Purely Infinitesimal Geometry (Excerpt)

Abstract

The real world, into which we have been placed by virtue of our consciousness, is not there simply and all at once, but is happening; it passes, annihilated and newly born at each instant, a continuous one-dimensional succession of states in time. The arena of this temporal happening is a three-dimensional Euclidean space. Its properties are investigated by geometry, the task of physics by comparison is to conceptually comprehend the real that exists in space and to fathom the laws persisting in its fleeting appearances. Therefore, physics is a science which has geometry as its foundation; the concepts however, through which it represents reality---matter, electricity, force, energy, electromagnetic field, gravitational field, etc.---belong to an entirely different sphere than the geometrical. This old view concerning the relation between the form and the content of reality, between geometry and physics, has been overturned by Einstein's theory of relativ-ity.1The special theory of relativity led to the insight that space and time are fused into an indissoluble whole which shall here be called the world; the world, according to this theory, is a four-dimensional Euclidean manifold---Euclidean with the modification that the underlying quadratic form of the world metric is not positive definite but is of inertial index 1. The general theory of relativity, in accordance with the spirit of modern physics of local action [Nahewirkungsphysik], admits that as valid only in the infinitely small, hence for the world metric it makes use of the more general concept of a metric [Maβbestimmung] based on a quadratic differential form, developed by Riemann in his habilitation lecture. | But what is new in principle in this is the insight that the metric is not a property of the world in itself, rather, spacetime as the form of appearances is a completely formless four-dimensional continuum in the sense of analysis situs.