Abstract

A long-standing problem in the philosophy of mathematics concerns the application of mathematics in reasoning about the physical world. Almost without exception philosophers considering applications of mathematics first provide an account of pure mathematics and then try to show how this is consistent with applied mathematics. But this unquestioned practice has obscured certain fundamental issues in the philosophical study of applied mathematics. In my dissertation, I develop the first classification of accounts of applied mathematics that is independent of views about pure mathematics. This new perspective allows the consideration of the viability of accounts of applied mathematics in their own right. ;My approach focuses on the kind of relation that an account posits between the mathematical world and the physical world. Roughly, there are three broad categories based on whether the criteria of identity of mathematical objects involve physical objects. In the first five chapters I argue against philosophers like Frege who see an essential or internal relation at work, and those such as Field who deny that there is any relation between mathematics and the physical world. I argue that 'internal' approaches, i.e. approaches that bring in physical objects in the criteria of identity of mathematical objects, tie mathematics too closely to the physical world. At the same time, 'no relation' approaches that see no relation between mathematics and the physical world cannot explain many important applications of mathematics. Indeed, I defend an 'external' account in terms of mappings, whereby the criteria of identity of mathematical objects do not involve physical objects. ;I present these mappings as external relations between mathematical objects and physical situations and argue that mathematics can be applied in science because of the similarities between mathematical structures, such as the real number line, and physical situations. These structural similarities obtain precisely when mappings with certain properties exist. A central part of the dissertation is devoted to spelling out what these mappings are. Throughout I emphasize that my 'structuralist' account of applied mathematics is independent of traditional metaphysical debates about pure mathematics and relies on minimal ontological assumptions. Thus it can be adopted by philosophers of mathematics of most persuasions, including traditional realists, structuralists about pure mathematics, and even some nominalists. I use this fact to argue that indispensability arguments for realism about mathematics are invalid . ;The last three chapters are meant to test the validity of my account by, first of all, discussing how to account for idealization within science from within my perspective , and then by looking at a series of case studies that develop and support my 'structuralist' account of applied mathematics . These case studies are drawn from fluid mechanics and analytic mechanics, and highlight the shortcomings of previous approaches