I was sitting at tea at CMS yesterday listening to Martin Hyland and Thomas Forster talking about set theories, and it struck me — not for the first time — how important a certain kind of informal discussion is to the mathematical enterprise.
You’d miss some fun (“fun” in a rather stretched sense, but you know what I mean) if you didn’t ever have face-to-face philosophical discussions. But you wouldn’t necessarily miss out on a lot of philosophy. Because in written philosophy, you do still get the to and fro of ideas, the false starts, the dead ends, the conjectures, the refutations — often in the writing of a single author as she wrestles with objections and counter-objections. The dynamic is there on the pages (not in its untidiest and rawest form, to be sure, but still very evident). With mathematics published in the approved conventional styles, on the other hand, you get the end-product, some results and their proofs. But the dynamic that led to them, the whys and the wherefores, can be very hidden, and informal commentary can often be very laconic (or altogether missing). So a few arm-waving remarks over tea that might never get into a written paper can make all the difference to your understanding of how some bits of maths fit together. Conversely, missing out on picking up the folklore of maths is arguably a much bigger loss than missing out on “live” philosophical discussions.