Note 84 — Introductory material

With this caution as to the meaning of apriority, we shall find, I think, that the conclusions of Erdmann's final chapter, on the principles of a theory of Geometry, are largely invalidated by the diversity and inadequacy of his tests of the à priori. He begins by asserting, in conformity with the quantitative bias noticed above, that the question as to the nature of geometrical axioms is completely analogous to the corresponding question of the foundations of pure mathematics (p. 138). This is, I think, a radical error: for the function of the axioms seems to be, to establish that qualitative basis on which, as we saw, all qualitative comparison must rest. But in pure mathematics, this qualitative basis is irrelevant, for we deal there with pure quantity, i.e. with the merely quantitative result of quantitative comparison, wherever it is possible, independently of the qualities underlying the comparison. Geometry, as Grassmann insists[98], ought not to be classed with pure mathematics, for it deals with a matter which is given to the intellect, not created by it. The axioms give the means by which this matter is made amenable to quantity, and cannot, therefore, be themselves deduced from purely quantitative considerations.