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

[Joe Shipman]

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.

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. 

Even the proofs FROM the Universes axiom were very far from accessible for an unconscionably long time by the standards of other branches of mathematics; there were enough people of sufficient eminence who had earned enough credibility that their vouching for the correctness of the results was accepted, and this acceptance was professionally appropriate, but there was practically no way for even a gifted mathematician to embark upon a self-study course to learn the subject on his own, especially if his French was less than fluent, and that contrasts sharply with almost every other branch of pure mathematics. There was no real doubt about the correctness of the big results, and all the Fields medals were well-deserved, but the situation is still unfortunate and worth criticizing.

This is a larger issue than the eliminability of the Universes axiom, but the number theorists ought to have at least ADDRESSED the eliminability issue; however, any of them who were inclined to making the subject better documented and pedagogically acceptable had bigger fish to fry because a much larger subset of mathematicians cared that their proofs from ZFC+UA get properly written up than cared about the elimination of UA.

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