Mathematical induction and explanation

Abstract

Marc Lange (2009) sets out to offer a ‘neat argument that proofs by mathematical induction are generally not explanatory’, and to do so without appealing to any ‘controversial premisses’ (2009: 203). The issue of the explanatory status of inductive proofs is an interesting one, and one about which – as Lange points out – there are sharply diverging views in the philosophy of mathematics literature. It may be that Lange is correct in his verdict that proofs by mathematical induction lack explanatory power. However, I think that his argument to this conclusion is too quick. Lange’s core argument may be reconstructed as follows: Lange’s argument, which has the form of a reductio, depends on the condition (appealed to in step (7)) that mathematical explanations cannot run in a circle. In other words it cannot be the case, for two mathematical facts A and B, both that A explains B and that B explains A. I have some doubts about the inevitability of this condition, especially since Lange’s argument is formulated in terms of partial explanation, but I shall not pursue them here. Instead I shall focus on what I take to be both the most central and the most problematic step in the above argument, step (4), which claims that a standard inductive proof of ∀nP(n) (with basis case n = 1) and an alternative inductive proof that proceeds ‘upwards and downwards’ from some other base case (such as n = 5) must be regarded as equally explanatory. Schematically, the two kinds of proof run as follows: P1P P(1)-based proof For any property P: if P(1), and for any natural number k, if P(k), then P(k + 1), then for any natural number, n, P(n). P5P P(5)-based proof For any property P: if P(5), and for any natural number k, if P(k), then P(k+1), and for any natural number k > 1, if P(k), then P(k – 1), then for any natural number, n, P(n). If the proofs by mathematical induction are explanatory, then the very similar proofs by the ‘upwards and downwards from 5’ rule are equally explanatory. There is nothing to distinguish them, except where they start. (2009: 209)