How do you figure things out?1308 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

        Gamestorming

Gamestorming      5

        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

               Andrius's quest to know everything      My ten minute video summary of my quest to know everything that I made in August, 2009.1376

        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

        Morgan D. Jones of The Thinker's Toolkit

Morgan D. Jones of The Thinker's Toolkit      Morgan D. Jones is the author of "The Thinker's Toolkit: Fourteen Skills for Making Smarter Decisions in Business and in Life", 1995, Random House. He teaches ways of structuring the analysis of problems so that we don't jump to conclusions. He leads Thinker's Workshop, helped found Analytic Prowess, writes fiction, and taught analytic methods in Georgetown University's Master of Science Foreign Service graduate program. He worked in the C.I.A. for 36 years, including as an intelligence analyst of Soviet military affairs and chief of the Analytic Training Branch. I take his work to be representative of how the C.I.A. and other intelligence agencies analyze issues. Morgan D. Jones does a great job surveying the analytic techniques and providing intuition on how and where to use them. I'm especially interested in discovering how his analytic techniques relate to my ways of organizing, visualizing information. In his book, he considers applications in business and every day life. I think they also reflect on the analytical mindset at the C.I.A. where he learned, worked and taught. By his own logic, I think it's important to appreciate that bias. I suppose it also make sense to supplement them with espionage techniques, often clandestine, for gathering facts, as well as principles for finding moral bearings or not.2092

        Making a Living

Making a Living      I apply the "ways of figuring things out" to my challenge of making a living. I draw on 60 anecdotes that I wrote up in 2006 about my "money mind". I thought this through in writing a letter to DaveGray, JayCross and others. We can start with: '''Working'''. Don't be idle, unproductive or destructive. Then there are two independent threads, Giving and Taking.

  • Giving: Give everything that I have. Appreciate what others have to give. Learn what is fair in giving. Invest in a culture of giving.
  • Taking: Accept what I am given. Maximize what I can take. Learn my limits. Steward what I am given.
(In business, especially corporations, there is a similar dichotomy between investors and managers). Such 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.) 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.) Humility (or Service) then takes six concrete forms that pair the four levels of Servant and Served. They 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.
The gap between Servant and Served, expressed in the six ways above, comes together as Conscientiousness.1098

        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

        Physics

Physics      John Harland and I talk about physics. How might we think of it in terms of "ways of figuring things out" and my overview of that? John and I were graduate students at UCSD in the math department where we received our Ph.D's. I have a B.A. in Physics and I think John does, too, but he certainly knows and thinks a lot more about physics than I do. I share my notes based on my understanding of ideas that John sparked or stated and I tried to make sense of in my system. 849

        Independent trials

Independent trials      26

        Avoiding evil

Avoiding evil      Avoiding what would keep us from learning. Avoiding evil, not shutting ourselves down 48

        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

               Don't cause the little ones who believe to stumble      Mark 9:38 John said to him, "Teacher, we saw someone who doesn't follow us casting out demons in your name; and we forbade him, because he doesn't follow us." 9:39 But Jesus said, "Don't forbid him, for there is no one who will do a mighty work in my name, and be able quickly to speak evil of me. 9:40 For whoever is not against us is on our side. 9:41 For whoever will give you a cup of water to drink in my name, because you are Christ's, most certainly I tell you, he will in no way lose his reward. 9:42 Whoever will cause one of these little ones who believe in me to stumble, it would be better for him if he were thrown into the sea with a millstone hung around his neck. 9:43 If your hand causes you to stumble, cut it off. It is better for you to enter into life maimed, rather than having your two hands to go into Gehenna, into the unquenchable fire, 9:44 'where their worm doesn't die, and the fire is not quenched.' 9:45 If your foot causes you to stumble, cut it off. It is better for you to enter into life lame, rather than having your two feet to be cast into Gehenna, into the fire that will never be quenchedâ€" 9:46 'where their worm doesn't die, and the fire is not quenched.' 9:47 If your eye causes you to stumble, cast it out. It is better for you to enter into the Kingdom of God with one eye, rather than having two eyes to be cast into the Gehenna of fire, 9:48 'where their worm doesn't die, and the fire is not quenched.'* 9:49 For everyone will be salted with fire, and every sacrifice will be seasoned with salt. 9:50 Salt is good, but if the salt has lost its saltiness, with what will you season it? Have salt in yourselves, and be at peace with one another."1256

               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

          Are you embracing what's right? or keeping others from embracing it?      Matthew 23:14 "But woe to you, scribes and Pharisees, hypocrites! Because you shut up the Kingdom of Heaven against men; for you don't enter in yourselves, neither do you allow those who are entering in to enter. 1565

          Be sure to not cause others to stumble      Matthew 17:24 When they had come to Capernaum, those who collected the didrachma coins came to Peter, and said, "Doesn't your teacher pay the didrachma?" 17:25 He said, "Yes." When he came into the house, Jesus anticipated him, saying, "What do you think, Simon? From whom do the kings of the earth receive toll or tribute? From their children, or from strangers?" 17:26 Peter said to him, "From strangers." Jesus said to him, "Therefore the children are exempt. 17:27 But, lest we cause them to stumble, go to the sea, cast a hook, and take up the first fish that comes up. When you have opened its mouth, you will find a stater coin. Take that, and give it to them for me and you."2002

          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

          Is Jesus causing others to stumble?      Jesus' is concerned that he himself not be the cause of others, such as John, to stumble. Matthew 11:2 Now when John heard in the prison the works of Christ, he sent two of his disciples 11:3 and said to him, "Are you he who comes, or should we look for another?" 11:4 Jesus answered them, "Go and tell John the things which you hear and see: 11:5 the blind receive their sight, the lame walk, the lepers are cleansed, the deaf hear, the dead are raised up, and the poor have good news preached to them. 11:6 Blessed is he who finds no occasion for stumbling in me." 1827

          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

          Jesus' words cause them to have sin      John 15:22 If I had not come and spoken to them, they would not have had sin; but now they have no excuse for their sin.2044

          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

          Mental toughness, confidence and concentration      But most beginners give up too soon, because they lack the mental toughness attributes of confidence and concentration. It is hard to work on a problem if you don't believe that you can solve it, and it is impossible to keep working past your "frustration threshold". ... You build upon your preexisting confidence by working at first on "easy" problems, where "easy" means that *you* can solve it after expending a modest effort. ... then work on harder and harder problems that continually challenge and stretch you to the limit ... Eventually, you will be able to work for hours single-mindedly on a problem, and keep other problems simmering on your mental backburner for days or weeks. ... developing mental toughness takes time, and maintaining it is a lifetime task. But what could be more fun than thinking about challenging problems as often as possible? pg.16, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1415

          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

          Vary the trials      The mouse in the trap ... threw himself violently against the bars, now on this side and then on the other, and in the last moment he succeeded in squeezing himself through ... We must try and try again until eventually we recognize the slight difference between the various openings on which everything depends. We must vary our trials so that we may explore all sides of the problem. Indeed, we cannot know in advance on which side is the only practicable opening where we can squeeze through. The fundamental method of mice and men is the same; to try, try again, and to vary the trials so that we do not miss the few favorable possibilities ... a man can vary his trials more and learn more from the failure of his trials than the mouse. pg.16, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc., quoting "Mice and Men" by George Polya, Mathematical Discovery, Volume II, 1965.1413

        Avoiding Evil

Avoiding Evil      Avoiding evil. Acknowledge what is evil or futile and stay away from it.30

          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

          Have rapid cycles of learning      When I started my laboratory, Minciu Sodas, in 1998, I was living month-to-month, even week-to-week, not certain what I would try next. I think I learned a lot this way because I had very rapid and flexible cycles of learning. I had long term rhythms in my life for what I knew was important, but a snappy cycle for responding to opportunities.1106

          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

          Not to destroy what I treasured      In 1992, I was very sad because the woman I had loved with all my heart had returned home to Lithuania and married a man she loved. I had so many feelings for her. Indeed, I stayed in love with her for about seventeen years. Early on, I thought, maybe I should kill my feelings for her. But I refused. They were my honest feelings that I felt for her. I wasn't going to dampen them. I wasn't going to look for her faults. I wasn't going to diminish myself, who I was and who I could be. I stayed true to them.1231

          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

          Not rushing up from being knocked down      Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: When you get knocked down, the first thing you do is try to compose yourself and not rush up. Don't jump up because of embarrassment because your equilibrium is still off. And if you jump up too fast, you're going to stumble. You're going to fall back. The key is to hopefully be near a rope which you can use as a brace to help you up.1898

          Stories of people who ended badly      Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: "I had no intentions whatsoever to become a professional fighter because I had heard horror stories about former boxers who made money but, in the end, ended up with nothing. I didn't want to be one of those guys."1886

          Try different lines of questioning      Her blood pressure was through the roof. I leafed through the chart and noted it had been controlled in the past. I began to dig to see if I could identify a cause. "How do you feel?" I probed. "All right," she mumbled with her Arkansas accent. "Are you having any chest pain or problems breathing?" "No suh." I tried another line of questioning before moving on. "Have you had any recent stress in your life?" Jackpot. Her eyes welled. Her voice remained emotionless. "My gran-chillin, got kilt. On my fron poich," she said. ... She cowered in panic on the other side of the door, inches from her grandchildren. "Ah was a-scared to open it. Ah din know it were them. Ah din know it were them," she repeated. ... Her blood pressure shot up. And remained high two weeks later. If I had not asked, she would not have told me. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.19341937

        Don't be content with invalid explanations

Don't be content with invalid explanations      We feel the need to find explanations for everything, regardless of whether the explanations are accurate. ... Explanations, by making sense of an uncertain world, apparently render the circumstances of life more predictable and thus diminish our anxieties about what the future may hold. ... The compulsion to explain everything drives our curiosity and our thirst for knowledge of the world. ... Knowing - finding an explanation for an event - is one of the most satisfying of human experiences. There is great comfort in recognizing and making sense out of the world. Doing so creates order and coherence, and, where there is order, there is safety and contentment. We are instantly aware of the loss of this inner feeling of safety and contentment the moment we don't recognize a pattern in a situation that confronts us. ... Unfortunately, our compulsion to explain things can ... get us in trouble. When presented with an event that has no particular meaning, we find one anyway, and we subconsciously don't care whether the explanation is valid. ... the explanations we give for things don't have to be true to satisfy our compulsion to explain things. ... being satisfied, we are then blithely content to move on to something else without seriously questioning our explanation. This is one of the reasons we humans don't give sufficient consideration to alternatives ... the principal cause of faulty analysis. The Thinker's Toolkit by Morgan D.Jones drawing from The Universe Within: A New Science Explores the Human MindThe Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich.2131

          Appreciate that mental shortcuts are at work      Subconscious mental activity continuously influences our analysis and even determines our conclusions. ... Most of what goes on in the mind involves "mental shortcuts" that occur without our knowledge and beyond our conscious control. ... We cannot "teach" the mind how to work; it works as it works, and taking shortcuts is one of its ways. These shortcuts are beyond our conscious control. ... And where those shortcuts lead our thinking is anyone's guess. The Thinker's Toolkit by Morgan D.Jones drawing from The Universe Within: A New Science Explores the Human MindThe Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich.2123

          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

          Be aware of the vividness effect      Information is vivid because it was acquired either traumatically or recently and thus has made a strong impression on our memory. Information that is vivid is therefore more easily remembered than pallid, abstract information and, for that reason, has greater influence on our thinking. This information can far exceed what it actually deserves. The Thinker's Toolkit by Morgan D.Jones drawing from The Universe Within: A New Science Explores the Human MindThe Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich.2128

          Be wary of imputing causality      Another kind of patterning is the tendency of the human mind to look for cause-and-effect relationships. ... We seem to view the world in terms of cause and effect, and we somehow instinctively know the difference between the two concepts. ... So we by nature strive to know why something has happened, is happening, or will happen and what the result was, is, or will be. The Thinker's Toolkit by Morgan D.Jones drawing from The Universe Within: A New Science Explores the Human MindThe Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich.2126

          Be wary of seeing patterns      We are driven to view the world around us in terms of patterns (especially cause and effect). The human mind instinctively views the world in terms of patterns, which it recognizes based on memories of past experiences. ... We see the patterns in situations ... in sequences of events ... our compulsion to see patterns can easily mislead us when we analyze problems. ... patterns can be captivating ... the mind can also easily miscontrue random events as nonrandom, perceiving a pattern where, in fact, none exists. ... when we want to see a particular pattern or expect to see it, or have become accustomed to seeing it, not only do we fill in missing information but our brain edits out features that don't fit the desired or familiar pattern. The Thinker's Toolkit by Morgan D.Jones drawing from The Universe Within: A New Science Explores the Human MindThe Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich.2124

          Counter the subconscious mind      Sometimes, when I offer to show people how to structure their analysis of problems, they say, "Sorry, but I'm awfully busy." "No, thank you. We don't use things like that here." "Very interesting, but it sounds a bit arcane." "I wouldn't think of using such Byzantine methods." The rejection is so spontaneous and emphatic that it could only be a mental knee-jerk reflex, not a reasoned response. What could be causing the unreasoning reflex? Because they have not learned and understood the techniques, they could not have performed any cost-benefit analysis on them. They therefore have not assimilated any data on which to base a reasoned judgment. The only conclusion one can draw is that their reactions are purely intuitive, meaning they come from the subconscious mind. And why would the subconscious mind reject these techniques out of hand? Because structuring one's analysis is fundamentally at odds with the way the human mind is accustomed to work. ... if we structure our analysis, the mind can't be free to mess around. That's why the subconscious rebels when it's asked to structure its thinking. The Thinker's Toolkit by Morgan D.Jones2119

          Don't begin with conclusions      We commonly begin our analysis of a problem by formulating our conclusions; we thus start at what should be the end of the analytic process. The Thinker's Toolkit by Morgan D.Jones2120

          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

          Don't commit analytical sins      Over the past several decades cognitive science has discovered that we humans are unknowingly victimized by instinctive mental traits that defeat creative, objective, comprehensive, and accurate analysis. As a result, we unwittingly, repeatedly, habitually commit a variety of analytical sins. He lists five analytic sins, which I think relate to the counterquestions, which serve to keep us from commiting them:

  • Beginning analysis by formulating conclusions. Is this the way things should be?
  • Focusing on our favorite solution and dismissing alternatives. What else should I be doing?
  • Confuse "discussing and thinking hard" with analyzing. Would it make any difference?
  • Focus on substance and not process. What do I have control over?
  • Have no concept of structural analysis. Am I able to consider the question?
  • ... How does it seem to me?
The Thinker's Toolkit by Morgan D.Jones2122

          Don't favor particular solutions      What else should I be doing? Our analysis usually focuses on the solution we intuitively favor; we therefore give inadequate attention to alternative solutions. The Thinker's Toolkit by Morgan D.Jones2121

          Not to be fixated on our chosen solution      Humans have a penchant to seek out and put store in evidence that supports their beliefs and judgments while eschewing and devaluing evidence that does not. ... "What am I supposed to be looking at? What's the point?" In other word's, what's the focus? ... focusing has immense downsides when we analyze problems. It tends to make us view problems one-dimensionally, meaning we tend to focus on (glom on to) the first solution that makes sense - that offers an explanation - as we interpret reality through the lens of our biases. Having focused on a solution, we are captivated by it and thus become uninterested in alternative solutions. Our fixation on our chosen solution therefore causes us to value evidence that supports that solution and to devalue, disbelieve, discredit, and discard evidence that does not. We therefore tend to accept at face value information that is consistent with our beliefs and to critically scrutinize and discount information that contradicts them. ... We tend to see in a body of evidence what, according to our mindset, we expect and want to see and tend not to see, again according to our mind-set, what we don't expect or don't want to see. The Thinker's Toolkit by Morgan D.Jones drawing from The Universe Within: A New Science Explores the Human MindThe Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich.2133

          Overcome stereotyping      Stereotyping is a form of patterning ... I have to struggle against this impression to see this person as he or she really is; I have to struggle against my own mind... The Thinker's Toolkit by Morgan D.Jones drawing from The Universe Within: A New Science Explores the Human MindThe Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich.2125

          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

          Trial-and-error      ...we mess up in our efforts to analyze problems much more often than we are willing to admit. Indeed, the road to humankind's impressive achievements has invariably been paved with failures, many of which have set back our accomplishments by decades and longer. For every forward stride there have been telling missteps, because the problem-solving approach that has proven most practical and effective for the human species is the trial-and-error method. ... In all human affairs, from marriage to marketing to management, success is generally built upon failure. And while some failures are justly attributable to bad luck, most result from faulty decisions based on mistaken analysis. The Thinker's Toolkit by Morgan D.Jones2118

        Working

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

          Be put into a new situation      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.1008

        Leaving

Leaving      155

          Leaving      452

        Good Will: Building a culture of faith

Good Will: Building a culture of faith      This is the Kingdom of Heaven where Jesus is the Son of Man. Jesus urges us to go through the narrow gate that leads to life. Look for the ways to address Maslow's hierarchy of needs: Cling to what you have; get more than what you need; avoid extremes. Jesus believes in positivity, prefers it to negativity as taught by the Father, as given by his aloofness and prophets and warnings and having them resisted and humiliated and killed. But Jesus grows in reflection of the Father's way and ultimately does his will. Light as opposed to darkness.1855

        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

        Believing 1-2-3-4

Believing 1-2-3-4      In real life, we discard the unessential to identify God which is deeper than our very depths, around such a core we allow for ourselves and others, we seek harmony of interests and we find a unity (Spirit) by which any person can serve as the center. These four frames are: believing; believing in believing; believing in believing in believing; believing in believing in believing in believing.29

        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

        Giving

Giving      1096

        Particle point of view

Particle point of view      Our measurement can take place within the frame of measurement. We have a natural frame of reference, for example, the center of mass. That center of mass can then be considered as balancing different masses, and integrating a system of masses, and ultimately defining a vector space. This is a static, spatial, nontemporal point of view. Every state has a location. We can speak of the state of a system. Analogously, in math we have a blank sheet with a natural frame, a center, a balance around that center, a polynomial algebra of constructions, and ultimately, a vector space where a basis makes explicit that every point can be the center. And in life, we can discard the unessential, presume only God, allow for both self and others, find harmony amongst our interests, and create a space for good Spirit. 851

        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

     Faith accepts God's love      2099

          Ask in prayer      Matthew 21:20 When the disciples saw it, they marveled, saying, "How did the fig tree immediately wither away?" 21:21 Jesus answered them, "Most certainly I tell you, if you have faith, and don't doubt, you will not only do what was done to the fig tree, but even if you told this mountain, 'Be taken up and cast into the sea,' it would be done. 21:22 All things, whatever you ask in prayer, believing, you will receive."2049

          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

          Believe that you have received      Mark 11:22 Jesus answered them, "Have faith in God. 11:23 For most certainly I tell you, whoever may tell this mountain, 'Be taken up and cast into the sea,' and doesn't doubt in his heart, but believes that what he says is happening; he shall have whatever he says. 11:24 Therefore I tell you, all things whatever you pray and ask for, believe that you have received them, and you shall have them. 2053

          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

          Certainty      Mark 11:22 Jesus answered them, "Have faith in God. 11:23 For most certainly I tell you, whoever may tell this mountain, "Be taken up and cast into the sea" and doesn't doubt in his heart, but believes that what he says is happening; he shall have whatever he says.1408

          Command the forces of nature      Matthew 8:26 He said to them, "Why are you fearful, O you of little faith?" Then he got up, rebuked the wind and the sea, and there was a great calm. 8:27 The men marveled, saying, "What kind of man is this, that even the wind and the sea obey him?" 1621

          Faith evokes duty      When we believe, we take ourselves to be believed in; and whatever obeys the belief, obeys out of duty. Thus faith evokes duty. Luke 17:6 The Lord said, "If you had faith like a grain of mustard seed, you would tell this sycamore tree, 'Be uprooted, and be planted in the sea,' and it would obey you. 17:7 But who is there among you, having a servant plowing or keeping sheep, that will say, when he comes in from the field, 'Come immediately and sit down at the table,' 17:8 and will not rather tell him, 'Prepare my supper, clothe yourself properly, and serve me, while I eat and drink. Afterward you shall eat and drink'? 17:9 Does he thank that servant because he did the things that were commanded? I think not. 17:10 Even so you also, when you have done all the things that are commanded you, say, 'We are unworthy servants. We have done our duty.'"2057

          God is glorified when Jesus' disciples bear fruit      See video commentary by Dee Guyton! John 15:7 If you remain in me, and my words remain in you, you will ask whatever you desire, and it will be done for you. 15:8 "In this is my Father glorified, that you bear much fruit; and so you will be my disciples.1375

          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

          Teach from authority      Matthew 7:28 It happened, when Jesus had finished saying these things, that the multitudes were astonished at his teaching, 7:29 for he taught them with authority, and not like the scribes. 1483

     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

          What does Jesus want?      Matthew 8:2 Behold, a leper came to him and worshiped him, saying, "Lord, if you want to, you can make me clean." 8:3 Jesus stretched out his hand, and touched him, saying, "I want to. Be made clean." Immediately his leprosy was cleansed. 8:4 Jesus said to him, "See that you tell nobody, but go, show yourself to the priest, and offer the gift that Moses commanded, as a testimony to them." 1600

          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

     Faith is irresistible      2095

          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

          What grows larger      Matthew 13:31 He set another parable before them, saying, "The Kingdom of Heaven is like a grain of mustard seed, which a man took, and sowed in his field; 13:32 which indeed is smaller than all seeds. But when it is grown, it is greater than the herbs, and becomes a tree, so that the birds of the air come and lodge in its branches."1980

          What spreads through      Matthew 13:33 He spoke another parable to them. "The Kingdom of Heaven is like yeast, which a woman took, and hid in three measures of meal, until it was all leavened."1981

          Why will it obey      Luke 17:6 The Lord said, "If you had faith like a grain of mustard seed, you would tell this sycamore tree, 'Be uprooted, and be planted in the sea,' and it would obey you.2056

     Persistence of faith      2097

          Having faith      Matthew 17:20 He said to them, "Because of your unbelief. For most certainly I tell you, if you have faith as a grain of mustard seed, you will tell this mountain, 'Move from here to there,' and it will move; and nothing will be impossible for you. 17:21 But this kind doesn't go out except by prayer and fasting."2046

          Must keep praying and not give up      Luke 18:1 He also spoke a parable to them that they must always pray, and not give up, 18:2 saying, "There was a judge in a certain city who didn't fear God, and didn't respect man. 18:3 A widow was in that city, and she often came to him, saying, 'Defend me from my adversary!' 18:4 He wouldn't for a while, but afterward he said to himself, 'Though I neither fear God, nor respect man, 18:5 yet because this widow bothers me, I will defend her, or else she will wear me out by her continual coming.'" 18:6 The Lord said, "Listen to what the unrighteous judge says. 18:7 Won't God avenge his chosen ones, who are crying out to him day and night, and yet he exercises patience with them? 18:8 I tell you that he will avenge them quickly. Nevertheless, when the Son of Man comes, will he find faith on the earth?"2060

          Not doubt       Matthew 21:20 When the disciples saw it, they marveled, saying, "How did the fig tree immediately wither away?" 21:21 Jesus answered them, "Most certainly I tell you, if you have faith, and don't doubt, you will not only do what was done to the fig tree, but even if you told this mountain, 'Be taken up and cast into the sea,' it would be done. 21:22 All things, whatever you ask in prayer, believing, you will receive."2048

          Not doubt in your heart      Mark 11:22 Jesus answered them, "Have faith in God. 11:23 For most certainly I tell you, whoever may tell this mountain, 'Be taken up and cast into the sea,' and doesn't doubt in his heart, but believes that what he says is happening; he shall have whatever he says. 11:24 Therefore I tell you, all things whatever you pray and ask for, believe that you have received them, and you shall have them. 2052

     Requirement to participate in the Kingdom of Heaven      2096

          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 greatest commandment?      22:34 But the Pharisees, when they heard that he had silenced the Sadducees, gathered themselves together. 22:35 One of them, a lawyer, asked him a question, testing him. 22:36 "Teacher, which is the greatest commandment in the law?" 22:37 Jesus said to him, "'You shall love the Lord your God with all your heart, with all your soul, and with all your mind.' 22:38 This is the first and great commandment. 22:39 A second likewise is this, 'You shall love your neighbor as yourself.' 22:40 The whole law and the prophets depend on these two commandments."2018

          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

          Who can enter the Kingdom of Heaven      John 3:5 ... "Most certainly I tell you, unless one is born of water and spirit, he can't enter into the Kingdom of God! 3:6 That which is born of the flesh is flesh. That which is born of the Spirit is spirit. 3:7 Don't marvel that I said to you, 'You must be born anew.' 3:8 The wind blows where it wants to, and you hear its sound, but don't know where it comes from and where it is going. So is everyone who is born of the Spirit."2080

          Who can or can't understand      The word "overcome" is also translated "comprehended". God (who understands) and Jesus (who comes to understand) are the same God because they understand the same thing. Thus the darkness can't understand the light because it is not God. Only those who are of God can understand the light. John 1:5 The light shines in the darkness, and the darkness hasn't overcome it1967

     Speak to their faith      2098

          Building on faith as an outlook for a culture      Jesus looks to people's faith as the test of whether they have a shared culture. And given that faith, he speaks to that culture in them. Matthew 9:2 Behold, they brought to him a man who was paralyzed, lying on a bed. Jesus, seeing their faith, said to the paralytic, "Son, cheer up! Your sins are forgiven you." 1654

          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

          Speaking to the good in others      I did a study of the Gospel of Mark to understand Jesus' emotional responses, his cognitive expectations and how he responded to them to get things done. He spoke to the good in others.1332

          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

               Judas asks if he will betray Jesus      Matthew 26:25 Judas, who betrayed him, answered, "It isn't me, is it, Rabbi?" He said to him, "You said it."1248

          Incapacity is caused by a demon which can be cast out      Matthew 9:32 As they went out, behold, a mute man who was demon possessed was brought to him. 9:33 When the demon was cast out, the mute man spoke. The multitudes marveled, saying, "Nothing like this has ever been seen in Israel!" 9:34 But the Pharisees said, "By the prince of the demons, he casts out demons." 1730

          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

               Zero is just a place holder      Note that zero is just a place holder and so we don't have to multiply or add by zero in the algorithms. Zero is a fiction (as a numerical value outside of the system) and those operations are placeholder operations (meaningful only with regard to the particular system). Gospel Math. 1849

          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

          Convenient notation      Think about convenient notation. pg.29, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1426

          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

          Invariants      An invariant, as the name suggests, is merely some aspect of a problem - usually a numerical quantity - that does not change, even if many other properties do change. pg. 102, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1643

               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

          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

          Motel room paradox      Three women check into a motel room which advertises a rate of $27 per night. They each give $10 to the porter, and ask her to bring back 3 dollar bills. The porter returns to the desk, where she learns that the room is actually only $25 per night. She gives $25 to the motel desk clerk, returns to the room, and gives the guests back each one dollar, deciding not to tell them about the actual rate. Thus the porter has pocketed $2, while each guest spent 10-1 = $9, a total of 2 + 3 x 9 = $29. What happened to the other dollar? ... The actual "invariant" here is not $30, but $27, the amount that the guests spend, and this will always equal the amount that the porter took ($2) plus the amount that went to the desk ($25). pg. 22, 102, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1647

          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

        Believing

Believing      Referencing the absolute within us. Isolate what is most essential, what grounds everything else, what I truly want, what I love most intensely. What do I truly want?36

          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

          What does everything depend on?      Conversing with: all that follows Jesus taught that the whole law and the prophets depends on "Love God with all of your heart, with all your soul, and with all your mind" and "Love your neighbor as yourself".1220

          What encompasses everything?      Conversing with: everything As a child, I wondered how to apply myself. What was most important? I valued knowledge and I thought it was the key to all things. I thought it was important to know as broadly, deeply and generally as possible. I thus chose my quest to know everything and apply that knowledge usefully. Similarly, we can arrive at our deepest value in life by asking, what value includes all of our other values?1221

        Rooting ourselves in our inner world

Rooting ourselves in our inner world      1669

          Having a complete desire to win      Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: But there are a few, that have that intestinal fortitude, and they want to win at any cost. That's the guy who's listening to whatever his corner has to say.1896

        Give everything away

Give everything away      1080

          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

          Doing what I love      393

          Doing what you want to do      445

        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 do people need to hear?      Matthew 10:7 As you go, preach, saying, 'The Kingdom of Heaven is at hand!'1762

          What God wishes for      Matthew 9:12 When Jesus heard it, he said to them, "Those who are healthy have no need for a physician, but those who are sick do. 9:13 But you go and learn what this means: 'I desire mercy, and not sacrifice,' for I came not to call the righteous, but sinners to repentance." 1718

          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

          What lets one read Scripture with fresh eyes      Matthew 13:51 Jesus said to them, "Have you understood all these things?" They answered him, "Yes, Lord." 13:52 He said to them, "Therefore every scribe who has been made a disciple in the Kingdom of Heaven is like a man who is a householder, who brings out of his treasure new and old things."1989

          What might have changed the course of history      Matthew 11:20 Then he began to denounce the cities in which most of his mighty works had been done, because they didn't repent. 11:21 "Woe to you, Chorazin! Woe to you, Bethsaida! For if the mighty works had been done in Tyre and Sidon which were done in you, they would have repented long ago in sackcloth and ashes. 11:22 But I tell you, it will be more tolerable for Tyre and Sidon on the day of judgment than for you. 11:23 You, Capernaum, who are exalted to heaven, you will go down to Hades. For if the mighty works had been done in Sodom which were done in you, it would have remained until this day. 11:24 But I tell you that it will be more tolerable for the land of Sodom, on the day of judgment, than for you." 1907

          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

        Respect - Implicitly Care

Respect - Implicitly Care      186

          Five-Fingered Consensus      gauge extent of consensus 379

          Help Me Understand      know what participants want to know 381

          Red:Green Cards      honor feedback 380

        Believing in believing

Believing in believing      Allowing for self and others. Listen to others and God, appreciate what they care about as given by their choices, words and actions. Take that to heart.37

          What would be reasonable      As a child, I had made an offer to God, which seemed reasonable, and I felt God's presence, and I wondered if I should have a sign to confirm this understanding. But then I thought it would be wrong to ask for a sign, to look for a sign, to demand a sign, as I might invent a sign, or depend on a sign, and be lead astray. So I told God that my sign would be that I have no sign, and that if he ever meant otherwise, he would give me a sign.1223

        Presuming others likewise

Presuming others likewise      1670

          Develop our own understanding likewise      Pamela McLean, June 10, 2011. Perhaps all we ever really do in dig deeper into our own understanding, and develop that, in response to someone else sharing their own explorations of understanding.1833

          Having someone who sounds urgent but not desperate      Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: The key is having someone who's composed, someone who has experience. Someone who wants you to get out there and go at the guy but who doesn't sound so desperate or frustrated. And Angelo Dundee was that guy, who was in my corner, who said the right things at the right time."1897

          Make eye contact that you are OK      Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: While you're doing that, make eye contact with the referee to give him [the] sense that you're OK.1899

          Trust the person you are improvising with      Fresh Air interview of Steve Coogan, June 9, 2011: Coogan tells Fresh Air's Dave Davies that playing a version of himself was harder than playing fictional characters, in part because of the way he had to think about himself in front of a camera. "My biggest fear, really, was that it would be self-indulgent," he says. "Actors play my parents and actors play various people I meet, but the only real people are Rob and I. So it's a very difficult thing to try and be creative in that type of environment. But if you trust the person you're with, then you know when you're improvising you'll be able to create stuff and come up with ideas."1873

        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

          Express appreciation through gifts      When I go traveling and staying with people it is good to have some kind of present to express my appreciation. So I try to think through and carry gifts that would be special but for people who I might not even know yet. So in Lithuania I sometimes buy small pieces of art or blank leather bound books.1069

          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

          Asking about intentions      416

          Asking our group      386

          Consider what others say      407

          Considering who knows      440

          Considering who needs to know what      442

          Get feedback from people      392

          Identify with the words of a song      395

          Knowing if someone is paid      449

          Listening to elders      527

          Listening to others      455

          Listening to what you have to say      402

          Notice what a person does      494

          Noting choices people could have made      446

          Noting choices people make      447

          Serving others      394

          Taking music and words to heart      403

          Think of others as equals      408

        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

          Dependence on circumstances      Matthew 9:14 Then John's disciples came to him, saying, "Why do we and the Pharisees fast often, but your disciples don't fast?" 9:15 Jesus said to them, "Can the friends of the bridegroom mourn, as long as the bridegroom is with them? But the days will come when the bridegroom will be taken away from them, and then they will fast.1723

          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

          How others value followers      Matthew 10:24 "A disciple is not above his teacher, nor a servant above his lord. 10:25 It is enough for the disciple that he be like his teacher, and the servant like his lord. If they have called the master of the house Beelzebul, how much more those of his household!1802

          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

          Men reward each other in this world      Matthew 6:1 "Be careful that you don't do your charitable giving before men, to be seen by them, or else you have no reward from your Father who is in heaven. 6:2 Therefore when you do merciful deeds, don't sound a trumpet before yourself, as the hypocrites do in the synagogues and in the streets, that they may get glory from men. Most certainly I tell you, they have received their reward.1599

          Showing good will and allowing others to condemn you      Jesus, John the Baptist and Paul show good will to others by satisfying their norms, at least to an extent, and yet allow others to condemn them even so. Matthew 11:16 "But to what shall I compare this generation? It is like children sitting in the marketplaces, who call to their companions 11:17 and say, 'We played the flute for you, and you didn't dance. We mourned for you, and you didn't lament.' 11:18 For John came neither eating nor drinking, and they say, 'He has a demon.' 11:19 The Son of Man came eating and drinking, and they say, 'Behold, a gluttonous man and a drunkard, a friend of tax collectors and sinners!' But wisdom is justified by her children." 1864

          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

          Unfairness towards the good son      Luke 15:25 "Now his elder son was in the field. As he came near to the house, he heard music and dancing. 15:26 He called one of the servants to him, and asked what was going on. 15:27 He said to him, 'Your brother has come, and your father has killed the fattened calf, because he has received him back safe and healthy.' 15:28 But he was angry, and would not go in. Therefore his father came out, and begged him. 15:29 But he answered his father, 'Behold, these many years I have served you, and I never disobeyed a commandment of yours, but you never gave me a goat, that I might celebrate with my friends. 15:30 But when this, your son, came, who has devoured your living with prostitutes, you killed the fattened calf for him.' 15:31 "He said to him, 'Son, you are always with me, and all that is mine is yours. 15:32 But it was appropriate to celebrate and be glad, for this, your brother, was dead, and is alive again. He was lost, and is found.'"1373

          What a disciple should expect      Matthew 10:24 "A disciple is not above his teacher, nor a servant above his lord. 10:25 It is enough for the disciple that he be like his teacher, and the servant like his lord. If they have called the master of the house Beelzebul, how much more those of his household!1801

          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 is one's cross      Matthew 10:38 He who doesn't take his cross and follow after me, isn't worthy of me. 1816

          What to expect of a disciple      Matthew 10:24 "A disciple is not above his teacher, nor a servant above his lord. 10:25 It is enough for the disciple that he be like his teacher, and the servant like his lord. If they have called the master of the house Beelzebul, how much more those of his household!1800

          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

          Whether to be afraid      Matthew 10:29 "Aren't two sparrows sold for an assarion coin? Not one of them falls on the ground apart from your Father's will, 10:30 but the very hairs of your head are all numbered. 10:31 Therefore don't be afraid. You are of more value than many sparrows.1809

          Who bears the brunt of hate      Matthew 10:22 You will be hated by all men for my name's sake, but he who endures to the end will be saved. 1796

          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

          Why did he come?      Matthew 20:24 When the ten heard it, they were indignant with the two brothers. 20:25 But Jesus summoned them, and said, "You know that the rulers of the nations lord it over them, and their great ones exercise authority over them. 20:26 It shall not be so among you, but whoever desires to become great among you shall be your servant. 20:27 Whoever desires to be first among you shall be your bondservant, 20:28 even as the Son of Man came not to be served, but to serve, and to give his life as a ransom for many."2017

     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

               Addition means combine like units, list different units      We can practice this principle by considering "units", but also numbers used as units ("tens", "thousands", "millions", "sevenths", "percentages"), imaginary units ("zillions"), any numbers ("twos", (-1)), unknown numbers ("X"), squares of unknown numbers ("X**2"). We can mix units such as 2x + 3y -5x + 4y and likewise. Gospel Math. 1847

          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

          Graphic Gameplan      agree on executable action plan 335

          RACI Matrix      clarify responsibility, improve morale 375

          Scenario Slider      identify and prepare for key scenarios 378

          Who/What/When Matrix      make clear commitments 377

          WhoDo      brainstorm, plan and prioritize actions 376

        Believing in believing in believing

Believing in believing in believing      Empathizing with their views. Appreciate people's commitments, their investigations, endeavors and positions, and integrate them.38

               Choo-choo train      One day, perhaps for my birthday or for Christmas, my parents got me a black plastic choo-choo train that I could sit on and pretend to ride around the apartment. I must have been three years old because we were still living at the Page Street apartments in Buena Park. It was the first impressive toy that I had and I treasured it because it was "mine", because it was given to "me", and it was an acknowledgment that I should "have" things. My parents invited over Robbie's family. He was a little boy who used to live at the apartments, with whom I had played sometimes, but didn't really know, as I basically played by myself. My mother and father asked that I show him my choo-choo train and let him ride it. I said No. I felt it was mine and I wanted to keep it that way. They insisted. They said I was bad. This disturbed me greatly. My parents said later that I refused to talk to them for three days. But I remember some of what I was thinking: that it was deceitful of them to give me things and later to say what I must do with them; it was the same as to take them from me; that thus no notion of property could ever be secure; and thus there could be no ground for civilization; and this made progress untenable; yet progress was our highest ideal. I think finally they allowed that I had not behaved badly. And then I relented that my parents were loving people and that I should not make too much of this. But I told myself that I would never get attached to things. 1850

          Writing letters in my mind      I remember writing many letters in my mind, notably while I would ride my bike through the orchards of California's Central Valley and towards the Sierra mountains, when I worked in Visalia in 2001 as a software developer. Often, I never had a chance to write these letters, so they kept evolving. 1851

        Integrating some of us

Integrating some of us      1671

          Honoring a request      She expressed no bitterness about her illness or about my inability to cure her. During her last hospitalization, she made one request. She lay in the hospital bed, her chest heaving to get air. We held hands. "Doctor Ansen?" Her eyes sparkled. "Ah want ta see yo chile. Please, can ah see yo chile?" We hatched a plan. On the next Saturday morning, I brought my son to County Hospital for her to see. She came to the sixth floor window of the Medical A building. I stood outside and held my infant son up over my head like a gift offering to the gods. She smiled and waved at us from the sixth floor window, oxygen tubes dangling from her nose. We waved back. I cried when she died a few days later. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.19341942

          Let each other offend each other      Fresh Air interview of Steve Coogan, June 9, 2011: So the two comedians decided to give each other the artistic license to offend each other while filming. "What makes it interesting is that there's an edge to it and a discomfort to it that makes it engaging," Coogan says. "It's not just a couple of actors saying, 'Get a load of me. I'm laughing at myself.' There are a couple of moments where I find Rob irritating — genuinely — and I respond naturally, but not the way that I would in reality."1875

          Revealing that you have been abused      Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: I saw an episode of Oprah and [former child TV star] Todd Bridges finally came forth and said that he was sexually abused. I hear people always say that when you surrender and admit these things, it's a sense of freedom too.1903

          What I miss      Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: "Of all the things I miss in boxing, I miss the preparation of a bout, I miss choreographing tactics and moves and things like that," he says. "I miss all of my guys, my entourage being around me and working out with me, getting in better shape.1893

        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

          Recognizing how people help or not      In general, different people helped or not in different ways, as they were able and felt comfortable.1051

          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

          Appreciating people      479

          Appreciating people's leadership      482

          Appreciating people's skills      481

          Realizing what others can do      466

          Recognizing what others could do      460

          Recognizing who should be involved      462

        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

          Anticipating      Matthew 17:24 When they had come to Capernaum, those who collected the didrachma coins came to Peter, and said, "Doesn't your teacher pay the didrachma?" 17:25 He said, "Yes." When he came into the house, Jesus anticipated him, saying, "What do you think, Simon? From whom do the kings of the earth receive toll or tribute? From their children, or from strangers?"2001

          Distinguish between people and their demons      Matthew 8:28 When he came to the other side, into the country of the Gergesenes, two people possessed by demons met him there, coming out of the tombs, exceedingly fierce, so that nobody could pass that way. 8:29 Behold, they cried out, saying, "What do we have to do with you, Jesus, Son of God? Have you come here to torment us before the time?" 8:30 Now there was a herd of many pigs feeding far away from them. 8:31 The demons begged him, saying, "If you cast us out, permit us to go away into the herd of pigs." 8:32 He said to them, "Go!" They came out, and went into the herd of pigs: and behold, the whole herd of pigs rushed down the cliff into the sea, and died in the water.1622

          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

          Give as was given to you      By giving freely as was freely given to us, we extend the giving of the giver, support it, confirm it and include ourselves in it. Matthew 10:8 Heal the sick, cleanse the lepers, and cast out demons. Freely you received, so freely give. 1780

          How Jesus treats us      Jesus treats us as equals, as we treat him. Matthew 10:32 Everyone therefore who confesses me before men, him I will also confess before my Father who is in heaven. 10:33 But whoever denies me before men, him I will also deny before my Father who is in heaven. 1810

          How will one be saved      Fleeing is key to being saved, and the variety of places to go, and the fact that there are more places than needed to "teach by making an example of". Matthew 10:22 You will be hated by all men for my name's sake, but he who endures to the end will be saved. 10:23 But when they persecute you in this city, flee into the next, for most certainly I tell you, you will not have gone through the cities of Israel, until the Son of Man has come. 1798

          Pray to God for vocation      Matthew 9:37 Then he said to his disciples, "The harvest indeed is plentiful, but the laborers are few. 9:38 Pray therefore that the Lord of the harvest will send out laborers into his harvest." 1736

          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 confirms the Kingdom of Heaven?      Matthew 10:8 Heal the sick, cleanse the lepers, and cast out demons. Freely you received, so freely give. 1779

          What does God control      Matthew 10:29 "Aren't two sparrows sold for an assarion coin? Not one of them falls on the ground apart from your Father's will, 10:30 but the very hairs of your head are all numbered. 10:31 Therefore don't be afraid. You are of more value than many sparrows.1808

          What happens to one who seeks or loses one's life      Matthew 10:39 He who seeks his life will lose it; and he who loses his life for my sake will find it. 1818

          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

          What is pleasing to God      The Kingdom of Heaven is hidden from the wise and understanding as part of the union of all through God's impartiality. It is not for those who understand (as God the Father) but who are coming to understand (as the Son of Man who is ultimately, upon understanding, the Son of God). Matthew 11:25 At that time, Jesus answered, "I thank you, Father, Lord of heaven and earth, that you hid these things from the wise and understanding, and revealed them to infants. 11:26 Yes, Father, for so it was well-pleasing in your sight. 1912

          What is the reward for leaving everything to follow Jesus      Matthew 19:27 Then Peter answered, "Behold, we have left everything, and followed you. What then will we have?" 19:28 Jesus said to them, "Most certainly I tell you that you who have followed me, in the regeneration when the Son of Man will sit on the throne of his glory, you also will sit on twelve thrones, judging the twelve tribes of Israel. 19:29 Everyone who has left houses, or brothers, or sisters, or father, or mother, or wife, or children, or lands, for my name's sake, will receive one hundred times, and will inherit eternal life. 19:30 But many will be last who are first; and first who are last.2009

          Who is blessed      Jesus thinks about who is blessed. He feels that those who aren't disappointed in him will be blessed. Matthew 11:6 Blessed is he who finds no occasion for stumbling in me.1828

          Who needs a physician?      Matthew 9:11 When the Pharisees saw it, they said to his disciples, "Why does your teacher eat with tax collectors and sinners?" 9:12 When Jesus heard it, he said to them, "Those who are healthy have no need for a physician, but those who are sick do. 9:13 But you go and learn what this means: 'I desire mercy, and not sacrifice,' for I came not to call the righteous, but sinners to repentance." 1720

          Who will be saved      Matthew 10:22 You will be hated by all men for my name's sake, but he who endures to the end will be saved.1797

          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

          Linear combination      ...the result was attained in two steps. First, we were lucky enough to spot a particularly accessible case, a special situation, and gave a solution well adapted, but restricted, to this special situation ... Then, by combining particular cases to which the restricted solution is applicable, we obtained the full, unrestricted solution, applicable to the general case... The first step deals with a particular case which is not only especially accessible, but also especially useful; we can appropriately call it a leading particular case: it leads the way to the general solution. The second step combines particular cases by a specific algebraic operation. ... n particular solutions, after being multiplied by given constants, are added to form the general solution. ... we add and subtract equations dealing with the special situation to obtain the general proof. Let us call the algebraic operation employed ... linear combination or superposition. We may use the terms introduced to outline our pattern: Starting from a leading special situation we attain the general solution by superposition of particular cases. "Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving" by George Polya, 1962, John Wiley & Sons.2252

        Include - Care

Include - Care      176

          Air Time Mastermind      attract solutions to pressing problems 336

          Brainwriting      include everybody in evolving an idea 340

          Button      pay attention to everyone 339

          Open Space      network around active participants and their purposes 338

          Quaker Conversation      invite everybody and let conversation flow 337

          Talking Chips      include every voice equally 341

        Believing in believing in believing in believing

Believing in believing in believing in believing      Fostering the spirit amongst us. Create space to include everybody as they are, both actively immersed and passively reflected, alternatively, so God's spirit may speak through them.39

          Dedicating my best hour      Starting around 1994, when I started to work from home as a software developer, I made sure to start my day by working on my philosophy for an hour or two. Even later, as I struggled and failed to make a living from my lab, Minciu Sodas, I always dedicated my best hour or two to my philosophy. My best hour is in the morning, when my mind is fresh and uncluttered with concerns. I pray to God, do some calisthenics, eat breakfast, check my emails to keep them off my mind, and then apply my mind to my philosophy, preferably to the deepest question that I can. My goal is to get a new idea every day. Then I feel that my day has gone well and it doesn't matter what else happens. I typically continue by reviewing, writing and sharing my strategy for applying myself and making a living. As the day wears on, I make some effort to make a living. But I don't let that have my best energies. I believe that we all have a right and duty to spend one or two hours each day of our best time to apply ourselves and do what we were created and inspired to do.1904

        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

          Accepting a person the way they are      520

          Keeping a person in mind      521

        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

        Time

Time      C1) Sequence C2) Poset with maximal or minimal elements C3) Least upper bounds, greatest lower bounds C4) Limits The act of ever getting a new sheet (blank or otherwise) makes for a countably infinite list. That is what we need for mathematical induction. Next, we may prefer some sheets as more noteworthy than others, which we ignore, so that some are most valuable. Such extremes are assumed by the extreme principle. An example is the square as the rectangle of a given perimeter that yields the most area. Next, we construct monovariants which say, in effect, that the only results which count are those that beat the record-to-beat, which yields sequences of increasing minimums, thus a greatest lower bound, or alternatively, a least upper bound. Finally, we allow such a boxing-in or boxing-out process to continue indefinitely, yielding (or not) a limit that may very well transcend the existing system (as the reals transcend the rationals). We thereby construct internal "time" which is fully fledged in that the continuum without gaps may be used to model space. 11

        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

        Wave point of view

Wave point of view       Our measurement can take place outside of our frames of measurement and thus link several such frames. This is a dynamic point of view where there is no distinction between the future and the past so that all is reversible. We can think of the wave in terms of where it starts and where it ends. It includes all paths between these two points. This is a deterministic, nonspatial point of view, which establishes time, an ideal continuum that is beyond the frames but thus relevant for us. Analogously, in math we may have a sequence of sheets, as with mathematical induction, some of which may be of ultimate importance, as with the extreme principle, thus allowing for boxing in with greatest lower bounds and least upper bounds, leading to limits that may transcend, go beyond what we can account for. Or in life, we can be open to care about everything, then care about our minds by which we care, then come up against our personal limits, then allow for an ideal (such as Jesus) that transcends our limits. 848

        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

          Favor the lost sheep of the House of Israel      Matthew 10:5 Jesus sent these twelve out, and commanded them, saying, "Don't go among the Gentiles, and don't enter into any city of the Samaritans. 10:6 Rather, go to the lost sheep of the house of Israel. 1761

          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

               Jesus healed Peter's mother-in-law      Matthew 8:14 When Jesus came into Peter's house, he saw his wife's mother lying sick with a fever. 8:15 He touched her hand, and the fever left her. She got up and served him. 1610

          People love as much as they are forgiven      Luke 7:41 ... A certain lender had two debtors. The one owed five hundred denarii, and the other fifty. 7:42 When they couldn't pay, he forgave them both. Which of them therefore will love him most?" 7:43 Simon answered, "He, I suppose, to whom he forgave the most." He said to him, "You have judged correctly." 7:44 Turning to the woman, he said to Simon, "Do you see this woman? I entered into your house, and you gave me no water for my feet, but she has wet my feet with her tears, and wiped them with the hair of her head. 7:45 You gave me no kiss, but she, since the time I came in, has not ceased to kiss my feet. 7:46 You didn't anoint my head with oil, but she has anointed my feet with ointment. 7:47 Therefore I tell you, her sins, which are many, are forgiven, for she loved much. But to whom little is forgiven, the same loves little.1367

          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

        Relax - Innovate

Relax - Innovate      177

          6-8-5      generate many rough ideas quickly 1693

          Break      challenge assumptions that constrain imagination 342

          Image-ination      get unstuck with new ideas 346

          Mission Impossible      let ideas settle, open room for new thoughts 344

          Object Brainstorm      suggest associations 343

          The Anti-Problem      get unstuck, note the obvious 345

        Caring

Caring      Being open to all outside us. Be open to any and all thoughts and associations. Note what captures my attention. Am I doing anything about this?40

          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

        Expose ourselves to new information

Expose ourselves to new information      The mother of all biases is the "mind-set" ... the distillation of our accumulated knowledge about a subject into a single, coherent framework or lens through which we view it. ... the summation or consolidation of all of our biases about a particular subject. ... feelings come upon us in a flash; we don't think about them before we sense them. Such is the power of mind-set. ... When we read a novel ... the author furnishes us bits of information ... At some point, our absorption and interpretation of these small details solidifies into a particular mind-set. ... mind-sets enable us to put events and information immediately into context without having to reconstruct from memory everthing that previously happened ... provide us instant insight into complex problems ... Being an aggregate of countless biases and beliefs, a mind-set represents a giant shortcut of the mind ... The influence a mind-set has on our thinking is thus magnified by many orders of magnitude over that of a simple bias ... There is only one way to change undesirable biases and mind-sets, and that is by exposing ourselves (our minds) to new information and letting the mind do the rest. Fortunately, the mind is, to a large degree, a self-changing system: give it new information, and it will change the bias. ... Most biases and mind-sets, however, are highly resistant to alteration and are changed only gradually, eroded away by repeated exposure to new information. The Thinker's Toolkit by Morgan D.Jones drawing from The Universe Within: A New Science Explores the Human MindThe Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich.2130

        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

          Drifting attention      401

          Keep us attentive      438

          Noting whether I pay attention      505

          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

          Figure things out soundly      The "eye" is how the body figures things out. We should choose to figure things out soundly. Matthew 6:22 "The lamp of the body is the eye. If therefore your eye is sound, your whole body will be full of light. 6:23 But if your eye is evil, your whole body will be full of darkness. If therefore the light that is in you is darkness, how great is the darkness! 1447

          Incompatibility      Matthew 6:24 "No one can serve two masters, for either he will hate the one and love the other; or else he will be devoted to one and despise the other. You can't serve both God and Mammon. 1448

          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

          Jesus wishes to see faith      Matthew 8:8 The centurion answered, "Lord, I'm not worthy for you to come under my roof. Just say the word, and my servant will be healed. 8:9 For I am also a man under authority, having under myself soldiers. I tell this one, 'Go,' and he goes; and tell another, 'Come,' and he comes; and tell my servant, 'Do this,' and he does it." 8:10 When Jesus heard it, he marveled, and said to those who followed, "Most certainly I tell you, I haven't found so great a faith, not even in Israel.1604

          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 preferred      Matthew 13:44 "Again, the Kingdom of Heaven is like a treasure hidden in the field, which a man found, and hid. In his joy, he goes and sells all that he has, and buys that field.1986

          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 must one give up      Luke 18:24 Jesus, seeing that he became very sad, said, "How hard it is for those who have riches to enter into the Kingdom of God! 18:25 For it is easier for a camel to enter in through a needle's eye, than for a rich man to enter into the Kingdom of God."2071

          What one is willing to give up for it      Matthew 13:45 "Again, the Kingdom of Heaven is like a man who is a merchant seeking fine pearls, 13:46 who having found one pearl of great price, he went and sold all that he had, and bought it.1987

          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

          Who is our master?      Matthew 23:10 Neither be called masters, for one is your master, the Christ.1555

          Who is the greatest in the Kingdom of Heaven      The greatest is the one who is most humble, thus most healing. 18:1 In that hour the disciples came to Jesus, saying, "Who then is greatest in the Kingdom of Heaven?" 18:2 Jesus called a little child to himself, and set him in their midst, 18:3 and said, "Most certainly I tell you, unless you turn, and become as little children, you will in no way enter into the Kingdom of Heaven. 18:4 Whoever therefore humbles himself as this little child, the same is the greatest in the Kingdom of Heaven. 18:5 Whoever receives one such little child in my name receives me, 18:6 but whoever causes one of these little ones who believe in me to stumble, it would be better for him that a huge millstone should be hung around his neck, and that he should be sunk in the depths of the sea. 1960

          Words are things brought of one's treasure      Matthew 12:33 "Either make the tree good, and its fruit good, or make the tree corrupt, and its fruit corrupt; for the tree is known by its fruit. 12:34 You offspring of vipers, how can you, being evil, speak good things? For out of the abundance of the heart, the mouth speaks. 12:35 The good man out of his good treasure brings out good things, and the evil man out of his evil treasure brings out evil things. 12:36 I tell you that every idle word that men speak, they will give account of it in the day of judgment. 12:37 For by your words you will be justified, and by your words you will be condemned."2043

          Your heart is where your treasure is      Matthew 6:19 "Don't lay up treasures for yourselves on the earth, where moth and rust consume, and where thieves break through and steal; 6:20 but lay up for yourselves treasures in heaven, where neither moth nor rust consume, and where thieves don't break through and steal; 6:21 for where your treasure is, there your heart will be also. 1435

     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

          What attracts people      Matthew 11:7 As these went their way, Jesus began to say to the multitudes concerning John, "What did you go out into the wilderness to see? A reed shaken by the wind? 11:8 But what did you go out to see? A man in soft clothing? Behold, those who wear soft clothing are in king's houses. 11:9 But why did you go out? To see a prophet? Yes, I tell you, and much more than a prophet.1853

        Poset with maximal or minimal elements

Poset with maximal or minimal elements      We may prefer some sheets as more noteworthy than others, which we ignore, so that some are most valuable. Such extremes are assumed by the extreme principle. An example is the square as the rectangle of a given perimeter that yields the most area. 18

               Four times a right triangle is the difference of two squares      A right triangle rotated four times brings us back to where we were. This yields a difference of two squares from which the Pythagorean theorem follows. Note also that sin(x) and cos(x) are equal to their fourth derivatives. Gospel Math. 1845

               Right triangles are more basic than circles      Consider a line segment of length AB and consider all of the right triangles with hypotenuse AB. Drop the altimeter Z so that the hypotenuse is X + Y. The radius is the average, the arithmetic mean (X+Y)/2 and the altimeter is the geometric mean of X and Y. This makes clear that the arithmetic mean is greater than the geometric mean because the radius is greater than the altimeter. It also shows that the square has greater area than other rectangles of the same perimeter. Then the right triangles form a semicircle without the two endpoints. Circles depend on distance and thus presume right triangles. Gospel Math. 1844

               When black and white points are interspersed...      Let B and W be finite sets of black and white points, respectively, in the plane, with the property that every line segment which joins two points of the same color contains a point of the other color. Prove that both sets must lie on a single line segment. ... The extreme principle comes to the rescue: Assume that the points do not all lie on a line. Then they form at least one triangle. Consider the triangle of smallest area. Two of its vertices are the same color, so between them is a point of the other color, but this forms a smaller triangle - a contradiction! pg.82, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1630

          Arithmetic-Geometric Mean Inequality      1998 x 2000 ? 1999 x 1999 Your intuition probably tells you that the question mark should be replaced with "<", for the left-hand side is the area of a rectangle that is not quite a square, while the right hand side is the area of a square with the same perimeter as the rectangle (namely 4x1999). It makes sense that given a fixed amount of fencing, the rectangle of maximum area is a square. ... (x+y)/2 >= square-root of xy. ... The arithmetic mean of two positive real numbers is greater than or equal to the geometric mean, with equality only if the numbers are equal. ... Here is a nice geometric proof of AM-GM. Let AC be the diameter of a circle, and let B be any point on the circle. Recall that ABC will be a right triangle. Now locate point D so that BD is perpendicular to AC. Then triangles ABD and BCD are similar; hence x/g = g/y. Thus g = square root of xy, the geometric mean of x and y. Indeed, that's why it is called a geometric mean! pg.193-194 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2191

          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

          Monotonizing      Always be aware of order and maximum/minimum in a problem, and always assume, if possible, that the elements are arranged in order ... Think of this as "free information". pg.83, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1632

          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

          Tangent-line approximation      The tangent-line definition of the derivative stems from its formal definition as a limit. ... This suggests a useful, but less well-known, application of the derivative, the tangent-line approximation to the function. ... In general, analyzing f(a+h) with its tangent-line approximation f(a) + hf'(a) is very useful, especially when combined with other geometric information, such as convexity. ... Prove Bernoulli's Inequality (1 + x)**alpha >= 1 + alpha x for x>= -1 and alpha >= 1, with equality when x = 0. ... For integer alpha, this can be proven by induction ... But induction won't work for arbitrary real alpha. Instead, define f(u) = u**alpha, and note that f'(u) = alpha u**(alpha - 1) and f''(u) = alpha(alpha-1)u**(alpha-2). ... Thus the graph y=f(u) is concave up...Therefore, the graph of y=f(u) lies above the tangent line for all u>=0. ... f(1+x) is always strictly greater than its linear approximation 1+alpha x, except when x=0, in which case we have equality. pg.296-297 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2241

          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

               Division algorithm      The extreme principle in number theory ... Let a and b be positive integers, b>=a. Then there exist integers q,r satisfying q>=1 and 0<=r pg.92, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1634

        Recombine - Explicitly Innovate

Recombine - Explicitly Innovate      178

          Forced Analogy      introduce new angles 347

          Heuristic Ideation Technique      generate ideas by recombining possibilities 348

        Caring about caring

Caring about caring      Growing our mind. Accumulate, select, organize and reorganize thoughts. Recognize influences, inject feeling, direct my thinking and create my own mind, my own personal language.41

          Atune my mind      In 2000 or so, I noted eight areas in my personal life where I wanted to improve myself: Be with God, foster my conscience, foster my willpower, foster my stewardship, be curious, serve others, support others' endeavors, be successful. Each morning, after I prayed, I would run through each of these and imagine how I might do that during the day. I didn't try to plan to actually do them because it would be too contrived to carve out so much time. Rather, I would imagine how I might do that, and then during the day I would be open to similar opportunities and seize them. Over several years this helped me change in these ways and I felt better about myself. Afterwards, I gave up this practice because it had achieved its purpose and it took up time, perhaps twenty minutes. 2259

          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

          Take a long walk      Sometimes I set aside a day, often Sunday, for an excursion by bus and/or a long walk. I may bring along a philosophical problem that I work on, along with relevant diagrams or notes which I add to along the way. On such trips I often try to spend time with God, too. The trip helps me spend longer time on a problem, wrap my mind around it, clears my mind a bit, perhaps includes some random inputs or insights, helps me feel peaceful and reach deeper into my feelings, balance them out.1700

          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

          Ask for something more substantial from participants      Fresh Air interview of Steve Coogan, June 9, 2011: The idea for The Trip was first proposed by Michael Winterbottom, who directed Coogan and Brydon in A Cock and Bull Story. Winterbottom wanted something "more substantial" from the two actors, Coogan says.1874

          Being invited by the right person      Fresh Air interview of Stephen Colbert, June 15, 2001: That's when Sondheim wrote Colbert a personal note. "[He said that] against his instincts, he had a good time on my show and would I consider playing Harry in Company?" he says. "And he ended the letter with the sentence 'You have a perfect voice for musical theater.' And I read it to my wife and she said, 'Boy, you have to do this. No one, let alone Stephen Sondheim is going to ask you to do Sondheim.' And I said, 'You're right, I have to do it.' "1935

          Changing the level of protein intake      Can the amount of protein that we eat change the activity of an enzyme? How does protein intake affect cancer initiation? Our first test was to see whether protein intake affected the enzyme principally responsible for aflatoxin metabolism, the mixed function oxidase (MFO). This enzyme is very complex because it also metabolizes pharmaceuticals and other chemicals, friend or foe to the body. Paradoxically, this enzyme both detoxifies and activates aflatoxin. It is an extraordinary transformation substance. At the time we started our research, we hypothesized that the protein we consume alters tumor growth by changing how aflatoxin is detoxified by the enzymes present in the liver. ... After a series of experiments, the answer was clear. Enzyme activity could be easily modified simply by changing the level of protein intake. pg.51, T.Colin Campbell, Thomas M.Campbell, The China Study, 2006. Health.1829

          Enlisting a team      Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: After deciding to go pro, Leonard enlisted Morton to bring others onto his team. Morton asked several men, including lawyer Mike Trainer and Angelo Dundee, Muhammad Ali's trainer, to work with Leonard.1888

          Hear stories from real life      I heard similar stories from my other patients .. He was killed when he pushed a girl out of the way of the bullets. My patient, the mother of the dead boy, climbed into bed with her mother, also my patient, and they held each other and cried together. Her two surviving children struggled at school. She developed diabetes and hypertension and some heart abnormalities. The grandmother's health deteriorated as well. How can these experiences not affect health and accelerate death in our patients? Each story left me, mouth agape, in shock and dismay. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.19341939

          Read some of a key person's public letters      Pamela McLean, June 10, 2011. I don't pretend to read all the group posts that land in my inbox - but I do try to dip in to yours now and again to sample your thoughts and activities.1835

          What keeps people interested      Fresh Air interview of J.J. Abrams, June 13, 2011: "Whether it's commercial, whether it's online, whether it's any number of magazines, by the time a movie comes out — and all it is is entertainment — people are force-fed information they didn't even want. ... So by the time you go to the movie, it doesn't have that power anymore."1872

        Control what our mind focuses on

Control what our mind focuses on      Third, structuring helps us to focus our analysis. The mind instinctively focuses. That's how the mind works, so it's going to focus whether we want it to or not. Therefore, we're better off to work with the mind than against it and, in so doing, control what it focuses on. If we don't, it will do its own focusing, and its shortcuts can lead us down the wrong path. The Thinker's Toolkit by Morgan D.Jones2114

          Focus      [Edumnd Bolles, A Second Way of Knowing] We shift attention from sensation to sensation, watching only part of the visual field, listening to only some sounds around us, savoring certain flavors among many. Our capacity to select and "impart intensity" to sensations prevents us from being slaves to the physical world around us. Reality is a jumble of sensations and details. Attention enables us to combine separate sensations into unified objects and lets us examine objects closely to be sure of their identity. ... The Thinker's Toolkit by Morgan D.Jones drawing from The Universe Within: A New Science Explores the Human MindThe Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich.2132

        Maximize what I can take

Maximize what I can take      1085

          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

          Make the most of wasted time      In graduate school I started collecting record albums because I worked as a teaching assistant and grading math exams was numbingly mindless. I also wanted to expand my horizons.1012

          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

          Appreciating attentiveness      509

          Being totally present      502

          Focusing on my role      483

          Imagining the proper mindset      463

          Put in feeling      397

          Putting to memory      487

          Tune in to the season      406

        Variational principle

Variational principle      Wikipedia: A variational principle is a scientific principle used within the calculus of variations, which develops general methods for finding functions which minimize or maximize the value of quantities that depends upon those functions. For example, to answer this question: "What is the shape of a chain suspended at both ends?" we can use the variational principle that the shape must minimize the gravitational potential energy. According to Cornelius Lanczos, any physical law which can be expressed as a variational principle describes an expression which is self-adjoint. These expressions are also called Hermitian. Such an expression describes an invariant under a Hermitian transformation. Felix Klein's Erlangen program attempted to identify such invariants under a group of transformations. In what is referred to in physics as Noether's theorem, the Poincaré group of transformations (what is now called a gauge group) for general relativity defines symmetries under a group of transformations which depend on a variational principle, or action principle. See: History of variational principles in physics and Brachistochrone, the curve of fastest descent907

          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

          Hertz's principle of least curvature      Wikipedia: Hertz's principle of least curvature is a special case of Gauss' principle, restricted by the two conditions that there be no applied forces and that all masses are identical.905

          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

          Being a blind guide      Matthew 23:23 "Woe to you, scribes and Pharisees, hypocrites! For you tithe mint, dill, and cumin, and have left undone the weightier matters of the law: justice, mercy, and faith. But you ought to have done these, and not to have left the other undone. 23:24 You blind guides, who strain out a gnat, and swallow a camel! 1564

          Can different values coexist      Matthew 10:29 "Aren't two sparrows sold for an assarion coin? Not one of them falls on the ground apart from your Father's will, 10:30 but the very hairs of your head are all numbered. 10:31 Therefore don't be afraid. You are of more value than many sparrows.1807

          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

          Having them answer      Mark 12:15 He said to them, "Whose is this image and inscription?" They said to him, "Caesar's." 1397

          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 is the nature of wisdom?      Wisdom approaches people where they are at, or where they claim to be at. Matthew 11:18 For John came neither eating nor drinking, and they say, 'He has a demon.' 11:19 The Son of Man came eating and drinking, and they say, 'Behold, a gluttonous man and a drunkard, a friend of tax collectors and sinners!' But wisdom is justified by her children." 1865

          What is the source of hate      Matthew 10:22 You will be hated by all men for my name's sake, but he who endures to the end will be saved. 1795

          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

          Whether they think Jesus is of God      Jesus wants to know how they think of him, whether or not he is from God, because that determines how he interacts with them. Luke 18:18 A certain ruler asked him, saying, "Good Teacher, what shall I do to inherit eternal life?" 18:19 Jesus asked him, "Why do you call me good? No one is good, except one—God. 18:20 You know the commandments: 'Don't commit adultery,' 'Don't murder,' 'Don't steal,' 'Don't give false testimony,' 'Honor your father and your mother.'"2067

          Who finds or loses life      Matthew 10:39 He who seeks his life will lose it; and he who loses his life for my sake will find it. 1817

          Who is greater      Matthew 11:11 Most certainly I tell you, among those who are born of women there has not arisen anyone greater than John the Baptizer; yet he who is least in the Kingdom of Heaven is greater than he. 1854

          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

          Who only kills the body      The body can be resurrected. Matthew 10:26 Therefore don't be afraid of them, for there is nothing covered that will not be revealed; and hidden that will not be known. 10:27 What I tell you in the darkness, speak in the light; and what you hear whispered in the ear, proclaim on the housetops. 10:28 Don't be afraid of those who kill the body, but are not able to kill the soul. Rather, fear him who is able to destroy both soul and body in Gehenna. 1806

          Why Jesus came      Jesus brings out every person on their own. Matthew 10:34 "Don't think that I came to send peace on the earth. I didn't come to send peace, but a sword. 10:35 For I came to set a man at odds against his father, and a daughter against her mother, and a daughter-in-law against her mother-in-law. 10:36 A man's foes will be those of his own household.1812

          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

          Working independently of his disciples      Matthew 11:1 It happened that when Jesus had finished directing his twelve disciples, he departed from there to teach and preach in their cities. 1822

        Least upper bounds, greatest lower bounds

Least upper bounds, greatest lower bounds      We construct monovariants which say, in effect, that the only results which count are those that beat the record-to-beat, which yields sequences of increasing minimums, thus a greatest lower bound, or alternatively, a least upper bound.19

               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

          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

          Growth rates of functions      It is important to understand the hierarchy of growth rates for the most common functions. The best way to learn about this is to draw lots of graphs. Any quadratic function of x will dominate any linear function of x, provided that x is "large enough". ... x**a will eventually dominate x**b provided that a > b > 0. ...if a is any positive number, and b > 1, then b**x eventually dominates x**a ... exponential functions grow faster than polynomial functions ... if a is any positive number, and b > 1, then x**a eventually dominates log(b)x. In summary, the hierarchy of growth rates, from slowest to highest, is logarithms, powers, exponents. pg.190-191 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2190

          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

          Monovariant      A monovariant is a quantity which may or may not change at each step of a problem, but when it does change, it does so monotonically (in only one direction). Another term used for monovariant is semi-invariant. Monovariants are often used in the analysis of evolving systems, to show that certain final configurations must occur, and/or determine the duration of the system. Many monovariant arguments also make use of the extreme principle (at least the well-ordering principle). pg.112, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. 1747

               John Conway's Checker Problem      Put a checker on every lattice point ... in the plane with y-coordinates less than or equal to zero. The only legal moves are horizontal or vertical "jumping". By this we mean that a checker can leap over a neighbor, ending up 2 units up, down, right, or left of its original position, provided that the destination point is unoccupied. After the jump is complete, the checker that was jumped over is removed from the board. ... Is it possible to make a finite number of legal moves and get a checker to reach the line y=5? pg.114, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. 1751

               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

               Reversing cards in a deck      The n cards of a deck ... are labeled 1,...,n. Starting with the deck in any order, repeat the following operation: if the card on top is labeled k, reverse the order of the first k cards. Prove that eventually the first card will be 1 (so no further changes occur). pg.112, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. 1750

        Appeal - Implicitly Innovate

Appeal - Implicitly Innovate      179

          Ethos, Logos, Pathos      identify weakness in presentation 349

          Heart, Hand, Mind      appeal to a person holistically 350

        Caring about caring about caring

Caring about caring about caring      Acknowledging what transcends our limits. Let go of my personal limits and live the vision of an unlimited God who can live through all.42

          Appreciate my vulnerabilities      I was in first or second grade when I got glasses. It turned out that my vision was very bad. I can hardly make out the features of people's faces. Although I resisted needing glasses, I appreciated that they were a great gift. I wondered, how would I have survived before glasses were invented? I might not be able to work at all, I might have to be a beggar. And so I appreciated that I was not master of my own circumstances, and thus not free to do as I pleased.2090

        Recognizing the limits of our openness

Recognizing the limits of our openness      1675

          Be immersed in the lives of those who suffer      The hand of institutional racism was invisible to most white people, including my friends, who tended to avoid institutions or neighborhoods that catered to black people out of fear for their own safety or discomfort. My weekly session with my patients in the General Medicine Clinic heightened my sensitivity to the issues of race in America. In 1906, W.E.B. DuBois said, "The problem of the twentieth century is the problem of the color line." I was a middle-class white man from a small city in upstate New York. I had never been in a position to understand the meaning of these words until I was immersed in the lives of my patients that revealed their truth so powerfully and so tragically. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.19341936

          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

          Have hang ups and neuroses      Fresh Air interview of Steve Coogan, June 9, 2011: "There is a certain insecurity and a certain kind of malcontentedness that [I] just channel into [my] work," he says. "Being creative means some of the things that bother you stop bothering you, because you exploit them creatively. So it's a kind of a strange process, but you need to have hangups and neuroses to be creative. If you're just in a state of nirvana, you're not going to be very interesting or funny."1876

          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

          Not wanting to listen to your trainer      Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: "When you're losing, you really don't want to hear what your trainer has to say. He may tell you the right thing, but you're so exhausted, you're so beat up, that your lungs are burning, your legs are tired. And you've given up1895

          Things getting out of control      Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: I shared the wealth. And I tried to give [my family] a home. I bought homes for most of my family members. And I bought cars. I bought 10 cars, they were Pintos. They were free [for them.] I just tried to help them because they're my brothers and sisters. But things just got out of control. Things became too frequent.1902

          What a typical person goes through      Just about every man had a scar from a knife or bullet wound. Almost every woman had lost a close family member to violence. The names of lost loves and relatives were tattooed onto the arms and in the memories of my patients. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.19341940

        Become aware of bias

Become aware of bias      We instinctively rely on, and are susceptible to, biases and assumptions. Bias - an unconscious belief that conditions, governs, and compels our behavior ... Immanuel Kant theorized that the mind is not designed to give us uninterpreted knowledge of the world but must always approach it from a special point of view ... with a certain bias. ... Acquiring a bias is not a conscious mental process ... we are stuck with biases whether we want them or not ... It is bias that enables us to repeat an action we have taken before without going through all of the mental steps that led to the original act. ... assumptions beget biases, biases beget assumptions ... much more often than not, our biases lead us to correct conclusions and reactions, and they do it exceedingly fast. ... the rapidity of this process and the fact that it is unconscious - and thus uncontrolled - have the unfortunate effect of strengthening and validating our biases at the expense of truth. The reason ... is that we tend to give high value to new information that is consistent with our biases, thus reinforcing them, while giving low value to, and even rejecting, new information that is inconsistent. New information that is ambiguous either is construed as consistent with our biases or is dismissed as irrelevant. ... because most biases are hidden from our consciousness, we aren't aware of their existence or of their effects ... The trouble with biases is that they impose artificial constraints and boundaries on what we think. The insidious part is that we aren't even aware that our thinking is restricted. The Thinker's Toolkit by Morgan D.Jones drawing from The Universe Within: A New Science Explores the Human MindThe Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich.2127

        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

          Knowing what belongs to me      456

          Knowing what I will help with      471

          Recognizing how I can help others      461

          Recognizing the kind of person I am      424

     My limits      103

          Being limited      458

          Not finding alternatives      454

          Realizing my limitations      459

          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

          God through Jesus and angles saves those who stumble      God, Jesus, angels look to protect and save those who stumble. Matthew 18:10 See that you don't despise one of these little ones, for I tell you that in heaven their angels always see the face of my Father who is in heaven. 18:11 For the Son of Man came to save that which was lost.1345

          How are we one with Jesus and God      Matthew 10:40 He who receives you receives me, and he who receives me receives him who sent me. 10:41 He who receives a prophet in the name of a prophet will receive a prophet's reward. He who receives a righteous man in the name of a righteous man will receive a righteous man's reward. 10:42 Whoever gives one of these little ones just a cup of cold water to drink in the name of a disciple, most certainly I tell you he will in no way lose his reward." 1819

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

          How does one receive      We receive by receiving on behalf of another, of God in them, by giving through them. Matthew 10:40 He who receives you receives me, and he who receives me receives him who sent me. 10:41 He who receives a prophet in the name of a prophet will receive a prophet's reward. He who receives a righteous man in the name of a righteous man will receive a righteous man's reward. 10:42 Whoever gives one of these little ones just a cup of cold water to drink in the name of a disciple, most certainly I tell you he will in no way lose his reward." 1820

          Note from whom God hides and to whom reveals      The significance of Jesus' miracles are hidden from John the Baptist, from Bethsaida and Capernaum, from the wise and understanding, but "infants" are delighted and uplifted, those who are fresh in faith, who don't want to justify the world, but want it bettered, by those who suffer and labor, who want healing. Matthew 11:25 At that time, Jesus answered, "I thank you, Father, Lord of heaven and earth, that you hid these things from the wise and understanding, and revealed them to infants. 11:26 Yes, Father, for so it was well-pleasing in your sight. 1910

          What are the consequences of not believing?      Matthew 7:26 Everyone who hears these words of mine, and doesn't do them will be like a foolish man, who built his house on the sand. 7:27 The rain came down, the floods came, and the winds blew, and beat on that house; and it fell—and great was its fall." 1497

          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 beyond this world will manifest in this world      Matthew 6:6 But you, when you pray, enter into your inner room, and having shut your door, pray to your Father who is in secret, and your Father who sees in secret will reward you openly. 1597

          What is needed for judgment      Jesus is the judge because he is a "son of man" and knows what it means to be human. Yet he seeks not his own will, but that of his Father, who sent him and gave him life. John 5:24 "Most certainly I tell you, he who hears my word, and believes him who sent me, has eternal life, and doesn't come into judgment, but has passed out of death into life. 5:25 Most certainly, I tell you, the hour comes, and now is, when the dead will hear the Son of God's voice; and those who hear will live. 5:26 For as the Father has life in himself, even so he gave to the Son also to have life in himself. 5:27 He also gave him authority to execute judgment, because he is a son of man. 5:28 Don't marvel at this, for the hour comes, in which all that are in the tombs will hear his voice, 5:29 and will come out; those who have done good, to the resurrection of life; and those who have done evil, to the resurrection of judgment. 5:30 I can of myself do nothing. As I hear, I judge, and my judgment is righteous; because I don't seek my own will, but the will of my Father who sent me.2085

          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 makes people believe      God the Father changes people's hearts. John 4:47 When he heard that Jesus had come out of Judea into Galilee, he went to him, and begged him that he would come down and heal his son, for he was at the point of death. 4:48 Jesus therefore said to him, "Unless you see signs and wonders, you will in no way believe." 4:49 The nobleman said to him, "Sir, come down before my child dies." 4:50 Jesus said to him, "Go your way. Your son lives." The man believed the word that Jesus spoke to him, and he went his way. 2084

          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

          Where did the good and the bad come from      Matthew 13:37 He answered them, "He who sows the good seed is the Son of Man, 13:38 the field is the world; and the good seed, these are the children of the Kingdom; and the darnel weeds are the children of the evil one. 13:39 The enemy who sowed them is the devil. The harvest is the end of the age, and the reapers are angels. 13:40 As therefore the darnel weeds are gathered up and burned with fire; so will it be at the end of this age. 13:41 The Son of Man will send out his angels, and they will gather out of his Kingdom all things that cause stumbling, and those who do iniquity, 13:42 and will cast them into the furnace of fire. There will be weeping and the gnashing of teeth. 13:43 Then the righteous will shine forth like the sun in the Kingdom of their Father. He who has ears to hear, let him hear.1985

          Who comes to the light      John 3:17 For God didn't send his Son into the world to judge the world, but that the world should be saved through him. 3:18 He who believes in him is not judged. He who doesn't believe has been judged already, because he has not believed in the name of the one and only Son of God. 3:19 This is the judgment, that the light has come into the world, and men loved the darkness rather than the light; for their works were evil. 3:20 For everyone who does evil hates the light, and doesn't come to the light, lest his works would be exposed. 3:21 But he who does the truth comes to the light, that his works may be revealed, that they have been done in God."2079

          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

          Who loves the world      It is the Father who loved the world and gave up his Son for it. John 3:14 As Moses lifted up the serpent in the wilderness, even so must the Son of Man be lifted up, 3:15 that whoever believes in him should not perish, but have eternal life. 3:16 For God so loved the world, that he gave his one and only Son, that whoever believes in him should not perish, but have eternal life. 3:17 For God didn't send his Son into the world to judge the world, but that the world should be saved through him. 2083

          Whom to fear and why      Fear God. God is the only one who can destroy completely. Matthew 10:26 Therefore don't be afraid of them, for there is nothing covered that will not be revealed; and hidden that will not be known. 10:27 What I tell you in the darkness, speak in the light; and what you hear whispered in the ear, proclaim on the housetops. 10:28 Don't be afraid of those who kill the body, but are not able to kill the soul. Rather, fear him who is able to destroy both soul and body in Gehenna. 1805

          Why is a reward received      Matthew 10:40 He who receives you receives me, and he who receives me receives him who sent me. 10:41 He who receives a prophet in the name of a prophet will receive a prophet's reward. He who receives a righteous man in the name of a righteous man will receive a righteous man's reward. 10:42 Whoever gives one of these little ones just a cup of cold water to drink in the name of a disciple, most certainly I tell you he will in no way lose his reward." 1821

          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

               Continuum hypothesis      Wikipedia: There is no set whose cardinality is strictly between that of the integers and that of the real numbers. ... The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that this continuum hypothesis can neither be disproved nor be proved using the axioms of Zermelo-Fraenkel set theory, the standard foundation of modern mathematics, provided ZF set theory is consistent.1174

          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

          Integrals      The fundamental theorem of calculus gives us a method for computing definite integrals. ... There is a lot of interplay between summation, integration, and inequalities; many problems exploit this. Compute the limit as n approaches infinity of the sum from k =1 to n of n/(k**2 + n**2). The problem is impenetrable until we realize that we are not faced with a sum, but the limit of a sum; and that is exactly what definite integrals are. So let us try to work backwards and construct a definite integral whose value is the given limit. ... Even finite sums can be analyzed with integrals. If the functions involved are monotonic, it is possible to relate integrals and sums with inequalities... pg.301-302 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2245

        Dream - Explicitly Care

Dream - Explicitly Care      180

          Cover Story      thinking big 351

          Customer, Employee, Shareholder      imagine futures from several perspectives 355

          Make a World      sketch a vision of a future state 354

          Storyboard      imagine and create possibilities 353

          The Comeback Album of The Decade Game      imagining a new life 352

        Caring about caring about caring about caring

Caring about caring about caring about caring      Acknowledging what transcends our limits. Let go of my personal limits and live the vision of an unlimited God who can live through all.43

          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

          Jesus focused on what is natural      Sometimes, as my debts grew, I would wonder if I should have foresworn from ever going into debt. However, I realized the positive aspects of my debts. And I recalled that Jesus discouraged saving, but had much to say about debtors, which he seemed to take as the natural human condition, and which makes sense, given that we're created by God and raised by our parents.1109

          Seek the most straightforward meaning      In reading the Scripture, especially, passages where the meaning is not obvious, I try to consider the most straightforward interpretation, even the most childlike one. It may not be the traditional interpretation because people may want to take away a different meaning than is intended. At the Last Supper, Jesus equated his bread with his body and his wine with his blood, and gave it to his disciples to eat and to drink. Taking him literally, and trying to understand what he might mean, most straightforwardly, I thought, "Given that this is his bread, he will eat it, and it will become his body. In that sense, it is his body already, by its destiny, its purpose. And so, if he gives what is truly his to them instead, he changes its course, and they are eating his body. In the parable of the Good Samaritan, by which Jesus explains the meaning of "neighbor" in "Love your neighbor", he asks, "Who was the neighbor to the Samaritan?" People don't pay attention to the fact that he defines the neighbor as the one who helped, not the one who was hurt.1951

          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

          What we're not taught in school      If I had not asked, she would not have told me. I might have just adjusted her medicine and had her return in three months. I did not learn how to treat this in medical school. There was no medicine for grief, for the inevitability of urban violence. I felt powerless. I mumbled my sympathy and asked her to return in a month to recheck her blood pressure. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.19341938

          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

        Establish rational, systematic frameworks

Establish rational, systematic frameworks      Fifth, by establishing rational, systematic frameworks within which to analyze problems, analytic structuring techniques enable us to impose our analytic will on our subconscious mind, preventing it from following the instinctive mental traits that lead to faulty analysis. The Thinker's Toolkit by Morgan D.Jones2116

        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

          Observe the paths that people take in life      In high school there was a student who worked long hours in the cafeteria. His work was often his explanation for what he was doing with his time. When he got paid he would ask for his wage to be given to him in rolls of quarters. Then he would throw his quarters down the sidewalk with the hope, it seemed, that people would run after them. I concluded that work could very well be a spiritual dead end.1045

        Making sense of fate

Making sense of fate      159

          Making sense of fate's intent      472

        Millstone hung around the neck

Millstone hung around the neck      In the gospels of Mark, Matthew and Luke, Jesus refers to a "huge millstone" hung around the neck, which I think is the "relative learning" of taking a stand, following through and reflecting, for it is very slow learning, yet it is better than evil. In Greek it is "mulos onikos", which is a donkey-powered millstone, and I take him to compare such a ponderous "relative learner" (or scientist) with an ass.1253

               Millstone in Luke      Luke 17:1 He said to the disciples, "It is impossible that no occasions of stumbling should come, but woe to him through whom they come! 17:2 It would be better for him if a millstone were hung around his neck, and he were thrown into the sea, rather than that he should cause one of these little ones to stumble. 17:3 Be careful. If your brother sins against you, rebuke him. If he repents, forgive him. 17:4 If he sins against you seven times in the day, and seven times returns, saying, 'I repent,' you shall forgive him."1257

               Millstone in Mark      Mark 9:38 John said to him, "Teacher, we saw someone who doesn't follow us casting out demons in your name; and we forbade him, because he doesn't follow us." 9:39 But Jesus said, "Don't forbid him, for there is no one who will do a mighty work in my name, and be able quickly to speak evil of me. 9:40 For whoever is not against us is on our side. 9:41 For whoever will give you a cup of water to drink in my name, because you are Christ's, most certainly I tell you, he will in no way lose his reward. 9:42 Whoever will cause one of these little ones who believe in me to stumble, it would be better for him if he were thrown into the sea with a millstone hung around his neck. 9:43 If your hand causes you to stumble, cut it off. It is better for you to enter into life maimed, rather than having your two hands to go into Gehenna, into the unquenchable fire, 9:44 'where their worm doesn't die, and the fire is not quenched.' 1255

               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

          What is lowly      12:14 Jesus, having found a young donkey, sat on it. As it is written, 12:15 "Don't be afraid, daughter of Zion. Behold, your King comes, sitting on a donkey's colt." 12:16 His disciples didn't understand these things at first, but when Jesus was glorified, then they remembered that these things were written about him, and that they had done these things to him. Zechariah 9:9 Rejoice greatly, daughter of Zion! Shout, daughter of Jerusalem! Behold, your King comes to you! He is righteous, and having salvation; lowly, and riding on a donkey, even on a colt, the foal of a donkey.1965

        Thread

Thread      T) Extend the domain F) Continuity R) Self-superimposed sequence These three frames are the cycle of the scientific method: take a stand (hypothesize), follow through (experiment), reflect (conclude). I imagine that they link B1, B2, B3, B4 with C1, C2, C3, C4 to weave all manner of mathematical ideas, notions, problems, objects. 12

        Relative Learning

Relative Learning      In real life, this is taking a stand, following through and reflecting, but it is important to avoid evil, keep varying and not fall into a rut of self-fulfillment.25

        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

          As with little, so with much      Mark 16:10 He who is faithful in a very little is faithful also in much. He who is dishonest in a very little is also dishonest in much. 16:11 If therefore you have not been faithful in the unrighteous mammon, who will commit to your trust the true riches? 16:12 If you have not been faithful in that which is another's, who will give you that which is your own?2055

          Doing the little things, but not the big ones      Matthew 23:23 "Woe to you, scribes and Pharisees, hypocrites! For you tithe mint, dill, and cumin, and have left undone the weightier matters of the law: justice, mercy, and faith. But you ought to have done these, and not to have left the other undone. 23:24 You blind guides, who strain out a gnat, and swallow a camel! 1563

          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

          What is more and less important      Matthew 23:23 "Woe to you, scribes and Pharisees, hypocrites! For you tithe mint, dill, and cumin, and have left undone the weightier matters of the law: justice, mercy, and faith. But you ought to have done these, and not to have left the other undone. 23:24 You blind guides, who strain out a gnat, and swallow a camel!2050

        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

          Appropriate new ideas      ...cultivating a confident attitude, so that when you see a beautiful solution, you no longer think, "I could never have done that", but instead think, "Nice idea! It's similar to ones I've had. Let's put it to work!" Learn to shamelessly appropriate new ideas and make them your own. There's nothing wrong with that; the ideas are not patented. If they are beautiful ideas, you should excitedly master them and use them as often as you can, and try to stretch them to the limit by applying them in novel ways. pg.20, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1419

          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

          Inequalities      I think inequalities become relevant when we patch together different solution spaces. Inequalities are important because many mathematical investigations involve estimation, optimizations, best-case and worst-case scenarios, limits, etc. Equalities are nice, but are really quite rare in the "real world" of mathematics. ... Of the many ways of proving inequalities, the simplest is to perform operations that create logically equivalent but simpler inequalities. More sophisticated variants include a little massage, as well. pg.189-191 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2189

          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

        Circumscribe - Consent

Circumscribe - Consent      183

          Pie Chart Agenda      agree quickly on agenda 364

          The 7Ps Framework      plan out a meeting 365

          Visual Agenda      increase interest in a meeting 363

        Take a stand

Take a stand      Taking a stand. Recognize what is important in the given scope.45

          Believing in my superiority      My mother thought that I might need glasses, but I resisted that idea, I suppose because I didn't wish to allow that I was imperfect in any way. One day we were driving home from Lithuanian Saturday school on Highway 5. I would read out loud the exit signs as they came up. My sister Rima managed to read our exit "Carmenita" before I did! How could that be? I realized that she saw it before me. That's when I admitted that I needed glasses.2091

        Extend application

Extend application      1676

          Embracing a role      I was at a crossroads. Ready or not, here I was, "Presenting Dr. Ansell." A "real" doctor. And while I felt like a poseur, a fraud, I decided that despite my insecurity and inexperience, I needed to act as if I knew what the hell I was doing. One month into my internship, on the West Side of Chicago, in a steamy corner of the fourth floor of Fantus clinic, at the County Hospital. An epiphany. I suppressed a wave of panic and shoved my doubts aside. Oh. I got it. I was a "real" doctor now. The patients expected no less. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.19341948

          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

          Appreciating my freedom      500

          Considering what I have to do      486

          Considering what I have to do      506

          Knowing what I can do at least      503

          Noting what I do wrong      504

          What rules do I not follow?      516

        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 coin      Mark 12:15 ... But he, knowing their hypocrisy, said to them, ´Why do you test me? Bring me a denarius, that I may see it. 12:16 They brought it.1394

               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

          Are you a hypocrite?      Matthew 7:3 Why do you see the speck that is in your brother's eye, but don't consider the beam that is in your own eye? 7:4 Or how will you tell your brother, 'Let me remove the speck from your eye;' and behold, the beam is in your own eye? 7:5 You hypocrite! First remove the beam out of your own eye, and then you can see clearly to remove the speck out of your brother's eye. 1475

          As you do, so God does to you      Mark 11:25 Whenever you stand praying, forgive, if you have anything against anyone; so that your Father, who is in heaven, may also forgive you your transgressions. 11:26 But if you do not forgive, neither will your Father in heaven forgive your transgressions."2054

          Calling out hypocrisy      Each test is a test of the tested, but also a test of the tester and their test. Mark 12:14 ... "...Is it lawful to pay taxes to Caesar, or not? 12:15 Shall we give, or shall we not give?" But he, knowing their hypocrisy, said to them, "Why do you test me? Bring me a denarius, that I may see it." 1398

          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

          Having the least bit      Matthew 17:20 He said to them, "Because of your unbelief. For most certainly I tell you, if you have faith as a grain of mustard seed, you will tell this mountain, 'Move from here to there,' and it will move; and nothing will be impossible for you. 17:21 But this kind doesn't go out except by prayer and fasting."2047

          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

          Let the bad and the good grow distinct      Matthew 13:24 He set another parable before them, saying, "The Kingdom of Heaven is like a man who sowed good seed in his field, 13:25 but while people slept, his enemy came and sowed darnel weeds also among the wheat, and went away. 13:26 But when the blade sprang up and brought forth fruit, then the darnel weeds appeared also. 13:27 The servants of the householder came and said to him, 'Sir, didn't you sow good seed in your field? Where did this darnel come from?' 13:28 "He said to them, 'An enemy has done this.' "The servants asked him, 'Do you want us to go and gather them up?' 13:29 "But he said, 'No, lest perhaps while you gather up the darnel weeds, you root up the wheat with them. 13:30 Let both grow together until the harvest, and in the harvest time I will tell the reapers, "First, gather up the darnel weeds, and bind them in bundles to burn them; but gather the wheat into my barn."'"1979

          Let the tester be tested by his testing      Every test is also a test of the tester, of their values and their intent. Jesus lets John be tested by his testing. Matthew 11:2 Now when John heard in the prison the works of Christ, he sent two of his disciples 11:3 and said to him, "Are you he who comes, or should we look for another?" 11:4 Jesus answered them, "Go and tell John the things which you hear and see: 11:5 the blind receive their sight, the lame walk, the lepers are cleansed, the deaf hear, the dead are raised up, and the poor have good news preached to them. 11:6 Blessed is he who finds no occasion for stumbling in me." 1826

          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

          What is one worthy of?      If we are laborers, then we should not have to depend on anything but our labor. Matthew 10:9 Don't take any gold, nor silver, nor brass in your money belts. 10:10 Take no bag for your journey, neither two coats, nor shoes, nor staff: for the laborer is worthy of his food. 1782

          What should we depend on?      If we are laborers, then we should not have to depend on anything but our labor. Matthew 10:9 Don't take any gold, nor silver, nor brass in your money belts. 10:10 Take no bag for your journey, neither two coats, nor shoes, nor staff: for the laborer is worthy of his food. 1781

          Who empathizes with whom?      Matthew 23:29 "Woe to you, scribes and Pharisees, hypocrites! For you build the tombs of the prophets, and decorate the tombs of the righteous, 23:30 and say, 'If we had lived in the days of our fathers, we wouldn't have been partakers with them in the blood of the prophets.' 23:31 Therefore you testify to yourselves that you are children of those who killed the prophets. 23:32 Fill up, then, the measure of your fathers. 1573

          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 our enemy      Matthew 10:21 "Brother will deliver up brother to death, and the father his child. Children will rise up against parents, and cause them to be put to death.1794

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

          Your measure will be used to measure for you      Luke 6:38 "Give, and it will be given to you: good measure, pressed down, shaken together, and running over, will be given to you. For with the same measure you measure it will be measured back to you."1354

        Continuity

Continuity      As in Polya's discussion of Descartes' universal method, we can apply continuity to consider the implications of a constraint or an equation. Polya asks about an iron ball floating in mercury, if we pour water on it, will the ball sink down or float up or stay the same? He answers this by first imagining that the water has no specific gravity (like a vacuum) and then increasing it continuously until it approaches and surpasses that of iron. Varying the variable is putting the constraint to the test, presuming that there is a solution point, just as we do and can in physical reality. At what points will the model break or hold? Continuity is the thread that we sew. 24

          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

          Interpret dynamically      To understand what a moving curtain is, we shall explore, in some detail, the most important idea of elementary calculus. ... What is the fundamental theorem of calculus (FTC), what does it mean, and why is it true? ... We start with the very useful define a function tool .. Let g(t) = integral from a to to of f(x) with regard to x. We choose the variable t on purpose, to make it easy to visualize g(t) as a function of time. As t increases from a, the function g(t) is computing the area of a "moving curtain" ... Notice that g(a) = 0. ... Differentiation is not just about tangent lines - it has a dynamic interpretation as instantaneous rate of change. Thus g'(t) is equal to the rate of change of the area of the curtain at time t. ... The area grows fast when the leading edge of the curtain is tall, and it grows slowly when the leading edge is short. It makes intuitive sense that g'(t) = f(t) since in a small interval of time delta-t, the curtain's area will grow by approximately f(t)delta-t. ... The crux move was to interpret the definite integral dynamically, and then observe the intuitive relationship between the speed that the area changes and the height of the curtain. This classic argument illustrates the critical importance of knowing as many possible alternate interpretations of both differentiation and integration. Note that the variable "t" is understood here as time and that is part of the necessary implicit context for understanding. pg.282-284 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2230

          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

        Perfect - Explicitly Validate

Perfect - Explicitly Validate      182

          Do, Redo & Undo      make mistakeproof 361

          Wizard of Oz      test the completeness of a design 362

        Follow through

Follow through      Following through. Live out fully the consequences of my principles.46

          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

               Black Hole as a Natural Laboratory      "Seeing a star get ripped apart by a black hole from almost 4 billion light-years away, that's a remarkable thing," says astronomer Dave Goldberg, co-author of A User's Guide to the Universe: Surviving the Perils of Black Holes, Time Paradoxes, and Quantum Uncertainty, who was not part of the studies. "We want to study black holes because they are tremendous natural laboratories for what happens to matter at very high energies." "Rare Cosmic Blast Traced to Black Hole" by Dan Vergano, USA TODAY, June 16, 20111950

          Destroying a prototype      Fresh Air interview of Michael Hiltzik about the Hoover Dam, June 8, 2010: "The federal government created two labs for the concrete to be tested; they enlisted the University of California [and] the University of Colorado to test other formulations. And they even built a test dam in a California river valley — they built it so it could be destroyed, so they could see what kinds of stresses and strains this dam would undergo."1877

          Practice with sparring partners similar to one's opponent      Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: Dundee, Morton and Trainer were responsible for managing Leonard's fights and for selecting sparring partners who were similar in height, speed and power to the opponents Leonard was about to face. These sparring partners worked with Leonard for weeks to prepare him for a match. ... To prepare, Leonard chose a sparring partner who idolized Duran. "He fought like Roberto Duran — he used his head and dirty tactics and what-have-you," he says. "And it made me more aware, from a defensive standpoint, so when I faced Duran, I was prepared."1889

          Practicing as a child      Fresh Air interview: When J.J. Abrams was 13, his grandfather gave him a Super 8 camera to shoot homemade movies. He started off making horror films — and killing off all of his relatives. "I would take anyone who was available — my sister, my mother, any friends — and I would kill them in crazy ways," he tells Fresh Air's Terry Gross. "We would do makeup effects. ... I would make blood and ask my mom if I could borrow her makeup — which didn't trouble her because she knew I was going to basically just kill someone with it. It was all ridiculous." Abrams went onto a successful film and television career, with credits including Alias, Lost, Fringe, Mission: Impossible III, Star Trek (2009) and Cloverfield.1868

        Hypothesis Testing

Hypothesis Testing      Hypothesis testing is primarily about analyzing individual hypotheses and so I list it here. Secondarily, it is also related to ranking. ...a hypothesis helps to narrow the scope of our thinking. It thus provides a framework - a mind-set - within which to analyze and interpret information ... it can have dramatically negative effects if it leads to satisficing - to focusing on one hypothesis to the exclusion of alternatives. ... [Hypothesis testing ranks] competing hypotheses by the degree to which relevant evidence is inconsistent. The favored hypothesis has the least inconsistent evidence, not the one with the most consistent evidence. ... Consistent evidence proves nothing, because evidence can, and usually does, support more than one hypothesis. ... Only inconsistent evidence has any real value in determining the credibility of a hypothesis and in ranking hypotheses by the degree of their credibility.

  • Construct a matrix. Label the first column "Evidence". Label the other columns to the right "Hypotheses", and enter descriptors of the hypotheses atop the columns. The hypotheses must be mutually exclusive .... need not be collectly exhaustive.
  • List "significant" evidence down the left-hand margin, including "absent" evidence. ... Include the absence of evidence one would expect if the hypothesis were true as well as the absence of evidence one would expect if it were untrue. ... Ask "[Public health professor Dr. Russell] Alexander's question": What evidence not included in the matrix would refute one or more of the hypotheses?
  • Working across the matrix, test the evidence for consistency [or inconsistency] with each hypothesis, one item of evidence at a time.
  • Refine the matrix. Add or reword hypotheses. Add "significant" evidence relevant to any new or reworded hypothesis and test it against all hypotheses. Delete, but keep a record of, evidence that is consistent with all of the hypotheses. It has no diagnostic value.
  • Working downward, evaluate each hypothesis.
  • Rank the hypotheses by the strength of inconsistent evidence.
  • Perform a sanity check.
the cardinal strength of hypothesis testing lies in its function, which is to disprove, not prove, hypotheses. Hypothesis testing mirrors the PROs, CONs, and FIXes technique ... in problem solving, inconsistent evidence plays the key role during the analytic phase while consistent evidence plays the key role in the summation and presentation phase. The Thinker's Toolkit by Morgan D.Jones1774

        Pursue

Pursue      1073

          Try to collect what I am due      I am due a rather large refund in Lithuania for value added tax which I have paid as a business, for my income comes from outside of the country, but I have accumulated expenses within the country. I had to request permission for a refund. Then they informed me that they would need to do a complete audit of all of the books for my business. Which is to say, not just the documents relevant to the value added tax, but absolutely everything. I told them that it would take me hundreds of hours to prepare. They said that they would give me plenty of time, six weeks.1014

        Practice

Practice      157

          Considering a principle's consequences      513

          Living with integrity      524

          Practicing our principles      525

          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

          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. 1569

          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

          Denouncing      Why does Jesus denounce the cities? Perhaps they are free to repent, perhaps others will learn from them, and perhaps he is confirming God's will. Matthew 11:20 Then he began to denounce the cities in which most of his mighty works had been done, because they didn't repent. 11:21 "Woe to you, Chorazin! Woe to you, Bethsaida! For if the mighty works had been done in Tyre and Sidon which were done in you, they would have repented long ago in sackcloth and ashes. 11:22 But I tell you, it will be more tolerable for Tyre and Sidon on the day of judgment than for you. 11:23 You, Capernaum, who are exalted to heaven, you will go down to Hades. For if the mighty works had been done in Sodom which were done in you, it would have remained until this day. 11:24 But I tell you that it will be more tolerable for the land of Sodom, on the day of judgment, than for you." 1908

          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

          Judge by the outcome that you hear and see      Matthew 11:2 Now when John heard in the prison the works of Christ, he sent two of his disciples 11:3 and said to him, "Are you he who comes, or should we look for another?" 11:4 Jesus answered them, "Go and tell John the things which you hear and see: 11:5 the blind receive their sight, the lame walk, the lepers are cleansed, the deaf hear, the dead are raised up, and the poor have good news preached to them. 11:6 Blessed is he who finds no occasion for stumbling in me." 1823

          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

          What will happen to the good and the bad      Matthew 13:37 He answered them, "He who sows the good seed is the Son of Man, 13:38 the field is the world; and the good seed, these are the children of the Kingdom; and the darnel weeds are the children of the evil one. 13:39 The enemy who sowed them is the devil. The harvest is the end of the age, and the reapers are angels. 13:40 As therefore the darnel weeds are gathered up and burned with fire; so will it be at the end of this age. 13:41 The Son of Man will send out his angels, and they will gather out of his Kingdom all things that cause stumbling, and those who do iniquity, 13:42 and will cast them into the furnace of fire. There will be weeping and the gnashing of teeth. 13:43 Then the righteous will shine forth like the sun in the Kingdom of their Father. He who has ears to hear, let him hear.1984

          What will happen to the good and the bad      Matthew 13:47 "Again, the Kingdom of Heaven is like a dragnet, that was cast into the sea, and gathered some fish of every kind, 13:48 which, when it was filled, they drew up on the beach. They sat down, and gathered the good into containers, but the bad they threw away. 13:49 So will it be in the end of the world. The angels will come forth, and separate the wicked from among the righteous, 13:50 and will cast them into the furnace of fire. There will be the weeping and the gnashing of teeth." 1988

          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 the cause of the result?      Matthew 6:28 Why are you anxious about clothing? Consider the lilies of the field, how they grow. They don't toil, neither do they spin, 6:29 yet I tell you that even Solomon in all his glory was not dressed like one of these. 6:30 But if God so clothes the grass of the field, which today exists, and tomorrow is thrown into the oven, won't he much more clothe you, you of little faith? 1456

          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 on the lookout for new ideas      Always be on the lookout for new ideas. Each new problem that you encounter should be analyzed for its "novel idea" content. pg.20, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1420

          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

          Catalyst      Sometimes telescoping won't work with what you start with, but the introduction of a single new term will instantly transform the problem. We call this the catalyst tool. ... Simplify the product (1 + 1/a)(1 + 1/a**2)(1 + 1/a**4) ... (1 + 1/a**(2**100)). Call the product P and consider what happens when we multiply P by 1 - 1/a. The "catalyst" is the simple difference of two squares formula (x-y)(x+y) = x**2 - y**2. (1 - 1/a)P = (1 - 1/a)(1 + 1/a)(1 + 1/a**2)(1 + 1/a**4) ... (1 + 1/a**(2**100)) = (1 - 1/a**2)(1 + 1/a**2)(1 + 1/a**4) ... (1 + 1/a**(2**100)) etc. = (1 - 1/a**201). pg.175 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2181

          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

          Generating functions      The crossover tactic of generating functions owes its power to two simple facts.

  • When you multiply x**m by x**n, you get x**(m+n).
  • "Local" knowledge about the coefficients of a polynomial or power series f(x) often provides "global" knowledge about the behavior of f(x), and vice versa.
The first fact is trivial, but it is the technical "motor" that makes things happen, for it relates the addition of numbers and the multiplication of polynomials. The second fact is deeper ... Given a (possibly infinite) sequence a0, a1, a2, ..., its generating function is a0 + a1 x + a2 x**2 + ... In general, we don't worry too much about convergence issues with generating functions. As long as the series converges for some values, we can usually get by ... The term "generatingfunctionology" was coined by Herbert Wilf, in his book of the same name. We urge the reader to at least browse through this beautifully written textbook, which among its many other charms, has the most poetic opening sentence we've ever read (in a math book). pg.143-144, 149 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2164

          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

          Telescope      A geometric sequence is exactly like an arithmetic sequence except that now the consecutive terms have a common ratio ... a, ar, ar**2, ar**3... The Gaussian pairing tool is no help for summing geometric series, because the terms are not additively symmetric. However, the wonderful telescope tool comes to the rescue .... S = a + ar + ar**2 + ... + ar**(n-1) and rS = ar + ar**2 + ar**3 + ... + ar**n. Observe that S and rS are nearly identical, and hence subtracting the two quantities will produce something really simple ... all terms cancel except for the first and the last. (That's why it's called "telescoping", because the expression "contracts" the way some telescopes do.) We have S - rS = a - ar**n and solving for S yields S = (a - ar**n)/(1-r). ... The important thing is to be aware of the possibility for telescoping, which is really just an application of the adding zero creatively tool. And quite often, a telescoping attempt won't work perfectly, but will reduce the complexity of a problem. pg.173-174 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2180

          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

        Reevaluate - Implicitly Validate

Reevaluate - Implicitly Validate      181

          How-Now-Wow Matrix      identify what is both new and achievable 356

          NUF Test      reality check for ideas 360

          Pre-Mortem      be upfront about risks 357

          Start, Stop, Continue      reevaluate activities 359

          SWOT Analysis      gauge and shape possibilities of success 358

        Reflect

Reflect      Reflecting. Allow for mistakes and solutions, recognize them and make good of them.47

          Appreciating that I am happy      As a child I realized that I was very happy, I was very blessed, and as far as I could see, I would always be happy, for certainly I had the means to be. I had a mother and a father, wise and loving. We lived in a wonderful time and place, the utopia of the Californian suburbs of the 1970s, where for miles and miles around all of the people lived more or less the same, with their needs met and each family having a little something more, and people living ever more comfortably in a society that was ever more just. I had a brilliant mind that, as far as I could tell, was limitless in its potential. I was a very good child, able to keep myself safe and busy, keen to do the right thing, as if I had a good seed and it could not be otherwise. I also had enormous will, an exacting conscience, and a fierce ambition to apply myself, especially my mind. I would always have people who cared about me. I appreciated the many aspects of my happiness and, consequently, I could consider, were they my doing?1298

          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

          Comparing effort and outcome      Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: When they announced the decision, I felt like I had given 100 percent, just for the wrong fight.1892

          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

          Look at yourself in the mirror      Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: It's a gradual progression of getting better and, as weeks go by, you look into the mirror and you see a different person. It evolves. One minute, you look kind of soft and then, within 6-8 weeks, your muscle and all that definition appears. The mirror doesn't lie. It tells you exactly what you are."1894

          Suffering from the actions of others      Fresh Air interview of J.J. Abrams: His justification for such secrecy has to do with a script for a Superman film he wrote 15 years ago, he explains. "It was leaked out of the studio, and someone reviewed the script online in great detail, and it became this big thing and ultimately the film didn't get made,1870

          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

        Plausible reasoning

Plausible reasoning      "How high can a giraffe fly?" Instantly our mind tells us the question is "illogical". ... how do we know? An educated guess is that the mind compares the new information about the giraffe with the old information the mind has stored away about the animal. ... it clashes violently, and the mind tells us it doesn't match, meaning it doesn't make sense. This is a far cry from reasoning or rational thinking, and it's certainly light-years away from formal logic. ... "Plausible" means "seemingly true at first glance." ... As Morton Hunt put it, "Natural [plausible] reasoning often succeeds even when it violates laws of logic. What laws then does natural reasoning obey?" He cited two: plausibility and probability. "In contrast to logical reasoning," said Hunt, "natural reasoning proceeds by steps that are credible [plausible] but not rigorous and arrives at conclusions that are likely [probable] but not certain. ... Because logic, to be effective, requires total consistency and total certainty, logical thinking is unsuited for dealing with the real world in which the only certainty (other than taxes and TV commercials) is ultimate death. But plausible reasoning requires neither consistency nor certainty. Plausibility strives for the gap, just as truth does. The Thinker's Toolkit by Morgan D.Jones drawing from The Universe Within: A New Science Explores the Human MindThe Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich.2129

        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

          Allowing failure      439

          Be ready to improvise      437

          Giving things time      464

          Making good of mistakes      443

          Waiting patiently      425

        Observe

Observe      915

          Extend our senses with improved scientific instruments      Edward Cherlin, 2011.04.24: We know that our models are reasonably complete and accurate at best in the areas we have been able to observe, and that every new addition to our senses in improved scientific instruments, going back to Galileo's first telescope, reveals surprises like the mountains of the moon, the constancy of the speed of light (interferometers) or neutrino oscillation (simple but quite large neutrino detectors).916

        Caring for = Believing in

Caring for = Believing in      This matches the two branches, Believing and Caring, completely and quite likely, inversely. For Believing leads to Caring, and Caring leads to Believing. And so matching the two branches as wholes leads to this perspective.44

        Sets of objects exist

Sets of objects exist      We can fuse the particle and wave points of view to work with a partial reality. For example, we can talk about a banana, an apple or an orange as well defined objects that mean something more than a random assemblage of half of the atoms in a banana with half of the atoms in the apple. We are then no longer talking about the symmetry of the universe. John says: A symmetry group commutes with the underlying symmetry of a particular phenomenon, its spacial symmetry, as the set of possible transformations, possible futures. Previously, we worked with the entire universe, and if we translated it abstractly in a symmetry group and then ran things forward in the translated frame, it was exactly isomorphic to what it would have been if we had not translated it. But now, as we want to compartmentalize the universe, then the price to pay is that we are not translating by the full symmetry group, but only by some part of it. This is analogous to having a tensor product and considering only one component, so that we have partial symmetry. We are going to treat one part of the universe as a compartment. This gives the reality to the symmetry group because otherwise it couldn't be measured. Andrius: This compartmentalization is also what allows us to define entropy and the one-way direction of time, which says that states drift away from deliberateness, which is expressed by the compartmentalization. Compartmentalization also indicates the philosophical gaps or boundaries that allow for measurement to take place, allow for objectivity, the separation of the observer and the observed. Analogously, in math we have symmetry groups, and in life we have meanings, the essence of what we wish to say, which me take to be absolute, in cases where we have fundamental agreement. 847

        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

          Allegiance to Jesus' name      What is Jesus' name? It is a substitute for him that shows who is accepting his culture. It is perhaps a unifier for the two branches. A name allows for ambiguity and resolves it to unity. 18:1 In that hour the disciples came to Jesus, saying, "Who then is greatest in the Kingdom of Heaven?" 18:2 Jesus called a little child to himself, and set him in their midst, 18:3 and said, "Most certainly I tell you, unless you turn, and become as little children, you will in no way enter into the Kingdom of Heaven. 18:4 Whoever therefore humbles himself as this little child, the same is the greatest in the Kingdom of Heaven. 18:5 Whoever receives one such little child in my name receives me, 18:6 but whoever causes one of these little ones who believe in me to stumble, it would be better for him that a huge millstone should be hung around his neck, and that he should be sunk in the depths of the sea. 1961

          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

          Children do as the parent does      1866

               Children of Wisdom      Wisdom approaches people where they are at, or where they claim to be at. Matthew 11:19 ... But wisdom is justified by her children." 1867

          Creating a tangible reference to an abstract question      Jesus calls a child to make his point. 18:1 In that hour the disciples came to Jesus, saying, "Who then is greatest in the Kingdom of Heaven?" 18:2 Jesus called a little child to himself, and set him in their midst, 18:3 and said, "Most certainly I tell you, unless you turn, and become as little children, you will in no way enter into the Kingdom of Heaven. 18:4 Whoever therefore humbles himself as this little child, the same is the greatest in the Kingdom of Heaven. 18:5 Whoever receives one such little child in my name receives me, 18:6 but whoever causes one of these little ones who believe in me to stumble, it would be better for him that a huge millstone should be hung around his neck, and that he should be sunk in the depths of the sea. 1959

          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

          Essential concepts      Jesus cares about law, justice, mercy faith. Matthew 23:23 "Woe to you, scribes and Pharisees, hypocrites! For you tithe mint, dill, and cumin, and have left undone the weightier matters of the law: justice, mercy, and faith. But you ought to have done these, and not to have left the other undone. 23:24 You blind guides, who strain out a gnat, and swallow a camel!2051

          God's will: the bread of life      Bread of life - DIFFERENCE/UNITY IN BELIEVING - guarantee eternity - to do the Father's will that all he draws to Jesus do indeed believe in him unwaveringly, thus have eternal life, as Jesus' gift is unwavering1976

          How are distinctions reconciled?      The last day means that there is no more alternation of night and day and so the two branches are reconciled. John 6:39 This is the will of my Father who sent me, that of all he has given to me I should lose nothing, but should raise him up at the last day. 6:40 This is the will of the one who sent me, that everyone who sees the Son, and believes in him, should have eternal life; and I will raise him up at the last day."1963

          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

          How to unify      Jesus unifies the two branches by laying down his life. John 10:14 I am the good shepherd. I know my own, and I'm known by my own; 10:15 even as the Father knows me, and I know the Father. I lay down my life for the sheep. 10:16 I have other sheep, which are not of this fold. I must bring them also, and they will hear my voice. They will become one flock with one shepherd. 10:17 Therefore the Father loves me, because I lay down my life, that I may take it again. 10:18 No one takes it away from me, but I lay it down by myself. I have power to lay it down, and I have power to take it again. I received this commandment from my Father."1968

          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

          Identifying himself as born of spirit, not born of woman      The Kingdom of Heaven is rejected violently by those who prefer Jesus to be dead. Thus the violent enter into the Kingdom of Heaven, by killing Jesus so that he could save them. 11:12 From the days of John the Baptizer until now, the Kingdom of Heaven suffers violence, and the violent take it by force.1859

          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

          Note from whom God hides and to whom reveals      The significance of Jesus' miracles are hidden from John the Baptist, from Bethsaida and Capernaum, from the wise and understanding, but "infants" are delighted and uplifted, those who are fresh in faith, who don't want to justify the world, but want it bettered, by those who suffer and labor, who want healing. Matthew 11:25 At that time, Jesus answered, "I thank you, Father, Lord of heaven and earth, that you hid these things from the wise and understanding, and revealed them to infants. 11:26 Yes, Father, for so it was well-pleasing in your sight. 1911

          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

          Speak from what they can know      John 3:10 Jesus answered him, "Are you the teacher of Israel, and don't understand these things? 3:11 Most certainly I tell you, we speak that which we know, and testify of that which we have seen, and you don't receive our witness. 3:12 If I told you earthly things and you don't believe, how will you believe if I tell you heavenly things?2082

          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 do they already know      Jesus is sensitive to what they already know, even as they ask. Luke 18:18 A certain ruler asked him, saying, "Good Teacher, what shall I do to inherit eternal life?" 18:19 Jesus asked him, "Why do you call me good? No one is good, except one—God. 18:20 You know the commandments: 'Don't commit adultery,' 'Don't murder,' 'Don't steal,' 'Don't give false testimony,' 'Honor your father and your mother.'"2069

          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 example does he give?      Matthew 20:24 When the ten heard it, they were indignant with the two brothers. 20:25 But Jesus summoned them, and said, "You know that the rulers of the nations lord it over them, and their great ones exercise authority over them. 20:26 It shall not be so among you, but whoever desires to become great among you shall be your servant. 20:27 Whoever desires to be first among you shall be your bondservant, 20:28 even as the Son of Man came not to be served, but to serve, and to give his life as a ransom for many."2016

          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

          What is good      What is good is what we believe in (what you believe is what happens) and/or what we care about (what you find is what you love). Luke 18:18 A certain ruler asked him, saying, "Good Teacher, what shall I do to inherit eternal life?" 18:19 Jesus asked him, "Why do you call me good? No one is good, except one—God. 18:20 You know the commandments: 'Don't commit adultery,' 'Don't murder,' 'Don't steal,' 'Don't give false testimony,' 'Honor your father and your mother.'"2066

          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 God      God is understanding. To know God is to come to understanding, thus is by and through Jesus, for Jesus is coming to understand. Jesus reveals God to those who believe, those of the Kingdom of Heaven. God is impartial, and gives judgment to Jesus, whom he teaches to be impartial. Jesus is God's impartial judgment, for Jesus is distinct and independent of God, thus impartial with regard to God, for Jesus came to understanding. To know God is to know his impartiality, aand to know his Son is likewise to know his impartiality. The impartiality of God manifests in his Son who manifests it, Matthew 11:27 All things have been delivered to me by my Father. No one knows the Son, except the Father; neither does anyone know the Father, except the Son, and he to whom the Son desires to reveal him. 1913

          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 does God listen to      John 9:30 The man answered them, "How amazing! You don't know where he comes from, yet he opened my eyes. 9:31 We know that God doesn't listen to sinners, but if anyone is a worshipper of God, and does his will, he listens to him.2089

          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

        Symmetry

Symmetry      Symmetry group We unify internal and external points of view, link time and space, by considering a group of actions in time acting on space. Some aspects of the space are invariant, some aspects change. Actions can make the space more or less convoluted. At this point, we have arrived at a self-standing system, one that can be defined as if it was independent of our mental processes. Our problem has become "a math problem". Analogously, in real life, after projecting more and more what we mean in general by people, including ourselves and others, we finally take us for granted as entirely one and the same and instead make presumptions towards a universal language by which we might agree absolutely.13

               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

               Designing algorithms with index cards      I have the student write on an index card, "I'm thinking of a number". Then they create subsequent index cards, each with their own instruction: "Double the number", "Add 5 to the number", "Halve the number", "Square the number" and so on. Then I have them input a number and get the output and write down all of the intermediate answers. They do this for several numbers. Then we can consider an abstract input X. We can discuss if the algorithm can be simplified, and can it be reversed. Basic Math.1955

          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

          Algebraic symmetry      Sequences can have symmetry, like this row of Pascal's Triangle: 1, 6, 15, 20, 20, 15, 6, 1 ... In just about any situation where you can imagine "pairing" things up, you can think about symmetry. pg.74, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1627

          Combination of techniques      We shall end the chapter with an exploration of the diophantine equation x**2 + y**2 = n ... where n is a prime p. Our exploration will use several old strategic and tactical ideas, including the pigeonhole principle, Gaussian pairing, and drawing pictures. The narrative will meander a bit, but please read it slowly and carefully, because it is a model of how many different problem-solving techniques come together in the solution of a hard problem. pg.274 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2229

          Complex Numbers      Complex numbers are the crossover artist's dream: like light, which exists simultaneously as wave and particle, complex numbers are both algebraic and geometric. You will not realize their full power until you become comfortable with their geometric, physical nature. This in turn will help you to become fluent at translating between the algebraic and the geometric in a wide variety of problems. ... We strongly urge you to read at least the first few chapters of our chief inspiration for this section, Tristan Needham's Visual Complex Analysis. This trail-blazing book is fun to read, beautifully illustrated, and contains dozens of geometric insights that you will find nowhere else. ... If z=a+bi, we define the conjugate of z to be z-bar = a-bi. Geometrically, z-bar is just the reflection of z about the real axis. Complex numbers add "componentwise" ... Geometrically, complex number addition obeys the "parallelogram rule" of vector addition ... Multiplication by the complex number rCisTheta is a counterclockwise rotation by Theta followed by stretching by the factor r. So we have a third way to think about complex numbers. Every complex number is simultaneously a point, a vector, and a geometric transformation, namely the rotation and stretching above! pg.131-134, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2162

          Exploit underlying symmetry in polynomials      Algebra problems with many variables or of high degree are often intractable unless there is some underlying symmetry to exploit. ... Solve x**4 + x**3 + x**2 + x**1 + 1 = 0 ... we will use the symmetry of the coefficients as a starting point to impose yet more symmetry, on the degrees of the terms. Simply divide by x**2 yielding x**2 + x + 1 + 1/x + 1/x**2 then make the substitution u := x + 1/x. pg. 75, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1629

          Fixed objects      When pondering a symmetrical situation, you should always focus briefly on the "fixed" objects which are unchanged by the symmetries. For example, if something is symmetric with respect to reflection about an axis, that axis is fixed and worthy of study (the stream in the previous problem played that role). pg. 72 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1585

               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

          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

          Harmony      An informal alternate definition of symmetry is "harmony". ... If you can do something that makes things more harmonious or more beautiful, even if you have no idea how to define these two terms, then you are often on the right track. pg. 70 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1580

          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

               Completing the square by trying to symmetrize      x**2 + a*x = x*(x + a) = (x + a/2 -a/2)*(x + a/2 + a/2) = (x + a/2)**2 - (a/2)**2 Above is a way to discover the completing-the-square formula by trying to symmetrize the terms, then adding zero creatively. pg. 163, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. 1381

          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

          The Gaussian pairing tool      Gauss, as a child, added up the sum 1 + 2 + 3 + ... + 100, presumably by pairing up the number 1 and 100, 2 and 99, 3 and 98, ... 50 and 51, yielding 50 pairs of 101 for a total of 5,050. Paul Zeitz notes this as an example of symmetry and calls it the Gaussian pairing tool. pg. 75, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1628

          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

        Distill - Implicitly Transform

Distill - Implicitly Transform      184

          Back of the Napkin      evoke informal, focused answers 366

          Draw the Problem      describe problems so that we care 370

          Elevator Pitch      sum up what is remarkable about your solution 367

          Pecha Kucha / Ignite      share, present concisely 369

          Poster Session      communicate ideas passionately with simple images 368

          The Pitch      focus on feasibility and viability 371

        Caring for = Believing in

Caring for = Believing in      Distilling the essence. Distill the key idea and communicate it simply and vividly.32

     Capture with a personal expression      1589

               Watching human cells floating on my eyes      As a nearsighted child, I noticed "floaters" that moved across my eye, chains of blocks. I learned about cells in my biology, and then, one day, sitting in a dentist's chair, I thought, the floaters must be chains of cells which I can see because they are close up, right on my eyeball. My mother said no. But later an optometrist confirmed that, yes, indeed they were! I thought it was wonderful that I could see human cells with my own naked eyes!1236

          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

          Considering inseparability of perspectives      I am hypothesizing, based on my results with just a few rooms, that for each room of the House of Knowledge there are two groups of ways, one that relates to the heart and another to the world. I'm wondering if those two groups are inseparable, if they somehow must go together and thus fit together, for otherwise there could be 48 rooms and not 24 rooms, but there are many structural reasons why there should be only and exactly 24 rooms, but especially because the omniscope has 24 perspectives, and also because it matches well with the number of divisions 24/3=8, topologies 24/2=12 and representations 24/4=6. Furthermore, inseperability would be an intriguing notion. It would also relate to what I call anti-structure or the division of everything into -1 perspectives, which is given by an opposite, as in the perspective of free will, where good and bad coexist and are inseparable, unable to stand alone.1304

          Notice inherent ambiguity      I noticed the ambiguity in Jesus' explanation of why he speaks plainly to his believers, but in parables to others, that "they might not understand, they might be damned", which is to say, that he not impinge on their freedom, but that they might be free to accept him or not. The word "might" or "can" can be understood in two ways, by fate or by free will. Similarly, the word "meaningful" in English and in Lithuanian, and I suppose in every language, tends towards two different meanings, one where a word in a system has meaning within it, and another where a life in a world has meaning beyond it.1953

          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

          Address the needs of loved ones      Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: When he got back to D.C.'s Maryland suburbs, where his family lived, he realized that his father was extremely sick. His mother had her own health issues and the family needed money — right away. "Janks Morton, who eventually became my trainer and mentor, told me: 'You turn pro, you make money. You can pay your father's hospital bills.' ... I turned professional for that reason."1887

          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

          Relationships with people who face hardship      I learned a lot from the patients. I was discovering the tools of medicine from them. Many of my patients and I grew up together. They had seen me become a father for the first time, and they consoled me when my father died. I had seen their children grow up, having children themselves. I had helped them through family crises, tragedies, diseases, and deaths. I had no idea in those first weeks and months of General Medicine Clinic how much I would grow from these relationships. I am on a first-name basis with many of my original patients. From them I gained insight into illness and the dignity with which people can face hardship that has helped me through difficult times in my life. I have taken care of three generations of some families, and have seen the destruction that poverty, poor diet, obesity, diabetes, and hypertension can unleash on a family's tree. I learned that sometimes giving hope or an embrace is as therapeutic as a drug. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.1934

          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

        Focus on one element at a time

Focus on one element at a time      Fourth, structuring focuses on one element at a time, which, compared to our instinctive scattershot approach of tackling all elements simultaneously, is more systematic, more thorough, and more productive of relevant ideas. The Thinker's Toolkit by Morgan D.Jones2115

          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

          Building on a message      510

          Realizing what we're communicating      441

          What is our message?      518

          What is the meaning of an activity?      519

        Conservation

Conservation      998

          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 angular momentum      Wikipedia: In a closed system angular momentum is constant. This conservation law mathematically follows from continuous directional symmetry of space (no direction in space is any different from any other direction). If a planet is found to rotate slower than expected, then astronomers suspect that the planet is accompanied by a satellite, because the total angular momentum is shared between the planet and its satellite in order to be conserved. The conservation of angular momentum is used extensively in analyzing what is called central force motion. If the net force on some body is directed always toward some fixed point, the center, then there is no torque on the body with respect to the center, and so the angular momentum of the body about the center is constant. 985

          Conservation of color charge      Wikipedia: In particle physics, color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD). ... Color charge is conserved, but the book-keeping involved in this is more complicated than just adding up the charges, as is done in quantum electrodynamics.987

          Conservation of electric charge      Wikipedia: In physics, charge conservation is the principle that electric charge can neither be created nor destroyed. The quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is always conserved. Charge conservation is a physical law that states that the change in the amount of electric charge in any volume of space is exactly equal to the amount of charge flowing into the volume minus the amount of charge flowing out of the volume. In essence, charge conservation is an accounting relationship between the amount of charge in a region and the flow of charge into and out of that region.986

          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 mass-energy      Wikipedia: In physics, mass-energy equivalence is the concept that the mass of a body is a measure of its energy content. In this concept the total internal energy E of a body at rest is equal to the product of its rest mass m and a suitable conversion factor to transform from units of mass to units of energy. If the body is not stationary relative to the observer then account must be made for relativistic effects where m is given by the relativistic mass and E the relativistic energy of the body.983

          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

          Conservation of weak isospin      Wikipedia: In particle physics, weak isospin is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction. ... The weak isospin conservation law relates the conservation of T3; all weak interactions must preserve T3. It is also conserved by the other interactions and is therefore a conserved quantity in general.988

          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

          Lorentz symmetry      Wikipedia: In standard physics, Lorentz symmetry is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space".991

        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

        Knowing

Knowing      In real life, I can say from my work on "good will exercises" that on any subject (such as "helping the homeless") there are two truths (of the heart and of the world) that pull in different directions. For example, "my help can make things worse" and "I should help those who need help". There are four tests that agree as to which truth is of the heart (the metalevel, the solution space) and which is of the world (the level, the problem space).33

        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

     Jesus felt moved      2026

          Be profoundly troubled      Matthew 26:36 Then Jesus came with them to a place called Gethsemane, and said to his disciples, "Sit here, while I go there and pray." 26:37 He took with him Peter and the two sons of Zebedee, and began to be sorrowful and severely troubled. 26:38 Then he said to them, "My soul is exceedingly sorrowful, even to death. Stay here, and watch with me."1244

          Compassion      9:36 But when he saw the multitudes, he was moved with compassion for them, because they were harassed and scattered, like sheep without a shepherd.1733

          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

     What is hidden will be revealed      2027

          All will be revealed      We should strive to understand of our own free will, for it will be revealed regardless, even after we have lost it. Luke 8:16 "No one, when he has lit a lamp, covers it with a container, or puts it under a bed; but puts it on a stand, that those who enter in may see the light. 8:17 For nothing is hidden, that will not be revealed; nor anything secret, that will not be known and come to light. 8:18 Be careful therefore how you hear. For whoever has, to him will be given; and whoever doesn't have, from him will be taken away even that which he thinks he has."1372

          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

          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

        Feel - Implicitly Innovate

Feel - Implicitly Innovate      185

          Bodystorming      sense what might work in the real world 372

          Mood Board      capture the feel of an idea 374

          Synesthesia      appreciate visceral, memorable aspects 373

        Knowing whether

Knowing whether      Knowing Whether one knows, through our feelings. Caring about Carer. Listen to my feelings, whether I am riled or at peace, and why I feel that way. The person who is riled is wrong! I used to be very bothered when I engaged the homeless. It was because I focused on the truth "my help can make things worse" as if that were the truth of the heart, the truth that I should be thinking. 49

          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

          Emotional devastation of loved ones      Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: When they announced the decision, I felt like I had given 100 percent, just for the wrong fight. But the devastation — the emotional devastation that went across the board to my family and friends — was unbelievable. I saw them crying. Everyone was crying but me."1891

          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

          Sweetest feeling      Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: "There is no sweeter feeling than when you throw the perfect punch. You get a signal — you get this little tingling sensation that shoots up and down your arm to let you know that you've hit the jackpot. And you know. You know right away that guy's gone."1901

        Argue the other side

Argue the other side      Ask your friend to identify an issue about which he or she has strong feelings. Ask your friend to state his or her position on the issue and to defend tat position with argumentation. ... ask him or her to make the opposite case. ... Most people can't do it effectively; some can't do it at all and become quite upset when asked to. They are so focused on their interpretation of the issue that they cannot adequately articulate the other side (or sides). The Thinker's Toolkit by Morgan D.Jones drawing from The Universe Within: A New Science Explores the Human MindThe Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich.2134

          Devil's Advocacy      Learned churchmen took the devil's position simply for argument's sake to challenge the rationale presented in the nomination for sainthood, the idea being that through this process the truth, perforce, would out. ... challenge the proposition not because they disbelieve it but simply to test its validity. Devil's advocacy does this by seeking, with either the same or other evidence, to prove the opposite of whatever the challenged view holds. ... Devil's advocacy is analytically useful because it by design focuses on a contrary or opposite viewpoint and in doing so activates all of the instinctive behaviors associated with focusing: viewing the problem one-dimensionally through the lens of biases connected with that viewpoint; eschewing alternative solutions; valuing evidence that supports that viewpoint; devaluing and discarding evidence that does not. Devil's advocacy thus engages the very same mental tactics employed by the primary advocate but applies them in opposition, thereby promoting objectivity. What's more, the devil's advocate goes a step further by seeking out and obtaining new evidence, evidence about which the prime advocate has no knowledge or curiosity or that was disbelieved and discredited. This further step is the secret weapon of devil's advocacy, the extra dimension that makes it a formidable analytic technique. ... Devil's advocacy will ... open your mind to new dimensions and perceptions of the problem, poking holes in fallacious self-serving arguments and stripping away thinly reasoned and thinly supported analysis. The Thinker's Toolkit by Morgan D.Jones1775

          Keep distinct the advocate and decision maker      The hallmark of focusing is the revered human institution of advocacy - taking a position on an issure, marshaling supporting evidence, and defending that position against the arguments of those holding (focusing on) opposing views. ... advocacy works. It is ... the foundation of our judicial system, in which the prosecution prosecutes, the defense defends, and neither side does both. ... So advocacy does work, when someone other than the advocator makes the decision - such as a judge or a jury, the electorate, the board of directors, or the boss, whoever that might be. But ... advocacy feeds and perpetuates our mind-sets, biases, beliefs, and prejudices. It thus nurtures our tendency to focus and, in doing so, defeats objectivity ... our full understanding of the problem. The Thinker's Toolkit by Morgan D.Jones drawing from The Universe Within: A New Science Explores the Human MindThe Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich.2135

          Let go of untrue beliefs      We let go of untrue beliefs by seeking the truth. We tend to cling to untrue beliefs in the face of contradictory evidence. ... many of our most cherished individual beliefs, upon careful examination, are simply untrue ... we tend to hold fast to these beliefs in the very face of incontrovertible contradictory evidence ... We simply rationalize away the disparity. ... Robert Abelson offers compelling insight. He contends that we humans treat belief like material possessions. We acquire and retain material possessions because of the functions they serve and the value they offer. ... our possessions make us feel good ... he notes the similarity between beliefs and possessions is revealed in our language. We have beliefs. We adopt, inherit, acquire, hold, and cling to them. The Thinker's Toolkit by Morgan D.Jones drawing from The Universe Within: A New Science Explores the Human MindThe Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich.2136

        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

          Stumbling upon the irony of a situation      I walked to the river and I stumbled across a statue of the Japanese consul who had written 6,000 visas by hand to Jews so they could escape the country and the Nazi threat. I sat down and cried. I thought there should be a national exam that all the bureaucrats should be required to pass regarding their ability and responsibility to be creative and proactive to do the right thing to the extent that they can.1021

        My state of mind

My state of mind      138

          How I feel      435

          My state of mind      492

          Pray with my music      389

          Recognizing how I was moved      398

          Reflecting on my mindset      422

          Something feels right or not      399

          Something was missing      423

          What I like      413

          What is touching me      436

        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

     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

          How can I show my authority? without forcing belief?      By having the force of character to heal the paralytic, Jesus suggests that he has the strength of character to forgive him as well. People who believe his character can appreciate his personal ability to personally forgive. (Why a personal Jesus is relevant). People who don't believe can think that he's referring to a divine mystical authority and not acknowledge his person. Matthew 9:5 For which is easier, to say, 'Your sins are forgiven;' or to say, 'Get up, and walk?' 9:6 But that you may know that the Son of Man has authority on earth to forgive sins..." (then he said to the paralytic), "Get up, and take up your mat, and go up to your house." 1658

          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 be related to      Allow people to be doomed completely. If they aren't interested, let them go. Matthew 10:14 Whoever doesn't receive you, nor hear your words, as you go out of that house or that city, shake off the dust from your feet. 10:15 Most certainly I tell you, it will be more tolerable for the land of Sodom and Gomorrah in the day of judgment than for that city. 1786

          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 children's ways      Matthew 11:16 "But to what shall I compare this generation? It is like children sitting in the marketplaces, who call to their companions 11:17 and say, 'We played the flute for you, and you didn't dance. We mourned for you, and you didn't lament.' 1860

          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

               Foxes have holes and birds have nests      8:20 Jesus said to him, "The foxes have holes, and the birds of the sky have nests, but the Son of Man has nowhere to lay his head." 1616

               Lilies of the field      Matthew 6:28 Why are you anxious about clothing? Consider the lilies of the field, how they grow. They don't toil, neither do they spin, 6:29 yet I tell you that even Solomon in all his glory was not dressed like one of these. 6:30 But if God so clothes the grass of the field, which today exists, and tomorrow is thrown into the oven, won't he much more clothe you, you of little faith? 1455

          Analogy with the unrighteous world      Luke 18:1 He also spoke a parable to them that they must always pray, and not give up, 18:2 saying, "There was a judge in a certain city who didn't fear God, and didn't respect man. 18:3 A widow was in that city, and she often came to him, saying, 'Defend me from my adversary!' 18:4 He wouldn't for a while, but afterward he said to himself, 'Though I neither fear God, nor respect man, 18:5 yet because this widow bothers me, I will defend her, or else she will wear me out by her continual coming.'" 18:6 The Lord said, "Listen to what the unrighteous judge says. 18:7 Won't God avenge his chosen ones, who are crying out to him day and night, and yet he exercises patience with them? 18:8 I tell you that he will avenge them quickly. Nevertheless, when the Son of Man comes, will he find faith on the earth?"2059

          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 sign will he give      Matthew 16:1 The Pharisees and Sadducees came, and testing him, asked him to show them a sign from heaven. 16:2 But he answered them, "When it is evening, you say, 'It will be fair weather, for the sky is red.' 16:3 In the morning, 'It will be foul weather today, for the sky is red and threatening.' Hypocrites! You know how to discern the appearance of the sky, but you can't discern the signs of the times! 16:4 An evil and adulterous generation seeks after a sign, and there will be no sign given to it, except the sign of the prophet Jonah."1994

     What distinguishes the cultures      2032

          Born of woman vs. born of spirit      Matthew 11:11 Most certainly I tell you, among those who are born of women there has not arisen anyone greater than John the Baptizer; yet he who is least in the Kingdom of Heaven is greater than he. 1856

          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

     What his disciples should not yet tell others      2031

          Confiding      How do we know of the devil's temptation of Jesus? There were no witnesses except for Jesus himself. The Holy Spirit may have informed the evangelists or Jesus may have told his disciples himself. If they were to know, then it seems that he might as well have told them himself. Why would he then tell about the temptations? Perhaps to say that they were the "down-to-earth" salt of the earth that he had chosen, that it was he who needed them. Jesus also seems to have confided much to John, his best friend, who seems to present Jesus' point of view as Jesus understood himself.1355

          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

          Who should know       Matthew 16:20 Then he commanded the disciples that they should tell no one that he was Jesus the Christ. 16:21 From that time, Jesus began to show his disciples that he must go to Jerusalem and suffer many things from the elders, chief priests, and scribes, and be killed, and the third day be raised up.2000

               A right triangle is half a rectangle      This morphism is the basis for the area of a right triangle, but also for all of trigonometry, and shows that a function need not be a formula, and shows how two domains - angles and ratios - can be linked, as by shapes. Gospel Math. 1846

               Recasting geometry/combinatorics as parity      Remove the two diagonally opposite corner squares of a chessboard. Is it possible to tile this shape with thirty-one 2 x 1 "dominos"? ... At first, it seems like a geometric/combinatorial problem with many cases and subcases. But it is really just a question about counting colors. The two corners that were removed wre both (without loss of generality) white, so the shape we are interested in contains 32 black and 30 white squares. Yet any domino, once it is placed, will occupy exactly one black and one white square. The 31 dominos thus require 31 black and 31 white squares, so tiling is impossible. pg. 60 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1508

          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

          e      Determine, with proof, the largest number which is the product of positive integers whose sum is 1976. ... Once again, we shall inappropriately apply calculus to a discrete problem. It makes intuitive sense for the numbers whose sum is 1976 to be equal (see the discussion of the AM-GM inequality...) But how large should these parts be? Consider the optimization question of finding the maximum value of f(x) = (S/x)**x, where S is a positive constant. An exercise in logarithmic differentiation (do it!) shows that S/x = e. Thus, if the sum is S each part should equal e and there should be S/e parts. Now this really makes no sense if S=1976 and the parts must be integers, and having a non-integral number of parts makes even less sense. But at least it focuses our attention on parts whose size is close to e=2.71828... Once we start looking at parts of size 2 and 3, the problem is close to a solution... pg.313 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2250

          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

          Fantasize an answer      When looking at the conclusion of the problem, especially for a "to find" problem, sometimes it helps to "fantasize" an answer. Just make something up, and then reread the problem. Your fantasy answer is most likely false, and rereading the problem with this answer in mind may help you to see why the answer is wrong, which may point out some of the more important constraints of the problem. pg.30, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1428

          Interpreting algebraic variables as coordinates      Whenever a problem involves several algebraic variables, it is worth pondering whether some of them can be interpreted as coordinates. pg. 59 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1505

          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

          Recasting geometry as algebra      Descartes' idea of recasting geometric questions in a numeric/algebraic form led to the development of analytic geometry, which then led to calculus. pg. 60 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1506

          Two different ways       Keeping a flexible point of view is a powerful strategy. This is especially true with counting problems where often the crux move is to count the same thing in two different ways. To help develop this flexibility, you should practice creating "combinatorial arguments". This is just fancy language for a story that rigorously describes in English how you count something. ... Pay attention to the building blocks of "algebra to English" translation, and in particular, make sure you understand when and why multiplication rather than addition happens, and vice versa. Examples include addition (or), multiplication (and), exponentiation, combination, permutation, distinct members, products of choices, sums of choices, complements of combinations. pg.208 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2201

     Make it easier      The easier problem may actually be the more informative, relevant, natural, instructive problem. If the given problem is too hard, solve an easier one. ... For example, if the problems involves big, ugly numbers, make them small and pretty. If a problem involves complicated algebraic fractions or radicals, try looking at a similar problem without such terms. At best, pretending that the difficulty isn't there will lead to a bold solution... At worst, you will be forced to focus on the key difficulty of your problem, and possibly formulate an intermediate question, whose answer will help you with the problem at hand. And eliminating the hard part of a problem, even temporarily, will allow you to have some fun and raise your confidence. pg.18, 31 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1417

     Wishful thinking       It is helfpul to try to loosen up, and not worry about rules or constraints. Wishful thinking is always fun, and often useful. For example, in this problem, the main difficulty is that the top boxes labeled A and C are in the "wrong" places. So why not move them around to make the problem trivially easy? ... Ask yourself, "What is it about the problem that makes it hard?" Then, make the difficulty disappear! You may not be able to do this legally, but who cares? Temporarily avoiding the hard part of a problem will allow you to make progress and may shed light on the difficulties. pg.18, 31 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1416

        Agree - Implicitly Understand

Agree - Implicitly Understand      174

          Campfire      learn from stories, share culture 334

          Graphic Jam      visualize abstract concepts, absorb complexity 332

          History Map      appreciate relevance of organization's culture 331

          Memory Wall      appreciate people, foster team 329

          Visual Glossary      create a shared language 330

          Welcome to My World       share our internal map of reality 333

        Knowing what

Knowing what      Knowing What one knows, its origins. Caring about Believer. Appeal to shared inner implicit inherent culture despite explicit experience of examples that suggest differently. The truth of the world is easy to point to, can be shown by examples, whereas the truth of the heart must already be in you, is evoked by analogy. It is easy to show examples that "my help can make things worse". But how can I show that I "should" help? I can't observe that, but rather, the notion must already be in me. Likewise, I can point to the surface structure of a problem, but as for the deep structure, I have to appeal to you that you are already familiar with it.50

          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

          Ask people why      Why did County patients tolerate these waits and abusive conditions? Our patients declared that they came because County had "the best doctors." This was not true. There is no way we were the best. We were young, uninitiated, and worse, unsupervised. But many of our patients had been turned away from other institutions or had family or friends with the same experience "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.19341946

          How can we tie communication firmly into context?      Pamela McLean, June 10, 2011. Maybe that is why we need story telling (and pattern languages) in our post-web world - so that communication is tied firmly into context through anecdotes.1837

          The beauty of a role model      Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: "I watched Muhammad Ali, how when he would speak, how it was such a thing of beauty. It sounded so wonderful. And I wanted to be like him."1884

          What allows generalizations to work effectively?      Pamela McLean, June 10, 2011. Maybe generalisations only work effectively when people are living in the same cultural groups or working in the same silos. 1838

        Fact and Judgment: Taxonomy of Problem Types

Fact and Judgment: Taxonomy of Problem Types      While problems come in all varieties, shapes, and sizes, each can be categorized in terms of the roles that fact and judgment play in analysis of the problem. ... there is an inverse relationship between the number of facts and the amount of judgment required to solve a problem. ... the fewer facts we have, the more judgment is required ... the more facts - the less judgment. From this relationship we can define four basic types of problems:

  • Simplistic: There is only one answer, no more. Who is the governor of New York?
  • Deterministic: There is only one answer, but the correct formula must be used. What is the area of a square whose side is 20 feet?
  • Random: Different answers are possible, and all can be identified. Which of the candidates will win the election?
  • Indeterminate: Different answers are possible but are conjectural, so not all can be identified. What are the prospects for U.S.-Russian relations?
...The greater the role of facts in analyzing a problem, the greater our confidence in the findings of that analysis; the fewer facts, the less confidence. ... As the role of judgment increases, so does the probability of error. Therefore, as the probability of error increases, our confidence in our conclusions must diminish. But the human mind does not willingly or usually lose confidence in its conclusions, no matter what the circumstances ... cognitive experiments have shown that, even when the explanation we come up with doesn't represent the evidence very well, we use the explanation anyway and feel comfortable doing so. What's more, we defend the explanation in the face of strong contrary evidence, which we rebut with irrelevancies and sophistries. That's just how the human mind works. The insight, then, is to be wary of conclusions that are based largely on judgment, not facts. The Thinker's Toolkit by Morgan D.Jones2110

        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

          Risk a relationship for a chance to grow closer      I had believed that it was a bad idea to work for a friend or relative as it could destroy our relationship which is much more precious than any work. But then I had the chance to work for my friend John Harland. And indeed there were times of stress but it really deepened our friendship and I learned that much more what a wonderful person he is.1047

        Study

Study      137

          Collect recordings and lyrics      412

          Expand my interests      409

          Listen to examples      411

          Looking at different aspects of a subject      405

          Study videos, concerts, services      388

        The first will be last and the last will be first

The first will be last and the last will be first      In this world, the truth of the world is prior to the truth of the heart; but in peace, the truth of the heart is prior to the truth of the world. Jesus may be relating the principle of reflectivity, which changes the direction of the twosome, so that reflection switches what is first and what is last. Thus we can never be certain, which will be first and which will be last, but must stay on our toes.1349

     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

          Believe that God hears you when you speak once      Matthew 6:7 In praying, don't use vain repetitions, as the Gentiles do; for they think that they will be heard for their much speaking. 1431

          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

          How would the Gentiles have reacted      Matthew 11:20 Then he began to denounce the cities in which most of his mighty works had been done, because they didn't repent. 11:21 "Woe to you, Chorazin! Woe to you, Bethsaida! For if the mighty works had been done in Tyre and Sidon which were done in you, they would have repented long ago in sackcloth and ashes. 11:22 But I tell you, it will be more tolerable for Tyre and Sidon on the day of judgment than for you. 11:23 You, Capernaum, who are exalted to heaven, you will go down to Hades. For if the mighty works had been done in Sodom which were done in you, it would have remained until this day. 11:24 But I tell you that it will be more tolerable for the land of Sodom, on the day of judgment, than for you." 1905

          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

          3-12-3 Brainstorm      fresh, incremental improvements 326

          Challenge Cards      identify, address challenges, improve product 328

          Plus/Delta      constructive feedback 327

        Knowing how

Knowing how      Knowing How one knows, contrasts. Believing in Carer. Improve results by taking a more fundamental point of view. The truth of the world follows from the truth of the heart, but not the other way around. If "I should help those who need help", then I won't want my help to make things worse. But if I simply don't want to make things worse, I will never help anybody.51

     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

          The path I want to take      Fresh Air interview of boxer Sugar Ray Leonard, June 6, 2011: "I wanted to win the goal medal and then go home and further my education in college," he says.1885

        Weigh one element against another

Weigh one element against another      Second, structuring allows us to compare and weigh one element against another. Instead of looking at a whole bowl of vegetable soup, we look at the soup's ingredients, one ingredient at a time. This identifies which factors and relationships are critical not only to our analysis but also to the concerns of those who will make use of our findings. The Thinker's Toolkit by Morgan D.Jones2113

        Money (compensation of difference)

Money (compensation of difference)      Compensation substitutes for disparity in contributions. How there is value.1076

          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

          Championing faith      419

          What brings people to God      415

          Whether my prayers were answered      421

        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

     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

     Jesus' role      2038

          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

          Appreciating the power of God      Mark 11:1 When they drew near to Jerusalem, to Bethsphage and Bethany, at the Mount of Olives, he sent two of his disciples, 11:2 and said to them, "Go your way into the village that is opposite you. Immediately as you enter into it, you will find a young donkey tied, on which no one has sat. Untie him, and bring him. 11:3 If anyone asks you, "Why are you doing this?" say, "The Lord needs him and immediately he will send him back here."1407

          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

          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

               Jesus took our infirmities      Jesus dealt with the human condition, including illness. Matthew 8:16 When evening came, they brought to him many possessed with demons. He cast out the spirits with a word, and healed all who were sick; 8:17 that it might be fulfilled which was spoken through Isaiah the prophet, saying: "He took our infirmities, and bore our diseases."1611

          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

          Reference point in Scripture      The coming of the Son of Man is for Jesus a reference point in Scripture. Matthew 10:22 You will be hated by all men for my name's sake, but he who endures to the end will be saved. 10:23 But when they persecute you in this city, flee into the next, for most certainly I tell you, you will not have gone through the cities of Israel, until the Son of Man has come. 1799

          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

          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

          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

          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

          Who was he sent to      Matthew 15:22 Behold, a Canaanite woman came out from those borders, and cried, saying, "Have mercy on me, Lord, you son of David! My daughter is severely demonized!" 15:23 But he answered her not a word.His disciples came and begged him, saying, "Send her away; for she cries after us." 15:24 But he answered, "I wasn't sent to anyone but the lost sheep of the house of Israel."1990

          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

     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

     The word and the foursome      2039

          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

          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

          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

          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

               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

          Design the Box      translate features into benefits 321

          Flip It      open up opportunities 324

          Product Pinnochio      improve the end user experience 325

          The Virtuous Cycle      shift focus from process to recurring value 323

          Understanding Chain      unfold your knowledge from your audience's point of view 322

        Knowing why

Knowing why      Knowing Why one knows, scopes. Believing in Believer. Invert my point of view by taking up another's wider point of view. Given a subject such as "helping the homeless", and the four questions Why? How? What? Whether?, then the heart considers a broader question than the world. The world asks, What is helpful? (what makes things better, not worse) but the heart asks Why are we helpful? (because we should). This makes for six types of issues.52

     Define a concept in terms of more fundamental concepts      556

          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

          Phenomena coincide in that they appear together      Conversing with: identity In phonology, semiotics and other structural disciplines, there is a principle that two phenomena are the same if they appear together, so that no change in meaning is ever signified should one stay the same, but the other change. 795

          Phenomena coincide in that they never appear together      Conversing with: continuity In phonology, semiotics and other structural disciplines, there is a principle that two phenomena are the same if they never appear together. For example, Superman and Clark Kent must be the same person because they are never seen together.794

     Relate scopes      1701

          Comparing levels of the foursome: Why? How? What? Whether?      Conversing with: scope of a statement On a given subject, one way to distinguish the truths of the heart and of the world is that, given four questions - whether, what, how, why - with why being the broadest - the heart asks the broader question. This also leads to six different pairs, thus categories into which the related topic and good will exercise fall into.680

          Cross check      Conversing with: truth I found four different ways to distinguish between the truths of the heart and of the world. Three of the ways corresponded with each other all of the time in the more than 40 good will exercises that I developed. These tests thereby affirmed each other: We can point to the truth of the world with examples from experience, but not the truth of the heart, which must be in us already; the truth of the world follows from the truth of the heart, but not the other way around; and given four questions, whether-what-how-why, why being the broadest, the truth of the heart asks a broader question than the truth of the world. The fourth test is that the person who is riled is wrong, and confuses the truths of the heart and the world, but in some cases they didn't and I had to ammend my conclusions regarding that test.723

        Levels

Levels      1683

        Problem Restatement

Problem Restatement      How we define a problem usually determines how we analyze it. ... We frequently discover, based on information and perceptions gained midway through the analysis, that the initial problem statement was far off the mark. ... narrow definition of a problem caused their analysis to be shortsighted, overlooking alternative and possibly more beneficial solutions. ... Every problem ... can be viewed from multiple conflicting perspectives. And what drives these differing perspectives? Biases, those unseen killers of objective truth, determine our perspective of any problem. That perspective in turn drives our analysis, our conclusions, and ultimately our recommendations. ... it makes good sense at the outset ... to deliberately strive to identify and examine our biases as they relate to the problem at hand. ... the human mind by design works to conceal the biases that drive our thinking ... introspection is impracticable ... I recommend an indirect approach, which is to restate (redefine) the problem in as many different ways as we can think of. We simply shift our mental gears into a divergent mode ... and start pumping out restatements without evaluating them. The key here ... is letting ideas flow freely, without attempting to justify them. Sometimes restating the problem is difficult because the original statement was poorly articulated. ... One can generally gain most of the benefits of restating a problem in five or ten minutes. ... A problem restatement session will rather quickly, almost magically, focus on the crux of a problem - the core issues - and reveal what the problem is really all about ... if, as often is the case, we are analyzing a problem for someone else's benefit, it is best to generate the problem restatements in that person's presence. Doing so in an open discussion will reveal our consumer's prime concerns and what he or she considers to be the key issues. This will facilitate reaching agreement at the start concerning what the problem is and what our analysis will aim to find out. Two basic approaches: 1) Paraphrase: Reword the problem without shifting its primary focus. 2) Make a 180-Degree Turn: Shift the problem's focus by viewing it from the opposite direction. ... Most important of all, restatements should, whenever possible, be put into writing so we - and our consumer ... can study them. ... the goal of problem restatement is to expand our thinking about the problem, not to solve it. ... A valuable tip when restating problems is to make them simple, positive, and in active voice. The mind works more easily and quickly with simple, positive, active-voice sentences than with complex, negative, passive-voice sentences. The Thinker's Toolkit by Morgan D. Jones1766

          Depicting elements visually      Finally, all of the structuring methods presented in this book are visual processes that involve writing or depicting elements of a problem or on a display board or computer screen, where we can see them. Why is seeing them important? By enabling the brain actually to see the words or numbers or depictions of the problem, we engage more brainpower in analyzing and solving the problem and so gain added insights. Indeed, when elements are seen visually, we often discover correlations we missed when we simply thought about them. The old adage - "A picture is worth a thousand words" - speaks to the power of engaging the brain's visual capabilities. The Thinker's Toolkit by Morgan D.Jones2117

          Note how our feelings govern our perceptions      Georgia O'Keeffe separated levels in her mind, emotion and perception. ...in [Edmund] Bolles's book A Second Way of Knowing: "Georgia [O'Keeffe] was suddenly struck by the realization that her feelings governed the way she saw the scene. It was a moment of transformation: the entire visual world, she realized, was dependent on the emotional world." Said Bolles: "That day she learned the artist's secret; what you perceive depends on who you are. Analytical thinkers have generally assumed that we perceive reality as it is; they then use a process of abstract reasoning to interpret that perception. O'Keefe realized that the perception is the interpretation. It rests on an internal reality that governs the meaning we find in our sensations." The internal reality Bolles speaks of is, in fact, controlled by the mental traits I've been talking about. ... We view the world through a dense veil of burdensome, thought-warping biases and mind-sets. The Thinker's Toolkit by Morgan D.Jones drawing from The Universe Within: A New Science Explores the Human MindThe Universe Within by Morton Hunt and How We Know What Isn't So by Thomas Gilovich.2137

          The Matrix      A matrix is typically used to compare and make explicit the internal structure of records, thus to uncover correlations, rather than to organize or visualize externally. I think the matrix is a way to restate a problem so that we can think of it on two different levels, the whole and the parts. This internal structuring yields a systemic boundary that opens the way for external restructuring. First, we think of as many different explanations as we can... We call that diverging. We then sort the explanations (cluster them) into definable groups, which we'll call categories, for want of a better term. ... We also discern basic explanations ... which I have listed below by category ... Let's examine each category ... [matrix analysis begins] that's what structuring analysis does: it enables us to analyze each element of a problem separately, systematically, and sufficiently. ... We can find out by determining whether there was a correlation ... To analyze this question, I have constructed chronologies using a matrix. ... a matrix is nothing more than a grid with as many cells as needed for whatever problem is being analyzed. ... a matrix enables us ... to separate elements of a problem; categorize information by type; compare one type of information with another; compare pieces of information of the same type; see correlations (patterns) among the information. ... putting the numerical data into matrix form has the effect of isolating the data so they can be analyzed more easily, both separately and in combination. ... is there a correlation ...? [...mapping out relationships...] ... which one of the following circumstances would be best ... possible outcomes ... facilitate discussion of the six possible scenarios (alternative outcomes) to ensure that the pros and cons of each were analyzed separately, systematically, and sufficiently. ... the first thing I do, when confronted with a problem, is to ask myself how I can represent the problem in a matrix. ... when I can portray it in a matrix, the problem immediately opens itself to analysis, like the petals of a flower opening up to reveal its inner parts. Moreover, displaying a matrix like this one on a screen to guide discussion at a meeting can be extremely helpful. ... By providing a visual means of focusing our mind on each option, one at a time, the matrix enables us to easily compare and rank... The Thinker's Toolkit by Morgan D.Jones1771

        Humility

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

          Learning what is not related      I remember as a child that our parents used to give us allowance. They made it very clear that the allowance was not for any work that we did. The purpose of the allowance was for us to learn how to spend thoughtfully and manage our money. We also had to help with some basic chores. And we were to do them because we had to, it was independent of our allowance. I took from them that our responsibility to do our work and our need to be thoughtful about money were two separate issues.1044

          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

          Noting how others do things      429

          Noting the standard      430

          Preferring the standard      431

        Symmetry breaking

Symmetry breaking      1029

          Explicit symmetry breaking      Wikipedia: Explicit symmetry breaking indicates a situation where the dynamical equations are not manifestly invariant under the symmetry group considered. This means, in the Lagrangian (Hamiltonian) formulation, that the Lagrangian (Hamiltonian) of the system contains one or more terms explicitly breaking the symmetry. Such terms can have different origins:

  • Symmetry-breaking terms may be introduced into the theory by hand on the basis of theoretical/experimental results, as in the case of the quantum field theory of the weak interactions, which is expressly constructed in a way that manifestly violates mirror symmetry or parity.
  • Symmetry-breaking terms may appear in the theory because of quantum-mechanical effects. One reason for the presence of such terms — known as "anomalies" — is that in passing from the classical to the quantum level, because of possible operator ordering ambiguities for composite quantities such as Noether charges and currents, it may be that the classical symmetry algebra (generated through the Poisson bracket structure) is no longer realized in terms of the commutation relations of the Noether charges. Moreover, the use of a "regulator" (or "cut-off") required in the renormalization procedure to achieve actual calculations may itself be a source of anomalies.
  • Finally, symmetry-breaking terms may appear because of non-renormalizable effects. Physicists now have good reasons for viewing current renormalizable field theories as effective field theories, that is low-energy approximations to a deeper theory (each effective theory explicitly referring only to those particles that are of importance at the range of energies considered). The effects of non-renormalizable interactions (due to the heavy particles not included in the theory) are small and can therefore be ignored at the low-energy regime. It may then happen that the coarse-grained description thus obtained possesses more symmetries than the deeper theory. That is, the effective Lagrangian obeys symmetries that are not symmetries of the underlying theory. These "accidental" symmetries, as Weinberg has called them, may then be violated by the non-renormalizable terms arising from higher mass scales and suppressed in the effective Lagrangian.
1026

          Spontaneous symmetry breaking      Wikipedia: Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state. For spontaneous symmetry breaking to occur, there must be a system in which there are several equally likely outcomes. The system as a whole is therefore symmetric with respect to these outcomes (if we consider any two outcomes, the probability is the same). However, if the system is sampled (i.e. if the system is actually used or interacted with in any way), a specific outcome must occur. Though we know the system as a whole is symmetric, we also see that it is never encountered with this symmetry, only in one specific state. Because one of the outcomes is always found with probability 1, and the others with probability 0, they are no longer symmetric. Hence, the symmetry is said to be spontaneously broken in that theory. Nevertheless, the fact that each outcome is equally likely is a reflection of the underlying symmetry, which is thus often dubbed "hidden symmetry", and has crucial formal consequences, such as the presence of Nambu-Goldstone bosons. When a theory is symmetric with respect to a symmetry group, but requires that one element of the group is distinct, then spontaneous symmetry breaking has occurred. The theory must not dictate which member is distinct, only that one is. 1027

               A ball on top of a hill      Wikipedia: A common example to help explain this phenomenon is a ball sitting on top of a hill. This ball is in a completely symmetric state. However, its state is unstable: the slightest perturbing force will cause the ball to roll down the hill in some particular direction. At that point, symmetry has been broken, because the direction in which the ball rolled has a visible feature that distinguishes it from all other directions. The "choice" of direction is immaterial, however, as any other direction would do, i.e. the system is still bearing traces of the symmetry of the hill, albeit now somewhat less apparent.1028

          Symmetry breaking      Wikipedia: Symmetry breaking in physics describes a phenomenon where (infinitesimally) small fluctuations acting on a system which is crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observer unaware of the fluctuations (or "noise"), the choice will appear arbitrary. This process is called symmetry "breaking", because such transitions usually bring the system from a disorderly state into one of two definite states. Since disorder is more symmetric, in the sense that small variations to it don't change its overall appearance, the symmetry gets "broken". Symmetry breaking is supposed to play a major role in pattern formation.1025

               Phase transition      Wikipedia: A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another. A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium certain properties of the medium change, often discontinuously, as a result of some external condition, such as temperature, pressure, and others. For example, a liquid may become gas upon heating to the boiling point, resulting in an abrupt change in volume. The measurement of the external conditions at which the transformation occurs is termed the phase transition point. The term is most commonly used to describe transitions between solid, liquid and gaseous states of matter, in rare cases including plasma.1030

        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

        Absolute Learning

Absolute Learning      In real life, we address our doubts (surface problems) with counterquestions (deep solutions). I may doubt, How do I know I'm not a robot? and because that has me question all of my experiential knowledge, I can't resolve that by staying in the same level as my problem. Instead, I ask a counterquestion that takes me to my metalevel: Would it make any difference? If there's a difference, then I can check if I'm a robot. If there's not a difference, then it's just semantic and I'm fine with being a robot (by analogy, #3 and #4 may actually be equivalent in some total order). My counterquestion in this case forced you to pin down your variable, like forcing an "arbitrary" epsilon to be fixed so that I could choose my delta accordingly. There are six doubts answered by six counterquestions:

  • 10 Do I truly like this? How does it seem to me?
  • 20 Do I truly need this? What else should I be doing?
  • 21 Is this truly real? Would it make any difference?
  • 32 Is this truly problematic? What do I have control over?
  • 31 Is this truly reasonable? Am I able to consider the question?
  • 30 Is this truly wrong? Is this the way things should be?
Consider also the qualities of signs:
  • 10 malleable: icon can change without thing changing
  • 20 modifiable: index can change without thing changing
  • 21 mobile: index can change without icon changing
  • 32 memorable: symbol can change without index changing
  • 31 meaningful: symbol can change without icon changing
  • 30 motivated: symbol can change without thing changing
And consider the "ten commandments". Note that these six "rooms" correspond to the six divisions of everything as generated by God taking up the counterquestions. The zeroth division corresponds to What do I truly want? and obeying God, and the seventh division corresponds to Am I doing anything about this? and caring=believing, living as a person-in-general, thus relating (Obeying) God the Father and (Believing) Jesus. Consider how the other secondary structures likewise arise in the house of knowledge from God's taking up the primary structures.34

          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

        Structuring analysis of problems

Structuring analysis of problems      Structuring analysis has us tend to our mind's weaknesses and counter them before they become evident. Each counterquestion thus addresses a different weakness. Perhaps they complement each other in pairs. "What else should I be doing?" is divergent and "Am I able to consider the question?" is convergent. In general, perhaps the competition between convergence and divergence is given by the pairs of levels. Convergence takes us from questions to answers, whereas divergence takes us from answers to questions. Which is the broader view, the view of the heart? I also note that each of the six analytic tools seems to have us analyze the complementary levels. For example, a decision-probability-utility tree is defined by whether-what but has us analyze why and how, whereas a causal flow diagram is based on whether-why but has us analyze what and how, and so on. All of us regularly make mistakes of judgment based on faulty analysis ... While some of these errors can be blamed on a lack of information or education, most occur because of the way our minds work. Our minds frequently mislead us, giving us a false understanding of events and circumstances and causing our analysis of events and circumstances to be flawed ... By learning about the mental barriers and pitfalls that impede effective analysis and acquiring the skills and techniques to overcome them, our batting average can be improved, and significantly so. ... The skills and techniques I speak of are what this book is about: ways of organizing, or structuring, our analysis of problems. This book explains what it means to structure analysis; identifies and describes the mental traits that tend to lead us astray; explains how structuring our analysis of problems defeats the ill effects of these traits; describes fourteen easily understood structuring techniques; and provides exercises through which the reader can begin to master them. ... The word analysis means separating a problem into its constitutent elements. Doing so reduces complex issues to their simplest terms. ... But other than showing, as Bertrand Russel said, whether the elements of a problem are sensible or nonsensical, what does structuring buy us? ...

  • First, structuring helps the mind make sense out of complex problems.
  • Second, structuring allows us to compare and weigh one element against another.
  • Third, structuring helps us to focus our analysis. The mind instinctively focuses ... we're better off to work with the mind than against it and, in so doing, control what it focuses on.
  • Fourth, structuring focuses on one element at a time, which, compared to our instinctive scattershot approach of tackling all elements simultaneously, is more systematic, more thorough, and more productive of relevant ideas.
  • Fifth, by establishing rational, systematic frameworks within which to analyze problems, analytic structuring techniques enable us to impose our analytic will on our subconscious mind, preventing it from following the instinctive mental traits that lead to faulty analysis.
  • Finally, all of the structuring methods presented in this book are visual processes that involve writing or depicting elements of a problem ... when elements are seen visually, we often discover correlations we missed when we simply thought about them.
...effective problem solving depends, in the end, not on how we structure our analysis but on the soundness of our thinking, and for that we have to use our mind. Structuring is not a substitute for thinking. It is rather a means of facilitating and empowering thinking. ... The only sure way I know to achieve objectivity when analyzing a problem is to structure the analysis, setting up at the outset a rigorous step-by-step process to which the subconscious is forced to adhere and which ensures our complete understanding of the problem and full consideration of all reasonable alternative solutions. The Thinker's Toolkit by Morgan D. Jones2093

          The analytic method      Bertrand Russell ... in 1901 wrote in his Principles of Mathematics that, with regard to interpreting the language of philosophic theories, the solution is the analytic method - the breaking down of language until a theory shows itself to be either a set of sensible substatements or just nonsense. In this way, he said, many philosophical "problems" just disappear. Russell's book changed the way English philosophy was conducted by establishing the "analytic approach" as the only reputable method of studying philosophical questions. The Thinker's Toolkit by Morgan D. Jones2094

        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

               The Business Equation by John Caswell      The diagram above presents my structuring of The Business Equation taught by John Caswell of GroupPartners.net It does seem to me to correlate with the methods from my "money mind" stories.1703

        Forces

Forces      Forces are expressions of causality, of the relationship between before and after. They allow for a system to break down into subsystems. Rules are applied in six different ways to link states before and after an event. In math, these are the kinds of subsystems that Implication forms. In life, these are the ways that we visualize. 854

        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

        Tree of variations

Tree of variations      Model truth (can distinguish possibilities). Weighted averages, moves in games. 10 Evolution: Hierarchy => Sequence (for determining weights). Examination of cases.60

               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

               Multiplication, division by 10 moves the number      One example of fractal multiplication that comes to mind is multiplication by 10. I noticed this year that the decimal point is not logically positioned. It should be at (under or over) the "ones" place. Then the system would be symmetric. The space to the left would be the "tens" and the space to the right would be the "tenths". A little (video) essay could explain that when we multiply or divide by 10, it is the number that actually moves, not the decimal point. It's like the fact that the Sun is at (or near) the center of the solar system and the Earth revolves around it, not the other way around, as it may seem. The way that the decimal point is placed is very destructive pedagogically, it makes people (like me) think that there is something between the digit; that there is some mysterious difference between the whole numbers and the fractional numbers; and most sadly, that (jaded) teachers don't see that there's something wrong with the system, as I remember thinking as a child; and that there's no reason to care. I suspected that the decimal point is where it is for typographical reasons; maybe because it arose along with printing? or derived from accounting notation? My point being that an adult who appreciates what I've just written (and more along these lines) would appreciate some thing very deep about math (including how to calculate 10% discounts by shifting numbers to the right). Gospel Math. 1839

          Algorithmic Proof      ... the argument below can easily be rewritten as an induction proof. But it is much more instructive to present a new type of argument, an algorithmic proof where we give a general recipe for the construction of an Eulerian path. Consider first a graph with exactly two odd-degree vertices, which we will call s and f. Let us try to naively draw an Eulerian path, starting from s. We will travel randomly ... But this doesn't quite work ... We would be stuck at vertex f, with no way to "backtrack" and traverse the other edges. In this case, let us temporarily remove the edges that we have traveled. We are left with the subgraph ... Since the original graph was connected, the subgraph "intersected" some of the edges that we removed ... Now let us apply the "naive" algorithm to the subgraph ... So now we can perform "reconstructive surgery" on our original path and get an Eulerian path for the entire graph. ... This method will work in general. We may have to repeat the "remove the traveled edges and travel the subgraph" step several times ... but since the graph is finite, eventually we will finish. pg.124-125, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2159

          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

          Examination of cases      One of the methods of proof is examination of cases, for example, considering odd and even separately, as in a proof of 1+2+3+...+ n = n(n+1)/2.1640

          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

               Repeatedly folding paper     

  • If a rectangular paper is folded in half, and half again, and yet again, and so on, and they are all considered repeatedly applied transformations of the same whole, then that is fractal multiplication, "recopying the whole", as with your paper snowflake.
  • If a paper is folded once, and then again, and again, but those actions are thought as taking place separately, and especially, if I'm focused on the labelled components (rather than the repeating whole), then it is label multiplication, "redistributing the multiple", as with your pie halves sliced in five slices each.
1765

        Empathize - Implicitly Understand

Empathize - Implicitly Understand      171

          Empathy Map      develop a customer profile 316

          Fishbowl      observe conversational dynamics 317

          Give-and-Take Matrix      map out actors' motivations, interactions 315

          Pain-Gain Map      understand a person's motivations and decisions 314

          Show and Tell      elicit stakeholders' views 318

          Show Me Your Values      understand employee's perceptions of what values are involved 319

          Stakeholder Analysis      identify stakeholders and how to engage them 320

        How does it seem to me?

How does it seem to me?      Tracking the whole, introspecting. Imagine how people see things based on their circumstances.53

     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

          Imagine another person's mind      Conversing with: mind In tutoring, I would often understand my students by trying to model their mind, and so imagine the obstacles that were keeping them from solving a problem.692

          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

          Learn from other's mistakes      Conversing with: menaces Noting other people's failings and failures, I took them as confirmation not to use drugs, not to manipulate other people or think of us as different. I appreciated the privileges of my life, my opportunities and my need to apply myself and my virtues.765

     Feeling out my own mind as a mind-in-general      Conversing with: mind Conversing with: mind549

          Feel through a specific case      Conversing with: characteristicness The truth of the world follows from the truth of the heart, but not the other way around.658

          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

          Tracking my thinking      Conversing with: introspection In watching my own mind and noting how I look at different cases I see that the direction of the twosome changes upon reflection. Our mind shifts readily from "opposites coexist" to "all things are the same" and not the other way around. We shift from same to different because "same" implies opposites (for things to be the same they must also be different) whereas "different" doesn't (different things are just different). But if we think about the words "different" and "same", and thereby distance ourselves from what they mean to our minds, as we so often do, then it seems that different involves opposites ("opposites coexist"), and same does not ("all things are the same").611

        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

          Serving people of a different background      My patients' lives were a window into a slice of American life I had never known — sharecroppers, wooden shacks on dusty backroads, back-breaking cotton picking for pennies a pound. Towns whose names littered civil rights history — Philadelphia, McComb, Indianola, Yahoo City, Little Rock, Montgomery, Birmingham. Life under Jim Crow. "Yes, suh. No, suh." The humiliation of survival in places where being black meant no chance for justice. The Illinois Central ride to Chicago. The promise of jobs. The disappointment of segregation and the urban violence that greeted them. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.19341944

          What is most enjoyable for viewers      Fresh Air interview of J.J. Abrams: it's literally wanting people to have a good time and to have a little bit of a surprising time. So whenever I'm trying to keep things quiet, it is 100 percent an effort to make the experience of actually seeing the movie or TV show more enjoyable for the viewers."1871

          Who had an impact on my growth?      I sometimes measure my life progress by thinking of the people who have had an impact on my growth as a human being. My parents, my wife, my children, my friends and colleagues. I number my patients among them. Health. Chicago. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved.1933

        The Decision/Event Tree

The Decision/Event Tree      A decision/event tree is a diagram that graphically shows choices and their outcomes at different junctures in alternative sequences or chains of events. Each sequence or chain of events is a separate scenario.

  • The branches of the tree are mutually exclusive...
  • The branches are collectively exhaustive...
[The decision/event tree]:
  • dissects a scenario into its sequential events.
  • shows clearly the cause-and-effect linkages...
  • shows which decisions or events are dependent on others.
  • shows where the linkages are strongest and weakest.
  • enables us to visually compare how one scenario differs from another
  • ...reveals alternatives we might not otherwise perceive and enables us to analyze them - separately, systematically, and sufficiently.
A decision/event tree displayed on a screen for all [to see is a way of structuring a] discussion to ensure that all options are given due consideration... consider each option separately and systematically and to narrow the choices by visually eliminating those that didn't meet the needs...
A tree is preferable to a matrix when there are unfolding scenarios. A matrix is preferable to a tree when there is a recurring structure, an internal structure, or when we're analyzing correlation between independent dimensions. The Thinker's Toolkit by Morgan D.Jones1772

          The Probability Tree      the most difficult analytic problems are of the random and indeterminate type, where, because facts are scarce, our analytic product depends heavily on subjective judgments. For that reason, there can be no certainty when dealing with random or indeterminate problems. ... Estimating is what we do when we run out of data. ... The two types of probability we encounter most often in analysis are mutually exclusive (the "or" type, where we add probabilities) and conditionally dependent (the "and" type, where we multiply them). ... There are three inviolable rules for constructing probability trees:

  • As with any decision/event tree, the events depicted must be mutually exclusive, meaning each event is distinct from the others.
  • Likewise, the events must be collectively exhaustive, meaning they must include all possible events in the scenario being analyzed.
  • The probabilities of the branches at each node ... must equal 1.0
The Thinker's Toolkit by Morgan D.Jones1776

          The Utility Tree      utility is the benefit that someone has received, is receiving, or expects to receive from some situation. ... It is the reason why that person has taken, is taking, or will take a certain action. ... humans, when confronted with a choice between alternative courses of action (alternative options), choose the course that offers them the greatest utility, and the person making the choice defines what that utility is. ... we call it self-interest. ... The purpose of utility analysis is to rank any number of options according to how they serve the decision maker's self-interest.

  • Construct a decision/event tree for each option [alternative course of action].
  • Identify the perspective of the utility analysis.
  • Assign a utility value to each option-outcome combination - each branch (scenario) of the tree - by asking the Utility Question: If we select this option, and this outcome occurs, what is the utility from the perspective of ...?
  • Assign a probability to each outcome. Determine or estimate this probability by asking the Probability Question: If this option is selected, what is the probability this outcome will occur? The probabilities of all outcomes for a single option must add up to 1.0
  • Determine the expected values by multiplying each utility by its probability and then adding the expected values for each option.
  • Determine the ranking of the alternative option.
  • Perform a sanity check. Always separate the analysis (including the discussion) of utility and probability. ... even more challenging is analyzing (forecasting) the decisions of others ... the harder people work to achieve goals that are in their self-interest, the greater is the probability of their achieving these goals. The Thinker's Toolkit by Morgan D.Jones1777

        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

          Deducing people's motives from their actions      I have a godson in Lithuania and I would give him coins I collected in my travels, but his father would take them away and say that he would keep them until he got older, so he wouldn't lose them. In time I caught on that he was too keen on the coins himself to ever give them to his son so I resigned myself, but I always bring him some from my travels. I always knew that the main aim of the collection was to educate myself, but it was a bit sad that this education was not relevant to anybody else, at least not directly.1011

          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

          Assuming intersubjectivity      489

          Caring what loved ones believe      417

          Considering how others may see things      444

          Imagine a listener      396

          Imagining what others may think      493

          What others may like      414

          Your state of mind      490

        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 we seek first?      Matthew 6:31 "Therefore don't be anxious, saying, 'What will we eat?', 'What will we drink?' or, 'With what will we be clothed?' 6:32 For the Gentiles seek after all these things; for your heavenly Father knows that you need all these things. 6:33 But seek first God's Kingdom, and his righteousness; and all these things will be given to you as well. 1460

          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

          Least Common Multiple      ...we define the least common multiple, or LCM, of a and b to be the least positive integer which is a multiple of both a and b. pg.245 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2217

          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

               Bicycle gearing      Bicycle gears are given by the formula: (number of front teeth / number of back teeth) x circumference of wheel = distance traveled per revolution. That's an example of the proportional scaling ("rescaling the whole", like with projecting a teddy bear ). You could imagine starting out with two gears (the gear fixed to the wheel, and the gear you are pedaling) each the size of the back wheel. Replacing those giant gears with smaller gears is an action of magnifying or shrinking the distance traveled. You could add more gears to further magnify or shrink that ratio. Adding teeth to the gears is changing the units (and that aspect could be considered "rescaling the multiple", as with skip counting, but that is the correspondence of the number of teeth to the circumference of the gear, NOT the relationship between the two gears). The fact that multiplication is taking place at two different levels makes it challenging to think about because the gear ratio is "abstract" and not concretely, directly related to the size of the wheel. See also Dmitri's game!1763

          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

          Triangulating, then coloring      The walls of a museum gallery form a polygon with n sides, not necessarily regular or even convex. Guards are placed at fixed locations inside the gallery. Assuming that guards can turn their heads, but do not walk around, what is the minimum number of guards needed to assure that every inch of wall can be observed? ... A coloring reformulation comes to the rescue: Triangulate the gallery polygon. pg. 43, 61 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1512

        Get to Know - Implicitly Care

Get to Know - Implicitly Care      170

          Ice Breaker      introduce people as creative spirits 313

          Low-Tech Social Network      get to know each other 312

          Trading Cards      embrace each other as self-defined 311

        What else should I be doing?

What else should I be doing?      Discerning the whole, choosing a data set. Engage a dataset with a full variety of roles. 54

          Recognize that I have nothing better to do      Conversing with: causes Why do I want to know everything? Why am I organizing the kingdom of heaven? The most relevant answer that I have found is simply that I have nothing better to do.796

     Deriving a structure from a complete variety of examples      Conversing with: phenomenon 566

          Allowing for inconsistency      I considered the variety of prayer. I noticed that they have us think of God inconsistently, as one who has fated everything in advance, one who fixes and manages the situation, and one who can guide us spontaneously.1709

          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

          Choose a fruitful data set      Conversing with: circumstances for a phenomenon In developing a theory, I often choose a dataset of examples that makes vivid the phenomenon I am looking for and represents the complete variety of possibility. I analyzed Jesus' feelings and expectations by studying episodes from the Gospel of Mark because it was the most emotional of the Gospel. I studied the content of Jesus' parables in the Gospel of Luke. I studied what is "good" in the Gospel of Matthew. I tried to chase down Jesus' algebra of expressions to understand his "I am..." statements in the Gospel of John. I studied Lithuanian folk tales for my narrative theory because they are old, engaging, consistent and pure. I studied how God is imagined in the first 40 psalms because I was considering topologies as the variety of ways that we can imagine and I thought of God as the most generic yet intimate of concepts. I studied Lakoff and Johnson's target spaces from Metaphors We Live By because they had noted an important phenomenon and their many examples seemed to capture the variety of possibility.657

          Collect ways of looking at something      Conversing with: variety inherent in a phenomenon I developed and conducted good will exercises and noted different tests for distinguishing the truth of the heart and the truth of the world. In general, three of the tests agreed with each other. I think the fourth and most important test, whether the person confuses the truths of the heart and the world, indicates whether the person has resolved the conflict or not. Similarly, I noted the different properties of everything, and more generally, the divisions of everything along with their various representations.660

          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

          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

     Manifesting the complete variety that a structure allows for      Conversing with: wholeness 559

          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

          Applying structural possibilities     *** What can structure mean? Conversing with: unfolding I studied the statements and truths that arose in developing good will exercises. I noted that given the four questions Why? How? What? Whether?, the heart would address a broader question than the world, with Why being the broadest question. This allowed me to categorize the exercises in terms of the pairs of levels, and indeed, they related to the counterquestions. This left in each case a pair of levels not involved in the counterquestion. I realized that they were relevant to the doubts which the counterquestions addressed. And the order in which they were taken up determined whether they led to positive or negative feelings.681

          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

        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

        Divergent Thinking

Divergent Thinking      Most of us were not endowed with an inherent ability to think divergently. ... Why is opening up such a struggle? ... we humans are the fastest criticizers and ridiculers on the planet ... Negative thoughts flood our minds and overwhelm those tiny rivulets of positive notions. The foremost challenge in analyzing problems is to think divergently, not just at crucial points in our analysis but at every stage of the process. ... I regard as unnecessary overkill the wild, frenzied brainstorming methods that some people advocate ... Nevertheless, if blitzkrieg methods suit your purpose, use them. ... I have witnessed many a brainstorming session in which intelligent people go through the motions of generating ideas but treat the entire process as kind of a game. ... To analyze a problem effectively, we need to be open to ideas, at any time and from any source, from the very outset to the very end of the analytic process. ... 1) The more ideas, the better. ... 2) Build one idea upon another. The power of brainstorming lies in the freedom and spontaneity of the process: one idea spawns another and another and another in rapid succession. Participants are encouraged to say aloud that the idea one is suggesting "builds on" someone else's idea. This statement, which rewards and flatters that someone else and reaffirms the connectivity of ideas, adds momentum to the process. ... 3) Wacky ideas are okay. ... open the way to new, practical ideas ... incite humor, which relaxes our paranoid grip on the mind's generative capabilities and liberates more ideas. This commandment, however, bothers most people ... What troubles people is not that an idea is silly or foolish but that they will feel and look silly and foolish suggesting it. ... 4) Don't evaluate ideas. Neither yours nor especially someone else's. This rule liberates people from their self-imposed restraints in generating ideas. ... In brainstorming, the practicality of ideas is irrelevant. ... When we've finished diverging (exhausted our reservoir of ideas), the next step is to focus by winnowing out the impractical stuff and clustering ideas that are similar. We are now in a convergent mode. [In his example, he came up with six clusters and a seventh cluster of impractical ideas. I think I could structure his six clusters.] ... In the third step, we select those ideas that are intuitively practical and promising. ... a further useful step one can take is to perform PROs-CONs-and-FIXes on each of the selected ideas to identify their strengths and weakness and to evaluate their feasibility. This step also provides an opportunity to revise the ideas, enhancing and fine-tuning them. ... Frequently, when I'm discussing a problem with someone, I'll switch to a divergent mode. But I know that if I don't tell the person I'm switching, he or she will get upset, thinking I'm either off on some wild tangent or that I'm really serious about some idea I've raised for discussion when, in fact, I'm only testing the analytic waters. So I always tell people when I'm brainstorming ... The Thinker's Toolkit by Morgan D.Jones1768

        Audit

Audit      What else should I be doing? Audit. The calm Servant provides the rich Served with a budget, alternatives, substitutions.1090

          Recognizing what I spend my resources on      When I was fifteen I visited Lithuania for a summer and I would go to the dollar store and the lady there would exchange my US money for coins from all sorts of countries, I was very excited about that. I even brought her a flower. Later it seemed kind of ridiculous how I had cared so much about those coins.1041

        Play our part

Play our part      134

          Considering who is responsible      418

          Emulating      426

          Learning by teaching      488

          Learning each role      484

          Learning every role      427

          Mastering a language      474

          Taking up a responsibility      382

     Our Duties      110

          Considering our duties      508

          Considering what we need worry about      514

        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 does one love more      Matthew 10:37 He who loves father or mother more than me is not worthy of me; and he who loves son or daughter more than me isn't worthy of me. 10:38 He who doesn't take his cross and follow after me, isn't worthy of me. 1814

          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

        Total order

Total order      Imply model (can order procedures) (Strong induction, decision making, total ranking, integers) 21 Canon: Sequence => Network (for determining priorities)70

               Well-ordering theorem      Wikipedia: Axiom 9: For any set X, there is a binary relation R which well-orders X. This means R is a linear order on X such that every nonempty subset of X has a member which is minimal under R. ... Given Zermelo-Fraenkel axioms 1-8, there are many statements provably equivalent to axiom 9, the best known of which is the axiom of choice (AC), which goes as follows. Let X be a set whose members are all non-empty. Then there exists a function f from X to the union of the members of X, called a "choice function", such that for all Y element of X one has f(Y) element of Y. Since the existence of a choice function when X is a finite set is easily proved from axioms 1-8, AC only matters for certain infinite sets. AC is characterized as nonconstructive because it asserts the existence of a choice set but says nothing about how the choice set is to be "constructed." The Well-ordering theorem seems tightly related to the concept of a total order. It's interesting that it's related to the axiom of choice.1167

          Bucket elimination      Wikipedia: Bucket elimination is a satisfiability algorithm. It can be defined as a reformulation of adaptive consistency. Its definitions uses buckets, which are containers for constraint, each variable having an associated bucket. A constraint always belongs to the bucket of its highest variable. The bucket elimination algorithm proceeds from the highest to the lowest variable in turn. At each step, the constraints in the buckets of this variable x_i are considered. By definition, these constraints only involve variables that are lower than x_i. The algorithm modifies the constraint between these lower variables (if any, otherwise it creates a new one). In particular, it enforces their values to be extendible to x_i consistently with the constraints in the bucket of x_i. This new constraint, if any, is then placed in the appropriate bucket. Since this constraint only involves variables that are lower than x_i, it is added to a bucket of a variable that is lower than x_i. This algorithm is equivalent to enforcing adaptive consistency. Since they both enforce consistency of a variable with all its parents, and since no new constraint is added after a variable is considered, what results is an instance that can be solved without backtracking. Since the graph of the instance they produce is a subgraph of the induced graph, if the induced width is bounded by a constant the generated instance is of size polynomial in the size of the original instance. As a result, if the induced width of an instance is bounded by a constant, solving it can be done in polynomial time by the two algorithms.938

          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

          Permutations      Permutations are the ways of reordering a collection of objects, all distinct. A permutation labels a total order. 2199

          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

               Cutting a stack of cheese slices      If I have a stack of 10 slices of cheese and I slice them all in two, then:

  • if I'm simply changing the units, so that I'm thinking now of 20 small slices rather than 10 large slices, so that 1 large slice = 2 small slices, then I'm skip counting, "rescaling the multiple".
  • if I'm thinking of each large slice as consisting of a left piece and a right piece, (a first piece and a second piece, distinguishable or "labelled" with a child's ID 10x2), then I'm "redistributing the multiple", as with "sets, per each".
1764

        Prioritize - Implicitly Commit

Prioritize - Implicitly Commit      169

          $100 Test      assign relative value 307

          20/20 Vision      agree how to rank initiatives 304

          Build the Checklist      order tasks or rank tasks 309

          Dot Voting      converge on ideas which capture most interest 306

          Forced Ranking      agree on priorities 305

          Impact and Effort Matrix      ground commitments 308

          Pro/Con List      identify key pros and cons to weight for decision 303

          Value Mapping      get feedback on what features are important 310

        Would it make any difference?

Would it make any difference?      Discerning the parts, contrasting dimensions. Compare and contrast what aspects are most important. Tease out contradictions.55

     Comparing instances of a structure      Conversing with: example 551

          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

               Properties of living systems      I compared various structures I thought were related to properties of life, of living systems, expressing them from different points of view.1664

          Compatibility with structures I know      In Scripture I sometimes find references to images and numbers that bring to mind the conceptual structures that I have been documenting. I noticed how the seven days of creation could mean events for God, thus the seven divisions of everything, which they match in number. And so I think of creation as an operation +1 of reflection. And I notice with interest that the creations of the first three days are governed by the creations of the next three days, as noted by bishop Skvireckas in his notes to his translation of the Bible into Lithuanian. Similarly, I notice that Ezekiel's chariot of God is carried by four creatures, like the four representations of the nullsome, or the four representations of everything. I notice that there are 24 elders in Revelations. Such coincidences spark my mind and encourage me to think that I may be on track, overall.1954

          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

          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

     Contrasting possibilities within a system      Conversing with: opposing view 544

          Appreciate a person's weakness      Conversing with: obstacles I once asked a colleague what their deepest value was and, when they told me, I was taken aback, because I thought it personally happened to be their weak point. As I thought about myself and others, I decided that a person's deepest value is both their strong point and their weak point.751

          Contrast dimensions     *** What are the components? Conversing with: tension in possibilities In analyzing emotional responses, I noted three dimensions: positive vs. negative, calm vs. riled, sensitive vs. insensitive. I later saw how these were relevant in the good will exercises. In analyzing the counterquestions, I saw how they related a perspective with a situation.669

          Finding the obstacle      Conversing with: weaknesses Recognize in the three-cycle (take a stand, follow through, reflect) where am I stuck and thus feel riled.678

          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

          Spanning the obstacle      Conversing with: strengths In designing good will exercises, I expected them to consist of three parts - taking a stand, following through, and reflecting - that we might assimilate the truth of the heart. I realized, from a particular example, and from contemplating the general situation, that we must be having difficulty with one of those three parts. Therefore, we should place that one in the middle of the exercise, so that the exercise might help us overcome that obstacle by spanning it from and to what we knew we could do.679

        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

          How one concept misrepresents another      Fresh Air interview of doctor David Ansell, June 15, 2001: Working at County, Ansell says, made him realize just how much the current payment system drives health care inequalities. "There's a misunderstanding that if you just go to the [emergency room], that's health care," he says. "It's not. ... And I don't think the public or politicians really understand that. Health. Chicago.1931

          Surveys of different populations      Many years later, colleagues of mine conducted door-to-door health surveys in Chicago's poorest neighborhoods. More than a third of those surveyed had higher rates of depression, asthma, hypertension and smoking than those in white communities. "County: Life, Death and Politics at Chicago's Public Hospital" by David Ansell M.D. Copyright 2011, David Ansell M.D. Published by Academy Chicago Publishers. All Rights Reserved. Health. Chicago.19341941

        Pairwise Ranking

Pairwise Ranking      The paramount feature of weighted ranking is its method of ranking each item against each other. ... We thus systematically compare each item with every other item. ... If, by chance, two items end up with the same number of marks because our analysis was inconsistent, we simply rank these two items head to head to break the tie. ... It doesn't permit us to take analytic shortcuts that shortchange our analysis. We must analyze and make a decision on each member of every pair. ... Ranking things in pairs is something the mind can do easily, quickly, and effectively. The mind does love to focus. ... Pair ranking also provides reliability. ... What happens is that the process delivers the rankings. ... As you practice pair ranking, you'll discover that sometimes you can quickly determine with certainty which item is going to be first or which last. When you do, make a note of it and don't include that item anymore as you rank the others. The Thinker's Toolkit by Morgan D.Jones1773

          Instinctive ranking      Now rank these fifteen movies from best to worst according to your personal likes and dislikes. ... The main defect in our instinctive method lies in our tendency ... to view problems one-dimensionally ... to focus on (glom on to) the first solution that makes sense. Thus, the moment we think of a reason - any sound, persuasive reason - to like or dislike one of the movies on our list, we tend to latch on to that reason, make our ranking decision on that basis, and move on without considering our other likes and dislikes, which may actually be more important to us. ... It is well and good to use stringent criteria in making decisions, but we should be sure we have considered all relevant criteria and have applied all of the criteria equitably to all of the decisions. ... By applying different criteria to different ranking decisions, we distort our analysis, which can lead to gravely disappointing outcomes that later make us wonder what went wrong. ... There are two inherent weaknesses in our instinctive method. First, we tend to apply different criteria to the different items being ranked. Second, we tend to regard all of our criteria as having equal importance to us ... For that reason, rankings generated with our instinctive method tend to be inconsistent and are thus unreliable. ... your mind doesn't distinguish between vital and nonvital matters when it comes to ranking. The Thinker's Toolkit by Morgan D.Jones2140

          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

          PROs-CONs-and-FIXes      ...humans are compulsively negative... negative thoughts defeat creative object thinking... Negative thoughts can quickly overwhelm and preempt positive consideration. A powerful but simple technique for dealing with this problem is called PROs-CONs-and-FIXes. This rigorous technique compensates for negative thinking by forcing us to identify the positives first. Only then are we allowed to indulge joyously in the negatives. But the technique goes a step further by examining the negatives and trying to think of actions that could be taken to "fix" them, either converting them into positives or, if that isn't feasible, eliminating them altogether. Those negatives (cons) that can't be fixed represent the price one must pay, the burden one must bear, if the thing being evaluated is to be adopted or accepted.

  1. For each option
    1. List all the PROs
    2. List all the CONs
    3. Review and consolidate the CONs, merging and eliminating
    4. Neutralize as many CONs as possible
  2. Compare the PROs and unalterable CONs for all options.
The Thinker's Toolkit by Morgan D.Jones1767

          The Utility Matrix      A matrix offers two important advantages over a tree for performing utility analysis. First, the relative differences in utility values of outcomes are more easily perceived in a matrix than in a tree. Second, arithmetic calculations are easier to perform. This is due, in part, to the different configurations of the two structuring devices: a tree is busy, sort of sprawls out, and is sometimes unsymmetrical, while a matrix is a compact, tightly organized, symmetrical unit. Also, the focus of a utility matrix is squarely on alternative outcomes, while a tree portrays whole scenarios in which outcomes are only a part. Depicting scenarios - the strong suit of trees - tends to overshadow and divert attention from the outcomes. ... Because it facilitates arithmetic operations ... a matrix is more suitable than a tree for analyzing a problem from different perspectives and with different classes of outcomes.

  • Construct a utility matrix.
  • Identify the perspective of the utility analysis.
  • Assign a utility value to each option-outcome combination - each cell of the matrix - by asking the Utility Question: If we select this option, and this outcome occurs, what is the utility from the perspective of...?
  • Assign a probability to each outcome. Determine or estimate this probability by asking the Probability Question: If this option is selected, what is the probability this outcome will occur? The probabilities of all outcomes for a single option must add up to 1.0
  • Determine the expected values by multiplying each utility by its probability and then adding expected values for each option.
  • Determine the ranking of the alternative options.
  • Perform a sanity check.
The Thinker's Toolkit by Morgan D.Jones1778

          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

          Communicate what I may do if I don't get my way      When I came to study in Lithuania the university administration placed me in the dormitory for foreigners. This was in the Soviet days and that dormitory was practically empty and isolated from the local students. I told the housing authority that they give me a place in the regular dormitory or I would find somebody to trade places with me. That shook them up and so grudgingly they found a room for me in the regular dormitory with a student of Soviet history.1023

          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

          Comparing how things have been      485

          Comparing us      491

          Comparing what I give and get      501

          Considering what is the best we can do      467

          Disagreeing      453

          Knowing what is valued more than money      450

        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

          Can we prolong our lives?      Matthew 6:27 "Which of you, by being anxious, can add one moment to his lifespan? 1453

          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

          What is anxious for itself?      Matthew 6:34 Therefore don't be anxious for tomorrow, for tomorrow will be anxious for itself. Each day's own evil is sufficient.1464

        Powerset lattice

Powerset lattice      Vary implication (can satisfy various conditions) 32 Chronicle: Sequence => Hierarchy (for determining solutions) 71

               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

               Distributing with box mathematics      We can distribute 23 x 15 using grid paper as (20 + 3) x (10 + 5). We can change the tens to hundreds or thousands or millions. Likewise we can distribute (2X + 3) x (X + 5). Compare with two digit multiplication. Gospel Math. 1848

          Box multiplication: Redistributing the set      A set can be redistributed. A set is anything which can be thought as multiple units, where each element is distinct. A set can be the rows of a chessboard or an array. It can be the breakdown of a length, for example, 2 feet and 1/2 foot. We multiply the set by applying the distributive law and multiplying each element of the set separately. And that multiplication can be noncommutative, which is to say, we can make sure that the element is followed by the action. Such multiplication typically looks like a set matched with another set, yielding "box multiplication", as when we multiply (3 feet + 1/4 foot)(2 feet + 1/2 foot) and get 4 products which we then sum up. This is also the standard computation of multiplication, for example, multiplying 2 digit numbers together. Multiplication thus gives the ways of matching units, multiple units times multiple units, as in box multiplication, accounting for all possibilities. Units times units means that conditions are satisfied, thus generating all of the solutions. Here the units are all known, they form sets, thus there are distinct terms in expressions with multiple units.1529

          Constraint satisfaction      Wikipedia: In artificial intelligence and operations research, constraint satisfaction is the process of finding a solution to a set of constraints that impose conditions that the variables must satisfy. A solution is therefore a vector of variables that satisfies all constraints.939

          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

          Multiply in a particular order      Simplify .... We could multiply out all the terms, but it would take a long time, and we'd probably make a mistake. We need a strategy. If this expression is to simplify, we will probably be able to eliminate radicals. If we multiply any two terms, we can use the difference of two squares formula and get expressions which contain only one radical. pg.166 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2174

          Polya's pattern of two loci      899

               Illustrative example      Euclid's first problem in his Elements is: In drawing an equilateral triangle, given the first side AB, how do we draw the other two? The solution is: to draw a circle c(A) around A of length AB and to draw a circle c(B) around B of length AB. The third point C of the equilateral triangle will be at a point where the two circles intersect. (There are two such points, above and below the line segment.) Polya notes that this solution is a particular example of a general pattern of "two locii", which is to say, we can often find a desired point by imagining it as the intersection of two curves. I note further that each curve may be thought of as a condition (X="points within a distance AB of A", Y="points within a distance AB of B"). The solution created four regions:

  • Solutions to both X and Y.
  • Solutions to X.
  • Solutions to Y.
  • Solutions to the empty set of conditions.
The solver's thought process leveraged a deep math structure: the powerset lattice of conditions: {{X,Y}, {X}, {Y}, {}}. The solver envisaged the solution as the union of two conditions. In this deep structure, there is no reference to triangles, circles, lengths, continuity or the plane, all of which turn out to be of superficial importance. Here the crux, the mental challenge of the problem, is expressed exactly by the powerset lattice. And, notably, that is a mathematical structure! Math is the deep structure of math! 900

          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

     Creative rethinking      Many creative methods have us rethink, reorganize, untangle, recombine the conditions that define a problem.909

               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

        Ground - Explicitly Care

Ground - Explicitly Care      168

          Context Map      show effects of context 302

          The 5 Whys      discover the root cause 300

          The Blind Side      acknowledge unknowns 301

        What do I have control over?

What do I have control over?      Focusing on structure, appreciating its purpose. Relate a structure with its context by making its purpose explicit. 56

     The purpose of my circumstances      Conversing with: possibility 562

          Address the causes of my bad mood      Conversing with: responsibility I wondered about the causes of my moods and concluded that sometimes they have no reason, and perhaps that is their purpose, to kick us out of our usual tracks.758

          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

          Distinguish what I have control over      Conversing with: irresponsibility As a child, I was a bit afraid of plants. My mother had a beautiful yard with many plants, including a section with dense ivy and large throated jack-of-pulpits. I was suspicious of plants because they seemed very patient, which could be a ruse. I didn't want to allay my suspicions simply based on what other people did or thought. Yet I realized that, if indeed plants might be malevolent, it was beyond my capabilities to establish that, and so I should accept what people thought.720

          Entertain a doubt      Conversing with: possibility As I child, I found plants to be suspicious, because of their possibly great patience, thus unpredictable and dangerous.756

          Moral imagination      Conversing with: how things should be I imagined what the end of the world must be like, the making tangible of the Holy Spirit, the return of Christ as the identity of all. As a child, I imagined what death must be like, a slowing down of time to allow for a review and untangling of all of our life.754

          What would others want me to do for us all?      As a child, I appreciated that I could have been born anybody. How could I serve us all? What would others want me to do? Even though I could not ask them, yet I understood that I could imagine what they would have me do, but especially, if they understood my situation. Ultimately, I decided that the best thing I could do would be to know everything and apply that knowledge usefully. As I did that, I kept mindful of others around the world, thinking how my thoughts might be fruitful for them, imagining them rooting for me, and even allowing me to get some rest and relaxation. As an adult, I have even met such people and we've actually helped each other.1301

     A structure's purpose      Conversing with: answer 554

          Applying a structure      Conversing with: applicability I knew that my mind could encompass six perspectives, but not seven or eight, as in the Lord's prayer, and this helped me figure out how to listen to God. I realized that if I tried to contemplate each line of the Lord's prayer, then this would overload my mind, flatten me out, and indeed I would find myself before God, as if in my world there was a rift that opened up above me.621

          Augmenting a structure with its purpose     *** What is the significance of a structure? Conversing with: meaningfulness I had noticed that the good will exercises could be organized into six groups based on the pairs of questions that they asked from whether? what? how? why?, the heart asking the broader question, with why? being the broadest. Then I noticed, starting with a few examples, that the questions could be answered yes or no, and that the heart would answer one way, and the world another way. Given a doubt, Is this truly wrong?, and the counterquestion, Is this the ways things should be?, the world says, "It is wrong, it s