Mathematical explanations are explanations in which
mathematics plays a fundamental role. The expression ‘mathematical explanation’
(ME) has two distinct, although connected, meanings: in relation to pure
mathematics ME denotes proofs that are able not only to demonstrate the truth
of a given mathematical statement, but also to explain why the
statement is true, whereas in connection with empirical sciences ME refers to
explanations of non-mathematical facts (physical, biological, social,
psychological) justified by recourse to mathematics.
Although the concept of ME has been the subject of
analysis at least since Aristotle’s distinction between apodeixis tou oti
and apodeixis tou dioti (Post. An. I.13), and has been dealt with a
few times over the course of the development of Western thought (e.g.
Descartes, Newton, and Bolzano), it is only since the 1970s that an intense
philosophical debate has sprung up regarding
the nature of ME. This debate, linked to the gradual diffusion of Quinean
epistemology (Steiner 1978)
and the development of the anti-foundationalist philosophy of mathematics (the
so-called ‘maverick’ tradition, Cellucci 2008), centers on the following
questions: Do mathematical explanations exist? If mathematical explanations
exist, can they be reduced to a single model or are they heterogeneous among
themselves? What implications does the comprehension of the concept of
mathematical explanation have for some of the most important problems of the contemporary
philosophy of science (e.g. indispensability arguments, inference to the best
explanation, and the theory of scientific explanation)?