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