[FOM] Fwd: invitation to comment

Andre.Rodin at ens.fr Andre.Rodin at ens.fr
Thu May 19 03:39:32 EDT 2011

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.


> Begin forwarded message:
> From: Vladimir Voevodsky <vladimir at ias.edu>
> Date: May 18, 2011 4:44:13 PM EDT
> To: Harvey Friedman <friedman at math.ohio-state.edu>
> Cc: Vladimir Aleksandrovich Voevodsky <vladimir at ias.edu>
> Subject: Re: invitation to comment
> Dear Harvey,
> such a comment will take some time to write ...
> To put it very shortly I think that in-consistency of Peano arithmetic
> as well as in-consistency of ZFC are open and very interesting
> problems in mathematics. Consistency on the other hand is not an
> interesting problem since it has been shown by Goedel to be impossible
> to proof.
> Vladimir.
> On May 17, 2011, at 2:45 PM, Harvey Friedman wrote:
> > Dear Professor Voevodsky,
> >
> > I have become aware of your online videos at
> http://video.ias.edu/voevodsky-80th
> >  and http://video.ias.edu/univalent/voevodsky. I was particularly
> > struck by your discussion of the "possible inconsistency of Peano
> > Arithmetic". This has created a lot of attention on the FOM email
> > list. As a subscriber to that list, I would very much like you to
> > send us an account of how you view the consistency of Peano
> > Arithmetic. In particular, how you view the usual mathematical proof
> > that Peano Arithmetic is consistent, and to what extent and in what
> > sense is "the consistency of Peano Arithmetic" a genuine open
> > problem in mathematics. It would also be of interest to hear your
> > conception of what foundations of mathematics is, or should be, or
> > could be, as it appears to be very different from traditional
> > conceptions of the foundations of mathematics.
> >
> > Respectfully yours,
> >
> > Harvey M. Friedman
> > Ohio State University
> > Distinguished University Professor
> > Mathematics, Philosophy, Computer Science
> >
> >
> >
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