Subject: Re: [HM] Bourbaki, theory, and problems
From: Volker Eisermann (eiserman@math.uni-bonn.de)
Date: Fri Mar 31 2000 - 05:05:40 EST
Dear List-members,
Colin McLarty encouraged me to put the following reply to his posting
on the list.
> 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?
>
As I have a similar impression as Houzel (maybe I would date the tendencie
back to the more concrete a little bit later), I am very interested in
the question, how to make a statement like this more precise and how
to collect evidence for (or against) it. So let me just make a few
(hasty) remarks.
Suggestion:
The study of one example for the proposed trend towards abstractnes and
the beginning trend backwards to more concret things might be best studied
by looking at the development of particular fields (algebraic geometry,
number theory, algebraic topology etc.)
Algebraic Geometry:
If you look at the developments of the objects of algebraic geometry:
real affine verieties, complex proj. varieties, varieties over arbitrary
fields, schemes we can interpret this as a trend towards abstractness.
In the 60th, I suppose, a great part of the work done in algebraic
geometry was about schemes.
Today, there seems to be much work in algebraic geometry that doesn't
use schemes (partly coming from physics, like mirror-symmetry).
(At the ICM 1998 none of the seven invited lectures on algebraic
geometry was about schemes).
Topology and related Geometry:
I think, it is fair to say that the study of low dimensional manifolds
is more concrete and the study of generalised cohomology-theories is
more abstract. In the late 70th, early 80th we see some very important
work on 3-dim. (e.g. Thurston) or 4-dim. (e.g. Donaldson, Freedman)
manifolds.
What about looking at greatest figures: of course that's an
oversimplification, but it may be ok if used just as a first
approximation:
propose that in the 60th the leading figure was Grothendieck
and in the 90th the leading figure was Witten, compare
both and conclude that Witten represents a much different style of
mathematics than Grothendieck. Than we have one (indeed only very
small, but anyway) kind of evidence, that something has changed.
> Smale's work on the Poincare conjecture could be one case of the
> trend, except that it was done in the 1960s.
>
On the other hand, Smale's work on the Poincare conjecture is more
general than the work of Freedman on Poinc. conj. in dim 4.
>> Those who wanted less are the familiar champions of problem-oriented
>> math including Erdos and Siegel. Often Hermann Weyl is quoted on this
>> side, when he said that mathematics was becoming too abstract and
>> would soon run out of motivation for its ideas. But Weyl is problematic
>> since the "abstractions" he complained about in 1931 had become good
>> old basic algebra by the 1950s or 60s (and he reputedly recanted).
(By the way: There are constant rumors that it was precisly this attitude
of Siegel that caused that Hirzebruch had'n got a professorship in
Goettingen. Goettingen therefore loosing its position as Germanys
mathematical center to Bonn.)
>> Lemermeyer and McClearly both talked about revival of interest in
>> some concrete problems in the 1970s and I'm sure they are right. I
>> appreciate learning the cases they describe. And easy access to powerful
>> computers certainly would contribute to this.
If they are not already there, I think, you should add knots to this list.
>> But at the same time the 1970s produced great theoretical projects.
>> McCleary mentioned Robert Fulton using cohomology of schemes to re-prove
>> (or first prove, or disprove) results in Italian algebraic geometry. I
>> could well be wrong here, as I am no expert, but I don't think anyone
>> would say that Fulton turned in any systematic sense away from theory
>> and towards problems in the 1970s.
That is a point where it may be useful to have more precise concepts of
concret, problem-oriented and abstract, theory-oriented at hand:
I think we should make a difference between
- using an general theory to solve concret problems
and
- working on the general theory itself (its foundations, its relations
to other theories, trying to make it more powerful).
The first I would count for the concrete side, only the second for the
abstract side.
There are many cases where I would hestitate, e.g. the classification of
finite simple groups.
So we need to develop a more precise concept of what we mean by saying a
theorem, a concept, a paper, a book, a talk, the life-work of a
mathematician, a part of mathematics, is concret/abstract, oriented towards
abstract theory /oriented towards concrete problems.
So, my suggestion is: making the concept precise enough that we can
make a rational (not necessarily unambiguous) decision if some
paper is abstract/concrete and than take a certain amount of material
(e.g the papers published in Annals of Mathematics, or the Invited
lectures at ICMs) and apply our precise concepts to them being as honest
as we can.
Best regards,
Volker Eisermann
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