Intereting Posts

Field extension of composite degree has a non-trivial sub-extension
Why is the graph of a continuous function to a Hausdorff space closed?
What kind of “symmetry” is the symmetric group about?
nonstandard example of smooth function which fails to be analytic on $\mathbb{R}$
“Advice to young mathematicians”
How a group represents the passage of time?
Evaluate $ \displaystyle \lim_{x\to 0}\frac {(\cos(x))^{\sin(x)} – \sqrt{1 – x^3}}{x^6}$
Calculate the Wronskian of $f(t)=t|t|$ and $g(t)=t^2$ on the following intervals: $(0,+\infty)$, $(-\infty, 0)$ and $0$?
Growth of Tychonov's Counterexample for Heat Equation Uniqueness
Exponential curve fit
Why is the Fejér Kernel always non-negative?
Sum of Independent Folded-Normal distributions
Relationship between covariant/contravariant basis vectors
Why $X$ independent from $(Y,Z)$ implies that $E(XY^{-1} | Z)= E(X) E(Y^{-1}|Z)$?
Which is bigger among (i) $\log_2 3$ and $\log _3 5$ (ii) $\log_2 3$ and $\log _3 11$.

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.

- Representation of Cyclic Group over Finite Field
- Comparing the deviation of a function from its mean on concentric balls
- Image of a math problem that was stated in Cuneiform, Arabic, Latin and Finally in modern math notation
- University-level books focusing on intuition?
- Why does $\sqrt{x^2}=|x|$?
- Modal set-theory
- A Generalized version of Inclusion-Exclusion Principle?
- Mathematical analysis - text book recommendation sought
- A book of wheels
- Why does the Hilbert curve fill the whole square?

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.

**Added:** For further elaborations and context, I strongly recommend to read both, Borel’s article I linked to and Weil’s original, as well as Weil’s 1940 letter to his sister Simone Weil.

While I’m at it, I cannot refrain from *insisting* that you read *The Apprenticeship of a Mathematician*—preferably in its French original *Souvenirs d’apprentissage*—in case you haven’t done so already.

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(From metaphysics to mathematics), Science 1960, 52-56;

la metaphysique aux mathematiques,

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.”Further:

“Fortunately for researchers, as the fogs clear away on some point, they

reappear on another. A major part of the Tokyo Colloquium [1955] 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 ([1972], 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.ARMAND BOREL

Professor Emeritus

School of Mathematics

Institute for Advanced Study

Princeton, NJ 08540

- $\mathcal{L}$ is very ample, $\mathcal{U}$ is generated by global sections $\Rightarrow$ $\mathcal{L} \otimes \mathcal{U}$ is very ample
- Prove $F(n) < 2^n$
- Borel hierarchy doesn't “collapse” before $\omega_1$
- 6-digit password – a special decoding method
- Radical extension and discriminant of cubic
- iid variables, do they need to have the same mean and variance?
- Where has this common generalization of nets and filters been written down?
- How to plot a phase portrait for this system of differential equations?
- Prove continuity for cubic root using epsilon-delta
- $\frac{(2n)!}{4^n n!^2} = \frac{(2n-1)!!}{(2n)!!}=\prod_{k=1}^{n}\bigl(1-\frac{1}{2k}\bigr)$
- Conjectured closed form for $\operatorname{Li}_2\!\left(\sqrt{2-\sqrt3}\cdot e^{i\pi/12}\right)$
- Random solving of a Rubik cube .
- How I can express $(x,y)∈G$ by using the $r$ independent points $P_1,P_2,\ldots,P_r$
- Cohomology with Coefficients in the sheaf of distributions
- A problem in Sigma algebra