FOM: High aspirations for FOM - A Modest Proposal*

[Dan Halpern]

Hi all,
I just got on this list thanks to Leo Harrington and John Addison. Started
receiving the mail on Wednesday. So please bear with me if I don't seem to
be quite with it.  And I probably "don't get it" in more ways than have been
suggested so far. Actually I've been out of mathematics for more than ten
years.  I now consider myself a software engineer with an unusual :-)
background in mathematical logic.  However, that hasn't interfered with a
long held propensity to muse on foundational problems. In fact it's probably
accentuated the tendency and I muse now in a broader but less technical
context. 

The first note I received was Harvey's ReplyToLou dated 10/29. I found it
provocative in many ways, the most interesting of which was the suggestion
that "there are spectacular surprises to come from FOM for the practice of
mathematics."  To me that reinforces a feeling that new approaches are
needed in FOM.  (I see that Vaughn Pratt is suggesting one and infer that
Harvey is as well.) Coincidentally, that night I watched a recording of the
Nova program "The Proof" about Wiles' odyssey proving Fermat's Last Theorem.
(Do see it if you can.  It was brilliant!) For me the program suggested what
might be a seminal problem for such a new approach. In a nutshell the
problem is: "What "proof" did Fermat have?"  He, almost certainly, did not
have what we would consider a proof. But what made his claim so intriguing
is that all his other claims of theorems, and now this one, turned out to be
true.  In one case he made a conjecture and said he wasn't able to prove it.
That conjecture turned out to be false. I don't know enough about him to
know what he had ever written down in the way of proofs for his other
claims.  Of course, gathering such information would be a first step in
attacking the problem.

What would an answer look like?  It might involve analysing the kind of
reasoning he was using and then finding a domain of problems for which the
reasoning is logically sound. Or it might involve a concept like "almost
logically sound". Some of the analysis might involve historical, or
sociological issues. Who knows - eventually it might involve personal issues
like spiritual belief or sexual inclination. That might catch the attention
of Steve's barber! It would certainly expand the horizons of FOM - an
expansion that may be necessary if Harvey's claim is to come true.

Another shape for the answer is contained implicitly in the following quote
by Shimura in talking about his collaborator, Taniyama: 
      "Taniyama was not a very careful person as a mathematician - he made 
      a lot of mistakes.  But he made mistakes in a good direction and so 
      eventually he got the right answer. I tried to imitate him but I found 
      out it is very difficult to make good mistakes."

In fact, answering the question about Fermat might be facilitated by talking
to Shimura about Taniyama. It boggles the mind - Shimura and Taniyama would
then have played a key role in proving the theorem and solving the mystery! 

:-)
Dan Halpern

Footnote:
* Another candidate for the subject line was: "It's how they do it, not what
they do, stupid", a takeoff on a slogan of the 1992 Clinton campaign for
President.  No personal aspersions intended! It's a way of suggesting a
change of focus toward more consideration of how mathematics is practiced as
opposed to the collection of theorems that result. Close to but not quite
what Pratt is suggesting.