by David Jordan

       Poincaré has been called the last mathematical universalist, and he certainly earns the title. He made his reputation with his thesis connecting the theory of automorphic functions to classical results in hyperbolic geometry. He was the first mathematician to attack the realm of differential equations and celestial mechanics by seeking what qualitative properties could be gleaned from a system even when it was non-integrable. He is considered the founder of algebraic topology, having invented (discovered? divined?) the fundamental group to aid in his work classifying 2-manifolds. One of his conjectures, involving a similar such classification for 3-manifolds has baffled mathematicians for the century following his death (in 1912), though quite recently a possible solution has arisen. Poincaré is further credited beside Einstein and Lorentz as a co-founder of the theory of special relativity. Perhaps most importantly his work in differential equations and the 3-body problem foretold the presently very active study of chaotic dynamical systems, and indeed it was Poincaré who provided the first mathematical definition of chaos.

       Henri was one of those rare people about whom it cannot be doubted that he lived a charmed life. He was born in 1854 to a Professor of Medicine, Léon, and a gifted teacher, Eugénie, who later became his tutor during a childhood bout with diphtheria. (Geez man, it seems like you gotta have a childhood illness to be a math genius these days! Stupid healthy immune system...) Henri's family was rather influential: his cousin would later become the Prime Minister of France, and would preside over the nation during World War I. Henri pretty much excelled at everything during high school (oh by the way the school he went to: its been renamed after him. How many people can say that, eh?)  Poincaré has been compared with Gauss in terms of mental acuity and originality, but fortunately for us, unlike Gauss Poincaré published prolifically: nearly 500 papers in his lifetime.
       Aside from his academic publications, Poincaré was an esteemed popularizer of science at a time when popular science authors were very rare. Poincaré's interests in physics and mathematics had two life-long counterparts: the studies of psychology, and of the philosophy of science. Intriguingly his contributions to the fields of psychology consisted primarily of consenting to being monitored by an analyst named Toulouse while he went about his day, and also publishing memoirs wherein he described the creative process leading up to his own major mathematical discoveries. Again it is fascinating to contrast this extroverted behavior to that of Gauss, notoriously reserved and fiercely protective of his inner world. In the field of philosophy of science, Poincaré opened up a can of worms when he observed that the Peano axioms had not been shown to be logically consistent and that indeed any proof that, say, the integers satisfied these axioms would have to rely upon some form of induction which was itself a Peano axiom. He concluded quite firmly that mathematics simply could not be reduced to pure logic, that it depended upon intuition, and that the end goal of formal rigor was to purify, enhance, and extend intuition, and not to replace it. This was a rather controversial viewpoint in the era of David Hilbert and his widely accepted campaign of formalization.