An Essay on the Foundations of Geometry

careless and popular utterances of Helmholtz and Riemann, and to have taken no trouble to understand even the foundations of mathematical Metageometry. By this neglect, much of what he says is rendered wholly worthless. To begin with, he regards, as the purpose of Helmholtz's fairy tale, the suggestion of a possible fourth dimension, whereas the real purpose was quite the opposite—to make intelligible a purely three-dimensional non-Euclidean space. Helmholtz introduced Flatland only because its relation to Sphereland is analogous to the relation of ours to spherical space[106]. But Lotze says: The Flatlanders would find no difficulty in a third dimension, since it would in no way contradict their own Geometry, while the people in Sphereland, from the contradictions in their two-dimensional system, would already have been led to it. The latter contention I have already tried to answer; the former has an odd sound, in view of the attempt, a few pages later, to prove à priori that all forms of intuition, in any way analogous to space, must have three dimensions. One cannot help suspecting that the Flatlanders, with two instead of three dimensions, would make a similar attempt. But to return to Lotze's argument: Neither analogy can be used, he says, to prove that we ought perhaps to set up a fourth dimension, since, for us, no contradictions or otherwise inexplicable phenomena exist. The only people, so far as I know, who have used this analogy, are Dr Abbot and a few Spiritualists—the former in joke, the latter to explain certain phenomena more simply explained, perhaps, by Maskelyne and Cooke. But although Lotze's conclusion in this matter is sound, and one with which Helmholtz might have agreed, his arguments, to my mind, are irrelevant and unconvincing. There is this difference, he says, between us and the Spherelanders: the latter were logically forced to a new dimension, and found it possible; we are not forced to it, and find it, in our space, impossible. I have contended that, on the contrary, nothing would force the Spherelanders to assume a third dimension, while they would find it impossible exactly as we find a fourth impossible—not logically, that is to say, but only as a presentable construction in given space. After a somewhat elephantine piece of humour, about socialistic whales in a four-dimensional sea of Fourrier's eau sucrée, Lotze proceeds to a proof, by logic, that every form of intuition, which embraces the whole system of ordered relations of a coexisting manifold, must have three dimensions. One might object, on à priori grounds, to any such attempt: what belongs to pure intuition could hardly, one would have thought, be determined by à priori reasoning[107]. I will not, however, develop this argument here, but endeavour to point out, as far as its obscurity will allow, the particular fallacy of the proof in question. Lotze's argument is as follows. In this discussion, though our terminology is necessarily taken from space, we are really concerned with a much more general conception. We assume, in order to preserve the homogeneity of dimensions, that the difference (distance) between any two elements (points) of our manifold—to borrow Riemann's word—is of the same kind as, and commensurable with, the difference between any other two elements. Let us take a series of elements at successive distances x such that the distance between any two is the sum of the distances between intermediate elements. Such a series corresponds to a straight line, which is taken as the x-axis. Then a series OY is called perpendicular to the x-axis OX, when the distances of any element y, on OY, from +mx and -mx are equal. By our hypothesis, these distances are comparable with, and qualitatively similar to, x and y. So long as OY is defined only by relation to OX, it is conceptually unique. But now let us suppose the same relation as that between OX and OY, to be possible between OY and a new series OZ; we then get a third series OZ perpendicular to OY, and again conceptually unique, so long as it is defined by relation to OY alone. We might proceed, in the same way, to a fourth line OU perpendicular to OZ. But it is necessary, for our purposes, that OZ should be perpendicular to OX as well as OY. Without this condition, OZ might extend into another world, and have no corresponding relation to OX—this is a possibility only excluded by our unavoidable spatial images. At this point comes the crux of the argument. That OZ, says Lotze, which, besides being perpendicular to OY, is also perpendicular to OX, must be among the series of OY's, for these were defined only by perpendicularity to OX. Hence, he concludes, there can only be even a third dimension if OZ coincides with one, and—as soon as OX is considered fixed—with only one, of the many members of the OY series. In this argument it is difficult—to me at any rate—to see any force at all. The only way I can account for it is, to suppose that Lotze has neglected the possibility of any but single infinities. On this interpretation, the argument might be stated thus: There is an infinite series of continuously varying OY's; to the common property of these, we add another property, which will divide their total number by infinity. The remaining OZ, therefore, must be uniquely determined. The same form of argument, however, would prove that two surfaces can only cut one another in a single point, and numberless other absurdities. The fact is, that infinities may be of different orders. For example, the number of points in a line may be taken as a single infinity, and so may the number of lines in a plane through any point; hence, by multiplication, the number of points in a plane is a double infinity, ∞2, and if we divide this number by a single infinity, we get still an infinite number left. Thus Lotze's argument assumes what he has to prove, that the number of lines perpendicular to a given line, through any point, is a single infinity, which is equivalent to the axiom of three dimensions. The whole passage is so obscure, that its meaning may have escaped me. It is obvious à priori, however, as I pointed out in the beginning, that any proof of the axiom must be fallacious somewhere, and the above interpretation of the argument is the only one I have been able to find. 95. The rest of the Chapter is devoted to an attack on spherical and pseudo-spherical space, on the ground that they interfere with the homogeneity of the three dimensions, and with the similarity of all parts of space. This is simply false. Such spaces, like the surface of a sphere, are exactly alike throughout. Lotze shows, here and elsewhere, that he has not taken the pains to find out what Metageometry really is. I hold myself, and have tried to prove in this Essay, that Congruence is an à priori axiom, without which Geometry would be impossible; but the wish to uphold this axiom is, as Lotze ought to have known, the precise motive which led Metageometry to limit itself to spaces of constant measure of curvature. We see here the importance of distinguishing between Helmholtz the philosopher and Helmholtz the mathematician. Though the philosopher wished to dispense with Congruence, the mathematician, as we saw in Chapter I., retained and strongly emphasized it. A little later Lotze shows, again, how he has been misled by the unfortunate analogy of Sphereland. A spherical surface, he says, he can understand; but how are we to pass from this to a spherical space? Either this surface is the whole of our space, as in Sphereland, or it generates space by a gradually growing radius. Such concentric spheres, as Lotze triumphantly points out, of course generate Euclidean space. His disjunction, however, is utterly and entirely false, and could never have been suggested by any one with even a superficial knowledge of Metageometry. This point is less laboured than the former, which, in all its nakedness, is thus re-stated in the last sentence of the Chapter: "I cannot persuade myself that one could, without the elements of homogeneous space, even form or define the presentation of heterogeneous spaces, or of such as had variable measures of curvature." As though such spaces were ever set up by non-Euclidean mathematics! In conclusion, Lotze expresses a hope that Philosophy, on this point, will not allow itself to be imposed upon by Mathematics. I must, instead, rejoice that Mathematics has not been imposed upon by Philosophy, but has developed freely an important and self-consistent system, which deserves, for its subtle analysis into logical and factual elements, the gratitude of all who seek for a philosophy of space. 96. The objections to non-Euclidean Geometry which have just been discussed fall under four heads: I. Non-Euclidean spaces are not homogeneous; Metageometry therefore unduly reifies space. II. They involve a reference to a fourth dimension. III. They cannot be set up without an implicit reference to Euclidean space, or to the Euclidean straight line, on which they are therefore dependent. IV. They are self-contradictory in one or more ways. The reader who has followed me in regarding these four objections as fallacious, will have no difficulty in disposing of any other critic of Metageometry, as these are the only mathematical arguments, so far as I know, ever urged against non-Euclideans[108]. The logical validity of Metageometry, and the mathematical possibility of three-dimensional non-Euclidean spaces, will therefore be regarded, throughout the remainder of the work, as sufficiently established. 97. Two other objections may, indeed, be urged against Metageometry, but these are rather of a philosophical than of a strictly mathematical import. The first of these, which has been made the base of operations by Delbœuf, applies equally to all non-Euclidean spaces. The second, which has not, so far as I know, been much employed, but yet seems to me deserving of notice, bears directly against spaces of positive curvature alone; but if it could discredit these, it might throw doubt on the method by which all alike are obtained. The two objections are: I. Space must be such as to allow of similarity, i.e. of the increase or diminution, in a constant ratio, of all the lines in a figure, without change of angles; whereas in non-Euclid, lines, like angles, have absolute magnitude. II. Space must be infinite, whereas spherical and elliptic spaces are finite. I will discuss the first objection in connection with Delbœuf's articles referred to above. The second, which has not, to my knowledge, been widely used in criticism, will be better deferred to Chapter III. Delbœuf. 98. M. Delbœuf's four articles in the Revue Philosophique contain much matter that has already been dealt with in the criticism of Lotze, and much that is irrelevant for our present purpose. The only point, which I wish to discuss here, is the question of absolute magnitude, as it is called—the question, that is, whether the possibility of similar but unequal geometrical figures can be known à priori[109]. In discussing this question, it is important, to begin with, to distinguish clearly the sense in which absolute magnitude is required in non-Euclidean Geometry, from another sense, in which it would be absurd to regard any magnitude as absolute. Judgments of magnitude can only result from comparison, and if Metageometry required magnitudes which could be determined without comparison, it would certainly deserve condemnation. But this is not required. All we require is, that it shall be impossible, while the rest of space is unaffected, to alter the magnitude of any figure, as compared with other figures, while leaving the relative internal magnitudes of its parts unchanged. This construction, which is possible in Euclid, is impossible in Metageometry. We have to discuss whether such an impossibility renders non-Euclidean spaces logically faulty. M. Delbœuf's position on this axiom—which he calls the