Dimensions of Mathematical Explanation

My dissertation, Dimensions of Mathematical Explanation, consists of three essays. The first paper—“Arithmetic, Set Theory, Reduction and Explanation”—argues that viewing the natural numbers and arithmetical operations as sets has no explanatory value. Thus, contrary to received wisdom, there are bona fide intertheoretic reductions that are nevertheless unexplanatory. In the second paper, “Mathematical Explanation Beyond Explanatory Proof”, I challenge the widespread view that proofs play an essential role in most cases of explanation in mathematics. In particular, I look closely at the explanation of the solvability of polynomial equations provided by Galois theory, which has often been thought to revolve around an explanatory proof. The third essay, “Viewing-As and Dependence in Mathematical Explanation”, explores the phenomenon of “viewing one mathematical object as another”, which lies at the heart of a distinctive type of explanation. I argue that such cases pose a problem for the popular ontic conception of explanation (which requires causal or metaphysical dependence relations between explanans and explanandum). I sketch an alternative “cognitivist” notion of explanation that makes better sense of these cases.