Frasi di Henri Poincaré

Origine: La scienza e l'ipotesi, p. 91

da La scienza e l'ipotesi

da Scienza e metodo

Origine: Citato in Daniele Corradetti, Matafisica del numero, Argonautiche, p. 45. ISBN 978-88-95299-16-7

Origine: La scienza e l'ipotesi, pp. 73-ss.

Origine: La scienza e l'ipotesi, p. 60

Origine: La scienza e l'ipotesi, p. 128

Origine: La scienza e l'ipotesi, p. 140

Origine: La scienza e l'ipotesi, p. 141

Origine: La scienza e l'ipotesi, p. 117

Douter de tout ou tout croire, ce sont deux solutions également commodes, qui l'une et l'autre nous dispensent de réfléchir.
Preface, Dover abridged edition (1952), p. xxii
Science and Hypothesis (1901)

Origine: Science and Hypothesis (1901), Ch. I. (1905) Tr. George Bruce Halstead
Contesto: 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.<!--pp.13-14

Originale: (fr) 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.
Origine: Science and Method (1908), Part I. Ch. 2 : The Future of Mathematics, p. 31

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.
Science and Method (1908)
Contesto: 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.

Origine: Science and Hypothesis (1901), Ch. VI: The Classical Mechanics (1905) Tr. https://books.google.com/books?id=5nQSAAAAYAAJ George Bruce Halstead
Contesto: 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.<!--pp.73-74

Part I. Ch. 1 : The Selection of Facts, p. 22
Science and Method (1908)
Contesto: 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.

Origine: Science and Hypothesis (1901), Ch. I: On the Nature of Mathematical Reasoning (1905) Tr. https://books.google.com/books?id=5nQSAAAAYAAJ George Bruce Halstead
Contesto: 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.<!--pp.5-6

"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
Contesto: 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.

Origine: Science and Hypothesis (1901), Ch. VI: The Classical Mechanics (1905) Tr. https://books.google.com/books?id=5nQSAAAAYAAJ George Bruce Halstead
Contesto: 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.<!--pp.73-74

Origine: Science and Hypothesis (1901), Ch. IX: Hypotheses in Physics, Tr. George Bruce Halsted (1913)
Contesto: 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.

Origine: The Value of Science (1905), Ch. 5: Analysis and Physics
Contesto: 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.<!--pp.76-77

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.
Origine: The Value of Science (1905), Ch. 11: Science and Reality

Origine: Science and Hypothesis (1901), Ch. I. (1905) Tr. George Bruce Halstead
Contesto: 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.<!--pp.10-12

Origine: Science and Hypothesis (1901), Ch. VI: The Classical Mechanics (1905) Tr. https://books.google.com/books?id=5nQSAAAAYAAJ George Bruce Halstead
Contesto: 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.<!--pp.73-74

Origine: Science and Hypothesis (1901), Ch. I. (1905) Tr. George Bruce Halstead
Contesto: 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.<!--pp.10-12

Origine: Science and Hypothesis (1901), Ch. IX: Hypotheses in Physics, Tr. George Bruce Halsted (1913)
Contesto: 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.

Origine: The Value of Science (1905), Ch. 11: Science and Reality
Contesto: 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.<!--p.142

Part II. Ch. 2 : Mathematical Definitions and Education, p. 131
Science and Method (1908)
Contesto: 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.

Origine: Science and Hypothesis (1901), Ch. I: On the Nature of Mathematical Reasoning (1905) Tr. https://books.google.com/books?id=5nQSAAAAYAAJ George Bruce Halstead
Contesto: 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.<!--pp.5-6

"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)
Contesto: 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.

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.
Origine: The Value of Science (1905), Ch. 2: The Measure of Time

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.
Origine: The Value of Science (1905), Ch. 10: Is Science artificial?

Origine: The Value of Science (1905), Ch. 11: Science and Reality

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.
Calcul des probabilités (2nd ed., 1912), p. 171

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.
Introduction, p. 13
The Value of Science (1905)

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.
Origine: Science and Hypothesis (1901), Ch. II: Dover abridged edition (1952), p. 20

Origine: Science and Hypothesis (1901), Ch. V: Experiment and Geometry (1905) Tr. https://books.google.com/books?id=5nQSAAAAYAAJ George Bruce Halstead

La sociologie est la science qui possède le plus de méthodes et le moins de résultats.
Part I. Ch. 1 : The Selection of Facts, p. 19
Science and Method (1908)