FOM: December 1 - December 22, 1998

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

1) Fermat's Last Theorem

I have in front of me a book, "Modular Forms and Fermat's Last Theorem"
(Springer 1997, eds. Cornell, Silverman, Stevens, ISBN# 0-387-94609-8)
which contains (in some sense) a "proof" of Wiles's Theorem ("Fermat's
Last Theorem").  It is nearly 600 pages long and chapters are written by
over two dozen contributors.  Chapters are:

I  An Overview of the Proof of Fermat's Last Theorem (Stevens)
II A Survey of the Arithmetic Theory of Elliptic Curves (Silverman)
III Modular Curves, Hecke Correspondences, and L-Functions (Rohrlich)
IV Galois Cohomology (Washington)
V Finite Flat Group Schemes (Tate)
VI Three Lectures on the Modularity of /rhobar_E,3 and the Langlands
Reciprocity Conjecture (Gelbart)
VII Serre's Conjectures (Edixhoven)
VIII An Introduction to the Deformation Theory of Galois Representations
IX Explicit Construction of Universal Deformation Rings (De Smit and
X Hecke Algebras and the Gorenstein Property (Tilouine)
XI Criteria for Complete Intersections (De Smit, Rubin, and Schoof)
XII l-adic Modular Deformations and Wiles's "Main Conjecture" (Diamond
and Ribet)
XIII The Flat deformation Functor (Conrad)
XIV Hecke Rings and Universal Deformation Rings (De Shalit)
XV Explicit Families of Elliptic Curves with Prescribed Mod N
Representations (Silverberg)
XVI Modularity of Mod 5 Representations (Rubin)
XVII An Extension of Wiles's Results (Diamond, appendix by Diamond and
XVIII Class Field Theory and the First Case of Fermat's Last Theorem
(Lenstra and Stevenhagen)
XIX Remark's on the History of Fermat's Last Theorem 1844 to 1984
XX On Ternary Equations of Fermat Type and Relations with Elliptic
Curves (Frey)
XXI Wiles's Theorem and the Arithmetic of Elliptic Curves (Darmon)

This is *exactly* the kind of textbook I like, which deals with a lot of
different subjects in a focused way by developing each of them only to
the point needed for the proof of some "big" result.  Unfortunately, it
is too hard for me because I am not an algebraic number theorist!

Let me say this differently: I took a course in algebraic number theory
in graduate school, which basically went through Serge Lang's "Red Book"
on the subject.  That is NOT enough of a prerequisite to understand this
book!  I would guess that only a third-year grad student who had been
studying algebraic number theory for two years already would really be
able to tackle this book and honestly say at the end that he understood
the proof.

To put it another way, this book would have to be combined with two
other substantial ones (e.g. Lang's and its successor for the
SECOND-year graduate algebraic number theory course) to have any chance
of being acceptable as a proof of FLT to a mathematician who is not a
specialist in the field (and even there we are assuming a solid general
mathematical background which might require one more book).  And that
"proof" would have about the minimum level of detail permissible and
require a great deal of work to verify.  It is an ambition of mine to
actually learn this proof someday, though it looks like it will have to
wait until I get an academic job.

2) The Four-color Theorem

This theorem is generally held to have a proof that is "too long" to be
humanly verifiable.  But the difficult part of the proof
is not all that bad -- one journal article of a few dozen pages conatins
sufficient information for any good mathematician to
satisfy himself that if two specific algorithms report success for some
input, then 4CT is true.  The two algorithms (a "discharging" algorithm
to generate an unavoidable set and a "reducing" algorithm to show each
graph in the unavoidable set is reducible) are also supplied along with
the input that is supposed to work, and the program code and input size
are also quite feasible.  The trace of the computer runs would be much
too large to check by hand, though the computer runs themselves only
take a few hours on a Sun workstation.

Which of these two "proofs" is REALLY longer?  Which is more reliable?
This is obviously a trick question, but I think the point is clear that
*some* mathematicians who actually understand both proofs will have more
confidence in 4CT [because the Number Theory is so difficult and they
can write their own versions of the algorithms to check], while *others*
will have more confidence in FLT [like the contributors to the Springer
book who have internalized all the background material and have worked
with it for many years].

3) The Classification of Finite Simple Groups

I'd like to know more about this proof.  My impression (anyone who
disagrees please correct me) is that the "length" of the proof includes
almost all the relevant background material (in other words it does not
build on a lot of previous mathematics).  In this way it resembles 4CT
rather than FLT.  Also, certain core pieces of the argument (like the
Feit-Thompson odd order theorem) are both very long and very
well-verified (in the case of the OOT, thousands of people have
carefully read the proof but no one has been able to shorten it by very
much, it is "essentially" long in some ill-defined sense), while other
pieces are long but not so well-verified.  The not-so-well-verified
pieces are somehow considered less central and if a mistake is found in
them the feeling is it can be worked around.

We need more precise concepts of  "depth" and "width" here, but the
impression I get is that the CFSG proof is *wide* but not deep, except
for a few pieces which are both deep and very well-verified.  The 4CT
proof is very *long* but almost all of it is extremely shallow (and it
isn't particularly wide because it's pretty uniform).  The FLT proof is
shorter than the other two but very *deep* (and wide too, though still
more unified than CFSG).

These are three excellent examples!  Most of the other "big" proofs I
can think of don't raise any serious issues because they are not too
long, not too deep, and not too wide.

-- Joe Shipman

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