From: scerir (scerir@libero.it)
Date: Sat Nov 23 2002 - 10:39:39 MST
Mathematician Renč Thom died recently.
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Thom.html
http://www.economist.com/people/displayStory.cfm?story_id=1441693
Here is an interesting letter, about mathematics ... and 'rigour'!
> Renč Thom
> Institute des Hautes Etudes Scientifiques
> 35 Rue de Chartres 91440
> Bures-sur-Yvette France
>
>
> Many thanks for your letter of May 21st with the enclosed article
> by Arthur Jaffe and Frank Quinn. I have many reasons to be
> interested in it, not only because I am personally implicated in the
> "Cautionary Tales". There, I can only confirm that the
> description of my evolution with respect to mathematics is fairly
> accurate. Before 1958 I lived in a mathematical milieu involving
> essentially Bourbakist people, and even if I was not particularly
> rigorous, these people -- H. Cartan, J.-P. Serre, and H. Whitney (a
> would-be Bourbakist) -- helped me to maintain a fairly acceptable
> level of rigor. It was only after the Fields medal (1958) that I
> gave way to my natural tendencies, with the (eventually disastrous)
> results which followed. Moreover, a few years after that, I became
> a colleague of Alexander Grothendieck at the IHES, a fact which
> encouraged me to consider rigor as a very unnecessary quality in
> mathematical thinking. I somewhat regret that the authors, when
> quoting my work in singularity theory, did not emphasize its
> positive aspects, namely, the transversality lemma (with respect to
> jet systems), the theory of stratified spaces (allowing for some
> anticipatory work by H. Whitney and S. Lojasiewicz), the
> characterization of "gentle maps" (those without blowing up), the
> II and III isotopy lemmas. All this was _written_ for the first
> time in my unrigorous papers. Of course many people (Milnor,
> Mather, Malgrange, Trotman and his school, McPherson, to quote just
> a few) may claim to have a large part in the rigorous presentation
> of this theory.
>
> This leads me to the Jaffe-Quinn paper itself, which involves a
> very important question, and provides, I think, the first occasion
> (apart from some solemn observations of S. Mac Lane) for an in-depth
> discussion on mathematical rigor. I do still believe that rigor is
> a relative notion, not an absolute one. It depends on the
> background readers have and are expected to use in their judgment.
> Since the collapse of Hilbert's program and the advent of Goedel's
> theorem, we know that rigor can be no more than a local and
> sociological criterion. It is true that such practical criteria
> may frequently be "ordered" according to abstract logical
> requirements, but it is by no means certain that these sociological
> contexts can be _completely_ ordered, even asymptotically.
>
> One main argument of the Jaffe-Quinn paper is that we have to know,
> when we want to use it for further research, if a published result
> may be considered as "firm" as another, whether its validity may
> be universally accepted. My feeling is that it is unethical for a
> mathematical researcher to use a result the proof of which he does
> not "understand" (except for the specific case where he wants to
> disprove the result). In principle, of course, understanding here
> means a thorough knowledge of all the arguments involved in the
> written proof. From this viewpoint, it may not be as necessary as
> is usually thought to classify all known truths in a universal
> library. But finally I think the proposal of the authors, to
> establish a "label" for mathematical papers with regard to their
> rigor and completeness, is an excellent idea.
>
> Rigor is a Latin word. We think of _rigor mortis_, the rigidity of
> a corpse. I would classify the (would-be) mathematical papers
> under three labels:
>
> (1) a crib, or baby's cradle, denoting "live mathematics",
> allowing change, clarification, completing of proofs,
> objection, refutation.
>
> (2) the tombstone cross. Authors pretending to full rigor,
> claiming eternal validity, may use this symbol as freely as they
> wish. This kind of work would constitute "graveyard mathematics".
>
> (3) the Temple. This would be a label delivered by an external
> authority, the "body of high priests". This body could initially
> be made up of the editors in chief of the "core" papers as
> suggested by Jaffe-Quinn. Its task would be to bestow the label
> at least on those papers with sufficient promise to justify close
> examination. Later on, the IMU could decide on a permanent
> procedure to establish the priestly body, allowing for a
> relatively quick turnover of people in charge, with equitable
> worldwide geographic representation. One might suppose that such
> an institution could last a very long time. Should it however
> eventually come to grief, the unattainable nature of absolute
> rigor would be thereby demonstrated.
>
> Let me end with a personal observation. The Jaffe-Quinn paper
> discusses at length the situation of mathematical physics, but does
> not seem to admit that the problem may arise in other disciplines
> for which (unlike physics) E. Wigner's phrase about the
> "unreasonable effectiveness of mathematics" is not valid. I
> strongly disagree with such a restriction. I see no reason why
> mathematics (even without computers and numerical computation)
> should not be applied in other disciplines, in biology for example.
> In particular I believe that there are in analytic continuation
> singular circumstances (unfoldings, for instance) where it may be
> applied in a qualitative way. (This echoes of course my catastrophe
> theory philosophy.) Papers written in this state of mind are not
> read by professional mathematicians, who see no need for
> communication with any other disciplines apart from physics. And
> they are not intelligible to people of the other speciality, who
> generally lack the necessary mathematical culture. As a result
> they remain practically unread. The case may be defended of papers
> which have to create their own readership; they are babies without
> parents.
For the context of the words above, see:
E.P. Wigner, "THE UNREASONABLE EFFECTIVENSS OF MATHEMATICS
IN THE NATURAL SCIENCES"
Communications in Pure and Applied Mathematics,
Vol. 13, No. I, February 1960
<http://nedwww.ipac.caltech.edu/level5/March02/Wigner/Wigner.html>
A Jaffe and F Quinn, "Theoretical mathematics": Toward a
cultural synthesis of mathematics and
theoretical physics,
Bull. Am. Math. Soc. 29 (1993) 1-13
<http://front.math.ucdavis.edu/math.HO/9307227>
M Atiyay et al., Responses to "Theoretical Mathematics",
Bull. Am. Math. Soc. 30 (1994) 178-207
<http://front.math.ucdavis.edu/math.HO/9404229>
A Jaffe and F Quinn, Response to comments on "Theoretical
mathematics",
Bull. Am. Math. Soc. 30 (1994) 208-211
<http://front.math.ucdavis.edu/math.HO/9404231>
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