceiling 41 rooks are placed on a 10x10 chessboard. Prove that there must exist 5 rooks, none of which attack each other. ... When you see the number 41 juxtaposed with 10, you know that the pigeonhole principle is lurking about, since 41 is just one more than 4x10. Once attuned to the pigeonhole principle, we cannot help but notice that [41/10] = 5, which is encouraging, for this is the number of rooks we seek. pg.95, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Factoring Here is a simple problem with a "complete" solution, illustrating one of the most important tactics: factoring. ... Find all right triangles with integer sides such that the perimeter and area are equal. ... xy/2 = x + y + square root of (x**2 + y**2) ... xy - 4x - 4y + 8 = 0 ... Now we do something clever: add 8 to both sides to make the left-hand side factor. We now have (x-4)(y-4) = 8, and since the variables are integers, there are only finitely many possibilities ... The only tricky step was finding the factorization. But this wasn't really hard, as it was clear that the original left-hand side "almost" factored. As long as you try to factor, it usually won't be hard to find the proper algebraic steps. The factor tactic is essential for finding solutions. pg.264-265 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Fundamental Theorem of Arithmetic ...every natural number greater than 1 can be factored completely into primes. ... In fact, this factorization is unique, up to the order that we write the primes. This property of unique factorization is called the Fundamental Theorem of Arithmetic (FTA). We call the grouping of factors into primes the prime-power factorization (PPF). An ugly, but necessary, notation is sometimes to write a number n in "generic" PPF: n = (p1**e1)(p2**e2)...(pt**et). ... If p is a prime and p divides ab, then p divides a or p divides b ... is the key idea that we need ... Here is a simple example of FTA-style reasoning ... Prove that if a monic polynomial has a rational zero, then this zero must in fact be an integer. ... Let u/v be a zero of this polynomial. The crux move: without loss of generality, assume that u and v are relatively prime. ... get rid of fractions, by multiplying both sides by v**n ... This gives us ... u**n = [a multiple of v] ... we must conclude that v = 1 or -1, ie., u/v is an integer. The assumption that u and v are relatively prime means that they taken to be distinct as atoms. pg.244-248 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Halt or repeat If there are only finitely many states as something evolves, either a state will repeat, or the evolution will eventually halt. pg.113, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Intermediate pigeonhole If you have p pigeons and h holes, then at least one of the holes contains at least [p/h] pigeons. pg.94, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.  Label multiplication: Redistributing the multiple A multiple can be redistributed. This is similar to skip counting but it is counting collections, and so it is counting everything up as if we were counting up each item in each collection. Thus it is recounting ordinals. We can think of it as a labeling process, where each group and subgroup is counted and labeled. Thus it proceeds from whole to parts, in the opposite direction as tallying, which proceeds from parts to whole. Partitioning Partitioning is a tactic that deliberately focuses our attention on addition, by breaking a complex problem into several smaller and simpler pieces. [Often used in tandem with encoding]. ... A partition of a set is a division of S into a union of mutually exclusive (pairwise disjoint) sets. ... if S has been partitioned by the Ai, we must have [the cardinality of S equals the sum of the cardinalities of the Ai] ... This leads to a natural combinatorial tactic: Divide the thing that we want to count into mutually exclusive and easy-to-count pieces. ... For example ... the number of subsets of S must be (n 0) + (n 1) + (n 2) + ... + (n n) ... The partitioning tactic makes a pretty utopian assumption, namely that the thing we wish to count can be nicely broken down into pairwise disjoint subsets. Reality is often messier... pg.213-214, 225 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc Partitions Given a positive integer n, a partition of n is a representation of n as a sum of positive integers. The order of the summands does not matter, so they are conventionally placed in increasing order. pg.151, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Pigeonhole principle If you have more pigeons than pigeonholes, and you try to stuff them into the holes, then at least one hole must contain at least two pigeons. ....sometimes also called the Dirichlet principle. pg.92, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. n+1 positive integers Show that among any n+1 positive integers, there must be two whose difference is a multiple of n. ... The penultimate step ... is realizing that the two desired numbers must have same remainder upon division by n. Since there are only n possible remainders, we are done. pg. 93, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Two points a mile away Every point on the plane is colored either red or blue. Prove that no matter how the coloring is done, there must exist two points, exactly a mile apart, which are the same color. ... imagine the vertices of an equilateral triangle with side length one mile. There are three vertices, but only two colors available! The pigeonhole principle tells us that two vertices must be the same color! pg.93, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Repeated application of pigeonhole principle ...This rather elaborate problem was a good illustration of both the pigeonhole principle and the wishful thinking strategy, i.e., not giving up. When you think that the problem can be attacked with the pigeonhole principle, first try to do the job neatly. Look for a way to define pigeons and holes that yields a quick solution. If that doesn't work, don't give up! Use the pigeonhole principle once again to gain a foothold, then try to use it again. Keep extracting information! pg.95, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. The Factor Tactic Multiplication rarely simplifies things. Instead you should Factor relentlessly. pg. 163, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. The principle of inclusion-exclusion The partitioning tactic makes a pretty utopian assumption ... We shall explore a number of ways of dealing with situations where sets overlap and overcounting is not uniform ... Sometimes the complement counting tactic fails us, because the complement is just as complicated as the original set. The principle of inclusion-exclusion (PIE) is a systematic way to handle these situations. In simplest form, PIE states that the number of elements in the union of two sets is equal to the sum of the number of elements in the sets minus the number of elements in their intersection. ... It is easy to see why this is true: Adding |A| and |B| overcounts the value of |A union B|. This overcounting is not uniform; we did not count everything twice, just the elements of A intersection B. Consequently, we can correct by subtracting |A intersect B|. ... In general, we conjecture The cardinality of the union of n sets = + (sum of the cardinalities of the sets) - (sum of the cardinalities of all possible intersections of two sets) + (sum of the cardinalities of all possible intersections of three sets) ... + [if n is odd] or - [if n is even] (the cardinality of the intersection of all n sets). It is pretty easy to see why the formula is true in an informal way. For example, consider the following [Venn] diagram, which illustrates the three-set situation ... It makes sense that in the general case, we would alternate between adding and subtracting ... We have reduced PIE, then, to proving the identity ... (r 0) - (r 1) + (r 2) - ... + (-1)**r (r r) = 0. ... Just expand 0 = (1-1)**r with the binomial theorem and you immediately get [it]! ... Whenever you see the words "at least" you should be alerted to the possibility of counting complements (1-1)**r is the binomial theorem expressing the Venn diagram of conditions acting on parity/balance. pg.226-229 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc |
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Structure - Understand
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Structure - Understand Affinity Map discover patterns relating categories Card Sort categorize Post-Up generate all relevant ideas Spectrum Mapping expose the range of positions The World Cafe recognize emergent patterns |
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Am I able to consider the question?
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Am I able to consider the question? Focusing on parts, sorting experience. Collect first hand experience and recognize patterns. Categorize examples from personal experience Conversing with: possibilities Express intuition in terms of principles Conversing with: right behavior I or others have intuition on some subject. I learned to express that in terms of a set of principles, typically seven and eight. With Joe Damal's help I formulated and analyzed my principles, how to engage enemies, and also his principles, how to organize people, and how to get things done. I likewise worked with Steve Bonzak to formulate and systematize his principles of kung fu.  Laying out thoughts Conversing with: thoughts In studying a question, I may collect a full variety of examples, group them and then lay out the groups as index cards on a table, and shuffle them around, looking for different dimensions that they may represent of particular aspects. In this way, I noted 12 ways that I connect with God, but other structures as well, such as the utility of an Irdakiss standard for import/export of data between tools for organizing thoughts. As I do this, a particular structure may come to mind, such as the 12 topologies, and so I will try to flesh it out. Sort first hand experience Conversing with: possible experience I have often collected examples from first hand experience and sorted them into groups. For example, I collected personal anecdotes where I had grappled with a doubt and addressed it with a counterquestion. Sorting these situations led to a system of seven counterquestions. Contents of Jesus' Parables What did Jesus have to say? I wrote out each of Jesus's parables from the Gospel of Matthew and then sorted them into groups based on the point that each parable was making. I reduced them to eight sayings which I then fleshed out structurally. Joe Damal and I noted principles of community organizing and grouped them into eight ways: Address problems of individuals, Stimulate public interest, Heighten awareness of values, Bring together community resources, Train ourselves and others to act as leaders, Bring out societal issues that need to be addressed, Recognize other opportunities for investigations, Challenge ourselves. Directions of the Good In what sense did Jesus use the concept good? What is good, what are the sources of good, where does the good come from? I made a copy of the Gospel of Luke and cut out each reference that he made to the good. I grouped them according to what was good. I found eight groups which I then structured. Failing to sort messages from God Once I failed miserably with this method. Almost every morning I listen to God and write down what he has to say. After several years, I tried to see if there were any patterns I could detect. However, the answers were so rich and intense that I kept getting dozens of dozens of groups, more and more as I added more data. I wasn't able to find an angle from which I could group his thoughts to me. He didn't seem too pleased, either. I sorted episodes from the Gospel of Mark according to my interpretation of Jesus' feelings and how he responded to them.  Twelve Topologies in Target Spaces for Metaphor In George Lakoff and Mark Johnson's book Metaphors We Live By, I wrote out all of their conceptual metaphors, and grouped them into twelve groups, which I then matched with the twelve topologies.  Twelve Topologies in the Psalms I wanted to better understand twelve topologies, which I think of as the vocabulary of the imagination. In the first 40 psalms, I copied out the instances where God is imagined, then grouped them according to the mental image used, and came up with twelve groups, which I then matched with the twelve topologies. Ways of engaging the violent As a good will exercise, I and Joe Damal noted our many experiences engaging the violent, which I distilled to eight principles for doing so. Ways of getting things done Joe Damal and I noted ways of getting things done and grouped them into eight ways. Why we change our minds Joe Damal and I asked dozens of Chicago youth, Did you ever change your mind? and Why? and grouped the reasons into eight categories, which I then helped him structure. Interpret examples in terms of existing categories Contrast types Conversing with: mind's circumstances I analyzed the six kinds of truths of the hearts and the world given by pairs from the four levels (why, how, what, whether), what is common to each type, which I then realized was related to the counterquestions. Describe behavior in terms of general principles Conversing with: behavior I considered Jesus' behavior, how he got things done to address his emotional responses based on his expectations. I concluded that he expected that we are all one and, as a general principle, he spoke to the good will in others. Note the behavior that characterizes each category Conversing with: character I looked in the Gospel of Mark for episodes where Jesus exhibited emotional responses and grouped them accordingly. Then for each group I considered Jesus' behavior, what expectations he had that generated that response, and how did he then address his feelings by getting things done.  Noticing patterns As I write up my ways of figuring things out that I've associated with the various rooms in the House of Knowledge, there seem to be two groups for each room. I've noticed a pattern. And how the groups relate, for example, the purpose of a structure (which may have no subject to experience it) vs. the purpose of my circumstances (which have a subject who lives them). I wonder if that is true, in general, and if that relates to the truths of the heart and the world. After writing out some of these examples in a diagram, I wonder whether these are two viewpoints on the same way, and whether both viewpoints are necessary aspects of each way of figuring things out, so that they are both in play in each way, to a greater or lesser extent, creating a tension, as with question and answer, or theory and practice, that is essential to every way, and that here draws on an internal structure (of the way of figuring things out) that is externalized. Ranking first hand experience Conversing with: value I have collected distinct answers from more than 600 people to the question, What is your deepest value in life which includes all of your other values? In 2006, I tried to make a map of these values by asking my lab's leaders to rank 30 values as to which they personally felt closer to, and which farther from. I thought of these different scales as sonar readings from which I could reconstruct the underlying geometry. |
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Fivesome: Why-What
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Fivesome: Why-What Generalized Fisher Discriminant Function Starting with the Fisher Linear Discriminant Function With Pseudoinverse Covariance Matrix, if the target values differ from the limiting values, then the smallest deviations are obtained for medium size weights, and the activation function becomes a non-linear function, as does the non-decreasing odd and non-constant function phi (see Sarunas paper). The minimization of the cost function yields a linear classifier very similar to generalized discriminant analysis. (See: Sarunas Raudys, Evolution and generalization of a single neurone: I. Single-layer perceptron as seven statistical classifiers", Neural Networks 11 (1998) 283-296) Collect examples how TIME IS MONEY is used in English George Lakoff and Mark Johnson collected examples from everyday language to investigate how metaphorical concepts apply broadly as mental frameworks and are not simply aspects of language. To get an idea of how metaphorical expressions in everyday language can give us insight into the metaphorical nature of the concepts that structure our everyday activities, let us consider the metaphorical concept TIME IS MONEY as it is reflected in contemporary English.
Choreographing individual moves Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: "I worked on certain moves that would be a factor in the contest — whether I had to bob and weave or constantly be mobile, all those things," he says. "I choreographed my fights in my head before I faced the opponent." Playing around with connecting two ideas Fresh Air interview: Abrams explains that the concept for Super 8 came from two different movie ideas he was playing with in his head: a Stand By Me-like ensemble film based on his own childhood obsession with filmmaking, and a thriller about the U.S. Air Force moving Area 51 and its contents to another destination. "I didn't have more than that, but I had a premise where I thought, 'Oh, that's kind of a fun ... movie,' " he says. "And while playing with the idea of connecting these two movies, suddenly they started to answer each other's questions." Why would somebody say that "Why would I make up epilepsy?" he screamed. "Give me my phenobarbital." I let loose in return. I might be young and inexperienced but I was not a pushover. I was nose to nose with my first patient — not what I had imagined when I opted for a career in primary care. My mind raced as I considered my options. Phenobarbital was a controlled substance. What if he was a drug user? I gazed at the pile of charts in front of me. It was hot. The air was muggy. I had patients waiting for me in the hospital when I was done with clinic. Why would anyone fake epilepsy? He had a point. I had no frame of reference. I could not afford to get bogged down. I took a deep breath. "What the fuck?" I mused. I took his word, wrote out a prescription for three months of a medication I had only read about in a pharmacology book. He grabbed the script out of my hand, as soon as I wrote it, eyebrows furrowed. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.1934 First, structuring helps the mind make sense out of complex problems. Most problems, even the ones we regard as fairly simple, are much too complex and ambiguous to analyze without some kind of structuring. I use structuring when I work a jigsaw puzzle. I group pieces of the same or similar color or texture together - the all-blue sky pieces, for example. I then arrange these pieces in subgroups according to their shape. This approach allows my mind to focus on the subgroup of pieces that will most likely fit, eliminating hundreds of alternatives from consideration. Were I not to group the pieces, I would be forced to continually scan the entire field of unused pieces to find the few that are likely candidates. Yet this is exactly how most of us work a problem. We take in the entire problem (the entire puzzle) with all its complex dimensions (all of its pieces) in one gulp and try to digest it. Structuring frees us from this trap. |
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Sorting
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Sorting - what I do in grouping jigsaw puzzle pieces - is the most basic structuring technique. ... Some of us never sort problems into their component parts. ... Others of us sort, but we do it incoherently or according to misleading assumptions about the nature and similarity of the parts. All of us tend to believe that only complex problems require sorting. The fact is, analysis of even the simplest problems benefits from simple sorting. For example: In preparing a grocery list, it facilitates shopping to group items on the list according to their location in the store. Otherwise, one's eyes are constantly scanning the whole list to ensure that items have not been missed as one moves from one aisle to another. |
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Cut one's losses
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Cut one's losses Am I able to consider the question? Cut one's losses. The motivated Servant provides the humble Served with closure, clarity. When I entered graduate school at UCSD I was able to live in student housing for two years as a primary tenant. The housing was quite affordable. But as the two years were ending, I forget the details, but I tried to get on the list for housing afterwards, and it was morally frustrating because I think I tried to get around the regulations somehow. When I was six years old my parents opened a bank account for my birthday and deposited 10 dollars. And the next year they deposited 10 dollars, too. When my father would go to the bank he would take my book with him and they would stamp the amount of money that I had which reflected the interest I had earned, a few cents and yet it excited me that my money could grow exponentially. However, as I grew older there was inflation, but even more significantly, I realized that the older I grew as a child, the greater amounts of money I dealt with, and so it was completely absurd to save money. Finally, my parents let me take out the money and spend it. I bought an electronics kit that I had wanted. But I realized that 10 dollars had much less significance to a 12 year old than to a 6 year old. That experience made me very skeptical about the value of saving. I felt negative and so I changed my principles. I once had a summer job where I was supposed to prepare students for the SAT exam. The business owner was very happy to hire me because of my own excellent scores and put my picture in the ethnic newspaper. But after a few lessons he let me go. I felt that students should learn from first principles and also that they should do homework as projects that they themselves wanted to do. Parents were unhappy that students weren't getting any homework and that they weren't being forced to do it. Somehow it didn't even click in me that I could ever teach students in such a mindless way or that I had actually been fired. |
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My life story
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My life story Know my personal heritage Recalling my own life story Reflecting on the past |
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What God has joined together, don't let man tear apart
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What God has joined together, don't let man tear apart Matthew 5:31 "It was also said, 'Whoever shall put away his wife, let him give her a writing of divorce,' 5:32 but I tell you that whoever puts away his wife, except for the cause of sexual immorality, makes her an adulteress; and whoever marries her when she is put away commits adultery. ... 19:3 Pharisees came to him, testing him, and saying, "Is it lawful for a man to divorce his wife for any reason?" 19:4 He answered, "Haven't you read that he who made them from the beginning made them male and female,* 19:5 and said, 'For this cause a man shall leave his father and mother, and shall join to his wife; and the two shall become one flesh?' 19:6 So that they are no more two, but one flesh. What therefore God has joined together, don't let man tear apart." God will not delay justice Luke 18:1 He also spoke a parable to them that they must always pray, and not give up, 18:2 saying, "There was a judge in a certain city who didn't fear God, and didn't respect man. 18:3 A widow was in that city, and she often came to him, saying, 'Defend me from my adversary!' 18:4 He wouldn't for a while, but afterward he said to himself, 'Though I neither fear God, nor respect man, 18:5 yet because this widow bothers me, I will defend her, or else she will wear me out by her continual coming.'" 18:6 The Lord said, "Listen to what the unrighteous judge says. 18:7 Won't God avenge his chosen ones, who are crying out to him day and night, and yet he exercises patience with them? 18:8 I tell you that he will avenge them quickly. Nevertheless, when the Son of Man comes, will he find faith on the earth?" God's command: the true vine The true vine - DIFFERENCE/UNITY IN FATE - Jesus cleans those who keep to his commandment to love one another as he loves them, thus asking of the Father in the Son's name, and bearing fruit; and those who don't, are burned away. How was it in the beginning? Matthew 19:7 They asked him, "Why then did Moses command us to give her a bill of divorce, and divorce her?" 19:8 He said to them, "Moses, because of the hardness of your hearts, allowed you to divorce your wives, but from the beginning it has not been so. Offspring of vipers Is the viper a reference to the serpent of Genesis who tempted Adam and Eve? Matthew 23:33 You serpents, you offspring of vipers, how will you escape the judgment of Gehenna? What is appropriate Matthew 15:25 But she came and worshiped him, saying, "Lord, help me." 15:26 But he answered, "It is not appropriate to take the children's bread and throw it to the dogs." What is the nature of people? Matthew 19:7 They asked him, "Why then did Moses command us to give her a bill of divorce, and divorce her?" 19:8 He said to them, "Moses, because of the hardness of your hearts, allowed you to divorce your wives, but from the beginning it has not been so. Who was responsible for the commandments? Matthew 19:7 They asked him, "Why then did Moses command us to give her a bill of divorce, and divorce her?" 19:8 He said to them, "Moses, because of the hardness of your hearts, allowed you to divorce your wives, but from the beginning it has not been so. Why are the commandments the ways they are? Matthew 19:7 They asked him, "Why then did Moses command us to give her a bill of divorce, and divorce her?" 19:8 He said to them, "Moses, because of the hardness of your hearts, allowed you to divorce your wives, but from the beginning it has not been so. |
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Directed graph
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Directed graph Vary truth (can add or remove circular behavior) (With or without cycles) 30 Tour: Network => Sequence (for determining paths) Axiom of regularity Wikipedia: Every non-empty set x contains a member y such that x and y are disjoint sets. This eliminates, for example, the possibility of there being a set X which is its own element X = {X}. (Note that the empty set has no members). This allows us to think of set inclusion relations in terms of directed graphs because then it is clear which is the element of which, although there can be cycles. Thus the axiom of regularity may be thought to allow directed graphs to make sense. Wikipedia: The axiom of regularity is arguably the least useful ingredient of Zermelo-Fraenkel set theory, since virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity ... However, it is used extensively in establishing results about well-ordering and the ordinals in general. Connectivity and Cycles By [cycle] we mean a closed path that "travels" along the edges. ... A graph is connected if every pair of vertices has a path between them. If a graph is not connected, one can always decompose the graph into connected components. ... A connected graph that contains no cycles is called a tree. ... For trees, the number of edges is exactly one less than the number of vertices. ... If a graph has e edges and v vertices and e>=v, then the graph must contain a cycle. pg.120-123, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.  Divide out Multiplication: Redistributing the whole A whole can be redistributed. This is long division. We can focus on cases where the remainder is zero, or we can simply keep dividing forever. This is like children (or pirates) "dividing out" money, "counting out" money ("one for me, one for you, ..."). Most of the whole is divided out; then more of it; then more and so on. This yields multiple units. (Number of cycles) x (Number of people x Units ) Eulerian Path Find the conditions on a connected graph (or multigraph) for which it is possible to travel a path that traverses every edge exactly once. Such paths are called Eulerian ... A connected graph (or multigraph) possesses an Eulerian path if and only if it has zero or exactly two odd-degree vertices. In the former case, the path is a closed path. In the latter, the path must begin and end at the odd-degree vertices. pg.123-124, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Hamiltonian Path The "dual" of an Eulerian path is a Hamiltonian path ... a path that visits each vertex exactly once. If the path is closed, it is called a Hamiltonian cycle. While Eulerian paths possess a "complete theory", very little is known about Hamiltonian paths. At present, the necessary and sufficient conditions for Hamiltonians paths are unknown. This is unfortunate, because many practical problems involve Hamiltonian paths. ... Many problems involving scheduling and optimization of network paths can be recast as searches for Hamiltonian paths. pg.125, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. Handshake Lemma In any graph, the sum of the degrees of all the vertices is equal to twice the number of edges. pg.121, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. |
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Clarify - Explictly Understand
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Clarify - Explictly Understand Argument Map acknowledge all positions and make them understandable Atomize unpack large, messy structures Boundary Matrix identify relevant links between two disciplines Business Model Canvas rethink a business model Force Field Analysis appreciate forces both for and against change Post the Path clarify the existing process Prune the Future survey potential for incremental change Speedboat identify obstacles to a goal SQUID have and track a methodical discussion Staple Yourself to Something clarify a process's impact on an object The 4Cs gather and organize information |
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Is this the way things should be?
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Is this the way things should be? Focusing on the whole, working out a structure. Work out how complicated examples fit together. Study an example in all its details Conversing with: wholeness Analyzing examples Conversing with: wholeness I studied the language of Argumentation by diagramming the principles that I took up over time as I pursued my business and laboratory, Minciu Sodas. Distinguish different kinds of nodes in a diagram In studying argumentation, I analyzed how my principles unfolded in organizing my Minciu Sodas laboratory. In my diagram, I used different colors to distinguish different kinds of "driving ideas":
Immersing myself in a single example I spent much time analyzing the folk story "Ragana and Jonukas" as a representative example of folk stories in general. I wanted to understand the details and be able to explain their relevance. Observe history in the making Conversing with: history World news is repetitive. It is typically reported before, during and after it happens. And in Lithuania's independence movement, I witnessed how people did not consider creative alternatives but looked to known solutions, namely, independence. Expressing the structure of my mind Conversing with: structure Define structure in terms of itself Conversing with: expression, language Structure is complete when it can define itself in terms of itself. This is the case with the threesome, and with the totality of structure, including the three languages. Describing structure in terms of structure I am encouraged, in analyzing a new structure, to find that it leverages or interfaces with structures that I know from before. As I was studying the virtues from Plato's Republic and from St.Paul's hymn to love, I linked them to the foursome, threesome, twosome, sixsome and their representations. Describing the structure of my mind's actions on structure As I note structures that my mind encounters, I study how my mind interacts with them, and I note the structures that describe such interactions. For example, I realized that my mind conceives a division of everything as one of two or four representations, and those representations are organized by the division of everything into two or four perspectives, respectively. Noting the limits of my mind In defining the divisions of everything, I noted the limits of my mind in terms of the perspectives that I am able to take up. Working out a structure Conversing with: airtightness I analyzed the eightfold way's structure, its various permutations and how its various pieces fit together, for example, the gradation of six levels which forms its spine and organizes the divisions of everything. |
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Foursome: Why-Whether
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Foursome: Why-Whether Minimum Empirical Error Classifier When the limit values are the targets, then for two distant classes it is possible to obtain very high weights upon applying the minimizing cost function. Then the activation function is acting as a hard limiting threshold function. If one avoids local minima by using global minimization techniques, then one obtains a classifier similar to the minimum empirical error classifier. (See: Sarunas Raudys, Evolution and generalization of a single neurone: I. Single-layer perceptron as seven statistical classifiers", Neural Networks 11 (1998) 283-296) Can the amount of protein that we eat change the activity of an enzyme? How does protein intake affect cancer initiation? Our first test was to see whether protein intake affected the enzyme principally responsible for aflatoxin metabolism, the mixed function oxidase (MFO). This enzyme is very complex because it also metabolizes pharmaceuticals and other chemicals, friend or foe to the body. Paradoxically, this enzyme both detoxifies and activates aflatoxin. It is an extraordinary transformation substance. At the time we started our research, we hypothesized that the protein we consume alters tumor growth by changing how aflatoxin is detoxified by the enzymes present in the liver. ... After a series of experiments, the answer was clear. Enzyme activity could be easily modified simply by changing the level of protein intake. pg.51, T.Colin Campbell, Thomas M.Campbell, The China Study, 2006. Health. Impact on ecological balance Fresh Air interview of Michael Hiltzik about the Hoover Dam, June 8, 2010: "When you dam a river, basically you reduce the flow downstream. That's going to affect wildlife habitats. In certain rivers, you're going to destroy the spawning grounds for fish like salmon, you're going to destroy wetlands — you're really interfering with a lot of the ecological balance when you build a dam. Looking at basic conditions and services Fresh Air interview of doctor David Ansell, June 15, 2001: The first time Dr. David Ansell went into the men's room at Cook County Hospital in Chicago, he immediately ran out. "It was so bad, I couldn't use it," he says. "I ran across the street and had to use the bathroom there. It was quite an introduction to my first day at County." Health. Chicago. Looking for reasons for discrepancies Fresh Air interview of doctor David Ansell, June 15, 2001: On the South Side of Chicago, the life expectancy of an African-American male is eight years lower than that of a Caucasian man, Ansell explains. "When you look at the reasons for it, at least half of this is [because of] heart disease and cancer and things that could be treated," he says. "One of the problems with our current system is segregating people by insurance status, which ends up limiting the options of care — especially when you get down to the specialty care that people need." Health. Chicago. What is fair Fresh Air interview of doctor David Ansell, June 15, 2001: The only fair way to do this is where people have a card that gets them in, where that card is accepted widely and broadly by everyone, and [giving people] choice," he says. "So you could go anywhere you want, you get the care you want, and choose your own doctors — and that would be some sort of universal plan — Medicare for all, single-payer. We need a system that really gives patients — poor or rich — adequate care." Health. Chicago. Pamela McLean, April 11, 2011: My hunch is that any ideas I might put forward on developing educational systems would probably be 99% ignored and 1% ridiculed - but I might have got the proportions wrong. |
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Causal Flow Diagramming
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 Even when we fully understand what "the problem" is, we all too often have a poor comprehension of what is causing it; lacking that vital knowledge, we resort to that timeworn problem-solving method called trial and error with its focus glued to the first plausible corrective option we think of. All events in life are the results, the outcomes, of previous events. Life itself is a collage of ecological cycles, all intertwining, mingling, overlapping, sharing, and competing, all linked together in countless ever-evolving chains of cause-and-effect relationships. Likewise, every problem we analyze is the product of a definable cause-and-effect system. ... identifiying that system's components and how they interact to produce the problem is essential to effective problem solving. ... illuminating these components and their interactions can validate or invalidate our initial assumptions about how the system works and uncover hidden biases and erroneous perceptions. ... What is causing this problem? How are the major dynamic factors interacting to produce this result?
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Assess
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Assess Is this the way things should be? Consult. The calm Servant provides the humble Served with overview, advice. This is a macroeconomic perspective. I worked as a tutor onsite for a business person who was idealistic yet calculating and even crass as to his business practices. He would encourage us not to take lunch breaks saying that students often didn't show, but in reality a nine hour day of tutoring is completely draining and in the long run unhealthy and an hour break every day is essential. He would hire graduate students as independent contractors but not inform them that they needed to set aside, on their own, 15% of their income for social security. He would say nothing to parents when they missed a class or two but only after they had fallen behind four or five of them would he remind them that they needed to pay for the missed classes, thus often trapping them into maintaining the relationship. He would have the tutors close and leave with the collected money in a dangerous neighborhood and hold them accountable for it if it was lost or stolen. He would pay tutors less than one third of what he was charging and have them deliver to the parents' great expectations. We took in a lady I knew from the streets, helped her quit drinking, and took her in for two years and she worked as a cleaning lady. But then I drove her out because I thought that she should move on, as she was not active in the club's activity with the youth, but just wanted them to stay away so the club would stay clean, and didn't want to do anything more with her life. In high school there was a student, quite bright, who worked after school. He said it was so that he could have a car and have insurance for it. And the reason that he needed a car was so that he could drive to work. That seemed to me the most absurd circle of logic that I was all the more careful to avoid any desires for which I would have to work. Basically, the expectation in Lithuania is that I hire an accountant at least half-time, which I feel is counterproductive but also immoral as it is contrived work that simply justifies the system. After the war in Soviet-occupied Lithuania our grandparents had very little money yet they still sent packages to their relatives back home including old suits and shoes and cloth. This made a big difference to the families' well being. Years later when my grandmother was sick in bed her nieces were very happy to take turns to come out and take care of her. So on the one hand they were illegal workers but on the other hand it was a beautiful reciprocity. |
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Working things out
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Working things out Our actions Letting behavior build Linking our behaviors Pattern Analyzing a pattern Noting a recurring idea Thinking in metaphors Solve problems Apportioning our efforts Appreciating what fits together Having things work in an orderly fashion Making things up Not messing up my hair Noting obstacles Recognizing problems Recognizing solutions Recognizing what is hard Reducing effort |
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Context
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Context O Context If you read the problem carefully, if you understand and follow the rules, then you can also relax them, bend them. You can thus realize which rules you imposed without cause. You can also change or reinterpret the context. These are the holes in the cloth that the needle makes. I often ask my new students, what is 10+4? When they say it is 14, then I tell them it is 2. I ask them why is it 2? and then I explain that it's because I'm talking about a 12-hour clock. This example shows the power of context so that we probably can't write down all of the context even if we were to know it all. We can just hope and presume that others are like us and can figure it out just as we do. Analogously, in real life, it's vital to obey God, or rather, to make ourselves obedient to God. (Or if not God, then our parents, those who love us more than we love ourselves, who want us to be alive, sensitive, responsive more than we ourselves do.) If we are able to obey, then we are able to imagine God's point of view and even make sense of it. |
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Obeying
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Obeying |
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Sanity Check
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...when we have finished using any analytic structuring technique, we always do a "sanity check" by asking ourselves, "Does this make sense?" If the analytic result is reasonable, we're probably okay. But if our intuition tells us something doesn't seem right, we should go back and reexamine or completely redo our analysis. |
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Identity
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Identity John says that the whole concept of identity is very flexible in physics, even shaky. What is a storm? Where does it go when it no longer exists? What is the reality of the kludge? You can't know where things come and go in physics. Once you write about it you are not talking about physics in the big picture. This is where there is slack and wiggle room. Identity is what allows for eternal nature. We acknowledge the storm and, though it comes and goes, yet its eternal nature jumps over into us. This is eternal life. In math, there is Context, which can change the meaning entirely, as when 10+4 turns out to be 2'o-clock on a clock. In life, if we obey God, then we can imagine God's point of view, and that opens up incredible freedom. |
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The will of God
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The will of God In the gospel of John, Jesus says that the will of God is eternal life. Fulfill all righteousness Matthew 3:13 Then Jesus came from Galilee to the Jordan to John, to be baptized by him. 3:14 But John would have hindered him, saying, "I need to be baptized by you, and you come to me?" 3:15 But Jesus, answering, said to him, "Allow it now, for this is the fitting way for us to fulfill all righteousness." Then he allowed him. Heaven confirms Matthew 3:16 Jesus, when he was baptized, went up directly from the water: and behold, the heavens were opened to him. He saw the Spirit of God descending as a dove, and coming on him. 3:17 Behold, a voice out of the heavens said, "This is my beloved Son, with whom I am well pleased." How people of good will act so that others not reject them Jesus and John choose their ways so that people might not reject them, although they do, even so. Matthew 11:16 "But to what shall I compare this generation? It is like children sitting in the marketplaces, who call to their companions 11:17 and say, 'We played the flute for you, and you didn't dance. We mourned for you, and you didn't lament.' 11:18 For John came neither eating nor drinking, and they say, 'He has a demon.' 11:19 The Son of Man came eating and drinking, and they say, 'Behold, a gluttonous man and a drunkard, a friend of tax collectors and sinners!' But wisdom is justified by her children." How to enter into life Matthew 19:17 ... But if you want to enter into life, keep the commandments." 19:18 He said to him, "Which ones?" Jesus said, "'You shall not murder.' 'You shall not commit adultery.' 'You shall not steal.' 'You shall not offer false testimony.' 19:19 'Honor your father and mother.'* And, 'You shall love your neighbor as yourself.'" Jesus wants the law to be obeyed Matthew 8:4 Jesus said to him, "See that you tell nobody, but go, show yourself to the priest, and offer the gift that Moses commanded, as a testimony to them." Love your neighbor as yourself Leviticus 19:18 "'You shall not take vengeance, nor bear any grudge against the children of your people; but you shall love your neighbor as yourself. I am Yahweh. ... Leviticus 19:33 "'If a stranger lives as a foreigner with you in your land, you shall not do him wrong. 19:34 The stranger who lives as a foreigner with you shall be to you as the native-born among you, and you shall love him as yourself; for you lived as foreigners in the land of Egypt. I am Yahweh your God. Love your neighbor as yourself Luke 10:25 Behold, a certain lawyer stood up and tested him, saying, "Teacher, what shall I do to inherit eternal life?" 10:26 He said to him, "What is written in the law? How do you read it?" 10:27 He answered, "You shall love the Lord your God with all your heart, with all your soul, with all your strength, and with all your mind;* and your neighbor as yourself." 10:28 He said to him, "You have answered correctly. Do this, and you will live." Love your neighbor as yourself "Love your neighbor as yourself" is the negative commandment that unites six of the commandments. It shows the freedom that God opens up for us. Thus it is how God is one with us. Noting how people change their standards to condemn others Matthew 11:16 "But to what shall I compare this generation? It is like children sitting in the marketplaces, who call to their companions 11:17 and say, 'We played the flute for you, and you didn't dance. We mourned for you, and you didn't lament.' 11:18 For John came neither eating nor drinking, and they say, 'He has a demon.' 11:19 The Son of Man came eating and drinking, and they say, 'Behold, a gluttonous man and a drunkard, a friend of tax collectors and sinners!' But wisdom is justified by her children." The lost child should go where his parents will look for him Luke 2:48 When they saw him, they were astonished, and his mother said to him, "Son, why have you treated us this way? Behold, your father and I were anxiously looking for you." 2:49 He said to them, "Why were you looking for me? Didn't you know that I must be in my Father's house?" 2:50 They didn't understand the saying which he spoke to them. The Son is not greater than the Father Matthew 22:45 "If then David calls him Lord, how is he his son?" What is an extenuating circumstance Jesus points to hunger as a special context, an extenuating circumstance, just as the temple is and he and his culture is. Matthew 12:3 But he said to them, "Haven't you read what David did, when he was hungry, and those who were with him; 12:4 how he entered into God's house, and ate the show bread, which was not lawful for him to eat, neither for those who were with him, but only for the priests? What makes things possible Matthew 19:24 Again I tell you, it is easier for a camel to go through a needle's eye, than for a rich man to enter into the Kingdom of God." 19:25 When the disciples heard it, they were exceedingly astonished, saying, "Who then can be saved?" 19:26 Looking at them, Jesus said, "With men this is impossible, but with God all things are possible." What to do to inherit eternal life To inherit is to be a child, a son of the Father. Luke 18:18 A certain ruler asked him, saying, "Good Teacher, what shall I do to inherit eternal life?" 18:19 Jesus asked him, "Why do you call me good? No one is good, except one—God. 18:20 You know the commandments: 'Don't commit adultery,' 'Don't murder,' 'Don't steal,' 'Don't give false testimony,' 'Honor your father and your mother.'" Who shows mercy Luke 10:29 But he, desiring to justify himself, asked Jesus, "Who is my neighbor?" 10:30 Jesus answered, "A certain man was going down from Jerusalem to Jericho, and he fell among robbers, who both stripped him and beat him, and departed, leaving him half dead. 10:31 By chance a certain priest was going down that way. When he saw him, he passed by on the other side. 10:32 In the same way a Levite also, when he came to the place, and saw him, passed by on the other side. 10:33 But a certain Samaritan, as he traveled, came where he was. When he saw him, he was moved with compassion, 10:34 came to him, and bound up his wounds, pouring on oil and wine. He set him on his own animal, and brought him to an inn, and took care of him. 10:35 On the next day, when he departed, he took out two denarii, and gave them to the host, and said to him, 'Take care of him. Whatever you spend beyond that, I will repay you when I return.' 10:36 Now which of these three do you think seemed to be a neighbor to him who fell among the robbers?" 10:37 He said, "He who showed mercy on him." Then Jesus said to him, "Go and do likewise." Who will enter into the Kingdom of Heaven? Matthew 7:21 Not everyone who says to me, 'Lord, Lord,' will enter into the Kingdom of Heaven; but he who does the will of my Father who is in heaven. Whose is what to give? Matthew 20:23 He said to them, "You will indeed drink my cup, and be baptized with the baptism that I am baptized with, but to sit on my right hand and on my left hand is not mine to give; but it is for whom it has been prepared by my Father." 10 + 4 = 2 I ask my students, What is 10+4? and they answer 14, and then I say, No, it is 2! Do you know why? Because I'm thinking about a clock. 10 o'clock plus 4 o'clock is 2 o'clock on a clock. What the example shows is that meaning ultimately depends on the context which we interpret. Any explanations that we write may also be misinterpreted. Thus there is no way to explicitly assure that somebody means what we mean. However, the context may indeed coincide in all that is relevant to us, either explicitly or implicitly. That's why existentialism is important, because it's important for us that our words and concepts be grounded in the questions relevant to our existence. Gospel Math. Axiom schema of replacement Wikipedia: Less formally, this axiom states that if the domain of a definable function f is a set, and f(x) is a set for any x in that domain, then the range of f is a subclass of a set, subject to a restriction needed to avoid paradoxes. I think this relates to the idea that context allows us to "substitute variables" in different ways and perhaps with different results, different meanings, thus yielding flexibility of interpretation. |
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Obeying
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Obeying Listen to God Ask God a question Conversing with: answer In 1985, at the University of Chicago, I asked God, what kind of mathematics was important, and I understood him to say: walks on trees made of the elements of the threesome, which I denoted at the time as I (individual action), C (communal existence), U (universal thought), and I thought related to an infinite binary tree, which would be the case if I, C, U refered to the edges. I never quite knew what to make of that although I did prove a result about counting walks on trees and my thesis could be used to think further about this. I asked God, why did I have feelings for a woman who was too young for me, and he said it was because I will marry her. Also, when I prayed in the mornings, I would ask God a philosophical question and , which were very intense and generally more helpful indirectly. Now I mostly just listen to God, what does he have to say. 1999.08.18: I asked God which questions I should think over so as to understand why good will makes way for good heart. He responded:
1999.08.18: In my personal work I want to do some investigations into a question that is of interest to God. I told him so and he responded: Why does Good Will make way for Good Heart? Check with God Conversing with: selflessness I check in with God every so often, including when I have doubts on how I may behave, what is allowed or not, how not to be selfish. Hear from God in a dream Conversing with: God's perspective In dreams, I heard God say, "Those things are which show themselves to be" and "This is the fundamental unit of information" (a tableaux of large and small slashes) I traveled with my friend to his home in China. He arranged for me to stay there for an additional six weeks with his family. There was no way for me to directly pay his family for food or anything as that would be inappropriate. However, he took care of that for me. The one thing that he set straight was that I would do my own laundry by hand because otherwise his mother would have to do it. So we agreed on that and it seemed to work out well. Listen to God Conversing with: God's wishes God told me, as I listened to him, to investigate God's love in the ways of figuring things out. Work in tune with God Conversing with: sensitiveness In China, I asked for God's help to write up together how to develop the "good will exercises", which I did with full attention to God. Write down what God says Conversing with: God's wisdom's noteworthiness I write down what God tells me each morning, including his answers to my questions, and I realize that he (or even my unconscious) is of much more powerful intelligence than I am. Appreciate constraints Obey God Conversing with: God's wisdom Do good. Go to church. Chastity. Respect my parents. Pulling structures together Conversing with: totality I considered how to pull together all the conceptual structures I found, in my overview. I likewise considered how the Beatitudes (and Maslow's hierarchy of needs) served as a framework for the language of Narration. I used to think that I should be generous to the government with my money and pay it willingly and err on the side of paying more taxes. I wouldn't take the standard meal deductions because they seemed unrealistically large. But after having such a hard time with my business and realizing that there is nobody to care for me, I started to take them. I figured that this was the system's way of supporting me. Value my life Conversing with: God's gift In China, after a conversation with Tong Zhu about the moral dilemmas of being a doctor, I realized that I was too precious to God to make a living as I had from a job that I did not think contributed to society, coding for a health care start-up serving a regulatory mandate of dubious merit. What does my unconscious allow? As a student, I used to think that I might fall in love with any woman who I found irresistibly beautiful and good. But then I realized that my unconscious thought otherwise. My unconscious resisted me having any real feeling for a woman who was not Lithuanian, or at least, didn't speak Lithuanian. I thought that if a woman truly loved me, then she could readily learn Lithuanian and live with me in Lithuania. Maybe that's true. But my unconscious mind thought otherwise. So I took to heart my unconscious. |
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A greater context in which everything can be reinterpreted and get unexpected meaning
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A greater context in which everything can be reinterpreted and get unexpected meaning Bradford Hansen-Smith, 2011.04.25: One of the aspects of figuring things out is to figure out how to support ourselves doing what we desire in a society that has limitation on what it has decided has meaning and will pay for. My suggestion is to look towards your thoughts about "Identity." ...there is Context, which can change the meaning entirely,..." The larger the context the greater is one ability to make connections that give meaning to individual purpose. All other areas of compartmentalization will by necessity be limited. This is what I think people are really talking about with the overused phrase of "thinking outside of the box." You know, the one that was never there until we constructed it, giving it an identity of "reality" which then becomes directive and instead of moving through it stays on our shoulders, limiting our range of vision and diminishing our choices. Meaning is not in explaining what we know, rather in finding the value that lies between those things we know. Alignment with that which is out of our range extends our understanding revealing greater symmetries and ways of solving problems, like ways of making a living without having to get into someone else's box. Be taught by others how life is for them My black male patients had all been stopped by the police for traffic violations — "Driving while black." They taught me the routine. Something I had never experienced myself. Flashing red lights. A floodlight blasts through the back window illuminating the interior of the car. Every black parent taught his or her children how to respond to a police stop. White kids were taught to trust the police. Black kids were taught to be cautious around the police. There was a routine that black men had learned to follow when stopped by the police. Open the window. Put your hands up. Easy does it. ... "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.1934 Pamela McLean, June 10, 2011. Even the simplest of words may conjour up completely different images to different people - or even to ourselves as we picture ourselves in different place. I think of being with my friends in Nigeria or back home in the UK and of some words that would conjour up very different pictures in the two situations - school, kitchen, breakfast, family, water, wedding, party, market, taxi, greetings, condolences, morning, evening, rain, sun ... the list could go on. Pamela McLean, June 10, 2011. So many of us are connecting at a distance with people whose cultures are dramatically different from our own. We try to communicate about ideas, roles, structures, feelings - all kinds of complicated things - and it is temping to believe we all have a shared understanding. Perhaps that never really happens. Guest contributor - Going out I've been really eeticxd to see PV's mentality change just in the past year from bringing people INTO the church, to encouraging the church to go OUT. This year for Christmas, rather than doing the same Christmas program its done for over a decade, we've themed this Christmas Christmas to GO (picture a chinese to-go box as the image . Everyone in the Northland who has ever wnted to see The Singing Christmas Tree has come to PV to see it EXCEPT those who are homeless or in prison. So where are we taking the message of Christ? To the homeless and to prisons. I'm pretty eeticxd about this I think this is one small way we see the Kingdom of God expanding in a real, and practical way. God wants his law to be followed with all our hearts and our souls As Perry Recker noted to me, God ends Deuteronomy with a blessing and curse, and they are like the two branches of the House of Knowledge. God wants us to obey him, but if we stray, then he wants us to return with all of our hearts and all of our souls. We are to follow the letter; but if we don't, won't, can't follow the letter, then even so we are to follow with our spirit. Deuteronomy 30:1 It shall happen, when all these things have come on you, the blessing and the curse, which I have set before you, and you shall call them to mind among all the nations, where Yahweh your God has driven you, 30:2 and shall return to Yahweh your God, and shall obey his voice according to all that I command you this day, you and your children, with all your heart, and with all your soul; 30:3 that then Yahweh your God will turn your captivity, and have compassion on you, and will return and gather you from all the peoples, where Yahweh your God has scattered you. 30:4 If your outcasts are in the uttermost parts of the heavens, from there will Yahweh your God gather you, and from there he will bring you back: 30:5 and Yahweh your God will bring you into the land which your fathers possessed, and you shall possess it; and he will do you good, and multiply you above your fathers. 30:6 Yahweh your God will circumcise your heart, and the heart of your seed, to love Yahweh your God with all your heart, and with all your soul, that you may live. 30:7 Yahweh your God will put all these curses on your enemies, and on those who hate you, who persecuted you. 30:8 You shall return and obey the voice of Yahweh, and do all his commandments which I command you this day. 30:9 Yahweh your God will make you plenteous in all the work of your hand, in the fruit of your body, and in the fruit of your livestock, and in the fruit of your ground, for good: for Yahweh will again rejoice over you for good, as he rejoiced over your fathers; 30:10 if you shall obey the voice of Yahweh your God, to keep his commandments and his statutes which are written in this book of the law; if you turn to Yahweh your God with all your heart, and with all your soul. God's children are those who obey his law which he gave them Deuteronomy 32:5 They have dealt corruptly with him. They are not his children, because of their defect. They are a perverse and crooked generation. 32:6 Do you thus requite Yahweh, foolish people and unwise? Isn't he your father who has bought you? He has made you, and established you. 32:7 Remember the days of old. Consider the years of many generations. Ask your father, and he will show you; your elders, and they will tell you. 32:8 When the Most High gave to the nations their inheritance, when he separated the children of men, he set the bounds of the peoples according to the number of the children of Israel. 32:9 For Yahweh's portion is his people. Jacob is the lot of his inheritance. |
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Conscientiousness
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Conscientiousness The gap between Servant and Served, expressed in the six ways of Serving, comes together as Conscientiousness. I remember as a child we had to weed our garden every so often. It wasn't especially fun. And we weren't rewarded for it. It was just something that we were supposed to do. But there was something nice about doing that work conscientiously and feeling like part of the team, our family. Also, it's my position that laws are in flux and they may well change in my favor as I get to them. I remember as a child our grandmother came from Chicago to visit. She went outside to broom. I think that we said that we could broom. But she made a point that we didn't know how to broom. Then she taught us how to broom. By the way, in California we did not use brooms, we would clear the pavement by driving everything away with water from the hose. When I started up a business in Lithuania I was very careful to understand and fulfill all of the rules to the letter. But at a certain point it became clear that for me to do the kind of business that I wanted to do it would be practically impossible. In particular, in Lithuania I am expected, as a sole proprietor, to have a full set of books for my business, including a cash book. And there were many other laws, such as having to write letters to myself before each trip as to my business purpose, and so on. Whereas in the United States it would be enough to have a shoe box of receipts. ... So my thinking is that I do as much as I can of the paperwork and keep attentive to it even though it's not possible for me to do absolutely all of what is required, so that I at least am making a good faith effort. I was anxious and I told God I was too anxious to work part-time on my philosophy and on the business too at that point in my life. I offered a deal with him to let me focus on the business and I would make the business model work for him, too. Then I felt more at ease. Our parents taught us to pick up money from the ground if we found it, even if it was just a penny. That it was a matter of respect for money. It didn't matter if it was worth picking up or not. It was important to treat money with respect. |
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Engage God
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Engage God Considering God's welfare Engage God Engaging God Hold God accountable Who is out to get me? Tom Wayburn: As I scrolled through your beautiful website ... something reminded me of Calvin's answer when Hobbes asked him if he believed in God. "Well, someone's out to get me," quoth Calvin. Presuming God's self-love Telling God |
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Not yet sorted
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Not yet sorted I'm collecting and sorting ever more ways of figuring things out. Please send them to me at Thank you! Andrius Kulikauskas Creative constraint Sometimes, as my debts grew, I would wonder if I should have foresworn from ever going into debt. But time and again I realized that the pressure of my debts kept me from sitting still, from stagnating, and forced me to be enormously creative in ways that I probably would not have been otherwise. I thought of it as a creative constraint that opened up my thinking, just as such constraints can in the creative arts. |
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More ways Jesus figured things out
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More ways Jesus figured things out Structures Jesus thought in terms of a variety of structures including the division into four (the parable of the sower, the tempations by the devil), the eightfold way (the Beatitudes, the Lord's prayer), the threesome (the donkey's millstone around your neck). He drew (pictures? diagrams?) in the sand. What is the nature of good? I did a study of the Gospel of Matthew to see how Jesus variously uses the concept "good". He talks of: good heart, good God, good gift, good quality, good person, good deed, good word, good news. Content of Jesus's sayings I considered Jesus's parables in Luke in terms of their content. I think he had eight different things to say. |
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Other ways: Math
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Other ways: Math Edward Cherlin: There are proofs that depend on the algebraic or topological nature of other mathematical structures, where you can't do induction because the underlying sets are not well-ordered. This is the essence of category theory and much of model theory. Epsilon induction Wikipedia: In mathematics,-induction (epsilon-induction) is a variant of transfinite induction, which can be used in set theory to prove that all sets satisfy a given property P[x]. If the truth of the property for x follows from its truth for all elements of x, for every set x, then the property is true of all sets. ... This principle, sometimes called the axiom of induction (in set theory), is equivalent to the axiom of regularity. Forcing (in set theory) Wikipedia In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1962, to prove the independence of the continuum hypothesis and the axiom of choice from Zermelo-Fraenkel set theory. Forcing was considerably reworked and simplified in the 1960s, and has proven to be an extremely powerful technique both within set theory and in other areas of mathematical logic such as recursion theory. ... Perhaps more clearly, the method can be explained in terms of Boolean-valued models. In it, any statement is assigned a truth value from some infinite Boolean algebra, rather than just a true/false value. Then an ultrafilter is picked in this Boolean algebra, which assigns values true/false to statements of our theory. The point is that the resulting theory has a model which contains this ultrafilter, which can be understood as a model obtained by extending the old one with this ultrafilter. By picking a Boolean-valued model in appropriate way, we can get a model that has the desired property. In it, only statements which must be true (are "forced" to be true) will be true, in a sense (since it has this extension/minimality property). Thought experiments in Computer Science Wikipedia's list of thought experiments in Computer science: Halting problem (limits of computability), Turing machine (limits of computability), Two Generals' Problem, Dining Philosophers (computer science) Thought experiments in Mathematics Thought experiments in Mathematics: Balls and vase problem (infinity and cardinality), Gabriel's Horn (infinity), Infinite monkey theorem (probability), Lottery paradox (probability), Sleeping beauty paradox (probability) |
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Other ways: Andrius
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Other ways: Andrius |
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Other ways
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Other ways Clinical trial Following the outcomes of distinct groups which use or do not use a treatment or device. Miscellaneous science experiments Wikipedia has 18 pages related to miscellaneous science experiments. Other ways: Acting My friend Rimas Morkunas is a wonderfully funny actor. I want to study books by: Constantin Stanislavski (Konstantinas Stanislavskis, , (Aktoriaus saviruosa), , Michail Chekhov, Lee Strysberg, Namerovich Danchenko (or Danchenka), Merholdas about biomechanics, Efroso theatre, Aristotle's poetics. An understanding heart to judge God's people, discern between good and evil Solomon says, 1st Kings 3:7 I am but a little child. ... 3:9 Give your servant therefore an understanding heart to judge your people, that I may discern between good and evil; for who is able to judge this your great people? Other ways: Bible characters I study ways of figuring things out illustrated by characters in the Bible. (I consider God and Jesus separately). Warned in a dream, Joseph withdrew to Galilee Matthew ... Being warned in a dream, he withdrew into the region of Galilee, 2:23 and came and lived in a city called Nazareth Ask all the chief priests and scribes Matthew 2:4 Gathering together all the chief priests and scribes of the people, he asked them where the Christ would be born. Ask others to bring word when they know Matthew 2:8 He sent them to Bethlehem, and said, "Go and search diligently for the young child. When you have found him, bring me word, so that I also may come and worship him." Be informed Genesis 9:22 Ham, the father of Canaan, saw the nakedness of his father, and told his two brothers outside. Be instructed by God Genesis 6:14 Make a ship of gopher wood. You shall make rooms in the ship, and shall seal it inside and outside with pitch. 6:15 This is how you shall make it. The length of the ship will be three hundred cubits, its breadth fifty cubits, and its height thirty cubits. 6:16 You shall make a roof in the ship, and you shall finish it to a cubit upward. You shall set the door of the ship in its side. You shall make it with lower, second, and third levels. Be warned in a dream Matthew 2:12 Being warned in a dream that they shouldn't return to Herod, they went back to their own country another way. Become pregnant Matthew 1:18 Now the birth of Jesus Christ was like this; for after his mother, Mary, was engaged to Joseph, before they came together, she was found pregnant by the Holy Spirit. Children are like parents Matthew 2:22 But when he heard that Archelaus was reigning over Judea in the place of his father, Herod, he was afraid to go there. John calls Pharisees and Saducees the offspring of vipers Matthew 3:7 But when he saw many of the Pharisees and Sadducees coming for his baptism, he said to them, "You offspring of vipers, who warned you to flee from the wrath to come? Confession Matthew 3:6 They were baptized by him in the Jordan, confessing their sins. Contrasting ourselves with Jesus Matthew 3:11 I indeed baptize you in water for repentance, but he who comes after me is mightier than I, whose shoes I am not worthy to carry. He will baptize you in the Holy Spirit. Endow names with meaning Genesis 5:28 Lamech lived one hundred eighty-two years, and became the father of a son, 5:29 and he named him Noah, saying, "This same will comfort us in our work and in the toil of our hands, because of the ground which Yahweh has cursed." Follow a star Matthew 2:9 They, having heard the king, went their way; and behold, the star, which they saw in the east, went before them, until it came and stood over where the young child was. Hear from an angel in a dream Matthew 1:20 But when he thought about these things, behold, an angel of the Lord appeared to him in a dream, saying, "Joseph, son of David, don't be afraid to take to yourself Mary, your wife, for that which is conceived in her is of the Holy Spirit. 1:21 She shall bring forth a son. You shall call his name Jesus, for it is he who shall save his people from their sins." An angel told Joseph to flee to Egypt Matthew 2:13 Now when they had departed, behold, an angel of the Lord appeared to Joseph in a dream, saying, "Arise and take the young child and his mother, and flee into Egypt, and stay there until I tell you, for Herod will seek the young child to destroy him." Angel told Joseph to return to Israel Matthew 2:19 But when Herod was dead, behold, an angel of the Lord appeared in a dream to Joseph in Egypt, saying, 2:20 "Arise and take the young child and his mother, and go into the land of Israel, for those who sought the young child's life are dead." Inquire Matthew 2:7 Then Herod secretly called the wise men, and learned from them exactly what time the star appeared. Jesus sorts out the good and the bad Matthew 3:12 His winnowing fork is in his hand, and he will thoroughly cleanse his threshing floor. He will gather his wheat into the barn, but the chaff he will burn up with unquenchable fire." Judge the tree by the fruit Matthew 3:7 But when he saw many of the Pharisees and Sadducees coming for his baptism, he said to them, "You offspring of vipers, who warned you to flee from the wrath to come? 3:8 Therefore bring forth fruit worthy of repentance! Look Genesis 8:13 It happened in the six hundred first year, in the first month, the first day of the month, the waters were dried up from the earth. Noah removed the covering of the ship, and looked. He saw that the surface of the ground was dried. Not having sex Matthew 1:24 Joseph arose from his sleep, and did as the angel of the Lord commanded him, and took his wife to himself; 1:25 and didn't know her sexually until she had brought forth her firstborn son. Not to shame Matthew 1:19 Joseph, her husband, being a righteous man, and not willing to make her a public example, intended to put her away secretly. Receive unexpected guests and presents Matthew 2:11 They came into the house and saw the young child with Mary, his mother, and they fell down and worshiped him. Opening their treasures, they offered to him gifts: gold, frankincense, and myrrh. See a star Genesis 2:1 Now when Jesus was born in Bethlehem of Judea in the days of King Herod, behold, wise men from the east came to Jerusalem, saying, 2:2 "Where is he who is born King of the Jews? For we saw his star in the east, and have come to worship him." See and touch John 20:24 But Thomas, one of the twelve, called Didymus, wasn't with them when Jesus came. 20:25 The other disciples therefore said to him, "We have seen the Lord!" But he said to them, "Unless I see in his hands the print of the nails, and put my hand into his side, I will not believe." 20:26 After eight days again his disciples were inside, and Thomas was with them. Jesus came, the doors being locked, and stood in the midst, and said, "Peace be to you." 20:27 Then he said to Thomas, "Reach here your finger, and see my hands. Reach here your hand, and put it into my side. Don't be unbelieving, but believing." 20:28 Thomas answered him, "My Lord and my God!" 20:29 Jesus said to him, "Because you have seen me, you have believed. Blessed are those who have not seen, and have believed." See that one was mocked Matthew 2:16 Then Herod, when he saw that he was mocked by the wise men, was exceedingly angry, and sent out, and killed all the male children who were in Bethlehem and in all the surrounding countryside, from two years old and under, according to the exact time which he had learned from the wise men. See what a star stands over Matthew 2:9 They, having heard the king, went their way; and behold, the star, which they saw in the east, went before them, until it came and stood over where the young child was. Send out a probe Genesis 8:6 It happened at the end of forty days, that Noah opened the window of the ship which he had made, 8:7 and he sent out a raven. It went back and forth, until the waters were dried up from the earth. 8:8 He sent out a dove from him, to see if the waters were abated from the surface of the ground, 8:9 but the dove found no place to rest her foot, and she returned to him into the ship; for the waters were on the surface of the whole earth. He put out his hand, and took her, and brought her to him into the ship. 8:10 He stayed yet another seven days; and again he sent the dove out of the ship. 8:11 The dove came back to him at evening, and, behold, in her mouth was an olive leaf plucked off. So Noah knew that the waters were abated from the earth. 8:12 He stayed yet another seven days, and sent out the dove; and she didn't return to him any more. The meaning behind a name Matthew 1:21 She shall bring forth a son. You shall call his name Jesus, for it is he who shall save his people from their sins." The writings of a prophet Matthew 2:5... They said to him, "In Bethlehem of Judea, for this is written through the prophet, 2:6 'You Bethlehem, land of Judah, are in no way least among the princes of Judah: for out of you shall come forth a governor, who shall shepherd my people, Israel.'" What God is able to do Matthew 3:9 Don't think to yourselves, 'We have Abraham for our father,' for I tell you that God is able to raise up children to Abraham from these stones. What happens to those who don't give fruit Matthew 3:10 "Even now the axe lies at the root of the trees. Therefore every tree that doesn't bring forth good fruit is cut down, and cast into the fire. What is at hand Matthew 3:1 In those days, John the Baptizer came, preaching in the wilderness of Judea, saying, 3:2 "Repent, for the Kingdom of Heaven is at hand!" What is known is meant to be known and followed Moses and the Jews. Deuteronomy 29:29 The secret things belong to Yahweh our God; but the things that are revealed belong to us and to our children forever, that we may do all the words of this law. What roles are appropriate for who? Matthew 3:13 Then Jesus came from Galilee to the Jordan to John, to be baptized by him. 3:14 But John would have hindered him, saying, "I need to be baptized by you, and you come to me?" Who watches over us Matthew 3:9 Don't think to yourselves, 'We have Abraham for our father,' for I tell you that God is able to raise up children to Abraham from these stones. Other ways: Buddhism Compassion for men of little virtue All these living beings, hearing such a statement, must indeed realize the thought of the difficulty of meeting a Buddha and cherish a longing and a thirst for him; thus they will cultivate the roots of goodness. Therefore the Tathagata, though he does not in reality become extinct, yet announces his extinction. From the book Buddhist Thought Through Later Centuries. Learning the difficulty of meeting a Buddha If the Buddha abides long in the world, men of little virtue - who do not cultivate the roots of goodness and are spiritually poor and mean, greedily attached to the five desires, and caught in the net of wrong reflection and false views - if they see the Tathagata constantly present and not extinct, will become puffed up and lazy and unable to conceive the idea that it is difficult to meet a Buddha, and be unable to develop a mind of reverence for him. Therefore the Tathagata tactfully says: "Know, Bhikshus! The appearance of a Buddha in the world is a rare occurence." Wherefore? In the course of countless hundreds, thousands, myriads, koits of kalpas, some men of little virtue may happen to see a Buddha, or none may see him. For this reason I say: 'Bhikshus! A Tathagata may rarely be seen!' All these living beings, hearing such a statement, must indeed realize the thought of the difficulty of meeting a Buddha and cherish a longing and a thirst for him; thus they will cultivate the roots of goodness. Therefore the Tathagata, though he does not in reality become extinct, yet announces his extinction. From the book Buddhist Thought Through Later Centuries. Other ways: Business What ways of figuring things out are relevant for business? Other ways: Chess I played a lot of chess in my youth. I want to share ways that I and others have figured things out. Assign values to chess pieces and add them up We can gauge which player has the material advantage by counting up and comparing the values of their remaining pieces. The usual scale is: Queen 9, Rook 5, Bishop or Knight 3, Pawn 1. This is not part of the rules, but simply a helpful method. Computer chess programs may use more subtle scales. The material advantage is important because it reflects greater opportunities, but also, because that relative advantage increases as pieces are traded off, so that exchanges become threats in and of themselves. Material advantage may be outweighted by the "initiative", which is to say, somebody may fall victim to a mating attack even though they have more pieces. Material advantage and initiative are like potential energy and kinetic energy. Other ways: Creative Arts Natalie d'Arbeloff investigates artistically with her book ''Designing With Natural Forms'', artistic investigations. See also: Experiments in Seeing. Allusion Wikipedia: "I don't use shamanism to refer to death, but vice versa - through shamanism, I refer to the fatal character of the times we live in. But at the same time I also point out that the fatal character of the present can be overcome in the future." Create an event score Wikipedia: The term "score" is used in exactly the sense that one uses the term to describe a music score: a series of notes that allow anyone to perform the work, an idea linked both to what Nam June Paik labeled the "do it yourself" approach and to what Ken Friedman termed "musicality." While much is made of the do it yourself approach to art, it is vital to recognize that this idea emerges in music, and such important Fluxus artists as Paik, Higgins, or Corner began as composers, bringing to art the idea that each person can create the work by "doing it." This is what Friedman meant by musicality, extending the idea more radically to conclude that anyone can create work of any kind from a score, acknowledging the composer as the originator of the work while realizing the work freely and even interpreting it in far different ways from those the original composer might have done. Event scores, such as George Brecht's "Drip Music", are essentially performance art scripts that are usually only a few lines long and consist of descriptions of actions to be performed rather than dialogue. Ring discussions Wikipedia: Another aspect of Beuys' pedagogy included open "ring discussions," where Beuys and his students discussed political and philosophical issues of the day, including the role of art, democracy, and the university in society. Some of Beuys' ideas espoused in class discussion and in his art-making included free art education for all, the discovery of creativity in everyday life, and the belief that "everyone [was] an artist." Other ways: Dance Twenty years or so ago I gave my sister a book on dance that was very philosophical. I think it had eight main concepts. The first was breathing. Other ways: David Ellison-Bey I'm collecting ways that my friend David Ellison-Bey has figured things out. He leads the Moorish Cultural Workshop. Living by principles David Ellison-Bey, 2011.04.27: Living by principles has us see beyond the hype. Having principles and living by them in practicing the best in you which has gotten you this far along, seeing beyond the hype. Look at a person of great character David Ellison-Bey, 2011.04.27: We can look at Mother Theresa, look at her life, how she dedicated her life. Did she have to fill out her income tax? Did she get anything from the diocese, even? Did she have to account for it? Have a secretary, financial secretary, recording secretary? But she was going around picking people off the street. What have we not figured out? David Ellison-Bey, 2011.04.27: ...and we still haven't figured out the money system out for the corruption and greed for the materials of life that's beyond many in their lifetimes of work, sweat and blood, etc. Other ways: God I'm collecting ways that God figures things out as evident from passages in the Bible. I'm also considering them from what I myself know of God personally and through my philosophy, for example, God's headlong nature, and his interest in love. Appreciate human nature Genesis Yahweh said in his heart, "I will not again curse the ground any more for man's sake, because the imagination of man's heart is evil from his youth; neither will I ever again strike everything living, as I have done. 8:22 While the earth remains, seed time and harvest, and cold and heat, and summer and winter, and day and night shall not cease." Assign a purpose Genesis 1:11 God said, "Let the earth yield grass, herbs yielding seed, and fruit trees bearing fruit after their kind, with its seed in it, on the earth"; and it was so. 1:12 The earth yielded grass, herbs yielding seed after their kind, and trees bearing fruit, with its seed in it, after their kind; and God saw that it was good. Assign a role Genesis 1:14 God said, "Let there be lights in the expanse of sky to divide the day from the night; and let them be for signs, and for seasons, and for days and years; 1:15 and let them be for lights in the expanse of sky to give light on the earth"; and it was so. 1:16 God made the two great lights: the greater light to rule the day, and the lesser light to rule the night. He also made the stars. 1:17 God set them in the expanse of sky to give light to the earth, 1:18 and to rule over the day and over the night, and to divide the light from the darkness. Be thorough Genesis 7:20 The waters prevailed fifteen cubits upward, and the mountains were covered. 7:21 All flesh died that moved on the earth, including birds, livestock, animals, every creeping thing that creeps on the earth, and every man. 7:22 All in whose nostrils was the breath of the spirit of life, of all that was on the dry land, died. 7:23 Every living thing was destroyed that was on the surface of the ground, including man, livestock, creeping things, and birds of the sky. They were destroyed from the earth. Only Noah was left, and those who were with him in the ship. 7:24 The waters prevailed on the earth one hundred fifty days. Bless with purpose Genesis 1:22 God blessed them, saying, "Be fruitful, and multiply, and fill the waters in the seas, and let birds multiply on the earth." Command Genesis 1:3 God said, "Let there be light," and there was light. God created the earth and seas 1:9 God said, "Let the waters under the sky be gathered together to one place, and let the dry land appear"; and it was so. God created the sky between the waters above and below 1:6 God said, "Let there be an expanse in the middle of the waters, and let it divide the waters from the waters." Consider all possibilities God's mind is unlimited. Thus God considers all possibilities, sensible or not. We are all thoughts in God's mind, like variations on a chess board, which may not be played (as Jesus, the winning variant, was played), yet are all relevant for winning the game. Counting the generations Matthew 1:17 So all the generations from Abraham to David are fourteen generations; from David to the exile to Babylon fourteen generations; and from the carrying away to Babylon to the Christ, fourteen generations. Divide one thing from out of another thing Genesis: God divided the light from the darkness. 1:5 God called the light "day," and the darkness he called "night." There was evening and there was morning, one day. Divide the earth from the sea Genesis 1:9 God said, "Let the waters under the sky be gathered together to one place, and let the dry land appear"; and it was so. Divide with an expanse Genesis 1:6 God said, "Let there be an expanse in the middle of the waters, and let it divide the waters from the waters." 1:7 God made the expanse, and divided the waters which were under the expanse from the waters which were above the expanse; and it was so. 1:8 God called the expanse "sky." There was evening and there was morning, a second day. Dividing everything into perspectives God thinks by dividing everything into perspectives. These divisions accord with his days of creation in Genesis. A day for God is an event, and that event is a division of everything. Ensure the survival of the species Genesis 6:20 Of the birds after their kind, of the livestock after their kind, of every creeping thing of the ground after its kind, two of every sort shall come to you, to keep them alive. Explain Genesis 1:27 God created man in his own image. In God's image he created him; male and female he created them. 1:28 God blessed them. God said to them, "Be fruitful, multiply, fill the earth, and subdue it. Have dominion over the fish of the sea, over the birds of the sky, and over every living thing that moves on the earth." 1:29 God said, "Behold, I have given you every herb yielding seed, which is on the surface of all the earth, and every tree, which bears fruit yielding seed. It will be your food. 1:30 To every animal of the earth, and to every bird of the sky, and to everything that creeps on the earth, in which there is life, I have given every green herb for food"; and it was so. Favor Genesis 6:8 But Noah found favor in Yahweh's eyes. Fill with life Genesis 1:20 God said, "Let the waters swarm with swarms of living creatures, and let birds fly above the earth in the open expanse of sky." 1:21 God created the large sea creatures, and every living creature that moves, with which the waters swarmed, after their kind, and every winged bird after its kind. God saw that it was good. Finish work Genesis 2:1 The heavens and the earth were finished, and all their vast array. 2:2 On the seventh day God finished his work which he had made; Genealogy Matthew 1:1 The book of the genealogy of Jesus Christ, the son of David, the son of Abraham. 1:2 Abraham became the father of Isaac. Isaac became the father of Jacob. ... Matthan became the father of Jacob. 1:16 Jacob became the father of Joseph, the husband of Mary, from whom was born Jesus, who is called Christ. Give dominion over Genesis 1:26 God said, "Let us make man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the birds of the sky, and over the livestock, and over all the earth, and over every creeping thing that creeps on the earth." 1:27 God created man in his own image. In God's image he created him; male and female he created them. Give names to things Genesis 1:5 God called the light "day," and the darkness he called "night." Name the earth and seas Genesis 1:10 God called the dry land "earth," and the gathering together of the waters he called "seas." Name the sky Genesis 1:8 God called the expanse "sky." God rejoices in both multiplying and destroying Deuteronomy 28:63 It shall happen that as Yahweh rejoiced over you to do you good, and to multiply you, so Yahweh will rejoice over you to cause you to perish, and to destroy you; and you shall be plucked from off the land where you go in to possess it. Headlong God does as God thinks. They are the same. Thus God is headlong. God may ask, would God be even if God wasn't? and the experiment proceeds directly. Judge a person's heart God instructed Samuel to anoint Jesse's son David. God said he chooses not as humans do, not on outward appearance, but on the heart. Even so, David was handsome. Samuel I 16:6 It happened, when they had come, that he looked at Eliab, and said, "Surely Yahweh's anointed is before him." 16:7 But Yahweh said to Samuel, "Don't look on his face, or on the height of his stature; because I have rejected him: for I see not as man sees; for man looks at the outward appearance, but Yahweh looks at the heart." 16:8 Then Jesse called Abinadab, and made him pass before Samuel. He said, "Neither has Yahweh chosen this one." 16:9 Then Jesse made Shammah to pass by. He said, "Neither has Yahweh chosen this one." 16:10 Jesse made seven of his sons to pass before Samuel. Samuel said to Jesse, "Yahweh has not chosen these." 16:11 Samuel said to Jesse, "Are all your children here?" He said, "There remains yet the youngest, and behold, he is keeping the sheep." Samuel said to Jesse, "Send and get him; for we will not sit down until he comes here." 16:12 He sent, and brought him in. Now he was ruddy, and withal of a beautiful face, and goodly to look on. Yahweh said, "Arise, anoint him; for this is he." Make according to kind Genesis 1:21 God created the large sea creatures, and every living creature that moves, with which the waters swarmed, after their kind, and every winged bird after its kind. ... 1:25 God made the animals of the earth after their kind, and the livestock after their kind, and everything that creeps on the ground after its kind. Make in God's own image Genesis 1:26 God said, "Let us make man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the birds of the sky, and over the livestock, and over all the earth, and over every creeping thing that creeps on the earth." 1:27 God created man in his own image. In God's image he created him; male and female he created them. Make provision Genesis 6:21 Take with you of all food that is eaten, and gather it to yourself; and it will be for food for you, and for them. Observe the thoughts of people's hearts Genesis 6:5 Yahweh saw that the wickedness of man was great in the earth, and that every imagination of the thoughts of his heart was only evil continually. Provide for the ecosystem Genesis 7:1 Yahweh said to Noah, "Come with all of your household into the ship, for I have seen your righteousness before me in this generation. 7:2 You shall take seven pairs of every clean animal with you, the male and his female. Of the animals that are not clean, take two, the male and his female. 7:3 Also of the birds of the sky, seven and seven, male and female, to keep seed alive on the surface of all the earth." Remember Genesis 8:1 God remembered Noah, all the animals, and all the livestock that were with him in the ship; and God made a wind to pass over the earth. The waters subsided. Remember my covenant with a token Genesis 9:8 God spoke to Noah and to his sons with him, saying, 9:9 "As for me, behold, I establish my covenant with you, and with your offspring after you, 9:10 and with every living creature that is with you: the birds, the livestock, and every animal of the earth with you, of all that go out of the ship, even every animal of the earth. 9:11 I will establish my covenant with you: all flesh will not be cut off any more by the waters of the flood, neither will there ever again be a flood to destroy the earth." 9:12 God said, "This is the token of the covenant which I make between me and you and every living creature that is with you, for perpetual generations: 9:13 I set my rainbow in the cloud, and it will be for a sign of a covenant between me and the earth. 9:14 It will happen, when I bring a cloud over the earth, that the rainbow will be seen in the cloud, 9:15 and I will remember my covenant, which is between me and you and every living creature of all flesh, and the waters will no more become a flood to destroy all flesh. 9:16 The rainbow will be in the cloud. I will look at it, that I may remember the everlasting covenant between God and every living creature of all flesh that is on the earth." 9:17 God said to Noah, "This is the token of the covenant which I have established between me and all flesh that is on the earth." Removing himself God thinks as he creates, and he creates by removing himself, thereby considering whether he would be even if he was not. Repeat what is important Genesis 1:26 God said, "Let us make man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the birds of the sky, and over the livestock, and over all the earth, and over every creeping thing that creeps on the earth." 1:27 God created man in his own image. In God's image he created him; male and female he created them. Rest Genesis: and he rested on the seventh day from all his work which he had made. 2:3 God blessed the seventh day, and made it holy, because he rested in it from all his work which he had created and made. See corruption Genesis 6:11 The earth was corrupt before God, and the earth was filled with violence. 6:12 God saw the earth, and saw that it was corrupt, for all flesh had corrupted their way on the earth. See everything Genesis 1:31 God saw everything that he had made, and, behold, it was very good. See righteousness Genesis 7:1 Yahweh said to Noah, "Come with all of your household into the ship, for I have seen your righteousness before me in this generation." See what God created Genesis 1:1 In the beginning God created the heavens and the earth. 1:2 Now the earth was formless and empty. Darkness was on the surface of the deep. God's Spirit was hovering over the surface of the waters. 1:3 God said, "Let there be light," and there was light. 1:4 God saw the light, and saw that it was good. Self-collaborate Note the use of the plural "us". Genesis 1:26 God said, "Let us make man in our image, after our likeness: and let them have dominion... Smell the pleasant aroma Genesis 8:20 Noah built an altar to Yahweh, and took of every clean animal, and of every clean bird, and offered burnt offerings on the altar. 8:21 Yahweh smelled the pleasant aroma. Yahweh said in his heart, "I will not again curse the ground any more for man's sake, because the imagination of man's heart is evil from his youth; neither will I ever again strike everything living, as I have done. 8:22 While the earth remains, seed time and harvest, and cold and heat, and summer and winter, and day and night shall not cease." Solildarity with similarity God has solidarity with man because God made man in his own image. Genesis 9:6 Whoever sheds man's blood, his blood will be shed by man, for God made man in his own image. Sorrow and regret Genesis 6:6 Yahweh was sorry that he had made man on the earth, and it grieved him in his heart. 6:7 Yahweh said, "I will destroy man whom I have created from the surface of the ground; man, along with animals, creeping things, and birds of the sky; for I am sorry that I have made them." Truth of the heart and truth of the world God thinks the truth of the heart, but God allows for the truth of the world. God sees how we relate them. Which one do we choose? Unfolding perspective God thinks by unfolding his perspective, as in my video summary, "I Wish to Know", generating I, then You, then Other. What fulfilled a prophecy Matthew 1:22 Now all this has happened, that it might be fulfilled which was spoken by the Lord through the prophet, saying, 1:23 "Behold, the virgin shall be with child, and shall bring forth a son. They shall call his name Immanuel"; which is, being interpreted, "God with us." Fulfilled: He will be called a Nazarene Matthew ... Being warned in a dream, he withdrew into the region of Galilee, 2:23 and came and lived in a city called Nazareth; that it might be fulfilled which was spoken through the prophets: "He will be called a Nazarene." Fulfilled: Out of Egypt I called my son Matthew 2:14 He arose and took the young child and his mother by night, and departed into Egypt, 2:15 and was there until the death of Herod; that it might be fulfilled which was spoken by the Lord through the prophet, saying, "Out of Egypt I called my son." Isaiah prophesied John the Baptist Matthew 3:3 For this is he who was spoken of by Isaiah the prophet, saying, "The voice of one crying in the wilderness, make ready the way of the Lord. Make his paths straight." Jeremiah's prophecy fulfilled Matthew 2:17 Then that which was spoken by Jeremiah the prophet was fulfilled, saying, 2:18 "A voice was heard in Ramah, lamentation, weeping and great mourning, Rachel weeping for her children; she wouldn't be comforted, because they are no more." What is compatible Genesis 6:1 It happened, when men began to multiply on the surface of the ground, and daughters were born to them, 6:2 that God's sons saw that men's daughters were beautiful, and they took for themselves wives of all that they chose. 6:3 Yahweh said, "My Spirit will not strive with man forever, because he also is flesh; yet will his days be one hundred twenty years." Who is God's beloved Son Matthew 3:17 Behold, a voice out of the heavens said, "This is my beloved Son, with whom I am well pleased." Who pleases Matthew 3:17 Behold, a voice out of the heavens said, "This is my beloved Son, with whom I am well pleased." Other ways: History Analyze Theories of History Wikipedia lists some 300 theories of history. I can analyze each theory to see what "way of figuring things out" each presumes. Historical method Wikipedia: The historical method comprises the techniques and guidelines by which historians use primary sources and other evidence to research and then to write histories in the form of accounts of the past. The question of the nature, and even the possibility, of a sound historical method is raised in the philosophy of history as a question of epistemology. Historiography Wikipedia has hundreds of pages related to historiography: Historiography is the study of historians and their work. It is writing about rather than of history. Historiography is meta-level analysis of descriptions of the past. The analysis usually focuses on the narrative, interpretations, worldview, use of evidence, or method of presentation of other historians. There is also a journal Historical Methods: A Journal of Quantitative and Interdisciplinary History. Cognitive preference for having an orientation in time The cognitive interest of human beings in having an orientation in time is one of five factors in Jörn Rüsen's "disciplinary matrix" of historiography. Empirical research methods Empirical research methods are one of five factors in Jörn Rüsen's "disciplinary matrix" of historiography. Forms of representation Forms of representation are one of five factors in Jörn Rüsen's "disciplinary matrix" of historiography. Jörn Rüsen's disciplinary matrix Jörn Rüsen's "disciplinary matrix" of historiography relates five factors: the cognitive interest of human beings in having an orientation in time; theories or "leading views" concerning the experiences of the past; empirical research methods; forms of representation; the function of offering orientation to society. See: "Jörn Rüsen's Theory of Historiography between Modernism and Rhetoric of Inquiry" by Alan Megill, Vol. 33, No. 1, Feb., 1994, History and Theory. Also: compiled by Zenonas Norkus. Leading views concerning the experiences of the past Theories or "leading views" concerning the experiences of the past is one of five factors in Jörn Rüsen's "disciplinary matrix" of historiography. Loanwords Loanwords (words borrowed by one language from another language) are studied as evidence that a concept (such as "books") traveled from one culture to another. Zigmas Zinkevicius studied Lithuanian words in this way. National psyche When I think of the behavior of the Russian nation, especially its brutally domineering attitude towards Lithuania and other neighboring countries, then I wonder, where did that come from? And so I trace it back to the Mongol occupation of the Russian principalities, which was remembered as very brutal. I imagine that the subsequent history of the Russian nation as a whole is a reaction to that traumatic experience, much as an adult's behavior can sometimes be explained as a reaction to childhood experiences. It seems that feelings of national solidarity can carry down attitudes for centuries, long beyond the source of those attitudes is relevant. I think the Serbian nation similarly experienced were traumatized by the Turks. I think the Spanish nation was traumatized by the Moors, and attributed its victory to Catholicism, and thus pursued with zeal the Inquisition and missionary work in the New World. The American South, Black American and even the urban North of the United States deal with trauma of slavery and the end of the plantation culture. National psyche is the mindset that is attributed to a nation rather than to the individuals of the nation. The latter would be the ethnic character or cultural influences on people. The Wikipedia article on National Psychology confuses these. Offering orientation to society The function of offering orientation to society is one of five factors in Jörn Rüsen's "disciplinary matrix" of historiography. Semiotic algebra of symbols Semiotics is the study of symbols and can be used to study history, for example, who reacted to whom, who chose to use who's symbols, and who chose to use different or opposed symbols, and who chose to join symbols of various origins. Other ways: Humor Other ways: Law I'm collecting legal theories for ways of figuring things out. I have the books "The Theory of Rules" by Karl N. Llewellyn, "The Legal Analyst" by Ward Farnsworth, "Truth, Error, and Criminal Law: An Essay in Legal Epistemology" by Larry Laudan, and "Thinking Like a Laywer: An Introduction to Legal Reasoning" by Kenneth J Vandevelde. ÀÈÆËËÁËÐÁØ Other ways: Medicine I think there may be clients or sponsors interested that I collect ways of figuring things out in medicine, nutrition, etc. Other ways: Music Other ways: Philosophy Acknowledge ruptures Wikipedia: The idea, here, is that a truth's invariance makes it genuinely indiscernible: because a truth is everywhere and always the case, it passes unnoticed unless there is a rupture in the laws of being and appearance, during which the truth in question becomes, but only for a passing moment, discernible. Such a rupture is what Badiou calls an event, according to a theory originally worked out in Being and Event and fleshed out in important ways in Logics of Worlds. Amend the truth Wikipedia about Badiou: While such knowledge is produced in the process of being faithful to a truth event, it should be noted that, for Badiou, knowledge, in the figure of the encyclopedia, always remains fragile, subject to what may yet be produced as faithful subjects of the event produce further knowledge. Analyze examples from literature Wikipedia about Badiou: He develops these ideas with examples from the prose of Samuel Beckett and the poetry of Stéphane Mallarmé and Fernando Pessoa (who he argues has developed a body of work that philosophy is currently incapable of incorporating), among others. Apply concepts external to one's method Wikipedia: Consequently, philosophy is, according to Badiou, a thinking of the compossibility of the several truth procedures, whether ..., or whether this is undertaken through the more traditionally philosophical work of addressing categories like truth or the subject (concepts that are, as concepts, external to the individual truth procedures, though they are functionally operative in the truth procedures themselves). Arrange concepts to spark a new insight Wikipedia about Badiou: His view of the link between philosophy and art is tied into the motif of pedagogy, which he claims functions so as to "arrange the forms of knowledge in a way that some truth may come to pierce a hole in them". Combine different methods Wikipedia: Consequently, philosophy is, according to Badiou, a thinking of the compossibility of the several truth procedures, whether this is undertaken through the investigation of the intersections between distinct truth procedures (the intersection of art and love in the novel, for instance), or whether this is undertaken through the more traditionally philosophical work of addressing categories like truth or the subject (concepts that are, as concepts, external to the individual truth procedures, though they are functionally operative in the truth procedures themselves). Consider methods as methods (or not) Wikipedia: Truth, for Badiou, is a specifically philosophical category. While philosophy's several conditions are, on their own terms, "truth procedures" (i.e., they produce truths as they are pursued), it is only philosophy that can speak of the several truth procedures as truth procedures. (The lover, for instance, does not think of her love as a question of truth, but simply and rightly as a question of love. Only the philosopher sees in the true lover's love the unfolding of a truth.) Consider what a concept denies Wikipedia: In Handbook of Inaesthetics Badiou coins the phrase "inaesthetic" to refer to a concept of artistic creation that denies "the reflection/object relation". Reacting against the idea of mimesis, or poetic reflection of "nature", Badiou claims that art is "immanent" and "singular". Distance oneself with theoretical terminology Wikipedia: For Badiou, when philosophy addresses the four truth procedures in a genuinely philosophical manner, rather than through a suturing abandonment of philosophy as such, it speaks of them with a theoretical terminology that marks its philosophical character: inaesthetics rather than art; metapolitics rather than politics; ontology rather than science; etc. Distinguish what is and what seems Wikipedia: The idea, here, is that a truth's invariance makes it genuinely indiscernible: because a truth is everywhere and always the case, it passes unnoticed unless there is a rupture in the laws of being and appearance, during which the truth in question becomes, but only for a passing moment, discernible. Embrace competing notions Wikipedia: Badiou at once embraces the traditional modernist notion that truths are genuinely invariant (always and everywhere the case, eternal and unchanging) and the incisively postmodernist notion that truths are constructed through processes. Faithfully anounce the truth Wikipedia about Badiou: The subject who chances to witness such an event, if she is faithful to what she has glimpsed, can then introduce the truth by naming it into worldly situations. According to a process or procedure that subsequently unfolds only if those who subject themselves to the glimpsed truth continue faithful in the work of announcing the truth in question, genuine knowledge is produced (knowledge often appears in Badiou's work under the title of the "veridical"). Found in one place only? Wikipedia: Reacting against the idea of mimesis, or poetic reflection of "nature", Badiou claims that art is "immanent" and "singular". It is immanent in the sense that its truth is given in its immediacy in a given work of art, and singular in that its truth is found in art and art alone. Intersect different methods Wikipedia: Consequently, philosophy is, according to Badiou, a thinking of the compossibility of the several truth procedures, whether this is undertaken through the investigation of the intersections between distinct truth procedures (the intersection of art and love in the novel, for instance)... Mediated by other or self? Wikipedia: Reacting against the idea of mimesis, or poetic reflection of "nature", Badiou claims that art is "immanent" and "singular". It is immanent in the sense that its truth is given in its immediacy in a given work of art, and singular in that its truth is found in art and art alone. Name the truth into the world Wikipedia about Badiou: The subject who chances to witness such an event, if she is faithful to what she has glimpsed, can then introduce the truth by naming it into worldly situations. Outstrip knowledge with faith Wikipedia: According to Badiou, truth procedures proceed to infinity, such that faith (fidelity) outstrips knowledge. (Badiou, following both Lacan and Heidegger, distances truth from knowledge.) Reconcile Wikipedia: The major propositions of Badiou's philosophy all find their basis in Being and Event, in which he continues his attempt (which he began in Théorie du sujet) to reconcile a notion of the subject with ontology, and in particular post-structuralist and constructivist ontologies.[9] A frequent criticism of post-structuralist work is that it prohibits, through its fixation on semiotics and language, any notion of a subject. Badiou's work is, by his own admission,[10] an attempt to break out of contemporary philosophy's fixation upon language, which he sees almost as a straitjacket. ... two elements mark the thesis of Being and Event: the place of ontology, or 'the science of being qua being' (being in itself), and the place of the event - which is seen as a rupture in being - through which the subject finds realization and reconciliation with truth. This situation of being and the rupture which characterizes the event are thought in terms of set theory, and specifically Zermelo-Fraenkel set theory (with the axiom of choice) ... Uncouple concepts Wikipedia: Badiou's theory of truth, exposited throughout his work, accomplishes this strange mixture by uncoupling invariance and self-evidence (such that invariance does not imply self-evidence), as well as by uncoupling constructedness from relativity (such that constructedness does not lead to relativism). Witness unexpectedly Wikipedia about Badiou: The subject who chances to witness such an event, if she is faithful to what she has glimpsed, can then introduce the truth by naming it into worldly situations. Other ways: poetry Writing Tom Mandel: Writing is my primary tool for investigating the world. Pamela McLean, April 11, 2011: My hunch is that any ideas I might put forward on developing educational systems would probably be 99% ignored and 1% ridiculed - but I might have got the proportions wrong. Pamela McLean, April 11, 2011: They will agree on areas for investigation and will work collaboratively. Pamela McLean, April 11, 2011: All our courses will be learner centred. Pamela McLean, April 11, 2011: The resources that they collect and the original work they produce will be shared using whatever is Dadamac's current system (at present that is through our standard spreadsheet structure - see Published Dadamac spreadsheets and docs - compiled 2011 for examples of the structure). Pamela McLean, April 11, 2011: Students will be based "within the Dadamac.net space". They will create a collection of work, things they have written, practical projects they have done, people they have learned from and so on. Over the years,as they study (and help others to study) they will develop evidence of work done, and a high trust network of fellow learners, teachers and collaborators. Pamela McLean, April 11, 2011: The benefit of a prototype is that people can see it and prod at it and give feedback of one kind and another. The feedback may not be helpful and it may not be kind (although it might be) and so it is tempting to keep the prototype hidden from public view until it is reasonably robust and well tested. It may seem preferable to simply share "the idea" of the prototype - how it will work, what it will look like, how useful it will be. However when people give feedback on an idea there are so many possible areas of complete misunderstanding that the subsequent discussions may be fruitless. Even if the prototype is on the rickety side it is something definite to point to and discuss - so it has advantages over an idea. ... So, here goes with my prototype 21st century education system. Pamela McLean, April 11, 2011: Fortunately, because we are an online and emergent organisation we don't need a big budget or a big launch - just a vision and gradual growth. Pamela McLean, April 11, 2011: I have some clear ideas of how educational systems could emerge during the 21st century (and how I would like them to). Pamela McLean, April 11, 2011: My hunch is that any ideas I might put forward on developing educational systems would probably be 99% ignored and 1% ridiculed - but I might have got the proportions wrong. Pamela McLean, April 11, 2011: You are welcome to join me. Pamela McLean, April 11, 2011: I appreciate that friends might support me - but as they are biased that probably wouldn't count. Pamela McLean, April 11, 2011: My hunch is that any ideas I might put forward on developing educational systems would probably be 99% ignored and 1% ridiculed - but I might have got the proportions wrong. Pamela McLean, April 11, 2011: In recognition of its roots in Dadamac - and because I like the sound of the word - I name it "The Dadamacadamy" (to pronounce it say "acadamy", say "macadamy" say "dada', then run those two together for "Dadamacadamy"). Pamela McLean, April 11, 2011: There will be no formal accreditation from Dadamac, but there will be references, and evidence, and an increasing network of clients and other employers ready to pay for the services of people with a good Dadamac profile. Pamela McLean, April 11, 2011: Fortunately, because we are an online and emergent organisation we don't need a big budget or a big launch - just a vision and gradual growth. Pamela McLean, April 11, 2011: Faculties will emerge according to the interests of people who study with us and the research that we do. Pamela McLean, April 11, 2011: The resources that they collect and the original work they produce will be shared using whatever is Dadamac's current system (at present that is through our standard spreadsheet structure - see Published Dadamac spreadsheets and docs - compiled 2011 for examples of the structure). Pamela McLean, April 11, 2011: If they do related studies (for example with our friends at http:peoples-uni.org) they will add that to their profile. Pamela McLean, April 11, 2011: The resources that they collect and the original work they produce will be shared using whatever is Dadamac's current system (at present that is through our standard spreadsheet structure - see Published Dadamac spreadsheets and docs - compiled 2011 for examples of the structure). Pamela McLean, April 11, 2011: The resources that they collect and the original work they produce will be shared using whatever is Dadamac's current system (at present that is through our standard spreadsheet structure - see Published Dadamac spreadsheets and docs - compiled 2011 for examples of the structure). Pamela McLean, April 11, 2011: Students will normally study in small online Special Interest Groups. Pamela McLean, April 11, 2011: Fortunately, because we are an online and emergent organisation we don't need a big budget or a big launch - just a vision and gradual growth. Pamela McLean, April 11, 2011: I'm already studying there, and teaching, and doing research. Pamela McLean, April 11, 2011: Fortunately, because we are an online and emergent organisation we don't need a big budget or a big launch - just a vision and gradual growth. Pamela McLean, April 11, 2011: Given our current areas of activity it seems likely that studies related to health, development, education, social-processes on the Internet, and ecology will come to the fore. Thought experiments Wikipedia lists miscellaneous thought experiments: Buttered cat paradox, Braitenberg vehicles (robotics, neural control and sensing systems) (some have actually been built), Doomsday argument (anthropic principle), Dyson sphere, The Lady, or the Tiger? (human nature), The Planiverse Levinthal's paradox Wikipedia: Levinthal's paradox is a thought experiment, also constituting a self-reference in the theory of protein folding. In 1969, Cyrus Levinthal noted that, because of the very large number of degrees of freedom in an unfolded polypeptide chain, the molecule has an astronomical number of possible conformations.... Therefore if a protein were to attain its correctly folded configuration by sequentially sampling all the possible conformations, it would require a time longer than the age of the universe to arrive at its correct native conformation. This is true even if conformations are sampled at rapid (nanosecond or picosecond) rates. The "paradox" is that most small proteins fold spontaneously on a millisecond or even microsecond time scale. This paradox is central to computational approaches to protein structure prediction. Thought experiments in Philosophy Wikipedia lists thought experiments: Artificial brain, Avicenna's Floating Man, Bellum omnium contra omnes, Big Book (ethics), Brain-in-a-vat (epistemology, philosophy of mind), Brainstorm machine, Buridan's ass, Changing places (reflexive monism, philosophy of mind), China brain (physicalism, philosophy of mind), Chinese room (philosophy of mind, artificial intelligence, cognitive science), Coherence (philosophical gambling strategy), Condillac's Statue (epistemology), Gettier problem (epistemology), Ḥayy ibn Yaqẓân (epistemology), Hilary Putnam's Twin Earth thought experiment in the philosophy of language and philosophy of mind, How many men? (taxation as theft), Inverted spectrum, Kavka's toxin puzzle, Mary's room (philosophy of mind), Molyneux's Problem (admittedly, this oscillated between empirical and a-priori assessment), Newcomb's paradox, Original position (politics), Philosophical zombie (philosophy of mind, artificial intelligence, cognitive science), Plank of Carneades, Prisoner's Dilemma, Ship of Theseus, The (concept of identity), Simulated reality (philosophy, computer science, cognitive science), Social contract theories, Survival lottery, The (ethics), Swamp man (personal identity, philosophy of mind), Shoemaker's "Time Without Change" (metaphysics), Ticking time bomb scenario (ethics), Teleport (metaphysics), Trolley problem (ethics), The Violinist (ethics), Utility monster (ethics), Zeno's paradoxes (classical Greek problems of the infinite) Plato's Cave Plato considers an elaborate scenario where people sit in a cave and view projections which they purport to be real. (He invented movies for the purpose of this scenario!) A person ventures outside the cave and discovers the real world, then goes back to the others and tries to tell them of it, but they deride him. |
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Other ways: Physics
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Other ways: Physics Brocken spectre Brocken spectre is a spectacular atmospheric phenomenon. Cloud chamber Cloud chambers are used to observe cosmic rays - elementary particles. Classifying Ehrenfest classification of phase transitions Wikipedia: Paul Ehrenfest classified phase transitions based on the behavior of the thermodynamic free energy as a function of other thermodynamic variables. Under this scheme, phase transitions were labeled by the lowest derivative of the free energy that is discontinuous at the transition. First-order phase transitions exhibit a discontinuity in the first derivative of the free energy with respect to some thermodynamic variable.[1] The various solid/liquid/gas transitions are classified as first-order transitions because they involve a discontinuous change in density, which is the first derivative of the free energy with respect to chemical potential. Second-order phase transitions are continuous in the first derivative (the order parameter, which is the first derivative of the free energy with respect to the external field, is continuous across the transition) but exhibit discontinuity in a second derivative of the free energy.[1] These include the ferromagnetic phase transition in materials such as iron, where the magnetization, which is the first derivative of the free energy with the applied magnetic field strength, increases continuously from zero as the temperature is lowered below the Curie temperature. The magnetic susceptibility, the second derivative of the free energy with the field, changes discontinuously. Under the Ehrenfest classification scheme, there could in principle be third, fourth, and higher-order phase transitions. Modern classification of phase transitions Wikipedia: In the modern classification scheme, phase transitions are divided into two broad categories, named similarly to the Ehrenfest classes:
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