2013 June 3
EDITOR'S NOTE: This week I will be posting an illustrated and (occasionally) annotated version of Henri Poincaré's 1906 essay The Relativity of Space. The title is misleading for a modern reader. Most visitors to this site are probably aware that Poincaré came very close to discovering special relativity before Einstein (and that a few people, among them Poincaré himself, have claimed that he did). Relativity in the Einsteinian sense, however, enters only peripherally into this essay; indeed, one reason for reading it is to understand better just what the word "relativity" meant to a physicist in the early 1900s.
The essay is of much more than purely historical significance, however. Poincaré -- whose literary skills were on a par with his mathematical insights -- here presents to a general audience (in fact, to psychologists!) many principal elements of his thought. To Poincaré, as to many a philosopher in the present century, all of science ultimately depends on neurology. This idea was rooted in Nineteenth Century soil -- the belief that sense impressions are the basis of all knowledge on the one hand, the claim that consciousness is a physiological phenomenon on the other. Poincaré however viewed the nervous system abstractly -- as what would later be called a "neural network", -- and readers familiar with artificial intelligence will immediately recognise many of the points he is making.
Henri Poincaré was a Colossus of Rhodes with one foot in the Nineteenth Century, the other in the Twenty-first. The intervening Twentieth, however, was never quite sure what to make of him. The Nineteenth Century was an age for grand systems of materialistic philosophy, alternatives to religion which united science with art and mind with nature; Poincareé, with his elegant prose-style and immense intellectual range, was among the last architects of these vast structures which narrow specialists in the Twentieth Century would so abhor. The Twenty-first Century, to judge from its early decades, will be the age when chaos, complexity, and non-linearity blur the border between quantitative and qualitative sciences; Poincaré was the prophet who announced that this was coming, and more. While the Twentieth Century never ignored him, it shared few of his interests. The area of research for which he was perhaps most noted in his lifetime -- celestial mechanics -- was thought in the 1900s by all save the specialists to be an essentially closed field. His approach to relativity was old-fashioned and conservative, remembered only because of the priority dispute with Einstein. He worked on quantum mechanics, but played no significant role in the revolution which defined Twentieth Century physics. He rejected set theory, Bourbakian rigour, Anglo-Saxon analytic philosophy -- the very foundations of Twentieth Century mathematical thought. The qualitative approach to differential equations which he invented was widely used, but with reluctance -- a second-best to actually solving them. His famous conjecture about the 3-sphere went unproved until, appropriately, 2003.
Today, however, Henri Poincaré is once again our contemporary. Whether one agrees with his ideas or considers them to be (ironically, in light of his critique of "vicious circles" in number theory) ultimately circular, one cannot deny their modern resonance.