FOM: rigor and intuition
Vladik Kreinovich
vladik at cs.utep.edu
Wed Feb 13 11:03:07 EST 2002
Dear Friends,
I think Matt has a point but he is just picking Gromov as a wrong example. An
example I would support is Witten, a math genius behind M-branes and other
exciting features of modern theoretical physicsetc. Witten has proven a lot of
interesting mathematical theorems AND he also has a great intuition, so he
publsihes semi-physical papers where there are no proofs for some statements,
only arguments in favor.
This is clearly NOT mathematics in the mathematician's sense of the word, but
it is called mathematics by applied people, because he is makijng reasonable
statements about mathematical propositions about which we do not know yet
whether they are true or false. This is indeed a very useful activity from the
viewpoint of applications, because physics cannot wait until we prove theorems,
they want some answers now. If this answer is a mathematical theorm great. If
no theorem is known, a good intuitive guess is better than nothing.
The difference between this example and Gromov's theorems is that whn you ask
Gromov about the missing details of his proofs he definitely has them, so in
principle, by asking Misha Gromov a lot of questions, we can reconstruct the
precise proof.
With Witten's non-proved results, there are steps in his arguments about which
he himself does not know how to make them into precise proofs - and he does not
hide this fact of course.
In this sense, Gromov's results are theorems, while Witten's unproved "results"
are not yet theorems.
Matt's good point is that mathematics is not only about theorems, Witten's
stateents are also very useful, and many such statements are eventually proven
and transformed into theorems.
Vladik
> I would say that rigor and formalisms provide a needed basis for
> understanding among mathematicians. If all geometers treated rigor and
> formalisms in the way that Gromov and Thurston do, I think that subject
> would collapse in misunderstandings among its researchers. Gromov and
> Thurston's ideas have been incorporated into the mainstream of geometry
> largely because other mathematicians have taken the time to work them out
> rigorously.
>
> It seems that the mathematical community is willing to tolerate low
> standards of rigor from people whose intuitions are unusually helpful, but
> holds most of us to higher standards of rigor. This seems to be a
> productive approach.
>
> --Matt
>
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