Given a system of arbitrarily many mass points that attract each according to Newton's law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series converges uniformly.
During his childhood he was seriously ill for a time with diphtheria and received special instruction from his mother, Eugénie Launois (1830–1897).
Poincaré was born on 29 April 1854 in Cité Ducale neighborhood, Nancy, Meurthe-et-Moselle into an influential family. His father Leon Poincaré (1828–1892) was a professor of Medicine at the University of Nancy. His younger sister Aline married the spiritual Philosopher Emile Boutroux. Another notable member of Henri's family was his cousin, Raymond Poincaré, a fellow member of the Académie française, who would serve as President of France from 1913 to 1920. Poincaré was raised in the Roman Catholic faith, but later left the religion. He became a freethinker, believing the universe to be sufficient truth and was said to be an atheist.
In 1862, Henri entered the Lycée in Nancy (now renamed the Lycée Henri-Poincaré in his honour, along with Henri Poincaré University, also in Nancy). He spent eleven years at the Lycée and during this time he proved to be one of the top students in every topic he studied. He excelled in written composition. His mathematics Teacher described him as a "monster of mathematics" and he won first prizes in the concours général, a competition between the top pupils from all the Lycées across France. His poorest subjects were music and physical education, where he was described as "average at best". However, poor eyesight and a tendency towards absentmindedness may explain these difficulties. He graduated from the Lycée in 1871 with a bachelor's degree in letters and sciences.
During the Franco-Prussian War of 1870, he served alongside his father in the Ambulance Corps.
The subject is clearly defined by Felix Klein in his "Erlangen Program" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced, as suggested by Johann Benedict Listing, instead of previously used "Analysis situs". Some important concepts were introduced by Enrico Betti and Bernhard Riemann. But the foundation of this science, for a space of any dimension, was created by Poincaré. His first article on this topic appeared in 1894.
Poincaré entered the École Polytechnique in 1873 and graduated in 1875. There he studied mathematics as a student of Charles Hermite, continuing to excel and publishing his first paper (Démonstration nouvelle des propriétés de l'indicatrice d'une surface) in 1874. From November 1875 to June 1878 he studied at the École des Mines, while continuing the study of mathematics in addition to the mining engineering syllabus, and received the degree of ordinary mining Engineer in March 1879.
After receiving his degree, Poincaré began teaching as junior lecturer in mathematics at the University of Caen in Normandy (in December 1879). At the same time he published his first major article concerning the treatment of a class of automorphic functions.
After defending his doctoral thesis on the study of singular points of the system of differential equations, Poincaré wrote a series of memoirs under the title "On curves defined by differential equations" (1881–1882). In these articles, he built a new branch of mathematics, called "qualitative theory of differential equations". Poincaré showed that even if the differential equation can not be solved in terms of known functions, yet from the very form of the equation, a wealth of information about the properties and behavior of the solutions can be found. In particular, Poincaré investigated the nature of the trajectories of the integral curves in the plane, gave a classification of singular points (saddle, focus, center, node), introduced the concept of a limit cycle and the loop index, and showed that the number of limit cycles is always finite, except for some special cases. Poincaré also developed a general theory of integral invariants and solutions of the variational equations. For the finite-difference equations, he created a new direction – the asymptotic analysis of the solutions. He applied all these achievements to study practical problems of mathematical physics and celestial mechanics, and the methods used were the basis of its topological works.
The Problem of finding the general solution to the motion of more than two orbiting bodies in the solar system had eluded mathematicians since Newton's time. This was known originally as the three-body Problem and later the n-body Problem, where n is any number of more than two orbiting bodies. The n-body solution was considered very important and challenging at the close of the 19th century. Indeed, in 1887, in honour of his 60th birthday, Oscar II, King of Sweden, advised by Gösta Mittag-Leffler, established a prize for anyone who could find the solution to the Problem. The announcement was quite specific:
Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the Problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). They introduced the small parameter method, fixed points, integral invariants, variational equations, the convergence of the asymptotic expansions. Generalizing a theory of Bruns (1887), Poincaré showed that the three-body Problem is not integrable. In other words, the general solution of the three-body Problem can not be expressed in terms of algebraic and transcendental functions through unambiguous coordinates and velocities of the bodies. His work in this area was the first major achievement in celestial mechanics since Isaac Newton.
In 1893, Poincaré joined the French Bureau des Longitudes, which engaged him in the synchronisation of time around the world. In 1897 Poincaré backed an unsuccessful proposal for the decimalisation of circular measure, and hence time and longitude. It was this post which led him to consider the question of establishing international time zones and the synchronisation of time between bodies in relative motion. (See work on relativity section below.)
Poincaré's work at the Bureau des Longitudes on establishing international time zones led him to consider how clocks at rest on the Earth, which would be moving at different speeds relative to absolute space (or the "luminiferous aether"), could be synchronised. At the same time Dutch theorist Hendrik Lorentz was developing Maxwell's theory into a theory of the motion of charged particles ("electrons" or "ions"), and their interaction with radiation. In 1895 Lorentz had introduced an auxiliary quantity (without physical interpretation) called "local time" and introduced the hypothesis of length contraction to explain the failure of optical and electrical experiments to detect motion relative to the aether (see Michelson–Morley experiment). Poincaré was a constant interpreter (and sometimes friendly critic) of Lorentz's theory. Poincaré as a Philosopher was interested in the "deeper meaning". Thus he interpreted Lorentz's theory and in so doing he came up with many insights that are now associated with special relativity. In The Measure of Time (1898), Poincaré said, " A little reflection is sufficient to understand that all these affirmations have by themselves no meaning. They can have one only as the result of a convention." He also argued that Scientists have to set the constancy of the speed of light as a postulate to give physical theories the simplest form. Based on these assumptions he discussed in 1900 Lorentz's "wonderful invention" of local time and remarked that it arose when moving clocks are synchronised by exchanging light signals assumed to travel with the same speed in both directions in a moving frame.
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 colleagues.
These monographs include an idea of Poincaré, which later became the base for mathematical "chaos theory" (see, in particular, the Poincaré recurrence theorem) and the general theory of dynamical systems. Poincaré authored important works on astronomy for the equilibrium figures of a gravitating rotating fluid. He introduced the important concept of bifurcation points and proved the existence of equilibrium figures such as the non-ellipsoids, including ring-shaped and pear-shaped figures, and their stability. For this discovery, Poincaré received the Gold Medal of the Royal Astronomical Society (1900).
Henri Poincaré did not receive the Nobel Prize in Physics, but he had influential advocates like Henri Becquerel or committee member Gösta Mittag-Leffler. The nomination archive reveals that Poincaré received a total of 51 nominations between 1904 and 1912, the year of his death. Of the 58 nominations for the 1910 Nobel Prize, 34 named Poincaré. Nominators included Nobel laureates Hendrik Lorentz and Pieter Zeeman (both of 1902), Marie Curie (of 1903), Albert Michelson (of 1907), Gabriel Lippmann (of 1908) and Guglielmo Marconi (of 1909).
In 1905 Henri Poincaré first proposed gravitational waves (ondes gravifiques) emanating from a body and propagating at the speed of light. "Il importait d'examiner cette hypothèse de plus près et en particulier de rechercher quelles modifications elle nous obligerait à apporter aux lois de la gravitation. C'est ce que j'ai cherché à déterminer ; j'ai été d'abord conduit à supposer que la propagation de la gravitation n'est pas instantanée, mais se fait avec la vitesse de la lumière."
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.
Poincaré's mental organisation was not only interesting to Poincaré himself but also to Édouard Toulouse, a Psychologist of the Psychology Laboratory of the School of Higher Studies in Paris. Toulouse wrote a book entitled Henri Poincaré (1910). In it, he discussed Poincaré's regular schedule:
In case the Problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was finally awarded to Poincaré, even though he did not solve the original Problem. One of the judges, the distinguished Karl Weierstrass, said, "This work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics." (The first version of his contribution even contained a serious error; for details see the article by Diacu and the book by Barrow-Green). The version finally printed contained many important ideas which led to the theory of chaos. The Problem as stated originally was finally solved by Karl F. Sundman for n = 3 in 1912 and was generalised to the case of n > 3 bodies by Qiudong Wang in the 1990s.
Einstein's first paper on relativity was published three months after Poincaré's short paper, but before Poincaré's longer version. Einstein relied on the principle of relativity to derive the Lorentz transformations and used a similar clock synchronisation procedure (Einstein synchronisation) to the one that Poincaré (1900) had described, but Einstein's paper was remarkable in that it contained no references at all. Poincaré never acknowledged Einstein's work on special relativity. However, Einstein expressed sympathy with Poincaré's outlook obliquely in a letter to Hans Vaihinger on 3 May 1919, when Einstein considered Vaihinger's general outlook to be close to his own and Poincaré's to be close to Vaihinger's. In public, Einstein acknowledged Poincaré posthumously in the text of a lecture in 1921 called Geometrie und Erfahrung in connection with non-Euclidean geometry, but not in connection with special relativity. A few years before his death, Einstein commented on Poincaré as being one of the pioneers of relativity, saying "Lorentz had already recognised that the transformation named after him is essential for the analysis of Maxwell's equations, and Poincaré deepened this insight still further ...."
Poincaré believed that arithmetic is a synthetic science. He argued that Peano's axioms cannot be proven non-circularly with the principle of induction (Murzi, 1998), therefore concluding that arithmetic is a priori synthetic and not analytic. Poincaré then went on to say that mathematics cannot be deduced from logic since it is not analytic. His views were similar to those of Immanuel Kant (Kolak, 2001, Folina 1992). He strongly opposed Cantorian set theory, objecting to its use of impredicative definitions.
In addition, Toulouse stated that most mathematicians worked from principles already established while Poincaré started from basic principles each time (O'Connor et al., 2002).
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. Poincaré held that convention plays an important role in physics. His view (and some later, more extreme versions of it) came to be known as "conventionalism". Poincaré believed that Newton's first law was not empirical but is a conventional framework assumption for mechanics (Gargani, 2012). He also believed that the geometry of physical space is conventional. He considered examples in which either the geometry of the physical fields or gradients of temperature can be changed, either describing a space as non-Euclidean measured by rigid rulers, or as a Euclidean space where the rulers are expanded or shrunk by a variable heat distribution. However, Poincaré thought that we were so accustomed to Euclidean geometry that we would prefer to change the physical laws to save Euclidean geometry rather than shift to a non-Euclidean physical geometry.
His research in geometry led to the abstract topological definition of homotopy and homology. He also first introduced the basic concepts and invariants of combinatorial topology, such as Betti numbers and the fundamental group. Poincaré proved a formula relating the number of edges, vertices and faces of n-dimensional polyhedron (the Euler–Poincaré theorem) and gave the first precise formulation of the intuitive notion of dimension.
Poincaré's two stages—random combinations followed by selection—became the basis for Daniel Dennett's two-stage model of free will.
