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Henri Poincaré

Geburtstag: 29. April 1854
Todesdatum: 17. Juli 1912
Andere Namen:Анри Пуанкаре

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Jules Henri Poincaré [pwɛ̃kaˈʀe] war ein bedeutender französischer Mathematiker, theoretischer Physiker, theoretischer Astronom und Philosoph.

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„All that is not thought is pure nothingness“

— Henri Poincaré
Context: 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 Ch. 11: Science and Reality

„The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms.“

— Henri Poincaré
Context: 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 Ch. I. (1905) Tr. George Bruce Halstead

„The principal aim of mathematical education is to develop certain faculties of the mind, and among these intuition is not the least precious.“

— Henri Poincaré
Context: 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.

„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.“

— Henri Poincaré
Context: 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)

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„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.“

— Henri Poincaré
Context: 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 Ch. VI: The Classical Mechanics (1905) [https://books.google.com/books?id=5nQSAAAAYAAJ Tr.] George Bruce Halstead

„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.“

— Henri Poincaré
Context: 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 Ch. I. (1905) Tr. George Bruce Halstead

„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.“

— Henri Poincaré
Context: 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

„It is only through science and art that civilization is of value“

— Henri Poincaré
Context: 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.<!--p.142 Ch. 11: Science and Reality

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„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.“

— Henri Poincaré
Context: 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 Ch. I. (1905) Tr. George Bruce Halstead

„When shall we say two forces are equal?“

— Henri Poincaré
Context: 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 Ch. VI: The Classical Mechanics (1905) [https://books.google.com/books?id=5nQSAAAAYAAJ Tr.] George Bruce Halstead

„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.“

— Henri Poincaré
Context: 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 Ch. 5: Analysis and Physics

„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.“

— Henri Poincaré
Context: 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

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