Note 196 — Introductory material

(1) Points. The antinomy of the point—which arises wherever a continuum is given, and elements have to be sought in it—is fundamental to Geometry. It has been given, perhaps unintentionally, by Veronese as the first axiom, in the form: "There are different points. All points are identical" (op. cit. p. 226). We saw, in discussing projective Geometry, that straight lines and planes must be regarded, on the one hand as relations between points, and on the other hand as made up of points[190]. We saw again, in dealing with measurement, how space must be regarded as infinitely divisible, and yet as mere relativity. But what is divisible and consists of parts, as space does, must lead at last, by continued analysis, to a simple and unanalyzable part, as the unit of differentiation. For whatever can be divided, and has parts, possesses some thinghood, and must, therefore, contain two ultimate units, the whole namely, and the smallest element possessing thinghood. But in space this is notoriously not the case. After hypostatizing space, as Geometry is compelled to do, the mind imperatively demands elements, and insists on having them, whether possible or not. Of this demand, all the geometrical applications of the infinitesimal calculus are evidence[191]. But what sort of elements do we thus obtain? Analysis, being unable to find any earlier halting-place, finds its elements in points, that is, in zero quanta of space. Such a conception is a palpable contradiction, only rendered tolerable by its necessity and familiarity. A point must be spatial, otherwise it would not fulfil the function of a spatial element; but again it must contain no space, for any finite extension is capable of further analysis. Points can never be given in intuition, which has no concern with the infinitesimal: they are a purely conceptual construction, arising out of the need of terms between which spatial relations can hold. If space be more than relativity, spatial relations must involve spatial relata; but no relata appear, until we have analyzed our spatial data down to nothing. The contradictory notion of the point, as a thing in space without spatial magnitude, is the only outcome of our search for spatial relata. This reductio ad absurdum surely suffices, by itself, to prove the essential relativity of space.