Clark, list,
Yes, it's quasi-empiricism, embraced by Chaitin, Wolfram, and others.
I remember years ago happening upon a passage in which Peirce said that,
in the case of mathematical questions proven undecidable, answers might
still be reached, or approached, through abductive and inductive
inference. This is one of those passages of which I neglected to make a
note at the time and I've since been unable to find it. It's possible
that I misunderstood it, but I remembered it because I had already
wondered what Peirce thought about mathematical undecidables of the kind
that mathematical platonism would hold to have a true answer (i.e., not
whether parallel lines meet, which is postulational, but whether, say, a
given sequence of a trillion digits appears an infinite number of times
in the decimal expansion of pi - well actually I'm unsure whether that's
a good example but I hope my idea is clear).
Best, Ben
On Jul 2, 2014, at 10:06 PM, Clark Goble <[email protected]> wrote:
> 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.
Correction. I just checked my bookshelf and my memory was *way* off.
Putnam wrote on this way back in the 1970s. Far before the Four Color
Theorem proof. My apologies. That’ll teach me to go by memory.
On 7/3/2014 12:06 AM, Clark Goble wrote:
On Jul 2, 2014, at 2:57 PM, Benjamin Udellmwrote:
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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