Open Access Dissertation
Date of Award
I defend modal nominalism in philosophy of mathematics - under which quantification over mathematical ontology is replaced with various modal assertions - against two sources of resistance: that modal nominalists face difficulties justifying the modal assertions that figure in their theories, and that modal nominalism is incompatible with mathematical naturalism.
Shapiro argues that modal nominalists invoke primitive modal concepts and that they are thereby unable to justify the various modal assertions that figure in their theories. The platonist, meanwhile, can appeal to the set-theoretic reduction of modality, and so can justify assertions about what is logically possible through an appeal to what exists in the set-theoretic hierarchy. In chapter one, I illustrate the modal involvement of the major modal nominalist views (Chihara's Constructibility Theory, Field's fictionalism, and Hellman's Modal Structuralism). Chapter two provides an analysis of Shapiro's criticism, and a partial response to it. A response is provided in full in chapter three, in which I argue that reducing modality does not provide a means for justifying modal assertions, vitiating the accusation that modal nominalists are particularly burdened by their inability to justify modal assertions.
Chapter four discusses Burgess's naturalistic objection that nominalism is unscientific. I argue that Burgess's naturalism is inadequately resourced to expose nominalism (modal or otherwise) as unscientific in a way that would compel a naturalist to reject nominalism. I also argue that Burgess's favored moderate platonism is also guilty of being unscientific. Chapter five discusses some objections derived from Maddy's naturalism, one according to which modal nominalism fails to affirm or support mathematical method, and a second according to which modal nominalism fails to be contained or accommodated by mathematical method. Though both objections serve as evidence that modal nominalism is incompatible with Maddy's naturalism, I argue that Maddy's naturalism is implausibly strong and that modal nominalism is compatible with forms of naturalism that relax the stronger of Maddy's naturalistic principles.
Schwartz, James S.j., "Nominalism In Mathematics - Modality And Naturalism" (2013). Wayne State University Dissertations. 795.