When we consider the new views that are forced upon us by the theoryof relativity, we find that certain refinements are necessary. The special theory of relativity has shown that there was no reason to dispute the status of those qualities thought to be primary by classical science so long as we restricted our investigations to the points of view of observers at rest with respect to the object; but that when observers in relative motion were considered, it became impossible to effect a synthesis of all the apparent shapes so as to obtain one sole impersonal or real spatial shape. Real shape was thus removed from the status of a primary quality, and there was no sense in speaking of the shape of an object in a world devoid of all observers. Similar conclusions applied to size, duration and all the other primary qualities mentioned, with the sole exception of electric charge, which retained its impersonal status. Of course, had the theory of relativity deprived us of the possibility of conceiving of a sufficient number of primary qualities, hence of an objective world, it would have led us to a stone wall. But we know that such is not the case, and that the theory has merely modified the list of the primary qualities of classical science, assigning the same fundamental rôle to others in their stead. These are given by the space-time intervals and, more generally, by space-time configurations and tensors. In other words, the objective universe disclosed by relativity is no longer one of shape and duration, or space and time, but one of space-time.
Yet, regardless of the actual list of qualities that are to be classed as primary, the significance of these qualities remains the same in either science in that they are deemed to represent entities to which an impersonal existence may be conceded, if for none other than methodological reasons.
Sufficient information has been given in this book to enable the reader to understand the similarity which exists between the comparative status of the objective and the private view, on the one hand, and between tensors and tensor components, on the other. Thus, to take an illustration from electromagnetics, we know that what our instruments detect at a given point in a given frame of reference are the electric- and magnetic-field intensities. When the frame is changed, the measured values of these quantities are modified. Hence here we have the analogue of the various apparent shapes of the cone. And just as the apparent shapes of the cone could be attributed to an objective or real three-dimensional cone (in classical science) or to a space-time configuration (in relativity), so now, in the theory of relativity, the electric and magnetic intensities can be regarded as the six variable components of a real space-time tensor, the electromagnetic-field tensorsubsisting in the objective space-time world. When we change frames, the tensor remains unaffected in the objective space-time world, but its six components in the space-and-time frame we happen to occupy are subjected to definite variations when our frame is changed; and for this reason fields that appear purely electrical when we are moving with a certain relative velocity will appear electromagnetic when our relative velocity is modified.
The general classification of objective, absolute or public realities, on the one hand, and of relative or private ones, on the other, might yet appear to present none but a methodological interest and to have no bearing on the problem of the genuinely real. But, assuming for argument’s sake, that the concept of the genuinely real has a meaning, it can scarcely be denied that if we ever hope to discover this genuine reality, it is assuredly more reasonable to seek it in those realities which we have cause to regard as public rather than in those which remain essentially private. For, as we have seen, a private reality, such as colour or apparent shape (or three-dimensional spatial shape in relativity), is indefinite until such time as the conditions of observation are specified; hence it is not self-supporting, but entails the mention of a relationship to something else. On the other hand, the generalised primary qualities transcend this specification of relationships, that are always of a contingent nature. Hence, quite aside from considerations of a methodological order based on convenience, there appears to be every reason to assume that the generalised primary qualities approximate more nearly than the secondary ones to the unattainable ideal of the genuinely real, or the metaphysical reality.
We have now to consider a further type of difficulty which would confront the metaphysician who claims colour to be a reality having an existenceper sein a world devoid of observers. In the preceding pages, when discussing the relativity of colour to the percipient, we have dwelt on the relativity of colour to motion, embodied in the Doppler effect. But it is well known that a distinction between primary and secondary qualities had imposed itself long before the Doppler effect was discovered. One of the reasons responsible for this differentiation is to be found in the fact that colour, in many cases, is also relative to position. Berkeley gave an illustration of this type of relativity when he wrote of a cloud which, though it looks crimson fromhere, is a dark mist when we penetrate its interior. But his illustration was a poor one, for when we enter the cloud we are no longer viewing its surface. It would not be difficult to improve upon Berkeley’s example, but for our present purpose it will not be necessary to dwell further on this type of relativity, so we shall examine another species of relativity, namely, relativity to gauge.
Suppose, then, that a green object is placed on the table. The metaphysician who states that colours are realities which would exist in a world devoid of percipients would presumably state that the object’s surface was green in its very essence. But it might well happen that, on viewing the surface with a powerful microscope, we should find it to be made of a patchwork of blue and yellow spots and that all appearance of green had vanished. Unless the metaphysician proceeds to ignore the microscopic view entirely, he will be placed in the dilemma of wondering whether the green quality or the yellow-and-blue one is to be ascribed to the object in the real world. Even so, his troubles would not be at an end, for we might conceive of an ultra-microscopic vision compared with which our erstwhilemicroscopic vision would be macroscopic, and so on indefinitely. Now the change in colour that accompanies a passage from the macroscopic to the microscopic suggests that changes of equal importance might ensue from each successive passage to the following microscopic view. Indeed, it might be that for organisms whose size was of the order of wave-lengths no colour would manifest itself at all. Inasmuch as it would be highly arbitrary to assert that the ultra-microscopic vision is in any wise less worthy of consideration than the more usual macroscopic one, it appears to be impossible to ascribe definite colours to existents in a world devoid of all observers.
It is true that in the present discussion we started from an example where the microscope enabled us to detect a difference in colouring; and cases might arise where even under the microscope the object would still appear green. But, as we said before, the microscopic vision is itself ultra-macroscopic when contrasted with the infinite regress of ultra-microscopic visions we can conceive of; hence, even in this case, the difficulty would be only postponed, not obviated. After all, our ability to pass from microscopic to macroscopic views is contained within comparatively narrow limits. The microscope brings us to the limits of microscopic vision; yet it does not even enable us to see some of the tinier micro-organisms. As for the ultra-microscope, it yields but a blur. Turning to the ultra-macroscopic vision, it appears to be best exemplified when we view the Milky Way; but this vision in turn would be microscopic as contrasted with others of which we might conceive. The upshot of this discussion is that when the metaphysician speaks of red as inhering in the flower of a geranium, he is making a statement that cannot withstand scientific criticism.
Contrast these ambiguities and difficulties with the primary quality of shape in classical science (not in relativity). For in classical science, if a coin be recognised as round as a result of a synthesis of certain microscopic points of view, it will remain round when precipients situated in any position, moving with any velocity, and observing it with any macroscopic vision co-ordinate their impressions.Owing to this comparative irrelevance of the percipient, there appears to be a certain justification (in classical science) for speaking of a round coin existing in a world devoid of percipients, whereas there is none for speaking in the same sense of the brown colour of the coin.
The next stage we have to consider deals with the reduction of secondary to primary qualities. Whereas, up to the present, our arguments based on the relativity of perceptions applied solely to such secondary qualities as colour and sound, we shall now be in a position to advance arguments holding for the remaining secondary qualities, such as heat and smell.
We said, when discussing the red light, that it was impossible to effect a synthesis of all the colours perceived by the variously moving observers and obtain therefrom one definite, impersonal colour. Nevertheless, we may succeed in obtaining objectivity if we find it possible to regard sensations of colour as due to certain specificperturbations suffered by the retina as a result of the impact of electromagnetic waves. According to the relative motion of the observers, the objectively real electromagnetic waves will impinge upon the retina with one frequency or another, and corresponding sensations of colour will ensue. Colour would thus manifest itself as the result of a relationship.
The realistic metaphysician might urge that this may be a convenient way of accounting for colour, but that it is artificial and that, in spite of all, colour should be considered a reality existingper se. But if this stand were adopted it should also be permissible to say that the fizziness of the Seidlitz powders already existsper sein the powders, and results in no wise from a relationship,i.e., from the chemical interaction of the two powders. Or, to take Galileo’s illustration, we should say that the tickling quality must be in the feather, since a feather passed over our skin may tickle us.
The metaphysician’s stand appears untenable, for it is impossible to deny the existence of relationships giving rise to effects which were non-existent before the relationships were established, and which often differ appreciably from their generating causes. And every time we sense a colour or a sound or a smell, a relationship is established between some objective entity and that particular sense-organ which is affected. The fact is that the human organism is a highly complex piece of workmanship and that as a rule it reacts differently (though sometimes in the same way) to the various perturbating influences; while at other times, so far as we know, it does not react at all and no sensations are registered. To speak of colour and sound as existing independently of the organism is thus as nonsensical as to discuss a one-sided triangle.
To take an example at random, let us consider a problem of heat. The sun emits rays which are commonly supposed to be luminous and hot. If, then, as the metaphysician asserts, hotness is inherent in the rays, we should expect the space between the earth and the sun to be flooded with heat, hence warm. But such is not the case. The temperature of interplanetary space is in the neighbourhood of absolute zero. On the other hand, the atmosphere becomes warm under the sun’s rays. Why this difference? Because the energy of the electromagnetic waves is transformed into molecular agitation when the waves impinge on matter, just as a spark is produced when two flints are struck together. The more rarefied the matter on which the waves impinge, the smaller will be the percentage of the electromagnetic energy transformed or degraded into molecular agitation. It is partly for this reason (owing to the progressive rarefaction of the atmosphere) that the temperature gradually falls as we rise to high altitudes in an airplane. Now science claims that on a hot day, when standing in the shade, we experience a sensation of heat, our sensation results from an interaction between the extremities of our sensory nerves and the impacts of the molecules constituting the atmosphere. The metaphysician will presumably balk at this statement, claiming that the heat we sense exists independently of us in the air and that we merely detect itspresence. But if his contention were accepted, we should also have to assume that the heat existed in the sun’s rays and had been transferred directly from them to the air. And we have shown that this view is untenable, since the light rays produce heat merely as a result of a transformation of their energy into one of another type (molecular agitation). And since it appears impossible to maintain that the heat was already in the rays prior to their encounter with matter, what justification is there for asserting that the heat was already in the air prior to its encounter with our sensory organs? In every case we are witnessing transformations of energy, and these transformations are always accompanied by degradation (except in the ideally reversible transformations). This fact is expressed by the law of entropy, according to which, though the energy is conserved in quantity, its potentiality, or quality, is lowered,i.e., degraded.
Now, to revert to the problem of colour, it is possible that when a light-wave impinges on our retina some chemical change is produced (as in the case of the photographic plate, where the molecule of chloride of silver is loosened); possibly, too, a liberation of electrons takes place by a photo-electric effect. At all events, it is more than likely, on the strength of other phenomena of which we know something definite, that by the time the energy of a supposedly red electromagnetic vibration has reached our brains it has been subjected to all sorts of transformations. Even if we feel justified in saying that redness hits our eye, what right have we to maintain that, contrary to all the known laws of physics, this redness is perceived by our consciousness without any transformation having taken place? And if we admit that physics is not a complete hoax, how do we propose to assert that the redness we sense is already present as redness in the electromagnetic radiations?
The metaphysician’s attitude will lead to still further difficulties when a number of other examples are considered. In the first place, science has discovered that the electromagnetic vibrations accompanied by colour constitute an insignificant minority. There exist electromagnetic vibrations of varying frequencies identical in all respects with the luminous ones, except that they do not happen to influence the human eye. If we adhere strictly to the ultra-realistic viewpoint, we shall have to assume that these invisible radiations contain no properties of colour. And yet, as we know, though invisible to the human eye, they are detected by the photographic plate and, in all probability, by a number of animals. Were the photographic plate possessed of consciousness, or, again, were our eye provided with a photographic plate in place of the human retina and optic nerve, etc., the colour red would be unknown; whereas a new colour, ultra-violet, would be sensed with ease.
It thus becomes impossible to escape the conclusion that our precise perception of colour is affected by the idiosyncrasies of the eye and organism. We need not dwell on a number of well-known illustrations, such as the effect produced by certain drugs. If, therefore, when defending the reality of colour, the metaphysician is referring to those colours which the human eye can detect, he is making the veryexistence of his realities dependent on the accidental circumstance that the human eye is sensitive to one particular range of vibrations and to no other. He can emancipate himself from this difficulty only by assuming that all electromagnetic waves possess colour, whether visible or invisible. This would lead him to ascribe colour to radio waves. But why stop even there? Why not attribute colour, sound and smell to everything while we are about it, and talk of the colour of sound waves and the sound or smell of electromagnetic ones? We should thus be led into complete intellectual anarchy. Then again, consider interference phenomena. Red light plus red light yields obscurity. If we consider redness as an objective reality, we must at least concede that positive redness and negative redness must exist, so that a mutual destruction becomes possible. In similar fashion, there would be positive and negative sound, and all kinds of other absurd notions.
That all these points should have appeared controversial in the days of the Greeks is only natural, since they knew very little of the transformations of energy, and nothing of the invisible radiations or of the Doppler effect or of interference bands, etc. But nowadays the ultra-realistic standpoint is quite untenable for the scientist. The reader acquainted with scientific thought may feel that this long discussion was quite unnecessary. Such would also have been our own opinion had it not been for the profusion of contemporary philosophical literature purporting to defend the reality of secondary qualities. Were it not for a few scientific words interspersed here and there in their writings, one would have no difficulty in believing that these productions were those of men who had lived more than two thousand years ago.
Another type of objection to the method of quantitative reduction is that a philosophy of nature must be based on a consideration of qualities as well as of quantities. But the critics overlook the fact that by postulating too many conditions they may be rendering a consistent philosophy quite impossible. The aim of a philosophy of nature should be to increase our understanding of nature. Vagueness has ever been an obstacle to knowledge; it prevents us from differentiating between concepts which present but a superficial similarity, and obscures the interconnection of phenomena. For this reason, science has had to reduce qualitative differences whenever possible, since it was they that interfered with the possibility of obtaining any precise co-ordination.
In a similar spirit, science has ever endeavoured to eradicate such statements as “nature abhors a vacuum” or “chemical elements possess various affinities.” Statements of this sort, just like the Aristotelian qualities, are worthless, for they lead nowhere. To clothe our ignorance in the guise of a phrase may flatter our vanity, but it does not advance us one iota. In short, although the scientist may not have taught us much about colour and light, he has at least increased our powers of prevision; and that is no mean achievement. Of course prevision is not everything. The enjoyment of the present is also of interest; and it might always be claimed that the quantitativereduction had eliminated beauty and noble aspirations from the world. However, if this were the case, why not add that it has also eliminated filth and misery? But such considerations are quite beside the point. There is no reason why a physicist should not derive just as much pleasure from the beauties of music as a man ignorant of the laws of acoustics. Hence, we must make up our minds once and for all and decide whether in our quest for knowledge we intend to be guided purely and simply by our feelings, or whether we intend to seek our inspiration in the abstract intellectual ideal of unity and simplicity with which theoretical science dwells.
Let us now return to a more scientific subject. We have seen that a theory of mathematical physics reduces to a rational co-ordination of a large number of facts and phenomena. The interconnections between these phenomena permit us to pass by a continuous chain of reasoning from one to another, and enable us thereby to anticipate the future or the past from a knowledge of the present. We have also seen that various syntheses of facts were possible; but that, in general, one particular synthesis stood out prominently, owing to its greater simplicity and to the smaller number of auxiliary hypotheses it contained. Thanks to the absence of disconnected hypothesesad hocin this simplest synthesis, we are able to find our way about without being bewildered at each turn by gaps and hiatuses. Owing to our ability to foresee and to foretell the future activities of nature, we feel that we have succeeded in unravelling her hidden scheme and in understanding her guiding motives. To this extent we consider ourselves justified in assuming that the knowledge we have uncovered or created corresponds to reality, although, of course, the reality contemplated remains essentially pragmatic. This pragmatic conception of reality has been stressed repeatedly by Poincaré. But it is by no means peculiar to the great mathematician and philosopher. We find it in the teachings of Euler and Riemann, and of all the great scientists of the last two centuries; in short, it is the philosophy of science.[140]
Obviously, there is nothing in the nature of an explanation in a scientific theory. Phenomena are not explained; they are merely interconnected, or described in terms of their mutual relations. As a matter of fact, there is no cause to be surprised at this failure of science to explain phenomena, for the failure arises not from the limitations of science, butfrom the limitations of the human mind itself. All we can ever do is to interpretin terms of,andin terms of.However far we go, we can never avoid anultimate unknowable. This perfectly obvious contention is sometimes obscured because the end of the series may be so far removed as to escape our attention; but this is merely a mental subterfuge, a form of mental laziness.
Thus, the Greeks wondered why the earth did not fall. An explanation was soon forthcoming. The giant Atlas holds it on his shoulders. But what, then, prevented the giant Atlas from falling together with the earth he was carrying? The nether regions on which he stands give him support. This is as far as the explanation went, for no theory seems to have been advanced explaining why the nether regions did not fall, together with Atlas and the earth.
We might consider more scientific examples; but we should find that the case was always the same.is described in terms of,in terms of,and so on. Consider the phenomenon of gravitation. Does any one really imagine that Newton or Einstein has ever attempted to explain gravitation? To say that gravitation is a property of matter or a property of space-time in the neighbourhood of matter is just as much of an explanation as to say that sweetness is a property of sugar; for in the last analysis, what is matter—what is space-time? If we say that matter is an aggregate of molecules, atoms, electrons, protons, what of it? What are electrons? What are protons? We can only confess our complete ignorance and, while attempting to reduce the number of these unknown fundamental entities to a minimum, content ourselves with describing the properties which appear to characterise them and the relationships that appear to connect them. Clearly, those who seek explanations will find no comfort in science. They must turn to metaphysics.
And yet, as a matter of fact, these rather gloomy conclusions are gloomy only because we are expecting too much. If we content ourselves with what we can obtain, we shall find that the descriptions of science are creative and fertile, and not sterile, as descriptions usually are. As we have already explained, a scientific description is creative in that it allows us to foresee and to foretell. A description that could not confer this power on us would be of little use. Thus, we might describe the petals and stamens of a flower, describe its colour, odour and general appearance. A description of this sort would be useful in enabling us to recognise the same plant when we came across it again. But apart from this, it would teach us nothing new. It would not tell us, for instance, whether the root was bulbous or fibrous. On the other hand, if as a result of prolonged botanical study we could establish certain recurrent relationships between the characters of flowers and the other characteristics of the plants, we might be able by a mere examination of the flower to anticipate the nature of the root. In palæontology it is the same. The discovery of a fossilised bone has often been sufficient to enable scientists to reconstruct the entire skeleton of the unknown animal. It is in this respect that a scientific description is creative and differs from the ordinary type of description which we encounter in everyday life.
Now we have already attempted to explain that among all the synthesesof experimental facts known to science some stood out prominently, owing to their extreme simplicity. In these privileged syntheses simple relationships were discovered, and the expression of these permanent relationships constituted the laws of nature. If the experimental facts we are seeking to co-ordinate are few in number, and if the relationships between phenomena, as established by experiment, are more or less crude and uncertain, we may be led to a certain definite synthesis which will appear to be the simplest. But if ultra-refined experiment leads us to more precise relationships at variance with our former crude ones, or, again, if a wider array of facts has become known, it may well happen that our original synthesis will be incapable of co-ordinating all these results unless we appeal to a number of artificial corrective hypotheses. In this case we may be compelled to widen our synthesis or even to reconstruct it entirely. We must realise, therefore, that a synthesis can never be considered final, for we never know what the future may hold in store.
It is the same with our hypotheses and theories. Theories which would appear plausible when a small number of facts are appealed to, often become untenable when a greater number are taken into account. Consider, for instance, Lesage’s theory of gravitation. Lesage had suggested that gravitation might be accounted for by assuming corpuscles to be rushing hither and thither with enormous speeds in all directions through space. Under this hypothesis two bodies, say the sun and earth, would screen each other mutually from the impact of some of the corpuscles. The residual action caused by the impact of the other corpuscles would then be to press the sun and earth together; in this way gravitation would be accounted for mechanically. Needless to say, Lesage’s hypothesis was nothing but a wild guess, but, even so, it was felt that there might still be some truth in it.
The astronomer Darwin attacked the problem and proved that gravitation might be accounted for in this way, not merely qualitatively (for this would have been quite insufficient), but quantitatively as well, provided the corpuscles possessed no elasticity whatever, hence did not rebound after impact. Thus far, then, everything was in order, and the theory was acceptable. But now let us take into consideration a few additional facts. Gravitation, as we know, exerts its action even when bodies are interposed. Hence we must assume that only an insignificant percentage of the corpuscles would be arrested by the earth’s surface. Now, when account is taken of the sizes of the molecules of matter constituting the earth, and of the vacant spaces which separate them, certain limitations are imposed on the size, mass and velocity of the corpuscles. When, furthermore, we realise that the earth moves round the sun without slowing down in any perceptible way, as though its motion encountered no friction, we are compelled to conclude that the density of the gas formed by the corpuscles cannot surpass a certain limit. When all these facts are taken into account, we find that Lesage’s theory cannot be countenanced, for calculation proves that under the impact of the corpuscles, the earth would become white-hot in a very short time. The same difficulty would arise were we to seekto save the hypothesis by substituting for the impacts of material corpuscles the pressure of invisible electromagnetic radiations (possibly of the nature of the Millikan rays). Rather than introduce further hypothesesad hoc, for the sole purpose of saving Lesage’s theory, Lorentz preferred to abandon it.
We see, then, that it is owing to the incessant accumulation of new facts that scientific theories are constantly submitted to change and revision. Until the advent of Einstein’s theory, however, although theories of mathematical physics had to be subjected to incessant modification, our basic concepts of space, time and matter had remained unaffected. Indeed, there was no reason to deviate from the original synthesis in this respect. Only when ultra-precise experiments were performed was it found impossible to retain the classical spatio-temporal foundation. As a result, Einstein was compelled to construct a totally new synthesis built on an entirely different foundation, that of space-time. Had it not been for our knowledge of ultra-precise experiments, there would of course have been no reason to modify the essential characteristics of the classical synthesis. From this we see that the synthesis which we adopt depends on the extent of our knowledge, or, what is more to the point, on the extent of our ignorance of nature.
There is very little similarity between the erection of a building and the creation of a scientific theory. When a building is erected, we know beforehand the type of structure we propose to erect, and we can lay our foundations accordingly. But in an empirical science such as physics, the structure is never completed. The superstructures are subject to incessant change as our experiments become more precise and as new phenomena are revealed. The result is that we can never lay our foundations once and for all with perfect security. At all times we must be prepared to abandon our theory, remodel the foundations and erect a totally new structure.
It would appear to be unnecessary to stress these points further, for they must appear obvious to those who are at all familiar with the achievements of science. Certain metaphysicians, however, hold that similar reconstructions should never endanger the traditional concepts of space and time. But an attitude of this sort is equivalent to removing the study of space and time from the control of experience; for all concepts which we approach empirically are necessarily subject to change as our experimentation increases in precision. Furthermore, if we adopt this attitude, why not go the limit and assert that the discoveries of the laboratory can be figured outa priori? The entire trend of modern science is hostile to the philosophy of thea priori. Non-Euclidean geometry undermined its foundations a century ago, and to-day the physical theory of relativity has completed the work. To assert in the name of logic or of reason, as so many philosophers have done, that space and time must be separate,[141]or that space cannot manifest variations of curvature from place toplace,[142]or that space-time must always be flat,[143]or that matter must be a substance which is conserved, is to impose narrow philosophical doctrines which are likely to cramp the entire synthesis, rendering a simple scheme of natural relations impossible. To quote Weyl:
“Matter was imagined to be a substance involved in every change, and it was thought that every piece of matter could be measured as a quantity, and that its characteristic expression as a ‘substance’ was the Law of Conservation of Matter which asserts that matter remains constant in amount throughout every change. This, which has hitherto represented our knowledge of space and matter, and which was in many quarters claimed by philosophers asa prioriknowledge, absolutely general and necessary, stands to-day a tottering structure. First, the physicists, in the persons of Faraday and Maxwell, proposed the ‘electromagneticfield’ in contradistinction to matter, as a reality of a different category. Then, during the last century, the mathematician, following a different line of thought, secretly undermined belief in the evidence of Euclidean geometry. And now, in our time, there has been unloosed a cataclysm which has swept away space, time and matter, hitherto regarded as the firmest pillars of natural science, but only to make place for a view of things of wider scope, and entailing a deeper vision.”
Summarising, we may say that in natural science it is essential to beware of fundamental premises which are claimed to be forced upon us by the requirements of logic or of crude perception.
Yet, on the other hand, it must be admitted that when we have succeeded in constructing some lofty scientific synthesis which takes in the facts of experience, our synthesis must always be such as to permit us to account for the facts of crude observation; for these, to the same extent as the results of ultra-precise experiment, constitute facts which must be co-ordinated. A theory such as Einstein’s must not go counter to the apparent separateness of space and time when phenomena are investigated only in the usual crude way by commonplace observation. In other words, the scientific synthesis must reduce to the primitive synthesis when too great accuracy is not imposed, so that the primitive synthesis must appear in the light of a first approximation. Einstein’s theory satisfies this general condition in every respect. As we have seen, for low velocities or for crude observation the classical separateness of space and time reasserts itself; hence no conflict exists between the relativity theory and commonplace observation. Similar conclusions apply to the classical addition of velocities. This also appears to be verified to a first approximation.
Now it might be contended that since all finality seems to be denied a scientific theory, it would be better to cease theorising and content ourselves with a bare accumulation of facts. But an attitude of this sort would be untenable, for scientific theories, though necessarilyinaccurate from a long-range point of view, are nevertheless inevitable if knowledge is to exist. A bare accumulation of disconnected facts would not constitute a science, precisely because, the facts being disconnected, they would afford us no means of foretelling the future sequence of events. Under the circumstances, it is better to be guided by a faulty synthesis which we can improve upon as experience demands, rather than be guided by nothing at all. Furthermore, co-ordinations and theories which have since turned out to be patently inaccurate, have yet been successful in leading to the discovery of new facts.
Thus, Newton’s law, to-day recognised as inaccurate, brought about the discovery of Neptune; it permitted astronomers to assert that it was the same comet which reappeared at definite intervals. Again, it was thanks to Newton’s law that Römer, in 1675, discovered the finite velocity of light, thereby exploding the belief that the existence of an event and our perception of it were simultaneous occurrences. All these discoveries would have been arrived at by other means at some later date; yet it must be agreed that, owing to the law of gravitation, they were considerably precipitated. Indeed, it was not until the nineteenth century that the finite velocity of light was established by direct terrestrial measurement.
In other cases, a scientific theory, whether true or false, directs physicists towards crucial experiments of which they might never have thought. Eddington’s astronomical expedition was prompted by Einstein’s discoveries; had the theory been unknown, it is probable that the bending of light, when finally observed, would have been regarded as an error of observation or as due to some casual disturbing influence. Under the circumstances it might never have become a permanent scientific fact; and it is the same with the Einstein shift-effect. Regardless, then, of whether Einstein’s theory will endure or be scrapped to-morrow, it has led at least, to the discovery of new facts.
Furthermore, we must remember that each successive theory retains certain elements of the one it has supplanted. It rises to a higher plane, but it rises from a rung that has been set in position by its predecessor. It is not in the physical hypotheses that we shall find these elements of permanence, but in the equations themselves; for it is these that constitute the relations of reality. The physical hypotheses, as originally formulated, are but convenient scaffoldings of a more or less temporary nature. This is especially noticeable in the successive theories of electricity and magnetism. From the initial theories of Coulomb and Ampère to the present ones of Larmor and Lorentz, as modified by Einstein’s discoveries, we pass through the intermediary ones of Maxwell, Helmholtz and Hertz. We may follow the equations through these transformations and see how they were supplemented by the adjunction of additional terms or by the refinement of those already present. Thus, Maxwell’s equations differ from Ampère’s by the adjunction of a new term. Lorentz accepts Maxwell’s equation of the free ether without modification; it is only whenmatter is present that refinements are superadded. Again, Einstein’s gravitational equations degenerate into Newton’s when the field is weak and the velocities are small.
When we realise that all these successive refinements in the theories were brought about by a desire to co-ordinate an ever-increasing number of facts, it would appear that the scientist has very little control over the course his theory will follow, since in every case it is the criterion of simplicity of co-ordination which guides him.
However, it may be of interest to examine whether, in addition to these formal considerations, there may not exist other reasons which prompt scientists towards their solutions. It might be claimed that the various thinkers start from different initial philosophical presuppositions, and that these presuppositions, rather than the number of facts co-ordinated, are responsible for the differences in the nature of the theories and conclusions. Opinions of this sort have been expressed by numerous philosophers who have devoted their attention to what they call the “metaphysics of science”; but we may hasten to say that the opinion appears to be entirely unwarranted. It is solely the criterion of simplicity of co-ordination which decides on the orientation of science. Where the personal idiosyncrasies of the thinker may come into play will be in the directing of his initial attempts of co-ordination along certain lines rather than along others; but if a simple co-ordination is found to be impossible along the lines originally chosen, other attempts in totally different directions will be undertaken in due course. The final result may issue in a co-ordination presenting a philosophical import totally different from the one preferred originally; and this change is due to the facts of the case and not to a change in the philosophic fancies. Of course, it is not easy for the layman to realise this state of affairs, for scientists do not usually write a diary, mentioning all their abandoned attempts. When a theory becomes widely discussed, it is generally presented in its final form, and the numerous hesitations and partial attempts that may have extended over a period of years are lost sight of entirely. And yet, if we revert to the original scientific papers, these hesitations and changes of attitude are apparent.
It would be tedious to give illustrations; they abound everywhere. However, we may quote a few paragraphs of Einstein’s from his paper, “Cosmological Considerations on the General Theory of Relativity.” We shall see that he hesitated a long time between a number of alternatives before even considering the finite universe. We read:
“In the present paragraph I shall conduct the reader over the road that I have myself travelled, rather a rough and winding road, because otherwise I cannot hope he will take much interest in the result at the end of the journey. At this stage, with the kind assistance of the mathematician J. Grommer, I investigated centrally symmetrical gravitational fields, degenerating at infinity in the way mentioned. But here it proved that for the system of the fixed stars no boundary conditions of the kind can come into question at all, as was also rightly emphasised by the astronomer de Sitter recently.... At anyrate, our calculations have convinced me that such conditions of degeneration for the’s in spatial infinity may not be postulated.
“After the failure of this attempt, two possibilities next present themselves.[144]... From what has been said it will be seen that I have not succeeded in formulating boundary conditions for spatial infinity. Nevertheless, there is still a possible way out, without resigning as suggested under (b). For if it were possible to regard the universe as a continuum which isfinite (closed) with respect to its spatial dimensions, we should have no need at all for any such boundary conditions.”
And it is the same in every other case. A number of attempts are made to co-ordinate the facts, and the simplest solution is the one which is accepted by the scientific world, until such time as some newly discovered experimental result conflicts with it.
Of course, all these statements would suffer from an incurable vagueness, were any argument to develop as to what constitutes simplicity. But whatever the reason may be, we find that the agreement among scientists is striking in this respect. There have never been two highly developed theories co-ordinating the same number of facts, in which by common consent one was not recognised as vastly simpler than the other; just as a mathematical expression containing one term is simpler than one containing two.
A case in point would be afforded by a comparison of Einstein’s and Newton’s co-ordination of gravitation and mechanics. If we are concerned solely with the facts of planetary motion and mechanics known to Newton, then Newton’s laws of mechanics and his law of gravitation are by far the simpler. But if we supplement the facts known to him with those more recently discovered, Einstein’s synthesis has the advantage, for Newton’s co-ordination would necessitate a number of additional hypothesesad hoc. In the same way, so far as our everyday experience is concerned, a non-rotating earth is the simpler solution, whereas, when the planetary motions are taken into consideration, the reverse holds true. It may happen, of course, that the simpler synthesis will conflict with certain philosophical prejudices, but objections of this sort will always give way eventually before the higher criterion of simplicity. Indeed, we know that men finally accepted Galileo’s stand.
If, then, we may assume that the criterion of simplicity presents no further difficulty, the problem reduces to determining whether or not the theoretical physicist introduces presuppositions and assumptions uncritically. We are here concerned solely with the physicist and theoretical physicist, not with the pure mathematician, whose procedure is entirely different. In order to clear up these points, we shall give a rapid survey of some of the basic presuppositions that are common to physicists. Although we fear that it will be impossible to avoid a certain amount of repetition, yet the importance of the subject is such that a mere matter of elegance of presentation will have to be sacrificed.
The purpose of science is, as we have said, to co-ordinate facts and to obtain thereby a common knowledge which will hold for all men. But our co-ordinations, deductions and inductions must be conducted according to rigid rules which all men will recognise. In the absence of such common rules, two men starting from the same premises would arrive at different conclusions; as a result, a common knowledge, and therefore a science, would be impossible. We shall assume that the rules of logical reasoning will serve our purpose in this respect.
Here, of course, it might be argued that we are unwittingly introducing an assumption when we state that beings who neglected to conform to the same rules of reasoning would fail to agree. Of course, if we adopt so hypercritical an attitude, there is nothing left to argue about. The instant we start to talk we are assuming from past experience that sounds will be emitted and that our words will convey at least some meaning to our neighbour. It is the same with the rules of logic. Regardless of what the ontological status of these rules may be, we may assume that in view of past experience it is convenient for men to abide by them if they wish to construct a common science.
Let us consider now what scientists mean by theuniformity of nature. The uniformity of nature may be interpreted in various ways. It may mean that nature is regulated by universal laws. In this form, it is obviously not a presupposition adopted uncritically. The primitive man does not feel the necessity for it; he prefers to believe in miracles. If the modern physicist accepts this uniformity it is because it has been confirmeda posterioriin so large a number of cases that it seems simpler to accept it than to reject it, at least until further notice. Furthermore, it is an assumption which is perfectly well recognised as of limited validity, and scientists are the first to suspect that after a more microscopic survey of nature it may prove untenable. Nature may not be rational and our logic may be unable to cope with it. If this be the case, statistical knowledge may be the only type possible, though even statistical knowledge may fail us.
We can also understand the uniformity of nature in a different sense. We may wish to imply that an experiment attempted on a Monday would yield the same result if repeated on a Tuesday. In other words, the passage of time, though affecting us in a number of ways (aging, etc.), can be disregarded in certain important cases. Suppose, for the sake of argument, that we were to dispute the legitimacy of this assumption; what would be the result? Obviously science would become impossible. A geometrical proof holding in Euclid’s time would no longer hold to-day; knowledge would be intransmissible. There would be no object in performing an experiment to-day, since to-morrow our conclusions would have lost all value. In fact, it would be futile to attempt any experiment at all, since every experiment is spread over a certain time. In a world of this sort, prevision would be quite impossible; a rule of action would be out of the question. Inasmuch as one of the most important functions of science is to serve us with a rule of action, we can realise that science would become unthinkable.This brings out an important difference between science and history, for example. History tells us of events past and gone which, for all we care, may never reoccur; but in science a phenomenon which could never occur again would be of little interest. In short, we must agree that the very existence of a common science proves that our assumption of the uniformity of nature, whether it be justified or notsub specie œternitatis, is justified at least in that limited field of experience which is ours. On no account, then, must our belief be confused with a presupposition adopted uncritically.
As a matter of fact, even were we assured that the uniformity of nature was a mistaken belief, we should continue to be governed by it just the same. We know perfectly well that the uniformity of our stream of terrestrial life does not hold; we know that though we have risen every day, yet at any instant we may die. But unless we wish to exist in a state of complete stagnation, we must act as though our life would continue, and so we plan for the future. Under the circumstances, it appears futile to seek any rational justification for the uniformity of nature, basing our arguments on probability considerations. The fact that things have existed until to-day does not prove they will exist to-morrow; but it suggests that we act as though they would.
Among the principles, special mention must be made of the “principle of sufficient reason.” In physical science, the application of this principle is subject to all manner of difficulties. For instance, when we assert that there is no reason for a body to movehererather thanthere, we are assuming that we possess at least a general knowledge of all the possible influences which happen to be present.
Obviously, our assumption is apt to lead us to erroneous conclusions. We mentioned a case in point when discussing the problem of space. Assuming space to be perfectly empty, the principle of sufficient reason suggested that it should be homogeneous and isotropic; but as we were by no means certain that physical space was truly empty, the application of the principle was of no great value.
Difficulties of this sort are by no means restricted to problems of physics. Even in pure mathematics it is not always easy to decide in what way the principle should be applied. The mathematical problems in which we are compelled to appeal to sufficient reason are those dealing with probabilities; and the following elementary illustration, discussed by Bertrand, will afford us a sample of the type of difficulties which we may encounter. Suppose, for instance, we are asked: “What is the probability that on tracing the straight line at random, the line-segment contained within a given circle should be greater than the side of the equilateral triangle inscribed in the said circle?”
Now, if we conceive of all possible straight lines intersecting the circle, sufficient reason tells us that any one of these lines is as likely to be traced as any other. Hence, the probability we are seeking will be given by the ratio of the number of different lines that satisfy our stipulated condition, to the total number of different lines that can be drawn to intersect the circle. Clearly, this lastnumber will be given by considering all the lines passing through a given point of the circumference and then conceiving this point as occupying in succession every position on the said circumference. Under these conditions the probability turns out to be ⅓. But we might, with equal justification, have proceeded in another way. We might have considered all the straight lines parallel to a given direction and then conceived of this direction as suffering in succession every possible orientation. In this case, the probability would work out as ½. This example warns us that a large measure of arbitrariness lies concealed even in the most elementary problems where sufficient reason is appealed to.
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 theprinciple of continuityand theprinciple of causalityas 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 tofield 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 notbelieve 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 usa priorias 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 immediatelywhen 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, themagneton, 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 wouldhave 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 introducedad 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 motionswe 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 theradius 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 theearth’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.