The analysis of matter

THE notion of causality has been greatly modified by the substitution of space-time for space and time. We may define causality in its broadest sense as embracing all laws which connect events at different times, or, to adapt our phraseology to modern needs, events the intervals between which are time-like. Now owing to the fact that the formula for is formally the same for time-like and for space-like intervals, there is no longer the difference that formerly existed between causal and geometrical relations. Geodesics are geometrical, but they are also the paths of material particles. It is hardly correct to say that a particle moves in a geodesic; it is more correct to say that a particle is a geodesic (though not all geodesics are particles). To say that a particle moves in a geodesic is to use language appropriate to the conception of a space which persists through time, involving the notion of a position which may be occupied either at one time or at another. We think, for example, that it is possible to move from to or from to ; but such a view is incompatible with the theory of space-time. According to that theory, every position of a body has a date, and it is impossible to occupy the same position at another date, since the date is one of the co-ordinates of the position. When we travel from to , the date is continually advancing; the return journey, having different dates, does not cover the same route. Thus geometry and causation become inextricably intertwined.

Dr A. A. Robb has laid stress upon the fact that, when two events have a space-like interval, there can be no direct causal relation between them. This means that, given two such events and , if any inference is possible from the one to the other, it must be by way of a common causal ancestor. Two men may see the sun at the same moment, so that the interval between their percepts is space-like; the inference that so-and-so is seeing the sun now arises from our knowledge of radiation, and requires that we should trace his percept and our own to a common ancestry in the sun. We may therefore distinguish time-like and space-like intervals by saying that the former occur where there is some direct causal relation, while the latter occur where both events are related to a common ancestor or a common descendant. And possibly the magnitude of the interval may be derivable from the magnitude of the causal relation. But if this is to be possible, it will be necessary to achieve considerable precision as to what we mean by causal relations.

As we saw in Part II., perception as a source of knowledge concerning physical objects would be impossible if there were not, in the physical world, semi-independent causal chains, or causal lines as we may call them. The light which comes to us from a printed page retains the structure of the page; if it did not, reading would be impossible. The retention is only approximate; it ceases at a distance from the book. And it ceases within the eye if we have defective vision. But where there is such failure, perception ceases—or rather, it fades away as the failure to preserve structure increases. Thus it is essential to perception as a source of knowledge that there should be in the world causal series which are, within limits, independent of the rest of the world.

Another point concerning causation emerges from the consideration of perception. A number of simultaneous percepts—e.g. the letters of a word which we read at a glance—are to be regarded as "co-punctual" in the sense of our two preceding chapters. Each of these percepts has its own causal antecedents, different from those of the other percepts. It is true that there may be mutual modification—e.g. a colour looks different in the neighbourhood of another colour from what it looks against a dark background. But this is recognized as "modification," i.e. as effecting a change from a norm, which must remain within limits if perception is to be successful. Thus the percipient is the meeting-place of a number of more or less independent causal series—as many, at least, as there are distinguishable elements in his total momentary perceptual field. But although these lines have converged upon him more or less independently, the totality of his percepts now becomes a causal unit, as is seen in mnemic phenomena. Given a number of simultaneous percepts, a percept very similar to one of them, occurring on a future occasion, recalls something similar to the others, or at least may do so; here the co-punctuality of the percepts is essential to the character of their total effect.

In the physical world, the same sort of thing must be supposed to occur, though to a less striking degree. According to the theory of Chapter XXVIII., any event in the physical world occupies a finite region of space-time, whose finiteness consists in the fact that the said event is compresent with events which are not compresent with each other. On the analogy of mnemic phenomena, a group of co-punctual events may have effects which would have been impossible if the events had not been co-punctual. That is the reason why physics is compelled to resort to points in stating its causal laws. Until we have a complete group of co-punctual events, i.e. a point, we cannot be quite sure as to the effect which will follow from any one of the events; such knowledge as we can have will be more or less approximate.

It is these two opposite laws, of approximately separable causal lines on the one hand, and interactions of co-punctual events on the other, which make the warp and woof of the world, both physical and mental. In this chapter, I want to attain more precision as to the separable causal lines.

The possibility of perception, as we have seen already, depends upon the occurrence in the physical world of processes which may be called "radiations," provided the word is used somewhat more widely than is customary. The processes commonly called radiations are, naturally, the most perfect examples. In these, when they are undisturbed, we have a condition of some kind which spreads outward from a centre, changing in an apparently continuous manner as it travels. Something may be met with on the way which alters the law of change, or even stops the radiation in some direction altogether; but in the absence of obstacles the process proceeds according to its own intrinsic laws. The public senses—sight, hearing, and smell—depend upon radiations, in a generalized sense in the case of smell. Bodily senses, including touch, are more analogous to electric currents in their manner of propagation: they travel along nerves, but not through air or empty space. The public senses, also, travel along nerves, but the disturbance in the nerves is a prolongation, with alterations, of a process in the world outside the percipient's body, which is not the case with the bodily senses. It is owing to the existence of radiations that we live in a common world, since this depends upon the fact that neighbouring percipients receive similar stimuli at about the same time. The physical account of radiations is, however, very different in different cases. In the case of smell, the emission theory is universally accepted: we smell a body because portions of it travel from it to the nose. In the case of sound, only a process, not actual matter, is transmitted, but the process is in matter. In the case of light, if we accept the undulatory theory, the process consists of a transverse vibration, which may be said to be in the æther if that brings comfort to the speaker, but is certainly not in ordinary matter. If we could accept the light-quantum theory, we should still suppose that there is some periodic process, such that the action during one period is (Planck's constant); the light consists of (so to speak) atoms, each of which is such a process. There is a great difference of physical importance between these three cases of smell, sound, and light; the first is quite unimportant physically, the second a somewhat late development from more fundamental principles, the third a corner-stone of physical theory.

In the ideal case of a radiation, a few observations should suffice to determine its centre, and then, its laws being known, we could infer the whole connected system of events which constitutes it, in so far as the events enter into physical laws. The case of light from a fixed star very nearly realizes the ideal. The places in the universe where the light encounters obstacles are very few, though unfortunately they include the places where we live. It is because this example of light in vacuo is so nearly perfect that we know as much as we do about astronomy.

Radiation independent of matter, however, is only one form of causal process in the physical world. Apart from quantum changes, there are at least two others which are of great importance: one is the motion of matter, and the other is the transmission of a process by matter. The difference involved is essentially one as to causal laws: one sort of causal connection between events makes us regard them as part of the history of one piece of matter, while another does not, but there is no more intimate connection between an electron at one time and the same electron at another time than between two parts of one light-ray. Let us consider for a moment the nature of the causal laws which define one piece of matter.

One prima facie difference is that the propagation of light is spherical (or conical, in the case of a directed beam), whereas the motion of matter is linear. The history of a piece of matter is a "world-line"; the history of a light-wave is not. This difference may no longer exist if some adaptation of the light-quantum theory can be made satisfactory; but, if so, we shall feel that the difference between light and matter has been much diminished. Another difference is the relative indestructibility of matter. One form of energy changes into another, but the energy represented by the proper mass of an electron or proton is not known to change into other forms, and apparently never does so under terrestrial conditions: it does not radiate at all in any circumstances that we can produce or observe. Then there is the fact that the velocity of a body relative to any observer is always less than that of light. But in spite of the doubt as to light-quanta, the main feature of the causal laws that constitute matter seems to be their linear rather than spherical character. It is this that enables us to locate a given piece of matter at a given time. The light emitted by a flash is, at a given moment, diffused over the surface of a sphere, but an electron is as concentrated at one time as at another, and does not tend to spread itself out. A unit of matter may, therefore, be appropriately defined as a "causal line."

Before pursuing this subject, however, it will be well to dispose of the other kind of causal process which we mentioned just now, namely the transmission of a process by matter. This is itself of two sorts, one illustrated by sound, the other by the conduction of an electric current. In the case of sound we have a radiation; in the other case we have a more or less linear process. In each case, however, actual pieces of matter move, and cause others to move. The former belongs to the notion of a "causal line," to which we shall return in a moment. The latter belongs to the causal laws as to the interactions of different pieces of matter, which I do not wish to consider until I have elicited the intrinsic causal laws which constitute the definition of one piece of matter. These, as we saw, have been somewhat obscured by the notion of substance, which made it plausible to take for granted certain connections between events at different times, which, for us, are causal, and demand explicit recognition. It is these intrinsic laws which replace substance that I wish to consider now, leaving the interactions between different pieces of matter for a later stage.

What, then, constitutes a "causal line"? In other words, what constitutes one electron? Before asking ourselves what makes us call an electron at one time the same as an electron at another time, it may be well to ask ourselves: What constitutes an electron at one time?

We must find some reality for the electron, or else the physical world will run through our fingers like a jelly-fish. There is the same sort of reason, however, for not regarding an electron as an ultimate particular as there was for refusing this status to a space-time point. The electron has very convenient properties, and is therefore probably a logical structure upon which we concentrate attention just because of these properties. A rather haphazard set of particulars may be capable of being collected into groups each of which has very agreeable smooth mathematical properties; but we have no right to suppose Nature so kind to the mathematician as to have created particulars with just such properties as he would wish to find. We have, therefore, to ask ourselves: Can we construct an electron out of events, in the same sort of way in which we constructed space-time points? To this inquiry we must now address ourselves, confining ourselves, at first, to the electron at one time.

When I speak of "electrons" in this discussion, I shall include "protons," since everything that is to be said about the one is to be said about the other also.

We do not know much about the contents of any part of the world except our own heads; our knowledge of other regions, as we have seen, is wholly abstract. But we know our percepts, thoughts, and feelings in a more intimate fashion. Whoever accepts the causal theory of perception is compelled to conclude that percepts are in our heads, for they come at the end of a causal chain of physical events leading, spatially, from the object to the brain of the percipient. We cannot suppose that, at the end of this process, the last effect suddenly jumps back to the starting-point, like a stretched rope when it snaps. And with the theory of space-time as a structure of events, which we developed in the last two chapters, there is no sort of reason for not regarding a percept as being in the head of the percipient. I shall therefore assume that this is the case, when we are speaking of physical, not sensible, location.

It follows from this that what the physiologist sees when he examines a brain is in the physiologist, not in the brain he is examining. What is in the brain by the time the physiologist examines it if it is dead, I do not profess to know; but while its owner was alive, part, at least, of the contents of his brain consisted of his percepts, thoughts, and feelings. Since his brain also consisted of electrons, we are compelled to conclude that an electron is a grouping of events, and that, if the electron is in a human brain, some of the events composing it are likely to be some of the "mental states" of the man to whom the brain belongs. Or, at any rate, they are likely to be parts of such "mental states"—for it must not be assumed that part of a mental state must be a mental state. I do not wish to discuss what is meant by a "mental state"; the main point for us is that the term must include percepts. Thus a percept is an event or a group of events, each of which belongs to one or more of the groups constituting the electrons in the brain. This, I think, is the most concrete statement that can be made about electrons; everything else that can be said is more or less abstract and mathematical.

We have arrived at the conclusion that an electron at an instant is a grouping of events; the question is: what sort of group is it? Obviously it includes all the events that happen where the electron is. If we may regard the electron as a material point, the events constituting an electron will have the two characteristic properties of points, viz. any five are co-punctual, and not all sub-classes of four events are co-punctual with any event outside the group. I do not know whether there is any valid ground for supposing that an electron is of finite size; none of the usual arguments seem at all conclusive, since they only show the forces developed in the neighbourhood of an electron. However, it is usual to assume a finite size, and for us the matter is one of indifference. If we assume a finite size, the events belonging to the electron can be grouped into many points, not only into one; in this case, the electron is a group of points, i.e. a class of classes of events. It will save circumlocution to speak of the electron as a point, and leave it to the reader to make the necessary verbal alterations for adaptation to the hypothesis of finite size. But it should be remembered that in Heisenberg's theory the electron is neither a point nor of finite size, since ordinary spatial conceptions are inapplicable to it. For the moment, we will, however, confine ourselves to the older theory of the electron.

If the electron is a point, it is a material point, and thus differs from points in empty space. This difference, I believe, does not consist in anything characteristic of the electron at an instant, but in its causal laws. What distinguishes a material point from a point of empty space-time is that we can recognize a series of earlier and later material points as all parts of the history of one electron. In the Newtonian theory, one could say the same of a point of absolute space; but with the abandonment of absolute space we have become unable to regard a point at one time as in any sense the same as a point at another time, except in the case of a material point. The existence of this connection may be taken as the definition of "matter"; and obviously the connection is causal.

In order to develop this further, we must return to the view suggested in connection with perception, that events occur, usually, in groups arranged about centres. These centres may be taken to be places where there is matter. It is found that, given events arranged about a centre at one time, there are generally similar events arranged about neighbouring centres at slightly earlier or later times. By taking the centre very small, and by continually diminishing the time-like interval concerned, this statement can be made more and more nearly true; in the limit, when stated in the language of differentials, it may be exactly true, except where quantum phenomena are concerned. In their case, continuity is not the criterion, at least not continuity in all respects. There is continuity in some respects, and in others there is a jump of a definite amount connected with the quantum theory. This case shows, however, that continuity is not the essence of material identity; the essence is inferribility of a group of phenomena at one time from a group at another, when both groups are arranged about centres.[68] The time must be very short, and the inference is only approximate, except in the limit, as the time tends towards zero. Moreover, the time of the group is not any of the times at which the several members of the group occur, but the calculated time at which the group began to be propagated from the centre. The centre is "where the piece of matter is," and the route of the piece of matter is determined by the differential equations which result from the above principle. But as to what are the actual events at the centre, we know nothing except what follows from the fact that our percepts and "mental states" are among the events which constitute the matter of our brains.

Thus each material unit is a causal line whose neighbouring points are connected by an intrinsic differential law. The simplest form of such a law is the first law of motion, from which it follows that if a body covers a given distance in a very short time, it will cover a very nearly equal distance in the next very short time. I conceive—though this is conjectural—that, given any event anywhere in space-time, there is usually some qualitatively very similar event in a neighbouring place in space-time, and that, if there is any measurable relation between the two events, the "velocity" of the change varies continuously, so that at a third neighbouring point there will be an event differing from the second by very nearly the same amount as that by which the second differed from the first, provided the interval between the second and third points is equal to that between the first and second. This, together with the fact that events can be grouped about centres by the sort of laws which we have called "perspective," seems to explain the utility of matter in stating the causal laws of the physical world. But there is need of caution owing to quantum phenomena, as explained in the preceding paragraph. Continuity is the rule, but it may have exceptions. So long as the exceptions are subject to ascertainable laws, they do not make the whole system impossible.

So far, I have said nothing about extrinsic causal laws, i.e. those which we naturally regard as exemplifying the influence of one piece of matter upon another. Einstein's theory of gravitation has thrown a new light upon these; but this is matter for a new chapter.