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Re: [HM] Rene THOM [1923-2002]
Posted:
Nov 21, 2002 9:30 PM
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Who was the adressee of Thom's letter?
>Dear All,
>
>I would like to recall one of the early spectacular successes of
>cobordism: its key role in the proof of the Hirzebruch-Riemann-Roch
>theorem. The details can be gathered, for example, from:
>
> F. Hirzebruch, The signature theorem: Reminiscences and recreation,
> in _Prospects in Mathematics_, Annals of Math. Studies 70, pp. 3-31,
> Princeton UP, 1971.
>
> F. Hirzebruch, Kunihiko Kodaira: Mathematician, Friend, and Teacher,
> Notices of the AMS, December 1998.
>
>
> On p. 1459:
>
> This is used for the proof of my Riemann-Roch theorem, which was
> completed on December 10, 1953, and announced in the Proceedings
> of the National Academy of Sciences (communicated by S. Lefschetz
> on December 21, 1953). I had to reduce everything to complex
> split manifolds where the structural group is the triangular
> group contained in the general linear group. Then the arithmetic
> genus can be expressed by virtual signatures which (by the
> signature theorem as a consequence of Thom's cobordism theory)
> can be expressed by characteristic classes. ...
>
>
> J.-P. Serre, Lettre a\ Armand Borel, ine/dit , avril 1953,
> Oeuvres I, pp. 243-250 and p. 588.
>
> F. Hirzebruch, Neue topologishe Methoden in der Algebraischen
> Geometrie, Ergebnisse der Mathematik und ihrer Grenzgebiete 9,
> Springer, 1956.
>
>
>Rene/ Thom was one of the architects of 20th Century Mathematics.
>He strove for depth and for what is really important. His presence
>made a huge difference.
>
>Best wishes,
>Emili Bifet
>
>
>PS There is no substitute for the original:
>
><begin quote>
>
> Rene/ Thom
> Institute des Hautes Etudes Scientifiques
> 35 Rue de Chartres 91440
> Bures-sur-Yvette France
>
> Dear Dick,
>
> 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 Go"del'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.
>
><end quote>
>
>For the context of the words above, see:
>
>
> (A Jaffe and F Quinn, ``Theoretical mathematics'': Toward a
> cultural synthesis of mathematics and
> theoretical physics,
> Bull. Am. Math. Soc. 29 (1993) 1-13.)
>
>
>
> (M Atiyay et al., Responses to ``Theoretical Mathematics'',
> Bull. Am. Math. Soc. 30 (1994) 178-207.)
>
>
>
> (A Jaffe and F Quinn, Response to comments on ``Theoretical
> mathematics'',
> Bull. Am. Math. Soc. 30 (1994) 208-211.)
>
--
Dr. Patricio Herbst
Assistant Professor of Mathematics Education
The University of Michigan
1350 School of Education Building
610 East University Avenue
Ann Arbor, MI 48109-1259
USA
phone 1-734-763-3745
fax 1-734-763-1368
e-mail pgherbst@umich.edu
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