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Home > Dictionary of Science Quotations > Scientist Names Index P > Henri Poincaré Quotes > Mathematics

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Henri Poincaré
(29 Apr 1854 - 17 Jul 1912)

French mathematician, physicist and astronomer , who is often described as the last generalist in mathematics.


Henri Poincaré Quotes on Mathematics (24 quotes)

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... I left Caen, where I was living, to go on a geologic excursion under the auspices of the School of Mines. The incidents of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go to 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-Eudidean 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 convenience sake, I verified the result at my leisure.
— Henri Poincaré
Quoted in Sir Roger Penrose, The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics (1990), 541. Science and Method (1908) 51-52, 392.
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Les mathématique sont un triple. Elles doivent fournir un instrument pour l'étude de la nature. Mais ce n'est pas tout: elles ont un but philosophique et, j'ose le dire, un but esthétique.
Mathematics has a threefold purpose. It must provide an instrument for the study of nature. But this is not all: it has a philosophical purpose, and, I daresay, an aesthetic purpose.
— Henri Poincaré
La valeur de la science. In Anton Bovier, Statistical Mechanics of Disordered Systems (2006), 161.
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Longtemps les objets dont s'occupent les mathématiciens étaient our la pluspart mal définis; on croyait les connaître, parce qu'on se les représentatit avec le sens ou l'imagination; mais on n'en avait qu'une image grossière et non une idée précise sure laquelle le raisonment pût avoir prise.
For a long time the objects that mathematicians dealt with were mostly ill-defined; one believed one knew them, but one represented them with the senses and imagination; but one had but a rough picture and not a precise idea on which reasoning could take hold.
— Henri Poincaré
La valeur de la science. In Anton Bovier, Statistical Mechanics of Disordered Systems (2006), 97.
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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
…...
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Every good mathematician should also be a good chess player and vice versa.
— Henri Poincaré
Science and Method (1914, 2003), 48.
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Everybody firmly believes in it [Nomal Law of Errors] because the mathematicians imagine it is a fact of observation, and observers that it is a theory of mathematics.
— Henri Poincaré
…...
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How is it that there are so many minds that are incapable of understanding mathematics? ... the skeleton of our understanding, ... and actually they are the majority. ... We have here a problem that is not easy of solution, but yet must engage the attention of all who wish to devote themselves to education.
— Henri Poincaré
Science and Method (1914, 2003), 117-118.
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If we wish to foresee the future of mathematics, our proper course is to study the history and present condition of the science.
— Henri Poincaré
Science and Method (1914, 2003), 25.
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In addition to this it [mathematics] provides its disciples with pleasures similar to painting and music. They admire the delicate harmony of the numbers and the forms; they marvel when a new discovery opens up to them an unexpected vista; and does the joy that they feel not have an aesthetic character even if the senses are not involved at all? … For this reason I do not hesitate to say that mathematics deserves to be cultivated for its own sake, and I mean the theories which cannot be applied to physics just as much as the others.
— Henri Poincaré
(1897) From the original French, “Et surtout, leurs adeptes y trouvent des jouissances analogues á celles que donnent la peinture et la musique. Ils admirent la délicate harmonie des nombres et des formes; ils s’émerveillent quand une découverte nouvelle leur ouvre une perspective inattendue; et la joie qu’ils éprouvent ainsi n’a-t-elle pas le caractère esthétique, bien que les sens n’y prennent aucune part?...C’est pourquoi je n’hésite pas à dire que les mathématiques méritent d’être cultivées pour elles-mêmes et que les théories qui ne peuvent être appliquées á la physique doivent l’être comme les autres.” Address read for him at the First International Congress of Mathematicians in Zurich: '‘Sur les rapports de l’analyse pure et de la physique', in Proceedings of that Congress 81-90, (1898). Also published as 'L’Analyse et la Physique', in La Valeur de la Science (1905), 137-151. As translated in Armand Borel, 'On the Place of Mathematics in Culture', in Armand Borel: Œvres: Collected Papers (1983), Vol. 4, 420-421.
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It is through it [intuition] 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.
— Henri Poincaré
…...
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It may be appropriate to quote a statement of Poincare, who said (partly in jest no doubt) that there must be something mysterious about the normal law since mathematicians think it is a law of nature whereas physicists are convinced that it is a mathematical theorem.
— Henri Poincaré
Quoted in Mark Kac, Statistical Independence in Probability, Analysis and Number Theory (1959), 52.
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Mathematical discoveries, small or great … are never born of spontaneous generation. They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious.
— Henri Poincaré
As given, without citation, in Eric Temple Bell, Men of Mathematics (1937), 548.
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Mathematicians attach great importance to the elegance of their methods and their results. This is not pure dilettantism. What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details. But this is exactly what yields great results, in fact the more we see this aggregate clearly and at a single glance, the better we perceive its analogies with other neighboring objects, consequently the more chances we have of divining the possible generalizations. Elegance may produce the feeling of the unforeseen by the unexpected meeting of objects we are not accustomed to bring together; there again it is fruitful, since it thus unveils for us kinships before unrecognized. It is fruitful even when it results only from the contrast between the simplicity of the means and the complexity of the problem set; it makes us then think of the reason for this contrast and very often makes us see that chance is not the reason; that it is to be found in some unexpected law. In a word, the feeling of mathematical elegance is only the satisfaction due to any adaptation of the solution to the needs of our mind, and it is because of this very adaptation that this solution can be for us an instrument. Consequently this esthetic satisfaction is bound up with the economy of thought.
— Henri Poincaré
In 'The Future of Mathematics', Monist, 20, 80. Translated from the French by George Bruce Halsted.
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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.
— Henri Poincaré
In Science and Hypothesis (1901, 1907). Translated by W.J.G. from the original French, “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.”
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Mathematics is the art of giving the same name to different things.
— Henri Poincaré
Henri Poincaré and George Bruce Halsted (trans.) The Foundations of Science: Science and Hypothesis (1921), 375.
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One would have to have completely forgotten the history of science so as not to remember that the desire to know nature has had the most constant and the happiest influence on the development of mathematics.
— Henri Poincaré
In Henri Poincaré and George Bruce Halsted (trans.), The Value of Science: Essential Writings of Henri Poincare (1907), 79.
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Only the privileged few are called to enjoy it [mathematics] fully, it is true; but is it not the same with all the noblest arts?
— Henri Poincaré
From 'The Relation of Analysis and Mathematical Physics', Bulletin American Mathematical Society (1899), 4 (1899), 248. As cited in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 181.
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So is not mathematical analysis then not just a vain game of the mind? To the physicist it can only give a convenient language; but isn’t that a mediocre service, which after all we could have done without; and, it is not even to be feared that this artificial language be a veil, interposed between reality and the physicist’s eye? Far from that, without this language most of the intimate analogies of things would forever have remained unknown to us; and we would never have had knowledge of the internal harmony of the world, which is, as we shall see, the only true objective reality.
— Henri Poincaré
From La valeur de la science. In Anton Bovier, Statistical Mechanics of Disordered Systems (2006), 3, giving translation "approximately" in the footnote of the opening epigraph in the original French: “L’analyse mathématique, n’est elle donc qu’un vain jeu d’esprit? Elle ne peut pas donner au physicien qu’un langage commode; n’est-ce pa là un médiocre service, dont on aurait pu se passer à la rigueur; et même n’est il pas à craindre que ce langage artificiel ne soit pas un voile interposé entre la réalité at l’oeil du physicien? Loin de là, sans ce langage, la pluspart des anaologies intimes des choses nous seraient demeurées à jamais inconnues; et nous aurions toujours ignoré l’harmonie interne du monde, qui est, nous le verrons, la seule véritable réalité objective.” Another translation, with a longer quote, beginning “Without this language…”, is on the Henri Poincaré Quotes" page of this website.
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The genesis of mathematical creation is a problem which should intensely interest the psychologist.
— Henri Poincaré
In 'Mathematical Creation', The Value of Science, collected in Henri Poincaré and George bruce Halsted (trans.), The Foundations of Science (1913), 383.
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The genesis of mathematical invention is a problem that must inspire the psychologist with the keenest interest. For this is the process in which the human mind seems to borrow least from the exterior world, in which it acts, or appears to act, only by itself and on itself, so that by studying the process of geometric thought, we may hope to arrive at what is most essential in the human mind
— Henri Poincaré
As translated in Arthur I. Miller, Imagery in Scientific Thought Creating 20th-Century Physics (1984, 2013), 307. Opening of Paper delivered at Conference at the Institut Général Psychologique, Paris, 'L’Invention Mathématique', published in Enseignment Mathématique (1908), 10, 357. From the original French, “La genèse do l’Invention mathématique est un problème qui doit inspirer le plus vif intérêt au psychologue. C’est l’acte dans lequel l’esprit humain semble le moins emprunter au monde extérieur, où il n’agit ou ne paraît agir que par lui-même et sur lui-même, de sorte, qu’en étudiant le processus de la pensée géométrique, c’est ce qu’il y a de plus essentiel dans l’esprit humain que nous pouvons espérer atteindre.”
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The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law. They reveal the kinship between other facts, long known, but wrongly believed to be strangers to one another.
— Henri Poincaré
From Lecture to the Psychological Society, Paris, 'Mathematical Creation', translation collected in James Roy Newman The World of Mathematics (1956), Vol. 4, 2043.
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Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the logic piano imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages. No more than these machines need the mathematician know what he does.
— Henri Poincaré
From 'Les Mathématiques et la Logique', Science et Méthode (1908, 1920), Livre 2, Chap. 3, Sec. 2, 157. English as in Henri Poincaré and George Bruce Halsted (trans.), 'Mathematics and Logic', Science and Method collected in The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method (1913), 451. From the French, “Ainsi, c’est bien entendu, pour démontrer un théorème, il n’est pas nécessaire ni même utile de savoir ce qu’il veut dire. On pourrait remplacer le géomètre par le piano à raisonner imaginé par Stanley Jevons; ou, si l’on aime mieux, on pourrait imaginer une machine où l’on introduirait les axiomes par un bout pendant qu’on recueillerait les théorèmes à l’autre bout, comme cette machine légendaire de Chicago où les porcs entrent vivants et d’où ils sortent transformés en jambons et en saucisses. Pas plus que ces machines, le mathématicien n’a besoin de comprendre ce qu’il fait”.
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What, in fact, is mathematical discovery? It does not consist in making new combinations with mathematical entities that are already known. That can be done by anyone, and the combinations that could be so formed would be infinite in number, and the greater part of them would be absolutely devoid of interest. Discovery consists precisely in not constructing useless combinations, but in constructing those that are useful, which are an infinitely small minority. Discovery is discernment, selection.
— Henri Poincaré
In Science et Méthode (1920), 48, as translated by Francis Maitland, in Science and Method (1908, 1952), 50-51. Also seen elsewhere translated with “invention” in place of “discovery”.
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Without this language [mathematics] most of the intimate analogies of things would have remained forever unknown to us; and we should forever have been ignorant of the internal harmony of the world, which is the only true objective reality. …
This harmony … 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.
— Henri Poincaré
From La Valeur de la Science, as translated by George Bruce Halsted, in 'The Value of Science', Popular Science Monthly (Sep 1906), 69 195-196.
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See also:
  • 29 Apr - short biography, births, deaths and events on date of Poincaré's birth.
  • The Value of Science: Essential Writings of Henri Poincaré, by Henri Poincaré. - book suggestion.

Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
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