Poincarae's Philosophy: From Conventionalism to Phenomenology

By Elie Zahar | Go to book overview

Introduction

Speaking about Kant, Schopenhauer rightly claimed that it is far easier to criticize a genius than to praise him: his errors, always finite in number, can be easily circumscribed while his contribution to truth remains both inexhaustible and unfathomable (Schopenhauer 1958, Vol.1, Appendix). If I say this, it is by way of an excuse for the criticisms which will be levelled at some paradoxical aspects of Henri Poincaré's philosophy. However, my aim, which is a constructive one, is that of clarifying and of then reconciling Poincaré's various theses about the foundations of mathematics and of the natural sciences.

With Hilbert, Poincaré was the greatest mathematician of his time and one of the greatest ever. In E.T. Bell's words, he was the last universalist in that he looked upon every branch of mathematics and of theoretical physics as part of his domain (Bell 1937, Chapter 28). Poincaré had a prodigious knowledge of the field of complex analysis on which he left an indelible mark; he founded the discipline of analysis situs or topology; he studied the theory of groups, the probability calculus, classical mechanics, optics, electromagnetism, and astronomy; he attacked the three-body problem and addressed the question of planetary stability, thereby creating chaos theory. In this context, he established a result which proved essential to the second law of thermodynamics. Together with Lorentz and simultaneously with Einstein, he discovered Special Relativity. We owe him the Principle of Lorentz-covariance and one of the first applications of a symmetry requirement to physical laws; also the concept of an integrated four-dimensional physical continuum which was later attributed exclusively—and hence wrongly—to Minkowski. He determined the relativistic transformation equations for the electromagnetic field and for the charge-density, thereby correcting a serious mistake made by Lorentz in his 1904 paper. Poincaré finally developed a Lorentz-covariant gravitational theory which yielded Newton's law as a limiting case for small velocities. It is worth adding that he achieved these breakthroughs concur-

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