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 counter-arguments 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 semi-formal 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 inference-making.
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 anti-realism,
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 meta-vocabulary, 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
