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
and in the context of the 15 solicited replies to Jaffe & Quinn
http://front.math.ucdavis.edu/math.HO/9404229
and finally in the context of the response of Jaffe & Quinn to Thurston and the
gang of 15
http://front.math.ucdavis.edu/math.HO/9404231
Jaffe and Quinn ask whether speculative mathematics is dangerous, urge caution,
and prescribe 3 types of self censorship that most, but not all, of the
distinguished responders think would stifle mathematics. J&Q frame their
argument by an analogy between theoretical and experimental physics on the one
hand and speculative and proved mathematics on the other. Several responders
raise objections to the analogy. I think the most damning is an observation by
Enrico Fermi that is misquoted by responder Daniel Friedan. J&Q see proof as
validating a conjecture and rendering it useful for posterity or as disproving
it and showing that the conjecture should not have been made. Fermi's comment
on physical experiment is "There are two possible outcomes: if the result
confirms the hypothesis, then you've made a measurement. If the result is
contrary to the hypothesis, then you've made a discovery."
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.
They are not alone in this judgment. One of the responders, Armand Borel,
thinks that the Thurston program is harmful. On the other hand, Saunders Mac
Lane makes a nice analogy between intuition and faith:
Mathematics requires both intuitive work (e.g., Gromov, Thurston) and
precision (J. Frank Adams, J.-P Serre).
In theological terms, we are not saved by faith alone, but by faith and
works.
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.
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." 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. 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."
Thurston admits that the field of low dimensional topology may need a more
intuitive approach than "more algebraic or symbolic fields." And this judgment
is borne out by several of the responders to J&Q, when discussing Poincaré.
About his mistakes, Morris Hirsch says, "Poincaré in his work on Analysis Situs
was being as rigorous as he could, and certainly was not consciously
speculative." At the other end of the spectrum, Benoit Mandelbrot says
In the recently published letters from Hermite, his mentor, to
Mittag-Leffler, there are constant complaints
about Poincar ́e’s unwillingness to heed well-intentioned advice and
polish and publish full proofs.
Concluding that Poincar ́e was incurable, Hermite and E. Picard (who
inherited his mantle) shunned Poincaré,
prevented him from teaching mathematics, and made him teach
mathematical physics, then astronomy.
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. I do agree that speculation is dangerous, but that is what makes it
so important.
-Roger
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