FOM: FLT, 4CT, CFSG: Depth, length, and width of proofs

[Joe Shipman]

Alexander Zenkin wrote:

> Dear Joe,
>
> first of all thanks very much for your brilliant analysis of the
> outstanding  solutions of the three great Theorems: FLT, 4CT and CFSG as to
> their "long, deep, and wide". But if it is possible, could you tell your
> opinion on the theorems, and especially on the theorems solutions as to
> their influence upon the mathematics as a whole or, maybe, upon any specific
> directions of mathematics?
>
> Thank you in advance,
> Alexander

FLT:  The theorem itself has no direct impact but the proof has a lot of
influence in Number Theory because Wiles developed a number of new techniques.

4CT: The theorem itself has no direct impact but the proof had a great deal of
philosophical and foundational importance (though no particular importance in
its field of graph theory).

CFSG: The theorem had a big impact in group theory because many open questions
could now be resolved (the classification of all finite groups with a property P
could for many P be reduced to questions about simple groups).  The proof was
such a hodgepodge that it is hard to assess its impact -- I am sure some of the
pieces involved important new techniques in group theory, but I think almost all
its influence outside of group theory is sociological/philosophical.

Thanks for the kind words!

-- JS