Note 50 — Introductory material

That some logical necessity is involved in these axioms might, I think, be inferred as probable, from their historical development alone. For the systems of Metageometry have not, in general, been set up as more likely to fit facts than the system of Euclid; with the exception of Zöllner, for example, I know of no one who has regarded the fourth dimension as required to explain phenomena. As regards the space-constant again, though a small space-constant is regarded as empirically possible, it is not usually regarded as probable; and the finite space-constants, with which Metageometry is equally conversant, are not usually thought even possible, as explanations of empirical fact[66]. Thus the motive has been throughout not one of fact, but one of logic. Does not this give a strong presumption, that those axioms which are retained, are retained because they are logically indispensable? If this be so, the axioms common to Euclid and Metageometry will be à priori, while those peculiar to Euclid will be empirical. After a criticism of some differing theories of Geometry, I shall proceed, in Chapters III. and IV., to the proof and consequences of this thesis, which will form the remainder of the present work.