I'm giving a talk on the ways of figuring things out in Mathematics.

It's based on a letter that I wrote about a comprehensive system that I organized of 24 such ways.

It needs to be 10 minutes long, 1,000 words.

Discovery in Mathematics: A System of Deep Structure

I will talk about how we may systematically study the ways of figuring things out in mathematics.

George Polya, in his book, "How to Solve It", considers Euclid's problem of how to construct an equilateral triangle. If we are given the side AB, how do we construct the other two? The solution is a recurring idea which Polya calls the "pattern of two loci". We think of there being two separate conditions. One side must extend a length AB from the point A. Another side must extend a length AB from the point B. We thus draw two circles of radius AB centered at A and B. The points where the two circles intersect are those where we can draw a third point C which satisfies both conditions so that our triagle is equilateral.

I realized that our minds solve this problem by imagining a powerset lattice of conditions. Circle A is one condition, circle B is another condition, and the intersection of A and B satisifies the union of these two conditions. Our minds have thus solved the surface problem (constructing a triangle) by considering a simpler, deeper structure (a lattice of conditions). This brings to mind linguist Noah Chomsky's work in syntax and architect Christopher Alexander's work on pattern languages.

I collected such problem solving patterns discussed in Paul Zeitz's book "The Art and Craft of Problem Solving" and other sources. Each distinct pattern makes use of a structure which is familiar to mathematicians and yet is not explicit but mental. We may consider those math structures to be cognitively "natural" which are used by the mind in solving math problems. I present to you 24 patterns which I identified and systematized them in a way which suggests they are complete.

The system distinguishes between

We may always start a fresh sheet (independent trials).

cognitive structures used on a single mental "sheet", as in algebra (center, balance, polynomials, vector spaces),

those that suppose an endless sequence of sheets, as in analysis (sequence, poset with maximal or minimal element, least upper bounds or greatest lower bounds, limits).

Sheets may be "stitched together" (extend the domain, continuity, self-superimposed sequence).

Algebraic and analytic approaches thus combine to give a complete, explicit system (symmetry group).

Within a system, we can relate an explicit sheet where it has fixed expression with an implicit mental sheet where it is imagined and worked on (truth, model, implication, variable).

We can then manipulate a problem explicitly as one of six visualizations of a graph structure (tree of variations, adjacency graph, total order, powerset lattice, decomposition, directed graph).

However, the meaning of any explicit expression such as 10+4= ultimately depends on what is implicit, whether we have in mind, for example, the integer Z or a clock Z12 (context).

George Polya's book "How to Solve It" showed how we can collect recurring cognitive "patterns" which mathematicians apply intuitively in a variety of settings. For example, in drawing an equilateral triangle, given the first side AB, how do we draw the other two? Using the "pattern of two loci", we draw circles of radius AB centered at A and B and find their intersections. I note here that, in solving this problem, our mind constructs a lattice of conditions on the solution: the plane (no condition), circle A (one condition), circle B (one condition) and the intersection of A and B (two conditions). Thus the surface problem (constructing a triangle) is solved in the mind by considering a simpler, deeper structure (a lattice of conditions). This brings to mind linguist Noah Chomsky's work in syntax and architect Christopher Alexander's work on pattern languages.

I collected such problem solving patterns discussed in Paul Zeitz's book "The Art and Craft of Problem Solving" and other sources. [1] Each distinct pattern makes use of a structure which is familiar to mathematicians and yet is not explicit but mental. We may consider those math structures to be cognitively "natural" which are used by the mind in solving math problems. I identified 24 patterns and systematized them in a way which suggests they are complete. I will present this system as drawn and described in more detail at [2].

The system distinguishes between cognitive structures used on a single mental "sheet", as in algebra (center, balance, polynomials, vector spaces), and those that suppose an endless sequence of sheets, as in analysis (sequence, poset with maximal or minimal element, least upper bounds or greatest lower bounds, limits). We may always start a fresh sheet (independent trials). Sheets may be "stitched together" (extend the domain, continuity, self-superimposed sequence). Algebraic and analytic approaches thus combine to give a complete, explicit system (symmetry group). Within a system, we can relate an explicit sheet where it has fixed expression with an implicit mental sheet where it is imagined and worked on (truth, model, implication, variable). We can then manipulate a problem explicitly as one of six visualizations of a graph structure (tree of variations, adjacency graph, total order, powerset lattice, decomposition, directed graph). However, the meaning of any explicit expression such as 10+4= ultimately depends on what is implicit, whether we have in mind, for example, the integer Z or a clock Z12 (context).

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Puslapis paskutinį kartą pakeistas 2016 birželio 19 d., 15:55