Title:

Mathematics and the world : explanation and representation

This thesis is about two ways in which we use mathematics to understand the nonmathematical world: in particular, mathematical explanation and mathematical representation. In chapters 1 and 2, I motivate the project by suggesting that, in addition to shedding light on the nature of explanation and representation, it is necessary to develop accounts of these two worldoriented uses of mathematics in order to evaluate competing considerations in favour of, and against, mathematical realism. In chapters 3 and 4 I discuss extramathematical explanation. In chapter 3, I consider and reject four recent accounts of mathematical explanation. In chapter 4 I discuss and endorse what I call the modal account of extramathematical explanation. I argue, in line with Jansson and Saatsi and contra Baron, Colyvan and Ripley that such an account does not require countenancing counterpossibles, I discuss in virtue of what a mathematical fact can play this role and I address whether or not extramathematical explanations are causal. In chapters 5 and 6 I discuss mathematical representation. In chapter 5 I consider two fundamental challenges to developing an account of mathematical scientific representation: the first is Callender and Cohen’s claim that there are no special problems of scientific representation and the second is a set of influential objections owing to Frigg and Suárez that take aim at accounts of representation that appeal to the notion of structural similarity. In chapter 6 I argue that two recent accounts of mathematical representation are, in fact, complementary and, more generally, that mathematical representation is a special kind of epistemic representation. I draw on some work from epistemology to address, and argue against, Pincock’s claim that in order to understand a mathematical representation one must believe its mathematical content.
