Poincaré, Jules Henri



Poincaré, Jules Henri (1854-1912), French physicist and one of the foremost mathematicians of the 19th century.


Poincaré was a cousin of the French statesman and author Raymond Poincaré. He was born in Nancy and educated at the École Polytechnique and the École Supérieur des Mines in Paris. He taught at the University of Caen from 1879 to 1881 and was lecturer at the University of Paris from then until 1885, when he became professor there of physical mechanics, mathematical physics (1886), and celestial mechanics (1896).


Poincaré made important original contributions to differential equations, topology, probability, and the theory of functions. He is particularly noted for his development of the so-called Fuchsian functions and his contribution to analytical mechanics. His studies included research into the electromagnetic theory of light and into electricity, fluid mechanics, heat transfer, and thermodynamics. He also anticipated chaos theory. Among Poincaré's more than 30 books are Science and Hypothesis (1903; trans. 1905), The Value of Science (1905; trans. 1907), Science and Method (1908; trans. 1914), and The Foundations of Science (1902-8; trans. 1913). In 1887 Poincaré became a member of the French Academy of Sciences and served at its president in 1906. He also was elected to membership in the French Academy in 1908.


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Poincaré, Henri



b. April 29, 1854, Nancy, Fr.

d. July 17, 1912, Paris




in full JULES-HENRI POINCARÉ, French mathematician, theoretical astronomer, and philosopher of science who influenced cosmogony, relativity, and topology and was a gifted interpreter of science to a wide public.



Poincaré was from a family distinguished by its contributions to government and administration. His first cousin was Raymond Poincaré, president of the French Republic during World War I. Poincaré was ambidextrous and was nearsighted; during his childhood he had poor muscular coordination and was seriously ill for a time with diphtheria. He received special instruction from his gifted mother and excelled in written composition while still in elementary school. Becoming deeply interested in mathematics during adolescence, he attended in 1872-75 the École Polytéchnique in Paris, where he easily won top honours in mathematics, but was undistinguished in physical exercise and in art. Poincaré had an unusually retentive memory for everything he read; moreover, he could visualize what he heard, a useful faculty because he could not clearly see at a distance the mathematical symbols that were on the blackboard. Throughout his life he was able to perform complex mathematical calculations in his head and could quickly write a paper without extensive revisions. He received, in 1879, a doctorate from the École Nationale Supérieure des Mines with a thesis on differential equations.



Following a brief appointment in mathematical analysis at the University of Caen, in 1881 Poincaré joined the University of Paris, where, during the rest of his life, he lectured and wrote prolifically--almost 500 papers--on mechanics and experimental physics, in all branches of pure and applied mathematics, and in theoretical astronomy. Changing his lectures every year, he would review optics, electricity, the equilibrium of fluid masses, the mathematics of electricity, astronomy, thermodynamics, light, and probability. Many of these lectures appeared in print shortly after they were delivered at the university. In his extensive writings on probability, Poincaré anticipated the concept of ergodicity that is basic to statistical mechanics.


Applying his mastery of analysis to the question of the solvability of algebraic equations, Poincaré developed before age 30 the idea of the automorphic function--one that is invariant under a group of transformations that are characterized algebraically by ratios of linear terms. He showed how these functions can be used to integrate linear differential equations with rational algebraic coefficients and how to express the coordinates of any point on an algebraic curve as uniform functions of a single algebraic variable (or parameter). Some of these automorphic functions he called Fuchsian, after the German mathematician Immanuel Lazarus Fuchs, who was one of the founders of the theory of differential equations; he found that they were associated with transformations arising in non-Euclidean geometry. In recognition of his fundamental contributions to mathematics, he was elected, in 1887, to membership in the Académie des Sciences in Paris.


In celestial mechanics, Poincaré made substantial contributions to the theory of orbits, particularly the classical three-body problem (for example, the system involving the Sun, Moon, and Earth). This was part of the "problem of n bodies" (planets, stars, etc.), set for a prize by King Oscar II of Sweden: given the present masses, velocities, motions, and mutual distances of "n bodies," how long will they remain stable in their present spatial relationships, or will their orbits change at some future date? In his solution, Poincaré developed powerful new mathematical techniques, including the theories of asymptotic expansions and integral invariants, and made fundamental discoveries on the behaviour of the integral curves of differential equations near singularities. He was awarded the prize in 1889, even though his solution to the problem was only partially correct; in the same year he was also made a knight of the French Legion of Honour.


Poincaré summarized his new mathematical methods in astronomy in Les Méthodes nouvelles de la mécanique céleste, 3 vol. (1892, 1893, 1899; "The New Methods of Celestial Mechanics"). Another result of this work, his Analysis situs ("Positional Analysis") in 1895, was an early systematic treatment of topology, which deals with properties of a system that endure when metric distortion occurs--that is, topology deals with the qualitative characteristics of spatial configurations that do not vary during cumulative transformations. He also contributed to the theory of numbers by demonstrating how the conception of binary quadratic forms, which was developed by the German mathematician Carl Friedrich Gauss, could be cast in geometric form. In 1904 he lectured at the St. Louis Exposition.


In mathematical analysis, Poincaré made important contributions to the theory of equilibrium of rotating fluid masses. In particular, he described the conditions of stability of the pear-shaped figures that played so prominent a part in the researches of later cosmogony, with reference to the evolution of celestial bodies. He attempted an application of these ideas to the stability of Saturn's rings and to the origin of binary stars. In 1906, in a paper on the dynamics of the electron, he obtained, independently of Albert Einstein, many of the results of the special theory of relativity. The principal difference was that Einstein developed the theory from elementary considerations concerning light signaling, whereas Poincaré's treatment was based on the full theory of electromagnetism and was restricted to phenomena associated with the concept of a universal ether that functioned as the means of transmitting light.


After Poincaré achieved prominence as a mathematician, he turned his superb literary gifts to the challenge of describing for the general public the meaning and importance of science and mathematics. Always deeply interested in the philosophy of science, he wrote La Science et l'hypothèse (1903; Science and Hypothesis), La Valeur de la science (1905; The Value of Science), and Science et méthode (1908; Science and Method), all of which reached a wide public of nonprofessionals. His works were translated into English, German, Hungarian, Japanese, Spanish, and Swedish. He emphasized the subconscious, while probing the psychology of mathematical discovery and invention. He was a forerunner of the modern intuitionist school in that he believed that some mathematical induction is a priori and independent of logic. In his view, sudden illumination, following long subconscious work, is a prelude to mathematical creation. But his greatest contribution to philosophy was to emphasize the role played in scientific method by convention--i.e., by the arbitrary choice of concepts.


Poincaré's prestige and influence increased in French science, and in 1906 he was elected president of the Académie des Sciences and in 1908 to membership in the Académie Française, the highest honour accorded a French writer.




Poincaré conjecture



Poincaré conjecture


This conjecture, formulated by the French mathematician Henri Poincaré, is a famous problem of 20th-century mathematics. It asserts that a simply connected closed three-dimensional manifold is a three-dimensional sphere. The term simply connected means that any closed path can be contracted to a point. William Thurston formulated a program for the classification of three-manifolds that included the Poincaré conjecture in the more general setting of three-manifolds with finite fundamental groups. In another direction, a higher dimensional analog of the Poincaré conjecture states that any closed n-manifold which is homotopy equivalent to the n-sphere must be the n-sphere. When n is 3 this is equivalent to the original formulation above. The higher dimensional conjecture was proved by Stephen Smale (1961) when n is at least 5, and by Michael Freedman (1982) when n is 4. However, the original three-dimensional case has defied all attempts to solve it and remains a cause célèbre in mathematics.