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Geometry, throughout the seventeenth and eighteenth centuries, re-mained, 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 oppo-sition to the common sense of their day. It was only through Kant, the creator of modern Epistemology, that the geometrical problem re-ceived a modern form. He reduced the question to the following hy-potheticals: 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 inerad-icably 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.
One of the great difficulties, throughout this controversy, is the extremely variable use to which these words, as well as the word em-pirical, are put by different authors. To Kant, who was nothing of a psychologist, à priori and subjective were almost interchangeable terms;1 in modern usage there is, on the whole, a tendency to confine the word subjective to Psychology, leaving à priori to do duty for Epistemol-ogy. If we accept this differentiation, we may set up, corresponding to the problems of these two sciences, the following provisional defini-tions: à priori applies to any piece of knowledge which, though perhaps elicited by experience, is logically presupposed in experience: subjective applies to any mental state whose immediate cause lies, not in the ex-ternal world, but within the limits of the subject. The latter definition, of course, is framed exclusively for Psychology: from the point of view of physical Science all mental states are subjective. But for a Sci-ence whose matter, strictly speaking, is only mental states, we require, if we are to use the word to any purpose, some differentia among mental states, as a mark of a more special subjectivity on the part of those to which this term is applied.