An Essay on the Foundations of Geometry

we have determined these conditions, which are à priori necessary to any quantitative science of space, does the second question arise: what inferences can we draw, as to space, from the observed results of this quantitative science, i.e. of this measurement of spatial figures? The conditions of measurement themselves, though not results of any conception of magnitude, will be à priori, if it can be shown that, without them, experience of externality would be impossible. After this initial protest against Riemann's general philosophical position, let us proceed to examine, in detail, his use of the notion of a manifold. 63. In the first place there is, if I am not mistaken, considerable obscurity in the definition of a manifold, of which an almost verbal rendering was given in Chapter I. What is meant, to begin with, by a general conception capable of various determinations? Does not this property belong to all conceptions? It affords, certainly, a basis for counting, but if continuous quantity is to arise, we must, surely, have some less discrete formulation. It might afford a basis, for example, for the distinction of points in projective Geometry, but projective Geometry has nothing to do with quantity. Something more fluid and flexible than a conception, one would think, is necessary as the basis of continua. Then, again, what is meant by a quantum of a manifold? In space, the answer is obvious: what is meant is a piece of volume. But how about Riemann's other continuous manifold, colour? Does a quantum of colour mean a single line in the spectrum, or a band of finite thickness? In either case, what are the magnitudes to be compared? And how is superposition necessary, or even possible? A colour is fixed by its position in the spectrum: two lines in the same spectrum cannot be superposed, and two lines in different spectra need not be—their positions in their respective spectra suffice, or even, roughly, their immediate sense-quality. The fact is, Riemann had space in his mind from the start, and many of the properties, which he enunciates as belonging to all manifolds, belong, as a matter of fact, only to space. It is far from clear what the magnitudes are which the various determinations make possible. Do these magnitudes measure the elements of the manifold, or the relations between elements? This is surely a very fundamental point, but it is one which Riemann never touches on. In the former case, the superposition which he speaks of becomes unnecessary, since the magnitude is inherent in the element considered. We do not require superposition to measure quantities corresponding to different tones or colours; these can be discovered by analysis of single tones or colours. With space, on the other hand, if we seek for elements, we can find none except points, and no analysis of a point will find magnitudes inherent in it—such magnitudes are a fiction of coordinate Geometry. The magnitudes which space deals with, as we shall see in Chapter III., are relations between points, and it is for this reason that superposition is essential to space-measurement. There is no inherent quality in a single point, as there is in a single colour, by which it can be quantitatively distinguished from another. Thus the conception of a manifold, as defined by Riemann, either does not include colours, or does not involve superposition as the only means of measurement. From this dilemma there is no escape. 64. But if "measurement consists in a superposition of the magnitudes compared" (p. 256), does it not follow immediately that measurement is logically possible only where such superposition leaves the magnitudes unchanged? And therefore that measurement, as above defined, involves, as an à priori condition, that magnitudes are unchanged by motion? This consequence is not drawn by Riemann; indeed he proceeds immediately (pp. 256–7) to consider what he calls a general portion of the doctrine of magnitude (Grössenlehre), independent of measurement. But how is any doctrine of magnitude possible, in which the magnitudes cannot be measured? The reason of the confusion is, that Riemann's definition of measurement is applicable to no single manifold except space, since it depends on the noteworthy property that what we measure in Geometry is not points, but relations between points, and the latter, though not the former, may of course be unaltered by motion. Let us try, in illustration, to apply Riemann's definition of measurement to colours. We must remember that motion, in dealing with the colour manifold, means—not motion in space but—motion in the colour manifold itself. Now since every point of the colour manifold is completely determined by three magnitudes, which are given in fact, and cannot be arbitrarily chosen, it is plain that measurement by superposition—involving, as it does, motion, and therefore change in these determining magnitudes—is totally out of the question. The superposition of one colour on another, as a means of measurement, is sheer nonsense. And yet measurement is possible in the colour-manifold, by means of Helmholtz's law of mixture (Mischungsgesetz); but the measurement is of every separate element, not of the relations between elements, and is thus radically different from space-measurement[81]. The elements are not, like points in space, qualitatively alike, and distinguished by the mere fact of their mutual externality. What we have, in colours, is three fundamental qualitatively distinct elements, out of certain proportions of which we can build up all the other elements of the manifold—each of the resulting elements having the same combination of qualitative diversity and similarity as the three original elements. But in space, what could we make of such a procedure? Given three points, how are we to combine them in certain proportions? The phrase is meaningless. If some one makes the obvious retort, that we have to combine lines, not points, my rejoinder is equally obvious. To begin with, lines are not elements. Metaphysically, space has no elements, being, as the sequel will show, mere relations between non-spatial elements. Mathematically, this fact exhibits itself in the self-contradictory notion of the point, or zero magnitude in space, as the limit in our vain search for spatial elements. But even if we allow the line to pass as the spatial element, what does the combination of three lines in definite proportions give us? It gives us, simply, the coordinates of a point. Here again we see a great difference between the colour and space-manifolds. In colours, the combination of magnitudes gives a new magnitude of the same kind; in space, it defines, not a magnitude at all, but a would-be element of a different kind from the defining magnitudes. In the tone-manifold, we should find still different conditions. Here, no one of the measuring magnitudes can vanish without the tone vanishing too, and all three are so bound up together, in the single resulting sensation, that none can exist without a finite quantity of the others. They are all qualitatively different, both from each other, and from any possible tone, being constituents of it, as mass and velocity are constituents of momentum. All these different conditions require to be examined, before a manifold can be completely defined; and until we have conducted such an examination in detail, we cannot pronounce as to the à priori or empirical nature of the laws of the manifold. As regards space, I have attempted such an examination in the third and fourth chapters of this Essay. 65. I do not wish to deny, however, the great value of the conception of space as a manifold. On the contrary, this conception seems to have become essential to any treatment of the question. I only wish to urge that the purely algebraical treatment of any manifold, important as it may be in deducing fresh consequences from known premisses, tends rather to conceal than to make clear the basis of the premisses themselves, and is therefore misleading in a philosophical investigation. For mathematics, where quantity reigns supreme, Riemann's conception has proved itself abundantly fruitful; for philosophy, on the contrary, where quantity appears rather as a cloak to conceal the qualities it abstracts from, the conception seems to me more productive of error and confusion than of sound doctrine. We are thus brought back to the point from which we started, namely, the falsity of Riemann's initial disjunction, and the consequent fallacy in his proof of the empirical nature of the axioms. His philosophy is chiefly vitiated, to my mind, by this fallacy, and by the uncritical assumption that a metrical coordinate system can be set up independently of any axioms as to space-measurement[82]. Riemann has failed to observe, what I have endeavoured to prove in the next chapter, that, unless space had a strictly constant measure of curvature, Geometry would become impossible; also that the absence of constant measure of curvature involves absolute position, which is an absurdity. Hence he is led to the conclusion that all geometrical axioms are empirical, and may not hold in the infinitesimal, where observation is impossible. Thus he says (p. 267): "Now the empirical conceptions, on which spatial measurements are based, the conceptions of the rigid body and the light-ray, appear to lose their validity in the infinitesimal: it is therefore quite conceivable that the relations of spatial magnitudes in the infinitesimal do not correspond to the presuppositions of Geometry, and this would, in fact, have to be assumed, as soon as it would enable us to explain the phenomena more simply." From this conclusion I must entirely dissent. In very large spaces, there might be a departure from Euclid; for they depend upon the axiom of parallels, which is not contained in the axiom of Free Mobility; but in the infinitesimal, departures from Euclid could only be due to the absence of Free Mobility, which, as I hope my third chapter will show, is once for all impossible. Helmholtz. 66. Helmholtz, like Riemann, was important both in the mathematics and in the philosophy of Geometry. From the mathematical point of view, his work has been already considered in Chapter I.; the consideration of his philosophy, which must occupy us here, will be a more serious task. Like Riemann, he endeavoured to prove that all the axioms are empirical, and like Riemann, he based his proof chiefly on Metageometry. He had an additional resource, however, in the physiology of the senses, which first led him to reject the Transcendental Aesthetic, and enabled him to attack Kant from the psychological as well as the mathematical side[83]. The principal topics, for a criticism of Helmholtz, are three: First, his criterion of the à priori; second, his discussion with Land as to the "imaginability" of non-Euclidean spaces; third—and this is by far the most important of the three—his theory of the dependence of Geometry on Mechanics. Let us discuss these three points successively. 67. Helmholtz's criterion of apriority is difficult to discover, as he never, to my knowledge, gives a precise statement of it. From his discussion of physical and transcendental Geometry[84], however, it would appear that he regards as empirical whatever applies to empirical matter. For he there maintains, that even if space were an à priori form, yet any Geometry, which aimed at an application to Physics, would, since the actual places of bodies are not known à priori, be necessarily empirical[85]. It seems the more probable that he regards this as a possible criterion, as it is adopted, in several passages, by his disciple Erdmann[86], and so strange a test could hardly be accepted by a philosopher, unless he had found it in his master. I have called this a strange test, because it seems to me completely to ignore the work of the Critical Philosophy. For if there is one thing which, one might have hoped, had been made sufficiently clear by Kant's Critique, it is this, that knowledge which is à priori, being itself the condition of possible experience, applies—and in Kant's view, applies only—to empirical matter. Helmholtz and Erdmann, therefore, in setting up this