The Analysis of Matter

REPEATEDLY, in previous discussions, we have come up against the problem of measurement. It is time to consider it on its own account, both how it is to be defined, and in what circumstances it is possible. In the first place, what do we mean by measurement? Clearly we do not mean any method of assigning numbers to a collection of objects; there must be properties of importance connected with the numbers assigned. We do not say that the books in the British Museum are "measured" by their press-marks. Given any collection whose cardinal number is less than or equal to , we can assign some or all of the real numbers as "press-marks" of the several members of the collection. Given any collection of terms, it can be arranged in a Euclidean or non-Euclidean space of any known sort with any finite number of dimensions, and when so arranged it will be amenable to the whole of metrical geometry. But the "distance" between two terms of the collection, when it is defined in this way, will, in general, be quite unimportant, in the sense that it will have only such properties as follow tautologically from its definition, not such further empirical properties as would make the definition valuable. So long as this is the case, there is no reason to prefer one to another of the various incompatible systems of distances which pure mathematics would allow us to assign. Let us take an illustration. In projective geometry we start from a set of axioms which say nothing about quantity, and do not even obviously involve order. But it is found that they do lead to an order, and that, by means of the order,[Pg 110] co-ordinates can be assigned to points. These co-ordinates have a definite projective meaning: they represent the series of quadrilateral constructions required to reach the point in question from certain given initial points, (I omit complications concerning limits; these are dealt with in the chapter "Projective Geometry" in The Principles of Mathematics.) In this case, it may seem doubtful whether we have measurement or not. We have assigned co-ordinates in a manner which preserves the order-relations of points, and it turns out that the ordinary distance between two points is a simple function of their projective co-ordinates, though the function is somewhat different according as space is Euclidean, hyperbolic, or elliptic. It is just because of this difference that we shall not say we have "measured" distances when we have introduced projective co-ordinates. These co-ordinates, for example, will not tell us, even approximately, how long it would take to walk from one place to another, and this is the sort of thing that measurement ought to tell us. What, then, is meant when it is said that, in the theory of relativity, there is a metrical relation of interval? Let us take up the matter at the point where Eddington leaves it. He suggests that all that is needed is "comparability" between two point-pairs, or, as he says, between two "displacements." (We may leave aside for the moment the question whether this is only to hold for point-pairs which are very near together.) This language seems somewhat vague; let us try to give it precision. Suppose that between two point-pairs there is sometimes, but not always, a symmetrical transitive relation . Then we can define as "the distance between and " the class of all point-pairs having the relation to (). If now instead of ) we write , we shall have: [Pg 111] From these two it follows that every pair of objects , in the field of is such that This seems to be as much as is strictly implied by Eddington's words, but it is certainly not all that we need. Nor does it become sufficient if we add: There must be a connection between distances and ordinal relations, there must be ways of adding 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