Foundations of Mathematics: Reliability and Clarity: The Explanatory Role of Mathematical Induction

While studies in the philosophy of mathematics often emphasize reliability over clarity, much study of the explanatory power of proof errs in the other direction. We argue that Hanna's distinction between 'formal' and 'acceptable' proof misunderstands the role of proof in Hilbert's program. That program explicitly seeks the existence of a justification; the notion of proof is not intended to represent the notion of a 'good' proof. In particular, the studies reviewed here of mathematical induction miss the explanatory heart of such a proof; how to proceed from suggestive example to universal rule. We discuss the role of algebra in attaining the goal of generalizability and abstractness often taken as keys to being explanatory. In examining several proofs of the closed form for the sum of the first n natural numbers, we expose the hidden inductive definitions in the 'immediate arguments' such as Gauss's proof. This connection with inductive definition leads to applications far beyond verifying numerical identities. We discuss some objections, which we find more basic than those in the literature, to Lange's general argument that proofs by mathematical induction are not explanatory. We conclude by arguing that whether a proof is explanatory depends on a context of clear hypothesis and understanding what is supposedly explained to who.