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Quick note: many of you are saying that this guy is totally out of touch with how math works, and that formal derivation of proofs is just fantasy. It's true th

Quick note: many of you are saying that this guy is totally out of touch with how math works, and that formal derivation of proofs is just fantasy. It's true that that's an extreme view, but it is in fact one that is held by some top mathematicians, even if a small minority. There are people who have built up research programs out of trying to make it a reality. See, for instance, Doron Zeilberger, who apparently describes himself as an "ultrafinitist": http://en.wikipedia.org/wiki/Doron_Zeilberger http://en.wikipedia.org/wiki/Doron_Zeilberger http://www.math.rutgers.edu/~zeilberg/OPINIONS.html http://www.math.rutgers.edu/~zeilberg/OPINIONS.html I'm not saying I agree with that, just that it's not lunacy. Carry on.

jsyedidia 18y ago
I agree. The original post is echoing many of the views expressed at more length by Sussman and Wisdom in their book "Structure and Interpretation of Classical Mechanics." There they showed that a computational expression of classical mechanics was much more explicit and rigorous than the standard mathematical or physical treatments. They also have a similar long paper on differential geometry from a computational perspective: http://groups.csail.mit.edu/mac/users/wisdom/AIM-2005-003.pdf http://groups.csail.mit.edu/mac/users/wisdom/AIM-2005-003.pd...
- cchooper 18y ago
  Their point in SICM was that traditional notation was not as ambiguous as their computational notation. That's because traditional notation is designed to be convenient rather than explicit. They are right, but this does not mean that physics lacks rigour. More explicit notations were always available and physicists, being more intelligent than computers, were capable of resolving the ambiguities to reveal the fully rigorous structure underneath. There's a difference between using a convenient notation and lacking rigour. If you are always capable of discerning the true formal equations behind a convenient notation then everything is OK. If you can't, or people can't agree on what the formal meaning should be, then there is indeed a problem. The fact that the authors could derive a computational notation that no physicist would disagree with is proof that a lack of rigour never existed in the first place.
  - kragen 18y ago
    Do you mean, "was not as unambiguous"?
    - cchooper 18y ago
      Indeed I do. Thanks.
  - jsyedidia 18y ago
    You write that "More explicit notations were always available and physicists, being more intelligent than computers, were capable of resolving the ambiguities to reveal the fully rigorous structure underneath." I doubt that's true. For example, here's what Piet Hut, now a professor of physics at the Institute for Advanced Studies at Princeton, writes about his experiences with classical mechanics as an undergraduate in his review of SICM available at http://www.ids.ias.edu/~piet/publ/other/sicm.html http://www.ids.ias.edu/~piet/publ/other/sicm.html : "Soon I went through the library in search of books on the variational principle in classical mechanics. I found several heavy tomes, borrowed them all, and started on the one that looked most attractive. Alas, it didn't take long for me to realize that there was quite a bit of hand-waving involved. There was no clear definition of the procedure used for computing path integrals, let alone for the operations of differentiating them in various ways, by using partial derivatives and/or using an ordinary derivative along a particular path. And when and why the end points of the various paths had to be considered fixed or open to variation also was unclear, contributing to the overall confusion. Working through the canned exercises was not very difficult, and from an instrumental point of view, my book was quite clear, as long as the reader would stick to simple examples. But the ambiguity of the presentation frustrated me, and I started scanning through other, even more detailed books. Alas, nowhere did I find the clarity that I desired, and after a few months I simply gave up. Like generations of students before me, I reluctantly accepted the dictum that `you should not try to understand quantum mechanics, since that will lead you astray for doing physics', and going even further, I also gave up trying to really understand classical mechanics! Psychological defense mechanisms turned my bitter sense of disappointment into a dull sense of disenchantment."
    - 18y ago
cchooper 18y ago
It's not the claim that mathematics sometimes lacks rigour that's bothering anyone. It's the claim that computer science, as a subject, is somehow more rigorous than mathematics. It's the false equivalence of programs with computer science that's wrong.

jsyedidia 18y ago

I agree. The original post is echoing many of the views expressed at more length by Sussman and Wisdom in their book "Structure and Interpretation of Classical Mechanics." There they showed that a computational expression of classical mechanics was much more explicit and rigorous than the standard mathematical or physical treatments. They also have a similar long paper on differential geometry from a computational perspective: http://groups.csail.mit.edu/mac/users/wisdom/AIM-2005-003.pdf http://groups.csail.mit.edu/mac/users/wisdom/AIM-2005-003.pd...

cchooper 18y ago
Their point in SICM was that traditional notation was not as ambiguous as their computational notation. That's because traditional notation is designed to be convenient rather than explicit. They are right, but this does not mean that physics lacks rigour. More explicit notations were always available and physicists, being more intelligent than computers, were capable of resolving the ambiguities to reveal the fully rigorous structure underneath. There's a difference between using a convenient notation and lacking rigour. If you are always capable of discerning the true formal equations behind a convenient notation then everything is OK. If you can't, or people can't agree on what the formal meaning should be, then there is indeed a problem. The fact that the authors could derive a computational notation that no physicist would disagree with is proof that a lack of rigour never existed in the first place.
- kragen 18y ago
  Do you mean, "was not as unambiguous"?
  - cchooper 18y ago
    Indeed I do. Thanks.
- jsyedidia 18y ago
  You write that "More explicit notations were always available and physicists, being more intelligent than computers, were capable of resolving the ambiguities to reveal the fully rigorous structure underneath." I doubt that's true. For example, here's what Piet Hut, now a professor of physics at the Institute for Advanced Studies at Princeton, writes about his experiences with classical mechanics as an undergraduate in his review of SICM available at http://www.ids.ias.edu/~piet/publ/other/sicm.html http://www.ids.ias.edu/~piet/publ/other/sicm.html : "Soon I went through the library in search of books on the variational principle in classical mechanics. I found several heavy tomes, borrowed them all, and started on the one that looked most attractive. Alas, it didn't take long for me to realize that there was quite a bit of hand-waving involved. There was no clear definition of the procedure used for computing path integrals, let alone for the operations of differentiating them in various ways, by using partial derivatives and/or using an ordinary derivative along a particular path. And when and why the end points of the various paths had to be considered fixed or open to variation also was unclear, contributing to the overall confusion. Working through the canned exercises was not very difficult, and from an instrumental point of view, my book was quite clear, as long as the reader would stick to simple examples. But the ambiguity of the presentation frustrated me, and I started scanning through other, even more detailed books. Alas, nowhere did I find the clarity that I desired, and after a few months I simply gave up. Like generations of students before me, I reluctantly accepted the dictum that `you should not try to understand quantum mechanics, since that will lead you astray for doing physics', and going even further, I also gave up trying to really understand classical mechanics! Psychological defense mechanisms turned my bitter sense of disappointment into a dull sense of disenchantment."
  - 18y ago

cchooper 18y ago

It's not the claim that mathematics sometimes lacks rigour that's bothering anyone. It's the claim that computer science, as a subject, is somehow more rigorous than mathematics. It's the false equivalence of programs with computer science that's wrong.