Note 62 — Introductory material

We must entirely dissent, therefore, from the disjunction which underlies Riemann's philosophy of space. Either the axioms must be consequences of general conceptions of magnitude, he thinks, or else they can only be proved by experience (p. 255). Whatever can be derived from general conceptions of magnitude, we may retort, cannot be an à priori adjective of space: for all the necessary adjectives of space are presupposed in any judgment of spatial quantity, and cannot, therefore, be consequences of such a judgment. Riemann's disjunction, accordingly, since one of its alternatives is obviously impossible, really begs the question. In formulating the axioms of metrical Geometry, our question should be: What axioms, i.e. what adjectives of space, must be presupposed, in order that quantitative comparison of the parts of space may be possible at all? And only when we have determined these conditions, which are à priori necessary to any quantitative science of space, does the second question arise: what inferences can we draw, as to space, from the observed results of this quantitative science, i.e. of this measurement of spatial figures? The conditions of measurement themselves, though not results of any conception of magnitude, will be à priori, if it can be shown that, without them, experience of externality would be impossible.