How do you figure things out?677 ways
Choose from the domains above.
Write to Andrius (MS@MS.LT) to add more ways!
Research & Culture & Business

               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".1594

               Please crowdfund my philosophical portrait of Jesus!      Thank you for contributing money and for spreading the word! 1596

        Andrius Kulikauskas's ways of figuring things out

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:

  1. Obey the teacher.
  2. If you can't obey the teacher, then believe the good student, follow their example.
  3. If you can't believe any student, then simply care about the subject.
  4. If you don't care much, then be honest so that you don't get stuck, and you can keep trying.
  5. If you are honest, then you will start to care, you will appreciate those who are believable, and you will find the Teacher (that is, God) who you can obey.

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.

  • What questions does it take on? (Examples)
  • What does it interest us in? (Surface issue)
  • How does the question create tension? (Evident emotion)
  • How does the answer relax tension? (Counterbalancing emotion)
  • What structure does it evoke? (Deep issue)
  • Who are we conversing with?
  • How do we understand the structure more broadly as us amongst others? (Truth of the world)
  • How do we supplement the structure with ourselves? (Truth of the world follows from the truth of the heart)
  • How does the new structure express God through God's point of view? (Truth of the heart)
  • How does tension arise anew?
  • Consider also: How we find and engage God? conditions and unconditionality, morality, faith, scope, commonality, culture, threads, multiplicity of perspectives, negation of assumptions, importance, education, progression, the will of God.
I am looking especially for representative ways, one for each room in the House of Knowledge. I'm also wondering, what are compositions of ways of figuring things out? And how do the ways of figuring things out relate to languages of how thing come to matter (argumentation), come to have meaning (verbalization), and come to happen (narration).

1

        Taxonomy

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:
  • Avoiding evil, not shutting ourselves down
One branch models our inner world:
  • Rooting ourselves in our inner world, believing
  • Presuming others likewise, believing in believing
  • Integrating some of us, believing in believing in believing
  • Fostering an environment for integration, believing in believing in believing in believing
Another branch models our outer world:
  • Being completely open to the outer world, caring
  • Focusing and maximizing our openness, caring about caring
  • Recognizing the limits of our openness, caring about caring about caring
  • Allowing for an ideal that transcends our limits, caring about caring about caring about caring
The two branches are loosely coupled by the scientific method:
  • Taking a stand, having a hypothesis, extending the applicability of what we take to be true
  • Following through, designing and doing an experiment, driving it to its "logical" conclusion, breaking the model or not
  • Reflecting, noting the outcome, generalizing it as a principle
The two branches, taken as wholes, are completely matched:
  • Allowing for a person-in-general, matching up the inner and outer worlds
Then we have a system where people etc. can substitute for each other. We can have valuation or truth or reality etc. And we can think of them as relating two perspectives, like a game player (within the system) and a game maker (beyond the system). There are four levels of knowledge at which game player is related to game maker:
  • 0) Whether. Game player and game maker are taken to be the same, they are conflated, as when we learn by feeling how we feel in our heart or gut.
  • 1) What. Game player knows what we learn from experience whereas game maker knows the innate model.
  • 2) How. Game player's knowledge is implied by the game maker's knowledge, but not the other way around.
  • 3) Why. Game player and game maker are taken to be different, taking up different perspectives from Whether-What-How-Why, with the game maker taking the broader perspective, Why being the broadest. (This yields the six pairings that come next.)
Inside the system, there are six more ways, each of which accords with a counterquestion (doubts such as "How do I know I'm not a robot?" are addressed by relevant counterquestions such as "Would it make any difference?"). Each counterquestion inserts a broader perspective (Why=God's perspective, How=person-in-general's perspective, What=person-in-particular's perspective) into a narrower situation (How=person-in-general's situation, What=person-in-particular's situation, Whether=world's situation). So a person-in-general lives out a person-in-particular's situation as that question "Would it make any difference?" Furthermore, each of these ways accords with a restructuring (visualization) for building up a system.
  • How does it seem to me? evokes evolution (hierarchy restructured by sequence) for determining weights
  • What else should I be doing? evokes atlas (network restructured by hierarchy) for determining connections
  • Would it make any difference? evokes canon (sequence restructured by network) for determining priorities
  • What do I have control over? evokes chronicle (sequence restructured by hierarchy) for determining solutions
  • Am I able to consider the question? evokes catalog (hierarchy restructured by network) for determining redundancies
  • Is this the way things should be? evokes tour (network restructured by sequence) for determining paths
(If you want to see concrete examples of these six, look at the Sermon on the Mount! Jesus makes use of them all in his "antitheses", "you have heard it said... but I say unto you...", for example, if you love only your friends, how are you different from the pagans?) I also used them in a concrete form to coach peacemakers to engage murderous gangs on the roads in Kenya, namely: be straightforward, be thorough, be vulnerable, let them win, let them teach you, stick to your principles. And then those last ways are treated as a unity, like the gap or slack between the perspective and situation:
  • There is a greater context in which everything can be reinterpreted and get unexpected meaning. (As when 10+4=2 because we're talking about a clock.)
Another example of the 6+4 model is the Ten Commandments (4 positive "do's" and 6 negative "don'ts") and another example is John Caswell's Business Equation: http://www.selflearners.net/uploads/businessequation.gif You may think it's not practical because it's very abstract and very involved. It may be way too much. It's much of my life's work, and it's how I'm relating my life's work. But you can see how I'm applying it to different domains and it becomes concrete: Math, Physics, Jesus. (In June, 2011, I am crowdfunding Jesus's philosophical portrait.) If you gave me a list of anecdotes, then I could sort them for you and you would see. Then the categories would become more concrete. And they may perhaps become concrete enough for other people to use, but certainly for you to use. Indeed, this is what I think I should be doing with myself at this time, what would be best for everybody that I do. I have a system which I think takes any domain and organizes it according to a universal language. This means that specialists from different domains can talk to each other. But especially, as Pamela McLean has pointed out, it means that interdisciplinary people who are "specialist generalists" and good at talking with specialists from different fields, such people are able to show what they do. I can and will make "philosophical portraits" of such people, both for free and for pay, starting with Jesus and Pamela. It's a great help for me whoever would have me do their portrait, survey their personal ways of "figuring things out". 1306

     Bob Lichtenbert, a Philosophical Portrait      2011.05 Andrius Kulikauskas: I'm creating a philosophical portrait of Bob Lichtenbert.1442

     Pamela McLean, a philosophical portrait      Andrius Kulikauskas, 2011.05.11: Pamela McLean of Dadamac agreed that I draw her philosophical portrait by documenting and sharing her ways of figuring things out. She's a pioneer! Thank you, Pamela!1271

        Dee Guyton's ways of figuring things out

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.2

        Measurement

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. 850

        Don't cause believers to stumble

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.1252

               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."1258

          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.'"1324

               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? 1325

          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.1374

          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.1344

          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.2172

          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.1412

          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.1414

          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.1423

          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.1424

          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. 1102

               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.1104

          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.1111

          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. 1105

          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.1110

          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.1103

        Not shutting ourselves down

Not shutting ourselves down      1142

          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."1136

          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 1587

          Assessing probability honestly      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. The Thinker's Toolkit by Morgan D.Jones2146

          Don't cling to one scenario      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. The Thinker's Toolkit by Morgan D.Jones2143

          Satisficing      "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. The Thinker's Toolkit by Morgan D.Jones2142

        Working

Working      Don't be idle! Don't be unproductive or destructive.1079

        Leaving

Leaving      155

        Space

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.10

        Our inner world

Our inner world      1667

        Convergence

Convergence      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. The Thinker's Toolkit by Morgan D.Jones2108

        What you believe is what happens

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.1250

          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. 1477

          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."1546

          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."2019

          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. 1486

          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."2072

     Faith accepts God's love      2099

          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.1432

          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."1366

     Faith commands from God's certainty      2103

          Assurance in his own path      Matthew 8:22 But Jesus said to him, "Follow me, and leave the dead to bury their own dead." 1618

          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. 1478

     Faith is for those who wish for change      2102

          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." 1732

          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 2:181726

     Persistence of faith      2097

     Speak to their faith      2098

          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. 1609

          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. 1729

          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.1728

               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. 1727

          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."2100

     Their mindset traps them      2101

          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, Inc2220

               Axiom of the empty set      Wikipedia: There is a set such that no set is a member of it.1162

               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.1645

          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.1522

               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.1523

          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.1521

          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.2168

          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.1650

          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, Inc2243

          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.1520

          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.1233

          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.1224

          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.1222

     What includes everything      Conversing with: everything 1226

        Rooting ourselves in our inner world

Rooting ourselves in our inner world      1669

          Give away what I have      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.1013

        My actions

My actions      140

        Replying to Scripture with Scripture

Replying to Scripture with Scripture      The greater context makes exceptions legitimate.1329

          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. 1481

          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.1406

          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.1919

          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. 1401

          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.1335

          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.1319

          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. 1482

          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?" 1624

          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. 1813

     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. 1924

          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. 1378

               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. 1380

          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.2192

          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, Inc2202

          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, Inc2209

          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. 1742

          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. 1746

          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.2171

          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. 1744

          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. 1379

          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, Inc2200

          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.1511

               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.1644

               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.1509

          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.1743

          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.2170

        Presuming others likewise

Presuming others likewise      1670

        Major factors, major issues

Major factors, major issues      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. The Thinker's Toolkit by Morgan D.Jones2106

        Appreciate others

Appreciate others      As we give, we appreciate what others can and do give, for we may have them give on our behalf.1081

          Handicapping so as to even out chances      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.1006

          Noting people's inclinations      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.1064

          Take family stories to heart      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.1071

          Take to heart my employer's wishes      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.1066

        Hear

Hear      136

        Accepting God's justice

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.1347

          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.1620

          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. 1909

          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. 1598

          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?1655

          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." 1615

          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. 1576

          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." 1719

          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. 1917

          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. 1916

     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." 1925

          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.2188

        Plan - Explicitly Commit

Plan - Explicitly Commit      175

        Integrating some of us

Integrating some of us      1671

        Consider the analysis underlying numbers

Consider the analysis underlying numbers      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. The Thinker's Toolkit by Morgan D.Jones2145

          Assume probability is equal for all outcomes      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. The Thinker's Toolkit by Morgan D.Jones2147

          Calculate probability      ... when we have all the data, as in a deterministic problem, calculate probability by arithmetic computation. The Thinker's Toolkit by Morgan D.Jones2149

          Conditionally dependent events      Conditionally dependent events are those in which the occurrence of one event depends upon the occurrence of another. Multiply probabilities of the "and" type. The Thinker's Toolkit by Morgan D.Jones2152

          Mutually exclusive events      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%. The Thinker's Toolkit by Morgan D.Jones2151

          Translate probability expressions into percentiles      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. The Thinker's Toolkit by Morgan D.Jones2148

        Learn what is fair

Learn what is fair      1082

          Realizing the basis for leadership      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.1062

          Who is helping or hurting whom      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.1019

        Value others

Value others      158

        Healing

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.1364

          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.1363

          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." 1735

          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.1362

          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.1496

          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." 1653

          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, Inc2213

        Include - Care

Include - Care      176

        Fostering an environment for integration

Fostering an environment for integration      1672

          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.1443

        The Analytic Power of a Group

The Analytic Power of a Group      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. The Thinker's Toolkit by Morgan D.Jones2111

        Invest in giving

Invest in giving      Invest in a culture of giving1083

          Recognizing who is there for me      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.1050

        Others' limits

Others' limits      164

        Good heart: Saving those who stumble

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.1346

        Our outer world

Our outer world      1668

        Divergence

Divergence      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. The Thinker's Toolkit by Morgan D.Jones2109

        Taking

Taking      1097

        What you find is what you love

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.1251

          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." 1603

          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." 1601

          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. 1487

        Sequence

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.17

               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.1161

          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.1440

               Prove n! > 2**n where integer n >= 3      Weitz, pg.511498

               Prove that the sum of the interior angles of any n-gon is 180(n-2) degrees      Weitz, pg.501499

          6-8-5      generate many rough ideas quickly 1693

          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!1235

     Noting what we find no analogue for      1532

          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.1227

        Being completely open to the outer world

Being completely open to the outer world      1673

          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."1714

        Accept what I am given

Accept what I am given      1084

          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.1368

        My awareness

My awareness      145

          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? 1450

          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. 1436

          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."1612

          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."1410

          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."2074

          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.1331

          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.1360

          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."2007

          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. 1559

          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.1465

          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. 1473

          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. 2020

          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.2070

     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.1724

          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. 1739

          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, Inc2240

          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.2196

          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.2193

          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.1633

          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.1631

          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. 2257

          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.1921

     Opening up possibilities      1923

          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.1922

          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.1225

          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.1299

          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. 1590

          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. 1591

        Focusing and maximizing our openness

Focusing and maximizing our openness      1674

          Be with the best person to be influenced by      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.1009

          Consider how my time is best spent      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.1016

          Maximizing joy      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.1022

        Be present

Be present      143

          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.908

          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.904

          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.903

          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.1357

          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. 1566

          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. 1567

          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. 1568

          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. 1614

          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. 1737

          How people think in their hearts      Matthew 5:22 But Jesus, perceiving their thoughts, answered them, "Why are you reasoning so in your hearts?1321

          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. 1558

          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.1323

          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? 1479

          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."2077

          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.1462

          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. 2013

          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."2078

          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. 1551

          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.1918

          Will this be relevant to people?      Matthew 23:36 Most certainly I tell you, all these things will come upon this generation. 1577

          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.1365

               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.2255

               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. 1748

          Algorithmic proof      AM-GM reformulated...We have managed to change two of the n numbers in such a way that

  • one number that originally was not equal to A became equal to A;
  • the sum of all n numbers did not change;
  • the product of the n numbers increased.
Since there are finitely many numbers, this process will end when all of them are equal to A; then the product will be maximal. This proof is called "algorithmic" because the argument used describes a concrete procedure which optimizes the product in a step-by-step way, ending after a finite number of steps. pg.195-196 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2194

          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, Inc2234

          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, Inc2235

          Finite Injury Priority Method      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".1660

          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.2254

          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.2182

        Appeal - Implicitly Innovate

Appeal - Implicitly Innovate      179

        Recognizing the limits of our openness

Recognizing the limits of our openness      1675

          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.1880

          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.19341945

        Learn my limits

Learn my limits      Learn my limits, typically by overstepping them.1086

          Ask for help with a formality      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.1060

          Ask neighbors for donations      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.1065

          Break something      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.1024

          Not doing as I was told to do      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.1068

        Applying myself

Applying myself      149

     My limits      103

          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."1613

          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.' 1492

          How can judgment be escaped?      Matthew 23:33 You serpents, you offspring of vipers, how will you escape the judgment of Gehenna?1575

          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.1490

          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."2073

          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." 1560

          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. 1572

          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.1554

          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." 1927

        Limits

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). 20

          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, Inc2233

          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, Inc2231

          Infinite Injury Priority Method      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".1661

          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.1232

          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."1881

          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.1441

        Steward what I am given

Steward what I am given      Our vocation.1087

          Consider what I need to do myself      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.1054

          Considering the ultimate outcome      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.1036

          Observe the care that people show others      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.1070

        Making sense of fate

Making sense of fate      159

               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.1254

        Work-in-parallel

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.) 1100

        Scientific method

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. 852

     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.877

          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." 1657

          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.1784

          The sender experiments by sending      Matthew 10:16 "Behold, I send you out as sheep in the midst of wolves. 1788

        Extend the Domain

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. 23

          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, Inc2249

          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, Inc2224

          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.2177

          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, Inc2248

          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, Inc2222

          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, Inc2204

          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.1642

          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, Inc2232

          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, Inc2223

          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.2186

        Extend application

Extend application      1676

          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."1900

        Frequency-and-experience

Frequency-and-experience      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. The Thinker's Toolkit by Morgan D.Jones2150

        Align

Align      1072

          Organize around the client's perspective      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.1046

        My duties

My duties      161

        Hypothesis

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.913

          Mapping observables and observations      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. 911

               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.1393

          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. 1790

          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.1353

          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. 1811

          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. 1467

          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. 1433

          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. 1789

          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. 1570

          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. 1358

          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? 1452

          Who is the greatest among us?      Matthew 23:11 But he who is greatest among you will be your servant.1556

          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, Inc2238

          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, Inc2239

          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.1649

          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.1418

          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.1666

        Drive to logical conclusion

Drive to logical conclusion      1677

        Pursue

Pursue      1073

        Practice

Practice      157

          Considering a principle's consequences      513

          Trying      457

        Experimental design

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.914

          Design experiments to rule models in or out      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. 912

          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." 1659

               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.1395

          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.1992

          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.' 1494

          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.1608

          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.1350

          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?"1623

          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. 1549

          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. 1815

          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. 1359

          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.1489

          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."2045

          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.1792

          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.1783

          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. 1793

          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. 1791

          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.2179

          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, Inc2246

          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.1641

          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.2165

          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, Inc2247

          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, Inc2214

          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.1427

          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.2166

          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, Inc2237

          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.2183

          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?1300

          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.1920

          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.1665

          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.1108

        Accept outcome

Accept outcome      1678

          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.1883

          Establish the validity of evidence      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:

  • Who or what was the source?
  • What was the source's access? How did the source obtain the information? Was that method plausible? ...
  • What is the source's reliability? ...
  • Is the information plausible? ...
The Thinker's Toolkit by Morgan D.Jones2144

        Take stock

Take stock      1074

          Learning from experience whether something was worthwhile      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.1040

          Note how things work out in practice      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.1039

        Let things unfold

Let things unfold      150

        Linking heaven to earth

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.1348

     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.1241

          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!1480

          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. 1552

          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."1399

          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. 1738

          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.' 1434

          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. 2003

          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."1326

          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.1553

          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.1605

          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.2025

          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.1327

          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."1328

          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."1405

          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.1606

          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."1999

          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.1915

          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.2010

          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." 1825

          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.2081

          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."2075

          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."2076

          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. 2005

          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. 1914

          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.1958

          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."1998

               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.1170

               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.1586

          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, Inc2244

          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.1582

               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.1584

               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.1583

          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.1646

          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.1581

          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.2163

          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.1524

          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.2178

          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.2175

          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.1579

          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, Inc2242

          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.2253

     Capture with a personal expression      1589

          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.1725

          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.1230

     Express in a universal language      1588

          Structure laboratory around endeavors based on metaphysical categories      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.1471

        Match inner and outer worlds through person-in-general

Match inner and outer worlds through person-in-general      1525

          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)1390

          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."1691

          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.1688

          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.1757

          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).1758

          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".1712

          Convergence and Divergence      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. The Thinker's Toolkit by Morgan D.Jones2107

        Live as if we are all one person

Live as if we are all one person      1088

          Earn money together and share it      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.1067

          Learn from studying your teachers      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.1007

          Look for others who might value what I value      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.1010

          Make myself vulnerable to others' standards of behavior      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.1057

        Our message

Our message      153

          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.981

     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.982

          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".984

          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.... 989

          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.990

        The prophets

The prophets      Jesus speaks of "the prophets and the law".1978

        Truth

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.14

        Game maker vs. Game player

Game maker vs. Game player      1679

          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.1704

        Valuation

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:

  • Peace of mind (A decision substitutes for the decision maker.)
  • Motivation (External motivation substitutes for internal motivation.)
  • Money (Compensation substitutes for disparity in contributions.)
  • Humility (Autonomy substitutes for morality)
In the case of "peace of mind", we conflate ourselves with our avatar, which is to say, as decision maker we are one with our decisions. Whereas, in "humility", we divorce ourselves from our avatar. If people truly want to hang themselves, or be fat, drunk, stupid, mean, sick, ugly, filthy, destructive, uptight, hurtful, if they truly want that, then we'll sell them the rope in whatever color they want. (Yes, I get stuck at that point. Perhaps I doubt if they are truly free.)1101

        Experiments and Theory

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. 853

        Feel moved

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.1338

          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.1803

          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.1804

          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.1983

     Jesus felt moved      2026

          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."1993

          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? 2086

          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.1439

               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.1438

          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.1234

     Confronting my own outlook      1593

        Gut feeling

Gut feeling      1680

          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."1882

          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.1755

        Peace of mind

Peace of mind      A decision substitutes for the decision maker. They are conflated. Whether there is value.1078

          Noting changes over time      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.1037

        My state of mind

My state of mind      138

        Do not take away their freedom

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.1343

          Telling only his disciples      Matthew 16:20 Then he commanded the disciples that they should tell no one that he was Jesus the Christ. 1330

     Distinguish what they say and do      2030

          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.1547

          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.1488

          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. 1561

          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.1548

     Jesus speaks so they are free to understand or not      2029

          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!"1370

          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?"2061

          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.1237

          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.1785

     Jesus thinks by analogy      2028

          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." 1824

          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.1396

          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?"2058

          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.1928

          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? 1451

          Looking for comparisons      Matthew 11:16 "But to what shall I compare this generation? 1861

          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.1722

     Miracles allow for belief and unbelief      2033

          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."1995

          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.1356

     What distinguishes the cultures      2032

          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.1857

          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.' 1493

          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. 1550

          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."1361

          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."2064

          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.1858

          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. 1485

          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." 1495

          What is the difference between cultures?      Matthew 10:16 "Behold, I send you out as sheep in the midst of wolves. 1787

          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. 1997

          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."1996

          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, Inc2251

          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.2154

          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.1648

          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, Inc2208

          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.1507

          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.2195

          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, Inc2205

        Agree - Implicitly Understand

Agree - Implicitly Understand      174

          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.2258

        Origin

Origin      1681

        Motivation (internal and external)

Motivation (internal and external)      External motivation substitutes for internal motivation. What is valued.1077

          Compare how different systems work      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. 1018

          Recognizing the impact of external motivation      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.1063

        Study

Study      137

     How God cares and judges      2036

               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.1956

          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.1607

          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? 1449

          Correction of our self-conception      Matthew 23:12 Whoever exalts himself will be humbled, and whoever humbles himself will be exalted. 1557

          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."2063

          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? 1454

     How will God reward      2035

          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.1342

          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. 1731

          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.1430

          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. 1429

          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."2011

          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. 1595

     What is first from which the rest follows      2034

          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. 1461

          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. 1562

          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."2015

     What sanctifies      2037

          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. 1571

          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. 1476

          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.1437

          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. 1383

          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.1519

        Improve - Explicitly Innovate

Improve - Explicitly Innovate      173

     Heightening the pull in opposite directions      1651

          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.1652

          How are two concepts different?      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. 1699

        Implication

Implication      1682

          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.1879

          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."1715

          Monetary encouragement      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. 1000

          Money as a measure of motivation      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, 2011.05.051002

        God's significance

God's significance      147

        See oneself in Scripture

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.1339

          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.1966

          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?"1402

          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"1403

          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?1246

          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." 1247

          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.1371

          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.1404

          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.1982

          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.1721

          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."2012

          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. 2088

          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.1320

          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."1245

          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.1734

          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."1662

          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.1316

          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.2065

          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."1318

          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." 1926

          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. 1964

          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." 1717

     Concepts from Scripture      2041

          Instruments of stumbling will be destroyed      Instruments of stumbling (such as Assyria) will be destroyed, not simply punished2042

          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.1962

     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.'1852

     Led by the Spirit      2040

          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.1333

          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.1334

          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.1317

               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.1165

          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.1422

          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.1502

          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, Inc2236

          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.1504

          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, Inc2212

          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.1425

          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.1421

          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.1510

          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.2256

        Invert - Explicitly Transform

Invert - Explicitly Transform      172

          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.1698

     Relate scopes      1701

        Levels

Levels      1683

        Humility

Humility      Autonomy substitutes for morality. Why there is value.1075

          Learning what is useful or not for whom      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.1005

        Other's actions

Other's actions      151

        The Law: You have heard it said..., but I say to you...

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.1239

               Jesus's antitheses      I'm trying to understand the structure of Jesus's antitheses, "You have heard it said... but I say unto you..."1716

               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.

  • covet people - discard what causes you to stumble - do not let yourself stumble; let God punish
  • murder - reconcile, fix before you are judged - let yourself be saved; let God punish
  • covet things - be rewarded by God for your unrewarded love - let yourself be saved; let God reward
  • lie - do not set yourself or others up to stumble by your words - do not cause others to stumble; let God determine
  • steal - do not cause others to stumble by resisting them - do not cause others to stumble; let God reward
  • adultery - do not cause others to stumble by righteousness - do not cause others to stumble; let God punish
1663

          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.1977

        Restructuring

Restructuring      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.)

  • 10 Evolution: Hierarchy => Sequence (for determining weights)
  • 20 Atlas: Network => Hierarchy (for determining connections)
  • 21 Canon: Sequence => Network (for determining priorities)
  • 32 Chronicle: Sequence => Hierarchy (for determining solutions)
  • 31 Catalog: Hierarchy => Network (for determining redundancies)
  • 30 Tour: Network => Sequence (for determining paths)
I expect that they relate 0 Truth, 1 Model, 2 Implication, 3 Variable as follows ... I expect that each geometry reflects a particular way that we're thinking about a variable. I expect them to illustrate the six qualities of signs... Consider the geometry suggested by (6 of the 8) axioms of Zermelo-Fraenkel set theory, for example, the power set axiom. These are axioms for restructuring.15

               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.2155

     Models of Multiplication      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?

  • The addition rule is at work, adding exponents. 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.
  • I think of rescaling as a product of actions that either make bigger (numerator) or make smaller (denominator). They are all multiplying against some unknown, acting upon it. I call the actions "multiplication drops", either "magnifying drops" (say, multiplying by 10) or "shrinking drops" (dividing by 10). So these are actions x actions x (object with units). Thus magnifying and shrinking can cancel out. Also, actions can be decomposed into component actions, into primes.
  • Repeated addition 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.
  • Multiplication can give 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.
  • Multiplication can be thought of as counting items that have been grouped where each group has the same number of items. For example, 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).
  • Dividing out, for example, money per person. This is like multiple units. (Number of cycles) x (Number of people x Units )
1531

          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.1834

               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".1706

               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.1707

               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.1713

        Experiencing

Experiencing      1684

     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:

  • (1) Euclidean Distance Classifier
  • (2) Regularized Linear Discriminant Analysis
  • (3) Standard Fisher Linear Discriminant Function
  • (4) Fisher Linear Discriminant Function With Pseudoinverse Covariance Matrix
  • (5) Generalized Fisher Discriminant Function
  • (6) Minimum Empirical Error Classifier
  • (7) Maximum Margin Classifier
In classifier design, the main objective is to obtain a classifier that results in the minimum number of misclassification errors. Sarunas Raudys thinks of the single layer perceptron as evolving durings its training process so that at any moment it may be thought of as identical or close to one of these classifiers. More than two hundred algorithms for statistical classification rules have been proposed in the literature on statistical pattern recognition and discriminant analysis. He considered those that are similar to the perceptron design algorithm. Sarunas Raudys proposed new complexity-control techniques:
  • target value control
  • moving of the learning data centre into the origin of coordinates
  • zero weight initialization
  • use of an additional negative weight decay term called "anti-regularization"
  • use of an exponentially increasing learning step
The particular type of classifier that results depends on: the data; the cost function to be minimized; the optimization technique and its parameters; the stopping criteria.

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)1391

               The Algebra of Copyright      An example of the 6+4 model is in my paper The Algebra of Copyright, 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.) 1705

        Service

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:

  • How does it seem to me? Appraise. The calm Servant provides the motivated Served with an appraisal, judgment, price.
  • What else should I be doing? Audit. The calm Servant provides the rich Served with a budget, alternatives, substitutions.
  • Would it make any difference? Negotiate. The motivated Servant provides the rich Served with choices, priorities, contracts.
  • What do I have control over? Sell. The rich Servant provides the humble Served with possibilities, favors, exceptions.
  • Am I able to consider the question? Cut one's losses. The motivated Servant provides the humble Served with closure, clarity.
  • Is this the way things should be? Consult. The calm Servant provides the humble Served with overview, advice.
1099

        We sin in our hearts

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. 1261

          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?"1619

          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 judged1969

               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".1168

          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.1639

          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.1526

        Empathize - Implicitly Understand

Empathize - Implicitly Understand      171

          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.

  • I can maintain a state Z. be; refuge for spirit
  • There are situations where I can address Z. do; opportunity to do good
  • Freedom is needed to resolve Z. think; good heart
  • Others have it different than me with regard to Z. one; existential question
  • I need to consider everybody regarding Z. all; collaborating fate
  • I can complete Z. many; peace
  • I focus on the essence of Z. object; knowledge from obedience
  • I admit my dependence regarding Z. process; duty to obey
  • I may wrongly pursue Z. subject; answered prayer
  • I am unrestrained with regard to Z. necessary; limits of mind
  • I take partial responsibility for Z. actual; inhuman wisdom
  • I must appreciate the opportunity of Z. possible; self-contradiction
They may relate to ways of figuring things out. 1695

          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.610

          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.639

     Empathetically modeling a particular mind's thinking      1411

          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.605

          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.653

        Onesome: What-Whether

Onesome: What-Whether      1513

          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)1384

        Appraise

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.1089

          Become aware that a formula is at work      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.1004

          Set a price      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.1058

        Empathize

Empathize      142

        Reconcile quickly

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. 1260

          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.1337

          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 culture1971

          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. 1474

               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.1841

               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.1164

          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. 1745

          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, Inc2216

          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. 1752

               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. 1754

               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. 1753

          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. 1527

          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.2161

        Get to Know - Implicitly Care

Get to Know - Implicitly Care      170

          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.1696

          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.1444

          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.1711

          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.1445

          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.620

          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?1446

          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?628

          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.1708

        Twosome: How-Whether

Twosome: How-Whether      1514

          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)1385

          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.1756

               Basket for Jerry      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! :)1760

        Play our part

Play our part      134

     Our Duties      110

        Be perfect.

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. 1262

          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.1352

          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." 1617

          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.1351

          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 receive1972

          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."2006

          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. 1466

          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.2087

          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.1459

          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. 1265

          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. 1484

          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." 1906

          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.1463

          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, Inc2228

          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.1501

          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.1528

        Prioritize - Implicitly Commit

Prioritize - Implicitly Commit      169

          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.593

          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.619

          Find correlations between structures      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.1472

          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".614

          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".629

        Threesome: How-What

Threesome: How-What      1515

          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)1386

          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...1759

          Multiple Classes of Outcomes      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.

  • 1) Identify the outcome classes and each one's range of outcomes and weight each class according to its importance.
  • 2) Construct for each class of outcomes a utility matrix with identical perspectives and options.
  • Perform Steps 3 through 6 with each matrix
  • 3) Assign utilities from 0 to 100 to the outcome of each option-outcome combination (each cell of the matrix).
  • 4) Assign a probability to the outcome of each option-outcome combination (each cell).
  • 5) Compute expected values.
  • 6) Add expected values for each option and enter the totals in the "Total EV" column.
  • 7) Construct a single merged matrix with the same options as in the classes-of-outcomes matrices.
  • 8) Enter opposite each option the total expected values for that option from the classes-of-outcomes matrices.
  • 9) Multiply the total expected values under each class of outcomes by the class's weight.
  • 10) Add the resulting products (weighted expected values) for each option and enter the sums in the "Total Weighted EV" column.
  • 11) Rank the options. (The one with the greatest total weighted expected value is the preferred option.)
  • 12) Perform a sanity check.
One can, of course, take multiple-outcome analysis one step further by analyzing each class of outcomes from multiple perspectives, but I strong recommend against it. Combining multiple-outcome and multiple-perspective analysis would spin the analytic web much too fine. Doing so is akin to studying subtleties - lesser factors and lesser issues ... they never play a significant role. The Thinker's Toolkit by Morgan D.Jones2153

          Multiple Perspective Utility Analysis      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.

  • 1) Identify and weight the perspectives to be analyzed.
  • 2) Construct an identical utility matrix for each perspective - same options, same outcomes.
  • Perform Steps 3 through 6 with each matrix
  • 3) From each matrix's particular perspective, assign utilities from 0 to 100 to the outcome of each option-outcome combination (each cell of the matrix)
  • 4) Assign a probability to the outcome of each option-outcome combination (each cell)
  • 5) Compute expected values.
  • 6) Add expected values for each option and enter the totals in the "Total EV" column.
  • 7) Construct a single merged matrix - the same options as in the perspective matrices.
  • 8) Enter opposite each option the total expected values for that option from the perspective matrices.
  • 9) Multiply the total expected values under each perspective by the perspective's total weight.
  • 10) Add the resulting products (weighted expected values) for each option and enter the sums in the "Total Weighted EV" column.
  • 11) Rank the options. (The one with the greatest total weighted expected value is the preferred option.)
  • 12) Perform a sanity check.
You may wonder why we don't perform step 9 (multiplying expected values by each perspective's weight) before transfering the total expected values from the utility matrix to the merged matrix. ... first, to view these values side by side, to compare and validate them and, second, to perform more easily sensitivity analysis by applying different combinations of weights to the criteria... The Thinker's Toolkit by Morgan D.Jones2104

          Weighted ranking      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).

  • List all of the major criteria for ranking.
  • Pair-rank the criteria.
  • Select the top several criteria and weight them in percentiles (their sum must equal 1.0)
  • Construct a Weighted Ranking Matrix and enter the items to be ranked, the selected criteria, adn the criteria weights.
  • Pair-rank all of the items by each criterion, recording in the appropriate spaces the number of "votes" each item receives.
  • Multiply the votes by the respective criterion's weight.
  • Add the weighted values for each item and enter the sums in the column headed "Total Votes".
  • Determine the final rankings and enter them in the last column, headed "Final Ranking". (The item with the most points is ranked highest.)
  • Perform a sanity check.
So what did weighted ranking buy me over instinctive ranking? ... confidence in the validity of the rankings Weighted ranking is good for eliminating subjective judgments, for comparing objectively. The Thinker's Toolkit by Morgan D.Jones2141

        Negotiate

Negotiate      Would it make any difference? Negotiate. The motivated Servant provides the rich Served with choices, priorities, contracts.1091

          Give priority to what is most real      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.1017

        Compare

Compare      154

        Let your 'Yes' be Yes and your 'No' be No

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. 1264

          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." 1656

          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 them1973

               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.1160

               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. 1382

          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, Inc2215

          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.1503

        Ground - Explicitly Care

Ground - Explicitly Care      168

          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.644

          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.640

          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:

  • I engage
  • I wait, in suspense
  • I believe
  • I rely
  • I love
  • I suffer
Subsequently, I related them to six of the ten commandments, and to Jesus' logic of the woes.1697

          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.641

          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.1302

        Foursome: Why-How

Foursome: Why-How      1516

          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)1387

        Sell

Sell      What do I have control over? Sell. The rich Servant provides the humble Served with possibilities, favors, exceptions.1092

          Argue my case to save my situation      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.1015

          Mapping out the constraints that incomes impose on expenses      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.1061

        Declaring my purpose

Declaring my purpose      135

          Knowing what I truly want      512

     Our purpose      109

          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. 1625

          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. 1626

          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?' 1458

          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? 1457

        Decomposition

Decomposition      Vary model (can variously combine factors) (Pigeonhole principle, partitions, factorizations, encoding, full range of outputs, principle of inclusion-exclusion). We can interpret factorizations in terms of "and" and "or", and thus create equations that construct and relate roots. (Or, And, method of undetermined coefficients, expansion, construction) 31 Catalog: Hierarchy => Network (for determining redundancies)72

               Axiom of union      Wikipedia: For any set F there is a set A containing every set that is a member of some member of F1163

               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.1843

               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.1842

          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.2198

          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, Inc2203

          Complete the square      2185

          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. 1740

          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.2173

          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.2169

          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.2176

          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, Inc2225

          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, Inc2218

          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. 1749

          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.1638

          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, Inc2221

          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.2197

          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.2167

          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.1635

               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.1636

          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. 1741

          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. 1377

          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, Inc2227

        Structure - Understand

Structure - Understand      167

        Am I able to consider the question?

Am I able to consider the question?      Focusing on parts, sorting experience. Collect first hand experience and recognize patterns. 57

          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. 597

          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.627

          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.1303

          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.634

          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.600

               Dimensions of organic work: Principles of community organizing      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.1540

               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.1536

               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.1541

               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.1534

               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.1533

               Ways of getting things done      Joe Damal and I noted ways of getting things done and grouped them into eight ways.1539

               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.1537

        Fivesome: Why-What

Fivesome: Why-What      1517

          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)1388

        Cut one's losses

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.1093

          Failing to get around the rules      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.1035

          Recognize the inapplicability of an idea      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.1003

        My life story

My life story      146

          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?"2062

          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?1574

          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."1991

          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. 1267

          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. 1269

        Directed graph

Directed graph      Vary truth (can add or remove circular behavior) (With or without cycles) 30 Tour: Network => Sequence (for determining paths) 73

          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.2157

          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.2184

          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.2158

          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.2160

          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, Inc2226

          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, Inc2219

        Clarify - Explictly Understand

Clarify - Explictly Understand      166

          Boundary Matrix      identify relevant links between two disciplines 1692

        Is this the way things should be?

Is this the way things should be?      Focusing on the whole, working out a structure. Work out how complicated examples fit together.58

          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.613

          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.633

          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.1542

          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":

  • Black is for my personal principles. Those are reasonings that might not apply to other people, but seem to be fixed in my life, they are the definition of who I am. I'm morally bound to be true to them, to the extent that they express my person.
  • Blue is for my personal opinions. They are the conclusions I have drawn from my own experiences in my own situation. They might actually be wrong, but I live with them as the best that I can draw from my real life experience so far. They are practical observations about the world I live in.
  • Red is for my personal decisions. They are the choices that I have made to try to solve the pressures and conflicts of my principles and opinions. Usually it's the only positive choice that I can think of. But it's a choice that is optional, it's a step that I could have simply not taken.
  • Green is for my personal inspirations. They are the extra ideas that weren't necessary but I wanted to throw into my life. They are the ideas that represent the most freedom that I exercise.
I think that assigning the colors helped me think further, how does one thought extend another thought?1694

          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.1544

          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.618

        Foursome: Why-Whether

Foursome: Why-Whether      1518

          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)1389

          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.1878

          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.1930

          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.1929

          Whether I am right      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.1278

        Assess

Assess      Is this the way things should be? Consult. The calm Servant provides the humble Served with overview, advice. This is a macroeconomic perspective.1094

          Avoiding vicious circles      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.1048

          Not to be complicit with a dysfunctional system      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.1053

          Recognizing justice in reciprocity      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.1042

        Working things out

Working things out      165

     Our actions      112

     Pattern      107

        Sanity Check

Sanity Check      ...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. The Thinker's Toolkit by Morgan D.Jones2105

          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. 1210

          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."1211

          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." 1863

          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.'"2004

          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.2021

          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."2022

          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.2024

          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.1322

          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."2008

          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.'"2068

          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. 1491

               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.1840

               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.1169

        Obeying

Obeying      59

          Ask God what I should think over so as to understand?      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:

  1. What captures attention and guides it?
  2. What drops down upon reality and bounces away in random paths?
  3. What is wound in one direction, and lives through spinning in the opposite direction?
  4. What falls as rain day and night until there sprout and grow plants that will bear fruit?
  5. What like a ray reflects off of society and does not return?
  6. What by its turning (in the direction of winding) commands our attention and then slips away to the side?
That's all. Go work. I didn't know what to make of these, but I thought they all seemed to point to gyvas rupestis, Lithuanian for living care, which demands continuous effort but is unconscious of it, and is almost the opposite of love. 1470

     Appreciate constraints      582

          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.1687

          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.1689

          Accepting our parents' values      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.1001

          Appreciate that the situation may change      Also, it's my position that laws are in flux and they may well change in my favor as I get to them.1055

          Learning from elders how to do things right      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.1043

          Making sure to do enough to show that I am trying      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. 1052

        Engage God

Engage God      139

                    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.1107

        More ways Jesus figured things out

More ways Jesus figured things out      1238

     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.1243

     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.1240

     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.1249

        Other ways: Math

Other ways: Math      819

               Alternatives to mathematical induction      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.812

          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.1578

          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).1159

     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)857

                    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?1690

          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."1173

               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 Nazareth1197

          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. 1183

          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."1186

          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.1122

          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. 1123

          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.1189

          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.1177

          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."1125

          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.1196

               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?1205

          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.1308

          Command      Genesis 1:3 God said, "Let there be light," and there was light.1119

               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. 1121

               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."1120

          Confession      Matthew 3:6 They were baptized by him in the Jordan, confessing their sins.1201

          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.1219

          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.1207

          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.1176

          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.1171

          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.1113

               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.1115

               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.1114

          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.1217

          Empirical research methods      Empirical research methods are one of five factors in Jörn Rüsen's "disciplinary matrix" of historiography.1310

          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.1131

          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.1124

          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;1133

          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. 1188

          Forms of representation      Forms of representation are one of five factors in Jörn Rüsen's "disciplinary matrix" of historiography.1311

          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.1175

          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. 1130

          Give names to things      Genesis 1:5 God called the light "day," and the darkness he called "night."1116

               Name the earth and seas      Genesis 1:10 God called the dry land "earth," and the gathering together of the waters he called "seas."1118

               Name the sky      Genesis 1:8 God called the expanse "sky."1117

          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. 1686

          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.1218

          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."1191

               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."1179

               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."1195

          Inquire      Matthew 2:7 Then Herod secretly called the wise men, and learned from them exactly what time the star appeared. 1185

          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."1208

          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."21

          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! 1204

          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: Istorika. Istorikos darbų rinktinė compiled by Zenonas Norkus.1307

          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.1309

          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.1314

          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. 1126

          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. 1128

          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.1313

          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.1182

          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. 1178

          Offering orientation to society      The function of offering orientation to society is one of five factors in Jörn Rüsen's "disciplinary matrix" of historiography.1312

          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.1190

          Removing himself      God thinks as he creates, and he creates by removing himself, thereby considering whether he would be even if he was not.1216

          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. 1129

          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.1134

          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."1172

          See everything      Genesis 1:31 God saw everything that he had made, and, behold, it was very good. 1132

          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. 1193

          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. 1187

          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. 1112

          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... 1127

          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.1315

          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."1180

          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.'"1184

          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?1214

          Unfolding perspective      God thinks by unfolding his perspective, as in my video summary, "I Wish to Know", generating I, then You, then Other.1215

          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."1181

               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."1198

               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."1192

               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."1200

               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."1194

          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.1203

          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. 1206

          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!" 1199

          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.1685

          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?"1209

          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."1213

          Who pleases      Matthew 3:17 Behold, a voice out of the heavens said, "This is my beloved Son, with whom I am well pleased."1212

          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.1202

     Other ways: Philosophy      1138

          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. 1147

          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. 1151

          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. 1157

          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). 1140

          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". 1156

          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).1137

          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.)1145

          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". 1153

          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.1141

          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.1146

          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.1144

          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"). 1150

          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. 1155

          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)...1139

          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. 1154

          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.1149

          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.)1152

          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) ...1158

          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).1143

          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.1148

     Other ways: poetry      1228

          Writing      Tom Mandel: Writing is my primary tool for investigating the world.1229

          Put forward ideas      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.1276

          Agree on areas for investigation      Pamela McLean, April 11, 2011: They will agree on areas for investigation and will work collaboratively.1281

          Centering around the learner      Pamela McLean, April 11, 2011: All our courses will be learner centred.1279

          Collect resources      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).1282

          Create a portfolio      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.1286

          Don't share an idea - show a prototype      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.1272

          Growth      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.1294

          How things will emerge and how I would like them to      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). 1274

          Imagine the outcome      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.1277

          Invite others      Pamela McLean, April 11, 2011: You are welcome to join me.1297

          Look out for bias      Pamela McLean, April 11, 2011: I appreciate that friends might support me - but as they are biased that probably wouldn't count.1273

          My hunch      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.1275

          Naming      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").1295

          Network of references      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.1288

          Our needs      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.1292

          People's interests      Pamela McLean, April 11, 2011: Faculties will emerge according to the interests of people who study with us and the research that we do.1289

          Produce original work      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).1283

          Related studies      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. 1287

          Share as spreadsheets      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).1284

          Share through an online system      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).1285

          Studying in small online special interest groups      Pamela McLean, April 11, 2011: Students will normally study in small online Special Interest Groups.1280

          Vision      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.1293

          What I have been doing      Pamela McLean, April 11, 2011: I'm already studying there, and teaching, and doing research.1296

          What is fortunate      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.1291

          What is likely based on how things currently are      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.1290

     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)834

               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.836

     Thought experiments in Physics      Wikipedia lists thought experiments in Physics: Galileo's ship (classical relativity principle) 1632, Galileo's Leaning Tower of Pisa experiment (rebuttal of Aristotelian Gravity), GHZ experiment (quantum mechanics), Heisenberg's microscope (quantum mechanics), Kepler's Dream (change of point of view as support for the Copernican hypothesis), Ladder paradox (special relativity), Laplace's demon, Maxwell's demon (thermodynamics) 1871, Monkey and the Hunter, The (gravitation), Moving magnet and conductor problem, Newton's cannonball (Newton's laws of motion), Popper's experiment (quantum mechanics), Quantum pseudo telepathy (quantum mechanics), Quantum suicide (quantum mechanics), Schrödinger's cat (quantum mechanics), Sticky bead argument (general relativity), Renninger negative-result experiment (quantum mechanics), Twin paradox (special relativity), Wheeler's delayed choice experiment (quantum mechanics), Wigner's friend (quantum mechanics)856

               Brownian ratchet      Wikipedia: Richard Feynman's "perpetual motion" machine that does not violate the second law and does no work at thermal equilibrium839

               Bucket argument      Wikipedia: Argues that space is absolute, not relational. Isaac Newton's rotating bucket argument (also known as "Newton's bucket") was designed to demonstrate that true rotational motion cannot be defined as the relative rotation of the body with respect to the immediately surrounding bodies. It is one of five arguments from the "properties, causes, and effects" of true motion and rest that support his contention that, in general, true motion and rest cannot be defined as special instances of motion or rest relative to other bodies, but instead can be defined only by reference to absolute space. Newton discusses a bucket filled with water hung by a cord. If the cord is twisted up tightly on itself and then the bucket is released, it begins to spin rapidly, not only with respect to the experimenter, but also in relation to the water it contains. Although the relative motion at this stage is the greatest, the surface of the water remains flat, indicating that the parts of the water have no tendency to recede from the axis of relative motion, despite proximity to the pail. Eventually, as the cord continues to unwind, the surface of the water assumes a concave shape as it acquires the motion of the bucket spinning relative to the experimenter. This concave shape shows that the water is rotating, despite the fact that the water is at rest relative to the pail. In other words, it is not the relative motion of the pail and water that causes concavity of the water, contrary to the idea that motions can only be relative, and that there is no absolute motion.840

               Einstein's box      Wikipedia: Einstein considers a box (called Einstein's box; see figure) containing electromagnetic radiation and a clock which controls the opening of a shutter which covers a hole made in one of the walls of the box. The shutter uncovers the hole for a time Δt which can be chosen arbitrarily. During the opening, we are to suppose that a photon, from among those inside the box, escapes through the hole. In this way a wave of limited spatial extension has been created, following the explanation given above. In order to challenge the indeterminacy relation between time and energy, it is necessary to find a way to determine with adequate precision the energy that the photon has brought with it. At this point, Einstein turns to his celebrated relation between mass and energy of special relativity: E = mc2. From this it follows that knowledge of the mass of an object provides a precise indication about its energy. The argument is therefore very simple: if one weighs the box before and after the opening of the shutter and if a certain amount of energy has escaped from the box, the box will be lighter.... Bohr showed that, in order for Einstein's experiment to function, the box would have to be suspended on a spring in the middle of a gravitational field. ... 843

               Feynman sprinkler      Wikipedia: The question of which way an inverse sprinkler would turn (so, with the sprinkler sucking the water in rather than pumping it out) was the subject of an intense and remarkably long-lived debate.845