Introduction: I lay out the broad contours of my thesis: a defence of mathematical nominalism, and nominalism more generally. I discuss the possibility of metaphysics, and the relationship of nominalism to naturalism and pragmatism. Chapter 2: I delineate an account of abstractness. I then provide counterarguments to claims that mathematical objects make a di erence to the concrete world, and claim that mathematical objects are abstract in the sense delineated. Chapter 3: I argue that the epistemological problem with abstract objects is not best understood as an incompatibility with a causal theory of knowledge, or as an inability to explain the reliability of our mathematical beliefs, but resides in the epistemic luck that would infect any belief about abstract objects. To this end, I develop an account of epistemic luck that can account for cases of belief in necessary truths and apply it to the mathematical case. Chapter 4: I consider objections, based on (meta)metaphysical considerations and linguistic data, to the view that the existential quantifier expresses existence. I argue that these considerations can be accommodated by an existentially committing quantifier when the pragmatics of quantified sentences are properly understood. I develop a semiformal framework within which we can define a notion of nominalistic adequacy. I show how our notion of nominalistic adequacy can show why it is legitimate for the nominalist to make use of platonistic “assumptions” in inferencemaking. Chapter 5: I turn to the application of mathematics in science, including explanatory applications, and its relation to a number of indispensability arguments. I consider also issues of realism and antirealism, and their relation to these arguments. I argue that abstraction away from pragmatic considerations has acted to skew the debate, and has obscured possibilities for a nominalistic understanding of mathematical practices. I end by explaining the notion of a pragmatic metavocabulary, and argue that this notion can be used to carve out a new way of locating our ontological commitments. Chapter 6: I show how the apparatus developed in earlier chapters can be utilised to roll out the nominalist project to other domains of discourse. In particular, I consider propositions and types. I claim that a unified account of nominalism across these domains is available. Conclusion: I recapitulate the claims of my thesis. I suggest that the goal of mathematical enquiry is not descriptive knowledge, but understanding.
