Essay

Poincaré, Henri (1854-1912)
French mathematician and scientist

Preliminary version of an article in Europe 1789-1914: Encyclopedia of the Age of Industry and Empire, 5 vols., John Merriman and Jay Winter, eds., Vol. 4, pp. 1804-1805, New York, Charles Scribner's Sons, 2006

Scott Walter 1

The son of a professor on the faculty of medicine in Nancy, France, Jules-Henri Poincaré (29 April 1854-17 July 1912) was a member of an influential family that included his cousin Raymond Poincaré (1860-1934), president of France during World War I, and his brother-in-law Étienne-Émile-Marie Boutroux (1845-1921), professor of philosophy at the Sorbonne. A graduate of the École Polytechnique, Poincaré quickly established his scientific credentials in the theory of functions and the qualitative theory of differential equations. His discovery in 1880 of the automorphic functions of one complex variable, which he used to solve second-order linear differential equations with algebraic coefficients, was widely hailed as a work of genius.
Starting in 1881 Poincaré taught mathematics at the University of Paris, becoming professor of mathematical physics in 1886 and a member of the Academy of Science in 1887. Two years later he was awarded the Grand Prize of Oscar II, king of Sweden (r. 1872-1907), for his study of a thorny question in celestial mechanics known as the "three-body problem": how do three masses behave under the influence of gravitation? A milestone in the history of both celestial mechanics and dynamics, Poincaré's prize paper contains the proof of Poincaré's Recurrence Theorem, which states (roughly) that a closed mechanical system with finite energy (like that of three bodies gravitating in empty space according to Newton's law) will return periodically to a state very close to its initial state. It also contains the first mathematical description of what is now known as chaotic motion.
Both in this work in dynamics and others in group theory, multiple integrals, and the theory of functions, Poincaré would let the problem conditions vary continuously, and observe what happens, a method that leads directly to questions of topology. This branch of mathematics is concerned with properties of figures that are invariant under homeomorphisms (or bicontinuous one-to-one transformations). In algebraic topology, one such topological invariant is the Euler characteristic, which for a convex polyhedron is given by Euler's formula: sum the numbers of faces and vertices, subtract the number of edges, and the result is always the same, F-E+V=2. Starting in 1895, Poincaré laid the foundations of algebraic topology (then called analysis situs), defining "Betti numbers," and inventing a number of tools, which he used to generalize Euler's theorem for polyhedrons.
From the late 1880s Poincaré engaged with James Clerk Maxwell's theory of electrodynamics, and helped introduce this theory to Continental readers. In 1896 he relinquished his chair in mathematical physics for another in mathematical astronomy and celestial mechanics, but maintained a lively interest in the newly discovered phenomena of x-rays, gamma-rays, and electrons. Most notably, Poincaré pointed out in 1900 that in order for the principle of relative motion to hold (i.e., the principle according to which physical phenomena are insensible to uniform rectilinear motion), it was necessary to refer time measurements not to the "true time" of an observer at rest with respect to a universal, motionless carrier of electromagnetic waves known as the ether, but to a "local time" devised by the Dutch physicist Hendrik Antoon Lorentz (1853-1928) as a mathematical shortcut. For Poincaré, local time was the time read by the light-synchronized clocks of observers in common motion with respect to the ether, corrected by the light signal's time of flight, but ignoring the effect of motion on light propagation.
There was more to this exchange of light signals than the synchronization of clocks. The simultaneity of two events is not determined by objective considerations, Poincaré observed in 1898, but is a matter of definition. Measurements of distance suffer the same underdetermination, such that there is no true geometry of physical space in Poincaré's view. According to Poincaré's conventionalist philosophy, scientists are often confronted with open-ended situations requiring a choice between alternative definitions of their objects of study. In virtue of this freedom of choice, which marks the linguistic turn in philosophy of science, Poincaré was often thought to be upholding a variety of nominalism, an error he denounced with vigor. The choice scientists have to make, Poincaré explained, is not entirely free, as scientists are guided by experimental facts. In line with this understanding of scientific activity, Poincaré deplored the logicist program of Bertrand Arthur William Russell (1872-1970), which sought an axiomatic foundation for mathematics.
Conventionalism gained greater recognition upon publication of La science et l'hypothèse (1902; Science and Hypothesis), whose readers included the young Albert Einstein (1879-1955). In the summer of 1905 Poincaré and Einstein independently proposed what was to be known as the special theory of relativity, and are generally considered to be the theory's cofounders (along with Lorentz), although the question of paternity continues to spark debate.
Poincaré was a phenomenally productive scientist, with more than five hundred scientific papers and twenty-five volumes of lectures to his name, spanning the major branches of mathematics, mathematical physics, celestial mechanics, astronomy, and philosophy of science. By 1900 he was widely acknowledged to be the world's foremost mathematician.

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Footnotes:

1 Henri-Poincaré Archives (CNRS), and Department of Philosophy, University of Nancy.