| Summary: | This thesis is about two ways in which we use mathematics to understand the non-mathematical 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 world-oriented uses of mathematics in order to evaluate competing considerations in favour of, and against, mathematical realism. In chapters 3 and 4 I discuss extra-mathematical 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 extra-mathematical 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 extra-mathematical 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.
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