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.