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, 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: 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.

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 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.

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 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. 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 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 number. Interval, as we have seen, is a relation, and smaller intervals are not parts of it. If and are equal intervals in a straight line, we say that the interval is double of each, and we think of it as the "sum" of and . But it is only by a convention, though an almost irresistible one, that we assign as the measure of a number double that which we assign as the measure of or of . And to say that is the "sum" of and is to say something very ambiguous, since the word "sum" has many meanings. When and are considered as vectors, we may say that is their sum even when they are not in one straight line. Again, given suitable definitions, we may say that the points between and are the sum (in the logical sense) of the points between and , and between and ; this will only hold if is a straight line. But the distance between and , considered as a relation, is not properly the "sum," in any recognized sense, of the distances , . Thus all geometrical quantities are "intensive." This shows that the distinction of intensive and extensive is unimportant.

In connection with interval, it is worth while to compare its formal characteristics with those of similarity. We saw that, in the generalized geometry with which Eddington ends, we want a relation of four neighbouring points, expressing the fact that they form a parallelogram. But we met with certain difficulties owing to the fact that this is only supposed to be possible for an infinitesimal quadrilateral, which is a figment of the mathematical imagination, and that it was not wholly easy to see how to substitute a procedure by means of limits. We were led to the suggestion that, instead of saying " is a parallelogram," we should have to say " is more nearly a parallelogram than ." Perhaps this could be somewhat simplified. Suppose we say: " is more nearly a parallelogram than ." And perhaps this could be still further simplified so as to take the form: " is more like than is." We here suppose that between any two points there is a relation, which we will not call distance, but (say) "separation," and that this relation, like a shade of colour, is capable of a greater or less resemblance to another of the same kind. In a Euclidean space, two finite separations finitely separated may be exactly similar in the relevant respects; we then have a finite parallelogram.

But in the generalized geometry that we are considering, we shall say that no two separations are exactly alike, though they are capable of indefinite approximation to exact likeness. Let us see how far this will take us.

In the case of similarity, we have a relation which is capable of degrees, and may be called "quasi-transitive"—i.e. if is very like , and is very like , then must be rather like . This is just the sort of thing required for Weyl's geometry. Consider four points, , , , , and suppose that is rather like . Take a series of points forming a continuous route from to , without loops; this can be done by purely ordinal methods to be explained later. Suppose that among these points there are some, such as which make more like than is. We may suppose that these points have a limit or last term, which we will call . We can then similarly proceed along to a point which gives more like than for any other point on . We have then done nearly as well as possible, if not quite, with the three points , , as starting-points. By means of suitable postulates, we could insure that a construction of the above sort, carried out repeatedly without changing the points , , , should at last end with a definite point such that is more like than any other distance from is. We may call the figure a "quasi-parallelogram." Now let , , ... , ... be a series of points on a route from to . Then proceed to take points , , ... between and on some route, and form the quasi-parallelograms having one corner at , one corner at and one at , the fourth being called .

If, as Weyl assumes, infinitesimal distances which have one end in common are comparable, this must be taken to mean that two small finite distances are capable of a resemblance which may be called "quasi-equality," which grows more nearly complete resemblance as the distance grows smaller. We may assume, as before, that, given a point and a definite route from to , there will be one definite point on this route such that is more nearly equal to than is any other distance by on the route in question. We shall then say that and are "quasi-equal." Take also ... quasi-equal, and , ... quasi-equal. In this way we can construct a co-ordinate mesh with axes , . And we can now construct what will be in effect straight lines through : take all the points which are the corners opposite to of quasi-parallelograms , for different initial points , subject to quasi-equality between and . These points may be regarded as forming the quasi-straight line whose equation is . (Irrationals can be dealt with by the usual methods.) This quasi-straight line will start from in a certain direction, and may, for differential purposes, be regarded as really a straight line. It is not worth while to proceed further, since it is obvious that we have the necessary material.

Degrees of similarity may be, in a sense, measured by quasi-transitiveness. Suppose that , , , ... each have quasi-equality with the next. It may or may not happen that has quasi-equality with . One may presume that this will happen if and are very small and is not very large. Similarly, or rather a fortiori, we cannot infer that has quasi-equality with . The larger the value of for which such an inference remains true, the closer is the resemblance between and or between and . It is to be assumed that, by continually diminishing and the number of steps for which the inference is permitted can be increased without finite limit.

If the above is in any degree valid, it would seem that, if space-time is continuous, spatio-temporal measurement depends theoretically upon qualitative similarity, capable of varying degrees, between relations of pairs of points. It is not suggested that the analysis cannot be carried further, but only that this is a valid stage in the process of explaining what is meant by the quantitative character of intervals and by their measurement as numerical multiples of units.