On the Facts Underlying Geometry

Abstract

My investigations on spatial intuitions in the field of vision induced me also to start investigations on the question of the origin and essential nature of our general intuitions of space. The question which then forced itself upon me, and one which also obviously belongs to the domain of the exact sciences, was at first only the following: how much of the propositions of geometry has an objectively valid sense? And how much is on the contrary only definition or the consequence of definitions, or depends on the form of description? In my opinion, this question is not to be answered all that simply. For in geometry we deal constantly with ideal structures, whose corporeal portrayal in the actual world is always only an approximation to what the concept demands, and we only decide whether a body is fixed^†, its sides flat and its edges straight, by means of the very propositions whose factual correctness the examination is supposed to show².

Helmholtz’ views have been criticised and augmented by S. Lie**, and portrayed by him with the resources of group theory. It turned out that in order to be able to preserve Helmholtz’ calculations one must express his axioms a little differently (note 15). They also contain dispensable components (note 24).

As Helmholtz, Lie considered motions of spaces (of rigid bodies). The new coordinates of a moving (material) point thereby become functions of the old ones. But Lie had to presuppose that these functions are differentiable. The group-theoretic axiomatic presentation was freed by Hilbert*** from both this and yet another restriction, though with his limiting himself to the planimetric case. He presupposes essentially the following: Two successive continuous transformations of the plane (thus differentiability not demanded) yield another one (the motions form a group). A rotation (motion with a point held fixed) takes a moving point into infinitely many others. To this there is added a further continuity requirement.

Recently, H. Weyl (Raum, Zeit, Materie [‘Space, time, matter’], 4th ed., Berlin, 1921, p. 124) has given group-theoretic axioms for manifolds far more general than Euclidean or Riemannian ones.