Mathematical physics vs Philosophy: Hegel, Pythagorean triples, Spinors and Clifford Algebras

Abstract

The german philosopher G.W.F. Hegel (1770-1831) in his Phenomenology of Spirit developed a negative conception of mathematics (for him, the pursuit of equality transforms objects into corpses and leaves them in an inert or dismembered state). This conception is however essentially based on a study of the Euclidean demonstration of the Pythagorean theorem which remains superficial. Not only are there many other proofs, but what is at stake in Pythagoras' theorem refers to complex structures unnoticed by Hegel and which he could not know: relationship between quadratic form and square of a linear form, geometric algebra, spinors and rotations in the space, all concepts of great importance in modern physics. But these are also very close, in fact, to what Hegel privileged: dialectical synthesis and movement. Thus, it is mathematics, and especially mathematical physics, which has now something hegelian, and maybe more than (contemporary) philosophy.