Chaos and Cosmos

Abstract

Felix Klein [1] once said that the concept of group is perhaps the most characteristic concept of 19^th century mathematics. However, in his Erlangen Program (1872), an extraordinarily effective advertisement for the group concept, Klein does not make any attempt to “sell” it to the natural scientists. At that time he was probably too absorbed with geometry to think of other applications, but already a few years later his friend and one-time collaborator, Sophus Lie, in the preface to the third part of his celebrated “Theorie der Transformationsgruppen” wrote “The principles of mechanics have a group theoretic origin”. Soon afterwards, in 1904, this was explicitly proved by G. Hamel [2]. Little by little under the influence of Klein and of his school it began to be recognized that invariance under the group of the automorphisms of space-time was the true criterion of objectivity for the physical laws. Already in 1910 Klein [3] explicitly stated that relativity was synonymous with invariance under a group and Klein’s school systematically investigated the physical consequences of invariance under both finite and infinite dimensional Lie groups [4].