Historians From France

"I should not complain of the labour of this work, if my materials were always derived from such books as the chronicle of honest Froissard...who read little, enquired much, and believed all. The original Memoirs of the marechal de Boucicault...add some facts, but they are dry and deficient, if compared with the pleasant garrulity of Froissard."

"Et scahiez que Anglois et Escoçoiz, quant ilz se treuvent en bataille ensamble, sont dures gens et de longue alainne, et point ne s'esparngnent, mais s'entendent de eulx mettre à oultranche, comment qu'il prende. Ilz ne ressamblent pas les Alemans qui font une empainte, et, quant ilz voient qu'ilz ne puellent rompre ne entrer en leurs ennemis, ilz s'en retournent tout à ung fais."

"Ce lévrier nommé Blemach…laissa le roy et s'en vint tout droit au duc de Lancastre, et luy fist toutes les contenances telles que en devant il faisoit au roy Richart, et luy assist ses deux pies sus les epaules et le commença moult grandement à conjouir. Adont le duc de Lancastre qui point ne congnoissoit le lévrier, demanda au roy et dist: "Mais que veult ce lévrier faire?"…"Cestuy lévrier vous recueille et festoie aujourd'huy comme roy d'Angleterre que vous serés, et j'en seray déposé.""

"Et, se venons tout d'un père et d'une mere, Adam et Eve, en quoi poent il dire ne monstrer que il sont mieux signeur que nous, fors parce que il nous font gaaignier et labourer ce que il despendent? Il sont vestu de velours et de camocas fourés de vair et de gris, et nous sommes vesti de povres draps. Il ont les vins, les espisses et les bons pains, et nous avons le soille, le retrait et le paille, et buvons l'aige. Ils ont le sejour et les biaux manoirs, et nous avons le paine et le travail, et le pleue et le vent as camps, et faut que de nous viengne et de nostre labeur ce dont il tiennent les estas."

"Ensi fu ceste bataille desconfite que vous avés oy, qui fu ès camps de Maupetruis à deux liewes de le cité de Poitiers, le vingt unième jour dou mois de septembre, l'an de grasce Nostre Signeur mil trois cens cinquante six. Si commença environ heure de prime, et fu toute passée à none; mès encores n'estoient point tout li Englès qui caciet avoient, retourné de leur cace et remis ensamble…Et fu là morte, si com on recordoit adonc pour le temps, toute li fleur de la chevalerie de France: de quoi li nobles royaumes fu durement afoiblis, et en grant misère et tribulation eschei, ensi que vous orés recorder chi après."

"Cils Jehan Balle avoit eut d'usage que, les jours dou diemence après messe, quant toutes les gens issoient hors dou moustier, il s'en venoit en l'aitre et là praiechoit et faissoit le peuple assambler autour de li, et leur dissoit: "Bonnes gens, les coses ne poent bien aler en Engletière ne iront jusques à tant que li bien iront tout de commun et que il ne sera ne villains ne gentils homs, que nous ne soions tout ouni.""

"Gautier, vous en irées à chiaus de Calais, et dirés au chapitainne, monsigneur Jehan de Viane, que vous avés tant travilliet pour yaus, et ossi ont tout mi baron, que je me sui accordés à grant dur à ce que la plus grant grasce qu'il poront trouver ne avoir en moy, c'est que il se partent de le ville de Calais six des plus notables bourgeois, en purs les chiés et tous, deschaus, les hars ou col, les clés de le ville et dou chastiel en leurs mains. Et de chiaus je ferai ma volonté, et le demorant je prenderai à merci."

"Considerés que c'est de pueple, quant il s'esmuet et esliève et il a puissance contre son seigneur, et par especial en Angleterre. Là n'y a-il nul remède, car c'est le plus périlleus poeuple commun qui soit au monde et le plus oultrageux et orgueilleux. Et de tous ceulx d'Angleterre Londriens sont chiefs."

"Li rois d'Engleterre et li sien, qui s'en venoient tout singlant, regardent et voient devers l'Escluse si grant quantité de vaissiaus que des mas ce sambloient droitement uns bos."

"His chapters inspire me with more enthusiasm than even poetry itself. And the noble canon, with what true chivalrous feeling he confines his beautiful expressions of sorrow to the death of the gallant and high-bred knight, of whom it was a pity to see the fall, such was his loyalty to his king, pure faith to his religion, hardihood towards his enemy, and fidelity to his lady-love! – Ah, benedicite! how he will mourn over the fall of such a pearl of knighthood, be it on the side he happens to favour, or on the other. But, truly, for sweeping from the face of the earth some few hundreds of villain churls, who are born but to plough it, the high-born and inquisitive historian has marvellous little sympathy."

"Lors respondi li rois et demanda au chevalier, qui s'appelloit messires Thumas de Nordvich: "Messires Thumas, mes filz est il ne mors ne atierés, ou si bleciés qu'il ne se puist aidier?" Cilz respondi: "Nennil, monsigneur, se Dieu plaist; mais il est en dur parti d'armes: si aroit bien mestier de vostre ayde." "Messire Thumas, dist li rois, or retournés devers lui et devers chiaus qui ci vous envoient, et leur dittes de par moy qu'il ne m'envoient meshui requerre pour aventure qui leur aviegne, tant que mes filz soit en vie. Et dittes leur que je leur mande que il laissent à l'enfant gaegnier ses esporons; car je voel, se Diex l'a ordonné, que la journée soit sienne, et que li honneur l'en demeure et à chiaus en qui carge je l'ai bailliet.""

"Koyré's exaltation of the "Platonic and Pythagorean" elements of the Scientific Revolution... was based on a demonstrably false understanding of how Galileo reached his conclusions. ...By avoiding consideration of nonmathematical sciences, Koyré reduced the Scientific Revolution to the ideas of Copernicus, Kepler, Galileo, and Newton."

"Koyré based his analysis on a narrow definition of science that focuses only on its purely theoretical aspects. He saw the Scientific Revolution as the advent and triumph of what he called the "mathematization of nature." At the same time he downplayed experimentalism as a relatively unimportant aspect of the new science."

"There is something for which Newton — or better to say not Newton alone, but modern science in general — can still be made responsible: it is splitting of our world in two. I have been saying that modern science broke down the barriers that separated the heavens and the earth, and that it united and unified the universe. And that is true. But, as I have said, too, it did this by substituting for our world of quality and sense perception, the world in which we live, and love, and die, another world — the world of quantity, or reified geometry, a world in which, though there is place for everything, there is no place for man. Thus the world of science — the real world — became estranged and utterly divorced from the world of life, which science has been unable to explain — not even to explain away by calling it "subjective". True, these worlds are everyday — and even more and more — connected by praxis. Yet for theory they are divided by an abyss. Two worlds: this means two truths. Or no truth at all. This is the tragedy of the modern mind which "solved the riddle of the universe," but only to replace it by another riddle: the riddle of itself."

"The belief in creation as the background of empiricomathematical [sic] science–that seems strange. Yet the ways of thought, human thought, in its search for truth are, indeed, very strange."

"The infinite Universe of the New Cosmology, infinite in Duration as well as Extension, in which eternal matter in accordance with eternal and necessary laws moves endlessly and aimlessly in eternal space, inherited all the ontological attributes of Divinity. Yet only those — all the others the departed God took with him... The Divine Artifex had therefore less and less to do in the world. He did not even have to conserve it, as the world, more and more, became able to dispense with this service..."

"What the founders of modern science … had to do, was not criticize and to combat certain faulty theories, and to correct or to replace them by better ones. They had to do something quite different. They had to destroy one world and replace it by another. They had to reshape the framework of our intellect itself, to restate and to reform its concepts, to evolve a new approach to Being, a new concept of knowledge, and a new concept of science — and even to replace a pretty natural approach, that of common sense, by another which is not natural at all."

"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."

"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!"

"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."

"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."

"(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.""

"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."

"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."

"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..."

"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)."

"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.""

"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."

"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."

"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."

"[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."

"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."

"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!""

"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."

"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."

"‘[…] 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!’"

"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."

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

"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?"

"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."

"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."

"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."

"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."

"... 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."

"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."

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

"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."

"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."

"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."

"[T]he entire course of French history since then has been influenced by something that happened in the eighteenth century” and asks “did France cease to be a great power not, as is usually thought, on 15 June 1815 on the field of Waterloo, but well before that, during the reign of Louis XV when the natural birth-rate was interrupted?"