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

[Timothy Y. Chow]

Joe Shipman wrote:
> 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.

I agree that it's a "fair criticism" but I also think that it would be 
fair for them to take the attitude, "Who cares?"  That is, if you really 
care about the math involved, then you should take the time to study it 
yourself.  If it's not important enough to you to devote that time, then 
what business do you have telling them that they need to document what 
axioms were used?  Are they also supposed to document every time they use 
Replacement?  From their point of view, if you have some *mathematical 
application* (as opposed to sociological custom, or logical hygiene for 
its own sake) that requires you to ensure that universes are not used, 
they know how to go about doing so and will be happy to collaborate with 
you, but why go to the trouble before the need arises?

> 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

I wonder if the situation is really that different in other branches of 
mathematics.  It's very difficult to get to the frontier of any area of 
mathematics purely by self-study; unless you're Ramanujan, you are 
probably going to have to submit yourself to many years of formal 
schooling, all the way through the Ph.D. level at least.  Even after that, 
I think that for most areas, it's hard to do significant research unless 
you maintain constant contact with other researchers.

We've had similar complaints on FOM about other famous theorems, such as 
the classification of finite simple groups and the Poincare conjecture. 
I don't see that the problem is specific to arithmetic geometry.  It seems 
to have more to do with the amount of machinery needed to prove a specific 
result, rather than the "branch of mathematics" per se.  If there's a lot 
of machinery and there's scant reward for making the proof more accessible 
then the proof is probably going to remain inaccessible for a long time.

> 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).

I'm with you that 'f.o.m. don't get no respect' and the McLarty deserves 
more kudos for the excellent work that he's done.

Tim