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. It was Albert Einstein however, who was prepared to reconstruct the entire edifice of physics, who produced the successful new relativity model.
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 his first paper on relativity. The topic of the paper was "partly kinematic, partly dynamic", and included the correction of Lorentz's proof related to the Lorentz transformation (actually named by Poincaré). About a month later Einstein published his first paper on relativity. Both continued publishing work about relativity, but neither of them would reference each others work. Not only did Einstein not reference Poincaré's work, but he claimed never to have read it! (It is not known if he eventually did read Poincaré's papers.) Einstein finally referenced Poincaré and acknowledged his work on relativity in the text of a lecture in 1921 called `Geometrie und Erahrung'. 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...
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.
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.
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 called 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. Translation: 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)
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.
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.
- Gold Medal of the Royal Astronomical Society of London (1900)
- Bruce Medal (1911)
Named after him
- Poincaré crater on the Moon
- Asteroid 2021 Poincaré
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:
- Science and Hypothesis, 1901.
- The Value of Science, 1904.
- Science and Method, 1908.
- Dernières pensées (Eng., "Last Thoughts"); Edition Ernest Flammarion, Paris, 1913.
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.
- 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)
Poincaré recurrence theorem