Subject: Re: [HM] "theoretical tools" vs "problems"
From: Jeremy Gray (j.j.gray@open.ac.uk)
Date: Sat Jan 22 2000 - 06:40:19 EST
Dear All
Colin's question is interesting.
Theory versus problems is an old debate, fortunately, and the opinion that
the balance has recently swung towards problems and away from grand theories
is widespread. Although I'm not sure if Aubin and Colin McLarty are
confining their attention to Bourbaki or looking more broadly, let me offer
some evidence for the current view.
First, my crude litmus test: if it looks like a grand cohomology theory
which, when constructed, might enable someone to do something, I'm going to
call it THEORY. If it's lots of strange examples that make a creaky kind of
sense but need to be understood better, the activity is PROBLEMS. Most
(nearly all?) mathematics is somewhere between these extremes. Maybe it's a
matter of locating it on a spectrum.
Even within Bourbaki I think one could argue that the seminars are less
tightly focused on algebraic geometry, algebraic number theory and matters
related to the Weil conjectures. But I haven't attempted a statistical
analysis, and besides I'm not sure I could say where theory ends and problem
solving begins. But around Bourbaki there was all the Douady-Hubbard work on
iteration theory, Mandelbrot sets and the like, which is in large part
PROBLEMS. A little further away there is ergodic theory, flows on manifolds,
anything to do with Erdos, differential equations (ordinary or partial) that
have a direct physical interpretation.
An amusing aside, my French sources tell me that analytic number theorists
chafed under Weil's suggestion that algebraic number theory was the real
number theory and algebraic number theory was analysis, and argued that
their style of work is directly linked to what Euler did. But algebraic
number theory certainly seems more like PROBLEMS.
There was also the French success with Fields Medals at the Zurich ICM, a
victory for partial differential equation theorists, sandwiched between the
more (is this too indiscreet?) Atiyah-style approach that prefers the
mathematics of Drinfelds and Wittens. Zurich medallists' work looked more
like PROBLEMS than THEORY.
Plus I think I detect, at least in the UK, a rise in what one might call
elementary combinatorics. I don't mean it's easy, one might even regard it
as a sign of maturity, but it's combinatorics for its own sake, as distinct
from combinatorics to solve problems in statistics, or to solve problems in
the cohomology of singular varieties (or what Richard Stanley does). I think
there is a feeling that grand abstract THEORY is not the only game in town
any more (only a fool would deny it real value, as witness Wiles's work). It
is possible to get away with arguing that THEORY has been over-done, but
harder than it was to get away with arguing that elementary combinatorics is
JUST problem-solving.
One final point, my feeling is that applied mathematics is inevitably more
PROBLEM focused, and I have the impression that that is where American money
is going these days, and not just because applied mathematics may be
intrinsically more expensive (because it uses more computer time).
Back to you all.
Best wishes
Jeremy
----------------------------- Original Message -----------------------------
From: Colin McLarty <cxm7@po.cwru.edu>
Sent: 17 January 2000 23:19
To: historia-matematica@chasque.apc.org
Subject: [HM] "theoretical tools" vs "problems"
David Aubin has written a terrific dissertation: A CULTURAL HISTORY
OF CATASTROPHES AND CHAOS: AROUND THE "INSTITUT DES HAUTES ETUDES
SCIENTIFIQUES," FRANCE. You can get it from University Microfilms, in fact,
though a glitch in their system leads them to tell some people it is not
available.
It includes a valuable discussion of Bourbaki (drawing on Beaulieu
and Corry and others) to contrast Bourbaki's ideas with those of Thom and
Ruelle. This section includes a claim I disagree with, and I wonder what
others will say. I believe a very similar claim is in Mehrtens
MODERNE-SPRACH-MATHEMATIK, perhaps not so specific about the timing, but I
do not find the citation right now.
Aubin takes some quotes from the mathematician Christian Houzel, and
agrees with them, and here I quote Aubin's own summary:
While the previous period [1950s and 1960s] was one that
had witnessed the development of powerful new theoretical
tools of great generality, he [Houzel] noted, the 1970s
were rather characterized by a tendency to revive an old
interest in more concrete problems. (p.39)
I wonder how to recognize this new trend in concrete examples. For
example, in the 1950s and 1960s Serre and especially Grothendieck aimed
their work very largely at one problem in number theory: the Weil
conjectures, estimating the number of solutions to a set of polynomials in
any finite field. Deligne finished the job, using Grothendieck's methods, in
1972. Are Serre and Grothendieck taken as pioneers in the new trend towards
concrete problems?
Smale's work on the Poincare conjecture could be one case of the
trend, except that it was done in the 1960s.
Rene Thom sees his work on catastrophe theory in the 1970s as
less "theoretical" than his work on cobordism in the 1950s. So he is no
counterexample to the trend. But is that enough to make the "trend"?
To take the most famous theorem of the past 50 years, Andrew Wiles
certainly set out to solve one problem of very "old interest", Fermat's Last
Theorem. But he succeeded by proving a far more general one, a step in the
Langland's program which is among the most sweeping grand projects ever
taken in mathematics--and which arose in the 1970s.
Can any trend away from theory and towards problems actually be found?
I will say, what should not need saying, that my objection to this
claim goes along with great appreciation of Aubin's and Mehrtens's work in
advancing more interpretive history of mathematics.
Colin McLarty
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