# Henri Poincaré: Wikis

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# Encyclopedia

Henri Poincaré

Jules Henri Poincaré (1854–1912). Photograph from the frontispiece of the 1913 edition of Last Thoughts.
Born 29 April 1854
Nancy, Meurthe-et-Moselle
Died 17 July 1912 (aged 58)
Paris
Residence France
Nationality French
Fields Mathematician and physicist
Institutions Corps des Mines
Caen University
La Sorbonne
Bureau des Longitudes
Alma mater Lycée Nancy
École Polytechnique
École des Mines
Doctoral students Louis Bachelier
Dimitrie Pompeiu
Mihailo Petrović
Other notable students Tobias Dantzig
Known for Poincaré conjecture
Three-body problem
Topology
Special relativity
Poincaré–Hopf theorem
Poincaré duality
Poincaré–Birkhoff–Witt theorem
Poincaré inequality
Hilbert–Poincaré series
Poincaré metric
Rotation number
Coining term 'Betti number'
Chaos theory
Sphere-world
Poincaré–Bendixson theorem
Poincaré–Lindstedt method
Poincaré recurrence theorem
Influences Lazarus Fuchs
Influenced Louis Rougier
George David Birkhoff
Notable awards RAS Gold Medal (1900)
Sylvester Medal (1901)
Matteucci Medal (1905)
Bolyai Prize (1905)
Bruce Medal (1911)
Signature
Notes
He was a cousin of Pierre Boutroux.

Jules Henri Poincaré (29 April 1854 – 17 July 1912) (French pronunciation: [ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe][1]) was a French mathematician, theoretical physicist, and a philosopher of science. Poincaré is often described as a polymath, and in mathematics as The Last Universalist, 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 and applied mathematics, mathematical physics, and celestial mechanics. He was responsible for formulating the Poincaré conjecture, one of the most famous problems in mathematics. 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 considered to be one of the founders of the field of topology.

Poincaré introduced the modern principle of relativity 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 Lorentz 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.

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 (Belliver, 1956). His father Leon Poincaré (1828–1892) was a professor of medicine at the University of Nancy (Sagaret, 1911). His adored younger sister Aline married the spiritual philosopher Emile Boutroux. Another notable member of Jules' family was his cousin, Raymond Poincaré, who would become the President of France, 1913 to 1920, and a fellow member of the Académie française.[2]

### Education

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 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" (O'Connor et al., 2002). However, poor eyesight and a tendency towards absentmindedness may explain these difficulties (Carl, 1968). 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. 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. He graduated in 1875 or 1876. He went on to study at the École des Mines, continuing to study mathematics in addition to the mining engineering syllabus and received the degree of ordinary 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 in a characteristically thorough and humane way.

At the same time, Poincaré was preparing for his doctorate in sciences 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 différences. 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.

The young Henri Poincaré

### Career

Soon after, he was offered a post as junior lecturer in mathematics at Caen University, but 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 Sorbonne). He was initially appointed as the maître de conférences d'analyse (associate professor of analysis) (Sageret, 1911). Eventually, he held the chairs of Physical and Experimental Mechanics, Mathematical Physics and Theory of Probability, and Celestial Mechanics and Astronomy.

Also in that same year, Poincaré married Miss Poulain d'Andecy. Together they had four children: Jeanne (born 1887), Yvonne (born 1889), Henriette (born 1891), and Léon (born 1893).

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, King of Sweden's mathematical competition for a resolution of the three-body problem concerning the free motion of multiple orbiting bodies. (See #The 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 (see Galison 2003). 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 the year 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 anti-Semitic colleagues.

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.

A former French Minister of Education, Claude Allègre, has recently (2004) proposed that Poincaré be reburied in the Panthéon in Paris, which is reserved for French citizens only of the highest honour.[3]

##### Students

Poincaré had two notable doctoral students at the University of Paris, Louis Bachelier (1900) and Dimitrie Pompeiu (1905).[4]

## 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, thermodynamics, potential theory, 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:

### The 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 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 nineteenth 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[7]). The version finally printed contained many important ideas which lead 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.

### Work on relativity

Marie Curie and Poincaré talk at the 1911 Solvay Conference.

#### 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. He had introduced in 1895 an auxiliary quantity (without physical interpretation) called "local time" $t^\prime = t-vx^\prime/c^2$, where $x^\prime = x - vt$ 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).[8] 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.[9] 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.[10]

#### Principle of relativity and Lorentz transformations

He discussed the "principle of relative motion" in two papers in 1900[10][11] 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.[12] 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.[13] 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 gave what is now known as the relativistic velocity-addition law.[14] 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[15]:

 “ 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 x2 + y2 + z2c2t2 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.[16] Poincaré’s attempt at a four-dimensional reformulation of the new mechanics was rejected by himself 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.[17] So it was Hermann Minkowski who worked out the consequences of this notion in 1907.

#### Mass-energy relation

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.[10] 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/c2. 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).[12] This time (and later also in 1908) he rejected[18] the possibility that energy carries mass and also the possibility, that motions in the ether can compensate the above mentioned problems:

 “ 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 γm, 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 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 = E/c2 that resolved[19] Poincare's paradox, without using any compensating mechanism within the ether.[20] 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.[21]

#### Poincaré and Einstein

Einstein's first paper on relativity was published three months after Poincaré's short paper,[15] but before Poincaré's longer version.[16] It relied on the principle of relativity to derive the Lorentz transformations and used a similar clock synchronisation procedure (Einstein synchronisation) that Poincaré (1900) had described, but was remarkable in that it contained no references at all. Poincaré never acknowledged Einstein's work on Special Relativity. Einstein acknowledged Poincaré 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 ...."[22]

#### Assessments

Poincaré's work in the development of special relativity is well recognised[19], though most historians stress that despite many similarities with Einstein's work, the two had very different research agendas and interpretations of the work.[23] 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 bring the relativity principle in accordance with classical physics, while Einstein developed a mathematically equivalent kinematics based on the new physical concepts of the relativity of space and time.[24][25][26][27][28] 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.[29]

## Character

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. He believed that logic was not a way to invent but a way to structure ideas and that logic limits ideas.

### Toulouse' characterisation

Poincaré's mental organisation was not only interesting to Poincaré himself but also to 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 (O'Connor et al., 2002).

His method of thinking is well summarised as:

"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.") Belliver (1956)

### Attitude towards Cantor

Poincaré was dismayed by Georg Cantor's theory of transfinite numbers, and referred to it as a "disease" from which mathematics would eventually be cured.[30]

### View on economics

Poincaré saw mathematical work in economics and finance as peripheral. In 1900 Poincaré commented on Louis Bachelier's thesis "The Theory of Speculation", saying: "M. Bachelier has evidenced an original and precise mind [but] the subject is somewhat remote from those our other candidates are in the habit of treating." (Bernstein, 1996, pp. 199–200) Bachelier's work explained what was then the French government's pricing options on French Bonds and anticipated many of the pricing theories in financial markets used today.[31]

Awards

Named after him

## 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 was the life of mathematics. Poincaré gives an interesting point of view in his book Science and Hypothesis:

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 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 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.

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. 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.[citation needed]

### 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.[32]

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." [33]

Poincaré's two stages - random combinations followed by selection - became the basis for Daniel Dennett's two-stage model of free will.[34]

## References

### Footnotes and primary sources

1. ^ [1] Poincaré pronunciation examples at Forvo
2. ^ The Internet Encyclopedia of Philosophy Jules Henri Poincaré article by Mauro Murzi — accessed November 2006.
3. ^ Lorentz, Poincaré et Einstein — L'Express
4. ^ Mathematics Genealogy Project North Dakota State University, Accessed April 2008
5. ^ McCormmach, Russell (Spring, 1967), "Henri Poincaré and the Quantum Theory", Isis 58 (1): 37–55, doi:10.1086/350182
6. ^ Irons, F. E. (August, 2001), "Poincaré's 1911–12 proof of quantum discontinuity interpreted as applying to atoms", American Journal of Physics 69 (8): 879–884, doi:10.1119/1.1356056
7. ^ Diacu, F. (1996), "The solution of the n-body Problem", The Mathematical Intelligencer 18: 66–70, doi:10.1007/BF03024313
8. ^ Lorentz, H.A. (1895), Versuch einer theorie der electrischen und optischen erscheinungen in bewegten Kõrpern, Leiden: E.J. Brill
9. ^ Poincaré, H. (1898), "La mesure du temps", Revue de métaphysique et de morale 6: 1–13  Reprinted as The Measure of Time in "The Value of Science", Ch. 2.
10. ^ a b c Poincaré, H. (1900), "La théorie de Lorentz et le principe de réaction", Archives néerlandaises des sciences exactes et naturelles 5: 252–278 . See also the English translation
11. ^ Poincaré, H. (1900), "Les relations entre la physique expérimentale et la physique mathématique", Revue générale des sciences pures et appliquées 11: 1163–1175 . Reprinted in "Science and Hypothesis", Ch. 9–10.
12. ^ a b Poincaré, Henri (1904), "L'état actuel et l'avenir de la physique mathématique", Bulletin des sciences mathématiques 28 (2): 302–324 . English translation in Poincaré, Henri (1905), "The Principles of Mathematical Physics", in Rogers, Howard J., Congress of arts and science, universal exposition, St. Louis, 1904, 1, Boston and New York: Houghton, Mifflin and Company, pp. 604–622  Reprinted in "The value of science", Ch. 7–9.
13. ^ Letter from Poincaré to Lorentz, Mai 1905
14. ^ Letter from Poincaré to Lorentz, Mai 1905
15. ^ a b Poincaré, H. (1905), "Sur la dynamique de l’électron", Comptes Rendus 140: 1504–1508  Reprinted in Poincaré, Oeuvres, tome IX, S. 489–493.
16. ^ a b Poincaré, H. (1906), "Sur la dynamique de l’électron", Rendiconti del Circolo matematico Rendiconti del Circolo di Palermo 21: 129–176, doi:10.1007/BF03013466  Partial English translation in Dynamics of the electron.
17. ^ Walter (2007), Secondary sources on relativity
18. ^ Miller 1981, Secondary sources on relativity
19. ^ a b Darrigol 2005, Secondary sources on relativity
20. ^ Einstein, A. (1905b), "Ist die Trägheit eines Körpers von dessen Energieinhalt abhängig?", Annalen der Physik 18: 639–643 . See also English translation.
21. ^ Einstein, A. (1906), "Das Prinzip von der Erhaltung der Schwerpunktsbewegung und die Trägheit der Energie", Annalen der Physik 20: 627–633, doi:10.1002/andp.19063250814
22. ^ Darrigol 2004, Secondary sources on relativity
23. ^ Galison 2003 and Kragh 1999, Secondary sources on relativity
24. ^ Holton (1988), 196-206
25. ^ Hentschel (1990), 3-13
26. ^ Miller (1981), 216-217
27. ^ Darrigol (2005), 15-18
28. ^ Katzir (2005), 286-288
29. ^ Whittaker 1953, Secondary sources on relativity
30. ^ Dauben 1979, p. 266.
31. ^ Dunbar, Nicholas (2000), INVENTING MONEY, JOHN WILEY & SONS, LTD, ISBN 0-471-49811-4
32. ^ Hadamard, Jacques. An Essay On The Psychology Of Invention In The Mathematical Field. Princeton Univ Press (1949)
33. ^ Science and Method, Chapter 3, Mathematical Discovery, 1914, pp.58
34. ^ Dennett, Daniel C. 1978. Brainstorms: Philosophical Essays on Mind and Psychology. The MIT Press, p.293

### Poincaré's writings in English translation

Popular writings on the philosophy of science:

• Poincaré, Henri (1902-1908), The Foundations of Science, New York: Science Press ; This book includes the English translations of Science and Hypothesis (1902), The Value of Science (1905), Science and Method (1908).
• 1892–99. New Methods of Celestial Mechanics, 3 vols. English trans., 1967. ISBN 1-56396-117-2.
• 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.

### General references

• 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.
• Bernstein, Peter L, 1996. "Against the Gods: A Remarkable Story of Risk". (p. 199–200). John Wiley & Sons.
• Boyer, B. Carl, 1968. A History of Mathematics: Henri Poincaré, John Wiley & Sons.
• Grattan-Guinness, Ivor, 2000. The Search for Mathematical Roots 1870–1940. Princeton Uni. Press.
• Dauben, Joseph (1993, 2004). "Georg Cantor and the Battle for Transfinite Set Theory" in Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, CA) (pp. 1–22). Internet version published in Journal of the ACMS 2004.
• Folina, Janet, 1992. Poincare and the Philosophy of Mathematics. Macmillan, New York.
• Gray, Jeremy, 1986. Linear differential equations and group theory from Riemann to Poincaré, Birkhauser
• Jean Mawhin (October 2005), "Henri Poincaré. A Life in the Service of Science" (PDF), Notices of the AMS 52 (9): 1036–1044
• Kolak, Daniel, 2001. Lovers of Wisdom, 2nd ed. Wadsworth.
• Murzi, 1998. "Henri Poincaré".
• O'Connor, J. John, and Robertson, F. Edmund, 2002, "Jules Henri Poincaré". University of St. Andrews, Scotland.
• 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)

### Secondary sources to work on relativity

• Cuvaj, Camillo (1969), "Henri Poincaré's Mathematical Contributions to Relativity and the Poincaré Stresses", American Journal of Physics 36 (12): 1102–1113, doi:10.1119/1.1974373
• Darrigol, O. (1995), "Henri Poincaré's criticism of Fin De Siècle electrodynamics", Studies in History and Philosophy of Science 26 (1): 1–44, doi:10.1016/1355-2198(95)00003-C
• Darrigol, O. (2000), Electrodynamics from Ampére to Einstein, Oxford: Clarendon Press, ISBN 0198505949
• Galison, P. (2003), Einstein's Clocks, Poincaré's Maps: Empires of Time, New York: W.W. Norton, ISBN 0393326047
• Giannetto, E. (1998), "The Rise of Special Relativity: Henri Poincaré's Works Before Einstein", Atti del XVIII congresso di storia della fisica e dell'astronomia: 171–207
• Giedymin, J. (1982), Science and Convention: Essays on Henri Poincaré’s Philosophy of Science and the Conventionalist Tradition, Oxford: Pergamon Press, ISBN 0080257909
• Goldberg, S. (1967), "Henri Poincaré and Einstein’s Theory of Relativity", American Journal of Physics 35 (10): 934–944, doi:10.1119/1.1973643
• Goldberg, S. (1970), "Poincaré's silence and Einstein's relativity", British journal for the history of science 5: 73–84, doi:10.1017/S0007087400010633
• Holton, G. (1973/1988), "Poincaré and Relativity", Thematic Origins of Scientific Thought: Kepler to Einstein, Harvard University Press, ISBN 0674877470
• Katzir, S. (2005), "Poincaré’s Relativistic Physics: Its Origins and Nature", Phys. Perspect. 7: 268–292, doi:10.1007/s00016-004-0234-y
• Kragh, H. (1999), Quantum Generations: A History of Physics in the Twentieth Century, Princeton University Press, ISBN 0691095523
• Miller, A.I. (1973), "A study of Henri Poincaré's "Sur la Dynamique de l'Electron", Arch. Hist. Exact. Scis. 10: 207–328, doi:10.1007/BF00412332
• Miller, A.I. (1981), Albert Einstein’s special theory of relativity. Emergence (1905) and early interpretation (1905–1911), Reading: Addison–Wesley, ISBN 0-201-04679-2
• Miller, A.I. (1996), "Why did Poincaré not formulate special relativity in 1905?", in Jean-Louis Greffe, Gerhard Heinzmann, Kuno Lorenz, Henri Poincaré : science et philosophie, Berlin, pp. 69–100
• Schwartz, H. M. (1971), "Poincaré's Rendiconti Paper on Relativity. Part I", American Journal of Physics 39 (7): 1287–1294, doi:10.1119/1.1976641
• Schwartz, H. M. (1972), "Poincaré's Rendiconti Paper on Relativity. Part II", American Journal of Physics 40 (6): 862–872, doi:10.1119/1.1986684
• Schwartz, H. M. (1972), "Poincaré's Rendiconti Paper on Relativity. Part III", American Journal of Physics 40 (9): 1282–1287, doi:10.1119/1.1976641
• Scribner, C. (1964), "Henri Poincaré and the principle of relativity", American Journal of Physics 32 (9): 672–678, doi:10.1119/1.1986815
• Zahar, E. (2001), Poincare's Philosophy: From Conventionalism to Phenomenology, Chicago: Open Court Pub Co, ISBN 081269435X
Non-mainstream
• Keswani, G.H., (1965), "Origin and Concept of Relativity, Part I", Brit. J. Phil. Sci. 15 (60): 286–306, doi:10.1093/bjps/XV.60.286
• Keswani, G.H., (1965), "Origin and Concept of Relativity, Part II", Brit. J. Phil. Sci. 16 (61): 19–32, doi:10.1093/bjps/XVI.61.19
• Keswani, G.H., (11966), "Origin and Concept of Relativity, Part III", Brit. J. Phil. Sci. 16 (64): 273–294, doi:10.1093/bjps/XVI.64.273
• Leveugle, J. (2004), La Relativité et Einstein, Planck, Hilbert — Histoire véridique de la Théorie de la Relativitén, Pars: L'Harmattan
• Whittaker, E.T. (1953), "The Relativity Theory of Poincaré and Lorentz", A History of the Theories of Aether and Electricity: The Modern Theories 1900–1926, London: Nelson

Cultural offices
Preceded by
Sully Prudhomme
Seat 24
1908–1912
Succeeded by
Alfred Capus

# Quotes

Up to date as of January 14, 2010

### From Wikiquote

To doubt everything or to believe everything are two equally convenient solutions; both dispense with the necessity of reflection.

Jules Henri Poincaré (), generally known as Henri Poincaré, was one of France's greatest mathematicians and theoretical physicists, and a philosopher of science.

## Sourced

A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same Nature.
• Le savant digne de ce nom, le géomètre surtout, éprouve en face de son œuvre la même impression que l'artiste ; sa jouissance est aussi grande et de même nature.
• A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.
• "Notice sur Halphen," Journal de l'École Polytechnique (Paris, 1890), 60ème cahier, p. 143
• C'est même des hypothèses simples qu'il faut le plus se défier, parce que ce sont celles qui ont le plus de chances de passer inaperçues.
• It is the simple hypotheses of which one must be most wary; because these are the ones that have the most chances of passing unnoticed.
• Thermodynamique: Leçons professées pendant le premier semestre 1888-1889 (1892), Preface
• La tâche de l'éducateur est de faire repasser l'esprit de l'enfant par où a passé celui de ses pères, en passant rapidement par certaines étapes mais en n'en supprimant aucune. À ce compte, l'histoire de la science doit être notre guide.
• The task of the educator is to make the child's spirit pass again where its forefathers have gone, moving rapidly through certain stages but suppressing none of them. In this regard, the history of science must be our guide.
• "La logique et l'intuition dans la science mathématique et dans l'enseignement" [Logic and intuition in thé science of mathematics and in teaching], L'enseignement mathématique (1899)
• La pensée ne doit jamais se soumettre, ni à un dogme, ni à un parti, ni à une passion, ni à un intérêt, ni à une idée préconçue, ni à quoi que ce soit, si ce n'est aux faits eux-mêmes, parce que, pour elle, se soumettre, ce serait cesser d'être.
• Thinking must never submit itself, neither to a dogma, nor to a party, nor to a passion, nor to an interest, nor to a preconceived idea, nor to whatever it may be, if not to facts themselves, because, for it, to submit would be to cease to be.
• Speech, University of Brussels (), during the festival for the 75th anniversary of the university's foundation; published in Œuvres de Henri Poincaré (1956), p. 152

### Science and Hypothesis (1901)

La Science et l'Hypothèse, English translation: Science and Hypothesis (1905), Dover abridged edition (1952)
• Douter de tout ou tout croire, ce sont deux solutions également commodes, qui l'une et l'autre nous dispensent de réfléchir.
• To doubt everything or to believe everything are two equally convenient solutions; both dispense with the necessity of reflection.
• Preface, Dover abridged edition (1952), p. xxii
• Les mathématiciens n'étudient pas des objets, mais des relations entre les objets ; il leur est donc indifférent de remplacer ces objets par d'autres, pourvu que les relations ne changent pas. La matière ne leur importe pas, la forme seule les intéresse.
• Mathematicians do not study objects, but the relations between objects; to them it is a matter of indifference if these objects are replaced by others, provided that the relations do not change. Matter does not engage their attention, they are interested in form alone.
• Ch. II, Dover abridged edition (1952), p. 20
• Comme nous ne pouvons pas donner de l'énergie une définition générale, le principe de la conservation de l'énergie signifie simplement qu'il y a quelque chose qui demeure constant.
• As we can not give a general definition of energy, the principle of the conservation of energy signifies simply that there is something which remains constant.
• Ch. X: Is Science artificial? as translated by George Bruce Halsted (1913)
• Le savant doit ordonner ; on fait la science avec des faits comme une maison avec des pierres ; mais une accumulation de faits n'est pas plus une science qu'un tas de pierres n'est une maison.
• The Scientist must set in order. Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house.
• Ch. 9: Hypotheses in Nature, as translated by George Bruce Halsted (1913)

### The Value of Science (1905)

Valeur de la Science (1905), as translated by George Bruce Halsted (1907)
• Le temps et l’espace... Ce n’est pas la nature qui nous les impose, c’est nous qui les imposons à la nature parce que nous les trouvons commodes.
• Time and Space... It is not nature which imposes them upon us, it is we who impose them upon nature because we find them convenient.
• Introduction, p. 13
• Il ne faut pas comparer la marche de la science aux transformations d’une ville, où les édifices vieillis sont impitoyablement jetés à bas pour faire place aux constructions nouvelles, mais à l’évolution continue des types zoologiques qui se développent sans cesse et finissent par devenir méconnaissables aux regards vulgaires, mais où un œil exercé retrouve toujours les traces du travail antérieur des siècles passés. Il ne faut donc pas croire que les théories démodées ont été stériles et vaines.
• The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new, but to the continuous evolution of zoologic types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past. One must not think then that the old-fashioned theories have been sterile or vain.
• Introduction, p. 14
• Cette harmonie que l’intelligence humaine croit découvrir dans la nature, existe-t-elle en dehors de cette intelligence ? Non, sans doute, une réalité complètement indépendante de l’esprit qui la conçoit, la voit ou la sent, c’est une impossibilité.
• Does the harmony the human intelligence thinks it discovers in nature exist outside of this intelligence? No, beyond doubt, a reality completely independent of the mind which conceives it, sees or feels it, is an impossibility.
• Introduction, p. 14
• Pour qu’un ensemble de sensations soit devenu un souvenir susceptible d’être classé dans le temps, il faut qu’il ait cessé d’être actuel, que nous ayons perdu le sens de son infinie complexité, sans quoi il serait resté actuel. Il faut qu’il ait pour ainsi dire cristallisé autour d’un centre d’associations d’idées qui sera comme une sorte d’étiquette. Ce n’est que quand ils auront ainsi perdu toute vie que nous pourrons classer nos souvenirs dans le temps, comme un botaniste range dans son herbier les fleurs desséchées.
• For an aggregate of sensations to have become a remembrance capable of classification in time, it must have ceased to be actual, we must have lost the sense of its infinite complexity, otherwise it would have remained present. It must, so to speak, have crystallized around a center of associations of ideas which will be a sort of label. It is only when they have lost all life that we can classify our memories in time as a botanist arranges dried flowers in his herbarium.
• Ch. 2: The Measure of Time
• Si toutes les parties de l’univers sont solidaires dans une certaine mesure, un phénomène quelconque ne sera pas l’effet d’une cause unique, mais la résultante de causes infiniment nombreuses ; il est, dit-on souvent, la conséquence de l’état de l’univers un instant auparavant.
• If all the parts of the universe are interchained in a certain measure, any one phenomenon will not be the effect of a single cause, but the resultant of causes infinitely numerous; it is, one often says, the consequence of the state of the universe the moment before.
• Ch. 2: The Measure of Time

### Science and Method (1908)

Science et méthode (1908), as translated by Francis Maitland (1914)
The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful.
• La sociologie est la science qui possède le plus de méthodes et le moins de résultats.
• Sociology is the science with the greatest number of methods and the least results.
• Part I. Ch. 1 : The Selection of Facts, p. 19
• Le savant n’étudie pas la nature parce que cela est utile ; il l’étudie parce qu’il y prend plaisir et il y prend plaisir parce qu’elle est belle. Si la nature n’était pas belle, elle ne vaudrait pas la peine d’être connue, la vie ne vaudrait pas la peine d’être vécue. Je ne parle pas ici, bien entendu, de cette beauté qui frappe les sens, de la beauté des qualités et des apparences ; non que j’en fasse fi, loin de là, mais elle n’a rien à faire avec la science; je veux parler de cette beauté plus intime qui vient de l’ordre harmonieux des parties, et qu’une intelligence pure peut saisir.
• The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. If nature were not beautiful it would not be worth knowing, and life would not be worth living. I am not speaking, of course, of the beauty which strikes the senses, of the beauty of qualities and appearances. I am far from despising this, but it has nothing to do with science. What I mean is that more intimate beauty which comes from the harmonious order of its parts, and which a pure intelligence can grasp.
• Part I. Ch. 1 : The Selection of Facts, p. 22
• C’est parce que la simplicité, parce que la grandeur est belle, que nous rechercherons de préférence les faits simples et les faits grandioses, que nous nous complairons tantôt à suivre la course gigantesque des astres, tantôt à scruter avec le microscope cette prodigieuse petitesse qui est aussi une grandeur, tantôt à rechercher dans les temps géologiques les traces d’un passé qui nous attire parce qu’il est lointain.
• It is because simplicity and vastness are both beautiful that we seek by preference simple facts and vast facts; that we take delight, now in following the giant courses of the stars, now in scrutinizing the microscope that prodigious smallness which is also a vastness, and now in seeking in geological ages the traces of a past that attracts us because of its remoteness.
• Part I. Ch. 1 : The Selection of Facts, p. 23
• Le but principal de l'enseignement mathématique est de développer certaines facultés de l'esprit et parmi elles l'intuition n'est pas la moins précieuse. C'est par elle que le monde mathématique reste en contact avec le monde réel et quand les mathématiques pures pourraient s'en passer, il faudrait toujours y avoir recours pour combler l'abîme qui sépare le symbole de la réalité.
• The principal aim of mathematical education is to develop certain faculties of the mind, and among these intuition is not the least precious. It is through it that the mathematical world remains in touch with the real world, and even if pure mathematics could do without it, we should still have to have recourse to it to fill up the gulf that separates the symbol from reality.
• Part II. Ch. 2 : Mathematical Definitions and Education, p. 128
• Variant translation: The chief aim of mathematics teaching is to develop certain faculties of the mind, and among these intuition is by no means the least valuable.
• C'est par la logique qu'on démontre, c'est par l'intuition qu'on invente.
• It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better.
• Part II. Ch. 2 : Mathematical Definitions and Education, p. 129
• La logique nous apprend que sur tel ou tel chemin nous sommes sûrs de ne pas rencontrer d'obstacle ; elle ne nous dit pas quel est celui qui mène au but. Pour cela il faut voir le but de loin, et la faculté qui nous apprend à voir, c'est l'intuition. Sans elle, le géomètre serait comme un écrivain qui serait ferré sur la grammaire, mais qui n'aurait pas d'idées.
• Logic teaches us that on such and such a road we are sure of not meeting an obstacle; it does not tell us which is the road that leads to the desired end. For this, it is necessary to see the end from afar, and the faculty which teaches us to see is intuition. Without it, the geometrician would be like a writer well up in grammar but destitute of ideas.
• Part II. Ch. 2 : Mathematical Definitions and Education, p. 130
• Toute définition implique un axiome, puisqu'elle affirme l'existence de l'objet défini. La définition ne sera donc justifiée, au point de vue purement logique, que quand on aura démontré qu'elle n'entraîne pas de contradiction, ni dans les termes, ni avec les vérités antérieurement admises.
• Every definition implies an axiom, since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have proved that it involves no contradiction either in its terms or with the truths previously admitted.
• Part II. Ch. 2 : Mathematical Definitions and Education, p. 131

## Misattributed

• Everybody firmly believes in it because the mathematicians imagine it is a fact of observation, and observers that it is a theory of mathematics.