Platonism

The PhiMath reading group is back for the academic year 2025/2026! We kick off with William Tait’s compelling exposition of the tension between Truth and Proof in Mathematics. Are mathematical proofs constructed or discovered by means of a proof? In the decade-long debate between constructivist and platonists, Tait defends Platonism by attacking some of Dummett’s main claims in favour of intuitionism. By adopting a similar approach towards the relaionship between language and reality as Dummet’s, he aims to downsize the accusations intuitionists wage against mathematical realists, arguing that proofs are merely representations of mathematical truth. The issues Tait brings up in this paper are extremely engaging for Intuitionists and Platonists alike, so come plenty and take part in the discussion!

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.

For our first session of 2025, we will engage with Robin Jeshion’s Intuiting the Infinite. He defends Charles Parsons’ Kantian appeal to mathematical intuition to address the access problem of Platonism: If mathematical objects are abstract objects, how can we gain knowledge of them?

Jeshion argues that intuition plays a fundamental role in justifying our knowledge of the infinitude of natural numbers, responding to key criticisms about the cogency of arbitrary objects, vague representation, and the role of spatial and temporal structures in mathematical thought.