First, I’ll mention that I’ve updated my blogroll – there’s been a real burst in math blogs over the summer, at least in part instigated by my friends at the Secret Blogging Seminar, but also by the spurt of Fields Medalists with blogs. (Are we up to 10% of the total number now?) I’ve also added a few philosophy blogs that I’ve been reading for a while, and a couple that I should have been reading, but of course I’m sure I’m missing others.
Anyway, there’s new math job search gossip stuff going on on the web – I think the discussion on that post is interesting and relevant across disciplines for people trying to figure out whether this is generally a good thing or not.
Tim Gowers discusses the way logarithms and other abstract things should be taught. He advocates a way that’s a bit more formalist than some others suggest, but it sounds reasonable to me. There’s also interesting discussion of formalism there in the comments, though some of it sounds more like structuralism to me. See for example Terence Tao’s comment, “I guess there is a fundamental transition in mathematical learning when one realises that what mathematical objects are (and how they are constructed) may be less important than what mathematical objects do (e.g. what properties they obey).”
Also, a discussion about the Axiom of Choice at The Everything Seminar (I may add that one to my links later too), focusing on a puzzle I first heard from my friend Lukas Biewald. There’s interesting discussion in the comments that reveals implicit ideas about platonism and formalism among mathematicians. I think the anti-platonist majority there should be a bit more careful though, because similar issues apply in arithmetic, thanks to Gödel’s results. I think we should be much more hesitant to say that the natural numbers are just something we make up than they are with the universe of ZFC (or a topos, or whatever), as I mentioned before.