[FOM] Fwd: invitation to comment

[Kevin Watkins]

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 view
expressed 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 to
establish a reversal for the Poincaré conjecture, in the sense that
over some weak base theory, the Poincaré conjecture implies Con(PA).

(I have no intuition for whether or not this is even possible in the
case of the Poincaré conjecture... but for the sake of argument, one
might admit the possibility that *some* acceptedly "mathematical"
result *might* eventually be shown to imply Con(PA) over a weak base
theory.)

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 philosophical
speculation or non-mathematical assumptions?

or conversely:

3. Would the proof of Con(PA), via the Poincaré conjecture (or some
substitute) come to be regarded as a purely mathematical result?

or alternatively:

4. Would this point of view regard it as inconceivable that it will
ever be shown that the Poincaré conjecture (or any "mathematical"
substitute) implies Con(PA), because the Poincaré conjecture has a
mathematical proof, while according to this point of view, Con(PA) can
never have a purely mathematical proof?

Kevin