My name is David Corfield and I'm very grateful to have been invited to join this blog as a contributor. With five years of blogging behind me at the n-Category Café, I relish the opportunity to talk with a new audience. My rate of blogging may have slowed, especially as my adminstrative load has increased - I'm now Head of Philosophy at the University of Kent - but I'm looking forward to writing some posts here.
I have interests in a variety of approaches to mathematical philosophy, including the statistical learning theory I picked up from my time at the Max Planck Institute for Biological Cybernetics in Tübingen, but the main idea I would like to promote to the audience here is that category theory is worth exploring as a resource for the mathematical philosopher.
I have recently published a couple of articles which examines the light category theory can throw on familiar infinite structures. In Understanding the Infinite I: Niceness, Robustness, and Realism, I look at the phenomenon where an infinite entity is defined by a universal property, and through this inherits 'for free' a range of other nice properties. In Understanding the Infinite II: Coalgebra, I look at the duality between minimally and maximally defined entities in the context of the duality between 'algebra' and 'coalgebra'.
Perhaps had I known of Shaughan Lavine (1994) Understanding the Infinite, Harvard University Press, I might have opted for a different title.
There's much to do to understand the relationship between category theory and the traditional foundational branches, which have drawn most philosophical attention. Recently, I posed a question on MathOverflow concerning category theory and Joel Hamkins' set theoretic multiverse. The answer by Joel there shows just the sort of joint investigation needed. A few years ago at the Café, we had a discussion on the relationship between category theory and model theory.
Category theory also has an interface with proof theory, but I know less about this. Something to look out for in the future is the new Homotopy type theory , and associated Univalent foundations.