208. Externality is not a relation, but an aspect of relations. Spatial order, owing to its reference to matter, is a real relation 198 209. Conclusion 199 INTRODUCTION. OUR PROBLEM DEFINED BY ITS RELATIONS TO LOGIC, PSYCHOLOGY AND MATHEMATICS. =1.= Geometry, throughout the 17th and 18th centuries, remained, in the war against empiricism, an impregnable fortress of the idealists. Those who held--as was generally held on the Continent--that certain knowledge, independent of experience, was possible about the real world, had only to point to Geometry: none but a madman, they said, would throw doubt on its validity, and none but a fool would deny its objective reference. The English Empiricists, in this matter, had, therefore, a somewhat difficult task; either they had to ignore the problem, or if, like Hume and Mill, they ventured on the assault, they were driven into the apparently paradoxical assertion that Geometry, at bottom, had no certainty of a different _kind_ from that of Mechanics--only the perpetual presence of spatial impressions, they said, made our experience of the truth of the axioms so wide as to seem absolute certainty. Here, however, as in many other instances, merciless logic drove these philosophers, whether they would or no, into glaring opposition to the common sense of their day. It was only through Kant, the creator of modern Epistemology, that the geometrical problem received a modern form. He reduced the question to the following hypotheticals: If Geometry has apodeictic certainty, its matter, _i.e._ space, must be _à priori_, and as such must be purely subjective; and conversely, if space is purely subjective, Geometry must have apodeictic certainty. The latter hypothetical has more weight with Kant, indeed it is ineradicably bound up with his whole Epistemology; nevertheless it has, I think, much less force than the former. Let us accept, however, for the moment, the Kantian formulation, and endeavour to give precision to the terms _à priori_ and _subjective_.