Henri Poincaré

French mathematician

• Born: April 29, 1854
• Birthplace: Nancy, France
• Died: July 17, 1912
• Place of death: Paris, France

One of the most important mathematicians of the late nineteenth century, Poincaré developed the theory of automorphic functions, did extensive work in celestial mechanics and mathematical physics, and was a codiscoverer of the special theory of relativity. His writing style was so clear that his books about the philosophy of science were read widely by the general public and translated into many languages.

Early Life

Henri Poincaré (pwahn-kah-ray) was born into one of the most distinguished families of France’s Lorraine region. His father, Leon, was a physician, and one of his cousins, Raymond, became president of the French Republic during World War I. Henri and his sister were adored by their mother, and she devoted herself to their education and rearing. When he was five, Henri contracted diphtheria, and the resulting weakness may have influenced his entire life. Because he was unable to join the other boys in their rough play, Henri was forced to entertain himself with intellectual pursuits. He developed a remarkable memory so that he could even cite page numbers for information in books that he had read many years earlier. In addition, because his eyesight was poor, he learned most of his classwork by listening, because he could not see the blackboard. Thus, he was forced to develop the ability to see spatial relationships in his mind at an early age.

Although he was a good student in his early years, there was no indication of his impending greatness until he was a teenager. He won first prize in a French national competition and in 1873 entered the École Polytechnique, where he exhibited his brilliance in mathematics. Upon his graduation, Poincaré entered the École des Mines in 1875 to study engineering. Although he was a careful student who did his work adequately, Poincaré spent much of his time pursuing mathematics as a recreation. He continued his practice of mathematics during his apprenticeship as a mining engineer.

Poincaré was not an extremely attractive man; he had thinning blond hair, wore glasses, and was short in stature; he was known for being absentminded and clumsy. Nevertheless, he maintained a happy personal life. He married at the age of twenty-seven, fathered four children, whom he adored, and never wanted for friends, because he was by nature humble and interested in other people.

Life’s Work

In 1879, Poincaré submitted the doctoral thesis in mathematics that he had written during his work as an engineer, and he received his degree that same year. The subject was the first of his great achievements: the theory of differential equations. His first appointment was as a lecturer of mathematical analysis at the University of Caen in 1879, and in 1881 he was invited to join the faculty at the University of Paris. He continued this appointment until his death in 1912, although by then his responsibilities had expanded to include mechanics and physics.

During his tenure, he was elected to the Académie des Sciences in 1887 and the Académie Française in 1908. this second appointment is most unusual for a mathematician, for it is given to honor literary achievements and is thus a sure indication of his lucid writing style. He was named president of the Académie des Sciences in 1906. Other awards included a Fellowship in the Royal Society in 1894, the Prix Poncelet, Prix Reynaud, and Prix Bolyai, and gold medals from the Lobachevsky Fund.

Much of Poincaré’s early work was in differential equations, a branch of calculus that is linked directly to the physical world. It was natural, then, for him to turn his attention from pure mathematics to physics and celestial mechanics. However, in his pursuit of solutions of physical and mechanical problems, he often created new tools of pure mathematics.

Poincaré was first drawn to celestial mechanics and astronomical physics by the classical three-body problem, which concerns the gravitational influence and distortions that three independent bodies in space would exert on one another; it held his interest throughout his life. Poincaré published partial results in his early years at the Sorbonne and later published work broadening the number of objects from three to any number. His results won for him a prize that had been offered by King Oscar II of Sweden.

In celestial mechanics, Poincaré was the first person to demand rigor in computations: He found the approximations used commonly at the time to be unacceptable, because they introduced obvious errors into the work. Consequently, more powerful mathematics had to be developed. This work was not centered on any one branch of mathematics but instead included calculus, algebra, number theory, non-Euclidean geometry, and topology. In fact, the field of topology was begun in large part with Poincaré’s study of orbits. He published much of this work in Les Méthodes nouvelles de la mécanique céleste (new methods in celestial mechanics), in three volumes between 1892 and 1899.

Poincaré’s other early achievement was in the theory of automorphic functions—a method for expressing functions in terms of parameters in mathematical analysis. These are functions that remain relatively unchanged though they are acted on by a series of transformations. He found that one class of these, which he called Fuchsian(for German mathematician Immanuel Fuchs), was related to non-Euclidean geometry, and this became an important insight. Indeed, there was some argument over priority in this development between Poincaré and German mathematician Christian Felix Klein ; however, scientific historians agree that Poincaré was the developer of these theories.

It seems that all branches of mathematics held Poincaré’s interest. Poincaré was essential to the development of algebraic geometry . Of particular importance is his development of a parametric representation of functions. For example, the general equation of a circle x² + y² = r² can be rewritten as two equations that describe the variables x and y in terms of some angle A. The equations x = r sine A and y = r cosine A are the equivalent of the original equation since x² + y² = r² sine²A + r² cosine²A = r² (sine²A + cosine²A), which in turn equals r² since sine² + cosine² = 1. Many problems can be solved using parameters that do not yield to any other methods.

Poincaré is equally important in physics . Although Albert Einstein is generally known for his theory of relativity, the special theory of relativity was discovered independently by Poincaré. He and Einstein arrived at the theory from completely different viewpoints, Einstein from light and Poincaré from electromagnetism, at about the same time (Einstein’s first work was published in 1905, and Poincaré’s was published in 1906). There can be no doubt that both men deserve a share of the credit. When Poincaré became aware of Einstein’s work, he was quite enthusiastic and supportive of the Swiss physicist even though most scientists were skeptical. Max Planck, who developed quantum theory, was another physicist who was recognized by Poincaré while he was being scorned by others. In addition, Poincaré developed the mathematics required for countless physical discoveries during the early twentieth century. An example is the wireless telegraph. He also developed the theory of the equilibrium of fluid bodies rotating in space.

Poincaré had a rare gift for a mathematician: He was able to write clearly and to make mathematics and science exciting to people whose educations were directed toward other fields. One of his most widely known works in the philosophy of science, Science et méthode (1908; Science and Method , 1914), is devoted to a study of how scientists and mathematicians create. Poincaré believed that some things in mathematics are known intuitively rather than from observation or from classic logic. His articles and books in the philosophy of science were avidly read and translated into most of the European languages and even into Japanese.

Poincaré continued in relatively good health until 1908, and in 1912 he died of an embolism following minor surgery. The church Saint-Jacques-de-Haut-Pas, the site of his funeral several days later, was filled with eminent persons from all fields who had come to pay a last tribute to his greatness.

Significance

Henri Poincaré was clearly one of the great mathematicians of his time. In fact, some believe that he had no peer. He won virtually every mathematical prize available, and he also won several scientific awards. His work entered every field of mathematics at the time, and he created at least one new branch called algebraic topology. His discoveries inspired other mathematicians for years after his death. In addition, Poincaré did first-rate work in celestial mechanics and was a codiscoverer of the theory of relativity.

The more than thirty books and five hundred papers that Poincaré published are a testament to his prolific career, especially since he died during his productive years. In addition, his writings on the philosophy of science sparked public interest in mathematics and the physical sciences and foreshadowed the intuitionist school of philosophy. These works have helped define the way human beings think about mathematical and scientific creation and will continue to do so for years to come. The practical applications of Poincaré’s work are numerous. Differential functions are the primary mathematics used in engineering and some of the physical sciences; his work in celestial mechanics was completely different from past works and altered the field’s course. In addition, he offered many new ideas in pure mathematics.

Perhaps the most articulate tribute to Poincaré was given in the official report of the 1905 Bolyai Prize written by Gustave Rados: “Henri Poincaré is incontestably the first and most powerful investigator of the present time in the domain of mathematics and mathematical physics.”

Bibliography

Barrow-Greene, June. Poincaré and the Three-Body Problem. Providence, R.I.: American Mathematics Society, 1996. Describes how Poincaré accidentally discovered chaos theory. Aimed at readers with an advanced level of mathematical knowledge.

Bell, E. T. “The Last Universalist.” In Men of Mathematics. New York: Simon & Schuster, 1937. This book is a series of twenty-nine chapters, each introducing a different mathematician from the early Greeks to the early twentieth century. Its account of Poincaré focuses on three areas: the theory of automorphic functions, celestial mechanics and mathematical physics, and the philosophy of science. Biographical information is also included.

Galison, Peter. Einstein’s Clocks, Poincaré’s Maps: Empires of Time. New York: W. W. Norton, 2003. Examines how Poincaré and Einstein created the modern conception of time through their ideas about relativity.

Nordmann, Charles. “Henri Poincaré: His Scientific Work, His Philosophy.” In Annual Report of the Board of Regents of the Smithsonian Institution. Washington, D.C.: Government Printing Office, 1913. Nordmann includes not only a summary of Poincaré’s work and philosophy as the title indicates but also a considerable amount of biographical information.

Poincaré, Henri. The Foundations of Science. Translated by George Bruce Halsted. New York: Science Press, 1913. Contains a preface by Poincaré and an introduction by Josiah Royce. Argues Poincaré’s philosophy of science.

‗‗‗‗‗‗‗. “The Future of Mathematics.” In Annual Report of the Board of Regents of the Smithsonian Institution. Washington, D.C.: Government Printing Office, 1910. This article represents Poincaré at his best. After a brief introduction, he guides the reader through most of the prominent fields of mathematics and predicts what he believed was to come. His explanations are excellent.

‗‗‗‗‗‗‗. Mathematics and Science: Last Essays. Translated by John W. Balduc. Reprint. Mineola, N.Y.: Dover, 1963. Another work in the philosophy of science.

Slosson, Edwin E. “Henri Poincaré.” In Major Prophets of Today. Freeport, N.Y.: Books for Libraries Press, 1968. Slosson chose several representatives from the modern era whom he viewed as having lasting prominence. His article on Poincaré includes biographical information as well as a discussion of Poincaré’s work in mathematics and philosophy.

Zahar, Eli. Poincaré’s Philosophy: From Conventionalism to Phenomenology. Chicago: Open Court, 2001. Traces the development of Poincaré’s philosophy, discussing his thoughts about general science, geometry, mathematics, relativity, and other subjects.