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

[Colin McLarty]

On Sun, Jan 7, 2018 at 2:23 PM, Joe Shipman <joeshipman at aol.com> wrote:

> Tim, although I mostly agree with what you say, and although I think we
> agree that the top mathematicians in the relevant area of mathematics may
> have “known” before McLarty’s work that ZFC proves FLT in a sense that
> would have taken an unreasonable amount of effort to document fully, it is
> still a fair criticism that they didn’t make any statement about the
> eliminability of the “Universes” axiom. After all, with that axiom it is
> very easy to prove Con(ZFC), but they don’t go around claiming to have
> proven ZFC is consistent even though it is an interesting statement of the
> same logical type (pi^0_1) as FLT.
>

The case for adequacy of ZFC was a bit clearer than that.  The experts saw
how modest paraphrase would eliminate universes, along with a number of
good general theorems on the tools.  In fact Deligne has told me he likes
the general tools as  way of thinking, though he prefers the idea of
eliminating universes from proofs in principle.  He says essentially this
in his essay "Quelques idees maitresses de l'oeuvre d'A Grothendieck" on
the IAS website as
https://publications.ias.edu/sites/default/files/77_Quelqyesidees.pdf.  I
note that the misselling of "quelques" in the URL is necessary.  If you
spell it correctly the link does not work.


What is really going on, I believe, is that those mathematicians who had
> taken the effort to familiarize themselves with the entire “Grothendieck
> technology”, because that was itself scandalously inadequately documented
> in refereed publications, were particularly ill-suited to look favorably on
> calls for effort to be expended on documenting proofs for non-specialists.
>

This seems to me pretty much right, though I would make more allowance for
the scale and novelty of the work.  Grothendieck complains about aspect of
it in his memoir *Recoltes et Semailles.*

McLarty’s work should itself have been published in a refereed journal much
> sooner


I can complain about the refereeing process but honestly it is mostly my
own fault this is not  published yet.  I will fix that.  I certainly
appreciate the support for my work.

best, Colin

; but his achievement is remarkably impressive and once it is fully
> published there ought to be a lot of pressure put on anyone who writes
> about results in this field to CITE IT (getting FLT down to third order
> arithmetic is nice but getting practically everything else they did down to
> finite order arithmetic is much more important).
>
> — JS
>
> Sent from my iPhone
>
> >> On Jan 6, 2018, at 5:17 PM, tchow <tchow at alum.mit.edu> wrote:
> >>
> >> On 2018-01-06 11:30, aa at post.tau.ac.il wrote:
> >> Just relying on the swear of somebody,
> >> be it even Goedel himself, is  against the spirit and moral
> >> of mathematics.
> > [...]
> >> 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.
> >
> > It's nice to see that there are still some idealists in the world.
> >
> > For my own part, I have long since given up on the criterion you state
> here.  How many mathematicians are there alive today?  Obviously depends on
> your definition, but let's say 100,000 as a conservative estimate.  So for
> something to be declared proved by your criterion, at least 50,000
> mathematicians should be able to read and verify for themselves the proof.
> "Should be able" presumably means that said mathematician should be able to
> do so with a "reasonable" amount of effort, I guess?  Not that I have to
> drop everything I'm doing and spend the rest of my life on the
> verification?  What's a reasonable amount of effort?  Maybe 500 hours?  I
> would say that if we adopt this criterion, then the vast majority of
> mathematics is "not proved."
> >
> > I would say that the probability p that I personally would be able to
> verify a randomly chosen theorem from the literature within 500 working
> hours is very small.  In most cases, I would not even know where to start.
> Well, I guess I would start with the journal article, but most likely I
> would get lost after at most a few sentences, and then I would not know
> what to do next.  I would probably start hunting around for a textbook that
> used some of the same keywords.  But I would not know whether I had picked
> the right textbook, or how much of the textbook I would really need to
> understand for the purpose at hand.  Moreover, I would probably need to
> understand multiple textbooks and papers.  I might even have to refresh my
> memory on things that I used to know but have forgotten, which would eat
> into my 500-hour budget.
> >
> > Now, maybe the problem is that I am a particularly under-educated
> mathematician.  Maybe most mathematicians have a much higher p value than
> mine.  But somehow I doubt it.  Suppose I had a big pot of money and could
> say to a randomly chosen mathematician, "I will triple your salary for the
> next 6 months if you can thoroughly referee this randomly chosen paper."  I
> might get some takers but I doubt that in most cases the mathematician
> would be able, at the end, to honestly declare that the paper had been
> thoroughly checked, including everything cited by the proof.  If it's not
> in the mathematician's field then chances are that 500 hours is not enough.
> >
> > Now maybe you're happy saying that only 0.01% of what is commonly
> considered proved is really proved.  Fine.  But returning to the subject at
> hand, by your criterion, it is clear that Fermat's Last Theorem *has not
> been proved*.  And if it hasn't been proved, then why are you fussing about
> whether "the proof" goes beyond ZFC?  What proof?  There is no proof.
> >
> > Finally note that the point of my citing Brian Conrad was *not* to claim
> that "the proof does not go beyond ZFC" is a piece of knowledge, but to
> assert that *your* contention that the proof *does* go beyond ZFC is mere
> belief.  You don't understand the proof yourself, so on what basis do you
> claim that it goes beyond ZFC?  You're basing that claim on hearsay.
> Hearsay is not knowledge.
> >
> > Tim
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