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

[tchow]

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