Readings

Reading meetings are the most common event hosted by our group. We read together/present a text, and freely discuss together.

Modal Structuralism

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?

Why is Burgess Not a Nominalist?

TBD and Online

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.

Intuiting the Infinite

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.

Infinity up on Trial: Reply to Feferman

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.

Why be a Height Potentialist?

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:

Structuralism in Mathematics: Sui Generis and Modalities

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

Structuralism in Mathematics: Categories

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.