Why proofs by mathematical induction are generally not explanatory
Proposed accounts of scientific explanation have long been tested against certain canonical examples. Various arguments are paradigmatically explanatory, such as Newton's explanations of Kepler's laws and the tides, Darwin's explanations of various biogeographical and anatomical facts, Wegener's explanation of the correspondence between the South American and African coastlines, and Einstein's explanation of the equality of inertial and gravitational mass. Any promising comprehensive theory of scientific explanation must deem these arguments to be genuinely explanatory. By the same token, there is widespread agreement that various other arguments are not explanatory examples so standard that I need only give their familiar monikers: the flagpole, the eclipse, the barometer, the hexed salt and so forth. Although there are some controversial cases, of course (such as 'explanations' of the dormi ti vevirtue variety), philosophers who defend rival accounts of scientific explanation nevertheless agree to a large extent on the phenomena that they are trying to save. Alas, the same cannot be said when it comes to mathematical explanation. Philosophers disagree sharply about which proofs of a given theorem explain why that theorem holds and which merely prove that it holds.1 This kind of disagreement makes it difficult to test proposed accounts of mathematical explanation without making controversial presuppositions about the phenomena to be saved.