Weil’s Letter From Prison

The great French mathematician André Weil spent the months of February-May 1940 in a prison in Rouen, as a result of what he referred to as “a disagreement with the French authorities on the subject of my military obligations”. Others might have called this “draft evasion”, and the story has something to do with why one of the most famous French mathematicians spent his post-war career not in France, but in Chicago and Princeton.

Weil’s sister was Simone Weil, who could variously be described as a moral, political and religious philosopher, an activist and mystic. She died in 1943 in England, from some combination of tuberculosis and starving herself out of sympathy with her compatriots in occupied France. During her brother’s prison stay they exchanged letters which were later published. One of these letters is a remarkable mathematical document that André Weil wrote to his sister, although she would have had little chance of understanding what he was talking about. It is reproduced in his collected works, and an English translation has just appeared in the latest Notices of the AMS.

The focus of Weil’s letter is the analogy between number fields and the field of algebraic functions of a complex variable. He describes his ideas about studying this analogy using a third, intermediate subject, that of function fields over a finite field, which he thinks of as a “bridge” or “Rosetta stone”. For function fields over a finite field, the analogies with number fields are quite close and many facts one knows about one subject can be used to make conjectures about what is true for the other. Some examples include the Riemann-Roch theorem and the Riemann hypothesis. After getting out of prison and leaving for the U.S., in 1941 Weil was able to prove the Riemann hypothesis for the function field case; of course for the number field case it remains an open problem.

For much more detail about this analogy, there’s an interesting textbook by Dino Lorenzini called An Invitation to Arithmetic Geometry.

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10 Responses to Weil’s Letter From Prison

  1. Eleggua says:

    I think that the criticism of Bourbaki is over the top. If you are interested in philosophy, in particular of the analytical type, then actually Bourbaki group helped developed some very interesting mathematics which lead to these developments:

    Cleared up the notion of ‘proof’ for a mathematician (i.e there are an ‘infinity’ of levels at which mathematics can be done : – Category theory is just one)

    Created the machinery that allowed 2nd order logic to be formalised. This lead to P Cohen proof of the undecidablity of continuum hypothesis.

    Also to the proof that Mathematics is sound, of course you have to belive in ordinals greater than the continuum.

    Finally and not least, A A Markov 1958 proof of the impossiblity of solving the homeomorphy problem for manifolds, dimension 4 and above. A result I believe to be the equivalent of the impossiblity of solving 5 deg poly by radicals. This result in my opinion is the REAL reason why everyone is doing category theory. I must point out that category theory has many enemies in mathematics, who feel that it is only useful for expositionary work, not actual creative stuff (except for Grothendieck)

    Essential the Bourbaki spirit now lives in combinatorial group theory and model theory. At the moment these subjects are at the fringe, but I believe that out of these 2 subject will come the mathematics for the 21 century like topology was for the 20th!

    an amateur Mathematican

  2. Peter says:

    Hi Danny,

    Bourbaki’s heyday was the fifties, part of the same over-emphasis on abstraction that lead to the “new math” disaster. By the 70s, Bourbaki’s influence had started to wane. One reason was that mathematicians had begun to lose interest in overly formalist approaches, another was that Grothendieck showed that very different foundations were needed (i.e. category theory vs. set theory). The influence of physics on mathematics also had an effect.

    These days, Bourbaki has little influence in math. Some of their books are actually pretty good, but mostly they are used as technical references, nobody tries to learn anything from them. For a long time now the group has stopped writing more.

    Mathematicians still generally do a terrible job of writing readable expository material, but mostly they are no longer doing overly formalistic things. But what they do is inherently different than what physicists do, largely because there is a strong culture of not allowing people to be imprecise, and insisting on absolute clarity of in arguments. Physicists would do well to learn something from this.

  3. D R Lunsford says:

    Peter,

    One of my best friends was educated in the heyday of “new math”, at Columbia and MIT. In high school of course he was at the top of his class, and in a class by himself technically. He related the following story. He learned calculus from Lang, was doing functional analysis as a high school student etc. etc. Then, he had to teach a course at MIT to engineers. He suddenly realized he couldn’t do a simple surface integral da capo. He stated to me, that he began to consider the entire axiomatic program embodied in Bourbaki to be a total sham, and sought to restructure his knowledge on intuitionist lines (Brouwer, Weyl). Needless to say this was a complete success and now this peson is a world-authority on the math and modeling of turbulence. This was in spite of, not because of, the French program.

    I think it can safely be argued that Bourbaki and its intellectual worldview of lofty abstractions and airy-fairy axiomatics has been a disaster for science, has attempted to rip math and physics apart from each other at the sternum, and has created the mental climate in which a ruse like string theory can develop in the first place.

    -drl

  4. Peter says:

    Hi Rafael,

    It’s not so much that prominent French Mathematicians were killed in WWI, but that many of the best math students were killed. Two-thirds of the students at the Ecole Normale Superieure died, this is the place that produced most mathematics teachers and researchers.

    For more about this and about Bourbaki, see

    http://planetmath.org/encyclopedia/NicolasBourbaki.html

  5. Rafael says:

    Hi Peter,

    Which French mathematicians were killed in WW I? Also was Bourbaki created to “collect” all known French mathematics?

    Thanks,

  6. D R Lunsford says:

    The parallel story of A. Raabe is worth mentioning. He was arrested in Krakow and died at Auschwitz before he could band together with other physicists for the purpose of making political statements or abstraction committees.

    He did very interesting work on relativistic rotators as a kid.

    http://arxiv.org/PS_cache/hep-th/pdf/0303/0303099.pdf

  7. Peter says:

    No, he would have been too young (born in 1906). Weil started his career at a time when much of the generation just older than him had been wiped out by WWI. This had a lot to do with why he and others of his generation ended up banding together to form Bourbaki. Also probably had a lot to do with why he was evading the draft. He had seen one generation of French mathematicians killed in WWI and didn’t want this to happen again, especially not to himself.

  8. Alejandro Rivero says:

    Now I think about, was Weil also in prison during WWI or was he too young?

  9. Peter says:

    Hi Daniel,

    Oops, typo in html. Fixed now. Thanks for pointing this out!

    Peter

  10. Hi Peter,

    I think that your link to the “English translation” is missing, at least as of now (10FEb05 @ 14:33:00h) it seems void. (Not that i can’t find the link otherwise… but, just a heads up. 😉

    Cheers,

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