Jules Henri Poincaré

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French mathematician and philosopher of science (1854–1912)

Poincaré was born at Nancy in eastern France and studied at the Ecole Polytechnique and the School of Mines. At first he had intended to become an engineer, but fortunately his mathematical interests prevailed and he took his doctorate in 1879 and then taught at the University of Caen. He was professor at the University of Paris from 1881 until his death.

As Poincaré is commonly referred to as the great universalist – the last mathematician to command the whole of the subject – an account of his work would have to cover the whole of mathematics. In pure mathematics he worked on probability theory, differential equations, the theory of numbers, and in his Analysis situs (1895; Site Analysis) virtually created the subject of topology. He was, however, hostile to the work on the foundations of mathematics carried out by Bertrand Russell and Gottlob Frege. The discovery of contradictions in their systems, disasters to Frege and Russell, was happily welcomed by Poincaré: “I see that their work is not as sterile as I supposed; it breeds contradictions.”

He also deployed the powerful weapons of modern mathematics against a number of problems in mathematical physics and cosmology. In 1887 Oscar II of Sweden offered a prize of 2000 krona for a solution to the question of whether or not the solar system is stable. Will the planets continue indefinitely in their present orbits? Or will some bodies move out of the system altogether, or collide catastrophically with each other? Poincaré published his answer in the monograph Sur les trois corps et les equations de la dynamique (1889; On the Three Bodies and Equations of Kinetics). The title refers to what is now known as the ‘three-body problem’: given three point masses with known initial positions and velocities, to work out their positions and velocities at any future time. The three-body problem had resisted all previous attempts to find a general solution. Poincaré also failed to find an analytical general solution, but he was awarded the prize for making significant advances in the ways of finding approximate solutions.

Poincaré also formulated a famous conjecture which, despite considerable effort and many false alarms, remains unsolved. To a topologist an ordinary sphere is a two-dimensional manifold (a 2-sphere) – two-dimensional because, although it looks like a three-dimensional solid, only its surface is significant. A loop placed on its surface can be shrunk to a point, or, in the language of topology, the 2-sphere is ‘simply connected’. This is seen as a defining property of a sphere. A torus, on the other hand, is not a sphere because not all loops placed upon it can be shrunk to points.

What about an n-sphere, the surface of an n+1-dimensional body? Poincaré's conjecture is that the n-sphere is the only simply connected manifold in higher dimensions, as the 2-sphere is the only simply connected 2-manifold. Stephen Smale proved in 1969 that the conjecture would hold for all dimensions n > 4, and in 1984 Michael Freedman added the case n = 4. The case of n = 3 remains a conjecture.

Poincaré, in such later books as Science and Hypothesis (1905), developed a radical conventionalism. The high-level laws of science, he argued, are conventions, adopted for ease and simplicity and not for ‘truth’. What would happen, he asked, if we found a very large triangle defined by light rays with angles unequal to 180°? As Euclidean geometry is so useful in countless other ways we would more likely sacrifice our physics to preserve our geometry and conclude that light rays do not travel in straight lines.

The French mathematician Jules Henri Poincaré (1854-1912) initiated modern combinatorial topology and made lasting contributions to mathematical analysis, celestial mechanics, and the philosophy of science.

Henri Poincaré was born at Nancy on April 29, 1854. His father was a physician. Henri attended elementary school and the lycée in Nancy and entered the école Polytechnique in Paris at the age of 18. There he demonstrated his brilliance in mathematics and also his phenomenal memory. Although his eyesight was poor, he never took notes in class, and after reading a book he could recall the page on which any statement occurred.

Strangely enough, at this time Poincaré seems not to have fathomed his own mathematical power, for in 1875 he entered the School of Mines with the intention of becoming an engineer. But 3 years later he qualified as a mining engineer and earned his doctorate of mathematical sciences with a thesis based on a difficult problem in differential equations. On the strength of this and other papers, he was appointed professor of mathematical analysis at Caen in 1879. Two years later he obtained a position at the University of Paris, and in 1886 he was made a professor there.

Poincaré's scientific output was prodigious and amazingly comprehensive. The generality and originality of his methods enabled him to master and then break new ground in mathematical physics, celestial mechanics, and nearly every branch of pure mathematics. It was Poincaré's style to emphasize qualitative solutions rather than quantitative recipes. He published several papers on the behavior of solutions and on the properties of integral curves. In 1895 he published Leçons sur le calcul des probabilités (Lessons on the Calculus of Probabilities). He was led to the study of the behavior of divergent and convergent series through his work in celestial mechanics. This led to further investigations of quadratic forms, integral invariants, and double intervals of periodic orbits. He also was actively involved in work on electromagnetic theory.

In 1906 Poincaré published a more general work, La Science et I'hypothèse (Science and Hypothesis), propounding a relativistic philosophy. In philosophy he advocated a variety of pragmatism which he called "conventionalism." "One does not ask, " he said, "whether a scientific theory is true, but only whether it is convenient." He thought that mathematical logic was barren, and when he heard that antinomies had crept into the logistic system of Bertrand Russell and Alfred North Whitehead he could barely conceal his glee. "Logistic is no longer barren, " he wrote, "it engenders antinomies."

Poincaré was elected to the Academy of Sciences in 1887, and he became president of that body in 1906. Two years later he was elected to the literary section of the French Institute, in recognition of his popular works on the philosophy and methods of science, which were widely read in France and translated into six languages. He was appointed a member of the Académie Française in 1909 and elected a foreign member of the Royal Society in 1894. Poincaré died in Paris on July 17, 1912.

Further Reading

Poincaré's views on the philosophy of science are best gained from his own works, especially his Science and Method, translated by Francis Maitland (1915). The best biography of Poincaré in English is in Eric T. Bell, Men of Mathematics (1937). See also volume 2 of Ganesh Prasad, Some Great Mathematicians of the Nineteenth Century: Their Lives and Their Works (2 vols., 1933-1934), and Tobias Dantzig, Henri Poincaré: Critic of Crisis (1954).

Additional Sources

Folina, Janet, Poincaré and the philosophy of mathematics, New York: St. Martin's Press, 1992.

Poincaré, Henri (1854-1912). French mathematician and an influential philosopher of science. Poincaré advanced the doctrine of conventionalism, arguing that the laws of mathematics and physics (e.g. those of Euclidean geometry or Newtonian mechanics) are neither true nor false, being freely created and adopted for their convenience, simplicity, and utility (La Science et l'hypothèse, 1903). A prolific innovator in numerous fields of pure and applied mathematics, he was convinced of the synthesizing and inventive power of intuition in mathematical induction. Although not in agreement with Einstein, he enunciated a theory of the relativity of time and space and, anticipating Heisenberg, of the interaction of the observer and the phenomenon observed.

His cousin Raymond was president of the Republic, 1913-20.

[Rhiannon Goldthorpe]

Poincaré, Jules Henri (1854-1912) French mathematician and philosopher. Poincaré's main philosophical interest lay in the formal and logical character of theories in the physical sciences. He is especially remembered for the discussion of the scientific status of geometry, in La Science et l'hypothèse (1902, trs. as Science and Hypothesis, 1905). The axioms of geometry are not analytic, nor do they state fundamental empirical properties of space. Rather, they are conventions governing the description of space, whose adoption is governed by their utility in furthering the purpose of description. Poincaré's conventionalism about geometry proceeded, however, against the background of a general realism about the objects of scientific enquiry, and he always insisted that there could be good reason for adopting one set of conventions rather than another. In his late Dernières Pensées (1912, trs. as Mathematics and Science: Last Essays, 1963), Poincaré attacks the logicist programme of Frege and Russell, denying that mathematics can be reduced to logic, and arguing that further intuition is always needed to derive the properties of numbers.

(1854–1912). French mathematician and philosopher, born at Nancy. The range of Poincaré's work was so great that the obituary number of the Revue de métaphysique et de morale (September 1913) devoted 130 pages — written by a philosopher, a mathematician, an astronomer, and a physicist — to outline his contributions. Bertrand Russell, in his introduction to the English translation of Poincaré's collected essays Science and Method (1908), contrasts Poincaré's writings on science with most professional philosophers as having 'the freshness of actual experience, of vivid contact with what he is describing', which shows, for example, in his vivid account of mathematical invention. As Poincaré says in his essay 'Mathematical discovery': 'This is a process in which the human mind seems to borrow least from the exterior world, in which it acts, or appears to act, only by itself and on itself, so that by studying the process of geometric thought we may hope to arrive at what is most essential in the human mind.' This leads him to ask how it is that many people do not understand mathematics, as it is founded on logic and principles common to us all. For his own discoveries Poincaré describes 'appearances of sudden illumination, obvious indications of a long course of previous unconscious work', and he gives interesting first-hand examples. He holds that most creative work consists of unconscious selection of possibilities, the selection being guided by subtle rules for rejection which may be worked out during conscious activity. The result is direction and purpose, with a minimum of thinking time wasted on dead ends. Analogies are important for guiding (conscious or unconscious) thinking, but Poincaré holds that the ultimate guide for creative mathematics is a sense of aesthetic elegance. This, however, does have the snag that it can bias thinking towards errors when the truth is less than elegant.

Poincaré was concerned with how sensation and perception relate to physics, or, rather, how physics is derived from our experience. Here his thinking in some ways paralleled that of Einstein, who derived some of his ideas from these deep questions of how knowledge can be gained, and how observations can suggest and test theories. Poincaré stressed the importance of hypotheses in science, and he was content to see truth as the unattainable end of a convergence.

Bibliography
• Poincaré, H. (1905). Science and Hypothesis.
• — —  (1908). Science and Method.

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