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

[David Fernandez Breton]

Arnon,


> Let me say it with even stronger words: in my opinion,
> a theorem can *really* be considered
> as proved in mathematics only when it reaches the stage in which its
> original
> proof has been  sufficiently simplified,  and then presented in textbooks
> in a way that most of the mathematicians can read and verify for
> themselves.
> If this cannot be done  or is not going to be done, then something
> very suspicious is going on.


I do believe that these things are not binary. There's no clear distinction
between a theorem  that can "really" be considered as proved, and one that
cannot. There are multiple degrees of certainty that we can have about the
truth of a theorem (possibly roughly depending on the set of people that
the proof is accessible to, in increasing order: just the person who
discovered the proof, a few "top" experts in the field, most experts in the
field, experts in the field plus many graduate students in the field, most
graduate students (not just from the field), etc.). And it's possible that
even when a theorem is (using your own words) "accessible to the
overwhelming majority of the mathematicians in the world, so that everyone
with Ph.D in mathematics [...] will be able to check the proof for
correctness", our certainty is still not 100%. 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). On the other hand, it is also conceivable that some gap is
consistently missed by many people for a while. 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?


> This is what happened in the past
> with every important mathematical theorem, and I find it very
> strange and very worrying that more than 20 years after the
> publication of Wiles's work, nothing like this has happened
> (as far as I know),  and the only significant progress that
> has been made is hidden in an unpublished
> work, the existence of which I could discover only after
> sending explicit questions to FOM.


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. While it's certainly possible that
eventually the mathematical community finds an "accessible" proof of FLT, I
wouldn't be surprised if this takes a long, long time. The fact that such a
proof hasn't been found after 20 years doesn't make me uneasy at all. And
in the meantime, everything that is not "accessible" to the general
mathematician is, by definition, only accessible to experts in the
corresponding field, and hence subject to the particular quirks of the
field. In the field of logic, it is very common to wait for a while after a
result has been found before submitting the corresponding paper for
publication, and in the meantime people learn about the result anyway (via
conference and seminar talks, and the arXiv). So the fact that Colin
McLarty's paper is unpublished doesn't strike me as particularly odd.

 David
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