INTRODUCTION. OUR PROBLEM DEFINED BY ITS RELATIONS TO LOGIC, PSYCHOLOGY AND MATHEMATICS.
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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.
Now the only mental states whose immediate causes lie in the external world are sensations. A pure sensation is, of course, an impossible abstraction—we are never wholly passive under the action of an external stimulus—but for the purposes of Psychology the abstraction is a useful one. Whatever, then, is not sensation, we shall, in Psychology, call subjective. It is in sensation alone that we are directly affected by the external world, and only here does it give us direct information about itself.
The à priori, since Kant at any rate, has generally stood for the necessary or apodeictic element in knowledge. But modern logic has shown that necessary propositions are always, in one aspect at least, hypothetical. There may be, and usually is, an implication that the connection, of which necessity is predicated, has some existence, but still, necessity always points beyond itself to a ground of necessity, and asserts this ground rather than the actual connection. As Bradley points out, "arsenic poisons" remains true, even if it is poisoning no one. If, therefore, the à priori in knowledge be primarily the necessary, it must be the necessary on some hypothesis, and the ground of necessity must be included as à priori. But the ground of necessity is, so far as the necessary connection in question can show, a mere fact, a merely categorical judgment. Hence necessity alone is an insufficient criterion of apriority.
To supplement this criterion, we must supply the hypothesis or ground, on which alone the necessity holds, and this ground will vary from one science to another, and even, with the progress of knowledge, in the same science at different times. For as knowledge becomes more developed and articulate, more and more necessary connections are perceived, and the merely categorical truths, though they remain the foundation of apodeictic judgments, diminish in relative number. Nevertheless, in a fairly advanced science such as Geometry, we can, I think, pretty completely supply the appropriate ground, and establish, within the limits of the isolated science, the distinction between the necessary and the merely assertorical.
I shall hope to have touched, with this discussion, on all the main points relating to the Foundations of Geometry.