Mathematical Accidents and the End of Explanation

A conspicuous difference between ‘traditional’ philosophy of science and ‘traditional’ philosophy of mathematics concerns the relative importance of the notion of explanation. Explanation has long featured centrally in debates in the philosophy of science, for at least two reasons. Firstly, explanation has been viewed as playing an important role in the methodology of science, due principally to the inductive character of scientific method. This has led to a focus on giving a philosophical model of scientific explanation, whose leading candidates have included Hempel’s deductive-nomological model, the causal model promoted by Lewis, van Fraassen’s pragmatic model, and the unification models of Kitcher and Friedman. Secondly, explanatory considerations have been an important feature of philosophical debates over scientific realism and antirealism. This has led to a focus on inference to the best explanation and the conditions under which this mode of inference can underpin robust ontological conclusions. By contrast, philosophical analysis of explanation in mathematics has – until very recently – been scattered and peripheral to the main debates in the philosophy of mathematics. This is partly a result of the traditional philosophical emphasis on the centrality of proof in mathematics, combined with the unstated assumption that issues of explanation are irrelevant to the regimentation and evaluation of purely deductive arguments. However there has recently been a significant increase of interest in mathematical explanation for reasons that are broadly analogous to those that have long held sway in the philosophy of science. Firstly, philosophers have begun to realize that explanatoriness is a genuine and significant feature of mathematical methodology. In particular, mathematicians frequently make a distinction between more and less explanatory proofs, and tend to value the former over the latter. Secondly, metaphysical debates concerning the existence of mathematical objects that arise from the QuinePutnam indispensability argument have started to focus more carefully on the putative explanatory role played by mathematics in science. These two strands of philosophical