FOM: was Weyl on Riemann surfaces fom?

[Martin Davis]

At 11:04 PM 1/13/98 -0500, simpson at math.psu.edu wrote:

>Martin, thanks for that clarification.  To a certain extent, I agree
>with both your original statement and your clarified statement.  But I
>have some extremely serious reservations.  We need to be very cautious
>here.
>
>For instance, in the light of your clarified statement, what do you
>make of Lou's example, Riemann surfaces?  The history is that there
>was something of a crisis in function theory (= the theory of analytic
>functions of one complex variable) because people were using
>questionable techniques involving multi-valued functions, etc.  Then
>Weyl came along and cleaned this up by giving a rigorous,
>set-theoretic definition of the concept of a Riemann surface.  Would
>you call this a contribution to f.o.m.?  Lou would.  I wouldn't.  I
>have no problem calling it a contribution to f.o.f.t., foundations of
>function theory.  But I definitely wouldn't call it a contribution to
>f.o.m.
>

Well, it's been 50 years since I learned about Riemann surfaces and even
studied Weyl's monograph. So this took some thought. Also, I've been
attending a conference at MSRI here in Berkeley, and was able to grab Lou
and get some clarification from him.

>From my point of view the question
     Was it fom?
is badly posed. Weyl's monograph was a straightforward contribution to
technical mathematics. However, viewed historically, it was part of a
*movement* that most definitely was fom. The movement has been called the
"arithmetization of analysis" There were contributions by Cauchy, Bolzano,
Weierstrass, Cantor and Dedekind. In the process various concepts that had
been understood on the basis of geometric and physical intuitions came to be
carefully analyzed and understood in purely arithmetic terms.

Riemann was, it goes without saying, a great genius, in some ways well ahead
of his time. But he was perfectly happy to use physical reasoning to justify
what he called Dirichlet's principle to obtain harmonic functions in
arbitrary domains. 

In my judgement this is not atypical of advances in fom. Rarely is it a
matter of an investigator setting out to analyze some fundamental concept.
Rather in the process of working on a technical problem, the investigator is
forced to dig deeper, and results may be obtained that compel new
understandings. Examples: Cantor's work on trigonometric series, led him to
transfinite iteration of the process of forming the derived set of a set of
points and thus to developing set theory. G\"odel's fundamental
contributions occurred in the context of very specific problems that had
been set by Hilbert.

Frege is the one contributor to fom I can think of who really did proceed by
setting forth to analyze concepts (logical reasoning, cardinality).

Martin