It requires a certain kind of philosophical temperament — which I seem to lack — to get worked up by the question “But do numbers really exist?” and excitedly debate whether to be a fictionalist or a modal structuralist or some other -ist. As younger colleagues gambol around cheerfully chattering about these things, wondering whether to be hermeneutic or revolutionary, I find myself sitting on the side-lines, slightly grumpily muttering under my breath ‘And who cares?’.

To exaggerate a bit, I guess there’s a basic divide here between two camps. One camp is primarily interested in analytical metaphysics, or epistemology, or the philosophy of language, and sees mathematics as a test case for their preferred Quinean naturalist line (or whatever). The other camp is puzzled by some internal features of the practice of real mathematics and would like to have a story to tell about them.

Well, if you’re tired of playing the ontology game with the first camp, then there’s actually quite a bit of fun to be had in the second camp, and maybe more prospect of making some real progress. In the broadest brush terms, here are just a few of the questions that bug me (leaving aside Gödelian matters):

- How should we develop/improve/augment/replace Lakatos’s model of how mathematics develops in his Proofs and Refutations?
- What makes a mathematical proof illuminating/explanatory? (And what are we to make of unsurveyable computer proofs?)
- Is there a single conceptual grounding for the standard axioms of set theory? (And what are we to make of the standing of various large cardinal axioms?)
- What is the significance of the reverse mathematics project? (Is it just a technical “accident” that RCA_0 is used a base theory in that project? Can some kind of conceptual grounding be given for that theory? Would it be more principled to pursue Feferman’s predicative project?)
- Is there any sense in which category theory provides new foundations/suggests a new philosophical understanding for mathematics?

There’s even a possibility that your local friendly mathematicians might be interested in talking about such things!

In addition to the mention of unsurveyable computer proofs, there is also the problem of unsurveyable human proofs, such as the classification of finite simple groups. There’s all sorts of interesting questions here. What is the ontological status of a theorem whose only known proof uses the classification theorem? What is the merit of current revisionist programs, which aim to generate a shorter, more uniform proof? How can we be sure that the classification is actually complete?

Tired of ontology? Yes, most definitely. In particular I find question five interesting. Do you have any recommended literature on the subject? (i.e., not on category itself, but philosophical discussion on its foundational value.)

For discussions of the foundational status(?) of category theory, I can’t really do better than suggest looking at Part C, “Articles with a philosophical bent”, of the biblio of the Stanford encyclopedia article Category theory. There’s more than enough to chew on there!