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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.

Another informal social gathering of our group for tea time! A good opportunity to get to know better the new members.

The topic: things that impressed and irritated us the most in the PoM during the 3 months the group runs. Of course, people can bring their own most and least favorite things regarding PoM outside of the group’s activity.

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.

For our next meeting on structuralism, we will be covering the two views championed by the authors of the textboock: Shapiro’s ante rem, or Sui Generis, structuralism and Hellman’s Modal Structuralism.

Main Readings

Chapter 5 and 6 in Hellman, G., & Shapiro, S. (2018). Mathematical Structuralism (Elements in the Philosophy of Mathematics). Cambridge: Cambridge University Press. doi:10.1017/9781108582933

For our second meeting on Structuralism in mathematics, we will be focusing on the category-theoretical side of the discussion. Stemming from work in algebraic topology, category theory has since the 1950’s become an indisposable tool for mainstream mathematics and is seen by some to encode the structuralist philosophy into mathematics itself. We will be reading and discussing some thoughts around the plausibility of using category-theoretic foundations for mathematics and whether this really follows the structuralist philosophy.

We are continuing our discussion about Higher-Order Logic (HOL), this time focusing specifically on arguments against adopting Higher-Order logics for various purposes. These arguments range from (HOL) being set theory in disguise, the (lack of) applicability of (HOL) for its intended purposes, and the serious metalogical issues the logic faces.