An essay on the foundations of geometry · Bertrand Russell
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_.
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.
These will be found, on the one hand, essential to the possibility of measurement in any continuum, and on the other hand, necessary properties of any form of externality with more than one dimension. They will, therefore, be declared _à priori_. (2) Those axioms which distinguish Euclidean from non-Euclidean spaces. These will be regarded as wholly empirical. The axiom that the number of dimensions is three, however, though empirical, will be declared, since small errors are here impossible, exactly and certainly true of our actual world; while the two remaining axioms, which determine the value of the space-constant, will be regarded as only approximately known, and certain only within the errors of observation[4]. The fourth chapter, finally, will endeavour to prove, what was assumed in Chapter III., that some form of externality is necessary to experience, and will conclude by exhibiting the logical impossibility, if knowledge of such a form is to be freed from contradictions, of wholly abstracting this knowledge from all reference to the matter contained in the form. I shall hope to have touched, with this discussion, on all the main points relating to the Foundations of Geometry.
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