[FOM] FLT Decisive by Normal Math Standards

[Harvey Friedman]

On Mon, Jan 8, 2018 at 5:44 PM, Arnon Avron <aa at tau.ac.il> 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.

ARNON WROTE:

>  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.

Generally speaking, FLT has only a rather limited kind of interest to
non mathematicians. That interest stems entirely from its particularly
high transparency and the simplest parts of its history. But it
doesn't come close to speaking (at least directly) to any intellectual
issues non mathematicians have. Furthermore, the level of certainty
behind FLT is far far far greater than what we generally see in any
subject other than mathematics, even in the hard sciences.
(Mathematical Physics, and Statistics, and computer science at the
theoretical level is included in mathematics, for this purpose, but
once one is seriously engaging with "reality", my remark applies).

CAN WE AND HOW CAN WE MAKE MATHEMATICAL RESULTS CERTAIN OR MORE
CERTAIN THAN THEY ARE NOW?

is a seriously interesting multidimensional foundational matter that I
have long wanted to do research on, and hope to get some serious time
to deal with it in 2019 (certainly not 2018).

> 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.

You and I think that the foundational assumptions underlying a
mathematical proof are of great importance, but mathematicians to my
knowledge, have never acknowledged any importance in this. Part of the
problem is that they remain deliberately unfamiliar with foundations,
and don't care. This is not new, and will continue until the dam
breaks (for them) in the new f.o.m. revolution ongoing this century.
>
> 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.
...

I'm sure you have heard the phrase "if it isn't broke don't fix it".
That is the nearly universal attitude of mathematicians. They did get
foundational at least briefly when division by zero and adding
divergent series and too much infinitesimal talk was making the
leading mathematicians of the time very nervous, in the 19th century.
All of this culminated by around 1920 with ZFC, and since then
mathematicians assume that there are no foundational problems, and
don't even bother to look when they *feel* good about proofs. This is
not going to change until the current f.o.m. revolution reaches a
significantly higher level of intensity.

> First of all, the state of things you describe make
> the possibility of a systematic error of
> all these referees rather plausible.

But compared to the non mathematical world, pure mathematics is
completely singularly in great shape, foundationally, and in terms of
certainty. Of course I am interested, at least as much as you, in
issues of certainty, witness the above in all caps. But I don't expect
hardly any normal mathematicians to care at all - at least until the
damn breaks later this century. You have to demonstrate a crisis
before they care.

Harvey Friedman