Note 179 — Introductory material

We have now seen in what the à priori element of Geometry consists. This à priori element may be defined as the axioms common to Euclidean and non-Euclidean spaces, as the axioms deducible from the conception of a form of externality, or—in metrical Geometry—as the axioms required for the possibility of measurement. It remains to discuss, in a final chapter, some questions of a more general philosophic nature, in which we shall have to desert the firm ground of mathematics and enter on speculations which I put forward very tentatively, and with little faith in their ultimate validity. The chief questions for this final chapter will be two: (1) How is such à priori and purely logical necessity possible, as applied to an actually given subject-matter like space? (2) How can we remove the contradictions which have haunted us in this chapter, arising out of the relativity, infinite divisibility, and unbounded extension of space? These two questions are forced upon us by the present chapter, but as they open some of the fundamental problems of philosophy, it would be rash to expect a conclusive or wholly satisfactory answer. A few hints and suggestions may be hoped for, but a complete solution could only be obtained from a complete philosophy, of which the prospects are far too slender to encourage a confident frame of mind.