## A system of deep structure in solving math problems

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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@ms.lt ... Paul Zeitz, I share with you my
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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@...
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Paul Zeitz,

I share with you my thoughts on the varieties of "deep structure" in
mathematical "frames of mind". Your book "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
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:
* 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
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) 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)

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,
reinterpret)

---------------

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"
(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 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

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
solutions.
* 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:
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.

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"
http://www.selflearners.net/ways/
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!
-----------------------------------------------------

I've posted my letter here:
http://www.gospelmath.com/Math/DeepStructure
http://t.co/IBCU0yj

Please think and write, How might we work together?

Andrius

Andrius Kulikauskas
http://www.selflearners.net
ms@...
(773) 306-3807