Explanatory Proofs and Beautiful Proofs

Abstract

This paper concerns the relation between a proof's beauty and its explanatory power — that is, its capacity to go beyond proving a given theorem to explaining why that theorem holds. Explanatory power and beauty are among the many virtues that mathematicians value and seek in various proofs, and it is impor-tant to come to a better understanding of the relations among these virtues. Mathematical practice has long recognized that certain proofs but not others have explanatory power, and this paper o↵ers an account of what makes a proof explanatory. This account is motivated by a wide range of examples drawn from mathematical practice, and the account proposed here is compared to other ac-counts in the literature. The concept of a proof that explains is closely intertwined with other important concepts, such as a brute force proof, a mathematical co-incidence, unification in mathematics, and natural properties. Ultimately, this paper concludes that the features of a proof that would contribute to its ex-planatory power would also contribute to its beauty, but that these two virtues are not the same; a beautiful proof need not be explanatory.