Hermann Weyl on Minkowskian space-time

Yvon Gauthier
University of Montreal

Abstract

If we suppose on one side that Minkowski had derived his world picture , what we call now Minkowki diagramms, from his work on number grids " Zahlengitter " in the geometry of numbers - which is his main achievement in number theory - there is no question of the mathematical nature of Minkowskian space-time. On the other side, if we insist on his postulate of the absolute world " Postulat der absoluten Welt " as it is termed in his celebrated lecture " Raum und Zeit " of 1908 (see 2, pp. 432-444), the Minkowskian world picture has physical significance, for Minkowski opposes the postulate of relativity (which Einstein wanted to call invariance), since, as he says, the meaning of the postulate is that phenomena are given in an absolute four-dimensional world of space and time. It is this world pstulate that Hermann Weyl will want to exploit in his own work on General Relativity (cf. 3).

Although Weyl uses freely the Minkowskian vocabulary of world, world-lines and world-points - even the Minkowskian " Substanz " to designate matter - he is more concerned with the metric structure of the world and its Riemannian geometry, which is curiously absent from Minkowski. It seems that Weyl refers to Minkowskian space-time as a convenient tool from Special Relativity Theory and as a philosophical aid to his own world picture which remains essentially Kantian (see 5).

Weyl pays due tribute to Minkowski on various occasions - see especially his major work Raum, Zeit, Materie, chap.21 ( cf. 4) but I want to put the emphasis on Weyl's original contribution to the Riemannian differential geometry of the four-dimensional manifold as different from Minkowski's coordinate representation of space-time. I shall end up with some philosophical remarks on the constructivist character of Weyl's endeavour in mathematics and physics (see 1).

REFERENCES

1. Gauthier, Y. Internal Logic. Foundations of Mathematics from Kronecker to Hilbert, Kluwer , Synthese Library , Dordrecht/Boston/London, 2002.
2. Minkowski, H. Gesammelte Abhandlungen, hrsg. v. D. Hilbert, Chelsea, New York, 1967.
3. Weyl, H. Gesammelte Abhandlungen, hrsg. v. K. Chandrasekharan, Springer-Verlag, Berlin, Heidelberg, New York, 1968.
4. Weyl, H. Space, Time, Matter, trans. by H.L. Brose, Dover Publications, 1950.
5. Weyl, H. Philosophy of Mathematics and Natural Science, Atheneum, New York, 1963.