I’ve been rereading a bunch of the stuff on Field in the last couple days (my exam is in about 14 hours – I hadn’t planned on publicizing this blog until afterwards, but I guess I ended up doing so early) and I’ve been struck by the thought that a lot of nominalist worries about the causal efficacy of mathematical objects could be put a bit more mildly.
In general I’m a believer in the Quine-Putnam argument, though I think that mathematics actually is dispensible, so it ends up being a nominalist argument. This means that if mathematics had turned out to be indispensible, then I’d have to believe in its entities and confront the epistemological worries. But I think they aren’t really a serious problem, because our theory can be confirmed as a whole. I do think there is a slight issue though, in that these numbers that enter into our physical theories aren’t located near the entities they’re correlated with, or connected to them in any obviously causal way. The appearance of abstract objects in a physical theory seems to me like it’s at least a minor point against the theory, because it’s basically a stronger version of the “action at a distance” that Einstein and the early modern scientists disliked so much.
Of course, if the best theory predicts action at a distance, then I guess we have to live with it, and if the best theory predicts action from nowhere then we have to live with that too. But it doesn’t mean I have to like it any more than I should have to like a theory that is essentially non-deterministic (like quantum mechanics) or has any of a number of other blemishes that I’d like to think an ideal scientific theory wouldn’t have.