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!