A little Proof, Dr Thurston! It aint Elementary!
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Who the Hell is Dr. Thurston? And how does he support his astonishing statement that mathematical proofs are "easier" than programming? I can respond only with my own experience. I was a mathematics scholar at the University of Cambridge (like Newton, but a little later). My supervisor and research director was a friend and colleague of Alan Turing. I always disappointed him in not fully understanding many of the established theorems of Analysis although, as with a computer today, I was able to use them very productively. I have vivid memories of studying in a medieval freezy student room in the wee post midnight hours, looking at two lines of Dedekind, and saying, "How in Hardy's name does that line follow that one?" Mathematics is a difficult subject, and, like playing the violin or composing sonnets, not everyone has the mind to do it. Programming is a beast of a different color. I have written and programmed a number of codes. Real ones; some used by NASA, others to design airfoils actually flown on record breaking aircraft, one, on wind turbine siting, sold here and in Europe at $25,000 a crack. They were tiresome to write, boring, detailed and demanding diligent programmers with more patience than I. But they involved no intellectual challenge beyond my limited apprehension. The above are just personal, factual statements. They PROVE nothing. But they do suggest from personal experience that Thurston's statement is BS. I mean, Theory crumbles before the Facts. Peter Lissaman, Da Vinci Ventures ============================================================ 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 |
Re: A little Proof, Dr Thurston! It aint Elementary!
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I think this is him:
http://en.wikipedia.org/wiki/William_Thurston Thanks Robert C [hidden email] wrote:
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Re: A little Proof, Dr Thurston! It aint Elementary!
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On 12/15/09 12:27 PM, Robert J. Cordingley wrote:
I think this is him:The essay that Russ mentioned only mentions programming in passing.. He doesn't say anything about it relative to `intellectual challenge', but he does talk a lot about what is the deep value of his enterprise. The message is in some sense that the rigor is not the end, it's the means. Some quotes: "When one considers how hard it is to write a computer program even approaching the intellectual scope of a good mathematical paper, and how much greater time and effort have been put into it to make it "almost" formally correct, it is preposterous to claim that mathematics as we practice it is anywhere near formally correct. Mathematics as we practice it is much less formally complete and precise than other sciences, but it is much less formally complete and precise for its content than computer programs. The difference has to do not just with the amount of effort: the kind of effort is qualitatively different. In large computer programs, a tremendous proportion of effort must be spent on myriad compatibility issues: making sure that all of the definitions are consistent, developing "good" data structures that have useful but not cumbersome generality, deciding on the "right" generality for functions, etc. The proportion of effort spent on the working part of a large program, as distinguished from the bookkeeping part, is surprisingly small. Because of the compatibility issues that almost inevitably escalate out of hand because the "right" definitions changes as generality and functionality are added, computer programs usually need to be rewritten frequently, often from scratch." and "The standard of correctness and completeness necessary to get a computer program to work at all is a couple orders of magnitude higher than the mathematical community's standard of valid proofs. Nonetheless, large computer programs, even when they have been very carefully written and very carefully tested, always seem to have bugs." ..and then he goes on to talk about how mathematics lacks a holistic sort of factoring process like goes on with large [evolving] programs. And many other interesting, reasonable remarks. ============================================================ 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 |
Re: A little Proof, Dr Thurston! It aint Elementary!
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In reply to this post by plissaman
Robert,
thanks for the additional quotations.
However, you made a slip of the fingers when you keyed in one of the passages. To head off needless controversy, I key it in correctly below. The capitalized word is where the slipup occured.
Easy slip to make because of the structure of the sentence.
n
Nicholas S. Thompson
Emeritus Professor of Psychology and Ethology,
Clark University ([hidden email])
http://www.cusf.org [City University of Santa Fe]
============================================================ 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 |
Re: A little Proof, Dr Thurston! It aint Elementary!
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On 12/15/09 2:10 PM, Nicholas Thompson wrote:
Sorry about that, I need a compiler to throw my random crap at. :-) While I'm at it, let me put in another plug for Coders At Work by Peter Seibel. Marcus ============================================================ 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 |
Re: A little Proof, Dr Thurston! It aint Elementary!
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In reply to this post by Nick Thompson
As noted below, Thurston’s interesting paper states that:
“Mathematics AS WE PRACTICE IT, is much more formally complete and precise than other sciences, but it is much less formally complete and precise for its content than computer programs.” If it were required that mathematical proofs be written in full formality, then some of them would be unreadable. Verifying a typical published proof might be similar to proving the correctness of a large computer program. Even worse, the basic ideas of the proof would no longer be visible. As Thurston points out, the basic ideas are what mathematicians actually work with in practice. Consider the theory of integrating Cos^n (kx) (for n a positive integer) as taught in a calculus class. Briefly, if n is odd, we can integrate this function by rewriting all but one factor of Cos(kx) in terms of Sin(kx) (using Cos^2(kx)=1-Sin^2(kx)) then making the substitution U = Sin(kx). If n is even, we can use a trigonometric identity to reduce the problem to integrating smaller powers of Cos(kx). [The identity is Cos^2(kx)=(1+Cos(2kx))/2.] Thus, we can reduce the problem to finding integrals of smaller and smaller even powers of Cos (kx) until we get to the second or first power, Cos^2(kx) or Cos(kx), both of which can readily be integrated. The above argument would be clear to mathematicians, and extremely good calculus students, and would enable them to compute certain integrals. Calculus students who are good, but not extremely good, would generally need to see a few examples before the process becomes clear. Weaker students would need to see more examples –maybe a lot more. Theoretically, the argument in the above paragraph would be an acceptable proof that, for example, the integral of Cos^n(kx) can always be carried out and has a certain form. Although the argument is informal and sketchy, it does give the main ideas, which is enough for mathematicians to convince themselves that the method is correct. A completely formal version of the argument would be like constructing a program to do the integration, then verifying that the program works. The main ideas would be lost. --John ________________________________________ From: [hidden email] [[hidden email]] On Behalf Of Nicholas Thompson [[hidden email]] Sent: Tuesday, December 15, 2009 4:10 PM To: [hidden email] Subject: Re: [FRIAM] A little Proof, Dr Thurston! It aint Elementary! Robert, thanks for the additional quotations. However, you made a slip of the fingers when you keyed in one of the passages. To head off needless controversy, I key it in correctly below. The capitalized word is where the slipup occured. "Mathematics as we practice it is much MORE formally complete and precise than other sciences, but it is much less formally complete and precise for its content than computer programs." Easy slip to make because of the structure of the sentence. n Nicholas S. Thompson Emeritus Professor of Psychology and Ethology, Clark University ([hidden email]<mailto:[hidden email]>) http://home.earthlink.net/~nickthompson/naturaldesigns/ http://www.cusf.org [City University of Santa Fe] ----- Original Message ----- From: Marcus G. Daniels<mailto:[hidden email]> To: [hidden email]<mailto:[hidden email]> Sent: 12/15/2009 1:09:23 PM Subject: Re: [FRIAM] A little Proof, Dr Thurston! It aint Elementary! On 12/15/09 12:27 PM, Robert J. Cordingley wrote: I think this is him: http://en.wikipedia.org/wiki/William_Thurston The essay that Russ mentioned only mentions programming in passing.. He doesn't say anything about it relative to `intellectual challenge', but he does talk a lot about what is the deep value of his enterprise. The message is in some sense that the rigor is not the end, it's the means. Some quotes: "When one considers how hard it is to write a computer program even approaching the intellectual scope of a good mathematical paper, and how much greater time and effort have been put into it to make it "almost" formally correct, it is preposterous to claim that mathematics as we practice it is anywhere near formally correct. Mathematics as we practice it is much less formally complete and precise than other scienc es, but it is much less formally complete and precise for its content than computer programs. The difference has to do not just with the amount of effort: the kind of effort is qualitatively different. In large computer programs, a tremendous proportion of effort must be spent on myriad compatibility issues: making sure that all of the definitions are consistent, developing "good" data structures that have useful but not cumbersome generality, deciding on the "right" generality for functions, etc. The proportion of effort spent on the working part of a large program, as distinguished from the bookkeeping part, is surprisingly small. Because of the compatibility issues that almost inevitably escalate out of hand because the "right" definitions changes as generality and functionality are added, computer programs usually need to be rewritten frequently, often from scratch." and "The standard of correctness and completenes s necessary to get a computer program to work at all is a couple orders of magnitude higher than the mathematical community's standard of valid proofs. Nonetheless, large computer programs, even when they have been very carefully written and very carefully tested, always seem to have bugs." ..and then he goes on to talk about how mathematics lacks a holistic sort of factoring process like goes on with large [evolving] programs. And many other interesting, reasonable remarks. ============================================================ 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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