Thurston vs Jaffe and Quinn
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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 ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
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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... ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
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In reply to this post by Roger Frye-3
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... Lee Rudolph ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
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