Academics From France

"Thought is only a flash between two long nights, but this flash is everything."

"Talk with M. Hermite. He never evokes a concrete image, yet you soon perceive that the more abstract entities are to him like living creatures."

"Point set topology is a disease from which the human race will soon recover."

"The ones who are preoccupied by logic are above all; to read their works, one is tempted to believe they have advanced only step by step, after the manner of a Vauban who pushes on his trenches against the place besieged, leaving nothing to chance. The others are guided by intuition and, at the first stroke, make quick but sometimes precarious conquests, like bold cavalrymen of the advance guard."

"One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. But the next morning ... I had only to write out the results, which took but a few hours. ... Just at this time I left Caen, where I was then living, to go on a geological excursion under the auspices of the school of mines. The changes of travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’ sake I verified the result at my leisure. ... Most striking at first is this appearance of sudden illumination, a manifest sign of long, unconscious prior work. The rôle of this unconscious work in mathematical invention appears to me incontestable ... Often when one works at a hard question, nothing good is accomplished at the first attack. Then one takes a rest, longer or shorter, and sits down anew to the work. During the first half-hour, as before, nothing is found, and then all of a sudden the decisive idea presents itself to the mind."

"La pensée ne doit jamais se soumettre, ni à un dogme, ni à un parti, ni à une passion, ni à un intérêt, ni à une idée préconçue, ni à quoi que ce soit, si ce n'est aux faits eux-mêmes, parce que, pour elle, se soumettre, ce serait cesser d'être."

"Tout le monde y croit cependant, me disait un jour M. Lippmann, car les expérimentateurs s'imaginent que c'est un théorème de mathématiques, et les mathématiciens que c'est un fait expérimental."

"Que l'on cherche à se représenter la figure formée par ces deux courbes et leurs intersections en nombre infini dont chacune correspond à une solution doublement asymplotique. ces intersections forment une sorte de treillis, de tissu, de réseau à maille infiniment serrées ; chacune de ces deux courbes ne doit jamais se recouper elle-même, mais elle doit se replier sur elle même de manière infiniment complexe pour venir recouper une infinité de fois toutes les mailles du réseau. On sera frappé de la complexité de cette figure, que je ne cherche même pas à tracer. Rien de plus propre à nous donner une idée de la complication du problème des trois corps et en général de tous les problèmes de la Dynamique où il n'y a pas d'intégrale uniforme et où les séries de Bohlin sont divergentes."

"We do not have and cannot have any means of discovering whether or not we are carried along in a uniform motion of translation."

"La tâche de l'éducateur est de faire repasser l'esprit de l'enfant par où a passé celui de ses pères, en passant rapidement par certaines étapes mais en n'en supprimant aucune. À ce compte, l'histoire de la science doit être notre guide."

"C'est même des hypothèses simples qu'il faut le plus se défier, parce que ce sont celles qui ont le plus de chances de passer inaperçues."

"Le savant digne de ce nom, le géomètre surtout, éprouve en face de son œuvre la même impression que l'artiste ; sa jouissance est aussi grande et de même nature."

"Later generations will regard set theory as a disease from which one has recovered."

"Later generations will regard Mengenlehre as a disease from which one has recovered."

"Science is a system of relations. Poincaré, saying so, says also, "It is before all a classification, a manner of bringing together facts which appearances separate, though they are bound together by some natural and hidden kinship...."It is in the relations alone that objectivity must be sought; it would be vain to seek it in beings considered as isolated from one another...."External objects...for which the word object was invented, are really objects and not fleeting and fugitive appearances, because they are not only groups of sensations, but groups cemented by a constant bond. It is this bond, and this bond alone, which is the object in itself, and this bond is a relation. "Therefore, when we ask what is the objective value of science, that does not mean: Does science teach us the true nature of things? but it means: Does it teach us the true relations of things?""

"I rely for my information about mathematical creation on such sources as Poincaré, who speaks of the mind seeming to act only of itself and on itself, selecting, making only the useful combinations, choosing, and finally being struck, as by strong light, with certainty."

"With the disappearance of the great French mathematician has disappeared the one man whose thought could carry all other thoughts, the one mind who, through a sort of rediscovery, could penetrate to its very depth all the knowledge which the mind of man can comprehend. And that is why the demise of this man at the peak of his intellectual strength is such a disaster. Discoveries will lag, groping efforts will be drawn out; for, the potent luminous brain will not be there to coordinate disjointed research, or to cast the daring plummet of a new theory into a world of obscure facts suddenly revealed by experience."

"One of the enduring legacies of Napoleon was the French system of grandes écoles, the elite schools that train the country's top technocratic and managerial students even today. ... Biographies of French mathematicians often begin with awed accounts of how well they did on the entrance tests, and how they fared on various national exams and competitions. Poincaré was no exception. He obtained the first prize in several national competitions, and was among the highest-ranking applicants to the École Polytechnique and the École Normale Supérieure in Paris, schools especially famous for the quality of their mathematics."

"Poincaré has justly emphasized the fact that we distinguish two kinds of alterations of the bodily object, "changes of state" and "changes of position." The latter, he remarked, are alterations which we can reverse by arbitrary motions of our bodies."

"Pure mathematicians... more than all others, have been led to realise how cautious we must be of the dictates of intuition and so-called common sense. They know that the fact that we can conceive or imagine a certain thing only in a certain way is no criterion of the correctness of our judgement. Examples in mathematics abound. ...Mathematicians, as a whole, refused to question the soundness of Einstein's theory on the sole plea that it conflicted with our traditional intuitional concepts of space and time, and we need not be surprised to find Poincaré... lending full support to Einstein when the theory was so bitterly assailed in its earlier days."

"Le plus grand hasard est la naissance d’un grand homme. Ce n’est que par hasard que se sont rencontrées deux cellules génitales, de sexe différent, qui contenaient précisément, chacune de son côté, les éléments mystérieux dont la réaction mutuelle devait produire le génie. On tombera d’accord que ces éléments doivent être rares et que leur rencontre est encore plus rare. Qu’il aurait fallu peu de chose pour dévier de sa route le spermatozoïde qui les portait ; il aurait suffi de le dévier d’un dixième de millimètre et Napoléon ne naissait pas et les destinées d’un continent étaient changées. Nul exemple ne peut mieux faire comprendre les véritables caractères du hasard."

"La logique parfois engendre des monstres. Depuis un demi-siècle on a vu surgir une foule de fonctions bizarres qui semblent s’efforcer de ressembler aussi peu que possible aux honnêtes fonctions qui servent à quelque chose. Plus de continuité, ou bien de la continuité, mais pas de dérivées, etc. Bien plus, au point de vue logique, ce sont ces fonctions étranges qui sont les plus générales, celles qu’on rencontre sans les avoir cherchées n’apparaissent plus que comme un cas particulier. Il ne leur reste qu’un tout petit coin."

"Toute définition implique un axiome, puisqu'elle affirme l'existence de l'objet défini. La définition ne sera donc justifiée, au point de vue purement logique, que quand on aura démontré qu'elle n'entraîne pas de contradiction, ni dans les termes, ni avec les vérités antérieurement admises."

"La logique nous apprend que sur tel ou tel chemin nous sommes sûrs de ne pas rencontrer d'obstacle ; elle ne nous dit pas quel est celui qui mène au but. Pour cela il faut voir le but de loin, et la faculté qui nous apprend à voir, c'est l'intuition. Sans elle, le géomètre serait comme un écrivain qui serait ferré sur la grammaire, mais qui n'aurait pas d'idées."

"C'est par la logique qu'on démontre, c'est par l'intuition qu'on invente."

"Le but principal de l'enseignement mathématique est de développer certaines facultés de l'esprit et parmi elles l'intuition n'est pas la moins précieuse. C'est par elle que le monde mathématique reste en contact avec le monde réel et quand les mathématiques pures pourraient s'en passer, il faudrait toujours y avoir recours pour combler l'abîme qui sépare le symbole de la réalité."

"Je ne sais si je n’ai déjà dit quelque part que la Mathématique est l’art de donner le même nom à des choses différentes."

"C’est parce que la simplicité, parce que la grandeur est belle, que nous rechercherons de préférence les faits simples et les faits grandioses, que nous nous complairons tantôt à suivre la course gigantesque des astres, tantôt à scruter avec le microscope cette prodigieuse petitesse qui est aussi une grandeur, tantôt à rechercher dans les temps géologiques les traces d’un passé qui nous attire parce qu’il est lointain."

"Le savant n'étudie pas la nature parce que cela est utile; il l'étudie parce qu'il y prend plaisir et il y prend plaisir parce qu'elle est belle. Si la nature n'était pas belle, elle ne vaudrait pas la peine d'être connue, la vie ne vaudrait pas la peine d'être vécue. Je ne parle pas ici, bien entendu, de cette beauté qui frappe les sens, de la beauté des qualités et des apparences; non que j'en fasse fi, loin de là, mais elle n'a rien à faire avec la science; je veux parler de cette beauté plus intime qui vient de l'ordre harmonieux des parties, et qu'une intelligence pure peut saisir."

"La sociologie est la science qui possède le plus de méthodes et le moins de résultats."

"All that is not thought is pure nothingness; since we can think only thought and all the words we use to speak of things can express only thoughts, to say there is something other than thought, is therefore an affirmation which can have no meaning. And yet—strange contradiction for those who believe in time—geologic history shows us that life is only a short episode between two eternities of death, and that, even in this episode, conscious thought has lasted and will last only a moment. Thought is only a gleam in the midst of a long night. But it is this gleam which is everything."

"It is only through science and art that civilization is of value. Some have wondered at the formula: science for its own sake; an yet it is as good as life for its own sake, if life is only misery; and even as happiness for its own sake, if we do not believe that all pleasures are of the same quality... Every act should have an aim. We must suffer, we must work, we must pay for our place at the game, but this is for seeing's sake; or at the very least that others may one day see."

"Now what is science? ...it is before all a classification, a manner of bringing together facts which appearances separate, though they are bound together by some natural and hidden kinship. Science, in other words, is a system of relations. ...it is in relations alone that objectivity must be sought. ...it is relations alone which can be regarded as objective. External objects... are really objects and not fleeting and fugitive appearances, because they are not only groups of sensations, but groups cemented by a constant bond. It is this bond, and this bond alone, which is the object in itself, and this bond is a relation."

"What is objective must be common to many minds and consequently transmissible from one to the other, and as this transmission can only come about by... discourse... we are even forced to conclude: no discourse no objectivity."

"Comme nous ne pouvons pas donner de l'énergie une définition générale, le principe de la conservation de l'énergie signifie simplement qu'il y a quelque chose qui demeure constant."

"All laws are... deduced from experiment; but to enunciate them, a special language is needful... ordinary language is too poor... This... is one reason why the physicist can not do without mathematics; it furnishes him the only language he can speak. And a well-made language is no indifferent thing; ...the analyst, who pursues a purely esthetic aim, helps create, just by that, a language more fit to satisfy the physicist. ...law springs from experiment, but not immediately. Experiment is individual, the law deduced from it is general; experiment is only approximate, the law is precise... In a word, to get the law from experiment, it is necessary to generalize... But how generalize? ...in this choice what shall guide us? It can only be analogy. ...What has taught us to know the true profound analogies, those the eyes do not see but reason divines? It is the mathematical spirit, which disdains matter to cling only to pure form."

"Mathematics have a triple aim. They must furnish an instrument for the study of nature. But that is not all: they have a philosophic aim and, I dare maintain, an esthetic aim. They must aid the philosopher to fathom the notions of number, of space, of time. And above all, their adepts find therein delights analogous to those given by painting and music. They admire the delicate harmony of numbers and forms; they marvel when a new discovery opens to them an unexpected perspective; and has not the joy they thus feel the esthetic character, even though the senses take no part therein? Only a privileged few are called to enjoy it fully, it is true, but is not this the case for all the noblest arts? This is why I do not hesitate to say that mathematics deserve to be cultivated for their own sake, and the theories inapplicable to physics as well as the others. Even if the physical aim and the esthetic aim were not united, we ought not to sacrifice either."

"We have not a direct intuition of simultaneity, nor of the equality of two durations. If we think we have this intuition, this is an illusion. We replace it by the aid of certain rules which we apply almost always without taking count of them. ...We ...choose these rules, not because they are true, but because they are the most convenient, and we may recapitulate them as follows: "The simultaneity of two events, or the order of their succession, the equality of two durations, are to be so defined that the enunciation of the natural laws may be as simple as possible. In other words, all these rules, all these definitions, are only the fruit of an unconscious opportunism.""

"(1) The rules applied are exceedingly various. (2) It is difficult to separate the qualitative problem of simultaneity from the quantitative problem of the measurement of time; no matter whether a chronometer is used, or whether account must be taken of a velocity of transmission, as... of light, because such a velocity could not be measured without measuring a time."

"Or... they observe an astronomic phenomenon... an eclipse of the moon, and... suppose... this... is perceived simultaneously from all points of the earth. That is not altogether true, since the propagation of light is not instantaneous; if absolute exactitude were desired, there would be a correction... according to a complicated rule."

"When navigators... determine a ... they must... calculate Paris time...[with] a chronometer set for Paris. The qualitative problem of simultaneity is made to depend upon the quantitative problem of the measurement of time."

"Roemer used eclipses of the satellites of Jupiter, and sought how much the event fell behind its prediction. But... this prediction [is] made... by... astronomic laws; for instance Newton's... [T]he velocity of light... is adopted, such that the astronomic laws compatible with this value may be as simple as possible."

"He has begun by supposing that light has a constant velocity... the same in all directions. This... could never be verified directly by experiment... The postulate... resembling the ... furnishes us with a new rule for the investigation of simultaneity."

"When an astronomer tells me that some stellar phenomenon, which his telescope reveals to him at this moment, happened... fifty years ago... I... ask... how he has measured the velocity of light."

"Si toutes les parties de l’univers sont solidaires dans une certaine mesure, un phénomène quelconque ne sera pas l’effet d’une cause unique, mais la résultante de causes infiniment nombreuses ; il est, dit-on souvent, la conséquence de l’état de l’univers un instant auparavant."

"[H]ave we really the right to speak of the cause of a phenomenon?"

"Behold... the only... [rule] we can follow: when a phenomenon appears... as the cause of another, we regard it as anterior. ...[T]herefore by cause... we define time; but...how do we recognize which is the cause and which the effect? We assume... the anterior fact, the antecedent, is the cause of the... consequent. It is then by time that we define cause. ...[S]hall we escape from this vicious circle?"

"Is not my present nearer my past of yesterday than the present of Sirius?"

"[This] shows, perhaps, why we have tried to put all physical phenomena into the same frame. But that can not pass for a definition of simultaneity, since this hypothetical intelligence, even if it existed, would be for us impenetrable. It is... necessary to seek something else."

"[Y]et when we speak of time... do we not unconsciously adopt this hypothesis... [and] put ourselves in the place of this imperfect god... [D]o not even the atheists put themselves in the place where god would be..?"