I just wasted the last hour on google looking in vain for an excerpt of Weil's writings describing the process of discovering mathematics. I believe he once beautifully described the feeling of loss that accompanies the realization that the discovery you made seems, in retrospect, trivial. Am I misremembering or just bad at googling? Thanks for your help.
You might have the following passage in mind:
It appears in A. Weil, De la métaphysique aux mathématiques, Science 60, p. 52–56 (see also Collected Papers II, p. 406–412). The excerpt and the reference are taken from a preprint of A. Borel on A. Weil, scan available here (from A. Knapp's homepage).
Edit: As Bill pointed out in his answer below, this preprint is published:
- In the Proceedings of the American Philosophical Society, Vol. 145, No. 1 (Mar., 2001), pp. 107–114 (jstor-link, may be behind a paywall).
- Reprinted and freely available as Mathematical Perspectives — André Weil Bull. Amer. Math. Soc. 46 (2009), 661–666.
Having aleady OCRed Borel's article on Weil (see Theo's reply), I cannot resist posting a larger excerpt here, since it provides much further context that I suspect will help readers to better appreciate Weil's "search for elegance, beauty and hidden harmonies". Perhaps it will help motivate some readers to join in such fruitful endeavors. I too strongly endorse the literatured cited by Theo. It can prove highly inspirational to budding mathematical minds. Edit: I just noticed that the AMS has a nicer typeset version of Borel's article from 2009 BAMS.
His output offers an extraordinary combination of foundational work, to secure a solid basis in some area, of often decisive contributions at the cutting edge, solving old or new problems, and of forays into unknown territory, in the form of problems or conjectures, guided by a seemingly infallible sense for the directions into which one should forge ahead.
Of course, I feel quite uncomfortable in making such a statement without backing it up in any way, so allow me to turn to the mathematicians to give an idea of these facets of his output in at least one area, algebraic geometry. The theorem he had proved in 1940 (see above) relied on some facts of algebraic geometry for some of which there was no solid reference. Moreover, the development of algebraic geometry, from "classical" (i.e. projective or affine complex varieties) to "abstract" (varieties over arbitrary fields), was also crying out for reliable foundations. It took him several years to supply them in a massive (and rather arid) treatise "Foundations of algebraic geometry" (1946), the only comprehensive basis for algebraic geometry for a number of years. Although dealing with a very general "abstract situation", he developed it in part in analogy with the theory of differentiable manifolds in differential geometry, and also with some constructions in algebraic topology. It was followed, among other items, by a monograph proving in full his 1940 result, by foundations for abelian varieties, fibre bundles in algebraic geometry, algebraic groups, the advocacy of the use of analytic fibre bundles in several complex variables, and in 1949, in a short Note, by a series of conjectures (soon called the Weil conjectures) which were to have an enormous impact on algebraic geometry. In particular, he postulated the existence of a cohomology theory in this set up, with properties allowing one to transcribe known arguments in algebraic topology, such as the Lefschetz fixed point theorem, a bold idea, unique to him, way ahead of its time. It was implemented some ten years later by A. Grothendieck (etale cohomology), and it took twenty-five years before Deligne proved the last, and by far hardest, of these conjectures, with far reaching consequences, not yet exhausted.
So far, I have said little of what has arguably been Weil's most abiding interest in mathematics: "Zeta functions". The first one was used by B. Riemann in 1857 to study the distribution of prime numbers among positive integers. The "Riemann hypothesis" about the zeroes of this function is still unproved and generally viewed as the Holy Grail of mathematics. The introduction of this function to study the discrete (the integers) in a continuous framework (real or complex numbers) was quite revolutionary and proved to be immensely fruitful. Zeta functions, with corresponding Riemann hypotheses, have proliferated in analysis, algebraic geometry and number theory, and have always been on Weil's mind. (His 1940 theorem dealt with one kind and his 1949 conjectures with generalizations of it.) He was convinced that the problem of the Riemann hypothesis, even in the original case, had to be attacked broadly. How broadly can be only explained in mathematical terms of course, but he drew an analogy with the Rosetta Stone, which seems to me so typical of his thought processes and of the aesthetic component in his approach to mathematics that I cannot resist trying to give an idea of it, as imprecise as it has to be. It is developed in a short article: De la metaphysique aux mathematiques, (From metaphysics to mathematics), Science 1960, 52-56; Collected Papers II, 406-412.
"Metaphysics", he explains, is meant here in the sense of the 18'th century mathematicians, when they spoke of, say, "the metaphysics of the theory of equations":
"... a collection of vague analogies, difficult to grasp and difficult to formulate, which nevertheless appeared to them to play an important role at certain moments in the research and discovery in mathematics".
and then he elaborates.
"Nothing is more fecund, all the mathematicians know it, than those obscure analogies, the blurred reflections from one theory to another ... nothing gives more pleasure to the researcher. One day the illusion drifts away, the premonition changes to a certitude: the twin theories reveal their common source before disappearing; as the Gita teaches it, knowledge and indifference are reached at the same time. The metaphysics has become mathematics, ready to form the subject matter of a treatise, the cold beauty of which cannot move us anymore."
"Fortunately for researchers, as the fogs clear away on some point, they reappear on another. A major part of the Tokyo Colloquium  was devoted to the analogies between number theory and the theory of algebraic functions. There we are still fully in metaphysics..."
"Algebraic functions" alludes here to a theory built up by Riemann by analytical, transcendental means. To link it to number theory, guided by "obscure analogies", is a problem which had fascinated Weil early on (as already hinted by the title of his Thesis), and he felt that progress was still scant by 1960. Meanwhile, a third topic had appeared: "algebraic curves over finite fields" (the subject matter of his 1940 theorem), which was easier to relate to the other two and thus served as an intermediary. These items and many generalizations or related results formed an enormous amount of mathematics naturally divided into three parts, each with its own framework, (in brief, transcendental, arithmetic and algebraico-geometric) and techniques. As Weil puts it, we are faced with a text in three parts (he calls them columns), each written in its own language, called by him Riemannian, arithmetic and Italian respectively, in analogy with the Rosetta Stone. However, there is an huge difference: the latter contains the same text in the three languages (or rather, assuming this, Champollion was able to decipher Egyptian hieroglyphic writing), while we have here only in each column fragments of what is hoped to be similar texts, once completed.
The task of the mathematicians, then, is to add translations of a given fragment into the other columns, to transform those obscure analogies into mathematics, and eventually build a dictionary which would allow one to pass from one column to the others. If it were sufficiently complete, then the Riemann hypothesis would be proved, Weil concludes, wondering how long mathematics will have to wait for a Champollion.
As an illustration of his outlook, let me mention a paper (, p. 249-64, in his Collected Papers III), where he formulates a statement in "Riemannian" language, the truth of which would imply that of the Riemann hypothesis (for many zeta functions), points out that it has an analogue in "Italian" which, in view of his earlier work, is a proven theorem, and comments that this provides for him, perhaps, the strongest evidence in favor of the original Riemann hypothesis, one of many examples of his unshakable belief in the unity and harmony of mathematics.
Weil was indeed fluent in the three languages and many of his works can be interpreted as contributions to the dictionary, but not all, though. In particular, as befits a man with his cultural interests, he had a strong commitment to the history of mathematics, which culminated in a history of number theory from 1800 B.C. to 1800 A.D. (from Hammurapi to Legendre). Much earlier it had been at the origin of the Historical Notes in Bourbaki, to which he was a main contributor until he retired.
As a mathematician, his work shows him to be at the same time an architect, a builder and a poet: an architect for fostering a global view of mathematics and striving to display its fundamental unity, a builder by his specific, often decisive, contributions to a great variety of topics and a poet by his search for elegance, beauty and hidden harmonies.
School of Mathematics
Institute for Advanced Study
Princeton, NJ 08540