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I would like to discuss an impression I had while reading Doron Zeilberger's 111th opinion. The impression that I had can be distilled in a simple question:

What is to point of proving rigorously things that physicists already can `explain'?

Doron seems of the opinion that there is usually no point. I guess in context one example Doron is referring to is the H-Theorem and Villani's work. For a moment I couldn't think of any convincing argument and momentarily became depressed as I am interested in rigorous mathematical physics.

After a little thought I came up with the following reason as to why it is worthwhile to rigorously prove things that physicists already know. Maybe the physicists knowledge about the object in question is not as deep as they think. After all if it were wouldn't a rigorous proof be easy? On the same note, the physicists may be able to predict some things in the model but their explanation for why it happens might be wrong. In the future they may, after physical experiments or theoretical arguments, realize this but the point with mathematical rigor is that after something is proven you know that your explanation is eternally correct.

In closing I would say that rigorous proofs in mathematical physics only validate the physicists reasoning most of the time. However, sometimes they show such reasoning to be faulty and as a result we gain more insight into the object in question.

Do you have any arguments for or against the above question?

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closed as not constructive by Bruce Westbury, José Figueroa-O'Farrill, Willie Wong, Peter Shor, Felipe Voloch May 7 '11 at 10:33

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Your post doesn't have much to do with Zeilberger's 111th Opinion. His punchline is that we should do more down-to-earth (i.e. programmable) discrete constructive exact mathematics rather than advanced complicated continuous unconstructive approximative mathematics. In other words, more algebra and combinatorics, and less analysis and statistics. This interpretation is in line with many of his other opinions, and I agree with much of it. (Although I would not equate "discrete" with "down-to-earth".) –  darij grinberg May 7 '11 at 9:20
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I wonder whether it is worse to have too few opinions, or too many. –  Angelo May 7 '11 at 11:04
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@Angelo, while this particular one is not among my favorites, I am at least glad that Zeilberger has so 'many opinions,' as I find (almost all of) them very interesting; which does not mean I completely share them, but I find them thought-provoking. (Of course, by 'many opinions' I mean the number of texts termed as such on his webpage, and not mutually contradictory opinions; as aluded to by darij grinberg, the general point of view expressed via them seems very consistent to me.) –  quid May 7 '11 at 11:29
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I personally felt vicariously embarrassed reading this opinion. The charge of boringness is weak -- the vast majority of humankind would find Zeilberger's stuff boring! –  Todd Trimble May 7 '11 at 13:34
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@unknown: It's okay not to be mainstream, provided you make a good argument. But in this case, his pronouncing the Fields Medalists' work as 'boring' is about as meaningful and informed as it would be coming from just about anyone else. I also find it churlish and rude. (But, enough of that for now.) –  Todd Trimble May 7 '11 at 21:06

1 Answer 1

That's probably not quite the right formulation of an issue. "Pure mathematics" has been defined as generating proofs, for around a century. Proofs come in various kinds: "cleaning the stables" and "insightful" are two that are relevant here. There are perfectly good reasons to point the idea of "proof" at physics-style "derivations". But the reasons are not usually ones physicists find solidly convincing.

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