Henri Poincaré

From Wikiquote
(Redirected from Henri Poincare)
Jump to navigation Jump to search
It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better.
To doubt everything or to believe everything are two equally convenient solutions; both dispense with the necessity of reflection.

Jules Henri Poincaré (29 April 185417 July 1912), generally known as Henri Poincaré, was one of France's greatest mathematicians and theoretical physicists, and a philosopher of science.


A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.
  • 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.
    • A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature. ...we work not only to obtain the positive results which, according to the profane, constitute our one and only affection, as to experience this esthetic emotion and to convey it to others who are capable of experiencing it.
      • "Notice sur Halphen," Journal de l'École Polytechnique (Paris, 1890), 60ème cahier, p. 143. See also Tobias Dantzig, Henri Poincaré, Critic of Crisis: Reflections on His Universe of Discourse (1954) p. 8
  • 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.
    • It is the simple hypotheses of which one must be most wary; because these are the ones that have the most chances of passing unnoticed.
      • Thermodynamique: Leçons professées pendant le premier semestre 1888–1889 (1892), Preface
  • 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.
    • The task of the educator is to make the child's spirit pass again where its forefathers have gone, moving rapidly through certain stages but suppressing none of them. In this regard, the history of science must be our guide.
      • "La logique et l'intuition dans la science mathématique et dans l'enseignement" [Logic and intuition in the science of mathematics and in teaching], L'enseignement mathématique (1899)
  • We do not have and cannot have any means of discovering whether or not we are carried along in a uniform motion of translation.
    • L’état actuel et l’avenir de la physique mathematique (1904)
  • 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.
    • Everyone is sure of this [that errors are normally distributed], Mr. Lippman told me one day, since the experimentalists believe that it is a mathematical theorem, and the mathematicians that it is an experimentally determined fact.
      • Calcul des probabilités (2nd ed., 1912), p. 171
  • 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.
    • Thought must never submit, neither to a dogma, nor to a party, nor to a passion, nor to an interest, nor to a preconceived idea, nor to whatever it may be, save to the facts themselves, because, for thought, submission would mean ceasing to be.
      • Speech, University of Brussels (19 November 1909), during the festival for the 75th anniversary of the university's foundation; published in Œuvres de Henri Poincaré (1956), p. 152
  • 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.
    • Poincaré, Henri. "Mathematical creation." The Monist (1910): 321-335. Translated from the French by George Bruce Halsted.
  • 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.
    • quoted in Jacques Hadamard, An essay on the psychology of invention in the mathematical field (1954), pp. 106.

Science and Hypothesis (1901)

La Science et l'Hypothèse, English translation: Science and Hypothesis (1905), Dover abridged edition (1952)
  • Douter de tout ou tout croire, ce sont deux solutions également commodes, qui l'une et l'autre nous dispensent de réfléchir.
    • To doubt everything or to believe everything are two equally convenient solutions; both dispense with the necessity of reflection.
      • Preface, Dover abridged edition (1952), p. xxii
  • 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.
    • Ch. I. (1905) Tr. George Bruce Halstead
  • 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.
    • Ch. I. (1905) Tr. George Bruce Halstead
  • 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.
    • Ch. I. (1905) Tr. George Bruce Halstead
  • 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.
    • Ch. I. (1905) Tr. George Bruce Halstead
  • 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.
    • Mathematicians do not study objects, but the relations between objects; to them it is a matter of indifference if these objects are replaced by others, provided that the relations do not change. Matter does not engage their attention, they are interested in form alone.
      • Ch. II: Dover abridged edition (1952), p. 20
  • 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?
    • Ch. V: Experiment and Geometry (1905) Tr. George Bruce Halstead
  • 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.
    • ... treatises on mechanics do not clearly distinguish between what is experiment, what is mathematical reasoning, what is convention, and what is hypothesis.
      • Ch. VI: The Classical Mechanics, Tr. George Bruce Halsted (1913)
  • 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.
    • The Scientist must set in order. Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house.
      • Ch. IX: Hypotheses in Physics, Tr. George Bruce Halsted (1913)
  • 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.
    • Ch. IX: Hypotheses in Physics, Tr. George Bruce Halsted (1913)
  • 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.
    • If, then, a phenomenon admits of a complete mechanical explanation, it will admit of an infinity of others, that will render an account equally well of all the particulars revealed by experiment.
      • Ch. XII: Optics and Electricity, as translated by George Bruce Halsted (1913)

The Value of Science (1905)

If all the parts of the universe are interchained in a certain measure, any one phenomenon will not be the effect of a single cause, but the resultant of causes infinitely numerous; it is, one often says, the consequence of the state of the universe the moment before.
Valeur de la Science (1905), as translated by George Bruce Halsted (1907)
  • 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.
    • Author's Essay Prefatory to the Translation: "The Choice of Facts," p.4
  • 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.
    • Time and Space … It is not nature which imposes them upon us, it is we who impose them upon nature because we find them convenient.
      • Introduction, p. 13
  • 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.
    • The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new, but to the continuous evolution of zoologic types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past. One must not think then that the old-fashioned theories have been sterile or vain.
      • Introduction, p. 14
  • 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.
    • Does the harmony the human intelligence thinks it discovers in nature exist outside of this intelligence? No, beyond doubt, a reality completely independent of the mind which conceives it, sees or feels it, is an impossibility.
      • Introduction, p. 14
  • 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 then 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.
    • Introduction, p. 14
  • 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.
    • For an aggregate of sensations to have become a remembrance capable of classification in time, it must have ceased to be actual, we must have lost the sense of its infinite complexity, otherwise it would have remained present. It must, so to speak, have crystallized around a center of associations of ideas which will be a sort of label. It is only when they have lost all life that we can classify our memories in time as a botanist arranges dried flowers in his herbarium.
    • Ch. 2: The Measure of Time
  • 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.
    • If all the parts of the universe are interchained in a certain measure, any one phenomenon will not be the effect of a single cause, but the resultant of causes infinitely numerous; it is, one often says, the consequence of the state of the universe the moment before.
      • Ch. 2: The Measure of Time
  • 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."
    • Ch. 2: The Measure of Time
  • 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.
    • Ch. 5: Analysis and Physics
  • 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.
    • Ch. 5: Analysis and Physics
  • 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.
    • As we can not give a general definition of energy, the principle of the conservation of energy signifies simply that there is something which remains constant.
      • Ch. 10: Is Science artificial?
  • 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.
    • Ch. 11: Science and Reality
  • 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.
    • Ch. 11: Science and Reality
  • 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.
    • Ch. 11: Science and Reality
  • 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.
    • Ch. 11: Science and Reality

Science and Method (1908)

Science et méthode (1908), as translated by Francis Maitland (1914)
The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful.

The page numbers below refer to the Francis Maitland translation text, not the original French text.

  • La sociologie est la science qui possède le plus de méthodes et le moins de résultats.
    • Sociology is the science which has the most methods and the least results.
      • Part I. Ch. 1 : The Selection of Facts, p. 19
  • 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.
    • The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. If nature were not beautiful it would not be worth knowing, and life would not be worth living. I am not speaking, of course, of the beauty which strikes the senses, of the beauty of qualities and appearances. I am far from despising this, but it has nothing to do with science. What I mean is that more intimate beauty which comes from the harmonious order of its parts, and which a pure intelligence can grasp.
      • Part I. Ch. 1 : The Selection of Facts, p. 22
  • 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.
    • It is because simplicity and vastness are both beautiful that we seek by preference simple facts and vast facts; that we take delight, now in following the giant courses of the stars, now in scrutinizing the microscope that prodigious smallness which is also a vastness, and now in seeking in geological ages the traces of a past that attracts us because of its remoteness.
      • Part I. Ch. 1 : The Selection of Facts, p. 23
  • 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.
    • I think I have already said somewhere that mathematics is the art of giving the same name to different things.
      • Part I. Ch. 2 : The Future of Mathematics, p. 31
  • 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.
    • The principal aim of mathematical education is to develop certain faculties of the mind, and among these intuition is not the least precious. It is through it that the mathematical world remains in touch with the real world, and even if pure mathematics could do without it, we should still have to have recourse to it to fill up the gulf that separates the symbol from reality.
      • Part II. Ch. 2 : Mathematical Definitions and Education, p. 128
      • Variant translation: The chief aim of mathematics teaching is to develop certain faculties of the mind, and among these intuition is by no means the least valuable.
  • C'est par la logique qu'on démontre, c'est par l'intuition qu'on invente.
  • 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.
    • Logic teaches us that on such and such a road we are sure of not meeting an obstacle; it does not tell us which is the road that leads to the desired end. For this, it is necessary to see the end from afar, and the faculty which teaches us to see is intuition. Without it, the geometrician would be like a writer well up in grammar but destitute of ideas.
      • Part II. Ch. 2 : Mathematical Definitions and Education, p. 130
  • 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.
    • Every definition implies an axiom, since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have proved that it involves no contradiction either in its terms or with the truths previously admitted.
      • Part II. Ch. 2 : Mathematical Definitions and Education, p. 131
  • 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.
    • The greatest chance is the birth of a great man. It is only by chance that two reproductive cells, of different sexes, met, each containing precisely those mysterious elements whose mutual reaction was to produce genius. We will agree that these elements must be rare and that their meeting is even rarer. That it would have taken little to deviate the spermatozoon that carried them from its course; it would have been enough to deviate it by a tenth of a millimeter and Napoleon would not have been born and the destiny of a continent would have been changed. No other example can better explain the true nature of chance.
      • Part I. Ch. 4 : Chance

Quotes about Poincaré

  • 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.
  • 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.
  • 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.
  • 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.
    • Paul Painlevé, Eulogy (c. 1912) as quoted by Tobias Dantzig, Henri Poincaré, Critic of Crisis: Reflections on His Universe of Discourse (1954)
  • 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.
  • 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?"


  • Later generations will regard Mengenlehre as a disease from which one has recovered.
  • Later generations will regard set theory as a disease from which one has recovered.
    • Ernst Hölder attributed this to Poincaré in 1924: "Poincaré at the Rome Congress (1908) went so far as to say …", but this is not an accurate summary of his remarks in "The Future of Mathematics". See Jeremy Gray, "Did Poincaré say ‘Set theory is a disease’?", The Mathematical Intelligencer, 13:19-22.
    • Also, probably a variant translation of Hölder's comment, "Point set topology is a disease from which the human race will soon recover."

See also

Philosophy of science
Concepts AnalysisA priori and a posterioriCausalityDemarcation problemFactInductive reasoningInquiryNatureObjectivityObservationParadigmProblem of inductionScientific methodScientific revolutionScientific theory
Related topics AlchemyEpistemologyHistory of scienceLogicMetaphysicsPseudoscienceRelationship between religion and scienceSociology of scientific knowledge
Philosophers of science PlatoAristotleStoicism
AverroesAvicennaRoger BaconWilliam of Ockham
Francis BaconThomas HobbesRené DescartesGalileo GalileiPierre GassendiIsaac NewtonDavid Hume
Immanuel KantFriedrich SchellingWilliam WhewellAuguste ComteJohn Stuart MillHerbert SpencerWilhelm WundtCharles Sanders PeirceHenri PoincaréPierre DuhemRudolf SteinerKarl Pearson
Alfred North WhiteheadBertrand RussellAlbert EinsteinOtto NeurathC. D. BroadMichael PolanyiHans ReichenbachRudolf CarnapKarl PopperW. V. O. QuineThomas KuhnImre LakatosPaul FeyerabendJürgen HabermasIan HackingBas van FraassenLarry LaudanDaniel Dennett

Wikipedia has an article about:
Wikisource has original works by or about:
Wikimedia Commons has media related to:

(by country)

AbelAnaxagorasArchimedesAristarchus of SamosAverroesArnoldBanachCantorCartanChernCohenDescartesDiophantusErdősEuclidEulerFourierGaussGödelGrassmannGrothendieckHamiltonHilbertHypatiaLagrangeLaplaceLeibnizMilnorNewtonvon NeumannNoetherPenrosePerelmanPoincaréPólyaPythagorasRiemannRussellSchwartzSerreTaoTarskiThalesTuringWeilWeylWilesWitten


123360eπFibonacci numbersIrrational numberNegative numberNumberPrime numberQuaternion


AbstractionAlgorithmsAxiomatic systemCompletenessDeductive reasoningDifferential equationDimensionEllipseElliptic curveExponential growthInfinityIntegrationGeodesicInductionProofPartial differential equationPrinciple of least actionPrisoner's dilemmaProbabilityRandomnessTheoremTopological spaceWave equation


Euler's identityFermat's Last Theorem

Pure math

Abstract algebraAlgebraAnalysisAlgebraic geometry (Sheaf theory) • Algebraic topologyArithmeticCalculusCategory theoryCombinatoricsCommutative algebraComplex analysisDifferential calculusDifferential geometryDifferential topologyErgodic theoryFoundations of mathematicsFunctional analysisGame theoryGeometryGlobal analysisGraph theoryGroup theoryHarmonic analysisHomological algebraInvariant theoryLogicNon-Euclidean geometryNonstandard analysisNumber theoryNumerical analysisOperations researchRepresentation theoryRing theorySet theorySheaf theoryStatisticsSymplectic geometryTopology

Applied math

Computational fluid dynamicsEconometricsFluid mechanicsMathematical physicsScience

History of math

Ancient Greek mathematicsEuclid's ElementsHistory of algebraHistory of calculusHistory of logarithmsIndian mathematicsPrincipia Mathematica


Mathematics and mysticismMathematics educationMathematics, from the points of view of the Mathematician and of the PhysicistPhilosophy of mathematicsUnification in science and mathematics