Re: [HM] 20th century mathematics and knot-theory


Subject: Re: [HM] 20th century mathematics and knot-theory
From: Colin McLarty (cxm7@po.cwru.edu)
Date: Fri Mar 24 2000 - 13:04:25 EST


        Wow a lot of ideas here.

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> 1. Conjecture: There have been two global revolutions in 20th
> century mathematics.
> The first one is concerned with: set theory, the axiomatic method,
> modern algebra (its most important proponent was Hilbert).
> The second one is concerned with: structures, category theory (its
> most important proponent was the Bourbaki group) (Both periods of
> change were usually subordinated under the concept of scientific
> modernity (e.g. in the work of Herbert Mehrtens).

        I don't recall Mehrtens saying much about category theory. But I
think Mehrtens is right to depict Bourbaki as the highpoint of Hilbert's
heritage in abstract axiomatic mathematics.

        But that is not all of Hilbert's heritage and another aspect,
especially due to Emmy Noether is the heavy reliance on morphisms. Actually,
she credits this to Dedekind, and indeed finds it in his work--but no one
before her had managed to recognize it there. This was formalized as
category theory.

        As to Bourbaki and category theory, you have to read Leo Corry
Modern algebra and the rise of mathematical structures (Birkhauser Verlag,
1996).

Individual Bourbakistes did a great deal with category theory--for example
Eilenberg co-invented it. Cartan did much, and taught it to Serre. Serre
likes to use the least possible aparatus, and so he minimizes appeal to
category theory, but obviously his work on spectral sequences and derived
functor cohomology is based on categorical ideas. All these people, but
especially, Serre I would guess, taught category theory to Grothendieck who
of course flew it to the moon and back. But Bourbaki as a group rejected
category theory and severely damaged their program by doing so. Grothendieck
later said they abandoned their ambition to produce a comprehensive
framework for mathematics as it is practiced, when they chose not to use
category theory.

        As I have argued before on HM, category theory is a part of the
"postmodern" mathematics which has broken out of the canonical bourbakiste
mold--without at all rejecting Bourbaki's methods as far as they go.
(Mehrtens recognizes only the more specifically problem-oriented moves
beyond Bourbaki as "post-modern". These also cannot do without Bourbaki's
methods in practice, though some have a sharply anti-Bourbakiste rhetoric.)

> 3. Conjecture: Algebraic topology was one of the central parts of
> mathematics of the first two periods. It had a low-dimensional origin,
> starting with Poincare and the classification problem of 3-manifolds,
> which is reflected in the book of Seifert and Threllfall. As Bourbaki
> mathematics conquered the world, algebraic topology became one of its
> most important parts.

        That is true of the Bourbaki members Eilenberg, Cartan, Serre,
Grothendieck, but not of the books published by Bourbaki as a group.

> Connected with this is a change in the nature of algebraic topology:
> its main objects are no longer low-dimensional manifolds, but topological
> spaces in arbitrary dimensions. Its most famous problem became the
> understanding of the homotopy groups of spheres.

        Well, maybe more famous are the Weil conjectures, which Weil and
Serre and Grothendieck understood as algebraic topology (perhaps
Grothendieck had the greatest faith in this view, and Weil the least).
Algebraic topology became cohomology, and all kinds of objects have
cohomology--topological spaces (in the point set sense), algebraic
varieties, groups, field extensions, everything.

        Grothendieck says that the proper objects of topology are not just
the point sets, but all toposes, precisely because cohomology applies
uniformly to all toposes.

        From that perspective we would at least have to say that Falting's
1986 Fields medal was also algebraic topology. And cohomological methods
appear throughout Wiles's proof of (part of) the Taniyama-Shimura conjecture
and so (all of) the Fermat theorem.

        I don't know much about knots. But I wonder if there is not a lot of
cohomology behind many of the later results that you give as showing a move
away from algebraic topology.

        Anyway, in some states of the US it is legal to shoot people for
holding obnoxious views on the nature of mathematics, and I hope nothing I
said here seems argumentative. I think you have a great project going.

best, Colin



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