Readings

For this session of PhiMath, we read Bob Hale’s paper “Properties and the Interpretation of Second-Order Logic”. He defends a deflationary conception of properties, which are the things we quantify over in second order logic. According to Hale, something is a property iff there could be a predicate that stands for it.

He looks into Quine’s claim that second-order logic is “set theory in sheep’s clothing”, which he understands as a broader charge motivated by the idea that second-order quantification carries heavy existential commitments. According to him, his deflationary interpretation of second-order logic can resist Quine’s critique. During the meeting, we consider Hale’s proposal and its implications for second-order logic.

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!

This time our classmates, Matteo and Josje, will presenet Panu Raatikainen’s Conceptions of truth in intuitionism.

We often summarize the intuitionist notion of truth as “truth as provability”, which marks a fundamental difference with the classical logicians. But in doing so, we overlook the fact that there are multiple competing conceptions of truth within intuitionism itself. Raatikainen presents a systematic overview of these different accounts, ranging from actualist to possibilist, and ultimately offers a critique of each, putting into question about the coherence of the intuitionist framework as a whole.

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.

For our first session of 2025, we will engage with Robin Jeshion’s Intuiting the Infinite. She 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.

In this session we will deal with a critique to some main ideas of Predicativists. They believe that, in some respects, arithmetic has some advantages that analysis and set-theory do not. Koellner puts this idea to the test.

Main Readings:

• Koellner, Peter (2016). Infinity up on Trial: Reply to Feferman. Journal of Philosophy 113 (5/6):247-260.

After talking about the infinite and Frege Arithmetic, it is time to turn to potentialism about set-theory. In this session we will turn to potentiality about the height of the cumulative hierarchy.

Main Readings:

• Soysal, Z. (2024). Why be a Height Potentialist?. Journal for the Philosophy of Mathematics, 1, 155–175. https://doi.org/10.36253/jpm-2938

Our second meeting of the year continues along the lines of potentialism. Now we turn to Frege Arithmetic and a genuinely potentialist account of it.

Welcome to PhiMath 2024! In our first meeting for this academic year, we will discuss a paper by Linnebo and Shapiro on the notion of Potential Infinity.