After a bit of a delay, my contribution to the Cambridge Elements in the Philosophy of Mathematics, co-edited by Stewart Shapiro and Penelope Rush, is now in press. The topic is Mathematics and Explanation. Here is a brief overview. I will post notes about the book’s four chapters in the coming weeks.
Summary: Some scientific explanations involve mathematics. Within mathematics, some proofs are said to explain. Do these practices tell us anything about the nature of explanation or mathematics? In this Element I divide this daunting topic into four parts. First, can any traditional theory of scientific explanation make sense of the place of mathematics in explanation? Each traditional theory that I will discuss is a monist theory because it supposes that what makes something a legitimate explanation is always the same (Ch. 1). Second, if traditional monist theories are inadequate, is there some way to develop a more flexible, but still monist, approach that will clarify how mathematics can help to explain (Ch. 2)? After considering the limitations of some recent flexible monist accounts, I examine the options for a pluralist approach. What sort of pluralism about explanation is best equipped to clarify how mathematics can help to explain in science and in mathematics itself? While a pluralist can allow that different sorts of explanations work differently, it still remains important to clarify the value of explanations (Ch. 3). Finally, how can the mathematical elements of an explanation be integrated into the physical world? Some of the evidence for a novel scientific posit may be traced to the explanatory power that this posit would afford, were it to exist. Can a similar kind of explanatory evidence be provided for the existence of mathematical objects, and if not, why not? (Ch. 4).