Last night, I went to the event for New Waves In Philosophy Of Mathematics, where I got to hear Roy Cook talk about neo-logicism, Augustin Rayo talk about mathematical trivialism, Oystein Linnebo talk about the individuation of natural numbers and Otavio Bueno talk about mathematical fictionalism. I was at the original phil of math conference at the University of Miami the spring before last that the volume came out of (in fact, I had to get up early both days to go to Panera Bread to grab coffee and whatnot for the conference participants), and I indexed the book this last summer, and, yeah, Otavio is my dissertation advisor, so most of what I heard wasn't particularly new to me, but much of was interesting.
Note, in particular, that "trivialism" means something in this context totally unrelated to the sense usually used in my particular subarea--it doesn't mean that everything is true, it means that mathematical statements are just trivially true. Rayo argued (not quite convincingly to my way of thinking, but forcefully and entertainingly) that there are two kinds of mathematical Platonism: the kind that people argue for and the kind that people attack, and that the latter has very little to do with the former. They come apart because the former is non-trivial. In particular, Benaceraff-style worries about our epistemic access to mathematical objects are irrelevant to the former kind, because mathematical objects aren't an extra ingredient of reality above and beyond the more mundane sorts of objects. By way of illustrating this, he said that from the perspective of his sort of trivialist Platonism, to assert that there are no numbers is to commit oneself to a contradiction. After all, if you say that "there are no numbers," that means the same thing as "the number of numbers is zero," which means that at least one number--zero!--does exist, and you've just contradicted yourself.
Now, as an argument against someone who denies the existence of numbers, this seems transparently question-begging--why, after all, should they agree that "there are no Xs" and "the number of Xs is zero" are equivalent?--but my sense from the discussion in the Q & A was that Rayo understands this and wasn't using it in that way, but just giving an illustration of the trivialist Platonist's stance. Compare to an argument about whether Batman is Bruce Wayne or Harvey Dent. If someone thinks Dent is Batman, they will deny that Wayne is Batman, but one would be begging the question against them in a fairly silly way if one argued that to deny that Wayne is Batman is to deny that Batman is Batman and commit themselves to a logical absurdity. (After all, both sides could pull the same trick.) Like I said, Rayo's position on phil of mathematics is pretty distant from mine, but it was an interesting presentation of his view, and he concluded with some good remarks on the notions of "bloated ontology" and the like.
This morning, I went to see the "Author Meets Critics" session on Hartry Field's book "Saving Truth From Paradox." Some interesting points came out of the exchange, even if as a matter of personal taste I would have liked a bit more back-and-forth on bigger-picture philosophical issues about the book--Is Field's basic approach to the paradoxes on the right track? Is denying instances of the Law of the Excluded Middle sufficiently less counter-intuitive than accepting counter-examples to the Law of Non-Contradiction to make the considerable deficit in simplicity between dialetheism and Field's paracomplete approach worthwhile? Does his "algebraic" picture really capture the intuitive notion of truth?, etc., etc.--rather than quite so much focus on the intimate technical details. That said, some philosophical interesting points came through, particularly in the Q&A, and I finally got a chance to meet the man, who's the external reader for my dissertation committee but who I hadn't actually met in person before today.
....and, speaking of meeting people, I finally got a chance to meet Colin Caret--the guy who does the Inconsistent Thoughts blog and a regular commenter here. I guess I really should have seized the opportunity to argue about dialetheism. Instead, I ended up just exchanging a few pleasantries about the conference and whatnot then going off to (in theory) grab coffee with him and J.C. Beall and a few others, but actually just let Colin buy me a cup of coffee while I boorishly isolated myself at a corner of the table to continue a discussion with Otavio about Philip K. Dick and quantum logic.
Then came lunch with Jody Azzouni and Mark Colyvan, who I got to bore at length with exactly why I don't think Graham Priest's "classical re-capture" works. Then I ducked bag to my hotel in Koreatown to print some stuff out, put some stuff away, and blog.*
More on Wednesday.
*Just to be a complete jackass, I was sorely tempted to do a couple of Tristam Shandy-style stabs at descriptive completeness there, like "Then I wrote that sentence. Then that one...." But I just wouldn't feel right, gentle reader, wasting your time like that.