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A system of deep structure in solving math problems

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  • Andrius Kulikauskas
    Hi Pamela, and all who love math, I share my theory to explain all of math! I appreciate our thoughts. Andrius, ms@ms.lt ... Paul Zeitz, I share with you my
    Message 1 of 1 , Apr 1, 2011
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      Hi Pamela, and all who love math, I share my theory to explain all of
      math! I appreciate our thoughts. Andrius, ms@...

      Paul Zeitz,

      I share with you my thoughts on the varieties of "deep structure" in
      mathematical "frames of mind". Your book "The Art and Craft of Problem
      Solving" has been profoundly helpful. I also share with Joanne Simpson
      Groaney ("Mathematics in Daily Life"), Alan Schoenfeld ("Learning to
      Think Mathematically..."), John Mason ("Thinking Mathematically"),
      Manuel Santos, and also Maria Druojkova (naturalmath.com) and the Math
      Future online group where I am active.

      I have been looking for the "deep ideas" in mathematics. George Polya's
      book "Mathematical Discovery" documents four patterns (Two Loci, the
      Cartesian pattern, recursion, superposition) of the kind I'm looking for
      (and which bring to mind architect Christopher Alexander's pattern
      languages). Your book documents dozens more. I've found Joanne
      Groaney's book helpful and I think the other writings I mention will
      also be in this regard.

      You note in your "planet problem", pg.63, that "on the surface" it is a
      nasty geometrical problem but "at its core" it is an elegant logical
      problem. This distinction brings to mind linguist Noah Chomsky's
      distinction between the surface structure and the deep structure of a
      sentence. In general, what might that deep structure look like? George
      Polya ends his discussion of the pattern of "superposition" or "linear
      combination" to say that it imposes a vector space. In an example he
      gives, the problem of "finding a polynomial curve that interpolates N
      points in the plane" is solved by "discovering a set of particular
      solutions which are a basis for a vector space of linear combinations of
      them". The surface problem has a deep solution, and the deep solution
      is a mathematical structure!

      In what follows, I discuss an illustrative example, I list 24 deep math
      structures, I consider how they form a system, and I sketch some future

      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!

      24 deep structures

      I list below 24 such deep structures which characterize the mathematical
      "frames of mind" by which we solve problems. I note in parentheses the
      related patterns, strategies, tactics, tools, ideas or problems. I have
      included every such that I have found in your book, as well as Polya's
      four patterns, "total order" and "weighted average" that I observed in
      Joanne Growney's book, and a few more that I know of. I preface each
      with a notation that I will reference later.

      A) Independent trials (Vary the trials, get your hands dirty, experiment
      with small numbers, collect scattered solutions, mental toughness,
      accumulate some data points, don't get hooked with one method, restate
      what you have formulated, apply what worked to new domains, add a little
      bit of noise)

      B1) Center (Blank sheet, what is so central that it is often left
      unsaid, origin of a coordinate system, natural or clever point of view,
      symmetrize an equation, average principle, choice of notation,
      convenient notation)
      B2) Balance (Parity, Z2: affirm-reject, multiplication by one, addition
      of zero, union with empty set, expansion around center)
      B3) Polynomials (Or, And, method of undetermined coefficients,
      expansion, construction)
      B4) Vector space (Superposition, linear combination, duality)

      C1) Sequence (Induction)
      C2) Poset with maximal or minimal elements (Extreme principle,
      squarishness, critical points - maximum, minimum, inflection, extremum
      C3) Least upper bounds, greatest lower bounds (Monovariants, algorithmic
      proof, optimization problem, world records: minimal times to beat keep
      C4) Limits (Taking a limit, boxing in or out, repeated bisection,
      derivative, diagonalization)

      T) Extend the domain (Eulerian math: Apply calculus ideas to discrete
      problems. Stitch together different systems. Define a function. Think
      outside of the box, outside of the Flatland. Generalize the scope of the
      F) Continuity (Vary the variable, existence of a solution, balancing
      point, appeal to physical intuition)
      R) Self-superimposed sequence (Recurrence relation as an automata,
      auto-associative memory of neurons as in Jeff Hawkins' "On
      Intelligence", generating function, telescoping tool, shift operator)

      C=B Symmetry group (Symmetry, invariant)

      0 Truth (Argument by contradiction, paradox of self-reference)
      1 Model (Wishful thinking, solve easier version, note familiar tools and
      concepts, reuse familiar solutions)
      2 Implication (Identify hypothesis and conclusion, penultimate step,
      work backwards, contrapositive)
      3 Variable (Classify the problem, is it similar to others, draw a
      picture, mental peripheral vision, without loss of generality)

      10 Tree of variations (Weighted averages, moves in games)
      20 Adjacency graph (Connectedness, coloring, triangulation of polygon)
      21 Total order (Strong induction, decision making, total ranking, integers)
      32 Powerset lattice (Polya's pattern of two loci, creativity: two monks,
      two ropes)
      31 Decomposition (Pigeonhole principle, partitions, factorizations,
      encoding, full range of outputs, principle of inclusion-exclusion)
      30 Directed graph (With or without cycles)

      O Context (Read the problem carefully, change the context, bend the
      rules, don't impose artificial rules, loosen up, relax the rules,


      I note that some problems and some concepts involve the application of
      two or more such deep structures. For example, the principle of
      inclusion-exclusion is equivalent to reorganizing (1-1)**N, where I
      imagine that multiplying out is Decomposition and canceling out is
      Balance (Parity). Or the "guards needed for a polygonal art gallery"
      problem I suppose involves triangulating the polygonal (creating an
      adjacency graph), coloring the vertices (so that no two colors are
      alike, thus parity) using three colors (total order distinguishing 3
      elements) and observing that (bijection) each vertex views the entire
      triangle (a consequence perhaps of squarishness and continuity).

      The deep structures above are the building blocks (and operations!?) of
      a grammar. The list above encourages me to believe that mathematical
      thinking, and indeed, all of mathematical theory and practice, may very
      well be expressed by such a grammar of what goes on in our minds!

      A system

      I organized the list by matching deep structures with "ways of figuring
      things out" that I have been collecting. I have noted about 200 ways
      that I have figured things out in my life ( http://www.selflearners.net
      ) and my quest to know everything (
      http://www.youtube.com/watch?v=ArN-YbPlf8M ). I have grouped them into
      24 "rooms" of a "house of knowledge": http://www.selflearners.net/ways/
      I have likewise grouped 90 Gamestorming business innovation games (
      http://www.gogamestorm.com/?p=536 ) and 148 ways that choir director Dee
      Guyton has figured things out in life, faith and music:
      http://www.selflearners.net/Notes/DeeGuyton Below, I discuss the math
      structures in groups, and briefly mention how they relate to "figuring
      thing out" in our lives. I treasure your discussion of Eulerian
      mathematics and, should I speculate too much, I ask your indulgence, as
      you write: "we have been deliberately cavalier about rigor... 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 a person's mathematical education"

      A) Independent trials
      We may think of our mind as "blank sheets", as many as we might need for
      our work. We shouldn't get stuck, but keep trying something new, if
      necessary, keep getting out a blank sheet. We can work separately on
      different parts of a problem. This relates also to independent events
      (in probability), independent runs (in automata theory) and independent
      dimensions (in vector spaces). If something works well, then we should
      try it out in a different domain. Sarunas Raudys notes that we must add
      a bit of noise so that we don't overlearn. Analogously, in real life,
      avoid evil, avoid futility.

      B1) Center B2) Balance B3) Polynomials B4) Vector 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". In real life, analogously, we
      discard the inessential 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.

      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". In real life,
      analogously, we can open our mind to all thoughts, we can collect and
      sort them by way of values, we can push ourselves to our personal
      limitations, and we can allow for an ideal person (such as Jesus) who
      transcends our limitations. These four frames are: caring; caring about
      caring; caring about caring about caring; caring about caring about
      caring about caring.

      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. 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. Next, 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.
      Finally, we can formulate what we have learned in general. We do this
      by considering a local constraint on values as a recurrence relation (on
      values a1, a2, ..., aN) and then superimposing the resulting sequence
      upon itself, as with a generating function, yielding a global
      relationship of the function with itself. This brings to mind the
      auto-associative memory that Jeff Hawkins discusses in his book "On
      Intelligence", where cortical columns use time-delay to relate patterns
      to themselves. If the model holds, then it can be tested further. This
      automata is the hand that makes the stitch. 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.

      C=B) 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.

      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.
      * Truth: Whether it is true? The two sheets may be conflated in which
      case we may interpret the problem as statements that we ourselves are
      making which may be true or false and potentially self-referential.
      Together they allow for proofs-by-contradiction where true and false are
      kept distinct in the level, whereas the metalevel is in a state of
      contradiction where all statements are both true and false. In my
      thinking, contradiction is the norm (the Godly all-things-are-true) and
      non-contradiction is a very special case that takes great effort, like
      segregating matter and anti-matter. Deep structure "solution spaces"
      allow us, as with Euclid's equilateral triangle, to step away from the
      "solution" and consider the candidate solutions, indeed, the failed
      * Model: What is true? The metalevel may simplify the problem at the
      level. Such a relationship may develop over stages of "wishful
      thinking" so that the metalevel illustrates the core of the problem.
      Ultimately, the metalevel gives the solution's deep structure and the
      level gives the problem's surface structure.
      * Implication: How is it true? The metalevel may relate to the level as
      cause and effect by way of a flow of implications. The metalevel has us
      solve the problem, typically by working backwards. The level presents
      the solution, arguing forwards.
      * Variable: Why is it true? The metalevel and the level may be distinct
      in the mind. Given the four levels (why, how, what, whether), the
      metalevel is associated with the wider point of view (why being the
      widest) and the level with a narrower point of view. We may think of
      them concretely in terms of the types of signs: symbol, index, icon,
      thing. The pairs of four levels are six ways to characterize the
      relationship. I believe that each way manifests itself through the
      relationship that we suppose for our variables: dependent vs.
      independent, known vs. unknown, given vs. arbitrary, fixed vs. varying,
      concrete vs. abstract, defined vs. undefined and so on. I need to study
      the variety that variables can express. I suppose that, mentally, the
      varying variables are active in both levels, whereas the fixed variables
      are taken to be in the level. The levels become apparent when, for
      example, we draw a picture because that distinguishes the aspects of our
      problem that our iconic or indexical or symbolic. Likewise, our mental
      peripheral vision picks up on aspects specific to a particular level.
      Analogously, 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):
      * 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. (Compare with Truth).
      * 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. (Compare
      with Model).
      * 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. (Compare with Implication).
      * 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. (Compare with Variable).

      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:
      10 Tree of variations: Model truth (can distinguish possibilities)
      20 Adjacency graph: Imply truth (can determine connectedness)
      21 Total order: Imply model (can order procedures)
      32 Powerset lattice: Vary implication (can satisfy various conditions)
      31 Decomposition: Vary model (can variously combine factors)
      30 Directed graph: Vary truth (can add or remove circular behavior)
      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:
      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
      Analogously, 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?

      O Context
      If you read the problem carefully, if you understand and follow the
      rules, then you can also relax them, bend them. You can thus realize
      which rules you imposed without cause. You can also change or
      reinterpret the context. These are the holes in the cloth that the
      needle makes. I often ask my new students, what is 10+4? When they say
      it is 14, then I tell them it is 2. I ask them why is it 2? and then I
      explain that it's because I'm talking about a 12-hour clock. This
      example shows the power of context so that we probably can't write down
      all of the context even if we were to know it all. We can just hope and
      presume that others are like us and can figure it out just as we do.
      Analogously, in real life, it's vital to obey God, or rather, to
      make ourselves obedient to God. (Or if not God, then our parents, those
      who love us more than we love ourselves, who want us to be alive,
      sensitive, responsive more than we ourselves do.) If we are able to
      obey, then we are able to imagine God's point of view and even make
      sense of it.

      Here's a link to my notes where I worked on the above:

      Implications in math

      Paul, I'm very excited to be able to think this way. I think I've
      suggested a framework that allows us to work with deep structures which
      express our mathematical thinking. These structures are to me very
      real. I think they do communicate the very real strategies, tactics,
      tools that you encompass with your book. Amazingly, these structures
      are all mathematical. This means that the surface problems we develop
      in math actually derive from and mirror the solutions already deep
      within us. Those solutions are supremely basic and pure as I've
      cataloged above. They likely ground all of math. They show that math
      unfolds from basic albeit deep notions. They make clear how math
      problems can be "classics" (memorably illustrating deep structures) or
      "junk food" (contrivances that destroy intuition). This framework
      suggests that we can analyze and foster the sense of beauty that guides

      Paul, I'm grateful for your decades of work. I'm glad that I can write
      to you and others as well. I share some further steps that call out for
      us to take.

      * We can collect, analyze and catalog thousands of math problems.
      * We can thus make and test hypotheses, even more so as we get feedback
      from others on how they like various problems.
      * We can work out the grammar of the deep structure. We can analyze the
      great mathematical discoveries. We can interview living mathematicians
      to learn how they think and try to model that. We can develop a
      universal method for solving math problems.
      * We should be able to construct, derive all mathematical objects from
      the deep structures. For example, you give a beautiful geometric proof
      of the fact that the arithmetic mean is greater than the geometric mean
      (pg.194) which suggests to me that: C2 (The Extreme principle) => most
      simply illustrated by the maximum of the quadratic (and key for area) =>
      "squarishness" (square is the most efficient rectangle) => half a
      rectangle is a right triangle => a right triangle is two copies of
      itself => the altitude A of the right triangle divides the hypotenuse C
      into X and Y and is their geometric mean => the possible right triangles
      with hypotenuse C draw out a folded circle with radius that is the
      average of X and Y. So this suggests a genealogy: square/rectangles =>
      right triangle => subdivided right triangle; folded circle => circle
      with center (when X=Y=A).
      * We can consider the methods of proof, which are I think distinct from
      the methods of discovery. I think there are six methods of proof and I
      hypothesize that they have us vary our trials between two sheets, namely
      at the gaps that the system leaves for God:
      ** A -> TFR: morphism (bridging from old domain to new domain)
      ** A -> C1: induction (initial case vs. subsequent cases)
      ** A -> C4: construction by algorithm (limit vs. members)
      ** A -> B=C: substitution (plug-in one system into another)
      ** A -> B1: examination of cases (separate sheets)
      ** A -> B4: construction (point becomes new center)
      * We can apply the system to try to solve some of the great outstanding
      problems, such as the Millenium problems.
      * We can study games, simple and complicated, in terms of the deep
      structures. What is fun about each of them? We can study chess.
      * We can involve all of the structures in a "game of math" which may
      have us shift back and forth between the deep structures and concrete
      problems that express them.
      * We can express the system and play the game with all manner of
      creative arts.
      * We can consider where math ideas come up in other disciplines. For
      example, the Gamestorming games involve ranking priorities, mapping
      adjacencies, sorting ideas and other relationships that helped me think
      through the system above.
      * We can develop a language for talking about such a game, a language
      that may ultimately help us talk by analogy about our daily lives, just
      as concepts from baseball or football are used in business or politics.
      * We can create a math book, videos and learning materials for adult
      self learners who'd like to make sense of the math they learned. I've
      been working on that here: http://www.gospelmath.com/Math/DeepIdeas

      Implications beyond math

      In my theory above, I've leveraged my work to know everything and to
      organize a culture (the kingdom of heaven) for the skeptical (the
      poor-in-spirit) by sharing and documenting ways of figuring things out,
      notably as games.

      I'm interested to apply the "house of knowledge"
      to other domains.

      * I've written out activities for organizing the kingdom of heaven.
      http://www.selflearners.net/Culture/ How are they related to the 24
      "frames of mind" in the house of knowledge?
      * I want to study more the gaps where God appears and why and how God
      becomes relevant.
      * I'd like to analyze other domains such as the historical method,
      scientific method, medicine, business, economics, the creative arts such
      as music and literature. I'd like to find funding for that. In
      particular, I imagine that I could work as a "resident blogger" for a
      domain (such as Gamestorming) and write, say, 24 posts, one for each
      deep structure.

      Thank you!

      All who read this, Thank you for reading this far!

      I've posted my letter here:

      Please think and write, How might we work together?


      Andrius Kulikauskas
      (773) 306-3807
      Twitter: @selflearners
      Chicago, Illinois
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