At a meeting some years ago of the Trinity Mathematical Society, Imre Leader and Thomas Forster gave introductory talks on “Does Mathematics need a Philosophy?” to a startlingly large audience, before a question-and-answer session. The topic is a very big one, and the talks were very short. After the event, I wrote up a few after-thoughts (primarily for maths students such as the members of TMS, though others might be interested …). I had occasion to revisit my remarks just recently. Rough and ready though they were, I’m happy enough to stand by their broad message, so here they are again, just slightly tidied up for new readers!
Imre did very briskly sketch a couple of philosophical views about mathematics, which he called platonism and formalism. And he suggested that mathematicians tend to be platonist in their assumptions about what they are up to (in so far as they presume that they are exploring a determinate abstract mathematical universe, where there are objective truths to be discovered) but they turn formalist when writing up their proofs for public consumption.
Now, platonism comes in various stripes, and we could argue the toss about which variety (if any) tends to be presumed by working mathematicians. And there’s a further issue about how far, if at all, the presumption of platonism is doing any mathematical work: is it just an idle philosophical wheel?