Jules Henri Poincaré influenced cosmogony, relativity, and topology and was a gifted interpreter of science to a wide public. Poincaré is the most important figure in the theory of differential equations and the mathematician who, after Newton, did the most remarkable work in celestial mechanics. When Gauss died in 1855, it was generally thought that there never again would be a universalist in mathematics—one who is at home in all branches, pure and applied. If anyone has since proved this view wrong, it is Poincaré, for he took all mathematics as his province.(1)

 

Bio

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In 1873, he entered the Ecole Polytechnique at the top of his class. His professor at the school is said to have referred to him as a “monster of mathematics.” After graduation, he followed courses in engineering at the Ecole des Mines and worked briefly as an engineer while writing his thesis for a doctorate in mathematics, which he obtained in 1879. Shortly afterward, he started teaching at the University of Caen, and in 1881 he became a professor at the University of Paris, where he taught and wrote prolifically until his untimely death in 1912. He wrote almost 500 papers—on mechanics and experimental physics, in all branches of pure and applied mathematics, and in theoretical astronomy. At the early age of thirty-three, he was elected to the Academie des Sciences, and in 1908 to the Academie Française, the highest honor accorded a French writer. The literary award was for books he wrote for the general public describing the meaning and importance of science and mathematics. He was also the recipient of innumerable prizes and honors both in France and abroad.

 

Mathematics–Algebra, Topology

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Before he was thirty years old, Poincaré became world famous with his epoch-making discovery of the “automorphic functions” of one complex variable (or, as he called them, the “fuchsian” and “kleinean” functions). Until the discovery of the higher homotopy groups in 1933, the development of algebraic topology was entirely based on Poincaré’s ideas and techniques.

 

Topology was not the invention of any one man.(2) Some topological problems are found in the work of Euler, Mobius, and Cantor, and even the word “topology” had been used in 1847 by J. B. Listing (1808–1882) in the title of a book, Vorstudien zur Topologie (Introductory Studies in Topology). But as a date for the beginning of the subject, none is more appropriate than 1895, the year in which Poincaré published his Analysis situs.(3) This book for the first time provided a systematic development. Topology is now a broad and fundamental branch of mathematics, with many aspects. It can, however, be subdivided into two fairly distinct subbranches: combinatorial topology and point-set topology. Poincaré had little enthusiasm for the latter. Combinatorial topology, or analysis situs, as it was then generally called, is the study of intrinsic qualitative aspects of spatial configurations that remain invariant under continuous one-to-one transformations. It is often referred to popularly as “rubber-sheet geometry,” for deformations of, say, a balloon, without puncturing or tearing it, are examples of topological transformations. A circle, for instance, is topologically equivalent to an ellipse. The dimensionality of a space is a topological invariant.

 

Special Theory of Relativity

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Poincaré became an expert in practically all parts of theoretical physics, and published more than seventy papers and books on the most varied subjects, with a predilection for the theories of light and of electromagnetic waves. On two occasions, he played an important part in the development of the new ideas and discoveries that revolutionized physics at the end of the nineteenth century. His remark on the possible connection between X-rays and the phenomena of phosphorescence was the starting point of H. Becquerel’s experiments that led him to the discovery of radioactivity. Secondly, Poincaré was active in the discussions concerning Lorentz’s theory of the electron from 1899 on. Poincaré was the first to observe that the Lorentz transformations form a group; and many physicists consider that Poincaré shares with Lorentz and Einstein the credit for the invention of the special theory of relativity.

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Popular Works on Science and Mathematics

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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. Poincaré’s popular works include Science and Hypothesis (1901), The Value of Science (1905), and Science and Method (1908). A quote from these writings is particularly relevant to this biographical history section. In 1908 he wrote, “The true method of foreseeing the future of mathematics is to study its history and its actual state.”

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Footnotes:

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(1) Carl B. Boyer, A History of Mathematics, 2nd Edition (New York, 1991), p. 600.

(2) Ibid., p. 603.

(3) Ibid., p. 603.