Note 75 — Introductory material

After a general introduction, and a short history of the development of Metageometry, Erdmann proceeds, in his second chapter, to discuss what are the axioms of Euclidean Geometry. The arithmetical axioms, as they are called, he leaves aside, as applying to magnitude in general; what we want here, he says, is a definition of space, for which the geometrical axioms are alone relevant. But a definition of space, he says—following Riemann—demands a genus of which space shall be a species, and this, since our space is psychologically unique, can only be furnished by analytical mathematics (p. 36). Now the space-forms dealt with by Geometry are magnitudes, and conceptions of magnitude are everywhere applied in Geometry. But before Riemann, only particular determinations of space could be exhibited as magnitudes, and thus the desired definition was impossible to obtain. Now, however, we can subsume space as a whole under a general conception of magnitude, and thus obtain, besides the space-intuition and the space-conception, a third form, namely, the conception of space as a magnitude (Grössenbegriff vom Raum, pp. 38–39). The definition of this will give us the complete, but not redundant, system of axioms, which could not be obtained by transforming the general intuition of space into the space-conception, for want of a plurality of instances (p. 40).