On 20 Dec 2009 at 9:41, Roger Frye wrote:
> I think it is important to look at William Thurston's paper
> ON PROOF AND PROGRESS IN MATHEMATICS
> http://front.math.ucdavis.edu/math.HO/9404236
> in the context of the very provocative article which stimulated it by Arthur
> Jaffe and Frank Quinn
> "THEORETICAL MATHEMATICS": TOWARD A CULTURAL SYNTHESIS OF MATHEMATICS
> AND THEORETICAL PHYSICS
> http://front.math.ucdavis.edu/math.HO/9307227
I agree strongly, and I'm very glad you've added this to the
required reading!
What follows are some disjointed comments, not all perhaps
entirely relevant.
...
>I think the most damning is an observation by Enrico Fermi that is
>misquoted by responder Daniel Friedan.
Highly irrelevant (but I think amusing) footnote: Daniel Friedan
is Betty's son, and got his Berkeley Ph.D. in the same cohort
with Roger Schlafly, Phyllis's son (both had been advised, one
officially, the other less officially, by one of my advisors,
Isadore Singer)--the after-ceremony festivities were (I have
been told) distinctly chilly.
> Part of what makes the article by J&Q so provocative is that they
> judge several living mathematicians. Here is what they say about
> Thruston:
> William Thurston´s "geometrization theorem" concerning structures on
> Haken three-manifolds
> is another often-cited example. A grand insight delivered with
> beautiful but insufficient hints,
> the proof was never fully published. For many investigators this
> unredeemed claim became
> a roadblock rather than an inspiration.
"The proof was never fully published" because Thurston never had it
in any way that he could fully explain to anyone else (and I am
not sure he is still claiming he actually did have it). However,
it is no longer a conjecture; it was proved a few years ago by
Grigory Perlman, who had a bit earlier proved its special
case, the 3-dimensional Poincare conjecture. Perhaps the key
ingredient of that proof (which has since been thoroughly
validated, refined, and somewhat extended, by several semi-
independent teams of mathematicians; and a detailed proof
published by my other advisor, John Morgan) was "geometrization
via Ricci flow", which was definitely Thurston's idea, but
which he didn't work on (at least, not publicly) nearly as
much as several other people, notably Richard Hamilton--who
nonetheless hadn't managed to push his results far enough
to settle the question. After Perlman put his (somewhat
sketchy) paper on the ArXiv, Columbia (where he and Morgan
are) had a working seminar dedicated to filling in all the
details. Hamilton (again, I am told) did not attend: instead,
he would find out before each meeting what the result to be
proved in that meeting would be, then he'd leave and work it
out for himself. When the time came that he couldn't do that,
he went back to Hawai'i to surf and lick his wounds.
Morgan is famous for really, really *understanding* things,
and then writing them up very, very correctly and clearly.
He told me once that Dennis Sullivan (who, like Thurston,
was famous in his youth for his amazing mathematical
fertility and ability to tie together many seemingly
unrelated ideas, and who also like Thurston didn't
in his youth always write up all the details of
his work--but who, unlike Thurston, was denied a Fields
Medal on those grounds, though his proof of the Adams
Conjecture actually *existed* unlike Thurston's proof
of the Geometrization Conjecture {of course, Thurston's
prize was officially for other work; but there *was* a
lot of controversy over his getting the medal when he
did, as I recall} said, after seeing _Amadeus_ on stage,
that he felt about Thurston like Salieri did about
Mozart: "he does mathematics like a cow pissing".
(Morgan and Sullivan are Texans.)
Later, and probably not coincidentally, Sullivan told
me once that it took him a long time to really appreciate
that in geometry you should *measure* things. In a way,
Thurston's biggest contribution to the mathematical
Zeitgeist is probably that he made so-called "geometric
topology" actually use *geometry*. (Sidebar: topology
of course comes in many flavors. "Geometric topology",
taken broadly, means topology that actually is interested
in the detailed point-and-subset-level structure of
topological spaces, without descending into the baroque
excesses known as "point-set topology". It is distinguished
from "algebraic topology", which is usually willing to
substitute for any given space another space "homotopy-
equivalent" to it, since they will have all the same
"algebraic" structures. It is also distinguished from
"pointless topology", one of John Kennison's areas of
expertise, which hovers even higher above the gross
details of topological spaces by literally doing away
with the points and only considering [what used to be]
the open sets of the space, and their relationships.)
>
> One of the responders, Armand Borel, thinks that the
> Thurston program is harmful.
And so it might have been. But it turned out not to be,
even in the overlap between Borel's own field and
geometric topology (namely, "rigidity theorems", on
which I could discurse if anyone wanted).
>
> What a breath of fresh air to read Thurston's honest and inspiring
> vision, which even J&Q appreciate, though with qualifications:
> Thurston himself may obtain satisfactory understanding through informal
> channels,
> but he is a mathematician of extraordinary power and should be very
> careful about
> extrapolating from his experiences to the needs of others.
Exactly so. (Jaffe and Quinn are pretty damned powerful, themselves,
by the way.)
> Thurston replies to J&Q's concerns indirectly by framing his own questions
> and by concentrating on the "positive rather than on the
> contranegative." He has learned from an early mistake of
> killing the field of foliations by proving everything in sight
> and documenting it "in a conventional, formidable mathematician´s
> style."
It was certainly believed, at that time, by many (e.g., Morgan)
that Thurston had indeed killed foliation theory. But it has
sprung back to life.
>Now he is more concerned with how humans personally understand
> mathematics.
>
> Thurston's answer seems to be that within a field, there is a common
> understanding of which published proofs can be trusted and which
> are known to be false.
I agree. So do lots of people, certainly lots of topologists.
Sullivan and Morgan used to quote folk-wisdom, and/or
Sammy Eilenberg, on Solomon Lefschetz, the man who [in his own
words] "planted the harpoon of algebraic topology in the whale
of algebraic geometry", that "He never stated an incorrect theorem,
or gave a correct proof.") But I think this is somewhat field
-dependent within mathematics.
>There is a flow of ideas that can take a long time to communicate
>even to an expert in a related field. But he trusts that flow more
>than he trusts formality: "Most mathematicians adhere to
>foundational principles that are known to be polite fictions."
I don't think "adhere to" is correct; they (we) really don't *care*
about foundations (except, as the saying goes, on Sundays), and
rather than "adhering to", "give lip service to".
> Thurston admits that the field of low dimensional topology
> may need a more intuitive approach than "more algebraic or
> symbolic fields."
Again, I don't think "need" is the _mot juste_. I think it is
(or has been) more capable of *benefiting* from intuition(s),
particularly from *well-trained* intuitions, than are (or
have been) some other, "more algebraic or symbolic fields".
That *may* be a result of the human brain (or some humans'
brains) being naturally equipped to *have* its intuitions
about "low dimensional topology" trained, than it is to
have its algebraic intuitions (such as they are) trained.
We do, after all, inhabit a world that *appears* to us to be
low-dimensional (rather than either high-dimensional
[as string theory has it] or 0-dimensional, that is,
discrete [as A New Kind of Science has it]). As Georg
Kreisel put it, in riposte to Eugene Wigner's famous
article-title "The Unreasonable Effectiveness of Mathematics
in the Physical Sciences", "Would it be obviously more
'reasonable' if we were not effective in thinking about
the external world in which we have evolved?"
(This is in a footnote, devoted to a digression on
Bourbaki, to Kreisel's 1978 review of Wittgenstein's
Lectures on the Foundations of Mathematics, published in
the Bulletin of the American Mathematical Society, volume
84, pages 79-90. Anyone who wants even *more* to read
ought to read that review, too. Kreisel, passim, is
in my opinion very sound in his account of how actual,
working mathematicians behave when it comes to "foundations".)
> What a shame that mathematicians who are not understood by their
> contemporaries (Fourier, Galois, Poincaré, etc.) are ignored.
> ON the other hand, we do need standards.
As you know, of course, the nice thing about standards is
that there are so many of them...
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