Part 4 — 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. 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 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 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 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 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.” 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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.

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 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 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 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 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 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 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 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 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 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.