Structuralism

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.

For our final meeting on structuralism, we cover the modal set-theoretic structuralism from Parsons and Linnebø. This rounds out the book we have been following and also our time with structuralism (for now).

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 now covering the theory of structuralism in mathematics. This first meeting will be an overview of the general position and its various parts together with specific focus on set-theoretic structuralism. Over the course of the coming meetings, we will spend some time on various conceptions of structuralism and their proponents.