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I'm rewriting and updating my talk Discovery in Mathematics: A System of Deep Structure.

I am providing more examples of the way things are figured out in mathematics, making my statements more detailed and precise, and drawing further conclusions.

Mathematics Expresses Discovery

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 triangle 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.