The analysis of matter

THE physics of Newton, considered as a deductive system, had a perfection which is absent from the physics of the present day. Science has two purposes, each of which tends to conflict with the other. On the one hand, there is a desire to know as much as possible of the facts in the region concerned; on the other hand, there is the attempt to embrace all the known facts in the smallest possible number of general laws. The law of gravitation accounted for all the facts about the motions of the planets and their satellites which were known in Newton's day; at the time, it exhibited the ideal of science. But facts and theories seem destined to conflict sooner or later. When this happens, there is a tendency either to deny the facts or to despair of theory. Thanks to Einstein, the minute facts which have been found incompatible with the natural philosophy of Newton have been fitted into a new natural philosophy; but there is not yet the complete theoretical harmony that existed while Newton was undisputed.

It is necessary to say something about the Newtonian system, since everything subsequent has arisen as an amendment to it, not as a fresh start. Most of the fundamental concepts of this system are due to Galileo, but the complete structure appears first in Newton's Principia. The theory is simple and mathematical; indeed, one of its main differences from modern theories is its belief (perhaps traceable to Greek geometry) that Nature is convenient for the mathematician, and requires little manipulation before his concepts become applicable.

The Newtonian system, stated with schematic simplicity, as, e.g. by Boscovitch, is as follows. There is an absolute space, composed of points, and an absolute time, composed of instants; there are particles of matter, each of which persists through all time and occupies a point at each instant. Each particle exerts forces on other particles, the effect of which is to produce accelerations. Each particle is associated with a certain quantity, its "mass," which is inversely proportional to the acceleration produced in the particle by a given force. The laws of physics are conceived, on the analogy of the law of gravitation, as formulæ giving the force exerted by one particle on another in a given relative situation. This system is logically faultless. It was criticized on the ground that absolute space and time were meaningless, and on the ground that action at a distance was inconceivable. This latter objection was sanctioned by Newton, who was not a strict Newtonian. But in fact neither objection had any force from a logical point of view. Kant's antinomies, and the supposed difficulties of infinity and continuity, were finally disposed of by Georg Cantor. There was no valid a priori reason for supposing that Nature was not such as the Newtonians averred, and their scientific successes afforded empirical, or at least pragmatic, arguments in their favour. It is no wonder, therefore, that, throughout the eighteenth century, the system of ideas which had led to the law of gravitation dominated all scientific thought.

Before physics itself had made any breaches in this edifice, there were, however, certain objections of an epistemological order. It will be worth while to consider these, since it is urged that the theory of relativity is not open to them, though I believe this claim to be only partially justified.

The most formidable and persistent attack was upon absolute space and time. This attack was initiated by Leibniz in the lifetime of Newton, especially in his controversy with Clarke, who represented Newton. In time, most physicists came to disbelieve in absolute space and time, while retaining the Newtonian technique, which assumed their existence. In Clerk Maxwell's Matter and Motion, absolute motion is asserted in one passage and denied in another, with hardly any attempt to reconcile these two opinions. But at the end of the nineteenth century the prevalent view was certainly that of Mach, who vigorously denied absolute space and time. Although this denial has now been proved to be right, I cannot think that before Einstein and Minkowski it had any conclusive arguments in its favour. In spite of the fact that the whole question is now ancient history, it may be instructive to consider the arguments briefly.

The important reasons for rejecting absolute space and time were two. First, that everything we can observe has to do only with the relative positions of bodies and events; secondly, that points and instants are an unnecessary hypothesis, and are therefore to be rejected in accordance with the principle of economy, which is the same thing as Occam's razor. It appears to me that the first of these arguments has no force, while the second was false until the advent of the theory of relativity. My reasons are as follows:

That we can only observe relative positions is, of course, true; but science assumes many things that cannot be observed, for the sake of simplicity and continuity in causal laws. Leibniz assumed that there are infinitesimals, although everything that we can observe exceeds a certain minimum size. We all think that the earth has an inside, and the moon a side which we cannot see. But, it will be said, these things are like what we observe, and circumstances can be imagined under which we should observe them, whereas absolute space and time are different in kind from anything directly known, and could not be directly known in any conceivable conditions. Unfortunately, however, this applies equally to physical bodies. The relative positions which we see are relative positions of parts of the visual field; but the things in the visual field are not bodies as conceived in traditional physics, which is dominated by the Cartesian dualism of mind and matter, and places the visual field in the former. This argument is not valid as against Mach, who argued that our sensations are actually part of the physical world, and thus inaugurated the movement towards neutral monism, which denies the ultimate validity of the mind-matter dualism. But it is valid as against all those for whom matter is a sort of Ding-an-sich, essentially different from anything that enters into our experience. For them, it should be as illegitimate to infer matter from our perceptions as to infer absolute space and time. The one, like the other, is part of our naive beliefs, as is shown by the Copernican controversy, which would have been impossible for men who rejected absolute space and time. And the remoteness from our perceptions is as much a discovery due to reflection in the one case as in the other.

It is impossible to lay down a hard-and-fast rule that we can never validly infer something radically different from what we observe—unless, indeed, we take up the position that nothing unobserved can ever be validly inferred. This view, which is advocated by Wittgenstein in his Tractatus Logico-Philosophicus, has much in its favour, from the standpoint of a strict logic; but it puts an end to physics, and therefore to the problem with which this work is concerned. I shall accordingly assume that scientific inference, conducted with due care, may be valid, provided it is recognized as giving only probability, not certainty. Given this assumption, I see no possible ground for rejecting an inference to absolute space and time, if the facts seem to call for it. It may be admitted that it is better, if possible, to avoid inferring anything very different from what we know to exist. Such a principle will have to be based on grounds of probability. It may be said that all inferences to something unobserved are only probable, and that their probability depends, in part, upon the a priori probability of the hypothesis; this may be supposed greater when we infer something similar to what we know than when we infer something dissimilar. But it seems questionable whether there is much force in this argument. Everything that we perceive directly is subject to certain conditions, more especially physiological conditions; it would seem a priori probable that where these conditions are absent things would be different from anything that we can experience. If we suppose as we well may—that what we experience has certain characteristics connected with our experiencing, there can be no a priori objection to the hypothesis that some of the things we do not experience are lacking in some characteristics which are universal in our experience. The inference to absolute space and time must, therefore, be treated as on a level with any other inductive inference.

The second argument against absolute space and time—namely, that they are unnecessary hypotheses—has turned out to be valid; but it is only in quite recent times that Newton's argument to the contrary has been refuted. The argument, as everyone knows, was concerned with absolute rotation. It is urged that, for "absolute rotation," we may substitute "rotation relatively to the fixed stars." This is formally correct, but the influence attributed to the fixed stars savours of astrology, and is scientifically incredible. Apart from this special argument, the whole of the Newtonian technique is based upon the assumption that there is such a quantity as absolute acceleration; without this, the system collapses. That is one reason why the law of gravitation cannot enter unchanged into the general theory of relativity. There are, of course, two distinct elements in the theory of relativity: one of them the merging of space and time into space-time—is wholly new, while the other—the substitution of relative for absolute motion—has been attempted ever since the time of Leibniz. But this older problem could not be solved by itself, because of the necessity for absolute acceleration in Newtonian dynamics. Only the method of tensors, and the new law of gravitation obtained in accordance with this method, have made it possible to answer Newton's arguments for absolute space and time. While, therefore, the contention that these are unnecessary would always have been a valid ground for rejecting them if it had been known to be true, it is only now that we can be confident of its correctness, since it is only now that we possess a mathematical technique which is in accordance with it.

Somewhat similar considerations apply to action at a distance, which was also considered incredible by Newton's critics, from Leibniz onwards, and even by Newton himself. There is one theory, which may well be true, according to which action at a distance is self-contradictory: this is the theory which derives spatio-temporal separation from causal separation. I shall say no more about this possibility at present, since it was not suggested by any of the opponents of action at a distance, all of whom considered spatial and temporal relations totally distinct from causal relations. From their point of view, therefore, the objection to action at a distance seems to have been little more than a prejudice. The source of the prejudice was, I think, twofold: first, that the notion of "force," which was the dynamical form of "cause," was derived from the sensations of pushing and pulling; secondly, that people falsely supposed themselves in contact with things when they pushed and pulled them, or were pushed and pulled by them. I do not mean that such crude notions would have been explicitly defended, but that they dominated the imaginative picture of the physical world, and made Newtonian dynamics seem what is absurdly called "intelligible." Apart from such mistakes, it should have been regarded as a purely empirical question whether there is action at a distance or not. It was in fact so regarded throughout the latter half or three-quarters of the eighteenth century, and it was generally held that the empirical arguments in favour of action at a distance were overwhelming.

Not wholly unconnected with the question of action at a distance was the question of the rôle of "force" in dynamics. In Newton, "force" plays a great part, and there seems no doubt that he regarded it as a vera causa. If there was action at a distance, the use of the words "central forces" seemed to make it somehow more "intelligible." But gradually it was increasingly realized that "force" is merely a connecting link between configurations and accelerations; that, in fact, causal laws of the sort leading to differential equations are what we need, and that "force" is by no means necessary for the enunciation of such laws. Kirchoff and Mach developed a mechanics which dispensed with "force," and Hertz perfected their views in a treatise[5] comparable to Euclid from the point of view of logical beauty, leading to the result that there is only one law of motion, to the effect that, in a certain defined sense, every particle describes a geodesic. Although the whole of this development involved no essential departure from Newton, it paved the way for relativity dynamics, and provided much of the necessary mathematical apparatus, particularly in the use of the principle of least action.

The first physical theory to be developed on lines definitely different from those of Newtonian astronomy was the undulatory theory of light. Not that there was anything to contradict Newton, but that the framework of ideas was different. Transmission through a medium had been made fashionable by Descartes, and unfashionable by the Newtonians; in the case of the transmission of light it was found necessary to revert to the older point of view. Moreover, the æther was never so comfortably material as "gross" matter. It could vibrate, but it did not seem to consist of little bits each with its own individuality, or to be subject to any discoverable molar motions. No one knew whether it was a jelly or a gas. Its properties could not be inferred from those of billiard balls, but were merely those demanded by its functions. In fact, like a painfully good boy, it only did what it was told, and might therefore be expected to die young.

A more serious change was introduced by Faraday and Maxwell. Light had never been treated on the analogy of gravitation, but electricity appeared to consist of central forces varying inversely as the square of the distance, and was therefore confidently fitted into the Newtonian scheme. Faraday experimentally and Maxwell theoretically displayed the inadequacy of this view; Maxwell, moreover, demonstrated the identity of light and electromagnetism. The æther required for the two kinds of phenomena was therefore the same, which gave it a much better claim to be supposed to exist. Maxwell's proof, it is true, was not conclusive, but it was made so by Hertz when he produced electromagnetic waves artificially and studied their properties experimentally. It thus became clear that Maxwell's equations, which contained practically the whole of his system, must take their place beside the law of gravitation as affording the mathematical formula for a vast range of phenomena. The concepts required for these equations were, at first, not definitely contradictory to the Newtonian dynamics; but by the help of subsequent experimental results contradictions emerged which were only removed by the theory of relativity. Of this, however, we shall speak in a later chapter.

Another breach in the orthodox system, of which the importance has only become fully manifest since the publication of the general theory of relativity, was the invention of non-Euclidean geometry. In the work of Lobatchevsky and Bolyai, although the philosophical challenge to Euclid was already complete, and the consequent argument against Kant's transcendental æsthetic very powerful, there were not yet, at least obviously, the far-reaching physical implications of Riemann's inaugural dissertation "Ueber die Hypothesen, welche der Geometric zu Grunde liegen." A few words on this topic are unavoidable at this stage, although the full discussion will come later.

One broad result of non-Euclidean geometry, even in its earliest form, was that the geometry of actual space is, at least in part, an empirical study, not a branch of pure mathematics. It may be said that empiricists, such as J. S. Mill, always based geometry upon empirical observation. But they did the same with arithmetic, in which they were certainly mistaken. No one before the non-Euclideans perceived that arithmetic and geometry stand on a quite different footing, the former being continuous with pure logic and independent of experience, the latter being continuous with physics and dependent upon physical data. Geometry can, it is true, be still studied as a branch of pure mathematics, but it is then hypothetical, and cannot claim that its initial hypotheses (which replace the axioms) are true in fact, since this is a question outside the scope of pure mathematics. The geometry which is required by the engineer or the astronomer is not a branch of pure mathematics, but a branch of physics. Indeed, in the hands of Einstein geometry has become identical with the whole of the general part of theoretical physics: the two are united in the general theory of relativity.

Riemann, who was logically the immediate predecessor of Einstein, brought in a new idea of which the importance was not perceived for half a century. He considered that geometry ought to start from the infinitesimal, and depend upon integration for statements about finite lengths, areas, or volumes. This requires, inter alia, the replacement of the straight line by the geodesic: the latter has a definition depending upon infinitesimal distances, while the former has not. The traditional view was that, while the length of a curve could, in general, only be defined by integration, the length of the straight line between two points could be defined as a whole, not as the limit of a sum of little bits. Riemann's view was that a straight line does not differ from a curve in this respect. Moreover, measurement, being performed by means of bodies, is a physical operation, and its results depend for their interpretation upon the laws of physics. This point of view has turned out to be of very great importance. Its scope has been extended by the theory of relativity, but in essence it is to be found in Riemann's dissertation.

Riemann's work, as well as that of Faraday and Maxwell, belongs, like the theory of relativity, to the development of the view of the physical world as a continuous medium, which has, from the earliest times, contested the mastery with the atomic view. Just as Newton caused absolute space and time to be embedded in the technique of dynamics, so Pythagoras caused spatial atomism to be embedded in the technique of geometry. Ever since Greek times, those who did not believe in the reality of "points" were faced with the difficulty that a geometry based on points works, while no other way of starting geometry was known. This difficulty, as Dr Whitehead has shown, exists no longer. It is now possible, as we shall see at a later stage, to interpret geometry and physics with material all of which is of a finite size—it is even possible to demand that none of the material shall be smaller than an assigned finite size. The fact that this hypothesis can be reconciled with mathematical continuity is a novel discovery of considerable importance; until recently, atomism and continuity appeared incompatible. There are, however, forms of atomism which have not hitherto been found easy to reconcile with continuity; and, as it happens, there is powerful experimental evidence in their favour. Just at the moment when Maxwell, supplemented by Hertz, appeared to have reduced everything to continuity, the new evidence for an atomic view of Nature began to accumulate. There is still an unreconciled conflict, one set of facts pointing in one direction, and another in another; but it is legitimate to hope that the conflict will be resolved before long modern atomism, however, demands a new chapter.