The Analysis of Matter

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 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 "[Pg 118] 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[Pg 119] 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[Pg 120] 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. [Pg 121] CHAPTER XIII MATTER AND SPACE COMMON sense starts with the notion that there is matter where we can get sensations of touch, but not elsewhere. Then it gets puzzled by wind, breath, clouds, etc., whence it is led to the conception of "spirit"—I speak etymologically. After "spirit" has been replaced by "gas," there is a further stage, that of the æther. Assuming the continuity of physical processes, there must be things happening between the earth and the sun when light travels from the sun to the earth; assuming the mediæval metaphysic of "substance," as all physicists did until recently, what is happening between the earth and the sun must be happening "in" or "to" a substance, which is called the æther. Apart from metaphysical interpretations, what we may be said to know (using this word somewhat liberally) is that processes occur where there is no gross matter, and that these processes proceed, at least approximately, in accordance with Maxwell's equations. There does not seem any necessity to interpret these processes in terms of substance; indeed, I shall argue that processes associated with gross matter should also be interpreted so as not to involve substance. There must, however, remain a difference, expressible in physical terms, between regions where there is matter and other regions. In fact, we know the difference. The law of gravitation is different, and the laws of electromagnetism suffer a discontinuity when we reach the surface of an electron or proton. These differences, however, are not of a metaphysical kind. To the philosopher, the difference between "matter" and "empty space" is, I believe, merely a difference as to the causal laws governing successions of events, not a difference expressible[Pg 122] as that between the presence or absence of substance, or as that between one kind of substance and another. Physics, as such, should be satisfied when it has ascertained the equations according to which a process takes place, with just enough interpretation to know what experimental evidence confirms or confutes the equations. It is not necessary to the physicist to speculate as to the concrete character of the processes with which he deals, though hypotheses (false as well as true) on this subject may sometimes be a help to further valid generalizations. For the present, we are confining ourselves to the standpoint of physics. Whether anything further can be known or fruitfully conjectured is a matter which we shall discuss at a later stage. We want, therefore, to consider the difference in physical formulæ which is described as that between the presence and absence of matter, and also to consider briefly the difficulties as to the interchanges of energy between matter and empty space. I say "empty space" or "æther" indifferently; the difference seems to be merely one of words. One way of approaching this subject is through the connection of mass with energy.[31] In elementary dynamics, the two are quite distinct, but nowadays they have become amalgamated. There axe two kinds of mass involved in physics, of which one may be called the "invariant" mass, the other the "relative" mass. The latter is the mass obtained by measurement, when the body concerned may be moving relatively to the observer; the former is the mass obtained when the body is at rest relatively to the observer. If we call the invariant mass and the relative mass , then, taking the velocity of light as unity, if is the velocity of the body relative to the observer, we have: [Pg 123] Thus increases as increases; if is the velocity of light, becomes infinite if is finite. In fact, the invariant mass of light is zero, and its relative mass is finite. Wherever