The Analysis of Matter

distances, and there must be ways of inferring new distances from a certain number of data, as in . If all these conditions are fulfilled, we can then proceed to ask whether our distances have any further important physical properties. The sort of relation that will not do is illustrated if we take to mean that and have the same apparent dimensions in the visual field of a certain observer—e.g. the diameters of the sun and moon will approximately have this relation, which is symmetrical and transitive, but physically unimportant. Let us see what is necessary in order to get a definition of distance which will have as many as possible of the properties possessed by distance in elementary geometry. [Pg 112] If we confine ourselves to three dimensions, we can at once define a plane: it will consist of all points equidistant from two given points. The points in this plane which are equidistant from two given points in it lie on a straight line; we may take this as the definition of a straight line. Thus given two points, , , we can define the middle point of it is the point on which is equidistant from and . We shall need an axiom to the effect that this point always exists and is always unique. Thus we can halve distances and double them: we shall of course define as half of . From this point onwards, the assignment of numerical measures to our distances offers no difficulty. It is therefore only necessary to scrutinize what has already been said. In ordinary Euclidean geometry, there is exactly one point on a plane which is equidistant from three given points on the plane; it is the centre of the circumscribed circle. In three dimensions, there is one point equidistant from four given points; in four, from five. This last holds also in the special theory of relativity, and even in the general theory so long as the distances concerned are small. If we take a point () near the origin, another point () is equidistant from this point and the origin if (where the have their values at the origin), which is a simple equation in . Four such equations give a unique set of values for (). Thus there is just one point equidistant from five given points close together. Moreover, a simple equation, which we may take to be that of the part of a plane near the origin, gives the locus of points near the origin and equidistant from it and a neighbouring point. In fact, as we should expect, for small distances everything proceeds as in elementary geometry, given the formula for . But the mere assumption that there is such a relation as between point-pairs does not yield these results, since it does not imply the interrelation of distances which is given by the formula for . Nevertheless, it does suffice theoretically as a basis of measurement, since, as we have seen, it enables us to halve distances and double them, and therefore to assign numbers to them. This shows that the geometry of relativity, even in its most general and abstract form, assumes a good deal more than the mere possibility of measurement, which, in itself, is of very little value. In itself, it does not lead to a geometry; this only results when there is some interconnection between different measures. It may be asked whether, when the geometry of relativity is generalized to the utmost, any genuinely quantitative element remains in its formulæ. We start with an ordered four-dimensional manifold, and we assign co-ordinates subject to[Pg 113] the sole restriction that their order-relations are to reproduce those of the given manifold. We then proceed to find formulæ (tensor-equations) which hold equally in all systems of co-ordinates satisfying the above condition. It might seem a possibility that such formulæ really express only ordinal relations, and that the sole advantage of co-ordinates lies in the fact that they provide names for the terms of a manifold of the required sort. (They do not provide names for all of them; the number of names is , and therefore only a vanishing proportion of real numbers can be named—i.e. expressed by means of a formula of finite complexity which employs integers.) This possibility requires investigation. The problem can be discussed equally well in two dimensions. In Gauss's theory of surfaces, a sphere and an ellipsoid, e.g. are distinguishable by the fact that there is an irreducible difference between the formulæ for which hold for the two surfaces when expressed in terms of two co-ordinates; this expresses the fact that the measure of curvature is constant in the case of the sphere, but not in the case of the ellipsoid. Yet from a purely ordinal point of view, such as that of analysis situs, the two figures are indistinguishable. What, exactly, is added to make the difference? This problem is essentially the same as that which arises in the general theory of relativity. In part, the answer in this case is simple. What is added is the comparability of distances in different directions. So long as our apparatus is purely ordinal, we can say of three points which have the order that is nearer to than is, but we cannot say anything analogous of three points which are not in a row—I do not say "in a straight line," because the concept involved is more general, as will appear later. But although this is part of the answer, it does not seem to be the whole, since our relation also enabled us to compare distances not having a common origin. [Pg 114] It seems that what distinguishes distance as required in geometry from such a relation as "subtending a given angle at a given point" is the absence of reference to anything external. When the distance between two points is equal to the distance between two others, we are supposed to have a fact which does not demand reference to some other point or points. In fact, this is the reason why the "interval" has been substituted for distance: the latter, as hitherto conceived, was found to depend upon the motion of the co-ordinate frame, and thus to be not an intrinsic geometrical relation. The distance, if it is to serve its purpose, must be a function of the two points exclusively, and must not involve any other geometrical data. Here, for relativity purposes, "geometry" includes "kinematics." The angle which two points subtend at a given point becomes a function of three points as soon as the given point is thought of as variable. There must be no such way of turning the distance between two points into a function involving other variables also. I am not sure, however, whether it is necessary to introduce this somewhat difficult consideration. In ordinary geometry, the points at a given distance from a given point lie on the surface of a sphere; but if we define the distance as the angle , where is a fixed point, the points at a given distance from lie on a cone. Now a sphere and a cone are distinguishable in analysis situs. Thus the above undesirable definition could be excluded by insisting that points at a given distance from a given point are to form an oval figure. In relativity theory, this is not true of points having zero interval from a given point; indeed, it is only true when the interval concerned is space-like. But it is possible to specify the characteristics, for analysis situs, of the three-dimensional surface of constant distance from a given point. These might be added to the postulate that distance exists. Whether, in some such way, we could overcome the apparent necessity for[Pg 115] distinguishing between a sphere and an ellipsoid, making the difference relative to the definition of distance, I do not feel sure, though obviously the question must be easily soluble. Every principle of measurement which is to be used in practice must be such that important empirical laws are connected with measures. There will always be an infinite number of ways of correlating numbers with the members of a class whose cardinal number is less than or equal to . Some of these may be important, but most must be unimportant. Some conditions can be laid down. In the first place, the members of the class concerned may be obviously capable of an order which is causally important. If we take all the patches of colour that ever have been or will be perceived, they have in the first place an order in space-time, which is obviously important causally; in this order, no two of them occupy the same position—i.e. the relations concerned are all asymmetrical. But they have also an order as shades of colour and as of varying brightness. In this order there are symmetrical transitive relations—e.g. between two patches of exactly the same shade. Physics professes to correlate also these further characteristics of colours with spatio-temporal quantities such as wave-lengths. This would not be plausible if continuous alterations of quality were not correlated with continuous alterations in the correlated physical quantities. Whenever we notice a qualitative series, such as that of colours of the rainbow, we assume that it must have causal importance, and we insist that numbers used as measures shall have the same order as the qualities which they measure. The former is a postulate, the latter a convention. Both have proved highly successful, but neither is an a priori necessity. There are orders which are obviously of no causal importance—e.g. alphabetical order among human beings. Human beings, like colours, have various orders that are causally important—the space-time order, order of height.[Pg 116] weight, income, intelligence as measured by Professor X's tests, etc. But alphabetical order would never be thought important; no one would hope to found a biometric calculus upon a system in which a human being had co-ordinates depending upon the alphabetical order of his name. Generally speaking, it would seem that the simplest relations are the most important. Here I am using a purely logical test of simplicity: taking propositions in which the given relation occurs, there will be some having the smallest number of constituents compatible with the mention of that relation; and again, a relation may be a molecular compound of other relations—i.e. a disjunction, conjunction, negation, or complex of all these. A relation which is molecular has always a certain definite number of atoms; a relation which is not molecular is called atomic, and has then a definite number of terms in the simplest propositions in which it occurs. An atomic relation is simpler in proportion to the fewness of its terms; a molecular relation, in proportion to the fewness of its atoms. There is much empirical reason to think that the laws of a science become more important and comprehensive as the relations involved become simpler. The relation of a man to his name is of immense complexity, whereas we may suppose that the relation upon which interval depends is fairly simple. And the qualitative order of colours alluded to above is also simple, so long as we are thinking of colours as given in perception, not as interpreted in physics. Such simple relations should, as far as possible, be the basis for systems of measurement. There is a traditional distinction between extensive and intensive quantities, which is somewhat misleading when taken seriously. The theory is that extensive quantities are composed of parts and intensive quantities are not. The only truly extensive quantities are numbers and classes. Where finite classes are concerned, the number of their terms may be taken[Pg 117] as a measure of them, and they have parts corresponding to all smaller numbers. But in geometry we are never concerned with quantities which have parts. The number of points in a volume, whether large or small, is always in the usual kinds of geometry; thus magnitude has nothing to do with