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
fourteenpage letter to Simone Weil, his sister, from
BonneNouvelle Prison in Rouen in March 1940,
sixtyfive 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
DedekindWeil 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 bracesandbrackets, {[ ]}. 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, 407436).
While this article was in proof, Philip Horowitz
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