Induction, Constructivity, and Grounding

This paper is divided into two parts, the first being a point of departure for the second. I will begin by discussing a well-known negative argument due to Mark Lange concerning the explanatory role of mathematical induction. In the first part of the paper, I offer yet another response to Lange’s argument and attempt to characterize the sort of explanatory role played by inductive proofs. That account depends on two structural principles about explanatory proof that look like a fragment of a constructive semantics for that concept. The remainder of the paper fills out this semantics and explores its consequences. It will be clear that this framework does not constitute a fully general characterization of the concept of mathematical proof; the question will be whether there is a natural class of proofs that it does characterize. My answer will be that it nicely describes what I shall call grounding explanatory proofs. A proof of this sort explains the sentence proved in terms of the grounds of the fact that it describes. I will conclude by briefly exploring the connections between grounding proofs and the notion of purity.