Henri Poincaré

Poincaré worked between 1854 and 1912 across mathematician, philosopher, astronomer, physicist and engineer and 5 further recorded occupations. Every statement below returns to a source shown in the evidence panel; where the record is silent or divided, the article says so instead of resolving it.

Henri Poincaré studied at Mines ParisTech, lycée Henri-Poincaré and École polytechnique. The recorded working language is French. The field of work recorded is algebraic geometry, mathematical analysis and geometry. Residence is recorded at France. Employment is recorded with University of Paris, École polytechnique and Corps des mines. Positions recorded include president of the Mathematical Society of France and seat 24 of the Académie française. Works named in the authority record are Poincaré conjecture, Poincaré group, Poincaré recurrence theorem, Poincaré disk model, Poincaré duality and Poincaré inequality. Distinctions recorded are Poncelet Prize, Sylvester Medal, Commander of the Legion of Honour and Gold Medal of the Royal Astronomical Society.

• Name — Henri Poincaré • Birth Name — Jules Henri Poincaré • Born — 29 April 1854 • Birthplace — Nancy, Meurthe-et-Moselle, France • Died — 17 July 1912 • Place of death — Paris, France • Relatives — Raymond Poincaré (cousin) Lucien Poincaré (cousin) • Fields — * Mathematics • physics • Spouse — Jeanne-Louise Poulain d'Andecy • Workplaces — * Corps des Mines • Caen University • La Sorbonne • Bureau des Longitudes • Education — * École Polytechnique • École des Mines • University of Paris (Doctorat) • Thesis Title — Sur les propriétés des fonctions définies par les équations différences • Thesis Url — https: patrimoine.sorbonne-universite.fr/idurl/1/1607 • Thesis Year — 1879 • Doctoral Advisor — Charles Hermite • Doctoral Students — * Louis Bachelier • Jean Bosler • Dimitrie Pompeiu • Mihailo Petrović Alas • Notable Students — * Tobias Dantzig • Théophile de Donder • Awards — * RAS Gold Medal (1900) • Sylvester Medal (1901) • Matteucci Medal (1905) • Bolyai Prize (1905) • Bruce Medal (1911) • Footnotes — He was an uncle of Pierre Boutroux.

Poincaré was born on 29 April 1854 in Cité Ducale neighborhood, Nancy, Meurthe-et-Moselle, into an influential French family. His father Léon Poincaré (1828–1892) was a professor of medicine at the University of Nancy. His younger sister Aline married the spiritual philosopher Émile Boutroux. Another notable member of Henri's family was his cousin, Raymond Poincaré, a fellow member of the Académie française, who was President of France from 1913 to 1920, and three-time Prime Minister of France between 1913 and 1929.

• Illustration — Plaque on the birthplace of Henri Poincaré at house number 117 on the Grande Rue in the city of Nancy During his childhood he was seriously ill for a time with diphtheria and received special instruction from his mother, Eugénie Launois (1830–1897). In 1862, Henri entered the Lycée in Nancy (now renamed the Lycée Henri-Poincaré in his honour, along with Henri Poincaré University, also in Nancy). He spent eleven years at the Lycée and during this time he proved to be one of the top students in every topic he studied. He excelled in written composition. His mathematics teacher described him as a "monster of mathematics" and he won first prizes in the concours général, a competition between the top pupils from all the Lycées across France. His poorest subjects were music and physical education, where he was described as "average at best". Poor eyesight and a tendency towards absentmindedness may explain these difficulties. He graduated from the Lycée in 1871 with a baccalauréat in both letters and sciences. During the Franco-Prussian War of 1870, he served alongside his father in the Ambulance Corps. Poincaré entered the École Polytechnique as the top qualifier in 1873 and graduated in 1875. There he studied mathematics as a student of Charles Hermite, continuing to excel and publishing his first paper (Démonstration nouvelle des propriétés de l'indicatrice d'une surface) in 1874. From November 1875 to June 1878 he studied at the École des Mines, while continuing the study of mathematics in addition to the mining engineering syllabus, and received the degree of ordinary mining engineer in March 1879. As a graduate of the École des Mines, he joined the Corps des Mines as an inspector for the Vesoul region in northeast France. He was on the scene of a mining disaster at Magny in August 1879 in which 18 miners died. He carried out the official investigation into the accident. At the same time, Poincaré was preparing for his Doctorate in Science in mathematics under the supervision of Charles Hermite. His doctoral thesis was in the field of differential equations. It was named Sur les propriétés des fonctions définies par les équations aux différences partielles. Poincaré devised a new way of studying the properties of these equations. He not only faced the question of determining the integral of such equations, but also was the first person to study their general geometric properties. He realised that they could be used to model the behaviour of multiple bodies in free motion within the Solar System. He graduated from the University of Paris in 1879.

• Illustration — The young Henri Poincaré in 1887 at the age of 33 After receiving his degree, Poincaré began teaching as junior lecturer in mathematics at the University of Caen in Normandy (in December 1879). At the same time he published his first major article concerning the treatment of a class of automorphic functions. There, in Caen, he met his future wife, Louise Poulain d'Andecy (1857–1934), granddaughter of Isidore Geoffroy Saint-Hilaire and great-granddaughter of Étienne Geoffroy Saint-Hilaire and on 20 April 1881, they married. Together they had four children: Jeanne (born 1887), Yvonne (born 1889), Henriette (born 1891), and Léon (born 1893). Poincaré immediately established himself among the greatest mathematicians of Europe, attracting the attention of many prominent mathematicians. In 1881 Poincaré was invited to take a teaching position at the Faculty of Sciences of the University of Paris; he accepted the invitation. During the years 1883 to 1897, he taught mathematical analysis in the École Polytechnique. In 1881–1882, Poincaré created a new branch of mathematics: qualitative theory of differential equations. He showed how it is possible to derive the most important information about the behavior of a family of solutions without having to solve the equation (since this may not always be possible). He successfully used this approach to problems in celestial mechanics and mathematical physics.

He never fully abandoned his career in the mining administration for mathematics. He worked at the Ministry of Public Services as an engineer in charge of northern railway development from 1881 to 1885. He eventually became chief engineer of the Corps des Mines in 1893 and inspector general in 1910. Beginning in 1881 and for the rest of his career, he taught at the University of Paris (the Sorbonne). He was initially appointed as the maître de conférences d'analyse (associate professor of analysis). Eventually, he held the chairs of Physical and Experimental Mechanics, Mathematical Physics and Theory of Probability, and Celestial Mechanics and Astronomy. In 1887, at the young age of 32, Poincaré was elected to the French Academy of Sciences he became its president in 1906, to honorary membership of the Manchester Literary and Philosophical Society, in 1892, and was elected to the Académie française on 5 March 1908. In 1887, he won Oscar II, King of Sweden's mathematical competition for a resolution of the three-body problem concerning the free motion of multiple orbiting bodies. (See three-body problem section below.) In 1893, Poincaré joined the French Bureau des Longitudes, which engaged him in the synchronisation of time around the world. In 1897 Poincaré backed an unsuccessful proposal for the decimalisation of circular measure, and hence time and longitude. It was this post which led him to consider the question of establishing international time zones and the synchronisation of time between bodies in relative motion. (See work on relativity section below.) In 1904, he intervened in the trials of Alfred Dreyfus, attacking the spurious scientific claims regarding evidence brought against Dreyfus. Poincaré was the President of the Société Astronomique de France (SAF), the French astronomical society, from 1901 to 1903.

• Illustration — The Poincaré family grave at the Cimetière du Montparnasse In 1912, Poincaré underwent surgery for a prostate problem and subsequently died from an embolism on 17 July 1912, in Paris. He was 58 years of age. He is buried in the Poincaré family vault in the Cemetery of Montparnasse, Paris, in section 16, close to the Rue Émile-Richard. A former French Minister of Education, Claude Allègre, proposed in 2004 that Poincaré be reburied in the Panthéon in Paris, which is reserved for French citizens of the highest honour.

Poincaré made many contributions to different fields of pure and applied mathematics such as: celestial mechanics, fluid mechanics, optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, Quantum mechanics, theory of relativity and physical cosmology. Among the specific topics he contributed to are the following: • algebraic topology (a field that Poincaré virtually invented) • the theory of analytic functions of several complex variables • the theory of abelian functions • algebraic geometry • the Poincaré conjecture, proven in 2003 by Grigori Perelman. • Poincaré recurrence theorem • hyperbolic geometry • Fuchsian groups • number theory • the three-body problem • the theory of diophantine equations • electromagnetism • special relativity • the fundamental group • In the field of differential equations Poincaré has given many results that are critical for the qualitative theory of differential equations, for example the Poincaré sphere and the Poincaré map. • Poincaré on "everybody's belief" in the Normal Law of Errors (see normal distribution for an account of that "law") • Published an influential paper providing a novel mathematical argument in support of quantum mechanics.

The problem of finding the general solution to the motion of more than two orbiting bodies in the Solar System had eluded mathematicians since Newton's time. This was known originally as the three-body problem and later the n-body problem, where n is any number of more than two orbiting bodies. The n-body solution was considered very important and challenging at the close of the 19th century. Indeed, in 1887, in honour of his 60th birthday, Oscar II, King of Sweden, advised by Gösta Mittag-Leffler, established a prize for anyone who could find the solution to the problem. The announcement was quite specific: Given a system of arbitrarily many mass points that attract each according to Newton's law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series converges uniformly. In case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was finally awarded to Poincaré, even though he did not solve the original problem. One of the judges, the distinguished Karl Weierstrass, said, "This work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics." (The first version of his contribution even contained a serious error; for details see the article by Diacu and the book by Barrow-Green). The version finally printed contained many important ideas which led to the theory of chaos. The problem as stated originally was finally solved by Karl F. Sundman for n = 3 in 1912 and was generalised to the case of n > 3 bodies by Qiudong Wang in the 1990s. The series solutions have very slow convergence. It would take millions of terms to determine the motion of the particles for even very short intervals of time, so they are unusable in numerical work.

• Illustration — Marie Curie and Poincaré talk at the 1911 Solvay Conference.

Poincaré's work at the Bureau des Longitudes on establishing international time zones led him to consider how clocks at rest on the Earth, which would be moving at different speeds relative to absolute space (or the "luminiferous aether"), could be synchronised. At the same time Dutch theorist Hendrik Lorentz was developing Maxwell's theory into a theory of the motion of charged particles ("electrons" or "ions"), and their interaction with radiation. In 1895 Lorentz had introduced an auxiliary quantity (without physical interpretation) called "local time" t^\prime = t-v x/c^2 \, and introduced the hypothesis of length contraction to explain the failure of optical and electrical experiments to detect motion relative to the aether (see Michelson–Morley experiment). Poincaré was a constant interpreter (and sometimes friendly critic) of Lorentz's theory. Poincaré as a philosopher was interested in the "deeper meaning". Thus he interpreted Lorentz's theory and in so doing he came up with many insights that are now associated with special relativity. In The Measure of Time (1898), Poincaré said, "A little reflection is sufficient to understand that all these affirmations have by themselves no meaning. They can have one only as the result of a convention." He also argued that scientists have to set the constancy of the speed of light as a postulate to give physical theories the simplest form. Based on these assumptions he discussed in 1900 Lorentz's "wonderful invention" of local time and remarked that it arose when moving clocks are synchronised by exchanging light signals assumed to travel with the same speed in both directions in a moving frame.

In 1881 Poincaré described hyperbolic geometry in terms of the hyperboloid model, formulating transformations leaving invariant the Lorentz interval x^2+y^2-z^2=-1, which makes them mathematically equivalent to the Lorentz transformations in 2+1 dimensions. In addition, Poincaré's other models of hyperbolic geometry (Poincaré disk model, Poincaré half-plane model) as well as the Beltrami–Klein model can be related to the relativistic velocity space (see Gyrovector space). In 1892 Poincaré developed a mathematical theory of light including polarization. His vision of the action of polarizers and retarders, acting on a sphere representing polarized states, is called the Poincaré sphere. It was shown that the Poincaré sphere possesses an underlying Lorentzian symmetry, by which it can be used as a geometrical representation of Lorentz transformations and velocity additions. He discussed the "principle of relative motion" in two papers in 1900 and named it the principle of relativity in 1904, according to which no physical experiment can discriminate between a state of uniform motion and a state of rest. In 1905 Poincaré wrote to Lorentz about Lorentz's paper of 1904, which Poincaré described as a "paper of supreme importance". In this letter he pointed out an error Lorentz had made when he had applied his transformation to one of Maxwell's equations, that for charge-occupied space, and also questioned the time dilation factor given by Lorentz. In a second letter to Lorentz, Poincaré gave his own reason why Lorentz's time dilation factor was indeed correct after all—it was necessary to make the Lorentz transformation form a group—and he gave what is now known as the relativistic velocity-addition law. Poincaré later delivered a paper at the meeting of the Academy of Sciences in Paris on 5 June 1905 in which these issues were addressed. In the published version of that he wrote: The essential point, established by Lorentz, is that the equations of the electromagnetic field are not altered by a certain transformation (which I will call by the name of Lorentz) of the form: • x^\prime = k\ell\left(x + \varepsilon t\right)\!,\;t^\prime = k\ell\left(t + \varepsilon x\right)\!,\;y^\prime = \ell y,\;z^\prime = \ell z,\;k = 1/\sqrt{1-\varepsilon^2}. and showed that the arbitrary function \ell\left(\varepsilon\right) must be unity for all \varepsilon (Lorentz had set \ell = 1 by a different argument) to make the transformations form a group. In an enlarged version of the paper that appeared in 1906 Poincaré pointed out that the combination x^2+ y^2+ z^2- c^2t^2 is invariant. He noted that a Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing ct\sqrt{-1} as a fourth imaginary coordinate, and he used an early form of four-vectors. Poincaré expressed a lack of interest in a four-dimensional reformulation of his new mechanics in 1907, because in his opinion the translation of physics into the language of four-dimensional geometry would entail too much effort for limited profit. So it was Hermann Minkowski who worked out the consequences of this notion in 1907.

Like others before, Poincaré (1900) discovered a relation between mass and electromagnetic energy. While studying the conflict between the action/reaction principle and Lorentz ether theory, he tried to determine whether the center of gravity still moves with a uniform velocity when electromagnetic fields are included. He noticed that the action/reaction principle does not hold for matter alone, but that the electromagnetic field has its own momentum. Poincaré concluded that the electromagnetic field energy of an electromagnetic wave behaves like a fictitious fluid (fluide fictif) with a mass density of E/c 2. If the center of mass frame is defined by both the mass of matter and the mass of the fictitious fluid, and if the fictitious fluid is indestructible—it's neither created or destroyed—then the motion of the center of mass frame remains uniform. But electromagnetic energy can be converted into other forms of energy. So Poincaré assumed that there exists a non-electric energy fluid at each point of space, into which electromagnetic energy can be transformed and which also carries a mass proportional to the energy. In this way, the motion of the center of mass remains uniform. Poincaré said that one should not be too surprised by these assumptions, since they are only mathematical fictions. However, Poincaré's resolution led to a paradox when changing frames: if a Hertzian oscillator radiates in a certain direction, it will suffer a recoil from the inertia of the fictitious fluid. Poincaré performed a Lorentz boost (to order v/c) to the frame of the moving source. He noted that energy conservation holds in both frames, but that the law of conservation of momentum is violated. This would allow perpetual motion, a notion which he abhorred. The laws of nature would have to be different in the frames of reference, and the relativity principle would not hold. Therefore, he argued that also in this case there has to be another compensating mechanism in the ether. Poincaré himself came back to this topic in his St. Louis lecture (1904). He rejected the possibility that energy carries mass and criticized his own solution to compensate the above-mentioned problems: In the above quote he refers to the Hertz assumption of total aether entrainment that was falsified by the Fizeau experiment but that experiment does indeed show that light is partially "carried along" with a substance. Finally in 1908 he revisits the problem and ends with abandoning the principle of reaction altogether in favor of supporting a solution based in the inertia of aether itself. But we have seen above that Fizeau's experiment does not permit of our retaining the theory of Hertz; it is necessary therefore to adopt the theory of Lorentz, and consequently to renounce the principle of reaction. He also discussed two other unexplained effects: (1) non-conservation of mass implied by Lorentz's variable mass \gamma m, Abraham's theory of variable mass and Kaufmann's experiments on the mass of fast moving electrons and (2) the non-conservation of energy in the radium experiments of Marie Curie. It was Albert Einstein's concept of mass–energy equivalence (1905) that a body losing energy as radiation or heat was losing mass of amount m = E/c 2 that resolved Poincaré's paradox, without using any compensating mechanism within the ether. The Hertzian oscillator loses mass in the emission process, and momentum is conserved in any frame. However, concerning Poincaré's solution of the Center of Gravity problem, Einstein noted that Poincaré's formulation and his own from 1906 were mathematically equivalent.

In 1905 Poincaré first proposed gravitational waves (ondes gravifiques) emanating from a body and propagating at the speed of light. He wrote: It has become important to examine this hypothesis more closely and in particular to ask in what ways it would require us to modify the laws of gravitation. That is what I have tried to determine; at first I was led to assume that the propagation of gravitation is not instantaneous, but happens with the speed of light.

Einstein's first paper on relativity was published three months after Poincaré's short paper, but before Poincaré's longer version. Einstein relied on the principle of relativity to derive the Lorentz transformations and used a similar clock synchronisation procedure (Einstein synchronisation) to the one that Poincaré (1900) had described, but Einstein's paper was remarkable in that it contained no references at all. Poincaré never acknowledged Einstein's work on special relativity. However, Einstein expressed sympathy with Poincaré's outlook obliquely in a letter to Hans Vaihinger on 3 May 1919, when Einstein considered Vaihinger's general outlook to be close to his own and Poincaré's to be close to Vaihinger's. In public, Einstein acknowledged Poincaré posthumously in the text of a lecture in 1921 titled "Geometrie und Erfahrung (Geometry and Experience)" in connection with non-Euclidean geometry, but not in connection with special relativity. A few years before his death, Einstein commented on Poincaré as being one of the pioneers of relativity, saying "Lorentz had already recognized that the transformation named after him is essential for the analysis of Maxwell's equations, and Poincaré deepened this insight still further....".

Poincaré's work in the development of special relativity is well recognised, though most historians stress that despite many similarities with Einstein's work, the two had very different research agendas and interpretations of the work. Poincaré developed a similar physical interpretation of local time and noticed the connection to signal velocity, but contrary to Einstein he continued to use the ether-concept in his papers and argued that clocks at rest in the ether show the "true" time, and moving clocks show the local time. So Poincaré tried to keep the relativity principle in accordance with classical concepts, while Einstein developed a mathematically equivalent kinematics based on the new physical concepts of the relativity of space and time. While this is the view of most historians, a minority go much further, such as E. T. Whittaker, who held that Poincaré and Lorentz were the true discoverers of relativity.

• Illustration — Topological transformation of a mug into a torus The subject is clearly defined by Felix Klein in his "Erlangen Program" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced, as suggested by Johann Benedict Listing, instead of previously used "Analysis situs". Some important concepts were introduced by Enrico Betti and Bernhard Riemann. But the foundation of this science, for a space of any dimension, was created by Poincaré. His first article on this topic appeared in 1894. His research in geometry led to the abstract topological definition of homotopy and homology. He also first introduced the basic concepts and invariants of combinatorial topology, such as Betti numbers and the fundamental group. Poincaré proved a formula relating the number of edges, vertices and faces of n-dimensional polyhedron (the Euler–Poincaré theorem) and gave the first precise formulation of the intuitive notion of dimension.

• Illustration — Title page to volume I of Les Méthodes Nouvelles de la Mécanique Céleste (1892) • Illustration — Chaotic motion in three-body problem (computer simulation) Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). They introduced the small parameter method, fixed points, integral invariants, variational equations, the convergence of the asymptotic expansions. Generalizing a theory of Bruns (1887), Poincaré showed that the three-body problem is not integrable. In other words, the general solution of the three-body problem can not be expressed in terms of algebraic and transcendental functions through unambiguous coordinates and velocities of the bodies. His work in this area was the first major achievement in celestial mechanics since Isaac Newton. These monographs include an idea of Poincaré, which later became the basis for mathematical "chaos theory" (see, in particular, the Poincaré recurrence theorem) and the general theory of dynamical systems. Poincaré authored important works on astronomy for the equilibrium figures of a gravitating rotating fluid. He introduced the important concept of bifurcation points and proved the existence of equilibrium figures such as the non-ellipsoids, including ring-shaped and pear-shaped figures, and their stability. For this discovery, Poincaré received the Gold Medal of the Royal Astronomical Society (1900).

After defending his doctoral thesis on the study of singular points of the system of differential equations, Poincaré wrote a series of memoirs under the title "On curves defined by differential equations" (1881–1882). In these articles, he built a new branch of mathematics, called "qualitative theory of differential equations". Poincaré showed that even if the differential equation can not be solved in terms of known functions, yet from the very form of the equation, a wealth of information about the properties and behavior of the solutions can be found. In particular, Poincaré investigated the nature of the trajectories of the integral curves in the plane, gave a classification of singular points (saddle, focus, center, node), introduced the concept of a limit cycle and the loop index, and showed that the number of limit cycles is always finite, except for some special cases. Poincaré also developed a general theory of integral invariants and solutions of the variational equations. For the finite-difference equations, he created a new direction – the asymptotic analysis of the solutions. He applied all these achievements to study practical problems of mathematical physics and celestial mechanics, and the methods used were the basis of its topological works. • Illustration — Saddle • Illustration — Focus • Illustration — Center • Illustration — Node

• Illustration — Photographic portrait of H. Poincaré by Henri Manuel Poincaré's work habits have been compared to a bee flying from flower to flower. Poincaré was interested in the way his mind worked; he studied his habits and gave a talk about his observations in 1908 at the Institute of General Psychology in Paris. He linked his way of thinking to how he made several discoveries. The mathematician Darboux claimed he was un intuitif (an intuitive), arguing that this is demonstrated by the fact that he worked so often by visual representation. Jacques Hadamard wrote that Poincaré's research demonstrated marvelous clarity and Poincaré himself wrote that he believed that logic was not a way to invent but a way to structure ideas and that logic limits ideas.

Poincaré's mental organisation was interesting not only to Poincaré himself but also to Édouard Toulouse, a psychologist of the Psychology Laboratory of the School of Higher Studies in Paris. Toulouse wrote a book entitled Henri Poincaré (1910). In it, he discussed Poincaré's regular schedule: • He worked during the same times each day in short periods of time. He undertook mathematical research for four hours a day, between 10 a.m. and noon then again from 5 p.m. to 7 p.m.. He would read articles in journals later in the evening. • His normal work habit was to solve a problem completely in his head, then commit the completed problem to paper. • He was ambidextrous and nearsighted. • His ability to visualise what he heard proved particularly useful when he attended lectures, since his eyesight was so poor that he could not see properly what the lecturer wrote on the blackboard. These abilities were offset to some extent by his shortcomings: • He was physically clumsy and artistically inept. • He was always in a rush and disliked going back for changes or corrections. • He never spent a long time on a problem since he believed that the subconscious would continue working on the problem while he consciously worked on another problem. In addition, Toulouse stated that most mathematicians worked from principles already established while Poincaré started from basic principles each time. His method of thinking is well summarised as:

Poincaré's famous lectures before the Société de Psychologie in Paris (published as Science and Hypothesis, The Value of Science, and Science and Method) were cited by Jacques Hadamard as the source for the idea that creativity and invention consist of two mental stages, first random combinations of possible solutions to a problem, followed by a critical evaluation. Although he most often spoke of a deterministic universe, Poincaré said that the subconscious generation of new possibilities involves chance. It is certain that the combinations which present themselves to the mind in a kind of sudden illumination after a somewhat prolonged period of unconscious work are generally useful and fruitful combinations... all the combinations are formed as a result of the automatic action of the subliminal ego, but those only which are interesting find their way into the field of consciousness... A few only are harmonious, and consequently at once useful and beautiful, and they will be capable of affecting the geometrician's special sensibility I have been speaking of; which, once aroused, will direct our attention upon them, and will thus give them the opportunity of becoming conscious... In the subliminal ego, on the contrary, there reigns what I would call liberty, if one could give this name to the mere absence of discipline and to disorder born of chance. Poincaré's two stages—random combinations followed by selection—became the basis for Daniel Dennett's two-stage model of free will.

• Poincaré–Andronov–Hopf bifurcation • Poincaré complex – an abstraction of the singular chain complex of a closed, orientable manifold • Poincaré duality • Poincaré disk model • Poincaré expansion • Poincaré gauge • Poincaré group • Poincaré half-plane model • Poincaré homology sphere • Poincaré inequality • Poincaré lemma • Poincaré map • Poincaré residue • Poincaré series (modular form) • Poincaré space • Poincaré metric • Poincaré plot • Poincaré polynomial • Poincaré series • Poincaré sphere • Poincaré–Einstein synchronisation • Poincaré–Lelong equation • Poincaré–Lindstedt method • Poincaré–Lindstedt perturbation theory • Poincaré–Steklov operator • Euler–Poincaré characteristic • Neumann–Poincaré operator • Reflecting Function

Here is a list of theorems proved by Poincaré: • Poincaré's recurrence theorem: certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state. • Poincaré–Bendixson theorem: a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere. • Poincaré–Hopf theorem: a generalization of the hairy-ball theorem, which states that there is no smooth vector field on a sphere having no sources or sinks. • Poincaré–Lefschetz duality theorem: a version of Poincaré duality in geometric topology, applying to a manifold with boundary • Poincaré separation theorem: gives the upper and lower bounds of eigenvalues of a real symmetric matrix B'AB that can be considered as the orthogonal projection of a larger real symmetric matrix A onto a linear subspace spanned by the columns of B. • Poincaré–Birkhoff theorem: every area-preserving, orientation-preserving homeomorphism of an annulus that rotates the two boundaries in opposite directions has at least two fixed points. • Poincaré–Birkhoff–Witt theorem: an explicit description of the universal enveloping algebra of a Lie algebra. • Poincaré–Bjerknes circulation theorem: theorem about a conservation of quantity for the rotating frame. • Poincaré conjecture (now a theorem): Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. • Poincaré–Miranda theorem: a generalization of the intermediate value theorem to n dimensions.

• Brouwer fixed-point theorem • Epistemic structural realism • French epistemology • History of special relativity • Institut Henri Poincaré, Paris • List of things named after Henri Poincaré • Relativity priority dispute

Henri Poincaré was born on 29 April 1854 in Nancy. Poincaré held the nationality of France. Married to: Jeanne-Louise Poulain d'Andecy. Recorded father: Émile-Léon Poincaré. Recorded child: Jeanne Poincaré and Léon Poincaré. Recorded sibling: Aline Boutroux.

Poincaré worked as mathematician, philosopher, astronomer and physicist, and the authority record catalogues a further 6 occupations under the same name, among them engineer, philosopher of science, topologist, university teacher, writer and polymath. Poincaré worked in French.

Poincaré died on 17 July 1912 in Paris. Poincaré is buried at Montparnasse Cemetery.

The record carries the affiliation itself; it makes no claim about Poincaré's standing within the field beyond what those bodies state.

Poincaré introduced group theory to physics, and was the first to study the group of Lorentz transformations. He also made major contributions to the theory of discrete groups and their representations.

• Illustration — First page of Science and hypothesis (1905) Poincaré had philosophical views opposite to those of Bertrand Russell and Gottlob Frege, who believed that mathematics was a branch of logic. Poincaré strongly disagreed, claiming that intuition was the life of mathematics. Poincaré gives an interesting point of view in his 1902 book Science and Hypothesis: Poincaré believed that arithmetic is synthetic. He argued that Peano's axioms cannot be proven non-circularly with the principle of induction (Murzi, 1998), therefore concluding that arithmetic is a priori synthetic and not analytic. Poincaré then went on to say that mathematics cannot be deduced from logic since it is not analytic. His views were similar to those of Immanuel Kant (Kolak, 2001, Folina 1992). He strongly opposed Cantorian set theory, objecting to its use of impredicative definitions. Moreover, in La logique de l'infini, Poincaré extended this criticism to Ernst Zermelo's axiomatization of set theory, arguing that Zermelo had not demonstrated that the axioms in his system were free from contradiction, and still required a prior intuition of Menge (set). However, Poincaré did not share Kantian views in all branches of philosophy and mathematics. For example, in geometry, Poincaré believed that the structure of non-Euclidean space can be known analytically. Poincaré held that convention plays an important role in physics. His view (and some later, more extreme versions of it) came to be known as "conventionalism". Poincaré believed that Newton's first law was not empirical but is a conventional framework assumption for mechanics (Gargani, 2012). He also believed that the geometry of physical space is conventional. He considered examples in which either the geometry of the physical fields or gradients of temperature can be changed, either describing a space as non-Euclidean measured by rigid rulers, or as a Euclidean space where the rulers are expanded or shrunk by a variable heat distribution. However, Poincaré thought that we were so accustomed to Euclidean geometry that we would prefer to change the physical laws to save Euclidean geometry rather than shift to non-Euclidean physical geometry.

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Popular writings on the philosophy of science: •; reprinted in 1921; this book includes the English translations of Science and Hypothesis (1902), The Value of Science (1905), Science and Method (1908). • 1905., The Walter Scott Publishing Co. • 1906., Athenæum • 1913. "The New Mechanics", The Monist, Vol. XXIII. • 1913. "The Relativity of Space", The Monist, Vol. XXIII. • 1913. • 1956. Chance. In James R. Newman, ed., The World of Mathematics (4 Vols). • 1958. The Value of Science, New York: Dover. On algebraic topology: • 1895.. The first systematic study of topology. On celestial mechanics: • 1890. • 1892–99. New Methods of Celestial Mechanics, 3 vols. English trans., 1967.. • 1905. "The Capture Hypothesis of J. J. See", The Monist, Vol. XV. • 1905–10. Lessons of Celestial Mechanics. On the philosophy of mathematics: • Ewald, William B., ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford Univ. Press. Contains the following works by Poincaré: • 1894, "On the Nature of Mathematical Reasoning", 972–981. • 1898, "On the Foundations of Geometry", 982–1011. • 1900, "Intuition and Logic in Mathematics", 1012–1020. • 1905–06, "Mathematics and Logic, I–III", 1021–1070. • 1910, "On Transfinite Numbers", 1071–1074. • 1905. "The Principles of Mathematical Physics", The Monist, Vol. XV. • 1910. "The Future of Mathematics", The Monist, Vol. XX. • 1910. "Mathematical Creation", The Monist, Vol. XX. Other: • 1904. Maxwell's Theory and Wireless Telegraphy, New York, McGraw Publishing Company. • 1905. "The New Logics", The Monist, Vol. XV. • 1905. "The Latest Efforts of the Logisticians", The Monist, Vol. XV. Exhaustive bibliography of English translations: • 1892–2017..

Poincaré had two notable doctoral students at the University of Paris, Louis Bachelier (1900) and Dimitrie Pompeiu (1905).

The work is documented at University of Paris, University of Caen Normandy, French Academy of Sciences, Bavarian Academy of Sciences and Humanities, Göttingen Academy of Sciences and Humanities in Lower Saxony, Saint Petersburg Academy of Sciences, Royal Swedish Academy of Sciences and Hungarian Academy of Sciences.

Awards • Oscar II, King of Sweden's mathematical competition (1887) • Foreign member of the Royal Netherlands Academy of Arts and Sciences (1897) • American Philosophical Society (1899) • Gold Medal of the Royal Astronomical Society of London (1900) • Commander of the Legion of Honour (1903) • Bolyai Prize (1905) • Matteucci Medal (1905) • French Academy of Sciences (1906) • Académie française (1909) • Bruce Medal (1911) Named after him • Institut Henri Poincaré (mathematics and theoretical physics centre) • Maison Poincaré, a mathematics museum in the 5th arrondissement of Paris • Poincaré Prize (Mathematical Physics International Prize) • Annales Henri Poincaré (Scientific Journal) • Poincaré Seminar (nicknamed "Bourbaphy") • The crater Poincaré on the Moon • Asteroid 2021 Poincaré • List of things named after Henri Poincaré Henri Poincaré did not receive the Nobel Prize in Physics, but he had influential advocates like Henri Becquerel or committee member Gösta Mittag-Leffler. The nomination archive reveals that Poincaré received a total of 51 nominations between 1904 and 1912, the year of his death. Of the 58 nominations for the 1910 Nobel Prize, 34 named Poincaré. Nominators included Nobel laureates Hendrik Lorentz and Pieter Zeeman (both of 1902), Marie Curie (of 1903), Albert Michelson (of 1907), Gabriel Lippmann (of 1908) and Guglielmo Marconi (of 1909). The fact that renowned theoretical physicists like Poincaré, Boltzmann or Gibbs were not awarded the Nobel Prize is seen as evidence that the Nobel committee had more regard for experimentation than theory. In Poincaré's case, several of those who nominated him pointed out that the greatest problem was to name a specific discovery, invention, or technique.

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