For this session, our dear classmate Edoardo will tell us all about Martin-Löf’s impact on the philosophy of math and logic. There is no specific text of focus; but those wanting to come extra-prepared can read Per Martin-Löf’s On the Meanings of the Logical Constants and the Justifications of the Logical Laws for some background on the central topics he will touch on.
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.