Mathematicians From France

"I am surprised at all the people in the high-tech industry focused on "making money"… If that's all they want to do, they should have a $100 printing press in their basements and they will truly "make money." Instead, if we focus all that energy on innovation, we'll change the world for the best."

"Awaiting me upon my return to Strasbourg were Henri Cartan and the course on "differential and integral calculus," which was our joint responsibility. ... One point that concerned him was the degree to which we should generalize Stokes' formula in our teaching. ... In his book on invariant integrals, Elie Cartan, following Poincare in emphasizing the importance of this formula, proposed to extend its domain of validity. Mathematically speaking, the question was of a depth that far exceeded what we were in a position to suspect. ... One winter day toward the end of 1934,1 thought of a brilliant way of putting an end to my friend's persistent questioning. We had several friends who were responsible for teaching the same topics in various universities. "Why don't we get together and settle such matters once and for all, and you won't plague me with your questions any more?" Little did I know that at that moment Bourbaki was born."

"I had also, unsuccessfully, looked for the works of Saint John of the Cross. The flashing beauty of his poems would probably have moved me more than did Saint Theresa, but it was not until much later that I came to know his work. I read a little of Saint Theresa and became quickly convinced that mystic thought is at bottom the same in all times and places: reading Suzuki's popular works on Zen was soon to confirm this conclusion. ... Speaking of a saint whose behavior was somewhat eccentric, one of the monks remarked gently: "But Christianity is madness" ("el cristianismo es una locum"). This perfectly orthodox statement often comes to mind when I think about my sister's life."

"[On meeting Raymond Paley] At first, we seemed to be on completely different wavelengths. Finally, it became apparent to me that he worked fruitfully only when competing with others: having the rest of the pack at his side spurred him to greater efforts as he tried to surpass them. In contrast, my style was to seek out topics that I felt exposed me to no competition whatsoever, leaving me free to reflect undisturbed for years. No doubt every scientific discipline has room for such differences of temperament. What does it matter if a given researcher is motivated primarily by hopes of winning the Nobel prize? Sometimes it seems to me that Ganesh, the Hindu god of knowledge, chooses the bait, noble or vulgar, best suited to each of his followers."

"Both the Jews and the brahmins of southern India are communities that, for twenty centuries, have devoted themselves tirelessly to the most abstract subtleties of grammar and theology. For the Jews it was the study of the Talmud, a task often passed down from father to son; for the brahmins, it was the Brahmanas and the Upanishads. It is hardly surprising that the younger generations, when their time came, turned toward the sciences, and preferably the most abstract among them: this trend was merely the natural extension of millennial traditions."

"Is it mere coincidence that in India Pāṇini's invention of grammar had preceded that of decimal notation and negative numbers, and that later on, both grammar and algebra reached the unparalleled heights for which the medieval civilization of the Arabic-speaking world is known?"

"I began to combine this ordinary form of touring with a specifically mathematical variety. I had formed the ambition of becoming, like Hadamard, a "universal" mathematician: the way I expressed it was that I wished to know more than non-specialists and less than specialists about every mathematical topic. Naturally, I did not achieve either goal."

"Hopf, back from Amsterdam, was teaching Brouwer's topology. He had helped arrange lodgings for me quite close to where he lived, rather far from the center of town, and together we would take the long tram ride to the university. One day I asked him what he would do when he got tired of topology. He replied in all seriousness: "But I'll never get tired of topology!""

"In comparison with the wise man, the saint is perhaps just a specialist - a specialist in holiness; whereas the wise man has no specialty. This is not to say, far from it, that Dehn was not a mathematician of great talent; he left behind a body of work of very high quality. But for such a man, truth is all one, and mathematics is but one of the mirrors in which it is reflected - perhaps more purely than it is elsewhere."

"Every mathematician worthy of the name has experienced, if only rarely, the state of lucid exaltation in which one thought succeeds another as if miraculously, and in which the unconscious (however one interprets this word) seems to play a role. In a famous passage, Poincaré describes how he discovered Fuchsian functions in such a moment. About such states, Gauss is said to have remarked as follows: "Procreare jucundum (to conceive is a pleasure)"; he added, however, "sed parturire molestum (but to give birth is painful)." Unlike sexual pleasure, this feeling may last for hours at a time, even for days. Once you have experienced it, you are eager to repeat it but unable to do so at will, unless perhaps by dogged work which it seems to reward with its appearance. It is true that the pleasure experienced is not necessarily in proportion with the value of the discoveries with which it is associated."

"‘[…] his correspondence with Digby, and, through Digby, with the English mathematicians WALLIS and BROUCKNER occupies the next year and a half, from January 1657 to June 1658. It begins with a challenge to Wallis and Brouckner, but at the same time also to Frenicle, Schooten “and all others in Europe” to solve a few problems, with special emphasis upon what later became known (through a mistake of Euler’s) as “Pell’s equation”. What would have been Fermat’s astonishment if some missionary, just back from India, had told him that his problem had been successfully tackled there by native mathematicians almost six centuries earlier!’"

"We should ask our fellow physicists to invent a principle of anti-interference, which would bring light out of two obscurities (Leray and Grothendieck)."

"About ancient mathematics (whether Greek or Mesopotamian) and medieval mathematics (Western or Oriental), the would-be historian must of necessity confine himself to the description of a comparatively small number of islands accidentally emerging from an ocean of ignorance, and to tenuous conjectural reconstructions of the submerged continents which at one time must have bridged the gaps between them."

"It is hard for you to appreciate that modern mathematics has become so extensive and so complex that it is essential, if mathematics is to stay as a whole and not become a pile of little bits of research, to provide a unification, which absorbs in some simple and general theories all the common substrata of the diverse branches if the science, suppressing what is not so useful and necessary, and leaving intact what is truly the specific detail of each big problem. This is the good one can achieve with axiomatics (and this is no small achievement). This is what Bourbaki is up to."

"An important point is that the p-adic field, or respectively the real or complex field, corresponding to a prime ideal, plays exactly the role, in arithmetic, that the field of power series in the neighborhood of a point plays in the theory of functions: that is why one calls it a local field."

"First rank scientists recruit first rank scientists, but second rank scientists tend to recruit third rank scientists, third rank scientists recruit fifth rank, and so on. If the director of the Department is genuinely interested in preserving the high quality of his Institute, he must exercise all of his power to put things in their right place, otherwise the deterioration process is destined to diverge indefinitely."

"In mathematics, perhaps more than in any other branch of knowledge, ideas spring fully armed from the creator's brain; thus mathematical talent usually reveals itself at a young age; and second-rate researchers play a smaller role than elsewhere, the role of a sounding board for a sound they do not contribute to shape."

"... the geometry over p-adic fields, and more generally over complete local rings, can provide us only with local data; and the main tasks of algebraic geometry have always been understood to be of a global nature. It is well known that there can be no global theory of algebraic varieties unless one makes them "complete", by adding to them suitable "points at infinity," embedding them, for example, in projective spaces. In the theory of curves, for instance, one would not otherwise obtain such basic facts as that the number of poles and zeros of a function are equal, of that the sum of residues of a differential is 0."

"God exists since mathematics is consistent, and the Devil exists since we cannot prove it."

"For our genius of a father did not limit himself to math. His brain was an octopus, the tentacles of which extended in all directions. He could scan Latin verse and Greek verse as well, and it was as if he were hearing Homer or Theocritus in person. Not to mention the fact that he read Greek from volumes filled with characters which in no way resembled the ones in our Greek grammar or book of excerpts from Greek literature. He also read Sanskrit, with its truly bizarre letters. He spoke Italian like Dante, Spanish like Cervantes, and so for almost every living language."

"When he was not busy doing math, he reads fat books covered with rough leather, on the pages of which can be seen very old, perfectly round holes dug out by medieval worms. Or else he passes through a museum while devoting himself to unbelievably profound notions about Van Gogh's paintings or Greek amphorae."

"Yes, but there is the problem. It would have been banal. Mediocre. We had been trained to despise everything which was not excellent. How disgusting to see the father of one of our classmates or friends on vacation playing cards or, even worse, sitting on the sofa watching television. We blush with shame for our unfortunate little friend."

"Sometimes my sister and I dream of having a run-of-the-mill father. He would make coffee and toss salads. He would not prefer his work to us, and he would tell us instead: "How pretty you look, my dear, tell me what you did today." He would speak to us with words of affectionate banality."

"In his thesis, Weil generalized Mordell's theorem on the finite generation of the group of rational points on an elliptic curve, to abelian varieties of any dimension. Weil then hoped to use this finite generation result for the rational points on the jacobian of a curve to go on to show that when a curve of genus > 1 is imbedded in its jacobian, only a finite number of the rational points of the jacobian can lie on the curve. Not finding a way to do this, he decided to call his proof of finite generation (the "theorem of Mordell-Weill") a thesis, despite Hadamard's advice not to be satisfied with half a result!"

"Alexandre Grothendieck was very different from Weil in the way he approached mathematics: Grothendieck was not just a mathematician who could understand the discipline and prove important results—he was a man who could create mathematics. And he did it alone."

"One day it occurred to me to measure the speed with which such rumors were propagated. All I had to do was to start one - the more preposterous the better - at one end of the stadium, and then hasten to the other end to await the results."

"(April 7) My mathematics work is proceeding beyond my wildest hopes, and I am even a bit worried - if it's only in prison that I work so well, will I have to arrange to spend two or three months locked up every year? In the meantime, I am contemplating writing a report to the proper authorities, as follows: "To the Director of Scientific Research: Having recently been in a position to discover through personal experience the considerable advantages afforded to pure and distinterested research by a stay in the establishments of the Penitentiary System, I take the liberty of, etc. etc.""

"Mozart's music, even at its most beautiful, often gives an impression of some being who, though very far above us in his incomprehensible serenity, nevertheless stops to remember us for a brief instant and comes within our reach, with gentle mockery and tender pity, to transcribe a fleeting message for us. But sometimes, in certain quartets and quintets, and in certain parts of The Magic Flute, this same being, without a thought for us, communicates with his fellow beings, and what we hear then is a world unknown to us, a world of which we are allowed only a furtive glimpse."

"Already while at the Ecole Normale, I had been deeply struck by the damage wreaked upon mathematics in France by World War I. This war had created a vacuum that my own and subsequent generations were hard pressed to fill. In 1914, the Germans had wisely sought to spare the cream of their young scientific elite and, to a large extent, these people had been sheltered. In France a misguided notion of equality in the face of sacrifice - no doubt praiseworthy in intent - had led to the opposite policy, whose disastrous consequences can be read, for example, on the monument to the dead of the Ecole Normale. Those were cruel losses; but there was more besides. Four or more years of military life, whether close to death or far away from it - but in any case far from science -, are not good preparation for resuming the scientific life: very few of those who survived returned to science with the keenness they had felt for it. This was a fate that I thought it my duty, or rather my dharma, to avoid."

"Kantian ethic, or what passes for it today, has always seemed to me to be the height of arrogance and folly. Claiming always to behave according to the precepts of universal maxims is either totally inept or totally hypocritical; one can always find a maxim to justify whatever behavior one chooses. I could not count the times (for example, when I tell people I never vote in elections) that I have heard the objection: "But if everyone were to behave like you..." - to which I usually reply that this possibility seems to me so implausible that I do not feel obligated to take it into account."

"In establishing the tasks to be undertaken by Bourbaki, significant progress was made with the adoption of the notion of structure, and of the related notion of isomorphism. Retrospectively these two concepts seem ordinary and rather short on mathematical content, unless the notions of morphism and category are added. At the time of our early work these notions cast new light upon subjects which were still shrouded in confusion: even the meaning of the term "isomorphism" varied from one theory to another. That there were simple structures of group, of topological space, etc., and then also more complex structures, from rings to fields, had not to my knowledge been said by anyone before Bourbaki, and it was something that needed to be said. As for the choice of the word "structure," my memory fails me; but at that time, I believe, it had already entered the working vocabulary of linguists, a milieu with which I had maintained ties (in particular with Émile Benveniste)."

"The major classic texts in analysis (Jordan, Goursat) which we had at first set out to replace aimed to set forth in a few volumes everything a beginning mathematician should know before specializing. At the end of the nineteenth century, such a claim could still be made seriously; by now it had become absurd. ... it soon became apparent that there was no alternative but to give up any idea of writing a text for college-level instruction. Above all it was important to lay a foundation that was broad enough to support the essential core of modern mathematics..."

"Otto Schmidt] called together the principal mathematicians in Moscow and Petrograd (later known as Leningrad) and spoke to them more or less as follows: "Whatever the regime, the work of mathematicians is too inaccessible to laymen for us to be criticized from the outside; as long as we stick together, we will remain invulnerable.""

"The primitive heat... in the interior of the earth would not increase the external temperature of space... for... the effect of this central heat has long since become insensible at the surface, although it may be very great at a moderate depth."

"Galileo and Hooke demonstrated that every sound is characterized by a precise number of vibrations per second, but the full understanding of even the simplest... vibrators requires... calculus. ...Leonard Euler, Daniel Bernoulli, Jean le Rond d'Alembert, and Joseph Louis Lagrange all studied the vibrating string.... In 1747 d'Alembert found... the wave equation and in 1753 Bernoulli stated the... decomposition of every motion of a string as a sum of elementary sinusoidal motions. Then at the beginning of the following century Fourier developed "harmonic analysis.""

"The formula and even the method was... used by Euler in... 1777, published... 1798. (Fourier... having failed to refer to earlier works...) ...[A]s with Bernoulli, Fourier had acquired an intimate understanding of the physical meaning of the problem. Over... two years... Fourier repeated all important experiments... carried out in England, France, and Germany and added experiments of his own. ...[S]triking ...confirmations of his new theory ...together with his overcoming ...difficulties advanced by the old masters. Fourier mentioned the motion of fluids... propagation of sounds and...vibrations of elastic bodies, as other applications ...fully aware of having opened up a new era for the solution or partial differential equations... It was the era of linearization that would dominate mathematics for the first half of the nineteenth century and... has remained important... The diffusion equation is a : Linear combinations of solutions are still solutions. It was not the first such equation... in history... but the method opened up enormous... possibilities."

"The novelty of his method... initially perplexed... mathematicians... from Lagrange to Laplace and Poisson. ...[P]ublication ...was ...delayed as many as fifteen years during which he ...defended, explained and extended [his work]."

"He ranks among the most important scientists of the 19the century for his studies in the propagation of heat... and the paternity of the expression ""—effet de serre—is attributed to Fourier."

"He started as a convinced Jacobin... and ended up a cautious liberal."

"It is true that M. Fourier had the opinion that the principal end of mathematics was the public utility and the explanation of natural phenomena; but such a philosopher as he is should have known that the unique end of science is the honor of the human mind, and that from this point of view a question of number is as important as a question of the system of the world."

"He carried on his elaborate investigations on the propagation of heat in solid bodies, published in 1822 in his work entitled La Theorie Analytique de la Chaleur. This work marks an epoch in the history of mathematical physics. "Fourier's series" constitutes its gem. By this research a long controversy was brought to a close, and the fact established that any arbitrary function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807 before the French Academy. The trigonometric series \sum_{n=0}^{n=\infty} (a_n\sin nx+b_n\cos nx) represents the function \phi(x) for every value of x if the coefficients a_n = \frac{1}{\pi}\int_{-\pi}^{\pi}\phi(x) \sin nx\,dx, and b_n be equal to a similar integral. The weak point in Fourier's analysis lies in his failure to prove generally that the trigonometric series actually converges to the value of the function."

"Fourier took a prominent part at his home in promoting the Revolution. Under the French Revolution the arts and sciences seemed for a time to flourish. ...The Normal School was created in 1795, of which Fourier became at first pupil, then lecturer. His brilliant success secured him a chair in the Polytechnic School, the duties of which he afterwards quitted, along with Monge and Berthollet, to accompany Napoleon on his campaign to Egypt. Napoleon founded the Institute of Egypt, of which Fourier became secretary. In Egypt he engaged not only in scientific work, but discharged important political functions. ...In 1827 Fourier succeeded Laplace as president of the council of the Polytechnic School."

"At the age of twenty-one he went to Paris to read before the Academy of Sciences a memoir on the resolution of numerical equations, which was an improvement on Newton's method of approximation. This investigation of his early youth he never lost sight of. He lectured upon it... he developed it... it constituted a part of a work entitled Analyse des equations determines (1831), which was in press when death overtook him. This work contained "Fourier's theorem" on the number of real roots between two chosen limits. Budan had published this result as early as 1807, but there is evidence to show that Fourier had established it before Budan's publication. These brilliant results were eclipsed by the theorem of Sturm, published in 1835."

"Fourier's analytical theory of heat (final form, 1822), devised in the Galileo-Newton tradition of controlled observation plus mathematics, is the ultimate source of much modern work in the theory of functions of a real variable and in the critical examination of the foundation of mathematics."

"In a military school directed by monks, the minds of the pupils necessarily waver only between two careers in life—the church and the sword. Like Descartes, Fourier wished to be a soldier; like that philosopher he would doubtless have found the life of a garrison very wearisome. But he was not permitted to make the experiment. His demand to undergo the examination for the artillery, although strongly supported by our illustrious colleague Legendre, was rejected with a severity of expression of which you may judge yourselves: "Fourier," replied the minister, "not being noble, could not enter the artillery, although he were a second Newton.""

"We shall now consider that peculiar heat which our globe had at the time of the formation of the planets, and which continues to be dissipated at the surface under the influence of the low temperature of the planetary space."

"The mobility of the air, which is rapidly displaced... and... rises when heated... diminish the intensity of the effects... [of] a transparent and solid atmosphere, but do not entirely change their character."

"[I]f all the strata of air of... the atmosphere... preserved their density with their transparency, and lost only the mobility... peculiar to them, this mass of air, thus become solid, on being exposed to the rays of the sun, would produce an effect the same in kind... [as] just described."

"It is difficult to know how far the atmosphere influences the mean temperature of the globe... It is to... M. de Saussure that we are indebted for a capital experiment which appears to throw... light on this... The theory of the instrument is... 1st... the acquired heat is concentrated, because it is not dissipated immediately by renewing the air; 2d, that the heat of the sun, has properties different from those of [invisible] heat... The rays... are transmitted in considerable quantity through the glass plates... They heat the air and the partitions which contain it. Their heat thus communicated ceased to be luminous, and preserves only the properties of non-luminous radiating heat. In this state it cannot pass through the plate of glass covering the vessel. ...It is necessary to consider attentively this order of facts, and the results of the calculus when we would ascertain the influence of the atmosphere and waters upon the thermometrical state of our globe."

"From the constitution of the solar system it is... probable that the temperature of the poles... are a little less than that of space... the same for all... although the distances from the sun may be unequal."