from the assumption that space—an aggregate of points—is a physical entity, an actual object of perception, the properties of which can be examined by physical means, and which can be measured in a way similar to that in which we measure matter or energy, i.e., as a quantity. My objection to this viewpoint is two-fold: In the first place it seems to me that space as conceived by this theory—in fact every conception of space as a quantity consisting of points, and for that reason also the current definition of space of pure geometry—degenerates into a logical circle and is therefore worthless as a definition—through failing to realise that nothing can be a relation and a relatum at the same time. As long as we do not define point independently of space, space defined as an aggregate of points becomes a relation between things (points) which in their turn are only relations between its different parts, i.e., between themselves; pure geometry and many philosophers are clearly conscious of this circle.
Secondly, even if we assume that we can arrive at a definition of point (as a space relatum—that which is spatially related) independently of space, and that we can remove this logical circle, Einstein’s space is, in so far as it remains a continuum of points, open to further objections:
(a) Either this continuum of points will be a thoroughly homogeneous one, and then it cannot be applied for the explanation of gravitation in Einstein’s way; or,
(b) It will be heterogeneous and then it becomes impossible to distinguish space from its “contents,” i.e., from matter and perceptions; space then becomes a superfluous higher structure of Experience, because the only property by which, if defined as a continuum of points, it can be distinguished from matter, is its homogeneity. The space-time of Einstein is a kind of medium, which by its heterogeneity acquires the properties of matter, becoming superfluous if we wish to describe the phenomena in terms of matter, and useless, if we wish to describe them in terms of something else.
45. Assuming that we have defined “point” independently of space (i.e., so that the definition of space is not presupposed in that of point) we have two possibilities open to us: either space is an aggregate of such points, or a relation between them. The followers of the sensationalist theory of space maintain that it is necessary to postulate points as the constituents of empty space: as those parts of space in which there is no matter. In dealing with this objection great caution is necessary: in substance it is a question whether empty space can be subject to physical, sense experience. Here, I think, the sensationalist theory of space
