Poincaré's Ding-an-Sich

Princeton's Graduate College
When I was a graduate student (1958-61), one of the philosophy majors came to me and told me what quantum mechanics is. I was not happy with his attitude and asked him whether he knows how to add waves. I asked specifically what happens when we add two sound waves with different frequencies. He then told me that this is a mathematical question, and that it does not carry its own substance. I thought he was hopeless and told him to go away.

If Bertrand Russell (1872-1970) of England had said the same thing to me about mathematics, I could not have told him to go away. Russel was one of the great thinkers of the past century and I like his history books. Russell told Herni Poincaré (1854-1912) that mathematics is only a tool of logic and nothing else, but Poincaré repeatedly disagreed.

Did Poincaré clearly define what mathematics is to Russell? We do not know, but we can try to figure out what Poincaré had in mind. We can do this not necessarily because we are smarter than he was, but because we are equipped with tons of progresses in physics made since Poincaré's death in 1912.

Let us go back to physics. Speaking of the role of mathematics in physics, Poincaré started with the three-by-three rotation matrix applicable to the three-dimensional space. He expanded it to four-by-four by augmenting the time variable. He then came up with five-by-five matrices in order to take into account the space-time translational variables. This is the way in which he completed the formulation of the Poincaré group.

Poincaré and Einstein

Poincaré Group after Einstein

When Einstein formulated his special relativity in 1905 and derived his energy-momentum relation, he did not take into consideration of the fact that particles have internal space-time symmetries. Then how can we define these symmetries?

Poincaré's Geometry and Topology

Two and Three


Other Poincaré Papers

copyright@2013,2015 by Y. S. Kim, unless otherwise specified.
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Why is he with Einstein?
The photo of the Poincaré sphere on this webpage came from Christian Brosseau's book entitled "Fundamentals of Polarized Light, A Statistical Optics Approach" (Wiley, New York, 1998). I am grateful to Professor Brosseau for sending me a copy of this book.