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{{Infobox scientist
|name              = Henri Poincaré
|other_names       = Jules Henri Poincaré 
|image             = Henri Poincaré-2.jpg
|image_size        = 230px
|caption           = <small>Jules Henri Poincaré (1854–1912).</small>
|birth_date        = {{birth date|df=yes|1854|4|29}}
|birth_place       = [[Nancy, France|Nancy]], Meurthe-et-Moselle, France
|death_date        = {{death date and age|df=yes|1912|7|17|1854|4|29}}
|death_place       = Paris, France
|residence         = France
|nationality       = French
|fields            = Mathematics and [[physics]]
|workplaces        = [[Corps des Mines]]<br>[[Caen University]]<br>[[University of Paris|La Sorbonne]]<br>[[Bureau des Longitudes]]
|education         = Lycée Nancy<br>(now {{Interlanguage link multi|Lycée Poincaré|fr}})
|alma_mater        = [[École Polytechnique]]<br>[[École des Mines]]<br>[[University of Paris]] ([[Doctorat|Dr]], 1879)
|thesis_title = Sur les propriétés des fonctions définies par les équations différences
|thesis_url   = http://iris.univ-lille1.fr/handle/1908/458
|thesis_year  = 1879
|doctoral_advisor  = [[Charles Hermite]]
|academic_advisors =
|doctoral_students = [[Louis Bachelier]]<br>[[Jean Bosler]]<br>[[Dimitrie Pompeiu]]<br>[[Mihailo Petrović]]
|notable_students  = [[Tobias Dantzig]]<br>[[Théophile de Donder]]
|known_for         = [[Poincaré conjecture]]<br>[[Three-body problem]]<br>[[Topology]]<br>[[Special relativity]]<br>[[Poincaré–Hopf theorem]]<br>[[Poincaré duality]]<br>{{nowrap|[[Poincaré–Birkhoff–Witt theorem]]}}<br>[[Poincaré inequality]]<br> [[Hilbert–Poincaré series]]<br> [[Poincaré metric]]<br> [[Rotation number]]<br> [[Betti number|Coining term 'Betti number']] <br>[[Bifurcation theory]]<br>[[Chaos theory]]<br>[[Brouwer fixed-point theorem]]<br>[[Sphere-world]]<br>[[Poincaré–Bendixson theorem]]<br>[[Poincaré–Lindstedt method]]<br>[[Poincaré recurrence theorem]]
|influences        = [[Lazarus Fuchs]]<br>[[Immanuel Kant]]<ref>[http://www.iep.utm.edu/poi-math/#H3 "Poincaré's Philosophy of Mathematics"]: entry in the [[Internet Encyclopedia of Philosophy]].</ref>
|influenced        = [[Louis Rougier]]<br>[[George David Birkhoff]]<br>[[Cubism]]
|awards            = {{nowrap|[[Gold Medal of the Royal Astronomical Society|RAS Gold Medal]] (1900)}} <br>[[Sylvester Medal]] (1901)<br>[[Matteucci Medal]] (1905)<br>[[Bolyai Prize]] (1905)<br>[[Bruce Medal]] (1911)
|signature         = Henri Poincaré Signature.svg
|footnotes         = He was an uncle of [[Pierre Boutroux]].
}}
'''Jules Henri Poincaré ''' ({{IPA-fr|ʒyl ɑ̃ʁi pwɛ̃kaʁe|lang}};<ref name="forvo">{{cite web|url=http://www.forvo.com/word/poincar%C3%A9/ |title=Poincaré pronunciation: How to pronounce Poincaré in French |publisher=forvo.com|accessdate=16 August 2015}}</ref><ref name="pronouncekiwi">{{cite web|url=http://www.pronouncekiwi.com/Henri%20Poincaré |title=Poincaré pronunciation: How to pronounce Poincaré in French |publisher=pronouncekiwi.com|accessdate=25 December 2015}}</ref> 29 April 1854&nbsp;– 17 July 1912) was a French mathematician, [[theoretical physicist]], engineer, and [[philosophy of science|philosopher of science]]. He is often described as a [[polymath]], and in mathematics as ''The Last Universalist'' by [[Eric Temple Bell]],<ref>{{cite book | first1=J. M. | last1=Ginoux | first2=C. | last2=Gerini | title=Henri Poincaré: A Biography Through the Daily Papers  | publisher=World Scientific | date=2013 | isbn=978-981-4556-61-3 | url = http://www.worldscientific.com/worldscibooks/10.1142/8956| doi=10.1142/8956 }}</ref> since he excelled in all fields of the discipline as it existed during his lifetime.

As a mathematician and physicist, he made many original fundamental contributions to [[Pure mathematics|pure]] and [[applied mathematics]], [[mathematical physics]], and [[celestial mechanics]].<ref>{{cite journal|author=Hadamard, Jacques|authorlink=Jacques Hadamard|title=The early scientific work of Henri Poincaré|journal=The Rice Institute Pamphlet|date=July 1922|volume=9|issue=3|pages=111–183|url=http://catalog.hathitrust.org/Record/100592035}}</ref> He was responsible for formulating the [[Poincaré conjecture]], which was one of the most famous [[unsolved problems in mathematics]] until it was solved in 2002–2003 by [[Grigori Perelman]]. In his research on the [[three-body problem]], Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern [[chaos theory]]. He is also considered to be one of the founders of the field of [[topology]].

Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the [[Lorentz transformations]] in their modern symmetrical form. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Dutch physicist [[Hendrik Lorentz]] (1853–1928) in 1905. Thus he obtained perfect invariance of all of [[Maxwell's equations]], an important step in the formulation of the theory of [[special relativity]]. In 1905, Poincaré first proposed [[gravitational wave]]s (''ondes gravifiques'') emanating from a body and propagating at the speed of light as being required by the Lorentz transformations.

The [[Poincaré group]] used in physics and mathematics was named after him.

==Life==
Poincaré was born on 29 April 1854 in Cité Ducale neighborhood, [[Nancy, Meurthe-et-Moselle]] into an influential family.<ref>Belliver, 1956</ref> His father Leon Poincaré (1828–1892) was a professor of medicine at the [[University of Nancy]].<ref>Sagaret, 1911</ref> His adored younger sister Aline married the spiritual philosopher [[Emile Boutroux]]. Another notable member of Henri's family was his cousin, [[Raymond Poincaré]], who would serve as President of France from 1913 to 1920, and who was a fellow member of the [[Académie française]].<ref name="IEP">[http://www.utm.edu/research/iep/p/poincare.htm The Internet Encyclopedia of Philosophy] Jules Henri Poincaré article by Mauro Murzi&nbsp;– Retrieved November 2006.</ref> He was raised in the Roman Catholic faith, but later left the religion. He became a freethinker, believing in the search for truth and was said to be an atheist.<ref>Joseph McCabe (1945). A Biographical Dictionary of Ancient, Medieval, and Modern Freethinkers. Haldeman-Julius Publications. Retrieved 10 April 2012.</ref><ref>Poincaré, Henri (1 January 1913). Dernières Pensées. p. 138. Retrieved 10 April 2012.</ref><ref>{{cite arxiv |eprint=1207.0759|last1=Weinstein|first1=Galina|title=A Biography of Henri Poincaré - 2012 Centenary of the Death of Poincaré|class=physics.hist-ph|year=2012}}</ref>

===Education===
[[File:Henri Poincaré maison natale Nancy plaque.jpg|thumb|right|200px| 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, Meurthe-et-Moselle|Nancy]] (now renamed the Lycée Henri Poincaré in his honour, along with the University of 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".<ref>O'Connor et al., 2002</ref> However, poor eyesight and a tendency towards absentmindedness may explain these difficulties.<ref>Carl, 1968</ref> He graduated from the Lycée in 1871 with a bachelor's degree in letters and sciences.

During the [[Franco-Prussian War]] of 1870, he served alongside his father in the Ambulance Corps.

Poincaré entered the [[École Polytechnique]] 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.<ref>F. Verhulst</ref>

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-lès-Jussey|Magny]] in August 1879 in which 18 miners died. He carried out the official investigation into the accident in a characteristically thorough and humane way.

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]]. Poincaré graduated from the [[University of Paris]] in 1879.

[[Image:Young Poincare.jpg|left|upright|thumb|The young Henri Poincaré]]

===First scientific achievements===
After receiving his degree, Poincaré began teaching as junior lecturer in mathematics at the [[Caen University|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 function]]s.

There, in Caen, he met his future wife, Louise Poulin d'Andesi (Louise Poulain d'Andecy) 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 of 1883 to 1897, he taught mathematical analysis in [[École Polytechnique]].

In 1881–1882, Poincaré created a new branch of mathematics: the 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]].

===Career===
He never fully abandoned his mining career to 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 de 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 [[University of Paris|Sorbonne]]). He was initially appointed as the ''maître de conférences d'analyse'' (associate professor of analysis).<ref>Sageret, 1911</ref> 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, and was elected to the [[Académie française]] in 1909.

In 1887, he won [[Oscar II of Sweden|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)

[[File:Poincaré gravestone.jpg|upright|thumb|The Poincaré family grave at the [[Cimetière du Montparnasse]]]]
In 1893, Poincaré joined the French [[Bureau des Longitudes]], which engaged him in the [[Clock synchronization|synchronisation of time]] around the world. In 1897 Poincaré backed an unsuccessful proposal for the [[Decimal degrees|decimalisation of circular measure]], and hence time and [[longitude]].<ref>see Galison 2003</ref> 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 1899, and again more successfully in 1904, he intervened in the trials of [[Alfred Dreyfus]]. He attacked the spurious scientific claims of some of the evidence brought against Dreyfus, who was a Jewish officer in the French army charged with treason by colleagues.

Poincaré was the President of the [[Société astronomique de France|Société Astronomique de France (SAF)]], the French astronomical society, from 1901 to 1903.<ref name=BSAF1911>[http://gallica.bnf.fr/ark:/12148/bpt6k9626551q/f616.item ''Bulletin de la Société astronomique de France'', 1911, vol. 25, pp. 581–586]</ref>

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 [[Cimetière du Montparnasse|Cemetery of Montparnasse]], Paris.

A former French Minister of Education, [[Claude Allègre]], proposed in 2004 that Poincaré be reburied in the [[Panthéon, Paris|Panthéon]] in Paris, which is reserved for French citizens only of the highest honour.<ref>[https://web.archive.org/web/20041127160356/http://www.lexpress.fr/idees/tribunes/dossier/allegre/dossier.asp?ida=430274 Lorentz, Poincaré et Einstein]</ref>

====Students====
Poincaré had two notable doctoral students at the University of Paris, [[Louis Bachelier]] (1900) and [[Dimitrie Pompeiu]] (1905).<ref>[http://www.genealogy.ams.org/id.php?id=34227 Mathematics Genealogy Project] North Dakota State University. Retrieved April 2008.</ref>

==Work==

===Summary===
Poincaré made many contributions to different fields of pure and applied mathematics such as: [[celestial mechanics]], [[fluid mechanics]], [[optics]], electricity, [[telegraphy]], [[capillarity]], [[Elasticity (physics)|elasticity]], [[thermodynamics]], [[potential theory]], [[Quantum mechanics|quantum theory]], [[theory of relativity]] and [[physical cosmology]].

He was also a populariser of mathematics and physics and wrote several books for the lay public.

Among the specific topics he contributed to are the following:
*[[algebraic topology]]
*[[several complex variables|the theory of analytic functions of several complex variables]]
*[[abelian variety|the theory of abelian functions]]
*[[algebraic geometry]]
*Poincaré was responsible for formulating one of the most famous problems in mathematics, the [[Poincaré conjecture]], proven in 2003 by [[Grigori Perelman]].
*[[Poincaré recurrence theorem]]
*[[hyperbolic geometry]]
*[[number theory]]
*[[N-body problem|the three-body problem]]
*[[diophantine equation|the theory of diophantine equations]]
*[[electromagnetism|the theory of electromagnetism]]
*[[Special relativity|the special theory of relativity]]
*In an 1894 paper, he introduced the concept of 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 homology sphere|Poincaré sphere]] and the [[Poincaré map]].
*Poincaré on "everybody's belief" in the [[q:Henri Poincaré|''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]].<ref name=McCormmach>
{{Citation
  | last =McCormmach
  | first =Russell
  | title = Henri Poincaré and the Quantum Theory
  | journal = Isis
  | volume = 58
  | issue = 1
  | pages = 37–55
  | date = Spring 1967
  | doi =10.1086/350182
}}</ref><ref name=Irons>
{{Citation
  | last =Irons
  | first =F. E.
  | title = Poincaré's 1911–12 proof of quantum discontinuity interpreted as applying to atoms
  | journal =  American Journal of Physics
  | volume = 69
  | issue = 8
  | pages = 879–884
  | date = August 2001
  | doi =10.1119/1.1356056
|bibcode = 2001AmJPh..69..879I }}</ref>

===Three-body problem===
The problem of finding the general solution to the motion of more than two orbiting bodies in the solar system had eluded mathematicians since [[Isaac Newton|Newton's]] time. This was known originally as the three-body problem and later the [[n-body problem|''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 of Sweden|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:

<blockquote>Given a system of arbitrarily many mass points that attract each [[inverse-square law|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 [[uniform convergence|converges uniformly]].</blockquote>

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<ref name=diacu>{{Citation
| author=Diacu, F. | year=1996 | title=The solution of the ''n''-body Problem | journal=The Mathematical Intelligencer | volume =18  | pages =66–70
| doi=10.1007/BF03024313
| issue=3}}</ref>). The version finally printed contained many important ideas which led to the [[chaos theory|theory of chaos]]. The problem as stated originally was finally solved by [[Karl F. Sundman]] for ''n''&nbsp;=&nbsp;3 in 1912 and was generalised to the case of ''n''&nbsp;>&nbsp;3 bodies by [[Qiudong Wang]] in the 1990s.

===Work on relativity===
[[Image:Curie and Poincare 1911 Solvay.jpg|thumb|right|[[Marie Curie]] and Poincaré talk at the 1911 [[Solvay Conference]]]]
{{main|Lorentz ether theory|History of special relativity}}

====Local time====
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"  <math>t^\prime = t-v x/c^2 \,</math><ref>{{Citation|title=A broader view of relativity: general implications of Lorentz and Poincaré invariance|volume=10|first1=Jong-Ping|last1=Hsu|first2=Leonardo|last2=Hsu|publisher=World Scientific|year=2006|isbn=981-256-651-1|page=37
|url=https://books.google.com/books?id=amLqckyrvUwC}}, [https://books.google.com/books?id=amLqckyrvUwC&pg=PA37 Section A5a, p 37]</ref>
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]]).<ref>{{Citation
| author=Lorentz, H.A. | year=1895 | title=[[s:de:Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern|Versuch einer theorie der electrischen und optischen erscheinungen in bewegten Kõrpern]] | place =Leiden| publisher=E.J. Brill}}</ref>
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 [[s:The Measure of Time|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.<ref>{{Citation
| author=Poincaré, H. | year=1898 | title=[[s:The Measure of Time|The Measure of Time]] | journal=Revue de métaphysique et de morale | volume =6 | pages =1–13}}</ref>
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.<ref name=action>{{Citation
| author=Poincaré, H. | year=1900 | title=[[s:fr:La théorie de Lorentz et le principe de réaction|La théorie de Lorentz et le principe de réaction]] | journal=Archives néerlandaises des sciences exactes et naturelles | volume =5  | pages =252–278}}. See also the [http://www.physicsinsights.org/poincare-1900.pdf English translation]</ref>

====Principle of relativity and Lorentz transformations====
He discussed the "principle of relative motion" in two papers in 1900<ref name=action /><ref>{{Citation
| author=Poincaré, H. | year=1900 | title= Les relations entre la physique expérimentale et la physique mathématique | journal=Revue générale des sciences pures et appliquées | volume =11  | pages =1163–1175 | url=http://gallica.bnf.fr/ark:/12148/bpt6k17075r/f1167.table}}. Reprinted in "Science and Hypothesis", Ch. 9–10.</ref>
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.<ref name=louis>{{Citation|author=Poincaré, Henri|year=1904/6|chapter=[[s:The Principles of Mathematical Physics|The Principles of Mathematical Physics]]|title=The Foundations of Science (The Value of Science)|pages=297–320|publisher=Science Press|place=New York}}</ref>
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.<ref name="univ-lorraine">{{cite web|url=http://henripoincarepapers.univ-lorraine.fr/chp/text/lorentz3.xml |archive-url=https://web.archive.org/web/20141006190045/http://henripoincarepapers.univ-lorraine.fr/chp/text/lorentz3.xml |dead-url=yes |archive-date=6 October 2014 |title=Letter from Poincaré to Lorentz, Mai 1905 |publisher=henripoincarepapers.univ-lorraine.fr|accessdate=16 August 2015}}</ref>
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.<ref name="univ-lorraine2">{{cite web|url=http://henripoincarepapers.univ-lorraine.fr/chp/text/lorentz4.xml |archive-url=https://web.archive.org/web/20141006134323/http://henripoincarepapers.univ-lorraine.fr/chp/text/lorentz4.xml |dead-url=yes |archive-date=6 October 2014 |title=Letter from Poincaré to Lorentz, Mai 1905 |publisher=henripoincarepapers.univ-lorraine.fr|accessdate=16 August 2015}}</ref>
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:<ref name=short>{{Citation | author=Poincaré, H. | year=1905 | title=[[s:fr:Sur la dynamique de l'électron (juin)|Sur la dynamique de l'électron]] ([[s:Translation:On the Dynamics of the Electron (June)|On the Dynamics of the Electron]]) | journal=Comptes Rendus | volume =140 | pages =1504–1508}} (Wikisource translation)</ref>

<blockquote>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:
::<math>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}.</math></blockquote>

and showed that the arbitrary function <math>\ell\left(\varepsilon\right)</math> must be unity for all <math>\varepsilon</math> (Lorentz had set <math>\ell = 1</math> 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 <math>x^2+ y^2+ z^2- c^2t^2</math> is [[Invariant (mathematics)|invariant]]. He noted that a Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing <math>ct\sqrt{-1}</math> as a fourth imaginary coordinate, and he used an early form of [[four-vector]]s.<ref name=long>{{Citation
| author=Poincaré, H. | year=1906 | title=[[s:fr:Sur la dynamique de l'électron (juillet)|Sur la dynamique de l'électron]] ([[s:Translation:On the Dynamics of the Electron (July)|On the Dynamics of the Electron]]) | journal=Rendiconti del Circolo matematico Rendiconti del Circolo di Palermo  | volume =21  | pages =129–176
| doi=10.1007/BF03013466}} (Wikisource translation)</ref> 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.<ref>Walter (2007), Secondary sources on relativity</ref> So it was [[Hermann Minkowski]] who worked out the consequences of this notion in 1907.

====Mass–energy relation====
Like [[Mass–energy equivalence#Electromagnetic rest mass|others]] before, Poincaré (1900) discovered a relation between mass and electromagnetic energy. While studying the conflict between the [[Newton's laws of motion|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.<ref name=action /> 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''<sup>2</sup>. 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).<ref name=louis /> This time (and later also in 1908) he rejected<ref>Miller 1981, Secondary sources on relativity</ref> the possibility that energy carries mass and criticized the ether solution to compensate the above-mentioned problems:

{{quote|The apparatus will recoil as if it were a cannon and the projected energy a ball, and that contradicts the principle of Newton, since our present projectile has no mass; it is not matter, it is energy. [..] Shall we say that the space which separates the oscillator from the receiver and which the disturbance must traverse in passing from one to the other, is not empty, but is filled not only with ether, but with air, or even in inter-planetary space with some subtile, yet ponderable fluid; that this matter receives the shock, as does the receiver, at the moment the energy reaches it, and recoils, when the disturbance leaves it? That would save Newton's principle, but it is not true. If the energy during its propagation remained always attached to some material substratum, this matter would carry the light along with it and Fizeau has shown, at least for the air, that there is nothing of the kind. Michelson and Morley have since confirmed this. We might also suppose that the motions of matter proper were exactly compensated by those of the ether; but that would lead us to the same considerations as those made a moment ago. The principle, if thus interpreted, could explain anything, since whatever the visible motions we could imagine hypothetical motions to compensate them. But if it can explain anything, it will allow us to foretell nothing; it will not allow us to choose between the various possible hypotheses, since it explains everything in advance. It therefore becomes useless. }}

He also discussed two other unexplained effects: (1) non-conservation of mass implied by Lorentz's variable mass <math>\gamma m</math>, Abraham's theory of variable mass and [[Walter Kaufmann (physicist)|Kaufmann]]'s experiments on the mass of fast moving electrons and (2) the non-conservation of energy in the radium experiments of [[Madame 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''&nbsp;=&nbsp;''E''/''c''<sup>2</sup> that resolved<ref name=darrigol>Darrigol 2005, Secondary sources on relativity</ref> Poincaré's paradox, without using any compensating mechanism within the ether.<ref>{{Citation|author=Einstein, A. |year=1905b |title=Ist die Trägheit eines Körpers von dessen Energieinhalt abhängig? |journal=Annalen der Physik |volume=18 |issue=13 |pages=639–643 |bibcode=1905AnP...323..639E |doi=10.1002/andp.19053231314 |url=http://www.physik.uni-augsburg.de/annalen/history/papers/1905_18_639-641.pdf |archive-url=https://web.archive.org/web/20050124051500/http://www.physik.uni-augsburg.de/annalen/history/papers/1905_18_639-641.pdf |dead-url=yes |archive-date=24 January 2005 |df=dmy-all }}. See also [http://www.fourmilab.ch/etexts/einstein/specrel/www English translation].</ref> 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.<ref>{{Citation|author=Einstein, A. |year=1906 |title=Das Prinzip von der Erhaltung der Schwerpunktsbewegung und die Trägheit der Energie |journal=Annalen der Physik |volume=20 |pages=627–633 |doi=10.1002/andp.19063250814 |issue=8 |bibcode=1906AnP...325..627E |url=http://www.physik.uni-augsburg.de/annalen/history/papers/1906_20_627-633.pdf |archive-url=https://web.archive.org/web/20060318060830/http://www.physik.uni-augsburg.de/annalen/history/papers/1906_20_627-633.pdf |dead-url=yes |archive-date=18 March 2006 |df=dmy-all }}</ref>

====Poincaré and Einstein====
Einstein's first paper on relativity was published three months after Poincaré's short paper,<ref name=short /> but before Poincaré's longer version.<ref name=long /> 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.<ref>{{cite web|title=The Collected Papers of Albert Einstein |url=http://einsteinpapers.press.princeton.edu/vol9-trans/52 |publisher=Princeton U.P. |accessdate=13 December 2014}}  See also this letter, with commentary, in {{cite journal |last=Sass |first=Hans-Martin |date=1979 |title=Einstein über "wahre Kultur" und die Stellung der Geometrie im Wissenschaftssystem:  Ein Brief Albert Einsteins an Hans Vaihinger vom Jahre 1919 |journal=Zeitschrift für allgemeine Wissenschaftstheorie |volume=10 |issue=2 |pages=316–319 |jstor=25170513 |language=de |doi=10.1007/bf01802352}}</ref> In public, Einstein acknowledged Poincaré posthumously in the text of a lecture in 1921 called ''Geometrie und Erfahrung'' 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 recognised that the transformation named after him is essential for the analysis of Maxwell's equations, and Poincaré deepened this insight still further ...."<ref>Darrigol 2004, Secondary sources on relativity</ref>

In 1905 Henri Poincaré first proposed [[gravitational waves]] (''ondes gravifiques'') emanating from a body and propagating at the speed of light.<ref>[http://www.academie-sciences.fr/pdf/dossiers/Poincare/Poincare_pdf/Poincare_CR1905.pdf] (PDF) Membres de l'Académie des sciences depuis sa création : Henri Poincare. Sur la dynamique de l' electron.  Note de H. Poincaré. C.R. T.140 (1905) 1504–1508.</ref>   ''"Il importait d'examiner cette hypothèse de plus près et en particulier de rechercher quelles modifications elle nous obligerait à apporter aux lois de la gravitation. C'est ce que j'ai cherché à déterminer ; j'ai été d'abord conduit à supposer que la propagation de la gravitation n'est pas instantanée, mais se fait avec la vitesse de la lumière."''

===Algebra and number theory===
Poincaré introduced [[group theory]] to physics, and was the first to study the group of [[Lorentz transformations]].<ref>Poincaré, Selected works in three volumes. page = 682</ref> He also made major contributions to the theory of discrete groups and their representations.
[[Image:Mug and Torus morph.gif|right|frame |50px |<center>Topological transformation of the torus into a mug <center>]]

===Topology===
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.<ref>J. Stillwell, Mathematics and its history. pages = 419–435</ref>

His research in geometry led to the abstract topological definition of [[homotopy]] and [[Homology (mathematics)|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.<ref>PS Aleksandrov, Poincaré and topology. pages = 27–81</ref>

===Astronomy and celestial mechanics===
[[File:N-body problem (3).gif|frame|left|150px |  <center> 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]].<ref>J. Stillwell, Mathematics and its history, [https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA254 page 254]</ref>

These monographs include an idea of Poincaré, which later became the base for mathematical "[[chaos theory]]" (see, in particular, the [[Poincaré recurrence theorem]]) and the general theory of [[dynamical system]]s.
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).<ref>A. Kozenko, The theory of planetary figures, pages = 25–26</ref>

===Differential equations and mathematical physics===
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).<ref>French: "Mémoire sur les courbes définies par une équation différentielle"</ref> 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.<ref>Kolmogorov, AP Yushkevich, Mathematics of the 19th century Vol = 3. page = 283 {{isbn|978-3764358457}}</ref><ref>Kolmogorov, AP Yushkevich, Mathematics of the 19th century. pages = 162–174</ref>

<gallery caption="The singular points of the integral curves">
   File: Phase Portrait Sadle.svg | Saddle
   File: Phase Portrait Stable Focus.svg | Focus
   File: Phase portrait center.svg | Center
   File: Phase Portrait Stable Node.svg | Node
</gallery>

===Assessments===
{{Further|History of special relativity|Relativity priority dispute}}
Poincaré's work in the development of special relativity is well recognised,<ref name=darrigol /> though most historians stress that despite many similarities with Einstein's work, the two had very different research agendas and interpretations of the work.<ref>Galison 2003 and Kragh 1999, Secondary sources on relativity</ref> 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 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.<ref>Holton (1988), 196–206</ref><ref>Hentschel (1990), 3–13</ref><ref>Miller (1981), 216–217</ref><ref>Darrigol (2005), 15–18</ref><ref>Katzir (2005), 286–288</ref>

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.<ref>Whittaker 1953, Secondary sources on relativity</ref>

==Character==
[[File:Henri Poincaré by H Manuel.jpg|thumb|right|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'' (intuitive), arguing that this is demonstrated by the fact that he worked so often by visual representation. He did not care about being rigorous and disliked logic. {{citation needed|date=October 2013}} (Despite this opinion, [[Jacques Hadamard]] wrote that Poincaré's research demonstrated marvelous clarity.<ref>J. Hadamard. L'oeuvre de H. Poincaré. Acta Mathematica, 38 (1921), p. 208</ref> and Poincaré himself wrote that he  <!-- TODO: Add Poincaré's opinion on rigorousness, see http://www.forgottenbooks.org/readbook/American_Journal_of_Mathematics_1890_v12_1000084889#233 — Each time I can I'm absolute rigour --> believed that logic was not a way to invent but a way to structure ideas and that logic limits ideas.)

===Toulouse's characterisation===
Poincaré's mental organisation was not only interesting 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).<ref>[http://name.umdl.umich.edu/AAS9989.0001.001 Toulouse, Édouard, 1910. ''Henri Poincaré'', E. Flammarion, Paris]</ref><ref name="google">{{cite book|title=Henri Poincare|author=Toulouse, E.|date=2013|publisher=MPublishing|isbn=9781418165062|url=https://books.google.com/books?id=mpjWPQAACAAJ|accessdate=10 October 2014}}</ref> 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&nbsp;a.m. and noon then again from 5&nbsp;p.m. to 7&nbsp;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 (O'Connor et al., 2002).

His method of thinking is well summarised as:

{{quote|text=''Habitué à négliger les détails et à ne regarder que les cimes, il passait de l'une à l'autre avec une promptitude surprenante et les faits qu'il découvrait se groupant d'eux-mêmes autour de leur centre étaient instantanément et automatiquement classés dans sa mémoire.'' (Accustomed to neglecting details and to looking only at mountain tops, he went from one peak to another with surprising rapidity, and the facts he discovered, clustering around their center, were instantly and automatically pigeonholed in his memory.)|sign=Belliver (1956)}}

===Attitude towards transfinite numbers===
Poincaré was dismayed by [[Georg Cantor]]'s theory of [[transfinite number]]s, and referred to it as a "disease" from which mathematics would eventually be cured.<ref name="daub266">Dauben 1979, p. 266.</ref>
Poincaré said, "There is no actual infinite; the Cantorians have forgotten this, and that is why they have fallen into contradiction."<ref>{{citation
|title=From Frege to Gödel: a source book in mathematical logic, 1879–1931
|first1=Jean
|last1=Van Heijenoort
|publisher=Harvard University Press
|year=1967
|isbn=0-674-32449-8
|page=190
|url=https://books.google.com/?id=v4tBTBlU05sC&pg=PA190}}, [https://books.google.com/books?id=v4tBTBlU05sC&pg=PA190 p 190]
</ref>

==Honours==
'''Awards'''
*Oscar II, King of Sweden's mathematical competition (1887)
*Foreign member of the [[Royal Netherlands Academy of Arts and Sciences]] (1897)<ref>{{cite web|url=http://www.dwc.knaw.nl/biografie/pmknaw/?pagetype=authorDetail&aId=PE00002358 |title=Jules Henri Poincaré (1854–1912) |publisher=Royal Netherlands Academy of Arts and Sciences |date= |accessdate=4 August 2015}}</ref>
*[[American Philosophical Society]] 1899
*[[Gold Medal of the Royal Astronomical Society]] of London (1900)
*[[Bolyai Prize]] in 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 center)
*[[Poincaré Prize]] (Mathematical Physics International Prize)
*[[Annales Henri Poincaré]] (Scientific Journal)
*Poincaré Seminar (nicknamed "[[Bourbaphy]]")
*The crater [[Poincaré (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]].<ref name="gray-biography">{{cite book|last1=Gray|first1=Jeremy|title=Henri Poincaré: A Scientific Biography|date=2013|publisher=Princeton University Press|pages=194–196|chapter=The Campaign for Poincaré}}</ref><ref>{{cite book|last1=Crawford|first1=Elizabeth|title=The Beginnings of the Nobel Institution: The Science Prizes, 1901–1915|date=25 November 1987|publisher=Cambridge University Press|pages=141–142}}</ref> The nomination archive reveals that Poincaré received a total of 51 nominations between 1904 and 1912, the year of his death.<ref name="nomination database">{{cite web|title=Nomination database|url=https://www.nobelprize.org/nomination/archive/list.php|website=Nobelprize.org|publisher=Nobel Media AB|accessdate=24 September 2015}}</ref> Of the 58 nominations for the 1910 Nobel Prize, 34 named Poincaré.<ref name="nomination database"/> 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).<ref name="nomination database"/>

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.<ref>{{cite journal|last1=Crawford|first1=Elizabeth|title=Nobel: Always the Winners, Never the Losers|journal=[[Science (journal)|Science]]|date=13 November 1998|volume=282|issue=5392|pages=1256–1257|doi=10.1126/science.282.5392.1256|url=http://www.sciencemag.org/content/282/5392/1256.%20full|bibcode = 1998Sci...282.1256C }}{{dead link|date=July 2016}}</ref><ref>{{cite journal|last1=Nastasi|first1=Pietro|title=A Nobel Prize for Poincaré?|journal=Lettera Matematica|date=16 May 2013|volume=1|issue=1–2|pages=79–82|doi=10.1007/s40329-013-0005-1|url=http://download.springer.com/static/pdf/888/art%253A10.1007%252Fs40329-013-0005-1.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2Fs40329-013-0005-1&token2=exp=1443095857~acl=%2Fstatic%2Fpdf%2F888%2Fart%25253A10.1007%25252Fs40329-013-0005-1.pdf%3ForiginUrl%3Dhttp%253A%252F%252Flink.springer.com%252Farticle%252F10.1007%252Fs40329-013-0005-1*~hmac=e3b3232b522ae46f6a45a2cd546f3ebca45f455c206c3af2d2c7dd78495c4ce1|accessdate=24 September 2015}}</ref> 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.<ref name="gray-biography"/>

==Philosophy==
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 (knowledge)|intuition]] was the life of mathematics. Poincaré gives an interesting point of view in his book ''[[Science and Hypothesis]]'':

{{quote|text=For a superficial observer, scientific truth is beyond the possibility of doubt; the logic of science is infallible, and if the scientists are sometimes mistaken, this is only from their mistaking its rule.}}

Poincaré believed that [[arithmetic]] is a [[Analytic/synthetic distinction|synthetic]] science. 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 and a posteriori|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 [[Impredicativity|impredicative]] definitions.

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 geometry|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).<ref>{{Citation|author= Gargani Julien|title=Poincaré, le hasard et l'étude des systèmes complexes|publisher=L'Harmattan|year=2012|page=124
|url=http://www.editions-harmattan.fr/index.asp?navig=catalogue&obj=livre&no=38754}}</ref> 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 a non-Euclidean physical geometry.<ref>{{Citation|title=Science and Hypothesis|first1=Henri |last1=Poincaré |publisher=Cosimo, Inc. Press|year=2007|isbn=978-1-60206-505-5 |page=50
|url=https://books.google.com/books?id=2QXqHaVbkgoC&pg=PA50}}</ref>

===Free will===
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.<ref>Hadamard, Jacques. ''An Essay on the Psychology of Invention in the Mathematical Field''. Princeton Univ Press (1945)</ref>

Although he most often spoke of a deterministic universe, Poincaré said that the subconscious generation of new possibilities involves [[Randomness|chance]].
<blockquote>
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.<ref>''Science and Method'', Chapter 3, Mathematical Discovery, 1914, pp.58</ref>
</blockquote>

Poincaré's two stages—random combinations followed by selection—became the basis for [[Daniel Dennett]]'s two-stage model of free will.<ref>Dennett, Daniel C. 1978. Brainstorms: Philosophical Essays on Mind and Psychology. The MIT Press, p.293</ref>

==See also==

===Concepts===
* [[Poincaré complex]] – an abstraction of the singular chain complex of a closed, orientable manifold.
* [[Poincaré duality]]
* [[Poincaré disk model]]
* [[Poincaré group]]
* [[Poincaré inequality]]
* [[Poincaré map]]
* [[Poincaré residue]]
* [[Poincaré series (modular form)]]
* [[Poincaré space]]
* [[Poincaré metric]]
* [[Hilbert–Poincaré series|Poincaré series]]
* [[Poincaré sphere (optics)|Poincaré sphere]]
* [[Poincaré–Lindstedt method]]
* [[Poincaré plot]]
* [[Poincaré half-plane model]]
* [[Poincaré–Steklov operator]]
* [[Poincaré–Lindstedt perturbation theory]]
* [[Poincaré–Lelong equation]]

===Theorems===
* [[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 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.

=== Other ===
{{Columns-list|colwidth=30em|
*[[French epistemology]]
*[[History of special relativity]]
*[[List of things named after Henri Poincaré]]
*[[Institut Henri Poincaré]], Paris
*[[Brouwer fixed-point theorem]]
*[[Relativity priority dispute]]
*[[Epistemic structural realism]]<ref>[http://plato.stanford.edu/entries/structural-realism/#Rel "Structural Realism"]: entry by James Ladyman in the ''[[Stanford Encyclopedia of Philosophy]]''</ref>
}}

==References==
* {{PlanetMath attribution|id=3793|title=Jules Henri Poincaré}}

===Footnotes and primary sources===
{{Reflist|colwidth=30em}}

===Poincaré's writings in English translation===
{{Wikisource|Science and Hypothesis}}

Popular writings on the [[philosophy of science]]:
*{{Citation
|author=Poincaré, Henri
|year=1902–1908
|title=The Foundations of Science
|place=New York
|publisher=Science Press
|url=https://archive.org/details/foundationsscie01poingoog}}; reprinted in 1921; This book includes the English translations of Science and Hypothesis (1902), The Value of Science (1905), Science and Method (1908).
* 1904. ''Science and Hypothesis,'' The Walter Scott Publishing Co.
* 1913. "The New Mechanics," The Monist, Vol. XXIII.
* 1913. "The Relativity of Space," The Monist, Vol. XXIII.
* 1913. {{Citation | title=Last Essays. |place=New York |publisher=Dover reprint, 1963 | url=https://archive.org/details/mathematicsandsc001861mbp}}
* 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. {{Citation |title=Analysis Situs
| url=http://www.maths.ed.ac.uk/~aar/papers/poincare2009.pdf}}. The first systematic study of [[topology]].

On [[celestial mechanics]]:
* 1892–99. ''New Methods of Celestial Mechanics'', 3 vols. English trans., 1967. {{isbn|1-56396-117-2}}.
* 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," 972–81.
** 1898, "On the Foundations of Geometry," 982–1011.
** 1900, "Intuition and Logic in Mathematics," 1012–20.
** 1905–06, "Mathematics and Logic, I–III," 1021–70.
** 1910, "On Transfinite Numbers," 1071–74.
* 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.

===General references===
* [[Eric Temple Bell|Bell, Eric Temple]], 1986. ''Men of Mathematics'' (reissue edition). Touchstone Books. {{isbn|0-671-62818-6}}.
* Belliver, André, 1956. ''Henri Poincaré ou la vocation souveraine''. Paris: Gallimard.
*[[Peter L. Bernstein|Bernstein, Peter L]], 1996. "Against the Gods: A Remarkable Story of Risk". (p.&nbsp;199–200). John Wiley & Sons.
* [[Carl Benjamin Boyer|Boyer, B. Carl]], 1968. ''A History of Mathematics: Henri Poincaré'', John Wiley & Sons.
* [[Ivor Grattan-Guinness|Grattan-Guinness, Ivor]], 2000. ''The Search for Mathematical Roots 1870–1940.'' Princeton Uni. Press.
* {{Citation|last=Dauben|given= Joseph|authorlink=Joseph Dauben|origyear=1993|year=2004|chapter=Georg Cantor and the Battle for Transfinite Set Theory|chapter-url=http://www.acmsonline.org/journal/2004/Dauben-Cantor.pdf|title=Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, CA)|pages=1–22}}{{dead link|date=July 2016}}. Internet version published in Journal of the ACMS 2004.
* Folina, Janet, 1992. ''Poincaré and the Philosophy of Mathematics.''  Macmillan, New York.
* [[Jeremy Gray|Gray, Jeremy]], 1986. ''Linear differential equations and group theory from Riemann to Poincaré'', Birkhauser {{isbn|0-8176-3318-9}}
* Gray, Jeremy, 2013.  ''Henri Poincaré: A scientific biography''. Princeton University Press {{isbn|978-0-691-15271-4}} 
*{{Citation |url=http://www.ams.org/notices/200509/comm-mawhin.pdf
|format=PDF|title=Henri Poincaré. A Life in the Service of Science
|author=Jean Mawhin |journal=Notices of the AMS
|date=October 2005 |volume=52 |issue=9 |pages=1036–1044 }}
* Kolak, Daniel, 2001. ''Lovers of Wisdom'', 2nd ed. Wadsworth.
* Gargani, Julien, 2012. ''Poincaré, le hasard et l'étude des systèmes complexes'', L'Harmattan.
* Murzi, 1998. "Henri Poincaré".
* O'Connor, J. John, and Robertson, F. Edmund, 2002, "Jules Henri Poincaré". University of St. Andrews, Scotland.
* [[Ivars Peterson|Peterson, Ivars]], 1995. ''Newton's Clock: Chaos in the Solar System'' (reissue edition). W H Freeman & Co. {{isbn|0-7167-2724-2}}.
* Sageret, Jules, 1911. ''Henri Poincaré''. Paris: Mercure de France.
* Toulouse, E.,1910. ''Henri Poincaré''.—(Source biography in French) at University of Michigan Historic Math Collection.
* {{cite book |title=Mathematics and Its History |edition=3rd, illustrated |first1=John |last1=Stillwell |publisher=Springer Science & Business Media |year=2010 |isbn=978-1-4419-6052-8 |url=https://books.google.com/books?id=V7mxZqjs5yUC}}
* [[F. Verhulst|Verhulst, Ferdinand]], 2012 ''Henri Poincaré. Impatient Genius''. N.Y.: Springer.
* ''Henri Poincaré, l'œuvre scientifique, l'œuvre philosophique'', by Vito Volterra, Jacques Hadamard, Paul Langevin and Pierre Boutroux, Felix Alcan, 1914. 
** ''Henri Poincaré, l'œuvre mathématique'', by [[Vito Volterra]].
** ''Henri Poincaré, le problème des trois corps'', by [[Jacques Hadamard]].
** ''Henri Poincaré, le physicien'', by [[Paul Langevin]].
** ''Henri Poincaré, l'œuvre philosophique'', by [[Pierre Boutroux]].

===Secondary sources to work on relativity===
* {{Citation | author=Cuvaj, Camillo | year=1969 | title= Henri Poincaré's Mathematical Contributions to Relativity and the Poincaré Stresses | journal=American Journal of Physics |pages=1102–1113 |volume=36 | issue=12|doi=10.1119/1.1974373|bibcode = 1968AmJPh..36.1102C }}
* {{Citation|author=Darrigol, O. |title=Henri Poincaré's criticism of Fin De Siècle electrodynamics  |year=1995 |journal=Studies in History and Philosophy of Science |volume=26|issue=1|pages=1–44|doi=10.1016/1355-2198(95)00003-C|bibcode=1995SHPMP..26....1D  }}
* {{Citation  | author=Darrigol, O. | year=2000 | title= Electrodynamics from Ampére to Einstein | place=Oxford |publisher=Clarendon Press |isbn=0-19-850594-9}}
* {{Citation|author=Darrigol, O. |title=The Mystery of the Einstein–Poincaré Connection| pages=614–626|url=http://www.journals.uchicago.edu/doi/full/10.1086/430652|doi=10.1086/430652|pmid=16011297  |year=2004 |journal=Isis|volume=95| issue=4}}
* {{Citation|author=Darrigol, O. |title=The Genesis of the theory of relativity |year=2005 |journal=Séminaire Poincaré|volume=1|pages=1–22|url=http://www.bourbaphy.fr/darrigol2.pdf|format=PDF |doi=10.1007/3-7643-7436-5_1|isbn=978-3-7643-7435-8 |bibcode=2006eins.book....1D }}
* {{Citation  | author=Galison, P. | year=2003 | title= Einstein's Clocks, Poincaré's Maps: Empires of Time | place=New York |publisher=W.W. Norton|isbn=0-393-32604-7}}
* {{Citation|author=Giannetto, E. |title=The Rise of Special Relativity: Henri Poincaré's Works Before Einstein |year=1998 |journal=Atti del XVIII congresso di storia della fisica e dell'astronomia |pages=171–207}}
* {{Citation  | author=[[Jerzy Giedymin|Giedymin, J.]] | year=1982 | title= Science and Convention: Essays on Henri Poincaré's Philosophy of Science and the Conventionalist Tradition | place=Oxford |publisher=Pergamon Press|isbn=0-08-025790-9}}
* {{Citation | author=Goldberg, S. | year=1967 | title= Henri Poincaré and Einstein's Theory of Relativity | journal=American Journal of Physics |pages=934–944 |volume=35 | issue=10|doi=10.1119/1.1973643|bibcode = 1967AmJPh..35..934G }}
* {{Citation | author=Goldberg, S. | year=1970 | title= Poincaré's silence and Einstein's relativity | journal=British journal for the history of science |pages=73–84 |volume=5 | doi=10.1017/S0007087400010633}}
*{{Citation  | author=Holton, G. | origyear=1973| year=1988 | chapter=Poincaré and Relativity| title= Thematic Origins of Scientific Thought: Kepler to Einstein | publisher=Harvard University Press|isbn=0-674-87747-0}}
* {{Citation | author=Katzir, S. | year=2005 | journal=Phys. Perspect. | title= Poincaré's Relativistic Physics: Its Origins and Nature |pages= 268–292 |volume=7 | doi=10.1007/s00016-004-0234-y | issue=3 |bibcode = 2005PhP.....7..268K }}
* {{Citation|author=Keswani, G.H., Kilmister, C.W. |year=1983 |journal=Brit. J. Phil. Sci. |title=Intimations of Relativity: Relativity Before Einstein |pages=343–354 |volume=34 |doi=10.1093/bjps/34.4.343 |issue=4 |url=http://osiris.sunderland.ac.uk/webedit/allweb/news/Philosophy_of_Science/PIRT2002/Intimations%20of%20Relativity.doc |deadurl=yes |archiveurl=https://web.archive.org/web/20090326084436/http://osiris.sunderland.ac.uk/webedit/allweb/news/Philosophy_of_Science/PIRT2002/Intimations%20of%20Relativity.doc |archivedate=26 March 2009 |df=dmy-all }}
* {{Citation | author=Kragh, H. | year=1999 | title= Quantum Generations: A History of Physics in the Twentieth Century |publisher= Princeton University Press|isbn=0-691-09552-3}}
* {{Citation | author=Langevin, P. | year=1913 | journal=Revue de métaphysique et de morale | title= L'œuvre d'Henri Poincaré: le physicien |page= 703 |volume=21|url=http://gallica.bnf.fr/ark:/12148/bpt6k111418/f93.chemindefer}}
* {{Citation | author=Macrossan, M. N. | year=1986 | journal=Brit. J. Phil. Sci. | title= A Note on Relativity Before Einstein |pages= 232–234 |volume=37| issue=2 |url=http://espace.library.uq.edu.au/view.php?pid=UQ:9560 | doi=10.1093/bjps/37.2.232}}
*{{Citation|author=Miller, A.I. |title=A study of Henri Poincaré's "Sur la Dynamique de l'Electron |year=1973 |journal=Arch. Hist. Exact Sci.|volume=10|pages=207–328|doi=10.1007/BF00412332|issue=3–5}}
* {{Citation | author=Miller, A.I. | year=1981 | title= Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) | place= Reading |publisher=Addison–Wesley |isbn=0-201-04679-2}}
*{{Citation| author=Miller, A.I. |contribution= Why did Poincaré not formulate special relativity in 1905? |year=1996 |editor1=Jean-Louis Greffe |editor2=Gerhard Heinzmann |editor3=Kuno Lorenz | title=Henri Poincaré : science et philosophie| pages=69–100|place=Berlin}}
* {{Citation | author=Schwartz, H. M. | year=1971 | title= Poincaré's Rendiconti Paper on Relativity. Part I | journal=American Journal of Physics |pages=1287–1294 |volume=39 | issue=7|doi=10.1119/1.1976641|bibcode = 1971AmJPh..39.1287S }}
* {{Citation | author=Schwartz, H. M. | year=1972 | title= Poincaré's Rendiconti Paper on Relativity. Part II | journal=American Journal of Physics |pages=862–872 |volume=40 | issue=6| doi=10.1119/1.1986684|bibcode = 1972AmJPh..40..862S }}
* {{Citation | author=Schwartz, H. M. | year=1972 | title= Poincaré's Rendiconti Paper on Relativity. Part III | journal=American Journal of Physics |pages=1282–1287 |volume=40 | issue=9| doi=10.1119/1.1986815|bibcode = 1972AmJPh..40.1282S }}
* {{Citation | author=Scribner, C. | year=1964 | title= Henri Poincaré and the principle of relativity | journal=American Journal of Physics |pages=672–678 |volume=32 | issue=9| doi=10.1119/1.1970936|bibcode =1964AmJPh..32..672S }}
* {{Citation | author=Walter, S. | year=2005 | editor=Renn, J. | contribution= Henri Poincaré and the theory of relativity | title=Albert Einstein, Chief Engineer of the Universe: 100 Authors for Einstein |pages=162–165 | place=Berlin | publisher=Wiley-VCH|contribution-url=http://scottwalter.free.fr/papers/2005-100authors-poincare-einstein-walter.html}}
* {{Citation | author=Walter, S. | year=2007 | editor=Renn, J. | contribution= Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905–1910 | title=The Genesis of General Relativity |pages=193–252 |volume=3 |place=Berlin | publisher=Springer|contribution-url=http://scottwalter.free.fr/papers/2007-genesis-walter.html}}
*{{Citation| author=Zahar, E. |year=2001 |title=Poincaré's Philosophy: From Conventionalism to Phenomenology |publisher=Open Court Pub Co|place=Chicago|isbn=0-8126-9435-X}}

;Non-mainstream
* {{Citation | author=Keswani, G.H., | year=1965| journal=Brit. J. Phil. Sci. | title= Origin and Concept of Relativity, Part I |volume=15| issue=60|pages=286–306 |doi=10.1093/bjps/XV.60.286}}
* {{Citation | author=Keswani, G.H., | year=1965 | journal=Brit. J. Phil. Sci. | title= Origin and Concept of Relativity, Part II|volume=16| pages=19–32| issue=61| doi=10.1093/bjps/XVI.61.19}}
* {{Citation | author=Keswani, G.H., | year=11966 | journal=Brit. J. Phil. Sci. | title= Origin and Concept of Relativity, Part III |volume=16|issue=64| pages=273–294| doi=10.1093/bjps/XVI.64.273 }}
* {{Citation  | author=Leveugle, J. | year=2004 |title=  La Relativité et Einstein, Planck, Hilbert—Histoire véridique de la Théorie de la Relativitén | publisher=L'Harmattan| place=Pars}}
* {{Citation | author=Logunov, A.A. | year=2004 | title= Henri Poincaré and relativity theory | pages=<!-- --> | arxiv=physics/0408077 |bibcode = 2004physics...8077L |isbn=5-02-033964-4}}
* {{Citation  | author=Whittaker, E.T. | year=1953 | title= A History of the Theories of Aether and Electricity: The Modern Theories 1900–1926| chapter= The Relativity Theory of Poincaré and Lorentz | place=London |publisher=Nelson}}

==External links==
{{commons|Henri Poincaré}}
{{wikiquote}}
{{wikisource author}}
* {{Gutenberg author |id=Poincaré,+Henri | name=Henri Poincaré}}
* {{Internet Archive author |sname=Henri Poincaré |sopt=w}} 
* {{Librivox author |id=4281}}
*[[Internet Encyclopedia of Philosophy]]: "[http://www.utm.edu/research/iep/p/poincare.htm Henri Poincaré]"—by Mauro Murzi.
*[[Internet Encyclopedia of Philosophy]]: "[http://www.iep.utm.edu/poi-math/ Poincaré’s Philosophy of Mathematics]"—by Janet Folina.
* {{MathGenealogy |id=34227}}
*[https://web.archive.org/web/20090930005045/https://www.informationphilosopher.com/solutions/scientists/poincare/ Henri Poincaré on Information Philosopher]
* {{MacTutor Biography|id=Poincare}}
*[http://henripoincarepapers.univ-nantes.fr/chronos.php A timeline of Poincaré's life] University of Nantes (in French).
*[http://henripoincarepapers.univ-nantes.fr Henri Poincaré Papers] University of Nantes (in French).
*[https://web.archive.org/web/20060627062431/https://www.phys-astro.sonoma.edu/BruceMedalists/Poincare/index.html Bruce Medal page]
*Collins, Graham P., "[http://www.sciam.com/print_version.cfm?articleID=0003848D-1C61-10C7-9C6183414B7F0000 Henri Poincaré, His Conjecture, Copacabana and Higher Dimensions,]" ''[[Scientific American]]'', 9 June 2004.
*BBC in Our Time, "[https://web.archive.org/web/20090424054425/http://www.bbc.co.uk/radio4/history/inourtime/inourtime.shtml Discussion of the Poincaré conjecture,]" 2 November 2006, hosted by [[Melvynn Bragg]]. 
*[http://www.mathpages.com/home/kmath305/kmath305.htm Poincare Contemplates Copernicus] at MathPages
*[https://www.youtube.com/user/thedebtgeneration?feature=mhum#p/u/8/5pKrKdNclYs0 High Anxieties&nbsp;– The Mathematics of Chaos] (2008) BBC documentary directed by [[David Malone (independent filmmaker)|David Malone]] looking at the influence of Poincaré's discoveries on 20th Century mathematics.

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