[FOM] Fwd: invitation to comment
joeshipman at aol.com
joeshipman at aol.com
Fri May 20 18:26:15 EDT 2011
The discussants are beginning to miss the point a little bit less, but
it is still much simpler than most have been supposing. Tim Chow is the
only person who seems to have followed up on the point I made
constructively, so I will elaborate here:
1) The situations for ZFC and PA are completely different and should
not be discussed together. As far as ZFC is concerned, Voevodsky's
remarks make sense, and specialists in foundations would do well to
emphasize connections between Con(ZFC) and more "purely mathematical"
statements such as the existence of a countably additive measure
defined on all subsets of the unit interval, or the Grothendieck
Universe axiom (until McLarty's recent work, would Voevodsky have
regarded Wiles's theorem as not having a purely mathematical proof?).
2) Any number of results Voevodsky would accept as purely mathematical,
for example the Robertson-Seymour graph minor theorem, not only use
principles in their standard proofs that can be used to prove Con(PA),
but have in fact been reversed so that it is a theorem in a very weak
theory that they themselves directly imply Con(PA). Therefore any talk
about Con(PA) not having a "purely mathematical proof" is simply wrong.
-- JS
-----Original Message-----
From: Kevin Watkins <kevin.watkins at gmail.com>
To: Foundations of Mathematics <fom at cs.nyu.edu>
Sent: Fri, May 20, 2011 2:54 pm
Subject: Re: [FOM] Fwd: invitation to comment
On Fri, May 20, 2011 at 2:00 AM, <Andre.Rodin at ens.fr> wrote:> 1. We
MAY possibly have a sound mathematical argument for> the Poincaré
conjecture.> Actually I believe we have already such an argument
(actually a proof) even I'm> not in a position to check details of
Perelman's proof. But I have confidence> to experts who cheked it.>> 2.
We MAY NOT possibly have a sound mathematical argument for> the
negation of the Poincaré conjecture. This is because it is no longer a>
conjecture but a theorem.>> 3. Perelman's proof of the Poincaré
conjecture IS a> mathematical proof proper, and DOES NOT involve some
further> non-mathematical assumptions.>> 4. Perelman's proof IS NOT a
mixture of mathematical reasoning> and philosophical speculation.>
Otherwise the mathematical community would not come to the consensus
about it> like it does not come to the consensus about foundations:
unlike mathematical> proofs philosophical speculations never result
into a consensus.Thank you... I see how that is consistent with the
point of viewexpressed in your earlier posting.If I haven't exhausted
your patience yet, I am wondering what,according to this point of view,
would change if someone were toestablish a reversal for the Poincaré
conjecture, in the sense thatover some weak base theory, the Poincaré
conjecture implies Con(PA).(I have no intuition for whether or not this
is even possible in thecase of the Poincaré conjecture... but for the
sake of argument, onemight admit the possibility that *some* acceptedly
"mathematical"result *might* eventually be shown to imply Con(PA) over
a weak basetheory.)In this case:1. Would the proof of the Poincaré
conjecture (or some substitute)cease to be regarded as a mathematical
proof proper?2. Would the proof come to be regarded as involving
philosophicalspeculation or non-mathematical assumptions?or
conversely:3. Would the proof of Con(PA), via the Poincaré conjecture
(or somesubstitute) come to be regarded as a purely mathematical
result?or alternatively:4. Would this point of view regard it as
inconceivable that it willever be shown that the Poincaré conjecture
(or any "mathematical"substitute) implies Con(PA), because the Poincaré
conjecture has amathematical proof, while according to this point of
view, Con(PA) cannever have a purely mathematical
proof?Kevin_______________________________________________FOM mailing
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