Nominalism

Note the reschedule!

For this session we will look at Modality and Structuralism, section 15 of: Charles Parsons, Mathematical Thought and Its Objects, 1st ed. (Cambridge University Press, 2007). As the title suggests, Parson explores an acocunt of structuralism trough modal notions; the role of necessity and possibility in mathematical discourse, and the ontology of mathematical objects.

Does mathematics require an ontological commitment to abstract objects or can modal formulations capture mathematical truth without such commitments? Can modal logic provide a foundation for mathematical necessity? How does this modal structuralism compare to eliminative forms of structuralism?

For this session, we will answer the title’s question with the short and sweet: John P. Burgess’s Why I Am Not a Nominalist, a broad overview against various forms of nominalism.

Burgess responds to nominalist attempts to dispense with abstract objects in mathematical and scientific discourse, challenging both instrumentalist and reconstructionist forms of nominalism, among others. Burgess purports to shift the burden of proof onto the nominalist rather than the realist, by arguing that nominalistic reconstructions need (and in his view fail) to account for the role of mathematics in science. His critique addresses Goodman, Quine, and Field, among others.