An Essay on the Foundations of Geometry

with some care, it seems desirable to discuss Lotze's strictures point by point. 91. The mathematical criticism begins (§ 131) with a somewhat question-begging definition of parallel straight lines. Two straight lines aα, bβ, according to this definition, are parallel when—a and b being arbitrary points on the two lines—if aα = bβ, then ab = αβ, where α, β are two other points on the two straight lines respectively. This definition—which contains Euclid's axiom and definition combined in a very convenient and enticing form—is of course thoroughly suitable to Euclidean Geometry, and leads immediately to all the Euclidean propositions about parallels. But it is perhaps more honest to follow Euclid's course; when an axiom is thus buried in a definition, it is apt to seem, since definitions are supposed to be arbitrary, as though the difficulty had been overcome, while in reality, the possibility of parallels, as above defined, involves the very point in question, namely, the disputed axiom of parallels. For what this axiom asserts is simply the existence of lines conforming to Lotze's definition. The deduction of the principal propositions on parallels, with which Lotze follows up his definition, is of course a very simple proceeding—a proceeding, however, in which the first step begs the question. 92. The next argument for the apriority of Euclidean Geometry has, oddly enough, an exactly opposite bearing, although it is a great favourite with opponents of Metageometry. Measurements of stellar triangles, and all similar attempts at an empirical determination of the space-constant are, according to Lotze, beside the mark; for any observed departure from two right angles, or any finite annual parallax for distant stars, would be attributed to some new kind of refraction, or, as in the case of aberration, to some other physical cause, and never to the geometrical nature of space. This is a strong argument for the empirical validity of Euclid, but as an argument for the apodeictic certainty of the orthodox system, it has an opposite tendency. For observations of the kind contemplated would have to be due to departures from Euclidean straightness, hitherto unknown, on the part of stellar light-rays. Such departure could, in certain cases, be accounted for by a finite space-constant, but it could also, probably, be accounted for by a change in Optics, for example, by attributing refractive properties to the ether. Such properties could only exist if ether were of varying density, if (say) it were denser in the neighbourhood of any of the heavenly bodies. But such an assumption would, I believe, destroy the utility of ether for Physics; a slight alteration in our Geometry, so slight as not appreciably to affect distances within the Solar System, would probably be in the end, therefore, should such errors ever be discovered, a simpler explanation than any that Physics could offer. But this is not the point of my contention. The point is that, if the physical explanation, as Lotze holds, be possible in the above case, the converse must also hold: it must be possible to explain the present phenomena by supposing ether refractive and space non-Euclidean. From this conclusion there is no escape. If every conceivable behaviour of light-rays can be explained, within Euclid, by physical causes, it must also be possible, by a suitable choice of hypothetical physical causes, to explain the actual phenomena as belonging to a non-Euclidean space. Such a hypothesis would be rightly rejected by Science, for the present, on account of its unnecessary complexity. Nevertheless it would remain, for philosophy, a possibility to be reckoned with, and the choice could only be decided upon empirical grounds of simplicity. It may well be doubted whether, in the world we know, the phenomena could be attributed to a distinctly non-Euclidean space, but this conclusion follows inevitably from the contention that no phenomena could force us to assume such a space. Lotze's argument, therefore, if pushed home, disproves his own view, and puts Euclidean space, as an empirical explanation of phenomena, on a level with luminiferous ether[103]. 93. Lotze now proceeds (§ 132) to a detailed criticism of Helmholtz, whom he regards as a typical exponent of Metageometry. It is possible that, at the time when he wrote, Helmholtz really did occupy this position; but it is unfortunate that, in the minds of philosophers, he should still continue to do so, after the very material advances brought about by the projective treatment of the subject. It is also unfortunate that his somewhat careless attempts to popularise mathematical results have so often been disposed of, without due attention to his more technical and solid contributions. Thus his romances about Flatland and Sphereland—at best only fairy-tale analogies of doubtful value—have been attacked as if they formed an essential feature of Metageometry. But to proceed to particulars: Lotze readily allows that the Flatlanders would set up Plane Geometry, as we know it, but refuses to admit that the Spherelanders could, without inferring the third dimension, set up a two-dimensional spherical Geometry which should be free from contradictions. I will endeavour to give a free rendering of Lotze's argument on this point. Suppose, he says, a north and south pole, N and S, arbitrarily fixed, and an equator EW. Suppose a being, B, capable of impressions only from things on the surface of the sphere, to move in a meridian NBS. Let B start from some point a, and finally, after describing a great circle, return to the same point a. If a is known only by the quality of the impression it makes on B, B may imagine he has not reached the same point a, but another similar point a′, bearing a relation to a similar to that of the octave in singing: he might even not arrange his impressions spatially at all. In order that this may occur, we require the further assumption, that every difference in the above-mentioned feelings (as he describes the meridian) may be presented as a spatial distance between two places. Even now, B may think he is describing a Euclidean straight line, containing similar points at certain intervals. Allowing, however, that he realizes the identity of a with his initial position, he will now seem, by motion in a straight line, to have returned to the point from which he started, for his motion cannot, without the third dimension, seem to him other than rectilinear. Up to this point, there seems little ground for objection, except, perhaps, to the idea of a straight line with periodical similar points—if B were as philosophical as, in these discussions, we usually suppose him to be, he would probably object to this interpretation of his experiences, on the ground that it regards empty space as something independent of the objects in it. It is worth pointing out, also, that B would not need to describe the whole circle, in order suddenly to find himself home again with his old friends. Accurate measurements of small triangles would suffice to determine his space-constant, and show him the length of a great circle (or straight line, as he would call it). We must admit, also, that so hypothetical a being as B might form no space-intuition at all, but as he is introduced solely for the purposes of the analogy, it is convenient to allow him all possible qualifications for his post. But these points do not touch the kernel of the argument, which lies in the statement that such a straight line, returning into itself after a finite time, would appear to B as an "unendurable contradiction," and thus force him, for logical though not for sensational purposes, into the assumption of a third dimension. This assertion seems to me quite unwarranted: the whole of Metageometry is a solid array in disproof of it. Helmholtz's argument is, it must be remembered, only an analogy, and the contradiction would exist only for a Euclidean. A complete three-dimensional Geometry has, we have seen in Chapter I., been developed on the assumption that straight lines are of finite length. A constant value for the measure of curvature, as our discussion of Riemann showed, involves neither reference to the fourth dimension, nor any kind of internal contradiction. This fact disproves Lotze's contention, which arises solely from inability to divest his imagination of Euclidean ideas. Lotze next attacks Helmholtz for the assertion that B would know nothing of parallel lines—parallel straight lines, as the context shows, he meant to say[104]. Lotze, however, takes him as meaning, apparently, mere curves of constant distance from a given straight line, which are part of the regular stock-in-trade of Metageometry. Parallels of latitude, in the geographical sense, would not—with the exception of the equator—appear to B as straight lines, but as circles. Great circles he would call straight, and this fact seems to have misled Lotze into thinking all circles were to be treated as straight lines. Parallels of latitude, therefore, though B might call them parallels, would not invalidate Helmholtz's contention, which applies only to straight lines. The argument that such small circles would be parallel, which we have just disposed of, is only the preface to another proof that B would need a third dimension. Let us call two of these parallels of latitude ln and ls, and let them be equidistant from the equator, one in the northern, one in the southern hemisphere. Consecutive tangent planes, along these parallels, converge, in the one case northwards, in the other southwards. Either B could become aware of their difference, says Lotze, or he could not. In the former case, which he regards as the more probable, he easily proves that B would infer a third dimension. But this alternative is, I think, wholly inadmissible. Tangent planes, like Euclidean planes in general, would have no meaning to B; unless, indeed, he were a metageometrician, which, with all his metaphysical and mathematical subtlety, the argument supposes him not to be—and to such a supposition Lotze, surely, is the last person who has a right to object. Lotze's attempted proof that this is the right alternative rests, if I understand him aright, on a sheer error in ordinary spherical Geometry. B would observe, he says, that the meridians made smaller angles with his path towards the nearer than towards the further pole—as a matter of fact, they would be simply perpendicular to his path in both directions. What Lotze means is, perhaps, that all the meridians would meet sooner in one direction than in the other, and this, of course, is true. But the poles, in which the meridians meet, would appear to B as the centres of the respective parallels, while the parallels themselves would appear to be circles. Now I am at a loss to see what difficulty would arise, to B, in supposing two different circles to have different centres[105]. We must, therefore, take the first alternative, that B would have no sort of knowledge as to the direction in which the tangent planes converged. Here Lotze attempts, if I have not misunderstood him, to prove a reductio ad absurdum: B would think, he says, that he was describing two paths wholly the same in direction, and then he might regard both paths as circles in a plane. It may be observed that direction, when applied to a circle as a whole, is meaningless; indeed direction, in all Metageometry, can only mean, even when applied to straight lines, direction towards a point. To speak of two lines, which do not meet, as having the same direction, is a surreptitious introduction of the axiom of parallels. Apart from this, I cannot conceive any objection, on B's part, to such a view—one should say must, not might. The whole argumentation, therefore, unless its obscurity has led me astray, must be pronounced fruitless and inconclusive. 94. After this preliminary discussion of Sphereland, Lotze proceeds to the question of a fourth dimension, and thence to spherical and pseudo-spherical space. As before, he appears to know only the more