Aspects of Mathematical Explanation: Symmetry, Unity, and Salience

Unlike explanation in science, explanation in mathematics has received relatively scant attention from philosophers. Whereas there are canonical examples of scientific explanations (as well as canonical examples of nonexplanations, such as “the flagpole,” “the eclipse,” and “the barometer”), there are few (if any) examples that have become widely accepted as exhibiting the distinction between mathematical proofs that explain why some mathematical theorem holds and proofs that merely prove that the theorem holds without revealing the reason why it holds. This essay offers some examples of proofs that mathematicians have considered explanatory (or not), and it argues that these examples suggest a particular account of explanation in mathematics (at least, of those explanations consisting of proofs). The essay compares its account to Steiner's and Kitcher's. Among the topics that arise are proofs that exploit symmetries, mathematical coincidences, brute-force proofs, simplicity in mathematics, merely clever proofs, and proofs that unify what other proofs treat as separate cases.