Subject: [HM] "theoretical tools" vs "problems"
From: Colin McLarty (cxm7@po.cwru.edu)
Date: Mon Jan 17 2000 - 18:19:00 EST
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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