An Essay on the Foundations of Geometry

à priori element in dynamics, is a further question. The reference to matter is necessitated by the homogeneity of empty space. For so long as we leave matter out of account, one position is perfectly indistinguishable from another, and a science of the relations of positions is impossible. Indeed, before spatial relations can arise at all, the homogeneity of empty space must be destroyed, and this destruction must be effected by matter. The blank page is useless to the geometer until he defaces its homogeneity by lines in ink or pencil. No spatial figures, in short, are conceivable, without a reference to a not purely spatial matter. Again, if Congruence is ever to be used, there must be motion: but a purely geometrical point, being defined solely by its spatial attributes, cannot be supposed to move without a contradiction in terms. What moves, therefore, must be matter. Hence, in order that motion may afford a test of equality, we must have some matter which is known to be unaffected throughout the motion, that is, we must have some rigid bodies. And the difficulty is, that these bodies must not only undergo no change due solely to the nature of space, but must, further, be unchanged by their changing relation to other bodies. And here we have a requisite which can no longer be fulfilled à priori: which, indeed, we know to be, in strictness, untrue. For the forces acting on a body depend upon its spatial relations to other bodies, and changing forces are liable to produce changing configuration. Hence, it would seem, actual measurement must be purely empirical, and must depend on the degree of rigidity to be obtained, during the process of measurement, in the bodies with which we are conversant. This conclusion, I believe, is valid of all actual measurement. But the possibility of such empirical and approximate rigidity, I must insist, depends on the à priori law that mere motion, apart from the action of other matter, cannot effect a change of shape. For without this law, the effect of other matter would not be discoverable; the laws of motion would be absurd, and Physics would be impossible. Consider the second law, for example: How could we measure the change of motion, if motion itself produced a change in our measures? Or consider the law of gravitation: How could we establish the inverse square, unless we were able, independently of Dynamics, to measure distances? The whole science of Dynamics, in short, is fundamentally dependent on Geometry, and but for the independent possibility of measuring spatial magnitudes, none of the magnitudes of Dynamics could be measured. Time, force, and mass are alike measured by spatial correlates: these correlates are given, for time, by the first law, for force and mass, by the second and third. It is true, then, that an empirical element appears unavoidably in all actual measurement, inasmuch as we can only know empirically that a given piece of matter preserves its shape throughout the necessary change of dynamical relations to other matter involved in motion; but it is further true that, for Geometry—which regards matter simply as supplying the necessary breach in the homogeneity of space, and the necessary term for spatial relations, not as the bearer of forces which change the configuration of other material systems—for Geometry, which deals with this abstract and merely kinematical matter, rigidity is à priori, in so far as the only changes with which it is cognizant—changes of mere position, namely—are incapable of affecting the shapes of the imaginary and abstract bodies with which it deals. To use a scholastic distinction, we may say that matter is the causa essendi of space, but Geometry is the causa cognoscendi of Physics. Without a Geometry independent of Physics, Physics itself, which necessarily assumes the results of Geometry, could never arise; but when Geometry is used in Physics, it loses some of its à priori certainty, and acquires the empirical and approximate character which belongs to all accounts of actual phenomena. 72. (3) This argument leads us to Land's distinction of physical and geometrical rigidity. The distinction may be expressed—and I think it is better expressed—by distinguishing between the conceptions of matter proper to Dynamics and to Geometry respectively. In Dynamics, we are concerned with matter as subject to and as causing motion, as affected by and as exerting force. We are therefore concerned with the changes of spatial configuration to which material systems are liable: the description and explanation of these changes is the proper subject-matter of all Dynamics. But in order that such a science may exist, it is obviously necessary that spatial configuration should be already measurable. If this were not the case, motion, acceleration and force would remain perfectly indeterminate. Geometry, therefore, must already exist before Dynamics becomes possible: to make Geometry dependent for its possibility on the laws of motion or any of their consequences, is a gross ὕστερον πρότερον. Nevertheless, as we have seen, some sort of matter is essential to Geometry. But this geometrical matter is a more abstract and wholly different matter from that of Dynamics. In order to study space by itself, we reduce the properties of matter to a bare minimum: we avoid entirely the category of causation, so essential to Dynamics, and retain nothing, in our matter, but its spatial adjectives[91]. The kind of rigidity affirmed of this abstract matter—a kind which suffices for the theory of our science, though not for its application to the objects of daily life—is purely geometrical, and asserts no more than this: That since our matter is devoid, ex hypothesi, of causal properties, there remains nothing, in mere empty space, which is capable of changing the configuration of any geometrical system. A change of absolute position, it asserts, is nothing; therefore the only real change involved in motion is a change of relation to other matter; but such other matter, for the purposes of our science, is regarded as destitute of causal powers; hence no change can occur, in the configuration of our system, by the mere effect of motion through empty space. The necessity of such a principle may be shown by a simple reductio ad absurdum, as follows. A motion of translation of the universe as a whole, with constant direction and velocity, is dynamically negligeable; indeed it is, philosophically, no motion at all, for it involves no change in the condition or mutual relations of the things in the universe. But if our geometrical rigidity were denied, the change in the parameter of space might cause all bodies to change their shapes owing to the mere change of absolute position, which is obviously absurd. To make quite plain the function of rigid bodies in Geometry, let us suppose a liquid geometer in a liquid world. We cannot suppose the liquid perfectly homogeneous and undifferentiated, in the first place because such a liquid would be indistinguishable from empty space, in the second place because our geometer's body—unless he be a disembodied spirit—will itself constitute a differentiation for him. We may therefore assume "dim beams, Which amid the streams Weave a network of coloured light," and we may suppose this network to form the occasion for our geometer's reflections. Then he will be able to imagine a network in which the lines are straight, or circular, or parabolic, or any other shape, and he will be able to infer that such a network, if it can be woven in one part of the fluid, can be woven in another. This will form sufficient basis for his deductions. The superposition he is concerned with—since not actual equality, but only the formal conditions of equality, are the subject-matter of Geometry—is purely ideal, and is unaffected by the impossibility of congealing any actual network. But in order to apply his Geometry to the exigencies of life, he would need some standard of comparison between actual networks, and here, it is true, he would need either a rigid body, or a knowledge of the conditions under which similar networks arose. Moreover these conditions, being necessarily empirical, could hardly be known apart from previous measurement. Hence for applied, though not for pure Geometry, one rigid body at least seems essential. 73. The utility, for Dynamics, of our abstract geometrical matter, is sufficiently evident. For having, by its means, a power of determining the configurations of material systems in whatever part of space, and knowing that changes of configuration are not due to mere change of place, we are able to attribute these changes to the action of other matter, and thus to establish the notion of force, which would be impossible if change of shape might be due to empty space. Thus, to conclude: Geometry requires, if it is to be practically possible, some body or bodies which are either rigid (in the dynamical sense), or known to undergo some definite changes of shape according to some definite law. (These changes, we may suppose, are known by the laws of Physics, which have been experimentally established, and which throughout assume the truth of Geometry.) One or more such bodies are necessary to applied Geometry—but only in the sense in which rulers and compasses are necessary. They are necessary as, in making the Ordnance Survey, an elaborate apparatus was necessary for measuring the base line on Salisbury Plain. But for the theory of Geometry, geometrical rigidity suffices, and geometrical rigidity means only that a shape, which is possible in one part of space, is possible in any other. The empirical element in practice, arising from the purely empirical nature of physical rigidity, is comparable to the empirical inaccuracies arising from the failure to find straight lines or circles in the world—which no one but Mill has regarded as rendering Geometry itself empirical or inaccurate. But to make Geometry await the perfection of Physics, is to make Physics, which depends throughout on Geometry, forever impossible. As well might we leave the formation of numbers until we had counted the houses in Piccadilly. Erdmann. 74. In connection with Riemann and Helmholtz, it is natural to consider Erdmann's philosophical work on their theories[92]. This is certainly the most important book on the subject which has appeared from the philosophical side, and in spite of the fact that, like the whole theory of Riemann and Helmholtz, it is inapplicable to projective Geometry, it still deserves a very full discussion. Erdmann agrees throughout with the conclusions of Riemann and Helmholtz, except on a few points of minor importance; and his views, as this agreement would lead one to expect, are ultra-empirical. Indeed his logic seems—though I say this with hesitation—to be incompatible with any system but that of Mill: there is apparently no distinction, to him, between the general and the universal, and consequently no concept not embodied in a series of instances. Such a theory of logic, to my mind, vitiates most of his work, as it vitiated Riemann's philosophy[93]. This general criticism will find abundant illustration in the course of our account of Erdmann's views. 75. After a general introduction, and a short history of the development of Metageometry, Erdmann proceeds, in his second chapter, to discuss what are the axioms of Euclidean Geometry. The arithmetical axioms, as they are called, he leaves aside, as applying to magnitude in general; what we want here, he says, is a definition of space, for which the geometrical axioms are alone relevant. But a definition of space, he says—following Riemann—demands a genus of which space shall be a species, and this, since our space is psychologically unique, can only be furnished by analytical mathematics (p. 36). Now the space-forms dealt with by Geometry are magnitudes, and conceptions of magnitude are everywhere applied in Geometry. But before Riemann, only particular determinations of space could be exhibited as magnitudes, and thus the desired definition was impossible to obtain. Now, however, we can subsume space as