Jules Henri Poincaré

Born Apr. 29, 1854, in Nancy; died July 17, 1912, in Paris. French mathematician. Member of the Académie des Sciences (1887).

Poincaré studied in Paris at the Ecole Polytechnique from 1873 to 1875 and at the Ecole Nationale Supérieure des Mines from 1875 to 1879. Beginning in 1886 he was a professor at the University of Paris. He became a member of the Bureau des Longitudes in 1893. Poincaré’s work in mathematics marked the end of the classical period and cleared the way for the development of the new mathematics, wherein not only quantitative relations but facts having a qualitative character are established.

An important group of studies by Poincaré dealt with the theory of differential equations. Poincaré studied the expansion of solutions of differential equations in terms of the initial conditions and small parameters, and he proved that certain series expressing the solutions of partial differential equations are asymptotic. His doctoral dissertation investigated the singular points of a system of differential equations. Poincaré then wrote a series of papers under the general title “On Curves Determined by Differential Equations” (1880). In these studies, he constructed the qualitative theory of differential equations, investigated the shape of integral curves in the plane, provided a classification of singular points, and investigated limit cycles, the distribution of integral curves on the surface of a torus, and certain properties of integral curves in n-dimensional space. He applied his investigations to the three-body problem and studied the periodic solutions of the problem and the asymptotic behavior of the solutions. He introduced methods using small parameters, fixed points, and variational equations and developed a theory of integral invariants.

Poincaré also wrote a number of important papers in celestial mechanics on the stability of motion and on the figures of equilibrium of a rotating fluid mass held together by the gravitation of its particles. His work in celestial mechanics was often characterized by a lack of rigor; for example, he made use of arguments by analogy. A. M. Liapunov was the first to make a rigorous investigation of the problems mentioned.

Poincaré was led by his investigation of ordinary differential equations with algebraic coefficients to the study of new classes of transcendental functions called automorphic functions. He proved the existence of automorphic functions with a given fundamental domain, constructed series for the functions, proved an addition theorem, and showed that algebraic curves could be uniformized. He used Lobachevskian geometry in developing his theory of automorphic functions. For functions of several complex variables his results include the construction of a theory of integrals analogous to the Cauchy integral and the demonstration that an everywhere meromorphic function of two complex variables can be represented as the quotient of two entire functions. These investigations and his work on the qualitative theory of differential equations drew Poincaré’s attention to topology. Poincaré introduced such fundamental concepts of combinatorial topology as Betti numbers and the fundamental group. He proved the Euler-Poincaré formula, which gives the relation between the number of edges, vertices, and faces (of any dimension) of an n-dimensional polytope. Finally, he gave the first intuitive formulation of the general concept of dimension.

In mathematical physics, Poincaré investigated the vibrations of three-dimensional continua. He also studied a number of problems in such areas as thermal conductivity, potential theory, and electromagnetic oscillations. He is the author of papers on the foundations of the Dirichlet principle, for which he developed the “sweeping out,” or balayage, process. Poincaré made a profound comparative analysis of the then current theories of optical and electromagnetic phenomena. In 1905 he wrote the paper “On the Dynamics of the Electron,” which was published the following year. In this paper he developed the mathematical consequences of the “relativity postulate” independently of A. Einstein.

Poincaré’s scientific work in the last decade of his life took place at a time when a revolution had begun in natural science. This fact undoubtedly determined his interest in that period in the philosophy of science and in the methodology of scientific cognition. A brief summary of his philosophical views follows.

The fundamental statements, that is, the principles and laws, of any scientific theory are neither synthetic a priori truths, as I. Kant had maintained, nor models or reflections of objective reality, as they had been for the 18th-century materialists. Rather, they are conventions, and the sole absolute criterion for these conventions is consistency. Which statement we choose from a set of possible statements is, in general, arbitrary if we ignore the circumstances under which the statement is used. In actual fact, however, we are guided by this latter consideration. The selection of fundamental principles or laws is thus limited, on the one hand, by the need of our minds for maximally simple theories and, on the other hand, by the necessity of the theories’ successful use. These requirements permit a certain freedom of choice because of their relative nature. This philosophical doctrine of Poincaré subsequently became known as conventionalism. A critique of Poincaré’s philosophical views was given by V. I. Lenin in Materialism and Empiriocriticism.