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

[Timothy Y. Chow]

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.

> In fact, it is my understanding that something like that happened to the 
> Italian school of geometry; it seems that everyone missed a flawed 
> argument for about a whole century, but unfortunately I don't know any 
> more details about this. Perhaps someone with the appropriate historical 
> knowledge can fill us in?

Probably you're thinking about this:

"Intuition and Rigor and Enriques's Quest" by David Mumford
http://www.ams.org/notices/201102/rtx110200250p.pdf

>  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.

Tim