# Henri Poincaré

**Jules Henri Poincaré** (April 29, 1854 – July 17, 1912), generally known as **Henri Poincaré**, was one of France's greatest mathematicians, theoretical scientists and a philosopher of science. Poincaré (pronounced (IPA) BrE: [ˈpwæŋ kæ reɪ]; AmE: [ˌpwɑːŋ kɑː ˈreɪ] [1]; Fr: [pwæ̃ ka ʁe]) is often described as the last "universalist" (after Gauss) capable of understanding and contributing in virtually all parts of mathematics.

As a mathematician and physicist, he made many original fundamental contributions to theoretical 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 and laid the foundations of modern chaos theory. Poincaré anticipated Albert Einstein's work and sketched a preliminary version of the special theory of relativity. The Poincaré group was named after him.

## Contents

## Life

Poincaré was born on April 29, 1854 in Cité Ducale neighborhood, Nancy, France 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.

### Education

During his childhood he was seriously ill for a time with diphtheria and received special instruction from his gifted mother, Eugénie Launois (1830-1897). He excelled in written composition.

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. 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 to absentmindedness may explain these difficulties (Carl, 1968). He graduated from the Lycée in 1871 with a bachelors 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 north east 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. Poincaré devised a new way of studying the properties of these functions. 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.

### Early career

Soon after, he was offered a post as junior lecturer in mathematics at Caen University. He never fully abandoned his mining career to mathematics however. 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).

### The three-body problem

In 1887, in honor of his 60th birthday, Oscar II, King of Sweden sponsored a mathematical competition with a cash prize for a resolution of the question of how stable is the solar system, a variation of the three-body problem. Poincaré pointed out that the problem wasn't correctly posed, and that nobody would succeed in giving a complete solution to it. His work was so impressive that in 1888 the jury recognized its value by allowing him the prize. He found that the evolution of such a system is often chaotic in the sense that a small perturbation in the initial state such as a slight change in one body's initial position might lead to a radically different later state. If the slight change isn't detectable by our measuring instruments, then we won't be able to predict which final state will occur. 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."

Weierstrass did not know how accurate he was. In Poincaré's paper, he described new mathematical ideas such as homoclinic points. The memoir was about to be published in *Acta Mathematica* when an error was found by the editor. This error in fact led to further discoveries by Poincare, which are now considered to be the beginning of chaos theory. The memoir was published later in 1890.

Also 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.

### Work on relativity

In 1893 he joined the French Bureau des Longitudes which engaged him in the synchronisation of time around the world. In 1897 he backed an unsuccessful proposal for the decimalisation of circular measure and hence time and longitude. This work led him to consider how clocks moving at high speed with respect to each other could be synchronised. In 1898 in “The Measure of Time” he formulated the Principle of relativity, according to which no mechanical or electromagnetic experiment can discriminate between a state of uniform motion and a state of rest. In collaboration with the Dutch theorist Hendrik Lorentz he went on to push the physics of the time to the limit to explain the behaviour of fast moving electrons as well as the failure of experiments to detect motion relative to the ether. He published the main features of special relativity before Einstein, and even introduced the 4-space notation that Hermann Minkowski became known for. It was Albert Einstein however, who first showed how the Lorentz transformations can be derived from a minimal number of assumptions and it is Einstein who is generally regarded as having produced the successful new Special relativity model. There is a growing minority opinion that Poincare deserves more credit for the development of Special Relativity than he has been given to date. [2] & [3].

Henri Poincaré and Albert Einstein had an interesting relationship concerning their work on relativity -- one might actually describe it as a lack of a relationship (Pais, 1982). Their interaction began in 1905, when Poincaré published a paper on relativity, in which he concluded that Lorentz finally had come up with a working theory that fully satisfies the Principle of Relativity. The title of the paper was "Sur la dynamique de l'électron", and included a correction of the Lorentz transformations (so named by Poincaré). A few weeks later Einstein submitted his first paper on relativity with as title "Zur Elektrodynamik Bewegter Körper", without acknowledgeing the earlier works of Lorentz and Poincaré. Both continued publishing work about relativity, but neither of them would reference each other's work. Not only did Einstein not reference Poincaré's work, but he claimed never to have read it! (It is now known that he in fact had read and discussed Poincaré's "La science et l'hypothese" of 1902). Einstein finally referenced Poincaré and acknowledged his work on relativity in the text of a lecture in 1921 called `Geometrie und Erfahrung'. Later in Einstein's life, he would comment on Poincaré as being one of the pioneers of relativity. Before Einstein's death, Einstein said:

*Lorentz had already recognized that the transformation named after him is essential for the analysis of Maxwell's equations, and Poincaré deepened this insight still further...*

Similarly, in a memoir on Poincaré's work on relativity theory, Lorentz commented :

*For certain of the physical magnitudes which enter in the formulae I have not indicated the transformation which suits best. This has been done by Poincaré, and later by Einstein and Minkowski. My formulae were encumbered by certain terms which should have been made to disappear. [...] I have not established the principle of relativity as rigorously and universally true. Poincaré, on the other hand, has obtained a perfect invariance of the electro-magnetic equations, and he has formulated "the postulate of relativity", terms which he was the first to employ.*

### Late career

Poincaré was responsible for formulating one of the most famous problems in mathematics. Known as the Poincaré conjecture, it is a problem in topology still not fully resolved today.

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 anti-Semitic colleagues.

In 1912 Poincaré underwent surgery for a prostate problem and subsequently died from an embolism on July 17 1912, aged 58. He is buried in the Poincaré family vault in the Cemetery of Montparnasse, Paris.

## Character

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' characterization

His mental organization was not only interesting to him 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 am and noon then again from 5 pm to 7 pm. He would read articles in journals later in the evening.

- He had an exceptional memory and could recall the page and line of any item in a text he had read. He was also able to remember verbatim things heard by ear. He retained these abilities all his life.

- 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 his lecturers were writing on the blackboard.

However, these abilities were somewhat balanced 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 worked on another problem.

In addition, Toulouse stated that most mathematicians worked from principle already established while Poincaré was the type that started from basic principle each time. (O'Connor et al., 2002)

His method of thinking is well summarized 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.* (He neglected details and jumped from idea to idea, the facts gathered from each idea would then come together and solve the problem.) (Belliver, 1956)

## Work

Among the specific topics he contributed to are the following:

- algebraic topology
- the theory of analytic functions of several complex variables
- the theory of abelian functions
- algebraic geometry
- number theory
- the three-body problem
- the theory of diophantine equations
- the theory of electromagnetism
- 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 sphere and the Poincaré map.

Poincaré made many contributions to different fields of applied mathematics such as: celestial mechanics, fluid mechanics, optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity and cosmology.

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

## Philosophy

Poincaré had the opposite philosophical views of Bertrand Russell and Gottlob Frege, who believed that mathematics were 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 (Murz, 2001), therefore concluding that arithmetic is *a priori* synthetic and not analytic. Poincaré then went on to say that mathematics can not be a deduced from logic since it is not analytic. His views were the same as those of Kant (Kolak, 2001). 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.

## Honors

**Awards**

- Gold Medal of the Royal Astronomical Society of London (1900)
- Bruce Medal (1911)

**Named after him**

- Poincaré crater on the Moon
- Asteroid 2021 Poincaré

## Publications

Poincaré's major contribution to algebraic topology was *Analysis situs* (1895), which was the first real systematic look at topology.

He published two major works that placed celestial mechanics on a rigorous mathematical basis:

*New Methods of Celestial Mechanics*ISBN 1563961172 (3 vols., 1892-99; Eng. trans., 1967)*Lessons of Celestial Mechanics*. (1905-10).

In popular writings he helped establish the fundamental popular definitions and perceptions of science by these writings:

*Dernières pensées*(Eng., "Last Thoughts"); Edition Ernest Flammarion, Paris, 1913.

## References

*This article incorporates material from Jules Henri Poincaré on PlanetMath, which is licensed under the GFDL.*

- Bell, Eric Temple (1986).
*Men of Mathematics*(reissue edition). Touchstone Books. ISBN 0671628186. - Belliver, André.
*Henri Poincaré ou la vocation souveraine*, Gallimard, 1956. - Boyer B. Carl.
*A History of Mathematics: Henri Poincaré*, John Wiley & Sons, inc., Toronto, 1968. - O'Connor, J. John & Robertson, F. Edmund, "Jules Henri Poincaré" University of St Andrews, Scotland (2002).
- Galison, Peter Louis (2003).
*Einstein's Clocks, Poincaré's Maps: Empires of Time*. Hodder & Stoughton. ISBN 034079447X. - Kolak, Daniel:
*Lovers of Wisdom (second edition)*, Wadsworth, Belmont, 2001. - Lorentz, H.A., Deux memoires de Henri Poincaré (1914), Acta Mathematica 38, 293, 1921.
- Pais, Abraham:
*Subtle is the Lord...*, Oxford University Press, New York, 1982. - Peterson, Ivars (1995).
*Newton's Clock: Chaos in the Solar System*(reissue edition). W H Freeman & Co. ISBN 0716727242. - Sageret, Jules.
*Henri Poincaré*, Mercvre de France, Paris, 1911. - E. Toulouse,
*Henri Poincaré*, Paris (1910) - (Source biography in French)

## See also

## External links

- A review of Poincaré's life and mathematical achievements - from the University of Tennessee at Martin, USA.
- A timeline of Poincaré's life University of Nancy ,(in French)
- "The Relativity of Space" - Poincaré's 1897 article in English translation.
- Henri Poincaré, His Conjecture, Copacabana and Higher Dimensions Article in Scientific American on the Poincare conjecture.
- The Poincaré Archives - Held at the University of Nancy, (in French)
- Poincaré article at The GAP Group, Centre for Interdisciplinary Research in Computational Algebra' University of St. Andrews' Scotland.
- Poincaré's obituary in
*The Times*. Times correspondent , Paris July 17th , 1912 - Online copy of "Science and Hypothesis" Poincaré's 1905 article hosted at Brock University, Ontario, Canada. - English translation

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