[FOM] FLT Decisive by Normal Math Standards

[James Smith]

Hi,

there is a similar debate on MO about the abc conjecture at the moment:

https://mathoverflow.net/questions/232087/have-there-been-any-updates-on-mochizukis-proposed-proof-of-the-abc-conjecture

Regards,

James


On 08/01/18 22:44, Arnon Avron wrote:
> 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.
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