On Jul 2, 2014, at 2:57 PM, Benjamin Udell <bud...@nyc.rr.com> wrote:
> Sung, list, I think you're getting into a thicket. Mathematicians have varied > on these questions. It’s interesting how there was a burst of activity on mathematical foundations in the early 20th century and then a lot of that dried up. It’s still a major topic in most introductions to philosophy of mathematics but until recently it seems more of a dead topic. The only interesting advance I could think of was Putnam’s perhaps Peirce inspired semi-empirical methods in mathematics paper. I rather enjoyed that paper but some noted that it was a bit after the fact given the reality of how computers were already being used in mathematical proof. As I recall Putnam’s paper came out somewhat after the Four Color Proof that brought a lot of attention to the role of computers in proofs. I think since then it’s just been accepted within mathematics that some unproven theorems appear to be true independent of proof and are treated as true. I’ve not kept up on the literature enough to know how that’s affected thinking about mathematical foundations. I have heard of late though that there’s starting to be more interest in mathematical foundations. Alas I’ve just not had time to follow the literature. I confess that I occasionally still enjoy reading some of those classic papers debating foundations and all the unease with different sorts of infinities in proofs. I’ve long thought that Peirce probably offers an unique approach that, other that Putnam touching somewhat on it, never really gets engaged with.
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