Under the microscope: Jules-Henri Poincaré, who lived from 1854 to 1912, was France's greatest mathematician, writes Prof William Reville.
He made fundamental contributions in several areas of mathematics, initiated new branches and sowed seeds that later blossomed into new branches, such as chaos theory, and foreshadowed Einstein's theory of relativity. He also wrote popular books on science and had definite ideas on how new ideas in mathematics are produced.
Jules-Henri was born in Nancy. His father, Léon, was professor of medicine at the University of Nancy, and his mother, Eugénie Launois, gave him special tuition. Jules-Henri's first cousin Raymond Poincaré was prime minister of France several times and president during the first World War.
Jules-Henri was near-sighted and had poor muscle co-ordination. He did not perform well at school in physical activities or music, but in most other areas he was a top student, and he was described by his maths teacher as a "monster" at the subject.
He entered the École Polytechnique in 1873 and graduated in 1875. He was ahead of all the other students in mathematics. He read widely and developed a remarkable memory by linking ideas visually. Poincaré went on to study at the École des Mines, graduating in 1879. He worked for a while in the mining industry, completing his PhD research during this time. He was appointed to a chair in mathematical physics and probability at the Sorbonne in 1886.
Poincaré had definite ideas about the nature of scientific and mathematical research, perhaps best summarised in his statement: "It is through logic that we prove, but through intuition that we discover." And again: "Logic therefore remains barren unless fertilised by intuition."
Poincaré also wrote: "The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. If nature were not beautiful it would not be worth knowing, and life would not be worth living."
Toulouse, director of the psychology laboratory at the School of Higher Studies in Paris, wrote a book in which he described Poincaré's way of thinking and working. Poincaré's method of working has been likened to a bee flying from flower to flower. He never spent long sessions working on a particular problem but would move from one problem to another, believing his subconscious would continue to work on the first while he consciously worked on the second. Indeed, he expected important ideas to come to him when he stopped working on a problem.
Poincaré had a wide knowledge of mathematics, physics and philosophy, which allowed him to attack problems from several angles. He made important contributions not only in mathematics but also in celestial mechanics, fluid mechanics and the special theory of relativity. Alas, I do not know enough mathematics to do more than list some of Poincaré's major contributions. They include developing the concept of automorphic functions, originating the field of algebraic topology and helping to develop number theory.
Poincaré's best-known work on celestial mechanics is on the three-body problem. In 1887 Oscar II, king of Sweden and Norway, held a mathematical competition to celebrate his 60th birthday. Poincaré submitted a paper on the three-body problem. The problem looks simple: how do you determine the motion of three bodies in space, interacting with each other through gravitational attraction, if you know their starting points? The solution is extremely sophisticated mathematically and defeated the greatest scientific minds of every age, even Isaac Newton.
Poincaré won the competition, but when the solution was sent to the journal Acta Mathematica a mistake was discovered. When the error was investigated Poincaré saw that in one case one of the three bodies moves in a seemingly random manner. It is not really random, because the motion can be described by mathematical equations, but it looks random. This phenomenon was later developed as chaos theory.
A chaotic system is one whose component parts obey simple rules but whose behaviour as a whole is so complex and irregular that it appears to be random unless you know a huge amount of hidden information about what it is doing. This information content is so huge that it is impractical to know it all. Exact prediction of a chaotic system is therefore very difficult, if not impossible, even though it is driven by simple rules. The weather is a familiar example of a chaotic system. Poincaré was the first to describe chaotic behaviour.
Poincaré also went a long way towards developing the special theory of relativity at the same time as Einstein. Einstein is usually given credit for the theory, however, and properly so, as he put his reputation on the line and unambiguously claimed objects get shorter when they move faster.
Poincaré was notoriously absent-minded. One day he had a visitor; Poincaré was immersed in his work and asked his guest to wait. He then forgot about his visitor for over an hour. Then, suddenly remembering him, he rushed out and said: "Go away!"
William Reville is associate professor of biochemistry and director of microscopy at University College Cork