Dissertation

Complete Dissertation

A copy of my full dissertation is available here.

Dissertation Abstract

When reading a proof, mathematicians want to be able to do more than just check the correctness of each of its steps. Indeed, as George Pólya¹ points out, they also want to grasp that the steps are “appropriate.” If mathematicians can grasp the appropriateness of each step in a proof, they often describe it as being “motivated.” Such proofs, unfortunately, have received little philosophical analysis. In this dissertation, I sharpen the definition of motivated proofs, evaluate their benefits and develop practical suggestions for producing them.

To accomplish this, I compare different proofs of the same number-theoretic theorems, including Fermat’s Little theorem, Wilson’s theorem and Dirichlet’s theorem for primes in an arithmetic progression. I suggest that a proof step be recognized as appropriate if and only if two conditions are met: (i) the intended reader can grasp the role it plays in the argument;² (ii) the intended reader can grasp factors that prompted the prover to introduce it. A proof is said to be motivated, then, if and only if all of its steps are recognized as appropriate. Motivated proofs thus convey more useful information to a reader than unmotivated ones. This gives them two main benefits: (i) they promote a deeper understanding of the mathematical argument; (ii) they foster more effective reuse of mathematical ideas. The case studies further reveal that three factors affect the degree to which a proof is motivated: (i) the mathematical content it contains; (ii) the manner that content is presented; (iii) the mathematical background of the intended audience.

Having sharpened the definition of motivated proofs, I then propose techniques for producing them. Generally, proofs that highlight information when it is relevant and hide it when it is not are better motivated than those that fail to do so. Consequently, one suggestion I make focuses on how a proof manages information, as reflected in, for example, the notation that it uses and how it is structured.

I conclude by sketching avenues for future work. For example, I observe that mathematicians often call other artifacts, such as definitions and theories, motivated, and suggest developing an account of motivation that applies to them, as well. Further, an account of motivation that applies to mathematical definitions and theories could be adapted to apply to scientific ones.

[1] George Pólya With, or without, motivation, The American Mathematical Monthly, 56(10): 684–691, 1949.
[2] Ibid, pg 568.