Subject: [HM] Bourbaki, theory, and problems
From: Colin McLarty (cxm7@po.cwru.edu)
Date: Tue Feb 01 2000 - 16:41:32 EST
I thank Franz Lemermeyer, John McCleary, and Jeremy Gray for very
thoughtful remarks on theory versus problems. This is an important question
to me for my work on understanding algebraic methods (specifically
homological methods) from Poincare through Grothendieck. I will urge some
ideas about it, but I hope it is clear that I am not sure of these ideas.
Jeremy Gray points out that, prima facie, the question of Bourbaki is
different from the question of abstract theory. As I read Mehrtens and Aubin
they both claim that historically Bourbaki's loss of influence was realized
as a trend from theory towards problems. Surely history of math ought to
include such large claims. But I think this particular claim is not so good.
It seems clear to me that Bourbaki is less influential today than in the
1960s, while I do not think there has been any large scale trend away from
theory and towards problems. (I also think Bourbaki style remains far more
influential than many people like to say. That is another issue.)
Bourbaki faced challenges from those who wanted less abstraction,
and also those who wanted more.
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).
The other side includes Grothendieck--and owes much to Serre even
though Serre does not favor the stress on large scale abstraction.
Grothendieck is more-than-Bourbakiste in two ways: First, Bourbaki regarded
their style as a means of summarizing math, not of doing research.
Grothendieck wanted all mathematics written this way as simultaneously
research report, detailed verification, and textbook. Second, where Bourbaki
reasons axiomatically about modules or topological spaces et c. (easily
formalized as sets in ZF), Grothendieck reasons axiomatically about
categories of modules or categories of spaces and even categories of
functors between such categories (formalized by adding Grothendieck
universes to ZF).
Grothendieck's high level abstraction produced many new concrete
structures. As an example, each etale sheaf on an arithmetic scheme is a
pretty small thing itself, with concrete arithmetic information. But
Grothendieck conceived of them in the general context of derived functor
cohomology--and he says it was in the same movement of thought which
produced toposes.
Of course, a trend towards concrete problems be the historically
decisive challenege to Bourbaki even if it was not the only challenge. But
I don't see such a trend. I am struck by the fact that throughout the late
19th century and all the 20th there are repeated claims that either:
1) Math has gotten too abstract and will suffer for it.
or
2) Math had recently gotten too abstract but now there is
a trend back to concrete problems.
Yet during this period math research as a whole has constantly
become more abstract, and never suffered from it. Abstract theory is a kind
of Hydra which is repeatedly killed and yet remains to be killed again. More
and more concrete problems are also solved, often using the new abstractions
and sometimes not.
I had asked whether Serre and Grothendieck count as pioneers in a
trend towards concrete problems, since they focussed on the Weil conjectures
during the 1950s and 1960s. Franz Lemermeyer gave a very helpful response to
me, though perhaps he and I will not entirely agree. Part of his response:
> I wouldn't think so - the main work here consisted in developing
>cohomology theories that would fit the predictions of Weil. I see
>this as a powerful tool of great generality. In a similar vein,
>I would count, say, Iwasawa theory to the general toolkit even
>though it can be used to answer concrete problems.
The abstract theory was developed with a specific concrete problem
in mind, and finally succeeded at solving it. I think we should not speak of
a trend away from the concrete towards abstraction (as people seek tools to
solve the problem) followed by a trend the other way (when people succeed).
Deligne was interested in concrete questions about theta functions
and the Ramanujan conjecture in the 1960s, and gave a proof assuming the
Weil conjectures. But he spent most of the 1960s working on etale cohomology
by which he finally proved the last Weil conjecture (and so the Ramanujan
conjecture) in 1972. And he remained interested in abstract ideas, such as
motives, after that. I do not believe he shifted his interests from abstract
to concrete or the other way. He lets each feed the other.
Lemermeyer and McClearly both talked about revival of interest in
s0ome 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.
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.
Lemermeyer concluded that we need caution in making such gneral
statements and I have to agree. But I hope I can be very cautious, for
example by testing my ideas here, and still support a very general claim:
mathematics at least since World War II has seen no serious trends opposing
abstract methods to concrete problems, but a constant growth of both in
lively interaction.
from very pretty snow-bound Cleveland, Colin
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