Part 4 — CHAPTER XXXVIII THE METHODOLOGY OF SCIENCE (continued 3)

As for the regulative norms which guide scientific discovery, we have seen that they reduce to a search for unity in diversity; to a desire to generalise, to detect uniformities or laws, and thus to economise effort, as Mach has aptly said. These same regulative norms are just as apparent in mathematics as in physics and have been heeded by all scientists from Democritus to the moderns.

We might also mention the principle of continuity and the principle of causality as understood by modern scientists. The principle of continuity implies the eradication of all action at a distance such as is exemplified in Newton’s law of gravitation. It is probable that a belief in the principle of continuity arises from the fact that in daily life all action seems to be transmitted by contact. Again, an object moves from “here” to “there” through a continuous series of intermediary positions, or at least when such is not the case we cease to recognise it as the same object. Newton himself felt the necessity of adhering to the principle of continuity, but in his day the development of theoretical science was not sufficiently advanced to permit a realisation of his hopes. For the same reason, physicists during the eighteenth and nineteenth centuries paid little heed to the principle. Particles were assumed to attract one another according to certain laws, but how the attraction was transmitted was never discussed.

Only when, as a result of Maxwell’s discoveries, electromagnetic induction was found to be propagated by continuous action through a medium, did the popularity of the principle increase. This leads us to field physics, to the discovery of a new category differing from matter, namely, the electromagnetic field. Einstein succeeded in placing gravitation on the same field basis as electromagnetics, in this way obviating Newton’s action at a distance. Even here, however, it was not necessarily the principle which guided him to his equations; it was the study of the facts co-ordinated in his theory which drove him to the principle. With such signal initial successes of field physics, it was only natural to seek to extend them still farther, and to interpret matter in terms of the field. To-day, the attempt seems to have been abandoned, but, once more, not owing to a change in the philosophical viewpoint, but owing to the tremendous technical difficulties that have been encountered. As Weyl tells us: “Meanwhile I have quite abandoned these hopes, raised by Mie’s theory; I do not believe that the problem of matter is to be solved by a mere field theory.”

The principle of causality as understood by Einstein and modern scientific writers refers to a phenomenological outlook. Thus, Newton’s absolute space is regarded as contrary to causality. To-day we should find that, generally speaking, the phenomenological outlook is the rule among physical scientists; and there is a very good reason for this preference. For so long as we can interpret sensible effects in terms of sensible causes, there is no advantage in invoking suprasensible ones about which anything can be postulated with impunity. True, it may be impossible in many cases to discover sensible causes, and it is not assumed that the whole world may be built up of sensible entities alone. Still, whenever suprasensible causes can be escaped, the procedure will always be to avoid them. It should be mentioned, however, that this last principle does not appeal with equal force to all physicists.

And now we come to a totally different type of assumptions, the so-called scientific hypotheses. These are never imposed on us a priori as inevitable; they constitute mere post-suppositions conceived of with a view to co-ordinating the known facts of experience. As such they are contingent on our knowledge of these facts, are never adhered to blindly, and may vary from time to time as this knowledge is increased or becomes more refined. Nevertheless, if the phenomena to which they apply seem more fundamental than the ones we happen to be studying, the hypotheses will automatically enter into our further co-ordinations, and will then be present in the light of assumptions. For instance, long before the kinetic theory of gases (based on a physical hypothesis) was established as firmly as it is to-day, all new phenomena pertaining to gases were interpreted in terms of the kinetic theory.

As we have said, the primary object of these physical hypotheses is to enable us to co-ordinate facts; the molecular hypothesis, the kinetic theory of gases, the quantum hypothesis, the ether, and the absoluteness or the relativity of space, are all cases in point. In a number of instances, the formulation of these physical hypotheses was arrived at as a result of highly technical theoretical considerations bearing on the equations of mathematical physics. By this we mean that their justification can be made clear only when a mathematical representation of the phenomena under consideration has been elaborated. In the absence of this mathematical representation, they would never even have suggested themselves and their utility could never have been anticipated. Planck’s hypothesis of the discontinuous nature of light-emission is of this sort.

In a number of other cases, physical hypotheses suggest themselves in a much simpler way, without our having to consider the mathematical treatment of the problem. Fresnel’s ether hypothesis, for example, was formulated as an obvious method of accounting for the wave nature of light which his celebrated experiment of the two mirrors seemed to render inevitable. Newton’s absolute space is another illustration of a physical hypothesis which appears to impose itself almost immediately when the dynamical facts of motion are taken into consideration.

It often happens that these physical hypotheses, conceived primarily for the purpose of co-ordinating facts, are subsequently proved to correspond to physical reality (atoms, electrons). In other cases no subsequent experiment has appeared to corroborate them. As an illustration, Ritz had imagined the existence of an atom of magnetism, the magneton, in order to account for the spectral series. Weiss also was led to a similar hypothesis when studying certain magnetic phenomena. Thus far, however, all attempts to isolate the magneton have failed, and it is quite possible that no such entity exists as a physical reality.[145] Nevertheless, although physical hypotheses often fail to be vindicated by further experiment, they may serve a useful purpose. Fresnel’s ether has been discarded, yet it was of use to Fresnel and his immediate successors.

To summarise, we see that a clear-cut distinction must be drawn between such tentative assumptions as physical hypotheses and the more basic assumptions that appear to be demanded by the very requirements of scientific knowledge. This does not mean that the latter assumptions present any absolute certainty; it simply means that they must necessarily be accepted, at least tentatively, unless we agree to throw up our hands and abandon all attempts at formulating a science. If, then, we note what appear to be sweeping changes in the scientific viewpoint, we must not attribute these changes to variations in the metaphysics of science; quite the reverse is true. The changes in the scientific viewpoint are necessitated by the discovery of new facts, and it is then these changes in the scientific attitude which are the cause of the variation in the scientist’s philosophical outlook upon the world. For this there should be no cause for surprise. Men might have philosophised about space, matter and infinity for ever and ever, giving logical definitions to their hearts’ content; but unless they had taken into consideration the facts which mathematics and physics and the sciences generally had revealed, we of to-day should be as ignorant of all these subjects as were our forefathers.

However, the writer realises full well that this general line of talk is of no great interest. The only way to understand the subject is to consider specific examples. For this purpose we have selected the Copernican system versus the Ptolemaic one, and finally the entire problem of the relativity of space and motion extending from Galileo to Einstein. We shall see that the vast changes that took place in the scientific outlook were brought about always in the same way—by the advancing pressure of facts.

Suppose, for instance, that we were starting science all over afresh, and that in conformity with our crude belief we assumed the earth to be fixed and the stars to be circling around it. After centuries of observation we should probably have discovered that the stars were not fixed to one another, for successive charts of the heavens would have shown that the shapes of the constellations were not immutable. Probably we should have ended by assuming that the stars were luminous bodies moving freely in infinite space. It would then have been only natural for us to follow the Greeks, and to assume that free bodies described circles, circular motion being the noblest of all motions. But, on the other hand, when we watched a stone gliding over a smooth surface of ice, it would become evident to us that the motion of free bodies on the earth’s surface appeared to be rectilinear. So here would be a duality in the laws of motion which would complicate our understanding of mechanics and our synthesis of natural phenomena.

Eventually some thinker would assert that the free motion of bodies was rectilinear, but that the stars, being attached to concentric spheres of crystal, were compelled to move in circles. In this way the duality in the laws of motion of free bodies would be removed; but it is questionable whether the cure would not be worse than the evil. For with our spheres of crystal an artificial hypothesis having no palpable connection with any other of the known phenomena would have been introduced ad hoc.

Experimenting still further, we might notice that on the surfaces of bodies rotating with respect to the earth, strange forces would make their appearance. We might eventually plot out the lay of these forces, and divide them into centrifugal and Coriolis forces. Later, a number of curious phenomena occurring on the earth’s surface would be discovered. We should notice that the trade winds and sea currents invariably followed slanting courses, that cyclones always whirled anti-clockwise in the northern hemisphere and clockwise in the southern hemisphere, that rivers flowing northward in the northern hemisphere had a tendency to eat into their east banks. The protuberance of the equator, and the decrease of weight, as we moved towards the tropics, would also be detected. Later still, curious experiments such as Foucault’s pendulum and the gyroscopic compass would be studied; and all these separate discoveries would remain disconnected unless we accumulated one hypothesis after another.

Finally, some genius would come along and say: “Why, all these phenomena will be in order if we assume the earth to be in rotation, the magnitude of this rotation being computed with reference to a system of axes fixed with respect to the stars. This rotation will entail the existence of a field of centrifugal and Coriolis forces on the earth’s surface as on all rotating bodies; and, as a result, all the strange, disconnected phenomena noted will fit quite naturally into a perfectly simple and consistent synthesis.”

Thus far we have seen that by assuming the earth to be in rotation and by referring motion to an extraterrestrial frame, a considerable simplification was introduced into our synthesis when purely mechanical experiments conducted on the earth’s surface were considered. But this simplification will be still more apparent when account is taken of the planetary motions. So long as in our description of these motions we refer our measurements to the earth frame, the wanderings of the various planets will appear very erratic. However, after long periods of observation certain uniformities would be disclosed, and we might succeed in constructing some highly complicated clockwork mechanism which would portray the motions of the sun, moon and planets. Such was indeed Ptolemy’s accomplishment. We might then, by accelerating the motion of our mechanism, foretell the positions of the sun, moon and planets at some future date. In this way the advent of eclipses might be forecast in a more or less accurate way. But the point is that this model could only yield us what we had put into it. We might predict the future positions of the planets because we had detected uniformity in their motions. But if perchance some stray comet were to wander into the solar system, it would be utterly impossible for us to anticipate its motion. Our powers of prevision would thus be extremely limited.

Then we have to consider another type of phenomenon first noticed by Hipparchus. We refer to the precession of the equinoxes, according to which the Pole Star appears to wander away gradually from the direction of true north, finally returning after having described a circle in the course of 28,000 years. To complete our model it would be necessary to take this phenomenon into consideration by assuming that the entire firmament was wobbling like a spinning top. All such phenomena and many others would appear as so many separate empirical discoveries, and no connection between them could be invoked unless one artificial hypothesis were piled up on another, until nature was deprived of all unity.[146] In short, we should merely have succeeded in describing phenomena just as we might describe the patterns on a butterfly’s wings, but our description would in the main be sterile. In a general way, prevision would be impossible; and any such discovery as that of Neptune by Leverrier, deduced from the peculiar motion of Uranus, would be completely out of the question, since no mathematical connection would have been found to exist between the motions of the various planets.

But just as all the difficulties we encountered in mechanics vanished when we substituted a description of motion in terms of the inertial frame (a frame fixed to the stars), so now, once again, by taking the same frame, we are enabled to overcome all the astronomical difficulties we have mentioned, and re-establish harmony and unity in nature. Kepler, as we know, referred the successive positions of the planets (as obtained from Tycho Brahe’s observations) to an inertial frame. He found that, as referred to this frame, each planet described an ellipse round the sun, the sun being situated at one of the foci of the ellipse. Kepler’s laws of planetary motions were then as follows:

1. The planets describe ellipses round the sun.

2. Their motions obey the law of areas, by which we mean that the radius vector joining the sun to a planet sweeps over equal areas in equal times.

3. The square of the time required for a planet to describe its orbit is proportional to the cube of the major axis of the ellipse described.

These are Kepler’s laws. They constitute a distinct advance over Ptolemy’s description, since in addition to the numerous advantages entailed by the choice of the inertial frame in terrestrial mechanics, these laws appear extremely simple and are capable of being expressed in concise mathematical form. But the great importance of Kepler’s laws is that they rendered Newton’s discoveries possible.

Newton argued that, since in an inertial frame free bodies appeared to follow rectilinear courses with constant speeds, it was probable that the planets were being acted on by forces which prevented them from following their natural courses. Now, Kepler’s second law, the law of areas, is compatible only with the existence of a force at all times directed along the line joining the sun and planet. Inasmuch as the planets described closed curves round the sun, this force was obviously one of attraction towards the sun. Kepler’s first and third laws, considered jointly, enabled Newton to determine the precise numerical law of solar attraction, the law of the inverse square. Further, the elliptical motion of the moon round the earth, as it appeared when referred to the inertial frame, connoted the existence of a similar law of attraction existing between moon and earth. It appeared probable, therefore, that we were in the presence of a general action of matter on matter, and Newton found that the phenomenon of weight could be accounted for, not only qualitatively, but also quantitatively, by assuming that matter always attracted matter. Hence, he was led to the formulation of his law of universal attraction.

It is easy to see that Newton’s description is more creative by far than Kepler’s, and still more so than Ptolemy’s. Thus, Kepler’s laws merely enable us to foretell the future positions of the planets, and nothing more. If we abided by these laws, we should probably assume that elliptical orbits were the only ones that could exist. Newton’s law, on the other hand, enables us to predict that other types of orbits (parabolas and hyperbolas) are equally capable of existence, and in this way the motions of certain comets are brought into the general synthesis. All we have to do is to note the instantaneous position and velocity of a body with respect to the sun, and we can foresee immediately the precise orbit it will follow. Hence, our powers of prevision, as deduced from Newton’s law, are much greater than they were in Kepler’s day.

Again, consider the phenomenon of the precession of the equinoxes. Prior to Newton’s discoveries this phenomenon was always a mystery, unconnected with all other phenomena. Now its cause is clear. It is due to the uneven pull exerted by the sun and moon on our planet, brought about by its radially unsymmetrical shape (illustrated by the protuberance of its equator). The result will be a periodic to-and-fro oscillation of the earth’s axis with respect to the sun. Owing to the earth’s, rotation this swinging motion will generate a gyroscopic couple which will cause our planet to wobble like a top. The net result is that even had the phenomenon of precession been unknown to astronomers, mathematicians would have anticipated it as an inevitable consequence of the earth’s unsymmetrical shape and of the law of gravitation. And it is the same in innumerable instances. Newton’s description has thus widened the scope of the general synthesis of nature. It has shown the intimate connection which exists between such apparently independent phenomena as the motions of cyclones, the phenomenon of the tides, the feeling of weight, planetary motions, the precession of the equinoxes, the annual variations of the angles of aberration, etc. It has allowed us to anticipate the existence of one phenomenon from that of another. All these discoveries were rendered possible by our choice of the inertial frame for the reference of motion; so we realise that a description in terms of the inertial frame is creative, whereas one in terms of the non-rotating earth leads us nowhere.

Of course, now that we have learned so much, thanks to the introduction of the inertial frame, there is nothing to prevent us from interpreting all our results and all our relationships backwards, as it were, in terms of a non-rotating earth. But by transplanting, in this way, all the results obtained from the first description into the new description, no really new description would have been obtained; all we should have accomplished would have been to obtain a parody on the first one. In the same way, a record run backwards on the pianola does not constitute a new tune, but merely a parody on the original one. Obviously, this is not the method we must follow when we seek to construct a new synthesis. We must not merely transpose the results of the old synthesis in terms of the new one; we must operate directly.

But then, in the problem we are here discussing, had the description in terms of the inertial frame been unknown, the motion of the sun with respect to the fixed earth would in itself have been so complex as to defy mathematical investigation. As judged from the earth, the sun would appear to circle round it every 24 hours (neglecting niceties); and as at different seasons of the year its distance from the earth would vary, its course would be a spiral expanding, and then contracting, every six months.

Under the circumstances, the law of areas would not be satisfied; that is to say, if we considered the line joining sun to earth, it would not sweep over equal areas in equal times. Now, a general theorem of mechanics proves that whenever a body is moving under the attraction of a central body, the law of areas is always verified; whence we must conclude that the sun could never be considered as rotating round the earth under the sole action of the earth’s attraction. Supplementary forces would have had to be brought into play, but their introduction would have been so arbitrary, and the mathematical difficulties so great that even Newton might have been nonplussed; and we should have remained as ignorant of the harmony of the heavens as were the Greeks.

In the examples discussed, we have limited ourselves to problems of extreme simplicity; but even in these simple problems we have seen that different descriptions, though equivalent in theory, were far from equivalent in practice. When we come to consider problems of greater complexity, such as those with which modern science deals, the uniqueness of one particular type of description stands out with increased definiteness, and it is this particular simple continuous description where everything is harmonious and connected that is deemed to constitute the real description. Only when we follow this unique description do we find it possible to forestall nature by forecasting her actions in advance, and as a result claim that we have discovered reality.

The conclusions to be drawn from the preceding discussions are then as follows: From a purely kinematical point of view, it is simpler to take an inertial frame as frame of reference when the planetary motions are considered; this is the essence of Copernicus’ discovery. Also, when dynamical phenomena are considered, it is simpler to select an inertial frame with respect to which rotation will be measured. In short, the inertial frame imposes itself as conducive to the simplest interpretation of all phenomena whether mechanical or astronomical, so that we may say that there exists a privileged frame of reference in space. Here we are merely stating facts which are in no wise subject to controversy.

We might, of course, content ourselves with this discovery and neglect to consider the “why” and “wherefore.” But if we wish to pass beyond, if we wish to seek a palpable cause for the existence of a privileged frame in space, we are driven to one of two conclusions. Either we must say that the existence of a privileged frame arises from the nature of space itself, in which case we are driven to a space which possesses dynamical properties per se; and this is absolute space in which an inertial frame is privileged because it is non-rotating. Or else we must assume that space itself has nothing to do with the matter, and that the existence of the privileged frame arises from extraneous conditions, from the presence of the stars, the will of the Supreme Being, the shape of the earth, the number of existing planets, or anything else we may care to imagine.

There is not the slightest doubt that so long as we restrict our attention to the limited number of facts known to Newton, the simplest solution is the first, and its adoption constitutes the essential characteristic of the Newtonian position. There is, however, no necessity to conceive of this absolute space in the light of a metaphysical reality. All we need say is: Everything occurs as though there existed an absolute background of space, hence as though absolute space existed. This was the attitude of classical science until the advent of Einstein’s theory. We hope to make these vital points clearer in the course of this chapter.

Prior to Newton, the problem of space and motion appears to have been any one’s guess. To mention only two of Newton’s contemporaries, Descartes guessed that motion was relative, while More guessed that motion and space were absolute. These guesses were unsupported by any kind of scientific evidence, one way or another. It is true that More, for instance, seems to have realised the necessity of considering scientific examples in support of his claims, but his illustrations were too poorly interpreted to be of any scientific interest. Thus, in a letter to Descartes, he notes that a man walking at a rapid pace experiences fatigue, whereas his friend lying in repose experiences none; whence More concludes that motion and space must be absolute. But had he considered a man standing motionless for hours while his friend was being borne away, sitting at ease in a carriage, one may wonder what conclusion he would then have reached.

A second argument presented by More has no greater merit. A simplified form of it would be as follows: Two trains pass each other, moving in opposite directions. If motion were relative, we could maintain that, as referred to the first train, a telegraph pole was speeding north, while, as referred to the second train, it was moving south. As the pole cannot be moving north and south all at the same time, relative motion cannot exist, and the pole must therefore be at rest in absolute space; whence More concludes that space must be absolute.

It is scarcely necessary to state that Newton’s arguments were of a totally different calibre, and even to-day it appears extremely difficult to refute them in their entirety. Newton was one of the earliest exponents of the scientific method. As he tells us himself:

“For the best and safest method of philosophising seems to be, first, diligently to investigate the properties of things and establish them by experiments, and then later seek hypotheses to explain them.... For hypotheses ought to be fitted merely to explain the properties of things and not attempt to predetermine them except so far as they can be an aid to experiments.”

These statements of Newton appear perfectly clear and merely reflect the spirit of scientific procedure. It has been held, however, by numerous philosophers who have written on the “metaphysics of science” that Newton departed from his avowed empirical method when he formulated his theory of absolute space and time. It is scarcely credible that any one at all conversant with mechanics could ever maintain an opinion of this sort, but as it appears to be widespread in certain quarters, it may be of interest to examine its claims.

We may state that the question is one of the utmost importance; for were it established that Newton’s absolute motion did not appear to be imposed by the dynamical facts of mechanics, it would be impossible to understand the philosophical significance of Einstein’s cylindrical universe.

Let us then return to Newton’s exposition. He starts out in the Principia by expressing his belief in absolute immovable space everywhere the same, and in absolute time. These statements are followed by a description of the experiment of the rotating bucket of water and others of a kindred nature. According to the critic, Newton evidently intended absolute space and time to be taken in the light of necessary presuppositions. But apart from the fact that in the order written the statements on absolute space and time precede any reference to experiment, there appears to be no basis for any such belief. Here we must remember that Newton was a scientist writing for the benefit of fellow scientists; and it is a method commonly followed by mathematicians to state a proposition and then show why the statement must be accepted as correct. Although the statement of the proposition precedes the proof, no one would be misled into believing that we were asked to accept the proposition in the light of a philosophical presupposition which might lead to controversy, and then regard the demonstration as an argument of secondary importance. And it is the same with absolute space and time. Just as Euclid follows up his statement of a geometrical proposition by a proper demonstration, so does Newton proceed to demonstrate the existence of absolute space by showing that absolute rotation is revealed by a number of experiments (by the experiment of the rotating bucket of water, among others). Had manifestations of absolute motion been unknown to Newton, had absolute motion eluded experiment, there never would have been any reason for him to postulate its existence in physics; and as a result its direct consequences, namely, absolute space and time, would have been reduced to meaningless hypotheses and no longer to necessary conclusions.

As a matter of fact, Newton mentions explicitly two methods of presentation which may be adhered to. Thus, we read in his Opticks:

“By this way of analysis we may proceed from compounds to ingredients, and from motions to the forces producing them; and in general from effects to causes, and from particular causes to more general ones, till the argument end in the most general. This is the method of analysis: and the synthesis consists in assuming the causes discovered, and established as principles, and by them explaining the phenomena proceeding from them, and proving the explanations.”[147]

As an illustration of the two methods, we may consider the case of the law of gravitation. Either we may say: The planets have been found to describe conics round the sun, with certain definite motions, and in order to account for these motions we must assume the existence of a solar attraction; or else we may say: The law of gravitation is such and such, and the proof of it is that the planets describe conics round the sun. From the standpoint of neatness of presentation, the deductive method is preferable, but from the standpoint of the chronological order of discovery, the first method describes the situation.

Let us not, then, waste any more time in dwelling on these perfectly obvious points which Newton explains time and again in answer to the criticisms of his contemporaries. The important point to decide lies elsewhere. Is it true that, as Newton imagined, the dynamical facts of motion, established empirically and illustrated in the experiment of the rotating bucket render absolute motion, hence absolute space, inevitable? Here we may be permitted to point out that a question of this sort must be left for scientists, trained in the school of rigorous thinking, to decide. It is indeed obvious that critics, like Berkeley and Kant, who possessed such hazy ideas of mechanics as to confuse velocity and acceleration, momentum and force, were too poorly equipped to express any opinions of value.[148] Turning, then, to the verdict of subsequent scientists, men of the calibre of Euler, Laplace and Poincaré, we find the conclusion unanimous. Absolute space is recognised by all as the simplest physical hypothesis that will account for the observed mechanical facts. To that extent it corresponds to reality. Even Einstein, to whom the downfall of the Newtonian position is due, recognises that with the facts at Newton’s disposal, his solution was the only one that could be defended.

We shall now consider these basic problems in greater detail, and endeavour to understand why it was that Newton was compelled to accept absolute motion. Inasmuch as the absoluteness or the relativity of motion cannot be settled by a priori reasoning, we are compelled to appeal to experiment. Suppose, then, we perform mechanical experiments in the interior of some gigantic rigid box. The facts disclosed by experiment will be as follows: So long as, with respect to this box, the stars appear to occupy fixed positions, the course of our experiments will remain unchanged regardless of whether we operate in this part or in that part of the box. In other words, our mechanical experiments offer us no means of deciding as to our location in space. We may infer therefrom that experiment suggests the relativity of position, which means that position can be defined only with respect to other bodies, or at least to observable points of reference. This relativity of position entails as a direct consequence the homogeneity of space. In precisely the same way we should find that the orientation of our box, together with that of the mechanical apparatus it contained, would also fail to manifest itself by any variation in the course of our experiment; whence we infer the relativity of orientation, or the isotropy of space.

But what about motion through space? The mere fact that position in space was relative might suggest at first blush that motion, being a mere change in position, should also be relative. But, here again, the problem we are dealing with is one of physics, not one of pure mathematics or of metaphysics. Now, if we perform our mechanical experiment first in a box which is non-rotating with respect to the stars, then in one which is in relative rotation, a considerable change can be detected in the course of the experiment. Yet the relativity of motion would imply that this change in the nature of the box’s relative motion should entail no perceptible difference; hence we are compelled to conclude that in contradistinction to position and orientation, motion cannot be relative. Motion through space is not a mere matter of point of view; we can detect it even in the absence of any perceptible landmarks in space.

A more thorough investigation of this problem led Newton to differentiate between two grand categories of motion: one of which appeared relative, in that it was impossible to detect it in the absence of landmarks in space; the other of which appeared absolute, since it was accompanied by physical disturbances and dynamical manifestations which could be detected and measured without our having to take landmarks in space into consideration. A train running smoothly along a straight track with constant speed is an illustration of the relative type of motion, while an abject rotating with respect to the stars, or a ship tossed at sea, affords us an illustration of the absolute type. In short, position, orientation and a certain type of motion manifested themselves as relative, whereas certain other types of motion manifested themselves as absolute. These were the facts of experiment, and they were summed up by the law of inertia, or again by the Galilean or Newtonian principle of relativity. It is this duality in the manifestations of motion that renders classical mechanics so unsatisfactory.

Two courses were open to Newton:

1. Either he might have assumed that space was absolute; that all motion was absolute, but that accelerated and rotationary motions alone could be detected by mechanical experiments;

2. Or else he might have assumed that space possessed a dual nature; absolute for rotationary motions, but relative for uniform translationary, or Galilean, motion.

Newton preferred the first alternative; and the Newtonian principle of relativity, which stressed the impossibility of detecting absolute velocity through mechanical experiments, acted as a damper, not on the absolute nature of space and motion, but on our ability to detect absolute velocity mechanically.

There were a number of reasons that most assuredly prompted him to this choice. In the first place, a duality in the nature of space and of motion was not easy to conceive of. Furthermore, a circumstance which influenced Newton’s successors was the fact that, after all, the Newtonian principle of relativity concerned solely mechanical phenomena. It was still possible that electromagnetic experiments would reveal the absolute velocity in which Newton believed but which had ever eluded science.

Nevertheless, many of Newton’s successors preferred to adopt the second alternative. They assumed that experiment had revealed a duality in the nature or structure of space and had proved its relativity for uniform translationary motion. They were not perturbed over the fact that electromagnetic and optical experiments might finally succeed in detecting absolute velocity; for they argued that were such experiments successful, all they would reveal would be velocity through the ether, not absolute velocity through empty space. According to this attitude the Newtonian principle of relativity emphasised no longer the inability of mechanical experiments to detect absolute velocity through empty space, but stressed the fact that absolute velocity was entirely meaningless.

Now the important point is the following: Whichever of the two previous attitudes we accept, that of Newton or that of many of his successors, space must still remain absolute in either case. For, in either case, space would be absolute for acceleration and rotation, regardless of what it might turn out to be for velocity. In much the same way, if an object is faintly coloured it is coloured; the faintness of the colouring cannot alter this fact. Hence, when we consider space, the entire question centres round the following problem: Is rotational motion truly absolute? If not, hence if centrifugal force cannot be attributed to rotation in space, whence does this force arise?

Of course, here a problem of extreme difficulty confronts us. When by varying external conditions at will we can produce variations in a magnitude, we may maintain that this magnitude is relative to surrounding conditions; for instance, the weight of a body varies with its distance from the earth. When, on the other hand, nothing that we can do appears to produce the slightest effect, we claim that the magnitude is absolute; mass in classical science was a case in point. Obviously, however, a test of this sort has a purely negative value for the simple reason that many of the external conditions lie beyond our power to vary. We cannot, for instance, annihilate the stars or the sun or the electrons. Thus, whereas a magnitude which has been established as relative will presumably remain a relative for all time to come, the same measure of assurance can no longer be claimed for our determinations of absolute quantities. Nevertheless, unless we are to fall into a state of complete agnosticism, we are compelled to establish a difference between quantities which appear absolute in the present state of our knowledge and those which are known to be relative. If we prefer, therefore, we may refer to absolutes as “relative-absolutes,” where the word “relative” implies the limitations of our present means of investigation.

It is possible to present Newton’s arguments in favour of absolute space in a number of different ways. The following illustrations may serve to clear up a few additional points. Consider a circular disk of gigantic proportions around whose central axis the stars would appear to be rotating in a clockwise direction with an angular velocity . We shall assume the disk to be inhabited by beings living around its centre; and we shall further suppose that dense clouds conceal the stars from their gaze, so that there would be no incentive for them even to suspect that their disk might be in rotation. These men would naturally refer all motion to their disk, just as, prior to Copernicus and Galileo, men were wont to refer all motions to the earth’s surface.

Suppose, now, that they were to perform mechanical experiments. They would soon discover that no object could remain motionless on the disk unless it were placed at the very centre or else fixed by artificial means. In particular, all objects originally fixed to some non-central point, then suddenly abandoned, would start moving away radially from the centre, then gradually follow a curved course, circling clockwise along an expanding spiral. In certain cases, however, a body might describe a true circle round the centre; but whenever this circular motion occurred, it would always be directed clockwise, and, furthermore, its angular velocity would invariably be given by a certain constant quantity co. Then again, if billiard balls were shot out in all directions with equal initial speeds from some non-central point, no two of their trajectories would be alike; not the slightest symmetry would be observed, and the clockwise motions would always be present. It may well be realised that for men who claimed space to be homogeneous and everywhere the same, this dissymmetry in the motions of bodies would be hard to account for.

Let us assume that just above the first disk a second one is rotating relatively to the first in a clockwise direction with angular velocity . The inhabitants of this second disk would find that bodies remained at rest wherever placed on their disk, that bodies set in motion would pursue straight courses with constant speeds, and, in short, that the asymmetry characteristic of the lower disk would give place to perfect symmetry of motion, no special direction being privileged. Furthermore, whereas the inhabitants of the lower disk would have the greatest difficulty in returning to the centre if peradventure they ever wandered away from it, and whereas they would always feel the action of unsymmetrical forces when they moved about, the inhabitants of the upper disk could move as they pleased and never be subjected to the action of forces.

Such would be the facts of the case. Now, if motion were relative, we could not assume that there existed any absolute difference between the motions of the two disks; either disk would be rotating, but only with reference to the other disk taken as standard. Yet here we see that whether or not a metaphysical difference is assumed to exist, it is quite certain that a vast physical difference is present in the conditions reigning on the two disks.

And so we are naturally led to enquire: What causes this dissymmetry in a homogeneous isotropic space? It is always preferable in science to search for causes in phenomena that are, so to speak, palpable, rather than in invisible agencies, for less guesswork is involved. In the present case, however, no visible cause can be countenanced. True, if the cloud: were to lift, it might strike the inhabitants of the lower disk that, with respect to their world, the stars were rotating clockwise with an angular velocity ; whereas the inhabitants of the upper disk would note that the stars appeared to be fixed. If our observers were firmly convinced of the relativity of motion, arguing that absolute motion in space was inconceivable, they might attempt to search for the cause of all these unsymmetrical phenomena in the state of relative rotation or rest of the stars.

But unless they were able to attribute a definite causal influence to this star rotation, the solution would be no solution at all. Although it now appears that they might have guessed right, yet in Newton’s time it would have been quite impossible to justify this attitude; not for philosophical reasons, but on account of the still undeveloped condition of mathematics and physics. This solution being denied them, then would be no other recourse but to appeal boldly to suprasensible absolute space and to recognise that the lower disk was rotating anti-clockwise in absolute space with an angular velocity , whereas the upper disk and stars were at rest. Henceforth, the causal influence would be attributed to the structure of absolute space itself; and the discrepancies noted above would be accounted for with perfect mathematical precision. (This illustration is, of course, merely an alternative way of presenting Newton’s experiment of the bucket of water.)

We are again driven to the same conclusions when we consider the problem of inertia. If here, there and everywhere are one and the same thing, why do we have to expend effort to move a body from here to there?

Again, if all motion were relative, the only significant type of motion would be motion relative to a frame of comparison. But then, if, as referred to a certain frame, the path of a free body were straight, with respect to another frame it would be crooked or curved. In other words there would be no sense in discussing the body’s absolute course, since by an appropriate choice of a frame of reference, we could make it any thing we pleased. Incidentally, the law of inertia would thereby be deprived of all meaning.

Consider, then, a number of free bodies taken at random, in relative motion with respect to one another. These bodies might ignore their mutual presences entirely (assuming even they were gifted with consciousness). If, therefore, a frame of reference were selected, there would be no reason to suppose that, as referred to this frame, the motions o the various bodies should be in any wise connected. And yet the law of inertia, deduced from experiment, proves that our anticipations would be incorrect.

For if we select a certain privileged frame, an inertial or Galilean frame, all the bodies will be found to describe straight lines with constant velocities as though there existed some secret understanding among them. Is it our frame that imposes this understanding on the unintelligent bodies? Do the bodies see the frame and agree to behave in the same way with respect to it? But the frame might be millions of miles away; furthermore, it only exists as a result of our free will; and the notions of the bodies with respect to one another would obviously remain the same even were some new frame to be chosen, or several different mes to be selected simultaneously. If we refuse to agree that it is absolute space itself which directs and guides the free bodies and which is thus the cause of this wonderful pre-established harmony, where else shall we look for a regulating cause? If we give up the search we fall back into the miraculous. Shall we appeal to the stars once more? But we have said that in Newton’s day this was impossible. There appears to be but one solution, and that is absolute space.

If we summarise all these results we see that whether we like it or not, the dynamical facts of motion and the law of inertia render an escape from absolute motion and space impossible. To be sure, as Einstein tells us, Newton might just as well have called his absolute space “Ether,” or anything else. The essential point was that something acting as an absolute background appeared to be required in order to account for the undeniable reality of centrifugal force generated by certain types of motion.

At all events, it was for the dynamical reasons outlined above, and for these alone, that Newton was driven to absolute space. So far as science is concerned, this is the only aspect of Newton’s discoveries that is of interest. Newton, however, after he had established the existence if absolute space scientifically, proceeded to weld it in with his theological ideas, identifying it with the existence of the Divine Being. This was, of course, his privilege, but these further speculations were extra-scientific and were placed on a par with his writings on the horned beast of Babylon or the bottomless pit of the Apocalypse.

We may be quite certain that had absolute space been based on considerations of a purely philosophical, theological or theosophical character, it would never have survived the criticisms of subsequent scientists, some philosophical schools would have championed it, others have attacked it, but scientists without exception would have rejected it as possessing no empirical or rational foundation. That Newton’s great authority was insufficient to ensure the acceptance of his ideas is seen when we consider the fate of his corpuscular theory of light or of his relief in the impossibility of constructing an achromatic lens. But in the case of absolute space, we find that precisely because it was based on the facts of experience, it survived till quite recently, when, under the attacks of Einstein, it began to crumble. Even so, however, we must remember that it was only thanks to the very recent discovery of new facts, unknown to Newton, that Einstein was able to overthrow the absolute doctrine, and, furthermore, as we shall see, the cornerstone of Newton’s mechanics, the absolute nature of rotation, is far from having been destroyed even to-day.

As Weyl puts it when discussing the dynamical facts of motion: “It is just these facts that, since the time of Newton, have forced us to attribute an absolute meaning, not to translation, but to rotation.” Again, in the following passage, Euler expresses the same opinion, amplifying it by stating that the metaphysical nature of space and time is irrelevant to Newton’s stand. We read:

“We do not assert that such an absolute space exists.”

And again elsewhere:

“What is the essence of space and time is not important, but what is important is whether they are required for the statement of the law of inertia. If this law can only be fully and clearly explained by introducing the ideas of absolute space and absolute time, then the necessity for these ideas may be taken as proved.”

Euler’s very clear statement brings out the pragmatic and anti-metaphysical attitude towards reality which is characteristic of scientific thought. From a failure to understand the significance of scientific hypotheses, the philosophers of the idealistic school felt it their duty to do away with absolute space. Had their attacks been limited to Newton’s theological expressions of opinion, there would have been no reason to criticise their arguments, for all theological opinions can be either defended or attacked. But their failure to grasp the deeper scientific reasons that drove Newton to absolute space, led them to the erroneous belief that mechanical phenomena might be accounted for equally well under the hypothesis of the relativity of space and motion. As a matter of fact, they should have realised that the scientific hypothesis of absolute space is quite independent of our philosophical preference for realism or idealism. In classical science space was absolute in the scientific sense for idealists and realists alike. Indeed, many of Newton’s scientific successors viewed nature in an idealistic way; yet they never saw fit to refute Newton’s absolute space and motion on this score.

And now let us consider one of the typical objections that have beer presented against Newton’s absolute space. It is claimed that if, as Newton tells us, space is to be perfectly homogeneous and of infinite extent, it follows that position in space, hence motion through space must be meaningless, since there would be no means of distinguishing one position from another. Now an argument of this sort is fair enough so long as we consider purely amorphous mathematical space; it ceases to present the same force when we consider a space, such as Newton’s which is assumed to possess a rigid structure. To prove this point, lei us assume that space, though homogeneous, is bounded by a sphere In this case, the homogeneity of space would not interfere (according to the critic) with the reality of motion, for we could always refer motion to the boundary.

Suppose, then, that the boundary were to be removed to infinity: would motion through space gradually lose all meaning? It appears quite unnecessary to adopt any such view. There is no reason why a local effect should not be developed by the passage of matter through a rigid homogeneous space, regardless of whether the boundary be at a finite distance or at infinity. To put it in a very crude way, there is no reason why the individual points of absolute space should not realise the presence or absence, hence the passage, of lumps of matter; and so the boundary would have no rôle to play.

Where motion would appear meaningless would be if we assumed matter to be continuous (not atomic) and considered a material rod of infinite length gliding along the straight line it defined. In this case (from the standpoint of the points of space) the motion of the rod might escape detection. In the general case, however, where lumps of matter, hence discontinuity (regardless of the molecular hypothesis), are concerned, there appears to be no reason to reject absolute motion through empty space on the ground that motion through a homogeneous and infinite medium would be logically meaningless. Furthermore, how the critic, having convinced himself that motion must be relative, proposes to account for the physical fact that of two flywheels in relative rotation, one will burst and the other remain unaffected, baffles the understanding. Either we must assume, with Mach, that the material universe exerts a definite causal influence on a body in rotation relative to it, or else we must accept Newton’s absolute space.

Berkeley’s criticisms appear to have arisen from some such argument as the one we have just discussed. Thus, he tells us that absolute motion cannot be imagined and for that reason should be banished from science. He then proceeds to point out that motion can only mean a displacement with respect to something sensible and that, space being suprasensible, motion through space is meaningless. Now it is perfectly correct to claim that men originally came to conceive of motion through visual experience by observing the motion of one body with respect to another. In other words, the phoronomic conception of motion is the natural one.

But when these points are granted, it still remains a fact that we cannot force our scientific co-ordinations into a mould which would satisfy too narrow a phenomenological attitude. In physics, as in pure mathematics, we are often confronted with conceptions which it may be difficult to imagine. Thus, although our understanding of continuity is derived from experience, for instance by passing our finger over the table, yet the concept of continuity has had to be extended and elaborated profoundly by mathematicians. Continuous curves with no tangents are not easy to imagine; yet we cannot deny their existence merely because it puts too great a strain on our imagination. Neither can we deny that there are as many points in a cube as on one of its sides, for Cantor has proved that such is indeed the case.

It is the same in physics. Experiment presents us with certain facts which we may often interpret in various ways. But we cannot ignore the facts merely because they happen to conflict with our own particular philosophy. All that we may do is to repeat the saying of a certain king of Spain, that had we been God, we should have constructed the universe with greater simplicity. Not being God, however, we must make the best of a bad job. Now, in the case of motion, any philosophy which starts to rule out absolute motion is forthwith confronted with the difficulty of accounting for centrifugal force. Berkeley, as might be expected, falls down on this point completely. His arguments reduce to a criticism of absolute space without affording us any better solution. Furthermore, his premises are scientifically incorrect.

For instance, when criticising Newton’s proof of absolute space as illustrated by the experiment of the rotating bucket of water, he remarks that owing to the earth’s velocity along its orbit, the particles of water cannot possibly be describing circles. And so Newton’s argument purporting to have detected absolute circular motion would thus be at fault. Berkeley’s argument reveals that same ever-recurring confusion between velocity and acceleration; he fails to realise that velocity through absolute space is not claimed by Newton to be detectible by mechanical experiments; the Newtonian principle of relativity states this point explicitly. Acceleration alone gives rise to forces. Now, so far as the particles of water are concerned, they possess exactly the same acceleration, whether they be whirling in circles or describing cycloids. Hence, Newton’s experiment discloses absolute acceleration, and this is all it was ever intended to disclose. A further argument presented by the same critic is based on the tangential force which he claims to be acting on the revolving water particles. He hopes thereby to account for centrifugal force without introducing absolute rotation. The reasoning is so obscure that we do not pretend to have fathomed it, but one thing is quite certain—the premises are totally incorrect; for there is no tangential force acting on uniformly revolving water particles, and centrifugal force manifests itself in this case. Obviously Berkeley is confusing momentum and force.

Other critics have speculated on the possibility of our space revolving in a more embracing one, this other in still another, etc. But what if it does! A hypothesis of this sort would never endanger the Newtonian position. Suppose, for argument’s sake, that our universe of space were likened to a bubble of ether rotating in a super-space; this bubble of ether would still represent absolute space. In every case the essential feature of absolute space and motion resides in the empirical fact that a certain definite set of dynamical axes appears to be present in nature; and the above-mentioned argument could never banish the existence of these absolute axes. All it could do would be to prove that nothing could be predicated of the motion of these axes in the super-space. But in any case the axes would be non-rotating in the ether bubble. This would thereby assume the position of absolute space as understood by science, and we should have to conceive of the more embracing space as rotating round it. Needless to say, the entire speculation is unscientific in the extreme since it reduces to a hypothesis ad hoc, beyond the control of experiment conceived of for the sole purpose of complicating a co-ordination o facts; whereas the only possible justification for this type of hypothesis is to permit us to introduce simplicity into our co-ordinations.

We may mention yet another type of argument because it brings out an important point of terminology. It is asserted, for instance, that acceleration must be relative since we measure it relatively to an inertial frame or to space. But here, of course, we have a mere confusion of words. The critic is assuming that absolute motion should mean motion with respect to nothing. But absolute motion in science does not mean this at all. It means motion which cannot be regarded as relative to something observable, such as matter; hence it becomes automatically motion with respect to something suprasensible, i.e., to space. Were it not for this interpretation placed on absolute motion in science, the expression would become meaningless, for motion with respect to nothing is itself nothing.

We may summarise the scientific attitude towards space and motion as follows: If motion can be detected otherwise than in relationship to things observable, or, more precisely, if a co-ordination of scientific facts renders it simpler to assume that such is the case, then motion, and hence space, must be considered absolute. If, on the other hand, no trace of absolute motion, as defined above, can be detected by experiment, two courses are open to us. Either we may assume that absolute motion has no scientific significance and that motion is always motion between observable existents, in which case motion and space are held to be relative; or else we may follow Lorentz’s method of dealing with the ether and claim that absolute motion would be detected were it not for a number of compensating physical effects which just happen to conceal its presence from our experiments. But if we adopt this very artificial attitude, it is imperative that we follow up our argument as Lorentz has done, and succeed in specifying the precise mode of action of these compensating effects and also their precise numerical magnitudes—and this would entail a considerable knowledge of advanced mathematics. So much for the scientific status of the problem of space and motion.

Now metaphysicians have a habit of confusing this scientific aspect of the absoluteness or relativity of motion and space with that other problem dealing with the essence of space. But, as Euler pointed out two centuries ago in the passage quoted previously, this metaphysical problem is of a totally different order, and has no bearing on the one that scientists are discussing. Even if we were prepared to attribute any meaning to such metaphysical inventions as Leibnitz’s monads, and agreed that space might be conceived of as the result of a relationship between them—even so, in view of the dynamical facts stressed by Newton, motion and space would still be absolute. It would be of no avail to hold that motion was relative since it was relative to the monads, for these, whether fictitious or real, are at all events suprasensible. Conversely, even were we to accept the metaphysician’s claim that, metaphysically speaking, space is an absolute existent, nevertheless, were it impossible to detect any trace of absolute motion, we should have to say that motion and space were scientifically relative, or else follow Lorentz’s method of dealing with the ether.

When, in addition to all these facts, we remember that the metaphysician’s theories reduce to mere expressions of opinion affording no possibility of proof or disproof, we can well understand that the scientific attitude towards space and motion has been governed solely by the empirical evidence.

And now the query will naturally be raised: If the dynamical facts of motion impose the belief in absolute rotation, how is it that Einstein, and even before him Mach, should have considered it possible to escape Newton’s solution? Let us first consider Mach’s argument. Mach’s aim was to co-ordinate the dynamical facts of motion in terms of sensible factors alone and thus obviate the introduction of that suprasensible entity, absolute space. And so he was led to conceive of rotation as rotation with respect to the universe, hence with respect to the stars. In contrast to Berkeley, however, Mach realised the great difficulty of accounting for centrifugal forces under this view. He, at least, made an attempt to solve the puzzle by attributing a direct dynamical influence to the relative rotation of the star-masses. But, since his attempt was not followed up mathematically, it was nothing more than a loose, unsupported suggestion. The fact is that in physical science the only convincing theories are those we can defend with quantitative arguments; mere undeveloped guesses are of no value.

Even Newton speculated on the cause of gravitation, attributing it to some species of ether pressure, yet in spite of his illustrious name these ideas were not even mentioned by subsequent scientists. Why? Because they afforded no quantitative treatment and were so vague as to be worthless. It is the same when we consider the hypotheses that have been suggested in order to account for the origin of the solar system: we find these to be extremely numerous and varied. But not one of them carried any weight until it had been submitted to mathematical investigation, and then, according to the results obtained, it was declared possible or impossible. And it is precisely this mathematical work that is the crux of the difficulty, calling for the genius of a Laplace or a Poincaré. In much the same way, we may guess that the sequence of prime numbers will eventually be included in a concise mathematical formula, or that some day interplanetary communications will be established. The difficulty resides not in the guess, not in the desire, but in its realisation. As a matter of fact, it is questionable whether loose guesswork has ever been of any use in science. Jules Verne’s idea of ships that travelled under water can scarcely be claimed to have contributed to the invention of submarines, any more than his story of a trip to the moon can be of much assistance in enabling us to set foot on our satellite. Taking a more scientific example, Democritus’ guess about atoms most certainly never advanced their discovery by a single hour.[149]

And so it is with Mach’s guess (we can scarcely dignify it with any other name). Indeed, in Mach’s day, it would have been quite impossible for any one to justify his idea, since the necessary material, i.e., space-time, was then unknown. When we realise the important rôle played by space-time in our attempts to avoid a belief in absolute rotation, we can well understand that the doctrine of the relativity of all motion would have been absurd in Newton’s day. In fact, any thinker prior to, say, the year 1900 could never have anticipated the discovery of space-time, for its sole justification arose from the negative experiments in optics and electrodynamics attempted at about that time. As for Newton, not only did he know nothing of the non-mechanical negative experiments, but in addition, the equations of electrodynamics had not been discovered in his day. Furthermore, even had he conceived of space-time through some divine inspiration, he could never have utilised it for the purpose of establishing the relativity of all motion. His ignorance of non-Euclidean geometry would have rendered the task impossible. In fact, space-time, in the seventeenth century, would have been a hindrance, and the sole result of its premature introduction into science would have been to muddle everything up and render the discovery of Newton’s law of gravitation well-nigh impossible.

And this brings us to another point which is often true in physical science. Premature discoveries are apt to do more harm than good. For instance, had the astronomers of the seventeenth century possessed more perfect telescopes, had they recognised that the planets (Mercury, in particular) did not obey Kepler’s laws rigorously, Newton’s law might never have been discovered. At all events, its correctness would have been questioned seriously and mathematicians might have lost courage and doubted their ability to discover natural laws. Leverrier, for example, might have lacked the necessary assurance to carry out his lengthy calculations leading to the discovery of Neptune. In short, physical science proceeds by successive approximations, and too rapid jumps in the accretion of knowledge are liable to be disastrous.

We may now follow up Einstein’s investigations step by step and see how Newton’s absolute rotation was gradually eliminated from science. The situation facing Einstein was somewhat different from the one that had confronted Newton. In Newton’s time, there was no a priori reason why motion through empty space should be regarded as absolute rather than relative. Whichever way experiment pointed would therefore be equally acceptable. But when motion through the ether was considered there was every reason to anticipate that it could not be meaningless. The fact was that the ether seemed to present the properties of an elastic material medium, so that it was difficult to anticipate a marked difference between motion through the ether and motion through matter. More important still, the equations of electromagnetics proved that phenomena should be affected by motion through the ether. It followed that when the negative experiments in electrodynamics were being performed, there was every reason to suppose that absolute velocity through the ether would be detected. Had this been the case, the stagnant ether would, in all probability, have been identified with Newton’s absolute space. And it might have been claimed that absolute velocity, which had always escaped mechanical detection, had been revealed at last by electromagnetic and optical tests.

Yet, as we know, absolute velocity, even through the ether, obstinately refused to reveal itself. The situation was similar to the one we mentioned when discussing space. There also, absolute velocity through space appeared to elude us, in spite of the fact that, owing to the absoluteness of rotation, space could not help but be absolute. But, in the case of space, this duality entailed by the Newtonian principle of relativity was accounted for immediately by the mathematical form of the equations of mechanics. The fundamental law of mechanics, stating that a force is equal to a mass multiplied by an acceleration, makes no mention of velocity; hence, absolute velocity is obviously irrelevant to mechanical processes. In the case of the ether, the elusiveness of velocity was much more disquieting; among other reasons, because the equations of electromagnetics contained a velocity explicitly in the form of the electric current.

And so, when account was taken of the supposedly semi-material nature the ether was thought to have, and when the lack of covariance of the equations of electromagnetics was considered, the course of least resistance obviously suggested that we assume velocity through the ether to be a reality, but that its effects were concealed by compensating influences. At any rate, we need not be surprised to find that chronologically this was the attitude first considered. It was, as we remember, the attitude championed by Lorentz. But when Lorentz had succeeded in accounting for the negative experiments under this view, his theory appeared so patently artificial that scientists recognised that something was wrong somewhere.

If, with Einstein, we adopt a very general attitude, neglecting to consider the why and wherefore of things, and if we restrict our attention solely to what experiment has revealed, we cannot fail to be struck by the following fact: Every time we have sought to detect absolute velocity, whether through space or through the ether, our attempts have failed. Even certain classical scientists, on the strength of mechanical experiments alone, had felt compelled to banish the thought that absolute velocity through space had any meaning. How much stronger, then, was the suspicion that some general principle was involved, when the same situation confronted us once more in the case of the ether! Under the circumstances, it can scarcely be said that Einstein followed a very revolutionary course when he postulated his special principle of relativity, claiming that absolute velocity through space or the ether was a meaningless concept. In so doing, he was merely stating in abstract form the result of experiment. Einstein’s special principle can be expressed, as follows:

“No experiment, regardless of its nature, whether mechanical, optical or electromagnetic, can ever enable us to detect our absolute velocity through space or ether.”

We see that the only difference between Einstein’s principle and the Newtonian principle of relativity is that henceforth the relativity of velocity holds for all manner of experiments, or, again, that ether and space are identified. Apart from this difference nothing is changed. As before, space or ether remains absolute, in that though relative for velocity, it still manifests itself as absolute for accelerated or rotationary motions, just as in Newton’s day.

Against Einstein’s stand, the metaphysician may object to absolute velocity being cast aside as meaningless, merely because no experiment seems capable of demonstrating its existence. But science, as we know, is not metaphysics; it is based on experience. An a priori rejection of absolute velocity would sin against the scientific method; but an absolute velocity which, though supposedly present, no experiment can reveal, and for which, in addition, no useful function can be found, plays no part in the workings of nature. Were future experiment to detect this absolute velocity, it would of course have to be reintroduced; but to retain it on general principles against the verdict of experiment would be very poor science.

We must be careful not to confuse the present situation with that entailed by atomism, for example. Even to-day we can scarcely say that atoms have been observed in the same sense that stones and tables have been observed; nevertheless, the majority of scientists accepted the existence of atoms years ago, because by postulating their presence a number of phenomena could be co-ordinated with simplicity. Thus, though eluding direct detection, atoms were demanded by indirect mathematical reasoning. Nothing similar is observed in the case of absolute velocity. We insist on this point because it is sometimes thought that the theory of relativity is essentially phenomenological. In a wide sense this is true, but it is most certainly not true if by phenomenalism we understand the word in its narrower sense. An out-and-out phenomenologist, such as Mach, would go so far as to deny the existence of atoms merely because they had never been observed by human eyes, regardless of whether it was useful to conceive of them for the purpose of co-ordinating empirical facts. This is not the attitude of science. But if by phenomenalism we mean the desire to free our understanding of things from unnecessary metaphysical notions which are in no wise demanded by experiment, then we are undoubtedly justified in claiming that not only the theory of relativity, but modern science itself, is essentially phenomenological. If these points are clearly understood, we see why it was that Einstein considered it necessary to rid physics of absolute velocity. In short, the exclusion of absolute velocity from science appears to be imposed by experiment; and the only course to adopt is to pursue this train of enquiry in a logical way and see where it will lead us.

We remember that the relativity of velocity, taken in conjunction with the equations of electromagnetics, indicated the invariance of the velocity of light in vacuo, as measured in any Galilean frame. From this, the Lorentz-Einstein transformations followed with mathematical logic. It is these transformations that entail, as we know, the relativity of duration, length and simultaneity. We cannot, therefore, regard the new notions as the result of some pipe dream or some divine inspiration; they were forced on Einstein by his transformation equations, and these, in turn, were derived from the initial principles, hence from ultra-refined experiment. Needless to say, it would have been highly arbitrary, and in fact absurd, to lay down the special principle of relativity before such time as the negative experiments had driven us to it. To proceed on the strength of some divine inspiration prior to the disclosures of experiment would have been to start out on a wild-goose chase. Here it cannot be emphasised too strongly that a belief in the relativity of velocity through the ether does not impose itself a priori; quite the reverse. To put it differently, there is no a priori reason why the equations of electromagnetics should preserve the same form regardless of the velocity of our frame of reference. We may feel more sympathetic towards one doctrine or another, but in the last analysis it is experiment, and experiment alone, which can guide science in such matters.

It is important to understand that at this stage Einstein had discovered only the Lorentz-Einstein transformations together with their inevitable relativistic consequences. There is not the slightest hint, in his writings, of a world of space-time. This great discovery is due to the mathematician Minkowski, who, in the year 1908, proved that the Lorentz-Einstein transformations connoted the existence of a four-dimensional space-time continuum of events. From then on, those concepts of classical science, a space of points and a time of instants, considered by themselves, faded into shadows. Here again, Minkowski’s discovery was purely mathematical. It issued from a simple application of the theory of groups, and no trace of philosophical prejudice can be found in his work.

In short, it was the mathematical equations, hence the ultra-precise experiments, that had rendered inevitable the new outlook of the world. As Minkowski tells us himself in his inaugural address: “The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength.” As for space-time, that amalgamated continuum, it had never even been suggested in the speculations either of scientists or of philosophers. To Einstein himself, the discovery of the four-dimensional space-time world was probably quite as much of a surprise as it was to the world at large; but there was no gainsaying the correctness of Minkowski’s arguments, which any one with an elementary knowledge of mathematics could verify. Thus, although space-time constitutes one of the most astounding philosophic revolutions ever witnessed, yet the procedure which led to its discovery reduces to the customary mathematical treatment of the empirical facts yielded by the experimenters.

Let us examine what bearing these new ideas will have on Newton’s absolute space and motion. Of course, to begin with, space-time taking the place of separate space and time, our outlook of the world is considerably modified. It is modified in the sense that distance, duration and simultaneity are no longer absolute. If, however, we view the problem of space from the standpoint of motion, that cornerstone of the Newtonian synthesis, we find very little change.

The same dynamical facts that had compelled Newton to recognise rotationary motion as absolute drove Einstein once more to the same conclusion. Hence, just as Newton, arguing from the standpoint of space, was compelled to accept absolute space, so now Einstein, arguing from the standpoint of space-time, was driven to absolute space-time. In short, whether we argue in terms of space or of space-time, the dynamical facts of motion appear to impose an absolute background in either case.

Yet, even when confining ourselves to the problem of motion, absolute space-time presents a great advantage over Newtonian absolute space. With absolute space, the difficulty was to account for the elusiveness, relativity or mythical nature of velocity through empty space (embodied in the Newtonian principle of relativity). But with absolute space-time this difficulty is overcome. For absolute space-time, while necessitating the absolute nature of rotation and acceleration, necessitates with equal inevitableness the relativity of velocity. And so the introduction of space-time has served to eliminate that displeasing dualistic feature which was characteristic of Newton’s space. From all this, we see that the fundamental characteristic of the Newtonian synthesis, namely, the presence of a suprasensible absolute frame with respect to which rotation and acceleration would be measured (the inertial frame), remained unaffected. A vindication of Mach’s ideas seemed to be as remote as ever.

Einstein, in the following words, summarises the situation as it stood before the general theory was elaborated:

“The principle of inertia, in particular, seems to compel us to ascribe physically objective properties to the space-time continuum. Just as it was necessary from the Newtonian standpoint to make both the statements, tempus est absolutum, spatium est absolutum, so from the standpoint of the special theory of relativity we must say, continuum spatii et temporis est absolutum. In this latter statement, absolutum means not only ‘physically real,’ but also ‘independent in its physical properties,’ having a physical effect, but not itself influenced by physical conditions.

“As long as the principle of inertia is regarded as the keystone of physics, this standpoint is certainly the only one which is justified.” [150]

Now, when it was realised that the real world was one of flat four-dimensional absolute space-time and no longer one of separate space and time, it became necessary to adjust the laws of nature to the new mould. The laws of electrodynamics found a ready place, and this was only natural, since space-time had been moulded on those very laws. Einstein then succeeded in modifying the laws of mechanics, rendering them compatible with space-time. There still remained, however, that most important law, the law of gravitation. This law, as given by Newton, was incompatible with space-time, so that the next step was to modify Newton’s law in a suitable way. Poincaré attacked the problem and obtained a solution in 1906. Further attempts were initiated by Abraham, Mie and Nordström.[151]

But all such attempts were soon to be overshadowed by Einstein’s own brilliant generalisation. He succeeded in establishing the long-sought fusion between mechanics and gravitation, obtaining thereby the most beautiful theory known to science. However, before following Einstein in his solution, we may state that Poincaré, as far back as 1906, had succeeded in establishing a most important point.[152] Prior to his investigations it had been held that Newton’s law of gravitation constituted a powerful argument against the space-time theory. For, on Laplace’s authority, it was believed that the observed planetary motions required that gravitation should be propagated with a speed many times greater than that of light; a fact difficult to reconcile with the maximum velocity required by relativity. Poincaré, however, proved that the force of gravitation could perfectly well be propagated with the speed of light and yet yield laws of planetary motions practically identical with those of Kepler. He proceeded to consider the various possible laws of gravitation compatible with the flat space-time theory though reducing to Newton’s law for slow velocities. One of these laws, in particular, was found to account for the precessional advance of Mercury’s perihelion.

The significance of these discoveries was to prove that the relativity theory had nothing to fear from the phenomenon of gravitation. But a still more important point had been established. Had Laplace’s contention been correct, were it a fact that gravitation spreads instantaneously throughout space, we should be faced with gravitational action at a distance, a most displeasing conception. Newton was averse to action at a distance, but it cannot be denied that his law of gravitation was a direct appeal to it. If, on the other hand, gravitation were propagated with a finite speed, we could conceive of it as a continuous action through a medium, similar to that of electromagnetic forces. Gravitation could then be connected with the methods of field physics inaugurated by Faraday and Maxwell. To-day, these investigations of Poincaré have none but a historical interest, for they have been superseded by Einstein’s brilliant solution of the problem of gravitation, rendered possible by the introduction of a variable space-time curvature. Nevertheless, it is well to note that should Einstein’s general theory be abandoned as a result of some crucial experiment, it would still be possible to preserve the special flat space-time theory by taking the law of gravitation given by Poincaré or Nordström. However, as we shall see, the special theory drives us to the general theory in such a variety of ways that there is little fear of science having to suffer the severe setback which a reversion to flat space-time would entail in our understanding of the unity of nature.

Let us now pursue the trail of subsequent discoveries and see how Einstein was led to abandon the idea of a rigidly flat absolute space-time acting as a container for matter, but in no wise modified by the presence of matter. According to his own acknowledgment, this important discovery was reached by at least three different lines of reasoning. We shall examine these reasonings separately, for they throw an interesting light on the methodology of the theory.

In the first place, Einstein remarked that spatial measurements permed on the surface of a disk (rotating in flat space-time) would necessarily yield non-Euclidean results, since, owing to the FitzGerald contraction, the same rod placed radially or transversely would vary in length. But in a rotating frame forces of inertia are active; hence it was suggested that forces of inertia were related to a non-Euclideanism of the space (not space-time) of the frame in which the forces were active. We shall see how this discovery will be of use later on. Incidentally, let us note that this first result follows as a mathematical necessity from the special theory; no new assumptions have been introduced. Expressed geometrically, it means that the observer on the rotating disk splits up flat space-time into a curved space and a curved time.

Now we pass to the second step. The most delicate physical experiments had established the identity of the two types of masses, the inertial and the gravitational. Classical science had regarded this identity as accidental, or at least unexplained. Einstein assumed that we were in the presence of a fact of very great significance in nature, a fact that would have to be taken into consideration in any theoretical co-ordination of knowledge. If we analyse the reasons for Einstein’s attitude we find that they reduce to a disinclination to believe that an identity of this sort could be a mere matter of chance. In the same way, if two men were to publish the same book, identical word for word, probability would suggest that one of them had copied the work of the other. If we accept the identity of the two masses as something more than a mere chance occurrence, we must assume that the great similarity between forces of inertia and of gravitation is due to the fact that these two types of forces are essentially the same. But we have seen that in regions of empty space where forces of inertia were active, space (though not space-time) was non-Euclidean. Hence it must follow that in a gravitational field near matter, space must also be non-Euclidean. But there is a marked difference between the distribution of forces of inertia and those of gravitation. Forces of inertia occurring far from matter can be cancelled by the observer’s changing his motion. Thus, centrifugal force on the disk can be made to vanish; all we have to do is to arrest the disk’s, hence the observer’s, rotation. On the other hand, we cannot get rid of the force of gravitation. It is true that in a falling elevator the gravitational pull would vanish as a result of the elevator’s motion; and we could no longer feel it, falling, as we should, together with the elevator (owing to the identity of the two masses). But we know that were the elevator extended enough the pull would reappear in distant places because of the radial distribution of the gravitational field round the earth. And so we must conclude that in the case of a gravitational field produced by matter, the non-Euclideanism of space can no longer be got rid of by merely varying our motion, hence by merely varying our method of splitting up space-time into our private space and time. It follows that the non-Euclideanism of space, present in a gravitational field, must come from a deeper source. In particular, it must arise from an intrinsic non-Euclideanism in the space-time which surrounds matter, since it is only a curved non-Euclidean space-time that can never be split up into a flat space and a flat time. And so we see that, around masses, space-time can no longer be flat, as it was in regions far removed from matter. The intimate connection between the presence of matter and a non-Euclideanism of space-time becomes apparent. Matter and, more generally, energy cause space-time to become curved.

And here we must note an important difference between the two arguments we have presented thus far. In the first case, that of the rotating frame or disk, the argument was, so to speak, irrefutable: it was imposed as a direct mathematical consequence of the Lorentz-Einstein transformations. But the second argument, that referring to the identity of the two types of masses, was based on an experimental fact. When we consider the wonderful discoveries that have issued at Einstein’s hands from this identity of the two masses, we may find it strange that Newton or some other scientist should have failed to attach any theoretical importance to it. Of course, the reason for Newton’s neglect to consider the matter is easily understood when we remember that he ignored space-time. Only with space-time, coupled with a knowledge of non-Euclidean geometry, could the full significance of this identity be understood. This same excuse, however, cannot hold for those scientists who in 1908 were just as well acquainted with the special theory as was Einstein himself. Why did three long years have to elapse between the discovery of space-time, in 1908, and Einstein’s identification of the two types of masses, in 1911? This brings us to a different subject of discussion.

The point we have been attempting to explain is that the co-ordinations of theoretical physicists are based primarily on experimental facts, not on pipe dreams. But when this point is granted, the genius of the individual scientist consists in singling out those particular facts which he suspects will yield the most interesting consequences. That Einstein should have picked out this identity of the two masses, whereas no one else had thought of it, is a tribute to the genius of Einstein; and that is all that can be said. It is claimed that the idea came to him suddenly when, walking along the street, he saw a man fall from a house-top. This incident is said to have directed his attention to the conditions of observation that would hold for an observer falling freely. The story is reminiscent of Newton’s apple. At any rate, regardless of how Einstein came to think of the identity of the two masses, the important point is that this identity constitutes an experimental fact—a fact which had till then appeared in the light of a miraculous coincidence. For this reason, if for no other, it might have been suspected that this marvellous coincidence concealed something important in nature, which we had not yet grasped.

Reverting to the motives which had driven Einstein to suspect that matter must modify the space-time structure, we have seen that they were of a totally different nature. The rotating-disk argument was exclusively mathematical; the equivalence of the two masses was based on a fact of experience. Neither of these two arguments was suggested by philosophy, theosophy or theology. To assert, then, that Einstein was influenced by the philosophy of Descartes or Leibnitz or Mme. Blavatsky or any one else would indicate a very poor knowledge of scientific method. Although the new views entail a marked change in our understanding of the relationships between extension and matter, yet there is no irresponsible guesswork in Einstein’s procedure. In particular, there is no desire to impose some special matter-moulding philosophy. If the theory is drifting towards certain philosophical conclusions, it is drifting with the current; and the current is represented by a rational co-ordination of the facts of experience.

We now come to the third argument advanced by Einstein in favour of his matter-modifying space-time theory. We have left it to the last, as it is more philosophical than the two preceding ones. We may condense it into the statement that where there is action there must also be reaction. Now, in rigidly flat space-time, the courses of free bodies must be conceived of as directed by the geometry or structure of space-time. Hence there exists an action of the space-time structure on the bodies. But then there should also exist a reciprocal action of the bodies on the structure. In much the same way, the electron affects the electromagnetic field and, inversely, the field affects the electron. Einstein expresses, in the following words, his dislike for a conception which would conflict with the reciprocity of action:

“In the first place, it is contrary to the mode of thinking in science to conceive of a thing (the space-time continuum) which acts itself, but which cannot be acted upon. This is the reason why E. Mach was led to make the attempt to eliminate space as an active cause in the system of mechanics. According to him, a material particle does not move in unaccelerated motion relatively to space, but relatively to the centre of all the other masses in the universe; in this way the series of causes of mechanical phenomena was closed, in contrast to the mechanics of Newton and Galileo. In order to develop this idea within the limits of the modern theory of action through a medium, the properties of the space-time continuum which determine inertia must be regarded as field properties of space analogous to the electromagnetic field. The concepts of classical mechanics afford no way of expressing this. For this reason Mach’s attempt at a solution failed for the time being.”

If we review the position of the theory as it now stands, we may summarise it as follows: Space-time is primarily an absolute four-dimensional continuum of events. When devoid of matter and energy, its structure is flat. When matter is present, the flat structure yields gently both around matter and in its interior. The yielding, however, is exceedingly slight. Though sufficient to account for gravitation, it is far too insignificant to be detected through direct measurements with rods (this holds at least in the case of the space around the sun). Disturbances of structure, caused by the sudden arrival or passage of matter, are propagated from place to place with the velocity of light, so that gravitation manifests itself as a continuous action through a medium, just like the action of the electromagnetic field. Action at a distance is thus avoided.

And now, what is the bearing of these discoveries on the problem of absolute rotation and absolute space-time? We see that space-time, by yielding slightly in the presence of matter, has lost that absolute rigidity which characterised Newton’s space. Owing to these varying distortions of the space-time background, brought about by variations in the matter distribution, the rotational velocity of a body suffers from a certain measure of indeterminateness. To this extent rotation is no longer absolute.[153] But when we review the sequence of deductions that has led us to this partial rejection of Newton’s belief in absolute rotation, we see that our discoveries have become possible only through the medium of space-time. Had space-time never been discovered, had we remained content with a separate space and time, it would have been quite impossible to establish this indeterminateness of rotation; for it would have been impossible to account for gravitation in terms of the variations in the structure of space alone. As a result, the spatial background would have remained rigid, and Newton’s position would have stood secure.

Even now, though the fluctuating space-time background has been discovered, the absolute nature of rotation has not been fully disproved. For in an empty universe, space-time would still preserve a well-defined rigid structure; hence, if we conceive of one single body introduced into this otherwise empty world, an absolute rotation of this body could always be detected. It would be betrayed by centrifugal forces; and these forces would arise because the particles of the body would be following world-lines which departed from the geodesics, or lines of least resistance, of the space-time structure. For this reason, rotation would still be absolute, since it would preserve a real meaning in a world devoid of matter. In short, it would have nothing to do with a rotation relative to the star-masses. Obviously we are still a long way from having vindicated Mach’s mechanics.

But when it is considered that Einstein’s theory is based on the relativity of velocity, then that it establishes, in a partial way at least, the relativity of acceleration, it cannot be denied that a logical extension of its triumphs would be for it to succeed in establishing the complete relativity of all motion and of inertia. Although Einstein never conceals his hope that this will be the ultimate outcome, there is no desire on his part to force the issue in defiance of facts.

It may, however, be of interest to ascertain what modifications in the theory would be required for the relativity of all motion and of inertia to be established. Inasmuch as it is the inherent structure of space-time persisting even in the absence of matter which is responsible for centrifugal force in an empty world, hence for absolute rotation and inertia, a vindication of Mach’s ideas would entail the following condition: It would be necessary to assume that in the absence of all matter, space-time would lose all trace of a structure and become amorphous. In other words, matter would have to create space-time and its structure (i.e., the -distribution, or the metrical field), and not merely modify locally a pre-existing structure. It is interesting to note that the relativity of rotation is thus leading us to a concept of space which could not have been anticipated a priori. Mach himself, when defending the relativity of motion, does not appear to have realised that the abandonment of Newton’s absolute space would entail a matter-created universe.

Now what Einstein has done has been to prove that it is mathematically possible to conceive of a universe in which the entire space-time structure (hence the entire -distribution, or metrical field) would be conditioned by matter. This, as we know, is the cylindrical universe. But, here again, we must remember that he has never asked us to accept this cylindrical universe merely because it entails the relativity of inertia and of rotation. However strongly the relativity of all motion may appeal to us, it reduces in fine to a mere philosophical fancy; and more imperative arguments must be presented before it can be entertained.