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.

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:

* 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)

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,

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

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

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.

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"

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!

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

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

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

Twitter: @selflearners

Chicago, Illinois