This was the man whose name became attached to one of the most famous conjectures in mathematics, only recently solved. It was originally a question Poincaré posed about three-dimensional spaces in which distance is not present. Geometry without distance is called topology, and Poincaré himself was the first to use algebraic methods in studying it. It was a subject dear to his heart because he rejected the idea of absolute distance in physics. Distance is merely what we measure, but how do we measure it, how do we know? He was absolutely right, and in the theory of relativity it turned out that physics admits no distance in the usual sense. It depends on the observer, and lengths appear foreshortened when travelling at great speed, leading to paradoxes like the one about a train on a railway track heading for a gap in the rails. Travelling at close to the speed of light, the train driver sees a shrunken gap that will not disturb the smooth ride, but an observer near the track sees a shrunken train smaller than the gap and heading for disaster. Who is right? Fortunately it's the train driver — the nearby observer has not understood that from the train's point of view the front and back cannot be over the gap at the same time.
Poincaré's research on the ideas underlying relativity theory emerged from his study of the new electromagnetism developed by the Scottish physicist James Clerk Maxwell, on which he became the pioneering French expert.
He had already made a huge mathematical breakthrough on one of the great problems of celestial mechanics: how to find a formula describing the motion of three bodies under their mutual gravitational attraction. Two bodies had been successfully tackled by Newton, and Poincaré's answer proved, rather surprisingly, that it was impossible for three. He showed that although most orbits would be stable, a tiny change, as happens in the real world, could yield unexpected instability. In modern terms this is like the proverbial flap of a butterfly's wings in Siberia creating unexpected changes to the weather in London.
The scope and depth of Poincaré's mathematical ability was surely greater than any physicist who followed him, but was he indeed a physicist or a mathematician? As a young graduate of the Ecole Polytechnique he joined the Corps de Mines, making important reports and recommendations on mining disasters, and remained a member of the corps all his life, rising to ever higher positions. He held academic chairs in astronomy, mathematics, physics, experimental mechanics, and even electrical theory after taking an interest in wireless telegraphy. He was at various times president of the main French scholarly societies in mathematics, physics, and astronomy, and was the man you wanted on important committees. No wonder they brought him in to the Dreyfus case, and he was a leading proponent of the unsuccessful French attempt to decimalise circular measure. Had there been a Nobel Prize in mathematics he would have won it hands down, and he very nearly got the physics prize in 1910, despite opposition from the experimentalists, particularly in Britain.
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