[FOM] FLT Decisive by Normal Math Standards

[Arnon Avron]

On Sun, Jan 07, 2018 at 04:41:55PM -0500, Harvey Friedman wrote:
> I would like to point out that FLT is considered a completely
> established mathematical theorem by what has come to be normal
> mathematical standards for a famous widely recognized statement.
> 
> Consider these ingredients.
> 
> 1. A proof was published in a major Journal after it was refereed with
> unusual care. That unusual care consisted of dividing the paper into
> many parts (perhaps 6 or so), with independent referees. The referees
> were well aware that it was important that the Journal - and therefore
> they - get it right.
> 
> 2. The process already passed a serious test by finding a serious
> error in the first round.
> 
> 3. After the fix and publication, and after having been looked at
> seriously by many experts in many seminars, no issues have arisen for
> many years.
> 
> This is much more than one has for an ordinarily fine result.

 We are not talking here about "an ordinarily fine result". Almost
all such results nowadays are of interest only for a very small group of
mathematicians in a certain area. As long as it remains so,
the people in that group can decide about the rules of their "game"
as they like. Who cares? But FLT, in contrast, is of interest to 
all of the mathematicians, and  to many, many, people that
are not mathematicians. Because of its simple formulation
and long challenging history, it has a lot of g.i.i. Accordingly,
the above criteria might suffice for publication in a journal
of algebraic geometry, but not for acceptance of FLT as an 
undoubted mathematical truth.
 
> Now perhaps people are not doubting this, but being rather focused on
> whether the proof was done in ZFC.
> 
> But the refereeing process probably didn't even touch this question.
> Rather, they probably focused totally on whether or not the proof
> meets the standard criteria that has been used in algebraic geometry
> and allied fields for  long periods of time.

FLT is not (only) a proposition of algebraic geometry, and it is
definitely not the property of the community of algebraic geometry.
Once this community has claimed to prove it, they should follow
the standard, eternal crfiteria of mathematics (at least mathematics
that deserves that name.) For example: making explicit in a very precise
way what are the assumptions that underline the alleged proof. 
 
> Probably none of the referees nor the author had any interest in the
> question of whether ZFC suffices. They generally have the point of
> view that mathematics is not built on any axiomatic framework and that
> axiomatic frameworks are a separate subject and have nothing to do
> with mathematical practice or the process by which mathematicians
> accept the validity of papers and results.

If you are right, and they are so careless, not minding
what are the mathematical assumptions that underlie their
proofs, then one should be *very* suspicious about their proofs.
First of all, the state of things you describe make
the possibility of a systematic error of
all these referees rather plausible. (The possibilty of a
`systematic error' is something that has been raised
by Kreisel in connection with the acceptance of Church Thesis -
acceptance which is by far more reliable than that of FLT!)
Second, no proof is more reliable than the assumptions on which
it is based. Period.  Moreover: Such assumptions always exist - and the first
thing a mathematician who is faithful to the real spirit
of mathematics should do, is to check what they are in a
given claimed proof, and whether 
s/he accepts them. What such a mathematician should *not*
do is to accept a proof of a theorem only because of the authority
of other people (especially if their standards, *according to what
you describe*, are rather low).

To sum up: if the algebraic geometers claim to have proved
something like FLT that is of great interest to people
outside their community, then it is about a time that they
show it - starting with clarifying what are the axioms
they adopt, and then presenting the proof in a way
that every mathemtician who is ready to devote her/his
time and energy to that can read and check the proof in a reasonable time.
(In the future there might be an additional demand:
that the proof will be fully formalized and checked
by a computerized proof-checker. This is slowly
becoming the norm in computer science.) 
I do not care if what is needed for that is to read a book of 200 
pages, or 600. (But the longer is the required book, the greater
are the chances that unspotted mistakes exist in the 
alleged proof). As long as this is not done, I shall
not believe that FLT has been proved, even if all other
mathematicians in the world would swear that it is. 


> This is the way the math community operates. 

This might indeed be what the math community has deteriorated
to in this post-modernist world.