Mathematicians From France

"I suppose the body to be nothing but a statue or machine made of earth, which God forms with the explicit intention of making it as much as possible like us."

"In order to seek truth, it is necessary once in the course of our life, to doubt, as far as possible, of all things."

"I could give here several other ways of tracing and conceiving a series of curved lines, each curve more complex than any preceding one, but I think the best way to group together all such curves and them classify them in order, is by recognizing the fact that all points of those curves which we may call "geometric," that is, those which admit of precise and exact measurement, must bear a definite relation to all points of a straight line, and that this relation must be expressed by a single equation. If this equation contains no term of higher degree than the rectangle of two unknown quantities, or the square of one, the curve belongs to the first and simplest class, which contains only the circle, the parabola, the hyperbola, and the ellipse; but when the equation contains one or more terms of the third or fourth degree in one or both of the two unknown quantities (for it requires two unknown quantities to express the relation between two points) the curve belongs to the second class; and if the equation contains a term of the fifth or sixth degree in either or both of the unknown quantities the curve belongs to the third class, and so on indefinitely."

"Thus, all unknown quantities can be expressed in terms of a single quantity, whenever the problem can be constructed by means of circles and straight lines, or by conic sections, or even by some other curve of degree not greater than the third or fourth. But I shall not stop to explain this in more detail, because I should deprive you of the pleasure of mastering it yourself, as well as of the advantage of training your mind by working over it, which is in my opinion the principal benefit to be derived from this science. Because, I find nothing here so difficult that it cannot be worked out by anyone at all familiar with ordinary geometry and with algebra, who will consider carefully all that is set forth in this treatise."

"Je pense, donc je suis."

"Le bon sens est la chose du monde la mieux partagée : car chacun pense en être si bien pourvu, que ceux même qui sont les plus difficiles à contenter en toute autre chose, n'ont point coutume d'en désirer plus qu'ils en ont."

"Mais apud me omnia fiunt Mathematicè in Natura"

"No more useful inquiry can be proposed than that which seeks to determine the nature and the scope of human knowledge. ... This investigation should be undertaken once at least in his life by anyone who has the slightest regard for truth, since in pursuing it the true instruments of knowledge and the whole method of inquiry come to light. But nothing seems to me more futile than the conduct of those who boldly dispute about the secrets of nature ... without yet having ever asked even whether human reason is adequate to the solution of these problems."

"The entire method consists in the order and arrangement of the things to which the mind's eye must turn so that we can discover some truth."

"So blind is the curiosity by which mortals are possessed, that they often conduct their minds along unexplored routes, having no reason to hope for success, but merely being willing to risk the experiment of finding whether the truth they seek lies there."

"Me tenant comme je suis, un pied dans un pays et l'autre en un autre, je trouve ma condition très heureuse, en ce qu'elle est libre."

"Mr. Clerselier has written me that you are expecting from him my Meditations... in order to present them to the queen of the land. ...If I had only been as wise as they say the savages persuaded themselves that the monkeys were, I never would have become known as a maker of books: Since it is said that they imagined that the monkeys could indeed speak, if they wanted to, but that they chose not to so lest they be forced to work. And since I had not the same prudence to abstain from writing, I now have neither as much liesure nor as much peace as I would have had if I had kept quiet. But since the mistake has already been made, and since I am now known by an infinity of people at the academy, who look askance at my writings and scour them for means of harming me, I do have great hope of being known to persons of great merit, whose power and virtue could protect me."

"M. Desargues puts me under obligations on account of the pains that it has pleased him to have in me, in that he shows that he is sorry that I do not wish to study more in geometry, but I have resolved to quit only abstract geometry, that is to say, the consideration of questions which serve only to exercise the mind, and this, in order to study another kind of geometry, which has for its object the explanation of the phenomena of nature... You know that all my physics is nothing else than geometry."

"What I have given in the second book on the nature and properties of curved lines, and the method of examining them, is, it seems to me, as far beyond the treatment in the ordinary geometry, as the rhetoric of Cicero is beyond the a, b, c of children."

"No doubt you know that Galileo had been convicted not long ago by the Inquisition, and that his opinion on the movement of the Earth had been condemned as heresy. Now I will tell you that all things I explain in my treatise, among which is also that same opinion about the movement of the Earth, all depend on one another, and are based upon certain evident truths. Nevertheless, I will not for the world stand up against the authority of the Church. ...I have the desire to live in peace and to continue on the road on which I have started."

"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."

"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."

"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."

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

"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."

"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."

"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."

"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."

"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."

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

"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."

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

"Douter de tout ou tout croire, ce sont deux solutions également commodes, qui l'une et l'autre nous dispensent de réfléchir."

"The very possibility of the science of mathematics seems an insoluble contradiction. If this science is deductive only in appearance, whence does it derive that perfect rigor no one dreams of doubting? If, on the contrary, all the propositions it enunciates can be deduced one from another by the rules of formal logic, why is not mathematics reduced to an immense tautology? The syllogism can teach us nothing essentially new, and, if everything is to spring from the principle of identity, everything should be capable of being reduced to it. Shall we then admit that the enunciations of all those theorems which fill so many volumes are nothing but devious ways of saying A is A! ...Does the mathematical method proceed from particular to the general, and, if so, how can it be called deductive? ...If we refuse to admit these consequences, it must be conceded that mathematical reasoning has of itself a sort of creative virtue and consequently differs from a syllogism."

"There is no science apart from the general. It may even be said that the very object of the exact sciences is to spare us these direct verifications."

"This procedure is the demonstration by recurrence. We first establish a theorem for n = 1; then we show that if it is true of n - 1, it is true of n, and thence conclude that it is true for all the whole numbers. ..Here then we have the mathematical reasoning par excellence, and we must examine it more closely. ...The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms. ...to arrive at the smallest theorem [we] can not dispense with the aid of reasoning by recurrence, for this is an instrument which enables us to pass from the finite to the infinite. This instrument is always useful, for, allowing us to overleap at a bound as many stages as we wish, it spares us verifications, long, irksome and monotonous, which would quickly become impracticable. But it becomes indispensable as soon as we aim at the general theorem... In this domain of arithmetic,.. the mathematical infinite already plays a preponderant rôle, and without it there would be no science, because there would be nothing general."

"We can not... escape the conclusion that the rule of reasoning by recurrence is irreducible to the principle of contradiction. ...Neither can this rule come to us from experience... This rule, inaccessible to analytic demonstration and to experience, is the veritable type of the synthetic a priori judgment. On the other hand, we can not think of seeing in it a convention, as in some of the postulates of geometry. ...it is only the affirmation of the power of the mind which knows itself capable of conceiving the indefinite repetition of the same act when once this act is possible. The mind has a direct intuition of this power, and experience can only give occasion for using it and thereby becoming conscious of it."

"But, one will say, if raw experience can not legitimatize reasoning by recurrence, is it so of experiment aided by induction? We see successively that a theorem is true of the number 1, of the number 2, of the number 3 and so on; the law is evident, we say, and it has the same warranty as every physical law based on observations, whose number is very great but limited. But there is an essential difference. Induction applied to the physical sciences is always uncertain, because it rests on the belief in a general order of the universe, an order outside of us. Mathematical induction, that is, demonstration by recurrence, on the contrary, imposes itself necessarily, because it is only the affirmation of a property of the mind itself."

"Les mathématiciens n'étudient pas des objets, mais des relations entre les objets ; il leur est donc indifférent de remplacer ces objets par d'autres, pourvu que les relations ne changent pas. La matière ne leur importe pas, la forme seule les intéresse."

"We see that experience plays an indispensable role in the genesis of geometry; but it would be an error thence to conclude that geometry is, even in part, an experimental science. If it were experimental it would be only approximative and provisional. And what rough approximation! ...The object of geometry is the study of a particular 'group'; but the general group concept pre-exists... in our minds. It is imposed on us, not as form of our sense, but as form of our understanding. Only, from among all the possible groups, that must be chosen... will be... the standard to which we shall refer natural phenomena. Experience guides us in this choice without forcing it upon us; it tells us not which is the truest geometry, but which is the most convenient. Notice that I have been able to describe the fantastic worlds... imagined without ceasing to employ the language of ordinary geometry."

"Is the position tenable, that certain phenomena, possible in Euclidean space, would be impossible in non-Euclidean space, so that experience, in establishing these phenomena, would directly contradict the non-Euclidean hypothesis? For my part I think no such question can be put. To my mind it is precisely equivalent to the following, whose absurdity is patent to all eyes: are there lengths expressible in meters and centimeters, but which can not be measured in fathoms, feet, and inches, so that experience, in ascertaining the existence of these lengths, would directly contradict the hypothesis that there are fathoms divided into six feet?"

"What is mass? According to Newton, it is the product of the volume by the density. According to Thomson and Tait, it would be better to say that density is the quotient of the mass by the volume. What is force? It, is replies Lagrange, that which moves or tends to move a body. It is, Kirchhoff will say, the product of the mass by the acceleration. But then, why not say the mass is the quotient of the force by the acceleration? These difficulties are inextricable. When we say force is the cause of motion, we talk metaphysics, and this definition, if one were content with it, would be absolutely sterile. For a definition to be of any use, it must teach us to measure force; moreover that suffices; it is not at all necessary that it teach us what force is in itself, nor whether it is the cause or the effect of motion. We must therefore first define the equality of two forces. When shall we say two forces are equal? It is, we are told, when, applied to the same mass, they impress upon it the same acceleration, or when, opposed directly one to the other, they produce equilibrium. This definition is only a sham. A force applied to a body can not be uncoupled to hook it up to another body, as one uncouples a locomotive to attach it to another train. It is therefore impossible to know what acceleration such a force, applied to such a body, would impress upon such an other body, if it were applied to it. It is impossible to know how two forces which are not directly opposed would act, if they were directly opposed. We are... obliged in the definition of the equality of the two forces to bring in the principle of the equality of action and reaction; on this account, this principle must no longer be regarded as an experimental law, but as a definition."

"... les traités de mécanique ne distinguent pas bien nettement ce qui est expérience, ce qui est raisonnement mathématique, ce qui est convention, ce qui est hypothèse."

"Le savant doit ordonner ; on fait la science avec des faits comme une maison avec des pierres ; mais une accumulation de faits n'est pas plus une science qu'un tas de pierres n'est une maison."

"If we study the history of science we see happen two inverse phenomena... Sometimes simplicity hides under complex appearances; sometimes it is the simplicity which is apparent, and which disguises extremely complicated realities. ...No doubt, if our means of investigation should become more and more penetrating, we should discover the simple under the complex, then the complex under the simple, then again the simple under the complex, and so on, without our being able to foresee what will be the last term. We must stop somewhere, and that science may be possible, we must stop when we have found simplicity. This is the only ground on which we can rear the edifice of our generalizations."

"It is often said that experiments should be made without preconceived ideas. That is impossible. Not only would it make every experiment fruitless, but even if we wished to do so, it could not be done. Every man has his own conception of the world, and this he cannot so easily lay aside. We must, for example, use language, and our language is necessarily steeped in preconceived ideas."

"Si donc un phénomène comporte une explication mécanique complète, il en comportera une infinité d’autres qui rendront également bien compte de toutes les particularités révélées par l’expérience."

"Scientists believe there is a hierarchy of facts and that among them may be made a judicious choice. They are right, since otherwise there would be no science..."

"One need only open the eyes to see that the conquests of industry which have enriched so many practical men would never have seen the light, if these practical men alone had existed and if they had not been preceded by unselfish devotees who died poor, who never thought of utility, and yet had a guide far other than caprice. As Mach says, these devotees have spared their successors the trouble of thinking."

"Le temps et l’espace... Ce n’est pas la nature qui nous les impose, c’est nous qui les imposons à la nature parce que nous les trouvons commodes."

"Il ne faut pas comparer la marche de la science aux transformations d’une ville, où les édifices vieillis sont impitoyablement jetés à bas pour faire place aux constructions nouvelles, mais à l’évolution continue des types zoologiques qui se développent sans cesse et finissent par devenir méconnaissables aux regards vulgaires, mais où un œil exercé retrouve toujours les traces du travail antérieur des siècles passés. Il ne faut donc pas croire que les théories démodées ont été stériles et vaines."

"Cette harmonie que l’intelligence humaine croit découvrir dans la nature, existe-t-elle en dehors de cette intelligence ? Non, sans doute, une réalité complètement indépendante de l’esprit qui la conçoit, la voit ou la sent, c’est une impossibilité. Un monde si extérieur que cela, si même il existait, nous serait à jamais inaccessible. Un monde si extérieur que cela, si même il existait, nous serait à jamais inaccessible."

"What we call objective reality is, in the last analysis, what is common to many thinking beings, and could be common to all; this common part, we shall see, can only be the harmony expressed by mathematical laws. It is this harmony... which is the sole objective reality, the only truth we can attain; and when I add that the universal harmony of the world is the source of all beauty, it will be understood what price we should attach to the slow and difficult progress which little by little enables us to know it better."

"The mathematician is born, not made."

"Pour qu’un ensemble de sensations soit devenu un souvenir susceptible d’être classé dans le temps, il faut qu’il ait cessé d’être actuel, que nous ayons perdu le sens de son infinie complexité, sans quoi il serait resté actuel. Il faut qu’il ait pour ainsi dire cristallisé autour d’un centre d’associations d’idées qui sera comme une sorte d’étiquette. Ce n’est que quand ils auront ainsi perdu toute vie que nous pourrons classer nos souvenirs dans le temps, comme un botaniste range dans son herbier les fleurs desséchées."