[FOM] Fwd: invitation to comment

[Kevin Watkins]

On Thu, May 19, 2011 at 3:39 AM,  <Andre.Rodin at ens.fr> wrote:
> I think the message is this. While for the in-consistency of PA and ZFC we may
> possibly have a sound *mathematical* argument (evidence) any attempted proof of
> the consistency of these systems will be not a mathematical proof proper but
> involve some further non-mathematical assumptions. So a mathematical proof of
> in-consistency of PA and/or ZFC will boost further mathematical research while
> the lack of such proof leaves us with this controversial mixture of
> mathematical reasoning and philosophical speculation that we call foundations.
>
> I'm sympathetic with this view, which does not imply, of course, that I think
> that the philosophical speculation is useless in general or that it is useless
> for maths. But we need a better separation of the two things. This is the
> principal task of *critical philosophy* in Kant's sense, to which I adhere.
>
> Andrei

I am an interested (layman, far from expert) observer of this discussion.

In trying to fully understand what you are proposing is the "message",
I think I would find it helpful, for contrast, to know what the
corresponding answers to the following would be according to this
point of view:

1. We [MAY / MAY NOT] possibly have a sound mathematical argument for
the Poincaré conjecture.
2. We [MAY / MAY NOT] possibly have a sound mathematical argument for
the negation of the Poincaré conjecture.
3. Perelman's proof of the Poincaré conjecture [IS / IS NOT] a
mathematical proof proper, and [DOES / DOES NOT] involve some further
non-mathematical assumptions.
4. Perelman's proof [IS / IS NOT] a mixture of mathematical reasoning
and philosophical speculation.

(Apologies if I am somehow missing the point...)

Kevin Watkins