CHAPTER XXXVIII THE METHODOLOGY OF SCIENCE

CHAPTER XXXVIII THE METHODOLOGY OF SCIENCE

AN opinion commonly expressed by philosophers is that the function of physicists should be to weigh, measure, tabulate, discover new chemical elements, new facts; and the aim of mathematicians, to solve equations. That so long as the scientist restricts himself to investigations of this sort, he is proceeding along strictly scientific lines and his efforts should be encouraged, but that when he proceeds to discuss the implications of his discoveries and to draw conclusions on the subject of space, time, energy, laws of nature, he is getting beyond his depth, and encroaching on the field of the metaphysician.

Bergson expresses this attitude very clearly in his book, “Duration and Simultaneity.” He informs us that the reason mathematicians have been so deceived by the significance of the theory of relativity, to the point of wishing to substitute space-time for separate space and time, lies in their lack of philosophical insight. They err by wishing to attribute a metaphysical significance to a concept which (according to Bergson) represents a mere mathematical fiction.

To these and kindred accusations, scientists (when they retort at all) will answer, with Kelvin: “Mathematics is the only true metaphysics.” Clifford, in his popular essays, voices the scientific attitude with increased emphasis when he writes: “The name philosopher, which meant originally ‘lover of wisdom,’ has come in some strange way to mean a man who thinks it his business to explain everything in a certain number of large books. It will be found, I think, that in proportion to his colossal ignorance is the perfection and symmetry of the system which he sets up; because it is so much easier to put an empty room tidy than a full one.” These opinions prove that there exists a definite misunderstanding between scientists and philosophers; a misunderstanding which might easily have been avoided had philosophers possessed a proper realisation of their inevitable limitations when discussing scientific matters. The simplest way to approach the source of the trouble appears to be to analyse the methods of scientists and ascertain in what respect they differ from those of philosophers.

The development of all the branches of physical science has proceeded roughly along the same lines. First we witness an accumulation of experimental and observational facts, furnished by crude observation and by the discoveries of the laboratory workers. Then, thanks to the efforts of the theoretical investigators, this raw material is co-ordinated into a consistent whole. In this way, out of a disconnected series of facts, a coherent doctrine or science is born. [Pg 374] History proves that with very few exceptions (illustrated by such men as Newton, Archimedes and also Hertz), the most brilliant experimenters had but little theorising ability; and that, vice versa, the ablest theoretical scientists made poor laboratory men. We are thus led to differentiate between two distinct types of scientists: the practical and the theoretical workers. For example, in physics, Maxwell, Planck and Einstein must be placed under the heading of theoretical investigators, whereas Hertz and Michelson are splendid illustrations of able experimenters.

In modern science, at least in physical science, these works of co-ordination can be attempted only through the medium of advanced mathematical analysis; hence the theoretical physicists must necessarily possess a profound knowledge of mathematics. Yet they are not, properly speaking, great mathematicians. Mathematics, for the theoretical physicists, constitutes but a tool, a means of arriving at a co-ordination of the physical facts. These men have never entered into the study of mathematics as an art in itself; they have never forged new mathematical instruments, never made mathematical discoveries. Thus, they cannot be classed with mathematicians of the calibre of Lagrange, Gauss, Riemann or Poincaré, to whom modern mathematics owes its existence. Just as we were compelled to make a distinction between experimenters and theoretical physicists, so once again we must make a distinction between theoretical physicists and pure mathematicians. Again, with very few exceptions, as exemplified by Fourier, Poisson, Poincaré, Minkowski and Weyl, pure mathematicians have rarely contributed directly to the advancement of theoretical physics, although indirectly, of course, their discoveries have been made use of by the physicists. Since they are neither great mathematicians nor able experimenters, what are we to call such men as Maxwell, Lorentz and Einstein?

If we concede that the name philosopher should apply to those who are concerned more especially with a harmonisation of the whole than with the seeking of individual facts, or, again, with a general view of things rather than with a restricted view, we must agree that the theoretical physicists must be called philosophers. They are, then, the philosophers of the inorganic world, just as the pure mathematicians might be called the philosophers of abstract relations.

Now, as we have said, the facts which these scientific philosophers are seeking to co-ordinate are of a restricted species; they are mathematical, physical and chemical in nature; hence it is clear that there is room for a more general type of philosopher—a super-philosopher, as it were—whose facts would comprise all the spheres of human knowledge, including consciousness, emotions and the relationships between mind and matter. The traditional philosophers—or shall we say the lay philosophers, since we are discussing scientific matters?—aspire to be placed in this category of thinkers.

It would appear, then, that theoretical scientists, and lay philosophers have much in common; they differ only in the scope of the facts they are seeking to co-ordinate. But here is where the first [Pg 375] breach arises. The theoretical scientist proceeds with the utmost caution and considers himself at liberty to theorise only after a sufficient number of facts have been established by experiment and observation; till then he remains silent. This cautious attitude is evidenced even in the most revolutionary theories, such as those of relativity and of the quanta. It was not one, nor two, nor even three of the negative experiments in electromagnetics that drove Einstein towards his revolutionary theory; it was the whole body of electrodynamics. Even so, he formulated his solution only after a number of more classical attempts to solve the same difficulties had failed.

But when we examine the procedure of the lay philosopher who discusses scientific matters, we see that his procedure is entirely different. Taking examples at random, we find a contemporary philosopher telling us that the spatio-temporal system has to be regarded as limited, since “all existent things, as distinguished from pure forms or orders, are finite.”[131] But the average scientist would retort that with the same degree of plausibility we might say that the spatio-temporal system must be infinite, since the concept of infinity exists. At all events, it is instructive to contrast these loose arguments with Einstein’s painstaking labours which led him to the finite universe.

Another example is afforded by Bergson’s comments on the invariant velocity of light. He explains that inasmuch as the corpuscular theory of light has been rejected, light is a propagation and not a transference of matter. It is, then, only natural (according to Bergson) that its speed should be invariant. In his own words: “Why should it be affected by a certain too human way of perceiving and conceiving of things?” Bergson apparently forgets that it is only light in vacuo that moves with an invariant speed, and, even so, only when far from matter. A ray of light passing through water is also a propagation, and yet its speed is not invariant, as Fizeau’s experiment proved seventy years ago. Also, we might with equal plausibility apply Bergson’s arguments to sound waves. They also constitute propagations, and yet, once again, their speed is not invariant. In short, it would appear that the philosopher, even when he interprets the facts correctly, appears to be ignorant of so large a number of other facts that his philosophical conclusions rest on no solid foundation.

So much for the facts; but the facts are not all, since the theoretical scientist’s chief aim is to co-ordinate these facts. But, here again, the philosopher’s criticisms would suggest that, being unable to follow the series of mathematical deductions which lead the theoretical scientist to his conclusions, he fails to understand the necessity for these conclusions, and accordingly considers them to be wild guesses over which he will invariably suggest his own as an improvement.

Possibly a few definite illustrations will bring out with greater clarity some of the points we are endeavouring to stress. Let us revert, for example, to space-time, which proves itself so offensive to Bergson. His position, as already stated, is that space-time is a pure mathematical fiction in no wise demanded by the theory [Pg 376] of relativity, which he accepts as sound physically, provided its interpretation is abandoned to lay philosophers. His arguments proceed somewhat as follows: You physicists have performed certain electromagnetic experiments; you mathematicians have deduced therefrom the Lorentz-Einstein transformations on purely formal grounds. Then you have plastered these transformations together and discovered a certain mathematical invariant. So far, well and good; now you must stop. It is for us philosophers to interpret these results. We will begin by ruling out your faulty inferences as to the existence of a four-dimensional continuum, space-time, of which space and time are mere abstractions. Follow our arguments carefully and we shall convince you. Possibly you mathematicians have not noticed that every observer is perfectly aware of a separate space and time wherever he goes, just as he is aware of a separate space and temperature. Were space-time one sole continuum, as you maintain, we could never be aware of space and of time, but only of space-time. This is contrary to the facts. Hence, space-time is a hoax, and your mathematics has led you astray.—Q. E. D.

But it is obvious that arguments of this sort can never have any merit, for the reason that they are based on a total misconception of the significance of dimensionality and manifolds. These are highly technical concepts which it requires more than a crude elementary knowledge of mathematics to grasp. However, even without going into details, we can easily convince ourselves of the fallacy of Bergson’s argument by applying it to the three-dimensional space of classical science. Thus, we know that wherever we are situated in three-dimensional space, we can always split it up into height, length and breadth; yet this does not preclude the fact that length, breadth and height are but abstractions from one single entity, namely, three-dimensional space. And, in the same way, as has been demonstrated by Minkowski, length, breadth, height and duration are but abstractions from four-dimensional space-time.

Another philosopher, Professor Broad, in his book, “Scientific Thought,” informs us: “(a) No matter what frame we choose, we shall need four independent pieces of information to place and date any instantaneous point-event. This fact is expressed by saying that Nature is a four-dimensional manifold; and nothing further is expressed thereby. (b) In whatever frame we choose we shall find that our four pieces of information divide into two groups; three of them are spatial and one is temporal. Thus we must be careful not to talk, or listen to, nonsense about ‘Time being a fourth dimension of Space.’”

Now it is agreed that whoever speaks of time as a fourth dimension of space is expressing himself very loosely. What we should say is: “Time is a fourth dimension of the space-time world.” But Broad’s argument does not suggest that it is this looseness of phraseology that offends him. His words would imply that it is the reference to time as a fourth dimension that must be branded as “nonsense.” His error in this respect is thus exactly the same as Bergson’s. As for his contention that the four-dimensional structure which [Pg 377] relativity ascribes to nature means no more than that events need four co-ordinates to be placed and dated, the statement is scarcely correct. Were this trivial piece of information all that Minkowski was conveying when he referred to the world as four-dimensional, his discovery would have excited neither admiration nor criticism. It would have passed unnoticed by scientists, as expressing a mere platitude that could have been no news even to a child. The point that philosophers so persistently fail to understand is the difference between an amorphous continuum, such as a manifold of sounds or colours, and a metrical continuum, i.e., one with which a definite geometry is associated. What Minkowski did was to prove that, contrary to the belief of classical science, the world was a four-dimensional “metrical” continuum, i.e., one with which a four-dimensional space and time geometry was associated. It was this novel aspect of the world that implied the revolutionary conception of the fusion of space and time. And it was this aspect that entitled Minkowski to speak of time as a fourth dimension in a profound sense as well as in the trivial sense which, though never disputed by classical science, was never stressed on account of its artificial nature. Indeed, had it not been for this interpretation placed upon his words by Minkowski himself and by all scientists, the four-dimensional world would have appeared to be as artificial as a four-dimensional space-temperature continuum. It is granted that the various meanings attributed to the word “dimension” by mathematicians may cause some trouble to beginners, but, on the other hand, unless the four-dimensional aspect of nature in the relativistic sense is understood, it is quite useless to philosophise on the more advanced aspects of the theory. Before we try to run, we should at least learn to stand on our feet.

To take another illustration at random, it is the same when Professor Broad discusses the law of gravitation. Notwithstanding his condemnation of the conception of space-time, as one sole four-dimensional continuum, he sees no inconsistency in discussing the geometry of this four-dimensional continuum. At any rate, after telling the reader that a space may be flat or curved like an egg or like a sphere, he decides (p. 224) that the law denotes a spherical curvature. Then, on a later page (p. 485), we are informed that the curvature is homaloidal. This taxes the imagination of the reader, for as “homaloidal” means flat, a space cannot be both homaloidal and spherical at one and the same time. However, it is to be presumed that Broad meant “homogeneous,” not “homaloidal,” and we shall interchange the two words accordingly. But even when amended in this way, his premises are totally incorrect, since the law , which is to account for gravitation, is neither spherical nor homogeneous nor homaloidal. This law represents a heterogeneous species of curvature; indeed, were the curvature spherical, it could never account for gravitation.

Starting from his mistaken premises, our philosopher soon falls into further confusion. After telling us (p. 224) that the supposedly [Pg 378] spherical law of curvature, , is able to account for gravitation, and for all the gravitational effects predicted by Einstein, he next informs us (p. 485) that this law cannot be the law of gravitation in the real world because the space round the sun is never completely empty, but is filled with tenuous cosmic matter, electromagnetic fields, and light waves. This casual presence of matter and electromagnetic fields would (always according to Broad) modify the law of spherical curvature, , causing it to become heterogeneous. But all these statements are utterly erroneous, since, as we have said, even in an ideally empty space-time, the law of curvature, , round the sun would still be heterogeneous, quite apart from the minute superadded effects that might be occasioned by a casual matter distribution. In other words, the heterogeneity is essential, not accidental.

We now come to the Professor’s philosophical conclusions. He notes that this passage from a space-time which is spherical and homogeneous in a gravitational field to one which is heterogeneous is brought about by the casual presence of matter round the sun. Einstein never stresses this point which Broad has brought to light, whence Broad concludes that this important transition from homogeneity to heterogeneity has been “slurred over.” As a philosopher he deprecates this loose treatment, pointing out that a totally new conception of space is involved, one that should be submitted to careful philosophical scrutiny.

Of course, this charge is totally unjustified and issues solely from our philosopher’s faulty understanding of the facts. The heterogeneity was never introduced in the casual way Broad believes. It was introduced by the front door the moment Einstein wrote out his gravitational equations , as any one familiar with the theory of spaces would have recognised immediately. Far from having been slurred over, this heterogeneity of space-time constitutes the very essence of Einstein’s general theory. Whereas in classical science the sun developed a field of force around itself in a perfectly flat space, in the general theory of relativity the sun develops a heterogeneous space-time curvature. The field of force is then but a manifestation of this curvature. As for the new conception that this heterogeneity forces upon our understanding of the world, it has been subjected to critical enquiry by Weyl and Einstein. Furthermore, it is not so new a conception as Broad appears to believe, for it is one we owe to Riemann following his epochal discoveries in non-Euclidean geometry seventy years ago. As Weyl tells us:

“Riemann rejects the opinion that had prevailed up to his own time, namely, that the metrical structure of space is fixed and inherently independent of the physical phenomena for which it serves as a background, and that the real content takes possession of it as of residential flats. He asserts, on the contrary, that space in itself is nothing more than a three-dimensional manifold devoid of all form; it acquires a definite form only through the advent of the material content piling it and determining its metric relations.” [Pg 379] And again, elsewhere:

“It is upon this idea, which it was quite impossible for Riemann in his day to carry through, that Einstein in our own time, independently of Riemann, has raised the imposing edifice of his general theory of relativity.” (We may mention that the reason Einstein was able to carry through Riemann’s ideas is because he applied them to space-time instead of to space alone.)

Of course, it is granted that Riemann’s discoveries and non-Euclidean geometry are not of easy access; yet, on the other hand, the man who ignores at least the implications of non-Euclidean geometry, is in no position to discuss Einstein’s theory or the problem of space even from a purely philosophical point of view. Indeed, it may be said that the philosophical importance of non-Euclidean geometry is even greater than its scientific importance. Such has always been the contention of Lobatchewski, Riemann and other great mathematicians.

So far as the writer is aware, the only philosopher who made any reference to non-Euclidean geometry, prior to Einstein’s discoveries, was Lotze; and Lotze expressed the hope that philosophy would never allow itself to be imposed upon by it. But a perusal of Lotze’s writings on the subject proves that he had a very superficial understanding of what was meant by “curvature.” His opinions seem to have been based on Helmholtz’s more or less successful attempts to popularise the new doctrine by easy illustrations. Unfortunately, the subject is too deep to be explained in any loose way. At any rate, the effect of this negative attitude on the part of philosophers has reacted to their disadvantage in that it has deprived them of a very powerful insight into the problem of space.

Our sole purpose in mentioning these few examples (which we might have multiplied ad infinitum) has been to show how wary we should be of criticising the conclusions of scientists before proceeding to acquire more than a superficial schoolboy knowledge of the physical and mathematical facts which they are endeavouring to co-ordinate. If, as a result of misinterpretation or ignorance on our part, we are acquainted with only a small number of these facts, the conclusions of scientists may well appear strange and unwarranted; but it should be remembered that had not all these additional facts, which we ignore, been known to the scientist, it is quite certain that he never would have been driven to the conclusions that offend our natural views so deeply.

Unfortunately, in many cases, these facts are not of an elementary character; they cannot be explained in an hour or so. More often than not, an appreciation of what they represent would require years of preliminary study, for they are often in the nature of conclusions derived from other facts through the medium of laborious mathematical analysis. These statements, of course, must not be construed as applying solely to theories of mathematical physics. They apply with equal force to the discussion of numerous concepts such as those of infinity, continuity and atomism, dimensionality, number, measurement, rigidity, etc. Discoveries in higher mathematics and in physics have thrown a new light on all these subjects. Students of advanced [Pg 380] mathematics know only too well how crude was their understanding of continuity and infinity before they were apprised of the mathematical discoveries of Riemann, Weierstrass, Cantor, Dedekind and du Bois Reymond. Furthermore, no one can have studied advanced mathematics without realising its intimate connection with problems of psychology, for mathematics has brought to light mind-forms of which we were only dimly conscious.

The thorough remodelling of our ideas of the universe which the discoveries of Planck and Einstein appear to be rendering inevitable, makes an understanding of these fundamental points imperative. For example, let us consider the quantum theory. As we shall see, its necessity arises only when we take into consideration a number of empirical discoveries pertaining to the various realms of physical science. Here, for instance, is a heated enclosure. We make a pinhole aperture in its wall and examine the colour and intensity of the light rays streaming out. The experimenter notes that as the temperature increases, the colour of the light rays passes gradually from red to white. He then studies the density of energy of the radiation emitted, and finds it proportional to the fourth power of the absolute temperature of the enclosure, regardless of the material of the enclosure. This empirical discovery is known as Stefan’s law. Then, by splitting up the light through a prism, he finds that the radiation which exhibits the maximum intensity is of a frequency which is proportional to the absolute temperature. This discovery is comprised in Wien’s displacement law. These are the facts of the case; the physicist has accomplished his task and may now retire. What is the significance of these facts? Were science to limit itself to the bare discovery and cataloguing of facts, there would be nothing more to do. But regardless of what philosophers may say to the contrary, science should not and does not limit itself to any such humble rôle. The discoveries of the experimenters are handed over to the theoretical investigators, and it is these who are called upon to interpret their significance.

In the present case, the outstanding names connected with the problem of radiation are those of Boltzmann, Wien, Rayleigh, Jeans and Planck, all of them theoretical physicists. Their object was to co-ordinate the facts of radiation with what little was known of the phenomenon of light emission. This task involved mathematical calculations which we should be unjustified in criticising unless we possessed an extensive knowledge of mathematics; hence we may assume that this part of the work is not disputed by the critic.

As a result of these calculations, it was found to be quite impossible to reconcile the existence of the facts disclosed by experiment with any continuous emission theory of light. Whenever it was assumed that the atoms in the heated enclosure emitted their radiations continuously, it always followed that, for any given temperature, the maximum intensity of the radiations should be found in the infinitely high frequencies, and not in the visible spectrum; whence it followed that a heated enclosure could never emit visible light. This was, of course, contrary to common experience. The literature on the [Pg 381] subject extends over a number of years. Various explanations of the discrepancy between theory and experience were suggested, but for one reason or another none was found satisfactory.

Then Planck noticed that if, contrary to all previous opinion, we assumed that the atoms in the heated enclosure emitted their radiations by discrete quanta, it was possible to obtain a mathematical formula of radiation in perfect agreement with experiment. According to Planck’s calculations, it was necessary to assume that light was emitted in bundles, or quanta, possessing an energy . In his expression, represents a universal constant called Planck’s constant, and represents the frequency of the light. Obviously, the greater the frequency of the light, the greater the value of its quantum of energy. These discontinuities in the light-energy emission necessitated the introduction of probability and entropy considerations into the theoretical treatment. Inasmuch as entropy is an abstruse concept drawn from thermodynamics, we see that the facts entering indirectly into the problem of radiation were considerably increased in number. Obviously, it would be quite impossible either to justify or to criticise Planck’s hypothesis by any general line of talk, for without the aid of the mathematical instrument, and a knowledge of thermodynamics and mechanics, no conceivable connection could be seen to exist between a discontinuous emission and the facts disclosed by experiment; hence the necessity of the hypothesis could not be gauged.

Thus far, however, nothing very revolutionary appeared to be involved. All we had to do was to assume that the emission of light by a heated atom was due to the breaking up of some intra-atomic structure. Nevertheless, on deeper investigation, even assuming Planck’s emission theory to be correct, a number of theoretical difficulties were noted. These are of so technical a nature that we shall not dwell on them. Suffice it to say that they relate in a general way to the exchange of energy between the various atoms and to the conditions of equilibrium. The names of Poincaré, Lorentz and Einstein are encountered at this stage, and the final result was that somewhere, somehow, discontinuities had to be introduced.

The next great advance was due to Einstein. He noticed that the well-known anomalies in the specific heats of bodies at very low temperatures could be explained, provided we assumed the discontinuity of energy. As applied to solids, this would connote that the vibrating systems could take up or emit energy only in definite quanta; and, as applied to gases, we should have to assume that the molecules could rotate only with definite frequencies. All these problems dealing with specific heats bring us into contact with the kinetic theory of gases, as also with other realms of physics, such as the problems of anomalous dispersion and selective reflection in optics. Again basing his deductions on the existence of quantum phenomena, Einstein deduced a formula for the photo-electric effect, and this formula was found to be in harmony with experiment. Finally, we may mention that Bohr’s atom, which is also based on the same idea of discontinuity, allowed him to [Pg 382] account for certain spectral series.

We now get to the important point. From the expression for the atom of energy or quantum , where is a constant and is the frequency of the radiation, it is obvious that there exist as many different types of quanta of energy as there exist different frequencies of radiation. There is no one unique type of quantum of energy in nature. That which is universal is not the quantum of energy , but the constant . It can be shown that Planck’s constant is not a mere number; it represents some definite abstract mathematical entity, and that entity is action.

We must assume, therefore, that there exist atoms of action in nature, just as there exist atoms of matter. But what is the deeper significance of this atom of action? First of all, has it any deep significance, or is its discovery on a par with that of some new variety of flower, or some new mineral? But to decide on a question of this sort we must possess a fairly thorough understanding of what is meant by action, as also of the part this important entity plays in science. Now, action is a highly abstruse concept taken from analytical dynamics. It would be absurd, therefore, to criticise the conclusions of scientists, whatever these might be, unless we were in a position to discuss the numerous facts which were involved in their arguments.

The verdict of scientists is that the atomicity of action entails a gigantic revolution in our understanding of nature. In the first place, the laws of mechanics, whether those of classical science or those of relativity, issue in a principle of action; so that by tampering with action, we are tampering with the basic laws of mechanics. Hence, it would appear that our insight into the laws of nature, even in spite of the great advance due to relativity, was still far too crude. We should have observed merely macroscopic average effects; the deeper laws, the underlying microscopic ones, would have escaped us completely. Conclusions of this sort had, of course, already been arrived at in other ways, as, for example, in the kinetic theory of gases; but the atomicity of action extends these views still farther. It suggests that change is always discontinuous; that a system passes from one state to another, not in a continuous way, but by a series of jerks or jumps. When we wish to decide how the jumps will follow one another, no exact laws can be formulated; and we are compelled to appeal to statistical considerations and probabilities. No rigid deterministic scheme is apparent in nature, or, in Weyl’s words, “no causality of physical nature which is founded on rigorously exact laws.” In the realm of the microscopic, we appear to be confronted with total chaos and anarchy. The past does not entail the present, as it would in a purely deterministic scheme. Free will appears to be rampant; and our sole means of prevision is to establish averages, just as a life-insurance company does when it fixes its premiums. Statistics and probability, blind chance and uncertainty, take the place of rigid determinism. Of course it may be that a further advance in our knowledge will enable us to rediscover a deterministic scheme beneath the chaos which is now [Pg 383] confronting us. But the point we wish to stress is that in the present state of our knowledge no definite stand can be assumed.

Even this is not all. There is every reason to suspect that the atomicity of action is closely related to the atomicity of matter, though what the precise connection may be remains a total mystery. Furthermore, when we realise that “action” contains the product of space and time, or, if we prefer, “elements of space-time,” and when we realise that what mathematicians call “the domains of probability” are now of finite extent instead of reducing to points, the possibility of space-time turning out to be atomic suggests itself strongly. There is no telling where we may be led, for our entire understanding of nature is at stake. As Weyl expresses it: “Above all, the ominous clouds of those phenomena that we are with varying success seeking to explain by means of the quantum of action, are throwing their shadows over the sphere of physical knowledge, threatening no one knows what new revolution.” As we have mentioned, it may be that space and time will turn out to be atomic, or, again, that we shall have to recognise them as approximate concepts which will have to be abandoned when the infinitely small is contemplated. In much the same way, the conception of temperature as due to an agitation of molecules loses its meaning when the molecules themselves are considered.

What the future may hold in store is any one’s guess at the present time. But one thing is certain: we are faced with a gigantic revolution; and the new ideas will undoubtedly conflict with the common-sense instinct which the rationalist often erroneously attributes to “reason.” But what if they do? Did not the existence of men walking upside down at the antipodes conflict with the crude common sense of our ancestors? Or, again, consider the less trivial illustration of the wave theory of light. When Fresnel defended it, Poisson pointed out that it would imply the existence of a bright shadow behind a body of a certain size situated at a certain distance from a wall. On the strength of this argument Poisson dissented from Fresnel’s views. Yet when the experiment was actually performed, the bright shadow was seen to be there. And why did Poisson regard his argument as valid? Because it was based on common sense; but we see that this common-sense instinct that was misleading him was by no means the product of reason; it was founded merely on partial and crude experience.[132] It is probable that the more we study of nature, the more the common sense of our day will be submitted to disagreeable jolts; and this is only natural, since the more refined our investigations, the farther we shall be wandering from the familiar world of common experience.

To revert to the new ideas that suggest themselves as a result of the quantum theory, we see that starting from facts which by common consent it is the function of the physicist to establish, we are gradually led through a series of theoretical deductions to considerations which [Pg 384] border on metaphysics. But whereas the metaphysician will claim a priori knowledge or else, whether he admits it or not, will be deducing his knowledge from the crudest facts of daily experience, the theoretical scientist will base his deductions on extremely accurate observations embracing all the phenomena known to science.

For this reason, these deductions will constitute knowledge; for knowledge springs from an accurate investigation of a large number of facts, not excluding those revealed by ultra-refined experiment. More precisely, a knowledge of nature consists in the co-ordination of all known facts, and this co-ordination must be performed so as to account for phenomena not merely in some vague qualitative way, but with the utmost accuracy. Only after this preliminary synthesis has been accomplished can general philosophical conclusions be forthcoming. Until then all we can do is guess, and past experience proves that ninety-nine times out of a hundred our guesses will be wrong.

In short, it is in the accuracy with which the facts are studied and co-ordinated, and it is in the number of facts considered, that the scientific method differs from all other methods. Therein resides its superiority. Had the great scientists confined themselves to a general line of talk based on a crude survey of a number of conspicuous facts, we might still to-day be defending Plato’s idea that earth, air, fire and water constituted the four elements, or Aristotle’s contention that the heavenly bodies described circles because circular motion was the most noble of motions, or, again, Kant’s belief that the axioms of geometry were in the nature of a priori synthetic judgments, and so forth.

Thus it may be realised that a discussion of the philosophical significance of the discoveries of physical and mathematical science must be left to the theoretical physicists and to the mathematicians. They alone, in view of their wide knowledge of facts and their mastery of the rigorous mathematical mode of thinking, are in a position to co-ordinate the apparently disconnected results furnished by experience and by reason. If, then, a super-philosophy is to be attained, it would appear that the most successful results would ensue from a work of collaboration between the scientists of the various branches of knowledge. Such collaborations are continually in progress. We need only mention the contributions of physicists like Einstein and Nernst to problems of physical chemistry; and physical chemistry is closely allied to the chemistry of colloids, hence to that of the living organisms. All these spheres of science, from mathematics to psychology, dovetail into one another, as terms like bio-chemistry, psycho-physics and physical chemistry indicate.

It might be urged that the results embodied in the special sciences do not embody all of nature; that there is mind and consciousness, that there are emotions, values in quality, religious instincts, and so on. But here it should be remembered that whatever transcends the sphere of the special sciences transcends it precisely because it is vague and only dimly apprehended. And where facts are so vague and poorly established as to be refractory to the scientific method, we generally find that there are so many different ways of co-ordinating them [Pg 385] loosely that almost any opinion can be expressed with the same degree of plausibility. As a result, the opinion of the wise man is on a par with that of the ignoramus.

If we wish to advance at all, it would seem that the best method would be to start by considering the facts we know something positive about, rather than reverse the procedure. The results embodied in the special sciences are so numerous and (relatively speaking) so well established, that we cannot afford to consider seriously any system of philosophy which would enter into conflict with them, or ignore them.

At any rate, granted that, in the present state of our knowledge, no co-ordination of facts can be anything but extremely fragmentary, it may be presumed that the theoretical scientists who have shown such marked ability in their co-ordinations of natural facts are the men best fitted to construct a philosophy of nature. We need not be surprised, therefore, to find that their study of nature has yielded them a philosophy which, with very few exceptions, they all share in common.

In the present chapter, we shall discuss the methodology together with the philosophic views that are accepted by the vast majority of theoretical physicists. In particular, the philosophy expressed will be exemplified by the attitude of the two leading scientists of our generation: Poincaré as a mathematician, and Einstein as a physicist.

The problem of knowledge, though generally considered independently of science proper, is yet so involved with scientific considerations (some of which are by no means elementary) that it appears necessary to approach it by the usual methods of scientific investigation. By knowledge we do not wish to imply our bare awareness of sensations, such as the worm and oyster probably share in common with human beings. Sensations constitute one of the original sources of knowledge, but knowledge entails considerably more; judgment and inference enter into knowledge, and knowledge is a construct or co-ordination in which our sensations enter merely as fundamental elements. For instance, when as a result of our visual and tactual sensations we recognise that a table is there before us, we are claiming knowledge; but the visual and tactual impressions by themselves do not constitute knowledge.

The natural uncritical view would be to assume that knowledge was given directly by the disclosures of perception. Thus, when we see the table, we know it to exist “there-now” in the spatio-temporal background. But if we analyse the situation, we cannot subscribe to this view, for our recognition of the table’s existence and position in space and time appears to have been brought about in a far more complicated manner than a cursory examination might lead us to suppose. What we really experience is an aggregate of colours with a well-defined contour. Only when we wish to ascribe a cause to these sensations does the idea of a table enter our consciousness. But here again, in this search for a cause, it is only in view of past experience, in view of an association of ideas, that we feel justified in postulating the existence of the table situated there-now in the spatio-temporal background. Our judgment may be correct, but it might also be erroneous, as it would [Pg 386] be had we been gazing at a skilful painting or at the image of a table in the mirror. Quite independently of such possible errors of judgment, the main point is that there is a decided difference between our awareness of the visual sensations and our belief in the presence of a table located there-now in space. Whereas the visual sensations do not entail as an inevitable corollary the existence of the table, yet our belief in its existence does presuppose our awareness of certain sense impressions or our memory thereof, since otherwise there would be no reason for us to postulate the table’s existence. For this reason we must credit to our awareness of sensations primacy over the so-called immediate disclosures of perception, just as we must concede to the recognition of experimental facts priority over the scientific theories constructed with a view to bringing about their co-ordination. Accordingly we shall refer to our awareness of sensations as facts.

A further illustration may make this point clearer. For example, we feel hot, and exclaim: “What a warm day!” Our awareness of a sensation of hotness constitutes a fundamental fact, but our claim that the day is warm is but an inference. We may test the warmth of the atmosphere with a thermometer, hence we may test the validity of our inference, but we cannot test our awareness of warmth. The feeling of warmth may be due to the temperature of the air, to fever or to sheer imagination, but, regardless of its cause, we know that we feel warm, and our assurance of this fact can neither be disputed nor analysed.[133]

Then again, we experience very similar sensations when our skin comes into contact with a substance at high temperature or with liquid air. If from this similarity in our sensations we should infer the identity of the temperatures involved, we should be led into error; but it would be our inferences that were at fault, and not the sensations, which subsist regardless of whether or not we choose to draw inferences therefrom.

It is scarcely necessary to mention that those philosophers who deny the legitimacy of the views just expressed, claiming our knowledge of external reality to be a matter of direct apprehension, find themselves in a very embarrassing position when such commonplace phenomena as mirror-images are considered. It is all very well for them to say that the results of visual perception may have to be corrected subsequently by varying the conditions of observation and taking into consideration all other perceptions. But the fact is that if this visual perception is to be regarded as revealing reality directly, i.e., as fundamental, its disclosures should be beyond dispute and it would [Pg 387] be meaningless to correct them. For we can correct a datum only in virtue of other criteria, which are then automatically regarded as more fundamental. In the present case the philosopher’s argument would indicate that the result of a co-ordination of perceptions in general, is to be taken as more fundamental than bare visual perception or bare auditory perception, etc. But inasmuch as the disclosures of no one of the individual perceptions that enter into the co-ordination can be accepted without a possible danger of correction, we must go back farther, behind the individual perceptions, to the sensations whose recognition is untainted by any trace of inference. We then arrive at our original conclusion, namely, that knowledge springs from a co-ordination not of perceptions, but of sensations.

Let us consider another example. The neo-realist would probably state that when we say, “I hear a bell,” no element of inference need enter into this knowledge. But a contention of this sort would of course be untenable, for the only reason the auditory sensation yields us any knowledge of the bell is because, in view of an association of ideas issuing from past experience, we have come to connect a certain class of auditory sensations with the presence of a bell. A man who had seen bells, but had never heard them clang would not attribute the sound to the bell any more than a man who had never encountered rattlesnakes or been told of their existence would ever say: “I hear a rattlesnake.” On the other hand, a man who had previously come into contact with these reptiles would be just as apt to say, “I hear a rattlesnake,” as the philosopher to say: “I hear a bell.” Obviously, past experience, and not direct recognition, is at the root of this knowledge. In the absence of past experience, all we could say would be, “I sense a sound,” and then try to describe the sound by reproducing it, as nearly as possible, with our tongue and lips.

With visual perceptions our conclusions will be exactly the same, but at first sight it might appear that the problem was slightly different, for even had we never seen a snake before, we should still be able to say, “I see a creature with a long, thin body,” and then proceed to coin some name for it. But it should be remembered that although we might never have seen a snake before, yet a snake is a particular instance of an object, and unless we had always been blind we should have seen objects ever since infancy. If, then, we wish to obtain a parallel to the case of the rattlesnake (as we applied it to auditory perceptions), we must start with a blind man who has felt objects but never seen any. Then, assuming he were suddenly endowed with eyesight, the question is whether he would identify immediately, by a mere act of intuition, the colours and shapes he would now see, with the table he had previously explored with his hands. This is precisely what we have every reason to doubt. We may now proceed to a more detailed discussion of the knowledge-problem.

Here it should be clearly understood that we are merely seeking to establish a hierarchy of knowledge following the psychological order according to which human knowledge appears to have arisen. We [Pg 388] may therefore consider the faculty of unreasoned sense-awareness as furnishing the initial data which it is necessary to accept before we are in a position to speculate any further. Together with our awareness of sensations we have to consider our awareness of thoughts or ideas. Whether, in the absence of sensations, thoughts would still have originated, is a question which does not concern us. At all events, an answer to it seems impossible, for we cannot study the thoughts of a living corpse.

Next we have to consider our awareness of the passage of time. This awareness appears to be closely allied with the faculty of memory. Indeed, it is only thanks to the memory of sensations or ideas that we can differentiate the present from the past. It is probable, therefore, that to a being devoid of all trace of memory, to a being sensing but the present, past and future would convey no meaning and the passage of time would be unthinkable. Again, whether, in the absence of our awareness of sensations or ideas, memory would have anything left to remember, is a question which need not detain us. The reason for this omission is, of course, that in the present survey we are considering normal human beings; and with these an awareness of sensations, ideas, and of the flowing of time is known to exist.

Our recognition of the simultaneity of two sensations is fundamental, in that it cannot be analysed further. The same fundamental nature must be credited to our recognition of a succession of sensations or ideas. For this reason a sense of simultaneity and succession (when referring to our awareness of sensations or ideas, and not, of course, to a simultaneity of spatially separated events) must serve as a basis for subsequent knowledge.

We have also to mention our judgments of constancy, invariancy or sameness. Thus, we recognise two sensations or two groups of sensations as identical, or, again, we judge them as subsisting unmodified through time. Although all these seemingly a priori basic sources of information are of a widely different nature, we may consider them as a whole in our future discussions.

Now, had men confined themselves to registering sense impressions and to sensing the flow of time, there could never have arisen such a thing as scientific knowledge or even crude commonplace knowledge. Somehow or other the initial facts had to be co-ordinated and interconnected. Only thus could a coherent form of knowledge leading to prevision become possible. It is to this synthesis of our sense impressions, supplemented by the fundamental forms of recognition we have mentioned, that we owe our belief in an external universe of space, matter, force and colour. In this universe, time was assumed to be flowing, events were regarded as exhibiting causal relationships; and our sense impressions were attributed to common causes existing in a public objective universe.

It is probable that a belief in causal connections arose from our ability to produce and to arrest certain sense impressions at pleasure through the conscious action of our will. Thus, we “will” a certain [Pg 389] effort (which turns out to correspond to the closing of our eyes), and the luminous impression ceases. It was then but a short step to extend our belief in causal connections to sequences of events in the outside world, even though in this case the human will was known to play no part. The same with force. Though suggested originally by our awareness of effort, it was soon exteriorised and credited with an existence in the outside world; and a similar process of exteriorisation would appear to have been responsible for our belief in physical time (no longer the “I” stream of our consciousness), in a time regarded as enduring in the outside universe regardless of our presence.

It does not seem to be of great interest to question whether these exteriorisations of the dehumanised concepts of causal connection, force and time were justified or not. In any case it cannot be denied that the exteriorisation of causal connections, for instance, has proved of inestimable service in allowing us to account for the routine of our experience. Therein resides its justification, since causality appears to be an indispensable condition for the sequence of phenomena to be intelligible, hence for prevision to be possible. But it would seem scarcely correct to state that the understanding imposes causality on an indifferent world; for, as we know, there are regions of science where, although causal connections may be suspected, none have yet been established.

Finally, we are led to enquire into the nature of this synthesis or co-ordination of facts which seems to be a prerequisite condition for knowledge to be possible. It is here that an analysis of the more sophisticated syntheses of science will be helpful in enabling us to understand how the mind proceeds. The reason for this is because scientific knowledge can be traced back to crude commonplace knowledge by a series of insensible gradations, the same incentives appearing to be active in all cases.

Now, when we consider the procedure of the scientist, we find that it consists in co-ordinating and linking together in a rational manner a number of experimental facts, with the maximum of simplicity. By “a rational way” we mean primarily “according to the rules of logic.” Irrespective of whether these rules are assumed to have been derived from experience or to reduce to a priori judgments, all normal men appeal to them, and all scientific theories, even the most revolutionary ones, are based on their acceptance. As for the criterion of simplicity, which enables us to select one co-ordination rather than another, it appears to be linked with our valuing of the expenditure of effort. Thus, even a dog finds it simpler to enter a house by the front door rather than clamber in through the back window. At any rate, inasmuch as, with small variations, all human beings agree unanimously on those co-ordinations which are to be regarded as the simplest, we may assume the urge towards simplicity to be fundamental. There are, of course, a number of other factors which enter into the construction of scientific syntheses, but for our present purpose the ones mentioned will suffice.

It will be noticed that in our list of fundamental facts, though our [Pg 390] awareness of the passage of time was considered to be unanalysable, no mention was made of our awareness of space and of objects, including our own human bodies, situated in space. The concept of space was assumed to have been generated in an indirect way when we sought to co-ordinate our sensory impressions with maximum simplicity. It would thus be in the nature of a mental construct arrived at a posteriori. By this we do not wish to imply that the concept of continuity and of extension may not have pre-existed in the mind in a latent form; all we wish to assert is that the necessity for appealing to the concept of space seems to have arisen from the totality of our sensory experience.

A co-ordination of our sensations will yield us at the same time empty space and what we will interpret as filled space representing material objects, including our own human bodies, which are then recognised as being situated in space. We cannot attempt to justify these views here, but those who are interested in the subject will find the ideas expounded in Poincaré’s masterly discussion of the problem in his book, “The Value of Science,” and more especially in “Last Thoughts.”

When it comes to differentiating and locating the portions of filled space occupied by bodies as against the empty portions, we again proceed in the same way, by co-ordinating sense impressions. Our belief in the existence of a table, for instance, is attributed to the fact that a simple co-ordination of the complex of our tactual, visual and muscular sensory impressions is possible only when we concede the existence of the table as a concrete reality, situated in the space before us. It is true that in common practice we may see a table and recognise it as such without touching it. But it is to be presumed that at earlier stages of our life we have exercised all our sensory faculties and have already recognised by a synthetic process the existence of space and of material objects. The table is then immediately recognised as existing, by a mere association of past impressions, without our having to explore it tactually.

Up to this point we have restricted our attention to space as mere extension. But space, as understood in common practice, implies considerably more: it represents a three-dimensional Euclidean continuum. When thus particularised, Kant’s arguments as to its a priori character are no longer tenable in the light of modern discovery; and we must assume that this special form we credit to space arises entirely from our co-ordination of sense impressions conducted in the simplest way possible. On no account may we consider three-dimensional Euclidean space to be imposed a priori either by sensibility or by the understanding.

These discussions on the empirical origin of space are not mere philosophic fancies having no bearing on science. They are in many respects vital; and it is generally conceded by scientists that the a priori doctrine of three-dimensional Euclidean space is one of the most pernicious teachings that philosophy has ever attempted to impose upon science. Similar arguments hold for “time,” when by “time” we are referring, not to our awareness of the time-stream in our consciousness, not to the “I” time, but to physical time or duration throughout the universe, which our consciousness has exteriorised [Pg 391] and projected into space. As Einstein remarks in a passage previously quoted:

“I am convinced that the philosophers have had a harmful effect upon the progress of scientific thinking in removing certain fundamental concepts from the domain of empiricism, where they are under our control, to the intangible heights of the a priori. This is particularly true of our concepts of time and space, which physicists have been obliged by the facts to bring down from the Olympus of the a priori in order to adjust them and put them in a serviceable condition.”

We may mention that these views on space as professed by the greatest scientists are in large measure to be attributed to the discoveries of non-Euclidean geometry supplemented by the investigations of the psycho-physicists. Still, in view of the difficulty of imagining hyperspaces and non-Euclidean spaces, the views presented might appear difficult to accept, and it might be held that three-dimensional Euclidean space imposes itself a priori regardless of experiment. But it should be noted that by the time men are of an age to philosophise, they have been subjected for so many years to beliefs based on inferences from experience, that the beliefs have remained, whereas the inferences, owing to the monotony of their repetition, have become second nature and appear intuitive.

And yet a moment of reflection should suffice to convince us that were three-dimensional Euclidean space an a priori condition of the understanding, it would have been quite impossible for mathematicians to wend their way through the non-Euclidean hyperspaces of relativity. Neither can three-dimensional space be considered to be imposed by sensibility, since, as Poincaré tells us, after a certain amount of perseverance, he was aided to a considerable degree by sensibility when investigating the problems of Analysis Situs of four dimensions.

This empirical origin of the spatial concept is stressed by Einstein in the following lines:

“We now come to our concepts and judgments of space. It is essential here, also, to pay strict attention to the relation of experience to our concepts. It seems to me that Poincaré clearly recognised the truth in the account he gave in his book, ‘La Science et l’Hypothèse.’ Among all the changes which we can perceive in a rigid body, those are marked by their simplicity which can be made reversibly by an arbitrary motion of the body; Poincaré calls these, changes in position. By means of simple changes in position we can bring two bodies into contact. The theorems of congruence, fundamental in geometry, have to do with the laws that govern such changes in position.”

When we realise that it is precisely these laws governing changes in position which govern our choice of a space among all those which the mathematician has to offer, we see how utterly dependent we are on experience when the problem of space is considered.

We may present these problems in a more vivid form. Suppose all we had ever seen of the world were given by its image in a reflecting spherical surface, such as a large door knob. The world of our visual perception would be very different from the one in which we [Pg 392] normally live; the shapes of objects would squirm in a variety of ways as we displaced them before the curved mirror. And yet, however different our world might appear from the one of common observation, we should eventually succeed in co-ordinating our perceptions. We should still conceive of an outside space, but this space would no longer be Euclidean. If, then, all of a sudden the mirror were to be removed, and we to behold the world as other men perceive it, we should be completely at sea, accustomed as we were to the laws of our non-Euclidean world. In fact, the situation would be very similar to that of the man who tries to ride a bicycle through a crowded thoroughfare while crossing his arms over the handle-bars. He probably would come to grief; and yet had he always ridden his bicycle in this peculiar way, he would find it just as hard to alter his habits and ride it in the normal way.

Summarising, we may say that a belief in an outside universe of space, matter and change is arrived at as a result of a synthesis of sense impressions. These conclusions, which apply to commonplace knowledge, will be substantiated further when we consider illustrations taken from the more advanced fields of knowledge of the scientist. As mentioned previously, it is impossible to draw a line and say: “Here scientific knowledge begins and commonplace knowledge ends.” And since the methods of the scientist are easier to dissect, a study of the scientist’s procedure cannot help but shed light on the more obscure problem of the genesis of commonplace knowledge.

In scientific syntheses we do not restrict ourselves to co-ordinating mere sense impressions; we must also co-ordinate scientific facts. But scientific facts are themselves the results of previous co-ordinations of other scientific facts, and these in turn are traceable to a co-ordination of sense impressions. A few examples taken at random from science will make these points clearer. Why, for instance, does the astronomer maintain that the sun is spherical?

It is, as we know, in order to account for the continued circular aspect of the solar disk, for the passage of sunspots, for their flattened appearance when nearing the sun’s edges, suggesting that they are seen in perspective, for the protuberance of its equator, for the Doppler effect exhibited on its equatorial rim, for the brilliancy of the planets when illuminated edgewise. It is also in order to render compatible the sun’s shape with its fluidic nature imposed by its high temperature. In other words, the aim of the scientist is to frame one single hypothesis which will permit him to co-ordinate this wide variety of facts. We may note that all the facts that the astronomer is seeking to co-ordinate presuppose a knowledge of space and of material objects situated in space. But of course we are assuming that by the time men began to worry about the shape of the sun, they had advanced beyond the primitive stage of recognising the existence of objects in space. Now, when we decide that the sun is spherical, our first argument is based on its circular aspect. In order to account for this, we appeal to probability, arguing that it is very improbable that the [Pg 393] sun should always turn the same face towards us. Of course the argument in itself does not carry much weight, since it is refuted in the case of the moon. Still, it serves as a suggestion, if nothing more.

Next consider the case of the planets. Were the sun a flat disk, it would appear strange that their brilliancy should remain appreciably the same regardless of their positions relatively to the sun. But this argument, be it noted, is highly sophisticated, for the natural view would be to assume that all bright points in the heavens shine of their own accord; and there would be no reason to differentiate between planets which reflected the solar light, and the stars which were in no wise dependent on the sun’s presence. It was only at a later stage that a differentiation between stars and planets became necessary. In short, the facts the astronomer is seeking to co-ordinate are of a highly sophisticated nature; it is only when we dissect them further and further, analysing the previous syntheses of science, that we are finally thrown back on our immediate awareness of sense impressions. We see, then, that science appears as an unending series of syntheses of other syntheses, but that in every case the synthetic method is the same.

Let us now pass to a less simple example, namely: “We know that molecules exist.” In the case of the existence of the table or the chair, all we needed was to co-ordinate certain immediate sense impressions. In the case of the shape of the sun, the procedure was more complicated, since we could not explore its surface with our hands and it was only by inference that we were led to believe we were viewing a spherical object from various positions in the course of a day or a year. But with molecules it is far worse, for no one has even seen or felt them. The inferences which we are led to make are based on others, these others again on others. The possibility of our co-ordination being proved incompatible with future discovery is therefore increased, and for this reason again our knowledge of molecules loses much of its certainty. Apart from questions of degree, however, this knowledge was arrived at in precisely the same way, by conceiving the simplest rational synthesis capable of co-ordinating a wide variety of facts of observation and experience.

It is probable that at a very remote stage in human history men noticed the difference in texture which existed between sand, which was grainy, and water, which appeared continuous and smooth. It would have been natural for them to wonder whether water would not turn out to be grainy if viewed microscopically. Some guessed one way, others another. There existed, however, a number of elementary facts of observation which had to be taken into consideration. For instance, a phenomenon on which the Greek thinkers laid due stress was the ability of wine and water to intermingle. The simplest manner of accounting for this was to assume that water and wine were formed of discrete particles which would exchange positions, much as two powders, one black and one white, would yield to a uniform grey mixture, when shaken together.

[Pg 394]

But Democritus went farther. Democritus appears to have been a man of exceptional scientific ability (as science went in those days); the geometrical solution of the volume of the pyramid and cone are attributed to him, and Pliny mentions that he spent his life among experiments. At any rate, he appears to have been the first thinker of antiquity (indeed, one of the very few) to display the scientific spirit, that of seeking unity in the various manifestations of nature by reducing quality to quantity. In this respect he initiated what was to become the necessary method of scientific investigation. Accordingly he suggested that all the elementary particles of matter were of the same substance. The qualitative differences which bodies reveal would then be due to differences in the shapes and sizes of their constituent elements or atoms. In this way unity was conceivable; but for this unity to endure, it was imperative that these elementary atoms should themselves constitute imperishable units. They could not be microcosms whose internal parts might suffer changes of position; hence they would have to be indivisible plena. As for cohesion, it was attributed to the atoms hooking on to one another.

As a scientific aspiration, Democritus’ scheme was perfect, but the trouble was that the facts known to him were too few in number. And so his theory was a crude guess at best; and it was only natural that a wider survey of facts should render it untenable. The co-ordination of facts known to modern science has proved, indeed, that atomism, as understood by Democritus, was untenable; for whereas beyond the atom of the Greeks there was no mystery, nothing further to look for, the atoms of matter are now known to be divisible. They differ qualitatively from each other, contain heterogeneities, are subject to change and decay. In other words, the atoms of modern science are new microcosms of baffling complexity, so that the appellation “atom” (meaning indivisible in Greek), retained by custom, is no longer appropriate. When we get beyond the atoms to the electrons and protons, we are in no position to assert that these constitute imperishable units. Further subdivision may be possible, and the existence of sub-electrons has been suggested. We do not know what would happen were it possible to cause a proton and an electron to coalesce—whether or not the result would be a complete annihilation of electricity and matter. Even if we adhere to the view that with these electrons and protons the ultimate atoms have been reached at last, we know so little about them that we cannot even be certain that they possess a definite size or shape. They may extend to infinity, they may reduce to mathematical points or singularities in the electromagnetic field; and this field itself is in no sense a substance. And so we see that atomism, as upheld by Democritus, is far from having been established by modern science. In the present state of our knowledge all we could do would be to guess, just as Democritus did in his day; but in view of the paucity of facts there are to guide us, no interest could be attached to our guesses.

Nevertheless, if by atomism we mean merely the tendency of matter and [Pg 395] electricity to congregate into entities of great stability, we are on safe ground and we may consider the doctrine proved. When understood in this more restricted sense, a wide variety of phenomena drive us to the atomic theory. In addition to the mixing of liquids, mentioned by the Greeks, we have to consider the diffusion of gases and of solutions, the compressibility of gases and the phenomenon of osmosis. All these phenomena appear to demand the existence of molecules or atoms. As an illustration let us consider the phenomenon of the compressibility of gases, studied by Boyle in the seventeenth century. We know that when a gas is compressed its volume is decreased. Yet its mass or weight remains unchanged. We cannot assume, therefore, that matter has vanished through compression; hence the simplest alternative is to suppose that the gas is made up of atoms floating or moving in the void. Compression will then result in crowding these atoms into smaller spaces while leaving their total number unchanged. Similar arguments may be advanced in the case of liquids. Thus, when we mix one pint of alcohol and one pint of water, we do not obtain two pints of the mixture, but appreciably less. The simplest way to account for this partial disappearance of volume is once again to assume that our liquids possess a grainy constitution and that vacant spaces exist between the grains or molecules. We may then assume that when water and alcohol are mixed, some of the molecules of water squeeze into the vacant spaces between the molecules of alcohol, or vice versa. Again, we find atomism cropping up in chemistry when Dalton sought to account for the empirical law of constant proportions. Nevertheless, although the corpuscular nature of matter seemed to impose itself if we wished to co-ordinate a large number of phenomena, something more was needed for this hypothesis to be accepted without reserve.

The major reason for the present-day belief in atomicity arises from the following considerations: A celebrated hypothesis due to Avogadro, the legitimacy of which we need not discuss here, suggested that if molecules existed, equal volumes of all gases maintained under the same conditions of pressure and temperature should always contain the same number. This number, called Avogadro’s number , was taken for the case where the volume selected was one cubic centimetre, the temperature zero centigrade, and the pressure that corresponding to 760 millimetres of mercury. Then it was shown as the result of highly complicated mathematical syntheses that, if molecules existed, a wide variety of phenomena should be influenced by the precise value of Avogadro’s number. The phenomena referred to deal with Brownian movements in fluids (Einstein), the viscosity of gases (Maxwell, Boltzmann, Einstein), the blue colour of the sky (Rayleigh, Keesom), equilibrium radiation (Planck), the specific heat of solids (Einstein), the phenomenon of critical opalescence (Smoluchowski), and many other phenomena which it is not necessary to mention. Accurate observations and experiments conducted on these phenomena should therefore permit us to deduce the value of Avogadro’s number. The results of all these experiments were in striking agreement, always [Pg 396] yielding the same value. As this number runs up into quintillions (), it was scarcely feasible to attribute the marvellous agreement to mere chance; hence we were forced to conclude that our suppositions were correct and that the existence of molecules had been established. It was then an easy matter to determine their masses and to obtain information as to their sizes and other characteristics.

Although we may be repeating ourselves unnecessarily, we must again draw attention to the fact that this knowledge of molecules, which is obtained through a co-ordination of experimental results, is in all respects (other than in degree of certainty) of the same type as our knowledge of the spherical shape of the sun or of our knowledge of space and of the table. On the other hand, it is essentially different from our awareness of feeling hot or tired.

Finally, let us consider a last example, namely, our belief that light is atomic. In this case our knowledge is still more uncertain. The justification for a belief in quanta was arrived at by Planck (as we have explained elsewhere), and arose from the peculiar phenomenon of black-body radiation. The mathematical treatment of problems of this sort is based on the calculus of probabilities. When calculations were conducted on the assumption of the continuous emission of light, we obtained Rayleigh’s law of radiation, which was refuted by facts. Planck noticed that by substituting discontinuous probabilities for continuous ones, the results of observation would be anticipated with great precision. When, a few years later, Nernst and then Einstein showed that the extension of these same discontinuities to the energy of molecular motions would account for the curious anomalies and variations of the specific heats of gases and solids, and when in addition Einstein succeeded in accounting with high precision for the photo-electric effect, Planck’s ideas appeared to gain in probability. The belief was enhanced still further when Bohr presented science with his model of the quantum-emitting atom.

On the other hand, there were a number of phenomena which the quantum theory was unable to explain. Light intensities inferior to a quantum had been observed, and the interference of light waves seemed to suggest continuity rather than discontinuity of emission. What is required is a more general synthesis, combining these two conflicting forms; and until a synthesis of this description has been formulated we can scarcely maintain that the existence of quanta has been established beyond dispute.[134]

The purpose of these different illustrations has been to show how slight and gradual is the transition from our commonplace knowledge of space and of the table to the loftiest forms of scientific knowledge. The methodology is ever the same, consisting in the formulation of a mental construct capable of co-ordinating in a rational and simple manner the sum total of our sense impressions. With the child and his [Pg 397] knowledge of space and of the table, this co-ordination is so simple that he obtains it without perceptible mental effort; whereas, in the more sophisticated cases of scientific knowledge, the synthesis is the result of abstruse mathematical speculations. Apart from this difference in degree, however, there exists no essential change.

And now let us examine the significance of these co-ordinations and investigate what they have revealed. We have seen that what we call reality reduces to the simplest co-ordination of the facts of observation, and that these facts, in the last analysis, represent co-ordinations of sense impressions. Hence we may say that it is these mind co-ordinations of sense impressions which yield us our knowledge of the so-called external universe. This scientific interpretation of reality may offend those who believe in the existence of some more absolute type of reality, the reality of the world in itself. To be sure, it appears natural enough for us human beings to assume that there must exist some sort of real world representing a reality of a more concrete category than the mental construct we have been discussing; a world which subsists in the absence of all human observers and in which the causes of our sense impressions originate. But what the nature of this real world of things in themselves may be, whether it could ever be described in human language, whether it is identical with the idea we have formed of it, are questions which are probably meaningless, and in any case are insoluble. As soon as we start arguing on these problems, it is a case of every man for himself; no one seems to be able to convince any one else. Fortunately, discussions of this kind are of no interest to science. In this chapter, the word reality will therefore be held to connote scientific reality, which means the simplest co-ordination of scientific facts, hence, ultimately, of sense impressions. That so restricted an interpretation of reality is ample for the needs of science issues from the following considerations:

The mere fact that men appear to understand one another when discussing the external world is sufficient proof that they have reached some common conception as to its nature. Hence, even if we assume that this so-called external universe, which each one of us believes he has discovered, represents but an individual dream which exists only in our respective minds, the fact remains that we must conceive of this dream as one shared in common by all men. As such, it manifests itself to all intents and purposes as an objective reality which we may regard as pre-existing to the observer who discovers it bit by bit.

It follows that the distinction between idealism and realism is purely academic in science, for our rule of action will be the same whichever of the two opposing philosophies we may prefer.[135] Thus the physicist who studies the properties of matter will proceed in precisely the same way, co-ordinating his results with the maximum of simplicity, regardless of whether he believes in matter as a metaphysical reality or as a mere mind construct. We have thus arrived at an understanding [Pg 398] of what is meant by the objective world of science.

Now it is obvious that had it been impossible to discover or create a common objective world the same for all men, of which the various observers would obtain private perspectives, science would have been quite impossible; for science deals with the general, not with the particular or the individual. Were it not for the fact that the objective world of John is also that of Peter, John and Peter would never agree on the most elementary subjects. An exchange of commonplace knowledge, and, to a still greater degree, of scientific knowledge, would be quite impossible. We must recognise, therefore, that the very existence of science proves that the co-ordinations of the sensations of John possess the same structure as those of Peter. This is all we can say. We have no means of discovering whether what one man sees as red the other might not see as blue were our observers to exchange eyes and brains while retaining their memory of past sensations. On the other hand, we can assert from experience that, for normal human beings, two objects which appear to be of the same colour to one observer will appear to be the same colour to the other.

When the importance to science of one same common objective world is realised, the necessity of ridding it of all purely individualistic appearances becomes obvious; and we are thereby led to the problem of illusions or hallucinations. By an illusory object we mean one that has only a private existence, yet whose appearance can be accounted for scientifically, in terms of other phenomena, to which we may concede a common objective existence. Thus, in the case of a mirage, if we insist upon saying that water lies before us in the desert, our objective world will differ from that of our friend who, standing where we see the water, will claim that no water is present. The unfortunate consequences of our error might be very great, as in the case of the dog who dropped the bone for its image.

Hence, either we must abandon a belief in a common objective universe and relinquish all attempts at framing a science, or else we must succeed in interpreting the differences in our opinions by appealing to some phenomenon, such as refraction, which all men will accept. In the case of the mirage, for instance, the introduction of refraction saves the objective universe, hence saves science. Finally, we may say that the bright band in the desert is a reality from the point of view of one particular individual, but that it is an illusion from the point of view of the generality; whereas refraction is a reality from the point of view of all.

In the case of hallucinations, we are dealing with delusions of another kind. Consider the fever-stricken man who sees snakes around him. Of course, the snakes, no more than the water in the desert, can have a place in the common objective world; so that in this respect the hallucination and the illusion appear analogous. Nevertheless, there exists an important difference between the two. Thus, in the case of the mirage, all we had to do was to stand at a given point to behold [Pg 399] the bright band in the desert as easily as any one else. A certain community in the illusion would still subsist. The hallucination, on the other hand, is essentially private. We cannot step into the patient’s place and suffer from his hallucination. Something more deep-seated is at stake than a mere change in our post of observation in a common objective world.

While it is true that the physiologist and brain specialist may succeed in accounting for a hallucination in terms of objective changes in our organic condition (blood pressure, etc.), yet even so we should be dealing with problems far more complex than those which the modern physicist is required to investigate. For all these reasons, while it is the aim of the physicist to account for illusions in terms of the common objective world and the laws of physics, the study of hallucinations belongs to another field of scientific research. In many cases the distinction we have established between a hallucination and an illusion lacks definiteness. For instance, the stars that we see when punched on the nose would not be considered the result of a hallucination, and yet, according to the division established above, the phenomenon would be more in the nature of a hallucination than of an illusion. Similar considerations would apply to dreams.

There exists a more popular interpretation of illusions. It results from a confusion between ignorance and illusion. Suppose, for instance, that we were to receive visual sensations which led us to assert that a table was present, but that on approaching the table we were to find that we could walk through it. If we, and we alone among men, could see the table, we should be justified in assuming, in view of past experience, that we were suffering from some hallucination, since any other alternative would deny us the possibility of conceiving of an objective world the same for all men. But if all who came into the room perceived the same table and failed, as we had done, to derive any tactual impressions from its presence, it would be taking too much for granted to assert that we had all of us been the victims of error. The mystery might be cleared up without difficulty were we to find it possible to account for this appearance in terms of phenomena already known; but if this procedure failed, the wiser course would be to agree that some new mysterious manifestation had been discovered. At all events, it would be wrong to maintain that the appearance must necessarily be the result of a collective illusion on the ground that its reality would conflict with the laws of matter. Our knowledge of natural laws is obtained only by generalising from experience; and where experience is incomplete, as it must always be, the laws can lay claim to no measure of certainty. For example, a man knowing nothing of atmospheric pressure might well assume that a balloon could never rise in the air without coming into conflict with the law of gravitation, or that a firefly could not emit light without getting burnt. The reason why the theoretical scientist is compelled to approach problems of a seemingly mysterious nature with great open-mindedness is because on [Pg 400] so many previous occasions he has been confronted with discoveries which yield nothing in strangeness to the one we have mentioned; and strangeness is not necessarily the product of illusion. More often than not, it is the result of ignorance and prejudice. In this respect, we have only to conceive of the surprise we should experience, had we never heard of an electromagnet, on discovering that distant objects flew towards it without being pulled by strings.

And now let us return to our objective world. We will assume that illusions have been barred therefrom; and the world we have thus obtained can be conceived of as existing in precisely the same way for all observers. If we assume that there is a real world of things in themselves, constituting the underlying cause of our sense experience, the point we wish to ascertain is whether scientific procedure can throw any light on its nature.

In the first place, when discussing the outside world as an existent reality, we must differentiate between its substance and its structure, or form. If, with science, we consider that our knowledge of the external universe can be arrived at only by a rational synthesis of facts of experience, we must recognise that substance escapes us completely; all we can hope to approach is structure, or relationships. Of course, a view of this sort presupposes that we are justified in asserting that knowledge can be acquired only by means of rational co-ordinations. If we dispute the legitimacy of this point the entire argument collapses. Hence the scientific argument can constitute no refutation of the views of the mystic. A man who “knows” of things through intuition or faith, or a man who tells us that he knows that in his previous incarnation he was Julius Cæsar, is a type of opponent with whom the scientist cannot even argue; for whatever can be neither proved nor disproved by observation or reason constitutes a form of knowledge which is meaningless, or at least completely irrelevant to science. If we did not place some kind of limitation on what we were to regard as knowledge, there would be no reason to prefer the opinions of a Newton or an Einstein to the ravings of an ignoramus or a lunatic; and human knowledge would become so conflicting as to lose all significance. It is not denied that intuition may often lead to discovery. Even in mathematics examples are numerous; but the fact remains that intuition alone has so often led men astray that unless its disclosures can be submitted to some kind of test, no reliance can be placed in them.

Furthermore, it is not asserted that intelligence is everything. We know that with animals and, still more, with insects, instinct plays a major rôle. But it is to be noted, first of all, that instinct is scarcely apparent in human beings. And in the second place, with bees, for example, which have selected the hexagonal shape for their combs, a choice which mathematical calculation proves to have been the best they could have selected, instinct has yielded exactly the same results that intelligence would have done; so that aside from the lack of consciousness which accompanies instinct, no difference is detected in the results to which it leads. Hence, it can scarcely be deemed [Pg 401] to yield a new form of knowledge. If these points be granted, we may return to our subject and enquire why it is that substance escapes us completely when by “knowledge” we refer to a rational co-ordination of facts and sense impressions.

Rational co-ordinations have their most perfect prototype in mathematical co-ordinations—in those of mathematical physics, for example. Not all co-ordinations can be constructed mathematically; chemistry, and of course, to a still greater degree, biology, afford us illustrations where mathematics is comparatively useless. Nevertheless, as the methods of co-ordination are essentially the same in all cases, the mathematical ordinations, in view of their greater clarity, can be studied to advantage as typical instances in this respect.

Now mathematical equations are nothing but relations, and from initial relations all we can deduce are other relations. In other words, our equations can never yield us more than we originally put into them. It follows that were all relationships in nature to be preserved and the substances changed, no observable difference could be detected; and we should never be able to differentiate between a whole class of worlds identical in structure but differing in substance. If, then, we discard the procedure of the mystic, or of the metaphysician who claims a knowledge that cannot be submitted to the control of experiment, we must recognise that substance escapes us completely and that our knowledge of the real world can at best reduce to a skeleton or structure.

An illustration given by Eddington presents the same problem in a more concrete manner. He assumes that in some future age the game of chess may be dead and forgotten; and he compares our position in respect to the real world with that of archaeologists who will have unearthed curious records of the game written in the usual obscure symbolism. Eventually they would succeed in reconstructing the moves of the different pieces, the two-dimensional ordering relation of the partitions, and the rules of the game. To this extent they could claim to have understood the game of chess. Yet certain aspects would escape them completely. The shape of the partitions, whether square, round or oblong, the aspect of the pieces, the nature of the board, whether of wood or of stone, would remain unknown. But it is to be noted that these elements of knowledge that escaped them would be irrelevant. The board might be of wood or stone, the partitions of diamonds or squares, and yet the game could be played as before. If, on the other hand, the ordering relation of the partitions were changed, if the permissible moves of the pieces were modified, the written records would make no sense. It is the same with the real world. The substance may change in nature, yet no difference will be perceived; but if the relationships are modified, a new world will divulge itself to our observation. To this extent, therefore, we are justified in saying that the most our observations can reveal reduces to the structure of the real world. As the French mathematician Bertrand once said, the age of the [Pg 402] captain, the number of the crew, the height of the mast can yield us no information about the position of the ship.

Structure, as outlined above, resolves itself into those mathematical relationships or equations which appear to account for the sequence of phenomena. But structure in the more usual sense would apply to the hidden mechanisms themselves, to the deep-seated, underlying causes. When, however, we wish to attain this more concrete knowledge of structure, we encounter a number of difficulties. As Poincaré points out, every time conservation of energy and the principle of least action are found to be satisfied (as is the case in electromagnetics, for example), an indefinite number of different mechanical models are possible; so that our knowledge of the structure of the world is vitiated by the fact that the substructure remains indefinite. Clifford expresses the same idea when he says:

“Whatever can be explained by the motion of a fluid can be equally well explained either by the attraction of particles or by the strains of a solid substance; the very same mathematical calculations result from the three distinct hypotheses; and science, though completely independent of all three, may yet choose one of them as serving to link together different trains of physical enquiry.”

In a certain measure, difficulties of this sort may prove temporary. It may happen that further experiment will enable us to determine which of the various hypotheses corresponds to concrete reality. Such, indeed, was the hope of the scientists of the eighteenth and early nineteenth centuries. And in our days the practical isolation of electrons and molecules is a proof that these hopes were not always unfounded. Yet, even so, we should have advanced but a step, and should again be faced with the problem of determining the structure of these electrons, and so on ad infinitum. It would be of no avail to say that our aim had been realised when the ultimate constituent elements were isolated. For either these supposedly ultimate elements would possess no structure, in which case we could never know anything further about them, or else they would present a structure; but then they would not constitute ultimate elements, since this structure would imply relationships between their various parts, and we should not be at the end of our journey.

With the discoveries of modern science, still other difficulties bar the route to success. It would now seem that as we wended our way into the microscopic, exact laws might vanish. If this be the case, the microscopic structures must ever elude us. Yet, even if we waive these latter difficulties, it is apparent that the scientific method can yield no knowledge of substance; the structure of reality is all we can approach.

But this reality, once again, is nothing but the expression of the simplest co-ordination of scientific facts; no genuine metaphysical reality is implied. If we discard the criterion of simplicity, we may conceive of our world in as many different ways as we please. Consider, for example, Einstein’s theory. One of the principal discoveries of [Pg 403] the theory has been to disclose unsuspected relationships between space and time, leading to the four-dimensional space-time structure of the universe. And yet, if we wished, we might account for all of Einstein’s previsions while retaining our traditional belief in a three-dimensional structure of space and in an unrelated time. Thus, Michelson’s experiment would be accounted for by Ritz’s or Stokes’ hypothesis; Bucherer’s experiment, by the dragging along of a certain mass of ether by the moving electron just as a ship drags along part of the water surrounding it; Fizeau’s experiment, by a modification of the ether in transparent bodies (and this was, in fact, Fresnel’s original explanation). The double bending of a ray of light, and Mercury’s motion, would be accounted for by a modification of the Newtonian law of gravitation; the Einstein shift-effect, by a modification in the intra-atomic structure of the solar atom, and so on. It is a pure waste of time to consider the various hypotheses that might be suggested, for we can invent as many different ones as we choose. Needless to say, all the simple beauty and unity of nature revealed by Einstein’s theory would be lacking; and prevision would be impossible, seeing that new postulates ad hoc would be needed at all further stages. But when all is said and done, what have simplicity, beauty, unity and prevision to do with the reality of the metaphysician?

Thus the reality of structure which the metaphysician would defend (quite aside from the reality of substance) presupposes his a priori belief in the inevitable simplicity of nature. But simplicity is, after all, but an expression of human appreciation. Furthermore, even if this view were contested, there would seem to be no reason to suppose that nature should prefer simplicity to complexity in the first place. As a matter of fact, what little we know of nature proves that the reverse is true when we cease to view phenomena in an approximate macroscopic way and adopt a more microscopic standpoint. To illustrate: There is no more simple law in physics than that of perfect gases; and yet we know that this apparent simplicity is due to our macroscopic observations and that it conceals the most bewildering chaos and uncertainty. It is the same when we consider numerous other phenomena, such as intra-atomic changes.

From this it follows that it is at least a questionable procedure to identify reality with simplicity of co-ordination. And if we discard this tentative identification, reality (even reality of structure) escapes us completely. Of course, the type of reality which we have been branding as elusive is the absolute reality, the “true being” of the metaphysician. The scientist also appeals to the word “reality,” but he employs it in a different sense. For him, reality is identified with simplicity of co-ordination, and he states his views explicitly, realising full well that a reality of this type is far from being absolute and that it is essentially pragmatic. It is for reasons such as these that the vast majority of scientists are agnostics at heart, not on account of any a priori predilection, but because [Pg 404] a proper understanding of the limitations of scientific knowledge leaves them no other alternative, refusing as they do to accept the knowledge of the mystic or the metaphysician as of any significance whatsoever. But scientific agnosticism must not be confused with that extreme form of idealism which denies the existence of any world apart from consciousness. It merely contents itself with stating that the objective world of science (that of space, time, matter, motion, in classical science, and of space-time, intervals and tensors, in relativity) is nothing but the embodiment of the simplest co-ordination of sense impressions, for which some unknowable supra-intelligible world is assumed to be responsible.

If, therefore, our minds worked differently, there would be no reason to assert that we should form the same conception of the world. We do not know whether, in the event of a Martian stepping down on earth, his mind would form the same mental construct of the universe as ours has done. For a Martian, there might be no such things as space, time and matter. Yet he might form a conception of the universe which would be perfectly intelligible to him. Those who believe in the metaphysical reality of space and time enduring in the world of things in themselves, under the plea that we can only conceive of existence in space and in time, appear to be guilty of the same error as those who, never having heard of fishes, would assert that no such creatures could exist, since they would be unable to breathe. Anthropomorphic arguments of this sort have no place in science.

In short, it is not denied that there may exist a world of things in themselves; all that is asserted is that a world of this sort defies all description. A few quotations from the writings of the leading men of science will make this attitude clearer. Thus, in Larmor’s book, “Æther and Matter,” we find: “Laws of matter are, after all, but laws of mind.” Again, in Poincaré we read: “Are the laws of nature, when considered as existing outside of the mind that creates or observes them, intrinsically invariable? Not only is the question insoluble; it is also meaningless. What is the use of wondering whether laws can vary with time in the world of things in themselves, when in a world of that type time may have no meaning? Of this world of things in themselves we can say nothing; we cannot even think of it. All we can discuss is how it would appear to minds similar to our own.” And, again, elsewhere: “The Bergsonian world has no laws. The world that may have laws is merely the more or less deformed image that scientists have conceived of it.” There is no need to give further quotations, for with slight variations the same philosophy is expressed by all scientists.

However, it cannot be emphasised too strongly that from a practical standpoint these questions are purely of academic interest. The physicist and the mathematical physicist are compelled to operate and reason as though they believed in the real existence of a real absolute objective universe, one of space and time, according to classical science; one of space-time, according to the theory of relativity. [Pg 405] In fact, as we have said, were it impossible to conceive of a common objective world, one existing independently of the observer who discovers it bit by bit, physical science would be impossible.

Incidentally, we may recall that such problems as the relativity or the absoluteness of space and motion have nothing in common with idealism and realism. Space may be subjective, yet absolute, or real and yet relative. It is important not to confuse the meaning of the words, “absolute,” and “relative” as used by physicists, with their meaning as understood by philosophers. Much of the difficulty that philosophers appear to have experienced in understanding the attitude of science seems to have arisen from confusions of this sort.

Now, if we revert to that rational synthesis of our sense impressions which yields us the objective world of science, there remains to discuss the methodology whereby this synthesis is achieved. In the more elementary cases, the procedure is one of commonplace, logical and inductive reasoning; but in the more advanced cases, the process is exclusively mathematical. There is a close resemblance between the procedure of development pursued by pure mathematics and that of theoretical physics. In either case the tendency has been a search for unity through progressive generalisation.

Mathematics is, of course, so vast a subject that it is impossible to expose even its characteristic aspects in a few pages; but for the purpose we have in view it will be its generalising aspect which we shall mention. Mathematics deals with generalities. It seeks to obtain general laws and general theories as opposed to particular results. For instance, the sum of the first two odd numbers is found to equal the square of 2; the sum of the first three odd numbers is found to equal the square of 3. This, however, does not tell us what the sum of the first 10,000 odd numbers would be. We might proceed to add up these numbers and discover the result; but the aim of mathematics is to obviate a laborious undertaking of this sort. It succeeds in this attempt by demonstrating the existence of a mathematical law according to which the sum of the first odd numbers is equal to however great may be. When a general law of this sort is established, we can apply its formula to any particular case and thus obtain a solution without further ado.

The procedure whereby this law is obtained is that of mathematical Induction. We verify the fact that the sum of the first two odd numbers is ; then we prove that if this is the case, the sum of the first three odd lumbers must necessarily be 32. Finally, we prove that if the sum of the first () odd numbers is (), then the sum of the first odd numbers must necessarily be . We assume that this law will hold however great may be, or, in other words, must always hold; and our mathematical law is established. In short, mathematics aspires to give us general laws in place of particular facts, or, again, to proceed from the particular to the general. A search for generality and unity constitutes the leit-motif of mathematics.[136] A few specific [Pg 406] examples may make this point clearer.

Suppose we start from the natural sequence of integers and labour under the impression that these integers alone constitute true numbers. We should notice that if we divided one of our numbers, say the number 10, by the number 5, we should obtain one of our integers, namely, 2. On the other hand, if we divided 10 by 4, the result would not yield one of our integers. Accordingly, we should be compelled to state that the division of one number by another might or might not yield a true number. In order to re-establish generality, mathematicians were compelled to assume the existence of fractional numbers, just [Pg 407] as actual or true as the integers. Thanks to this introduction it became possible to assert that the division of one number by another would always yield a true number. Now suppose we attempted to extract square roots. We should find that whereas the square roots of some of our true numbers were themselves true numbers (i.e., ), yet in the majority of cases we should obtain no such result (i.e., ). Hence, we should have to say that in certain cases the square root of a true number would be a true number, whereas in other cases it would correspond to nonsense. Once again, in order to obtain generality, we should have to incorporate these seemingly nonsensical magnitudes along with the true numbers, obtaining thereby the so-called irrational numbers. In a similar way negative numbers would have to be introduced in order to confer sense in all cases on the subtraction of one true number from another. Imaginary numbers would follow when we wished to generalise the significance of the square root of a number, whether positive or negative; then complex numbers, when we wished to confer generality on the significance of the addition of numbers, whether real or imaginary. A further generalisation would yield quaternions and hyper-complex numbers generally. Following still another line of generalisation, Cantor introduced transfinite numbers into mathematics.

A further illustration would be afforded by geometry. Thus, if in a plane we trace a straight line and a second-order curve (circle, ellipse, parabola or hyperbola), it may happen that the line intersects the curve in two points, but it may also happen that no intersection takes place. Generality can be re-established, however, provided we introduce imaginary points and consider them as true points. With this extended concept of a point we may say that the straight line always intersects the second-order curve in two true points, whether real or imaginary. Further generalisations require the introduction of ideal points, points at infinity, imaginary lines and angles.

Also, when we consider a second-order curve, it is necessary to state the positions of five of its points for the curve to be determined without ambiguity. But a circle is also a second-order curve; yet, in the case of a circle, when we state the positions of three of its points the circle is completely determined. Hence, our rule lacks generality and cannot be claimed to hold for all second-order curves, whether ellipses, parabolas, hyperbolas or circles. Once again, however, when imaginary points are introduced, this duality disappears and our rule becomes general. It is found that all circles, without exception, pass through the two same imaginary points called the circular points at infinity. When we take these two imaginary points into consideration we are able to state that all second-order curves, whether circles or ellipses, etc., are determined when five of their points are given. It is also interesting to note, as Cayley and Klein have shown, that when imaginary points, lines and angles are introduced, the various geometries (non-Euclidean [Pg 408] and Euclidean) can be treated in terms of projective geometry. We see that in all cases mathematicians attempt to replace the word sometimes by the word always, the particular by the general, thereby revealing unity where diversity once held sway.

Let us mention yet another geometrical example. It is always possible to draw a triangle circumscribed to a circle, that is to say, one whose three sides are tangent to the circle. Also it is always possible to inscribe a triangle in a circle; which means that the three summits lie on the circle. Now, Euler noticed that if two circles were drawn at random, it was impossible in the majority of cases to draw a triangle which would be inscribed in one circle while circumscribed to the other. When, however, one such triangle could be found, an indefinite number of others could also be drawn.

Poncelet then gave a celebrated generalisation of this theorem, and Jacobi showed its intimate connection with elliptical functions. We may understand the nature of this generalisation as follows:

A circle is a particular instance of a conic, and a triangle a particular instance of a polygon; whence Poncelet extended Euler’s theorem, which deals with circles and triangles, to one dealing with conics and polygons. In this illustration, again, we witness the same tendency, progress towards generalisation.

Possibly it is in the theory of functions that the most beautiful examples will be found. Mathematical functions are magnitudes whose values vary with that attributed to the variable they contain. We can construct as many different types of functions as we please by annexing additional terms, and the functions thus constructed differ in their behaviour from one another.

For instance, if we call the value of the function and represent the variable by , then is a mathematical function ( and being assigned constants). For every value of a definite value corresponds for . A more complicated function would be given by

It was soon discovered that, if instead of annexing merely a large number of successive terms to the expression of our function we added an infinite number of terms selected in an appropriate manner, we often obtained functions presenting exceedingly simple properties; so simple, indeed, as to warrant our coining separate names for them. These various functions may be likened to different beings, possessing various peculiarities and tendencies. Certain mathematicians (Borel, for instance) have gone so far as to differentiate between wholesome functions and contaminated ones; and the appellation “pathology of functions” has been introduced into mathematics. As an example of the [Pg 409] very simplest type of functions, we may mention those given by the unending series where 3! stands for for . These functions were known, respectively as, “sine ” (or sin ), “cosine ” (or cos ) and the exponential function .

It was then found that the functions sin and cos presented very rigid relationships as though they belonged to the same family. The name circular functions was given to the family; the relation we have in view being given by On the other hand, no such relationships appeared to exist as between these circular functions and the exponential function . But the entire aspect of the problem changed when Euler appealed to imaginary magnitudes, such as ; where stands for and stands for any real number, as before. Of course an imaginary number, as its name indicates, was conceived of by its inventors as an unreal magnitude, a mere mathematical fiction having nothing but a symbolic significance. And yet, as we shall see, without imaginary numbers some of the most practical industrial problems could never have been solved.

As we have said, Euler substituted imaginary numbers for the real ones in the functions we have discussed, and discovered that when this was done a surprising connection was found to exist between these erstwhile unrelated functions. The relation was In this formula we have only to make , and we obtain Thus, not only has the introduction of imaginary numbers revealed an unsuspected relationship between what were widely different types of functions, but the two numbers and , taken from totally different regions of mathematics, from geometry and from logarithms, are suddenly found to manifest extraordinary relationships.

[Pg 410]

But this introduction of imaginary numbers into mathematical analysis was soon to lead to still more wonderful discoveries. Cauchy undertook a general study of functions in which the variable would be a complex number, that is to say, a real number plus an imaginary one. He uncovered thereby a new universe of functions, called functions of a complex variable, possessing the strangest properties; and it was seen that our older functions were but particular instances of these more general types. But for these new functions to possess the essential attributes of our older ones, a certain mathematical restriction had to be placed upon them. The restrictions turned out to be expressed by the same mathematical relationship which defines the potential distribution in Newton’s law of gravitation, namely, Laplace’s equation.[137] This equation crops up again in the theory of conformal representation, in the theory of heat, in the theory of elasticity, etc. Unsuspected relationships thus appear to be springing up on all sides. Incidentally we may note that Riemann’s great discoveries in the theory of functions take their start from this analogy existing between the functions of a complex variable and the laws of the potential distribution around matter.

The theory of the functions of a complex variable constitutes one of the most extensive domains of pure mathematics, with which the names of some of the greatest mathematicians are associated, such as Cauchy, Riemann, Weierstrass, Jacobi, Abel, Hermite and Poincaré. Numerous problems pertaining to the ordinary functions receive a very rapid solution when we consider these functions as restricted cases of the more general functions of a complex variable. As time went on, a number of these functions manifesting strange yet simple properties were brought to light, always by the same process of successive generalisation. First came the elliptical functions discovered by Abel and Jacobi, then the modular functions of Hermite, and finally the automorphous functions of Poincaré. Once again the discovery of these new functions revealed unexpected relationships between domains of mathematics which had appeared to be totally estranged; and Poincaré established surprising relationships between the automorphous functions, certain problems of the theory of numbers, and non-Euclidean geometry.

Incidentally, the popular belief that mathematicians, by appealing to imaginary magnitudes and the like, are abandoning the world of reality for one of shadows, is belied by the fact that one of the most elementary mechanical problems, namely, the oscillatory motion of an ordinary pendulum, can be solved mathematically only by an appeal to elliptical functions, in spite of the fact that imaginary quantities are part and parcel of their make-up.[138] It is the same with a [Pg 411] number of electrical problems of widespread industrial importance; likewise, with Einstein’s law of gravitation, the orbits of the planets cannot be calculated unless we appeal to elliptical functions. Similar considerations would apply to Poincaré’s automorphous functions. However, this point being granted, it is of course obvious that the mathematician is not primarily interested in knowing whether his abstract speculations have any counterpart in the real objective world, any more than the poet wishes to know whether his dreams will come true. The sole object of his investigations is to explore all rational possibilities and to co-ordinate into one consistent whole various mathematical edifices which at first blush might appear to be totally estranged from one another.

We have insisted particularly on this generalising tendency of mathematics, as it is the one which will play a prominent part when we consider the methodology of mathematical physics as applied to the world of reality. Here we must recall that however mysterious it may seem, nature appears to be amenable to mathematical investigation and to be governed by rigid mathematical laws, at least to a first approximation. So far as scientists are concerned, this belief is not the outcome of religious or philosophical presuppositions. Rather is it a belief which is forced upon our minds by the triumphs of theoretical physics, the first grand example of which was afforded by Newton’s celestial mechanics. As soon as this susceptibility to law was recognised in nature, the avowed aim of science was to discover the unknown laws and in this manner allow us to foresee and to foretell, hence also to forestall, and to cease living in a world of unexpected miracles. In those realms where laws were finally established, science displaced superstition; and wherever, for one reason or another, laws could not be found, superstition continued to reign supreme. A case in point is afforded by meteorology, which in spite of recent progress remains an extremely backward science. As a result an astronomer, who would never think of praying for a solar eclipse, might still pray for a cloudless day favourable for his observations.

Now, in order to render nature amenable to mathematical treatment, it is necessary that we should succeed in reducing the various natural phenomena to common terms. This is done by seeking differences of quantity beneath differences of quality. When and only when this quantitative reduction has been accomplished can science proceed with its investigations, its deductions and inductions. In the case of the objective universe of physics, this process of reducing quality to quantity leads us to conceive of the objective universe as one of electromagnetic vibrations and of molecules, atoms and electrons acting on one another according to fixed laws, rushing hither and thither in space or vibrating round fixed points. The objective universe of [Pg 412] science is thus noiseless, lightless, odourless. All the qualities and values are conceived of as arising from the interactions existing between our sense organs and the motions of the outside world, just as the impact of a body would reveal itself as sound to a gong gifted with consciousness.

This quantitative reduction is, however, but a first step; and even after it has been accomplished, nature might still defy mathematical investigation. The fact is that mathematics consists in compounding the similar with the similar; wide heterogeneities even among quantities would render an appeal to human mathematics useless. Thus, the game of chess, where the various pieces can move in widely different ways, is not subject to mathematical treatment. The fact that theoretical science is possible proves that similarity and unity can be found in nature.

Then again, nature must be simple, or at least simple to a first approximation. Theoretically, simplicity cannot exist in nature, since the whole influences the part and the part influences the whole. But in certain cases it has been found permissible to neglect a number of influences owing to their minuteness, and science has thus been able to progress. It is because in meteorology this restriction of active influences to a minimum appears impossible that long-range weather prediction is any man’s guess.

But there is yet another condition which must be found in nature, and that is continuity. Mathematics is capable of attacking problems where discontinuities are present, but the technical difficulties are very great; and this is the chief reason why the theory of numbers presents such obstacles. On the other hand, where we are dealing with problems of continuity the mathematician feels more at ease; and that superb mathematical instrument known as the differential equation becomes applicable. Incidentally we can understand why it was that Planck’s discovery of quantum phenomena, or discontinuous jumps in the processes of nature, was such unwelcome news to theoretical physicists; the differential equation had lost its power.

Even now we are not at the end of our difficulties. Assuming that nature manifests unity, simplicity and continuity, we can only investigate her secrets provided we are faced with mathematical problems that we can solve. Even such relatively simple problems as that of the motion of four bodies attracting one another under Newton’s law, have thus far defied all attempts. Again, our differential equations are of a very simple species. They suffice only for the simplest type of problems, when, for instance, the future position of a body moving under the action of a given force depends solely on its circumstances of motion in its initial position. But what if the future position of the body were to depend on its entire past history? Phenomena of this type, subjected as they are to hereditary influences, are well known in physics, being illustrated by the hysteresis, or fatigue, of metals; and of course in biology they are the rule. Such problems can be attacked to-day, at any rate in the most elementary cases, following Volterra’s discoveries in integro-differential equations. This opens up a new domain of science, that known as [Pg 413] “hereditary mathematics.”

When we consider all these technical difficulties, and realise that solutions of these problems have been obtained thanks only to the most abstract speculations of pure mathematicians, to the invention or discovery of mathematical entities which at first sight would appear to be totally estranged from the world of reality, we must not belittle the practical importance of these seemingly unreal entities.

As a matter of fact, it is little short of a miracle, in view of the insignificance of our mathematical ability, that theoretical physics should have been possible at all. Had our solar system contained two suns such as are present in the double-star systems, or had one of the planets been comparable in size to the sun, it is safe to say that Newton, in spite of his genius, would never have discovered his law of gravitation; for he would have been thrown back on the tremendous problem of the three bodies.

Leaving aside these technical difficulties, the fact remains that in a number of cases mathematics has been found applicable to nature at least in an approximate way, and as a result mathematical laws have been discovered. We shall now proceed to show how these theories of mathematical physics have evolved. To begin with, a large number of empirical facts are first discovered by the experimenter. These facts are then co-ordinated into a consistent whole, general relations or laws are established, and in this way we obtain various groupings of phenomena, such as electricity, magnetism, optics, mechanics, chemistry, biology, etc. In certain cases it may be impossible to get beyond the initial stage; though experimental data are accumulated, the co-ordination of facts and the discovery of general relations and laws may defy analysis. In such cases we can scarcely regard our knowledge as constituting a science.

The next task is to co-ordinate and establish relations between these various realms of science. Confining ourselves to physics, we find that, first of all, relationships were discovered between electric and magnetic phenomena; then between light propagation and the vibrations of elastic solids (Fresnel, MacCullagh); then between electromagnetic phenomena and optical ones (Maxwell); then between heat and molecular motion (Maxwell, Boltzmann). But in spite of these lofty syntheses, one mysterious influence appeared to remain estranged from all the others; this was gravitation. It has been one of the triumphs of the relativity theory to succeed in establishing the connections between gravitation and optics or electricity.

Finally, an amalgamation of all these various realms is sought for in the expression of a gigantic universal law, the principle of action, governing one unique mathematical world-function, the function of action of the universe, which the theories of Weyl and Eddington appear to suggest.

In short, we see that the development of theoretical physics has followed that of pure mathematics in its generalising characteristics. In either case the breaking down of barriers, the discovery of unity [Pg 414] in diversity, has been the guiding motive. And yet, in spite of this mathematisation of physics exemplified in the works of the theoretical physicists, physics is not mathematics, and truth in physics is not the same as truth in mathematics.

In the first place, physics progresses by successive approximations and does not attain its goal at one stroke, as is often the case with mathematics. Thus, in the days of Galileo it would have been correct to say: The centre of gravity of a projectile moving in vacuo describes a parabola. Half a century later, Newton recognised that this statement was only approximate. The correct statement then became: Under ideal conditions of isolation, the centre of gravity of the projectile will describe an ellipse, the earth’s centre being situated at one of its foci. But, according to relativity, Newton’s statement in turn is approximate. We must now say that the trajectory lies along an ellipse whose axis is slowly rotating. There is every reason to believe that even Einstein’s mechanics is but an approximation to truth; and Schrödinger’s wave mechanics is already supplanting it in the infinitely small. Indeed, there is every reason to suppose that however far we go, we shall always be dealing with approximations. Here, then, is an essential difference between physics and mathematics. Thus far, it would appear that absolute truth might exist but that our present means of investigation had not allowed us to attain it. But we shall see that more is at stake than a temporary admission of failure.

Consider, for instance, the constants of physics, such as , the gas constant, or , Planck’s constant; and contrast them with the constants of mathematics, such as or . Whereas the constants of mathematics can be calculated to any degree of approximation we choose, the values we assign to the constants of physics can never be considered absolutely rigorous. It is not merely because physical observations are necessarily inaccurate or because conditions of observation may not be perfectly ideal owing to the presence of contingent influences; other reasons of a deeper order are involved. The fact is that we are by no means certain that nature is amenable to rigorous mathematical laws and that the so-called constants of physics represent other than average values. The triumphs of theoretical physicists suggest that a mathematisation of physics is permissible as a first approximation; but a closer microscopic survey might prove that this appearance of mathematical purity and simplicity in nature was due to our crude macroscopic survey of phenomena. For instance, it is a well-known fact that if conditions are sufficiently chaotic, the chaos will generate simplicity when we view things from a macroscopic standpoint; it is only when we wish to view things microscopically that the chaos appears and mathematical methods become impossible.

As we mentioned earlier in the chapter, the kinetic theory of gases furnishes us with an apt illustration. The molecules are rushing hither and thither, bounding off one another in the most capricious [Pg 415] way.[139] It would be quite impossible to foresee the history of a given molecule; and yet, thanks to this very chaos, the macroscopic laws of perfect gases are exceedingly simple. We may infer, therefore, that the constants of physics which enter into our macroscopic laws represent mere average effects and differ essentially from those of pure mathematics. More generally, even under ideal conditions of observation the world of physics can never be assimilated to the world of pure mathematics.

The simplification which ensues from a macroscopic survey of nature allows us to understand the reason for those periodic swings which take place in theoretical science between the physics of the general principles and the atomistic viewpoint. If we view nature atomistically, it will be our desire to interpret phenomena in terms of attractions and repulsions between molecules, atoms or electrons. Following the success of Newton’s treatment of planetary motions, the scientists of the eighteenth and early nineteenth century endeavoured to pursue this method by reducing the entire world of physical science to attractions and repulsions between discrete particles. All sorts of laws were appealed to: inverse-cube laws and laws involving still higher powers. But nature was not so simple as scientists had hoped, and paid very little heed to the difficulties of mathematical analysis. So there arose an opposing school of scientists, who substituted general principles embodying wider concepts such as energy, entropy, action, for the atomistic theories of their predecessors.

The argument of this new school of thought ran somewhat along the following lines: “By concentrating more and more upon your world of atoms and molecules you are losing sight of those general principles which experiment has revealed. Would it not be better, therefore, to abandon your hopelessly complex speculations, and accept as a starting point these fundamental principles, which can be deduced from macroscopic investigation? We grant that by so doing you will be substituting a macroscopic for a microscopic view of nature; but by reason of the mathematical difficulties into which you have been led and the numerous hypotheses you have had to make in order to compensate for your ignorance of the microscopic mechanisms, it might be the safer course to content yourselves, for the time being, at any rate, with the macroscopic viewpoint.”

We may illustrate the new viewpoint somewhat as follows: A life-insurance company does not know and has no means of foretelling when one of its clients will die; yet, when instead of considering the case of an individual client (which would correspond to the microscopic or atomistic viewpoint) we consider men as a whole (the macroscopic point of view), certain general death statistics can be established. By attaching too much importance to one individual, we should lose sight of this general principle, and premiums could never be fixed. The analogy is far from perfect; but it is helpful in that it shows [Pg 416] us that in certain cases, at least, the macroscopic point of view can yield us the information we require. A better illustration would be given in the following very free translation of a passage from Poincaré. As he tells us:

“Suppose that we are in the presence of some machine. The primary and terminal gears are alone observable, while the intermediary systems of transmission are concealed. We have no means of knowing whether the motion is transmitted by means of gears or straps, by means of pistons or by other means. Does this signify that we can never understand anything about the engine unless we take it to pieces? We know that such is not the case, for the principle of the conservation of energy enables us to determine the most important point. We observe, for example, that the last wheel turns ten times more slowly than the first, both these wheels being visible. We may infer therefrom that if a couple is applied to the first wheel it will balance a ten times greater couple applied to the last one. This information is obtained without our having to consider the mechanism whereby this state of equilibrium is realised and without our having to know how the various forces will balance one another inside the machine. When we consider the universe, similar arguments will apply. Most of its workings are beyond the sphere of observation; but by observing those motions which we can perceive, we can, thanks to the principle of the conservation of energy, derive conclusions which will remain true regardless of the structural details of those parts which are invisible.”

The tendency of science was therefore to lose interest in the microscopic mechanisms and to concentrate on the general principles. It was not long, however, before a return to the atomistic attitude was again forthcoming; but this time our knowledge was more advanced, and greater success was the result. We witness this return to the atomistic procedure in Lorentz’s theory of the electronic structure of matter and electricity. Marvellous anticipations resulted from this epochal theory, and it seemed as though a great advance had been made. But, once more, difficulties arose; and here we are referring to those mysterious phenomena which were the original starting point of the theory of relativity.

Einstein again reverts to general principles. Instead of endeavouring to account for the mysterious negative experiments by ascribing all kinds of curious properties to the electrons and to matter, he accepts these negative experiments as significant of some general principle, namely, the relativity of Galilean motion through the ether, entailing the invariance of the velocity of light.

Recent theories on the nature of the atom furnish instances of a similar sort. Thus, Bohr’s theory of the atom has atomistic tendencies in that it ascribes definite paths and motions to the electrons moving round the nucleus. In this way it was able to attribute to helium certain spectral lines which before then had been ascribed to hydrogen. Accurate experiments have since confirmed the correctness of Bohr’s anticipations. But insuperable difficulties were soon forthcoming when attempts were made to extend the theory to the heavier atoms. [Pg 417] Half-quantum numbers and spinning electrons were introduced with the result of complicating the theory considerably.

So Born and Heisenberg abandoned the atomistic outlook, stressing the necessity of following a strictly phenomenological procedure. The motions and orbits of the electrons were disregarded entirely, since these had never been observed, and nothing but the repartition, polarisation and intensities of the spectral lines was taken into account. This new departure, known as the matrix method, led to remarkable results, removing many of the difficulties which had beset Bohr’s atom.

Still more recently we have witnessed a return to the hidden-mechanism viewpoint with Schrödinger’s wave mechanics. We can scarcely refer to it as an atomistic method, since waves take the place of discrete particles. Nevertheless, from the standpoint of methodology, the general idea involved is the same—that of basing our deductions on things that cannot be observed.

This dual tendency in science, that of atomism versus the general principles, that of the microscopic versus the macroscopic, must not be thought to arise from any peculiarities in the philosophies of the various scientists. It is due to the circumstances under which they happen to find themselves. Thus, we see a theoretical scientist such as Einstein adopting a phenomenological attitude when investigating the difficulties attendant upon the negative experiments; and we see the same scientist, Einstein, adopting an atomistic attitude when studying the problems of Brownian movements, the problems of radiation, or the quantum theory of specific heats. In the same way, when we climb a ladder we raise our left leg, then our right one. It is not because we conclude we were wrong in raising our left leg; it is because we cannot progress unless we allow the right leg to catch up with the left one. So it is in science; and we may be quite assured that in the years to come, as in the past, we shall continue to witness these periodic swings from the macroscopic to the microscopic view, from the general principles to atomism. We must realise, however, that as each swing takes place we are advancing higher and higher towards that unattainable ideal, perfect knowledge.

And now let us revert to the common objective world of classical science, to the world of separate space and time in which molecules, atoms and electrons move and vibrate, and in which other types of realities, such as electromagnetic fields, are present. Let us recall once again that this real objective universe is the world as the scientist must assume it to be if he wishes to co-ordinate the complex of his experiences (reducing in the final analysis to sense impressions) in a simple and consistent manner. Whether or not this objective universe can be identified with the real world of the metaphysician is a subject we need not discuss further, since it is of no interest to science. The point we wish to investigate is somewhat different. It is to be noticed that the so-called secondary qualities, such as colour, sound and smell, have been banished as self-supporting entities. They are now ascribed to the reactions of our brain to the [Pg 418] neural disturbances which occur when our sense organs are submitted to the action of the realities of the common objective world.

The criticism is often directed against the scientific attitude that by reducing quality to quantity we are eliminating values, substituting, for example, electromagnetic vibrations for the colour red. Of course, it is scarcely necessary to state that, aside from all philosophical considerations, the reduction of quality to quantity is a prerequisite condition if science is to exist. To illustrate: In Newton’s day the vibratory nature of light was unknown. Red light differed from green light, but this qualitative difference manifested itself as an irreducible fact for which it was impossible to account. Under the circumstances, if the observer were to rush towards a red light or to move away from it, it was quite impossible for science to anticipate what effects would arise. As soon, however, as Fresnel discovered the vibratory nature of light, red light was found to differ from green light owing to its slower rate of vibration: prevision then became attainable. It was possible to anticipate that were we to approach a red lamp with sufficient speed it would appear green, that with greater speed it would appear violet, and that with still greater speed it would become invisible. Likewise, were we to recede from the light with sufficient speed it would also cease to appear visible. This was the celebrated Doppler-Fizeau effect, which astronomical observations soon succeeded in detecting; it is thanks to this effect that we are able to determine the radial speed of approach or of recession of the stars.

We may presume, therefore, that the practical utility of the scientist’s constant endeavour to reduce differences of quality to differences of quantity is not called into question by the critic, and that his objections are directed solely against the supposedly unjustified philosophical outlook which this reduction of quality to quantity has exercised on the scientific mind. But, in order to investigate the problem, we must proceed to a more detailed discussion of the reasons that compelled scientists to differentiate between the so-called primary and secondary qualities.

We shall first restrict our attention to the primary qualities of classical science, and shall furthermore consider the matter solely from the standpoint of visual perceptions. For instance, let us assume that we are viewing what would commonly be called a cone. When we view this supposedly conical object, we perceive it with various shapes according to the relative position we may occupy: we may see it as a circle, or as a triangle with a base which is either straight or elliptical; and so, by calling the object a cone, we are giving it a name which corresponds to none of its many perceived shapes. The conical shape is thus never perceived directly, but is the resultant of a synthesis of all the private views we may obtain when viewing the object from various directions. Now the question arises: Which of the various shapes should we call the real shape of the object? One of the many shapes it manifests when viewed from successive positions? Or that shape at which we have arrived in an indirect manner as a result of a co-ordination of private views? Of course the answer to [Pg 419] this question will depend on the meaning we wish the word “reality” to convey. If by “reality” we wish to refer to what we apprehend directly, it would be absurd to refer to the conical shape as the real shape, since this shape is never apprehended directly, but merely inferred. The trouble, however, with this tentative interpretation of reality would be that we could never attribute any definite shape to an object, for there would be no reason to favour the shape as seen from here rather than the shape as seen from there. On the other hand, if we ascribe to the word “reality” the alternative meaning, i.e., that pertaining to the entity issuing from a synthesis of private views, it becomes possible (so classical science believed) to attribute a definite shape to the object, a shape transcending the relative situation and motion of the percipient. And so, according to classical science, shape was no longer indefinite, no longer a matter of point of view: it became impersonal, and we were enabled to conceive of the object as having a definite shape even in a world devoid of all observers. This was the objective space-and-time world of classical science. It would appear, then, that if we wish to retain the word “reality” in the two different capacities mentioned, we should at least underline the difference in meaning by referring to a “private reality” in the first case and to a “common or objective reality” in the second.

We now come to the so-called secondary qualities—colour and sound. Here, for instance, is what would commonly be called a red light. We apprehend it as red wherever we may be situated, so that, in contrast to the case of apparent shape, it might appear as though we were justified in claiming that the light was really red, i.e., red in an objective world devoid of all observers. But if now, instead of occupying a succession of various positions at rest with respect to the luminous source, we move towards or recede from the light with sufficient speed, it will change colour. Colour, when considered from an impersonal objective standpoint, is thus just as indefinite as apparent shape. Can we at least combine these various colours, as perceived by the various observers, into one common colour, of which our private perceptions would constitute but different perspectives? Needless to say, the task is quite impossible. In other words, there exists in the case of colour no parallel to the objective cone of classical science, no possibility of speaking of objective colour: colour remains private. Here, then, is a first reason for differentiating colour from real objective three-dimensional shape, or, again, secondary from primary qualities.

In the preceding discussion of the primary qualities of classical science, we have considered solely objective shape. But, in a more extended sense, we must class with the primary qualities all those which may be deemed to manifest an impersonal existence. With this more general understanding, we find that the most important of the primary qualities of classical science were given by such concepts as shape, size, duration, mass, force and electric charge.

When we consider the new views that are forced upon us by the theory [Pg 420] of 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 tensor subsisting 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.

[Pg 421]

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 existence per se in 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 from here, 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 erstwhile [Pg 422] microscopic 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 specific [Pg 423] perturbations 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 existing per se. But if this stand were adopted it should also be permissible to say that the fizziness of the Seidlitz powders already exists per se in 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 its [Pg 424] presence. 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 very [Pg 425] existence 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 quantitative [Pg 426] reduction 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 hypotheses ad hoc in 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, but from the limitations of the human mind itself. All we can ever do is to interpret in terms of , and in terms of . However far we go, we can never avoid an [Pg 427] ultimate 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 syntheses [Pg 428] of 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 seek [Pg 429] to 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 hypotheses ad 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 out a priori? The entire trend of modern science is hostile to the philosophy of the a 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 to [Pg 430] place,[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 as a priori knowledge, absolutely general and necessary, stands to-day a tottering structure. First, the physicists, in the persons of Faraday and Maxwell, proposed the ‘electromagnetic field’ 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 necessarily [Pg 431] inaccurate 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 when [Pg 432] matter 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 any [Pg 433] rate, 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 is finite (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 hypotheses ad 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.

[Pg 434]

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 the uniformity 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 confirmed a posteriori in 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. [Pg 435] 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 not sub 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 move here rather than there, 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 last [Pg 436] number 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 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 [Pg 437] 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 [Pg 438] 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 [Pg 439] 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 [Pg 440] 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 [Pg 441] 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 [Pg 442] 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.

[Pg 443]

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 [Pg 444] 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 [Pg 445] 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 [Pg 446] 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 [Pg 447] 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 [Pg 448] 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 [Pg 449] 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 [Pg 450] 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 [Pg 451] 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.

[Pg 452]

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 [Pg 453] 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 [Pg 454] 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.

[Pg 455]

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.

[Pg 456]

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.

[Pg 457]

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 [Pg 458] 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 [Pg 459] 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 [Pg 460] 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 [Pg 461] 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 [Pg 462] 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 [Pg 463] 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 [Pg 464] 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 [Pg 465] 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 [Pg 466] 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 [Pg 467] 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 [Pg 468] 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 [Pg 469] 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.

Such arguments were forthcoming when the gravitational equations were studied more closely. It was seen that they entailed those peculiar forces which Mach’s relativity of rotation compels us to attribute to the stars. Over and above these motives, Einstein was guided by a desire to avoid the conception of a gradually impoverished nuclear universe of stars. When, in addition, account was taken of the low star-velocities, he found himself driven once more to the cylindrical universe. Also, we may recall that the possibility of reducing matter to electricity and gravitation necessitates the same cylindrical universe.

When all these facts are taken into consideration, one can realise that the finite universe, entailing as it does the relativity of rotation, rests on solid ground. We do not say that Mach’s idea is vindicated, since the cylindrical universe follows from the stability of the star distribution, and this may be a mistaken premise. All that is maintained is that the complete relativity of motion is mathematically possible, a point which had never been established by Mach or by any one else. Indeed, as we have said, Einstein does not insist that we accept the cylindrical universe. He states expressly that the theory of relativity does not require it as an inevitable necessity; it merely suggests it as a possibility. If we assume that our stellar universe is concentrated into a nucleus surrounded by infinite space, the quasi-Euclidean infinite universe must take the place of the cylindrical one. With this alternative universe, matter still modifies space-time locally, but can never create it in toto. As Einstein tells us: “If the universe were quasi-Euclidean, then Mach was wholly in the wrong in his thought that inertia, as well as gravitation, depends upon a kind of mutual action between bodies.”

In short, we realise that the only means of coming to any decision on the form of the universe, hence incidentally on the problem of the relativity of rotation, must be through the medium of astronomical observation; in particular, by measuring the velocities of the stars in the Milky Way and globular clusters.

[Pg 470]

There exists also a third type of universe, de Sitter’s hyperbolical universe, which also is mathematically possible. In de Sitter’s universe, just as in the quasi-Euclidean one, matter can never create space-time in toto, and as a result Mach’s idea must be abandoned. But, here again, it is not a philosophical dislike for a matter-created universe which guides de Sitter to the hyperbolical universe; it is the restoration of absolute time in the cylindrical universe, and also the lack of invariance of the boundary conditions, which disturb him. This lack of invariance restricts the covariance of the natural laws, hence, from a purely mathematical standpoint, goes counter to the general principle of relativity. De Sitter criticises the cylindrical universe in the following words:

“It should be pointed out that this relativity of inertia is only realised by making the time practically absolute. It is true that the fundamental equations of the theory, the field equations, and the equations of motion ... remain invariant for all transformations. But only such transformations for which at infinity can be carried out without altering the values of (I)”.[154]

These discussions on the form of the universe bring to light another important aspect of the methodology of science. We may notice that the accord which was practically unanimous in the earlier part of the theory now gives way to dissenting opinions. The reason for this is of course obvious. In the earlier part, experimental verifications were soon forthcoming and permitted us to decide matters one way or another. But when we are dealing with the form of the universe, verifications are extremely difficult; they depend on tedious astronomical observations which may require centuries to be completed. Hence, the only method of advance is to explore all mathematical possibilities in the hope that somehow or other we may be led to an experimentum crucis. De Sitter places the emphasis on the mathematical principle of relativity. Einstein adopts the attitude of the physicist and prefers to stress the dynamical aspect of relativity. Weyl and Eddington adhere more closely to the pure physics of the field, following the procedure inaugurated by Maxwell and Faraday.

But in all cases the driving force behind these attempts has been a desire to co-ordinate mathematically a maximum number of experimental facts in the simplest way possible. These facts have been collected from the various realms of physical science, and, indeed, it is owing to their wide variety that the conclusions obtained are compelling. Furthermore, it should be noted that the facts appealed to are the results of highly refined experiments. This again is important, since we have seen that a difference one way or another of a tiny fraction of an angle or a length might be sufficient to overthrow an entire theory.

Summing up these discoveries, we see that a matter-created universe entailing the complete relativity of all motion to the universal masses appears to be mathematically possible. If this solution is the correct one, the cornerstone of Newton’s absolute space is finally destroyed. If, on the other hand, the cylindrical universe should [Pg 471] prove untenable, we could scarcely claim that an absolute background to motion had been disposed of. In this measure, at least, Newton’s ideas would stand.

In short, we see that this problem of the relativity of rotation, of the relationships between matter and extension, of the finiteness of the universe, is not one for philosophy and a priori arguments to solve. A mathematical co-ordination of the facts developed by ultra-precise experiment can alone yield us an answer. As Weyl tells us:

“The historical development of the problem of space teaches how difficult it is for us human beings entangled in external reality to reach a definite conclusion. A prolonged phase of mathematical development, the great expansion of geometry dating from Euclid to Riemann, the discovery of the physical facts of nature and their underlying laws from the time of Galileo together with the incessant impulses imparted by new empirical data, finally, the genius of individual great minds, Newton, Gauss, Riemann, Einstein, all these factors were necessary to set us free from the external, accidental, non-essential characteristics which would otherwise have held us captive.”

To be sure, as our knowledge accumulates and our mathematical ability increases, we may be led to revise former conclusions. This is why the conclusions of Einstein differ from those of Newton, just as those of our descendants may differ from ours of to-day. But the main point to grasp is that these variations in our philosophical outlook are brought about by our increase of knowledge both mathematical and physical; and this increase requires centuries: it cannot be obtained overnight. Indeed, had Einstein lived in Newton’s time and ignored the mathematical and physical discoveries of the present day, he could never have improved upon Newton’s solution. Inversely, had Newton lived in our time and been acquainted with the facts known to modern science, it is very possible that he would have created the theory of relativity.

In our analysis of the methodology of scientific theories we have attempted to show that the deductions of the great scientists were always based on experiment and not on wild guesses. But this does not mean that other factors have not also come into play. For instance, we mentioned that a proper choice of facts was of supreme importance. Whereas a certain choice may lead to nothing of interest, some other choice may issue in a great discovery. A case in point was afforded by Einstein’s attitude towards the well-known identity of the two masses.

But even this is not all. In mathematics, as in architecture, certain co-ordinations are beautiful, others top-heavy or unsymmetrical. When, therefore, physical phenomena are translated into mathematical formulæ, the theoretical physicist will always endeavour to obtain beautiful equations rather than awkward ones. Of course, since he must restrict himself to a slavish translation of physical results, his initiative in this respect may be extremely limited. Nevertheless, in certain cases it may be possible to add an unobtrusive term to the equations, though this term may not actually be demanded by experiment. If this [Pg 472] additional term renders the equations more beautiful, there will be every incentive to retain it. It was an urge of this sort that guided Maxwell in his discovery of the equations of electromagnetics. The experimental data known in his day could not establish whether a certain additional term was necessary or not. Nevertheless, Maxwell introduced it because it beautified his equations, and was thus led to anticipate the existence of electromagnetic waves. As we know, subsequent experiment has justified Maxwell’s deep vision. It is much the same with the theory of relativity. One of the reasons for its great appeal to mathematicians is the extreme harmony, beauty and simplicity which it permits us to bestow on many of the equations of physics.

However, if we leave these aesthetic urges aside, we may say that the methodology that has yielded us the theory of relativity is the same methodology that has yielded all the great scientific discoveries. It is the methodology of Galileo, Newton and Maxwell: First, ascertain experimental facts; then, as needs be, frame tentative hypotheses or scaffoldings for the sole purpose of co-ordinating these facts into a consistent whole with a maximum of simplicity. Only when the theory has succeeded in accumulating a sufficient number of facts can its philosophical implications be studied. It follows that any attempt to reverse the normal order and posit the philosophy before the science will result in hampering future discovery by subordinating accuracy of treatment to loose guesswork. The history of human thought is full of discarded philosophic prejudices swept away by the onward march of science. Every new scientific discovery reveals aspects of nature, or even of the mind, as in pure mathematics, which we never suspected before. To-day, both the discoveries of relativity and the quantum hypothesis are case in point. It is precisely because the philosophy of a theory of mathematical physics can only be attained a posteriori, coming as a crowning achievement, that the philosophy which is beginning to disentangle itself from Einstein’s discoveries is still in an embryonic stage. It is not that a scientific philosopher of sufficient scope could not be found; it is because there are many scientific aspects of the subject which are still in doubt. The following passage from Einstein will explain what we mean:

“From the present state of theory it looks as if the electromagnetic field as opposed to the gravitational field rests upon an entirely new formal motif, as though nature might just as well have endowed the gravitational ether [metrical field[155]] with fields of quite another type, for example with fields of a scalar potential instead of fields of the electromagnetic type.

“Since according to our present conceptions the elementary particles of matter are also in their essence nothing else than condensations of the electromagnetic field, our present view of the universe presents two realities which are completely separated from each other conceptually, although connected causally, namely, gravitational ether [metrical field] and electromagnetic field, or—as they might also be [Pg 473] called—space and matter.

“Of course it would be a great advance if we could succeed in comprehending the gravitational field and the electromagnetic field together as one unified conformation. Then, for the first time, the epoch of theoretical physics founded by Faraday and Maxwell would reach a satisfactory conclusion. The contrast between ether [metrical field] and matter would fade away and through the general theory of relativity the whole of physics would become a complete system of thought like geometry and the theory of gravitation.”

We have quoted this passage at length, because it appears to us that by reading between the lines it is possible to realise the deeper philosophical problems pertaining to our knowledge of nature, hence to knowledge in general, which still obscure the significance of relativity.

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