An Essay on the Foundations of Geometry

the standpoint of general logic, the laws of thought and the categories, with the indispensable conditions of their applicability, will be alone à priori; but from the standpoint of any special science, we may call à priori whatever renders possible the experience which forms the subject-matter of our science. In Geometry, to particularize, we may call à priori whatever renders possible experience of externality as such. It is to be observed that this use of the term is at once more rationalistic and less precise than that of Kant. Kant would seem to have supposed himself immediately aware, by inspection, that some knowledge was apodeictic, and its subject-matter, therefore, à priori: but he did not always deduce its apriority from any further principle. Here, however, it is to be shown, before admitting apriority, that the falsehood of the judgment in question would not be effected by a mere change in the matter of experience, but only by a change which should render some branch of experience formally impossible, i.e. inaccessible to our methods of cognition. The above use is also less precise, for it varies according to the specialization of the experience we are assuming possible, and with every progress of knowledge some new connection is perceived, two previously isolated judgments are brought into logical relation, and the à priori may thus, at any moment, enlarge its sphere, as more is found deducible from fundamental postulates. 58. (2) Kant's arguments for the apriority of space. Having now discussed the logical canon to be used as regards the à priori, we may proceed to test Kant's arguments as regards space. The argument from Geometry, as remarked above, is upset by Metageometry, at least so far as those properties are concerned, which belong to Euclid but not to non-Euclidean spaces; as regards the common properties of both kinds of space, we cannot decide on their apriority till we have discussed the consequences of denying them, which will be done in Chapter III. As regards the two arguments which prove that space is an intuition, not a concept, they would call for much discussion in a special criticism of Kant, but here they may be passed by with the obvious comment that infinite homogeneous Euclidean space is a concept, not an intuition—a concept invented to explain an intuition, it is true, but still a pure concept[74]. And it is this pure concept which, in all discussions of Geometry, is primarily to be dealt with; the intuition need only be referred to where it throws light on the functions or the nature of the concept. The second of Kant's arguments, that we can imagine empty space, though not the absence of space, is false if it means a space without matter anywhere, and irrelevant if it merely means a space between matters and regarded as empty[75]. The only argument of importance, then, is the first argument. But I must insist, at the outset, that our problem is purely logical, and that all psychological implications must be excluded to the utmost possible extent. Moreover, as will be proved in Chapter IV., the proper function of space is to distinguish between different presented things, not between the Self and the object of sensation or perception. The argument then becomes the following: consciousness of a world of mutually external things demands, in presentations, a cognitive but non-inferential element leading to the discrimination of the objects presented. This element must be non-inferential, for from whatever number or combination of presentations, which did not of themselves demand diversity in their objects, I could never be led to infer the mutual externality of their objects. Kant says: "In order that sensations may be ascribed to something external to me ... and similarly in order that I may be able to present them as outside and beside one another, ... the presentation of space must be already present." But this goes rather too far: in the first place, the question should be only as to the mutual externality of presented things, not as to their externality to the Self[76]; and in the second place, things will appear mutually external if I have the presentation of any form of externality, whether Euclidean or non-Euclidean. Whatever may be true of the psychological scope of this argument—whose validity is here irrelevant—the logical scope extends, not to Euclidean space, but only to any form of externality which could exist intuitively, and permit knowledge, in beings with our laws of thought, of a world of diverse but interrelated things. Moreover externality, to render the scope of the argument wholly logical, must not be left with a sensational or intuitional meaning, though it must be supposed given in sensation or intuition. It must mean, in this argument, the fact of Otherness[77], the fact of being different from some other thing: it must involve the distinction between different things, and must be that element, in a cognitive state, which leads us to discriminate constituent parts in its object. So much, then, would appear to result from Kant's argument, that experience of diverse but interrelated things demands, as a necessary prerequisite, some sensational or intuitional element, in perception, by which we are led to attribute complexity to objects of perception[78]; that this element, in its isolation may be called a form of externality; and that those properties of this form, if any such be found, which can be deduced from its mere function of rendering experience of interrelated diversity possible, are to be regarded as à priori. What these properties are, and how the various lines of argument here suggested converge to a single result, we shall see in Chapters III. and IV. 59. In the philosophers who followed Kant, Metaphysics, for the most part, so predominated over Epistemology, that little was added to the theory of Geometry. What was added, came indirectly from the one philosopher who stood out against the purely ontological speculations of his time, namely Herbart. Herbart's actual views on Geometry, which are to be found chiefly in the first section of his Synechologie, are not of any great value, and have borne no great fruit in the development of the subject. But his psychological theory of space, his construction of extension out of series of points, his comparison of space with the tone and colour-series, his general preference for the discrete above the continuous, and finally his belief in the great importance of classifying space with other forms of series (Reihenformen[79]), gave rise to many of Riemann's epoch-making speculations, and encouraged the attempt to explain the nature of space by its analytical and quantitative aspect alone[80]. Through his influence on Riemann, he acquired, indirectly, a great importance in geometrical philosophy. To Riemann's dissertation, which we have already discussed in its mathematical aspect, we must now return, considering, this time, only its philosophical views. Riemann. 60. The aim of Riemann's dissertation, as we saw in Chapter I., was to define space as a species of manifold, i.e. as a particular kind of collection of magnitudes. It was thus assumed, to begin with, that spatial figures could be regarded as magnitudes, and the axioms which emerged, accordingly, determined only the particular place of these among the many algebraically possible varieties of magnitudes. The resulting formulation of the axioms—while, from the mathematical standpoint of metrical Geometry, it was almost wholly laudable—must, from the standpoint of philosophy, be regarded, in my opinion, as a petitio principii. For when we have arrived at regarding spatial figures as magnitudes, we have already traversed the most difficult part of the ground. The axioms of metrical Geometry—and it is metrical Geometry, exclusively, which is considered in Riemann's Essay—will appear, in Chapter III., to be divisible into two classes. Of these, the first class—which contains the axioms common to Euclid and Metageometry, the only axioms seriously discussed by Riemann—are not the results of measurement, nor of any conception of magnitude, but are conditions to be fulfilled before measurement becomes possible. The second class only—those which express the difference between Euclidean and non-Euclidean spaces—can be deduced as results of measurement or of conceptions of magnitude. As regards the first class, on the contrary, we shall see that the relativity of position—by which space is distinguished from all other known manifolds, except time—leads logically to the necessity of three of the most distinctive axioms of Geometry, and yet this relativity cannot be called a deduction from conceptions of magnitude. In analytical Geometry, owing to the fact that coordinate systems start from points, and hence build up lines and surfaces, it is easy to suppose that points can be given independently of lines and of each other, and thus the relativity of position is lost sight of. The error thus suggested by mathematics was probably reinforced by Herbart's theory of space, which, by its serial character, as we have seen, appeared to him to facilitate a construction out of successive points, and to which Riemann acknowledges his indebtedness both in his Dissertation and elsewhere. The same error reappears in Helmholtz, in whom it is probably due wholly to the methods of analytical Geometry. It is a striking fact that, throughout the writings of these two men, there is not, so far as I know, one allusion to the relativity of position, that property of space from which, as our next chapter will shew, the richest quarry of consequences can be extracted. This is not a result of any conception of magnitude, but follows from the nature of our space-intuition; yet no one, surely, could call it empirical, since it is bound up in the very possibility of locating things there as opposed to here. 61. Indeed we can see, from a purely logical consideration of the judgment of quantity, that Riemann's manner of approaching the problem can never, by legitimate methods, attain to a philosophically sound formulation of the axioms. For quantity is a result of comparison of two qualitatively similar objects, and the judgment of quantity neglects altogether the qualitative aspect of the objects compared. Hence a knowledge of the essential properties of space can never be obtained from judgments of quantity, which neglect these properties, while they yet presuppose them. As well might one hope to learn the nature of man from a census. Moreover, the judgment of quantity is the result of comparison, and therefore presupposes the possibility of comparison. To know whether, or by what means, comparison is possible, we must know the qualities of the things compared and of the medium in which comparison is effected; while to know that quantitative comparison is possible, we must know that there is a qualitative identity between the things compared, which again involves a previous qualitative knowledge. When spatial figures have once been reduced to quantity, their quality has already been neglected, as known and similar to the quality of other figures. To hope, therefore, for the qualities of space, from a comparison of its expression as pure quantity with other pure quantities, is an error natural to an analytical geometer, but an error, none the less, from which there is no return to the qualitative basis of spatial quantity. 62. We must entirely dissent, therefore, from the disjunction which underlies Riemann's philosophy of space. Either the axioms must be consequences of general conceptions of magnitude, he thinks, or else they can only be proved by experience (p. 255). Whatever can be derived from general conceptions of magnitude, we may retort, cannot be an à priori adjective of space: for all the necessary adjectives of space are presupposed in any judgment of spatial quantity, and cannot, therefore, be consequences of such a judgment. Riemann's disjunction, accordingly, since one of its alternatives is obviously impossible, really begs the question. In formulating the axioms of metrical Geometry, our question should be: What axioms, i.e. what adjectives of space, must be presupposed, in order that quantitative comparison of the parts of space may be possible at all? And only when