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24 Keys For Solving Math Problems


The ways of figuring things out in mathematics

House of Knowledge

Ways of figuring things out in mathematics


  • See more conceptual structures.
  • Get overview of all of math.
  • Distinguish natural, preaxiomatic math.

Study the Ways of Figuring Things Out in Mathematics

I am most encouraged by my study in 2011 of the ways of figuring things out in mathematics which I shared in this letter to the Math Future online group. Here is an extended version of the results which I presented in 2016 at the Lithuanian Mathematics Association Conference: Discovery in Mathematics: A System of Deep Structure.

The basic idea came in considering George Polya's "pattern of 2 loci" by which he solves Euclid's first problem of how to construct an equilateral triangle. We solve this problems in our minds by constructing a lattice of conditions. Given two points, the third point that we want to construct must satisfy two conditions, namely, it must be on both of two circles centered on the two other points, thus it must be at their intersection. The solution is clear as soon as our minds apply the relevant structure, namely, the lattice of conditions.

This example suggests a distinction between deep structure - the natural mathematical structures which we use in our mind to solve problems, and surface structure - the contrived math by which we describe problems on paper. I thus surveyed the problem solving patterns taught by Paul Zeitz, George Polya and others in their books.

Ways of figuring things out in other fields:

  • neuroscience, chess, physics, biology, Jesus, Gaon of Vilna, my philosophy.

Mathematical Goal

  • Overview how all of the branches of math unfold, the questions and answers, problems and solutions, challenges and theorems, concepts (structures, objects, relations, constants), ideas.

Philosophical Goal

  • Language of absolutes - Cognitive frameworks
  • Most interesting, most fruitful and most speculative would be for me to look for connections between the philosophical structures I work with and what seem to be related mathematical structures.

The kinds of variables

Analyze How We Use Variables. My understanding is that there are four levels of knowledge (whether, what, how, why) and that in thinking in a mathematical system we establish a level (surface structure) and a metalevel (deep structure). Our use of variables plays off this distinction. I thus made a diagram of the roles that we imagine variables to play.

I need to validate this further. I expected there to be six kinds of variables but instead found evidence for twelve kinds. It seems that, on the one hand, variables are used to solve problems, but on the other hand, variables are used to create problems.

Indeed, this points to a key aspect missing in my research so far. Much of advanced mathematics is about abstraction, the creation of frameworks. But it is clear that this process of abstraction is not arriving at cognitive foundations but rather is growing ever more rich, complex and distant.

  • Amounts and units. Combine like units, list differing units. Type theory.
  • Extending the domain.
  • Trigonometry: Every rectangle is half a triangle.
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This page was last changed on January 23, 2022, at 10:35 PM