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See: Math Discovery This is a letter that I wrote on April 2, 2011 to Paul Zeitz, to the Math Future group and others. Andrius Kulikauskas 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. http://groups.google.com/group/mathfuture/ 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 projects. 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:
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: affirmreject, 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 principle) C3) Least upper bounds, greatest lower bounds (Monovariants, algorithmic proof, optimization problem, world records: minimal times to beat keep increasing) 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 problem.) F) Continuity (Vary the variable, existence of a solution, balancing point, appeal to physical intuition) R) Selfsuperimposed sequence (Recurrence relation as an automata, autoassociative 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 selfreference) 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 inclusionexclusion) 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, reinterpret) I note that some problems and some concepts involve the application of two or more such deep structures. For example, the principle of inclusionexclusion is equivalent to reorganizing (11)**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 ( ). 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" (pg.312). 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 recordtobeat, which yields sequences of increasing minimums, thus a greatest lower bound, or alternatively, a least upper bound. Finally, we allow such a boxingin or boxingout 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) Selfsuperimposed 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 autoassociative memory that Jeff Hawkins discusses in his book "On Intelligence", where cortical columns use timedelay 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 selffulfillment. 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 selfstanding 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.
case we may interpret the problem as statements that we ourselves are making which may be true or false and potentially selfreferential. Together they allow for proofsbycontradiction 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 allthingsaretrue) and noncontradiction is a very special case that takes great effort, like segregating matter and antimatter. Deep structure "solution spaces" allow us, as with Euclid's equilateral triangle, to step away from the "solution" and consider the candidate solutions, indeed, the failed solutions.
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.
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.
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):
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).
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 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).
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 graphlike 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.)
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:
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:
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 12hour 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: http://www.gospelmath.com/Math/SolutionSpaces 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 inquiry. 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.
from others on how they like various problems.
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.
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).
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:
problems, such as the Millenium problems.
structures. What is fun about each of them? We can study chess.
have us shift back and forth between the deep structures and concrete problems that express them.
creative arts.
example, the Gamestorming games involve ranking priorities, mapping adjacencies, sorting ideas and other relationships that helped me think through the system above.
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.
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 poorinspirit) by sharing and documenting ways of figuring things out, notably as games. I'm interested to apply the "house of knowledge" http://www.selflearners.net/ways/ to other domains.
http://www.selflearners.net/Culture/ How are they related to the 24 "frames of mind" in the house of knowledge?
becomes relevant.
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. 
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