Science-Driven Mathematical Explanation

Abstract

The vast majority of philosophical work on explanation has concerned itself with scientific explanation. Aside from the obvious importance of science, another factor sometimes cited in support of this partiality is that there is ‘a substantial continuity between the sorts of explanations found in science and at least some forms of explanation found in more ordinary non-scientific contexts’ (Woodward 2009, p. 2). The idea seems to be that, by focusing on explanation in science, philosophers will be able to isolate and analyze features of explanatory practice that hold more generally. A notable exception to the above claim of continuity is explanation in mathematics. This topic was entirely ignored in the ‘first wave’ of work by analytic philosophers on explanation in the 1950’s and 1960’s. The situation has changed considerably over the past couple of decades, and there is now a significant amount of philosophical attention being paid specifically to the issue of mathematical explanation. Moreover, even philosophers whose main focus is on scientific explanation often acknowledge the existence of explanations within pure mathematics. Nonetheless, the predominant view is that mathematical explanation is qualitatively different both from scientific explanation and from explanation in ‘ordinary non-scientific contexts’. Thus the editors of a recent volume on interdisciplinary approaches to explanation write that ‘your explanation of why you’ll be home late for dinner and a mathematician’s proof of a theorem share very little’ (Wilson & Keil 2000, p. 91). There are good reasons for analyzing mathematical explanation separately from scientific explanation. Adopting this approach acknowledges the clear intuitive differences between these two spheres of explanatory practice, while also allowing theorists in one sphere to proceed unencumbered by potential counterexamples from the other. However, there is a fairly obvious problem with this neat picture, and that is that the spheres of mathematical practice and scientific practice frequently overlap. It is all very well emphasizing the qualitative differences between scientific explanation and its mathematical counterpart, but what about scientific explanations that make use of mathematics? Following Paolo Mancosu, I shall define a mathematical explanation in science (MES) to be ‘an explanation in natural science carried out by essential appeal to mathematical facts’ (Mancosu 2008, p. 135). An example of an MES that has been discussed at some length in the literature concerns the life-cycle of the North American