[FOM] What is the current state of the research about proving FLT?

[David Fernandez Breton]

>
> David Fernandez Breton wrote:
> >  For one, there's always the possibility that there is an inconsistency
> >  somewhere in the relevant axiom system (whichever it is that is
> necessary
> >  to carry out the corresponding proof).
>
> I don't believe that this is a legitimate source of uncertainty.
>
> Consider some "easy" proof such as the irrationality of sqrt(2).  If we
> were to formalize this proof in some axiomatic system S and then were to
> discover that S is inconsistent, then we would simply shrug and discard S.
> The inconsistency of S would not affect our belief in the correctness of
> the proof that sqrt(2) is irrational.  We would just come up with some
> other system S' in which we could formalize the proof.  And if S' turns
> out to be inconsistent, we would come up with S''.  Throughout, our
> certainty in the correctness of the proof of the irrationality of sqrt(2)
> would remain unmoved.
>

That's fair, but that's too elementary of a statement. As soon as we start
writing more "theoretically-heavy" statements (think some basic theorem in
measure theory, or in Banach space theory), it seems to me that these are
less likely to be considered "simply true, no matter what" and more likely
to be re-evaluated in the light of some hypothetical inconsistency. In any
case, I'm not too concerned about inconsistencies either (I don't think an
inconsistency will be found in my lifetime, even in something like ZFC+a
proper class of measurables), except for the fact that if we're
(unreasonably) strict, we'll never be 100% sure that there isn't one. My
larger point was just that, for a number of different reasons (some more
powerful than others), we will never attain this ideal of absolute
certainty that Arnon seemed to be advocating for.


> Probably you're thinking about this:
>
> "Intuition and Rigor and Enriques's Quest" by David Mumford
> http://www.ams.org/notices/201102/rtx110200250p.pdf


Perhaps. I'll have a look... thanks a lot for this reference!


> >  I don't know that this has actually happened with "every" important
> >  mathematical theorem, but at least in the case of FLT, keep in mind that
> >  we're talking about a problem that remained open for almost 400 years,
> and
> >  a proof that occupied about 200 pages.
>
> For the purposes of Arnon's concerns, the proof is much longer than 200
> pages, because we have to include all the pages of everything that the
> papers
> of Wiles and Taylor-Wiles rely on.
>

Completely fair. So, all in all, I'd say it's not odd (and certainly not
scandalous) that we still don't have any "accessible" proof of FLT.

   David
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