fea-weil - A 1940 Letter of Andr Weil on Analogy in...

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A 1940 Letter of André Weil on Analogy in Mathematics Translated by Martin H. Krieger 334 N OTICES OF THE AMS V OLUME 52, N UMBER 3 F or André Weil, “having a disagreement with the French authorities on the subject of [his] military ‘obligations’ was the rea- son [he] spent February through May [of 1940] in a military prison.” When he was released, he went into the service. Weil wrote this fourteen-page letter to Simone Weil, his sister, from Bonne-Nouvelle Prison in Rouen in March 1940, sixty-five years ago this month. (Keep in mind that the letter was not written for a mathematician, even though Simone could not understand most of it.) I first heard of the letter from a small passage translated in a book by D. Reed ( Figures of Thought ; London: Routledge, 1995). At the time I was trying to understand the range of solutions to the Ising model in mathematical physics, and in going to Weil’s letter I found poignant his exposition of a threefold analogy out of Riemann and Dedekind, one that proves to organize a great deal of disparate material. Moreover, I had just begun to appreciate the significance of the Langlands Program for my problem. [See the “Notes Added in Proof” to Mar- tin H. Krieger, Constitutions of Matter: Mathemati- cally Modeling the Most Everyday of Physical Phe- nomena (Chicago: University of Chicago Press, 1996), pp. 311–312.] Eventually, in chapter 5 of Doing Mathematics , I worked out the analogy and provided an exposition of the Weil letter. A recent Notices article (“Some of what mathematicians do”, November 2004, pp. 1226–1230) summarizes the argument of that book, including what I called the Dedekind-Weil analogy. The Weil letter is a gem, of wider interest to the mathematical and philosophical community, con- cerned both with the actual mathematics and with how mathematicians describe their work. I pro- vided a translation from the French in the book’s appendix. I am grateful to the editor of the Notices; publication herein will allow for an even wider au- dience. The letter is from André Weil, Oeuvres Scien- tifiques, Collected Papers , volume 1 (New York: Springer, 1979), pp. 244–255. The translation aims to be reasonably faithful, not only to the meaning but also to sentence structure. Brackets are in the Oeuvres Scientifiques text. Braces indicate foot- notes therein. My editorial insertions are indicated by braces-and-brackets, {[ ]}. It is slightly revised, as taken from Martin H. Krieger, Doing Mathe- matics: Convention, Subject, Calculation, Analogy (Singapore: World Scientific, 2003), pp. 293–305. In the notes to the Oeuvres Scientifiques , Weil indicates that he was wrong then about the influence of the theory of quadratic forms in more than two vari- ables and that Hilbert is explicit about the analogy in his account of the Twelfth Problem (for which see David Hilbert, “Mathematical problems”, Bul- letin of the American Mathematical Society 37 , 2000, 407-436). While this article was in proof, Philip Horowitz
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fea-weil - A 1940 Letter of Andr Weil on Analogy in...

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