Towards a philosophical account of explanation in mathematics

All proofs show that their conclusions are true; some also explain why they are true. But what makes a proof (or argument) explanatory, if it is? That is the central question of my thesis. I begin by identifying four accounts of scientific explanation which look like they might be useful for the intra-mathematical case, assessing the prospects for extending each account to the mathematical case. I examine whether we could get to a general result about mathematical explanation while drawing only on general assumptions about explanation. I argue that this methodology is flawed and that we need to pay serious attention to specific examples from mathematical practice, not just to general assumptions. I examine two existing accounts of intra-mathematical explanation: first, Steiner’s 1978 account. I propose a new and sympathetic reading that provides a better understanding of his account than can be found in the existing literature. Although Steiner’s account seems to focus on ontic aspects of explanation, I show how (my extension of) Steiner’s proposal can also account for what I take to be the primary epistemic function of an explanation, namely, to help us see why the fact to be explained is true. Second, I examine Lange’s 2017 account, which focuses on salient features. Of the features proposed by Lange, I suggest that symmetry is the best candidate for a feature of mathematical explanation, and I argue that we should see symmetry as an objective mathematical property that may have the propensity to appear salient to creatures like us in certain contexts. I argue that it is philosophically fruitful to play close attention to candidate examples of mathematical explanation, and in Chapter 5 I present an in-depth case study of a proof in Galois theory and propose a positive account of its explanatory value.