[FOM] Voevodsky corr051711,11:12AM-052211,11:05AM

[Monroe Eskew]

> On May 21, 2011, at 2:27 PM, Vladimir Voevodsky wrote:
>
> Pretty much just that. Because the "accepted mathematical principles" are
> themselves no more reliable in terms of consistency than PA.

In my opinion, this line of the correspondence is very revealing.  He
is saying more than just the fact that PA has less consistency
strength than many other "accepted mathematical principles."  He is
saying that he does not accept various proofs of the consistency of PA
because he does not accept the reliability of the underlying
principles.  Voevodsky rejects what most would consider ordinary
mathematical reasoning.  Further, he rejects it not just for some
philosophical reasons, but because he believes it will likely lead to
paradox, according to his statements in the video lecture.  This is of
course a bold hypothesis.  From my perspective, these views are very
radical.

But exactly how radical are they?  I wonder, is this case
idiosyncratic, or is it indicative of a growing methodological and
conceptual divergence between algebraists and the rest of
mathematicians?  Are we at a point where there is a serious lack of
common ground?  This seems like a very important and interesting
research question for the history and philosophy of mathematics, and
an alarming possibility for f.o.m.

Best,
Monroe