An essay on the foundations of geometry · Bertrand Russell
CHAPTER II.
Lotze. =85.= Lotze's argument as regards Geometry[100]--which follows a metaphysical argument as to the ontological nature of space, and assumes the results of this argument--consists of two parts: the first discusses the various meanings logically assignable (pp. 233-247) to the proposition that other spaces than Euclid's are possible, and the second criticizes, in detail, the procedure of Metageometry. The first of these questions is very important, and demands considerable care as to the logical import of a judgment of possibility. Although Lotze's discussion is excellent in many respects, I cannot persuade myself that he has hit on the only true sense in which non-Euclidean spaces are possible. I shall endeavour to make good this statement in the following pages. =86.= Lotze opens with a somewhat startling statement, which, though philosophically worthy to be true, does not appear to be historically borne out. Euclidean Geometry has been chiefly shaken, he says, by the Kantian notion of the exclusive subjectivity of space--if space is only our private form of intuition, to which there exists no analogue in the objective world, then other beings may have other spaces, without supposing any difference in the world which they arrange in these spaces (p. 233). This certainly seems a legitimate deduction from the subjectivity of space, which, so far from establishing the universal validity of Euclid, establishes his validity only after an empirical investigation of the nature of space as intuited by Tom, Dick or Harry. But as a matter of fact, those who have done most to further non-Euclidean Geometry--with the exception of Riemann, who was a disciple of Herbart--have usually inherited from Newton a naïve realism as regards absolute space. I might instance the passage quoted from Bolyai in Chapter I., or Clifford, who seems to have thought that we actually see the images of things on the retina[101], or again Helmholtz's belief in the dependence of Geometry on the behaviour of rigid bodies. This belief led to the view that Geometry, like Physics, is an experimental science, in which objective truth can be attained, it is true, but only by empirical methods. However, Lotze's ground for uncertainty about Euclid is a philosophically tenable ground, and it will be instructive to observe the various possibilities which arise from it.
If space is only a subjective form--so Lotze opens his argument--other beings may have a different form. If this corresponds to a different world, the difference, he says, is uninteresting: for our world alone is relevant to any metaphysical discussion. But if this different space corresponds to the same world which we know under the Euclidean form, then, in his opinion, we get a question of genuine philosophic interest. And here he distinguishes two cases: _either_ the relations between things, which are presented to these hypothetical beings under the form of some different space, are relations which do not appear to us, or at any rate do not appear spatial; _or_ they are the same relations which appear to us as figures in Euclidean space (p. 235). The first possibility would be illustrated, he says, by beings to whom the tone or colour-manifolds appeared extended; but we cannot, in his opinion, imagine a manifold, such as is required for this case, to have its dimensions homogeneous and comparable _inter se_, and therefore the contents of the various presentations constituting such a manifold could not be combined into a single content containing them all. But the possibility of such a combination is of the essence of anything worth calling a space: therefore the first of the above possibilities is unmotived and uninteresting. Lotze's conclusion on this point, I think, is undeniable, but I doubt whether his argument is very cogent. However, as this possibility has no connection with that contemplated by non-Euclideans, it is not worth while to discuss it further.
The second possibility also, Lotze thinks, is not that of Metageometry, but in truth it comes nearer to it than any of the other possibilities discussed. If a non-Euclidean were at the same time a believer in the subjectivity of space, he would have to be an adherent of this view. Let us see more precisely what the view is. In Book II., Chapter I., Lotze has accepted the argument of the Transcendental Aesthetic, but rejected that of the mathematical antinomies: he has decided that space is, as Kant believed, subjective, but possesses nevertheless, what Kant denied it, an objective counterpart. The relation of presented space to its objective counterpart, as conceived by Lotze, is rather hard to understand. It seems scarcely to resemble the relation of sensation to its object--_e.g._ of light to ether-vibrations--for if it did, space would not be in any peculiar sense subjective. It seems rather to resemble the relation of a perceived bodily motion to the state of mind of the person willing the motion. However this may be, the objective counterpart of space is supposed to consist of certain immediate interactions of monads, who experience the interactions as modifications of their internal states.
Such interactions, it is plain, do not form the subject-matter of Geometry, which deals only with our resulting perceptions of spatial figures. Now if Lotze's construction of space be correct, there seems certainly no reason why these resulting perceptions should not, for one and the same interaction between monads, be very different in beings differently constituted from ourselves. But if they were different, says Lotze, they would have to be utterly different--as different, for example, as the interval between two notes is from a straight line. The possibility is, therefore, in his opinion, one about which we can know nothing, and one which must remain always a mere empty idea. This seems to me to go too far: for whatever the objective counterpart may be, any argument which gives us information about it must, when reversed, give us information about any possible form of intuition in which this counterpart is presented. The argument which Lotze has used in his former chapter, for example, deducing, from the relativity of position, the merely relational nature of the objective counterpart, allows us, conversely, to infer, from this relational nature, the complete relativity of position in any possible space-intuition--unless, indeed, it bore a wholly deceitful relation to those interactions of monads which form its objective counterpart.
But the complete relativity of position, as I shall endeavour to establish in Chapter III., suffices to prove that our Geometry must be Euclidean, elliptic, spherical or pseudo-spherical. We have, therefore, it would seem, very considerable knowledge, on Lotze's theory of space, of the manner in which what appears to us as space _must_ appear to any beings with our laws of thought. We cannot know, it is true, what _psychological_ theory of space-perception would apply to such beings: they might have a sense different from any of ours, and they might have no sense in any way resembling ours, but yet their Geometry would have points of resemblance to ours, as that of the blind coincides with that of the seeing. If space has any objective counterpart whatever, in short, and if any inference is possible, as Lotze holds it to be, from space to its counterpart, then a converse argument is also possible, though it may give some only of the qualities of Euclidean space, since some only of these qualities may be found to have a necessary analogue in the counterpart.
But what is this possibility? A thing is possible, according to Bradley (Logic, p. 187), when it would follow from a certain number of conditions, some of which are known to be realized. Now the conditions to which a form of externality must conform, in order to be affirmed, are: first, of course, that it should be experienced, or legitimately inferred from something experienced; but secondly, that it should conform to certain logical conditions, detailed in Chapter III., which may be summed up in the relativity of position. Now what Metageometry has done, in any case, is to suggest the proof that the second of these conditions is fulfilled by non-Euclidean spaces. Euclid is affirmed, therefore, on the ground of immediate experience alone, and his truth, as unmediated by logical necessity, is merely assertorical, or, if we prefer it, empirical. This is the most important sense, it seems to me, in which non-Euclidean spaces are possible. They are, in short, a step in a philosophical argument, rather than in the investigation of fact: they throw light on the nature of the grounds for Euclid, rather than on the actual conformation of space[102]. This import of Metageometry is denied by Lotze, on the ground that non-Euclidean logic is faulty, a ground which he endeavours, by much detail and through many pages, to make good--with what success, we will now proceed to examine.
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].
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 _l{n}_ and _l{s}_, 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.
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.
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.
Ideas discussed in this chapter
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