An essay on the foundations of geometry. Bertrand Russell

that such a view places our results, as to the à priori, at the mercy of empirical psychology. How serious this danger is, the controversy as to Kant's pure intuition sufficiently shows.

      5. I shall, therefore, throughout the present Essay, use the word à priori without any psychological implication. My test of apriority will be purely logical: Would experience be impossible, if a certain axiom or postulate were denied? Or, in a more restricted sense, which gives apriority only within a particular science: Would experience as to the subject-matter of that science be impossible, without a certain axiom or postulate? My results also, therefore, will be purely logical. If Psychology declares that some things, which I have declared à priori, are not subjective, then, failing an error of detail in my proofs, the connection of the à priori and the subjective, so far as those things are concerned, must be given up. There will be no discussion, accordingly, throughout this Essay, of the relation of the à priori to the subjective—a relation which cannot determine what pieces of knowledge are à priori, but rather depends on that determination, and belongs, in any case, rather to Metaphysics than to Epistemology.

      6. As I have ventured to use the word à priori in a slightly unconventional sense, I will give a few elucidatory remarks of a general nature.

      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.

7. There are two grounds, I think, on which necessity may be sought within any science. These may be (very roughly) distinguished as the ground which Kant seeks in the Prolegomena, and that which he seeks in the Pure Reason. We may start from the existence of our science as a fact, and analyse the reasoning employed with a view to discovering the fundamental postulate on which its logical possibility depends; in this case, the postulate, and all which follows from it alone, will be à priori. Or we may accept the existence of the subject-matter of our science as our basis of fact, and deduce dogmatically whatever principles we can from the essential nature of this subject-matter. In this latter case, however, it is not the whole empirical nature of the subject-matter, as revealed by the subsequent researches of our science, which forms our ground; for if it were, the whole science would, of course, be à priori. Rather it is that element, in the subject-matter, which makes possible the branch of experience dealt with by the science in question[2]. The importance of this distinction will appear more clearly as we proceed[3].

      8. These two grounds of necessity, in ultimate analysis, fall together. The methods of investigation in the two cases differ widely, but the results cannot differ. For in the first case, by analysis of the science, we discover the postulate on which alone its reasonings are possible. Now if reasoning in the science is impossible without some postulate, this postulate must be essential to experience of the subject-matter of the science, and thus we get the second ground. Nevertheless, the two methods are useful as supplementing one another, and the first, as starting from the actual science, is the safest and easiest method of investigation, though the second seems the more convincing for exposition.

9. The course of my argument, therefore, will be as follows: In the first chapter, as a preliminary to the logical analysis of Geometry, I shall give a brief history of the rise and development of non-Euclidean systems. The second chapter will prepare the ground for a constructive theory of Geometry, by a criticism of some previous philosophical views; in this chapter, I shall endeavour to exhibit such views as partly true, partly false, and so to establish, by preliminary polemics, the truth of such parts of my own theory as are to be found in former writers. A large part of this theory, however, cannot be so introduced, since the whole field of projective Geometry, so far as I am aware, has been hitherto unknown to philosophers. Passing, in the third chapter, from criticism to construction, I shall deal first with projective Geometry. This, I shall maintain, is necessarily true of any form of externality, and is, since some such form is necessary to experience, completely à priori. In metrical Geometry, however, which I shall next consider, the axioms will fall into two classes: (1) Those common to Euclidean and non-Euclidean spaces. 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.

      FOOTNOTES:

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      [1] Cf. Erdmann, Axiome der Geometrie, p. 111: "Für Kant sind Apriorität und ausschliessliche Subjectivität allerdings Wechselbegriffe."

[2] I use "experience" here in the widest possible sense, the sense in which the word is used by Bradley.

[3] Where the branch of experience in question is essential to all experience, the resulting apriority may be regarded as absolute; where it is necessary only to some special science, as relative to that science.

[4] I have given no account of these empirical proofs, as they seem to be constituted by the whole body of physical science. Everything in physical science, from the law of gravitation to the building of bridges, from the spectroscope to the art of navigation, would be profoundly modified by any considerable inaccuracy in the hypothesis that our actual space is Euclidean. The observed truth of physical science, therefore, constitutes overwhelming