Part 4 — CHAPTER XXXVIII THE METHODOLOGY OF SCIENCE (continued 2)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

When we consider the new views that are forced upon us by the theory of relativity, we find that certain refinements are necessary. The special theory of relativity has shown that there was no reason to dispute the status of those qualities thought to be primary by classical science so long as we restricted our investigations to the points of view of observers at rest with respect to the object; but that when observers in relative motion were considered, it became impossible to effect a synthesis of all the apparent shapes so as to obtain one sole impersonal or real spatial shape. Real shape was thus removed from the status of a primary quality, and there was no sense in speaking of the shape of an object in a world devoid of all observers. Similar conclusions applied to size, duration and all the other primary qualities mentioned, with the sole exception of electric charge, which retained its impersonal status. Of course, had the theory of relativity deprived us of the possibility of conceiving of a sufficient number of primary qualities, hence of an objective world, it would have led us to a stone wall. But we know that such is not the case, and that the theory has merely modified the list of the primary qualities of classical science, assigning the same fundamental rôle to others in their stead. These are given by the space-time intervals and, more generally, by space-time configurations and tensors. In other words, the objective universe disclosed by relativity is no longer one of shape and duration, or space and time, but one of space-time.

Yet, regardless of the actual list of qualities that are to be classed as primary, the significance of these qualities remains the same in either science in that they are deemed to represent entities to which an impersonal existence may be conceded, if for none other than methodological reasons.

Sufficient information has been given in this book to enable the reader to understand the similarity which exists between the comparative status of the objective and the private view, on the one hand, and between tensors and tensor components, on the other. Thus, to take an illustration from electromagnetics, we know that what our instruments detect at a given point in a given frame of reference are the electric- and magnetic-field intensities. When the frame is changed, the measured values of these quantities are modified. Hence here we have the analogue of the various apparent shapes of the cone. And just as the apparent shapes of the cone could be attributed to an objective or real three-dimensional cone (in classical science) or to a space-time configuration (in relativity), so now, in the theory of relativity, the electric and magnetic intensities can be regarded as the six variable components of a real space-time tensor, the electromagnetic-field tensor subsisting in the objective space-time world. When we change frames, the tensor remains unaffected in the objective space-time world, but its six components in the space-and-time frame we happen to occupy are subjected to definite variations when our frame is changed; and for this reason fields that appear purely electrical when we are moving with a certain relative velocity will appear electromagnetic when our relative velocity is modified.

The general classification of objective, absolute or public realities, on the one hand, and of relative or private ones, on the other, might yet appear to present none but a methodological interest and to have no bearing on the problem of the genuinely real. But, assuming for argument’s sake, that the concept of the genuinely real has a meaning, it can scarcely be denied that if we ever hope to discover this genuine reality, it is assuredly more reasonable to seek it in those realities which we have cause to regard as public rather than in those which remain essentially private. For, as we have seen, a private reality, such as colour or apparent shape (or three-dimensional spatial shape in relativity), is indefinite until such time as the conditions of observation are specified; hence it is not self-supporting, but entails the mention of a relationship to something else. On the other hand, the generalised primary qualities transcend this specification of relationships, that are always of a contingent nature. Hence, quite aside from considerations of a methodological order based on convenience, there appears to be every reason to assume that the generalised primary qualities approximate more nearly than the secondary ones to the unattainable ideal of the genuinely real, or the metaphysical reality.

We have now to consider a further type of difficulty which would confront the metaphysician who claims colour to be a reality having an existence per se in a world devoid of observers. In the preceding pages, when discussing the relativity of colour to the percipient, we have dwelt on the relativity of colour to motion, embodied in the Doppler effect. But it is well known that a distinction between primary and secondary qualities had imposed itself long before the Doppler effect was discovered. One of the reasons responsible for this differentiation is to be found in the fact that colour, in many cases, is also relative to position. Berkeley gave an illustration of this type of relativity when he wrote of a cloud which, though it looks crimson from here, is a dark mist when we penetrate its interior. But his illustration was a poor one, for when we enter the cloud we are no longer viewing its surface. It would not be difficult to improve upon Berkeley’s example, but for our present purpose it will not be necessary to dwell further on this type of relativity, so we shall examine another species of relativity, namely, relativity to gauge.

Suppose, then, that a green object is placed on the table. The metaphysician who states that colours are realities which would exist in a world devoid of percipients would presumably state that the object’s surface was green in its very essence. But it might well happen that, on viewing the surface with a powerful microscope, we should find it to be made of a patchwork of blue and yellow spots and that all appearance of green had vanished. Unless the metaphysician proceeds to ignore the microscopic view entirely, he will be placed in the dilemma of wondering whether the green quality or the yellow-and-blue one is to be ascribed to the object in the real world. Even so, his troubles would not be at an end, for we might conceive of an ultra-microscopic vision compared with which our erstwhile microscopic vision would be macroscopic, and so on indefinitely. Now the change in colour that accompanies a passage from the macroscopic to the microscopic suggests that changes of equal importance might ensue from each successive passage to the following microscopic view. Indeed, it might be that for organisms whose size was of the order of wave-lengths no colour would manifest itself at all. Inasmuch as it would be highly arbitrary to assert that the ultra-microscopic vision is in any wise less worthy of consideration than the more usual macroscopic one, it appears to be impossible to ascribe definite colours to existents in a world devoid of all observers.

It is true that in the present discussion we started from an example where the microscope enabled us to detect a difference in colouring; and cases might arise where even under the microscope the object would still appear green. But, as we said before, the microscopic vision is itself ultra-macroscopic when contrasted with the infinite regress of ultra-microscopic visions we can conceive of; hence, even in this case, the difficulty would be only postponed, not obviated. After all, our ability to pass from microscopic to macroscopic views is contained within comparatively narrow limits. The microscope brings us to the limits of microscopic vision; yet it does not even enable us to see some of the tinier micro-organisms. As for the ultra-microscope, it yields but a blur. Turning to the ultra-macroscopic vision, it appears to be best exemplified when we view the Milky Way; but this vision in turn would be microscopic as contrasted with others of which we might conceive. The upshot of this discussion is that when the metaphysician speaks of red as inhering in the flower of a geranium, he is making a statement that cannot withstand scientific criticism.

Contrast these ambiguities and difficulties with the primary quality of shape in classical science (not in relativity). For in classical science, if a coin be recognised as round as a result of a synthesis of certain microscopic points of view, it will remain round when precipients situated in any position, moving with any velocity, and observing it with any macroscopic vision co-ordinate their impressions. Owing to this comparative irrelevance of the percipient, there appears to be a certain justification (in classical science) for speaking of a round coin existing in a world devoid of percipients, whereas there is none for speaking in the same sense of the brown colour of the coin.

The next stage we have to consider deals with the reduction of secondary to primary qualities. Whereas, up to the present, our arguments based on the relativity of perceptions applied solely to such secondary qualities as colour and sound, we shall now be in a position to advance arguments holding for the remaining secondary qualities, such as heat and smell.

We said, when discussing the red light, that it was impossible to effect a synthesis of all the colours perceived by the variously moving observers and obtain therefrom one definite, impersonal colour. Nevertheless, we may succeed in obtaining objectivity if we find it possible to regard sensations of colour as due to certain specific perturbations suffered by the retina as a result of the impact of electromagnetic waves. According to the relative motion of the observers, the objectively real electromagnetic waves will impinge upon the retina with one frequency or another, and corresponding sensations of colour will ensue. Colour would thus manifest itself as the result of a relationship.

The realistic metaphysician might urge that this may be a convenient way of accounting for colour, but that it is artificial and that, in spite of all, colour should be considered a reality existing per se. But if this stand were adopted it should also be permissible to say that the fizziness of the Seidlitz powders already exists per se in the powders, and results in no wise from a relationship, i.e., from the chemical interaction of the two powders. Or, to take Galileo’s illustration, we should say that the tickling quality must be in the feather, since a feather passed over our skin may tickle us.

The metaphysician’s stand appears untenable, for it is impossible to deny the existence of relationships giving rise to effects which were non-existent before the relationships were established, and which often differ appreciably from their generating causes. And every time we sense a colour or a sound or a smell, a relationship is established between some objective entity and that particular sense-organ which is affected. The fact is that the human organism is a highly complex piece of workmanship and that as a rule it reacts differently (though sometimes in the same way) to the various perturbating influences; while at other times, so far as we know, it does not react at all and no sensations are registered. To speak of colour and sound as existing independently of the organism is thus as nonsensical as to discuss a one-sided triangle.

To take an example at random, let us consider a problem of heat. The sun emits rays which are commonly supposed to be luminous and hot. If, then, as the metaphysician asserts, hotness is inherent in the rays, we should expect the space between the earth and the sun to be flooded with heat, hence warm. But such is not the case. The temperature of interplanetary space is in the neighbourhood of absolute zero. On the other hand, the atmosphere becomes warm under the sun’s rays. Why this difference? Because the energy of the electromagnetic waves is transformed into molecular agitation when the waves impinge on matter, just as a spark is produced when two flints are struck together. The more rarefied the matter on which the waves impinge, the smaller will be the percentage of the electromagnetic energy transformed or degraded into molecular agitation. It is partly for this reason (owing to the progressive rarefaction of the atmosphere) that the temperature gradually falls as we rise to high altitudes in an airplane. Now science claims that on a hot day, when standing in the shade, we experience a sensation of heat, our sensation results from an interaction between the extremities of our sensory nerves and the impacts of the molecules constituting the atmosphere. The metaphysician will presumably balk at this statement, claiming that the heat we sense exists independently of us in the air and that we merely detect its presence. But if his contention were accepted, we should also have to assume that the heat existed in the sun’s rays and had been transferred directly from them to the air. And we have shown that this view is untenable, since the light rays produce heat merely as a result of a transformation of their energy into one of another type (molecular agitation). And since it appears impossible to maintain that the heat was already in the rays prior to their encounter with matter, what justification is there for asserting that the heat was already in the air prior to its encounter with our sensory organs? In every case we are witnessing transformations of energy, and these transformations are always accompanied by degradation (except in the ideally reversible transformations). This fact is expressed by the law of entropy, according to which, though the energy is conserved in quantity, its potentiality, or quality, is lowered, i.e., degraded.

Now, to revert to the problem of colour, it is possible that when a light-wave impinges on our retina some chemical change is produced (as in the case of the photographic plate, where the molecule of chloride of silver is loosened); possibly, too, a liberation of electrons takes place by a photo-electric effect. At all events, it is more than likely, on the strength of other phenomena of which we know something definite, that by the time the energy of a supposedly red electromagnetic vibration has reached our brains it has been subjected to all sorts of transformations. Even if we feel justified in saying that redness hits our eye, what right have we to maintain that, contrary to all the known laws of physics, this redness is perceived by our consciousness without any transformation having taken place? And if we admit that physics is not a complete hoax, how do we propose to assert that the redness we sense is already present as redness in the electromagnetic radiations?

The metaphysician’s attitude will lead to still further difficulties when a number of other examples are considered. In the first place, science has discovered that the electromagnetic vibrations accompanied by colour constitute an insignificant minority. There exist electromagnetic vibrations of varying frequencies identical in all respects with the luminous ones, except that they do not happen to influence the human eye. If we adhere strictly to the ultra-realistic viewpoint, we shall have to assume that these invisible radiations contain no properties of colour. And yet, as we know, though invisible to the human eye, they are detected by the photographic plate and, in all probability, by a number of animals. Were the photographic plate possessed of consciousness, or, again, were our eye provided with a photographic plate in place of the human retina and optic nerve, etc., the colour red would be unknown; whereas a new colour, ultra-violet, would be sensed with ease.

It thus becomes impossible to escape the conclusion that our precise perception of colour is affected by the idiosyncrasies of the eye and organism. We need not dwell on a number of well-known illustrations, such as the effect produced by certain drugs. If, therefore, when defending the reality of colour, the metaphysician is referring to those colours which the human eye can detect, he is making the very existence of his realities dependent on the accidental circumstance that the human eye is sensitive to one particular range of vibrations and to no other. He can emancipate himself from this difficulty only by assuming that all electromagnetic waves possess colour, whether visible or invisible. This would lead him to ascribe colour to radio waves. But why stop even there? Why not attribute colour, sound and smell to everything while we are about it, and talk of the colour of sound waves and the sound or smell of electromagnetic ones? We should thus be led into complete intellectual anarchy. Then again, consider interference phenomena. Red light plus red light yields obscurity. If we consider redness as an objective reality, we must at least concede that positive redness and negative redness must exist, so that a mutual destruction becomes possible. In similar fashion, there would be positive and negative sound, and all kinds of other absurd notions.

That all these points should have appeared controversial in the days of the Greeks is only natural, since they knew very little of the transformations of energy, and nothing of the invisible radiations or of the Doppler effect or of interference bands, etc. But nowadays the ultra-realistic standpoint is quite untenable for the scientist. The reader acquainted with scientific thought may feel that this long discussion was quite unnecessary. Such would also have been our own opinion had it not been for the profusion of contemporary philosophical literature purporting to defend the reality of secondary qualities. Were it not for a few scientific words interspersed here and there in their writings, one would have no difficulty in believing that these productions were those of men who had lived more than two thousand years ago.

Another type of objection to the method of quantitative reduction is that a philosophy of nature must be based on a consideration of qualities as well as of quantities. But the critics overlook the fact that by postulating too many conditions they may be rendering a consistent philosophy quite impossible. The aim of a philosophy of nature should be to increase our understanding of nature. Vagueness has ever been an obstacle to knowledge; it prevents us from differentiating between concepts which present but a superficial similarity, and obscures the interconnection of phenomena. For this reason, science has had to reduce qualitative differences whenever possible, since it was they that interfered with the possibility of obtaining any precise co-ordination.

In a similar spirit, science has ever endeavoured to eradicate such statements as “nature abhors a vacuum” or “chemical elements possess various affinities.” Statements of this sort, just like the Aristotelian qualities, are worthless, for they lead nowhere. To clothe our ignorance in the guise of a phrase may flatter our vanity, but it does not advance us one iota. In short, although the scientist may not have taught us much about colour and light, he has at least increased our powers of prevision; and that is no mean achievement. Of course prevision is not everything. The enjoyment of the present is also of interest; and it might always be claimed that the quantitative reduction had eliminated beauty and noble aspirations from the world. However, if this were the case, why not add that it has also eliminated filth and misery? But such considerations are quite beside the point. There is no reason why a physicist should not derive just as much pleasure from the beauties of music as a man ignorant of the laws of acoustics. Hence, we must make up our minds once and for all and decide whether in our quest for knowledge we intend to be guided purely and simply by our feelings, or whether we intend to seek our inspiration in the abstract intellectual ideal of unity and simplicity with which theoretical science dwells.

Let us now return to a more scientific subject. We have seen that a theory of mathematical physics reduces to a rational co-ordination of a large number of facts and phenomena. The interconnections between these phenomena permit us to pass by a continuous chain of reasoning from one to another, and enable us thereby to anticipate the future or the past from a knowledge of the present. We have also seen that various syntheses of facts were possible; but that, in general, one particular synthesis stood out prominently, owing to its greater simplicity and to the smaller number of auxiliary hypotheses it contained. Thanks to the absence of disconnected hypotheses ad hoc in this simplest synthesis, we are able to find our way about without being bewildered at each turn by gaps and hiatuses. Owing to our ability to foresee and to foretell the future activities of nature, we feel that we have succeeded in unravelling her hidden scheme and in understanding her guiding motives. To this extent we consider ourselves justified in assuming that the knowledge we have uncovered or created corresponds to reality, although, of course, the reality contemplated remains essentially pragmatic. This pragmatic conception of reality has been stressed repeatedly by Poincaré. But it is by no means peculiar to the great mathematician and philosopher. We find it in the teachings of Euler and Riemann, and of all the great scientists of the last two centuries; in short, it is the philosophy of science.[140]

Obviously, there is nothing in the nature of an explanation in a scientific theory. Phenomena are not explained; they are merely interconnected, or described in terms of their mutual relations. As a matter of fact, there is no cause to be surprised at this failure of science to explain phenomena, for the failure arises not from the limitations of science, but from the limitations of the human mind itself. All we can ever do is to interpret in terms of , and in terms of . However far we go, we can never avoid an ultimate unknowable. This perfectly obvious contention is sometimes obscured because the end of the series may be so far removed as to escape our attention; but this is merely a mental subterfuge, a form of mental laziness.

Thus, the Greeks wondered why the earth did not fall. An explanation was soon forthcoming. The giant Atlas holds it on his shoulders. But what, then, prevented the giant Atlas from falling together with the earth he was carrying? The nether regions on which he stands give him support. This is as far as the explanation went, for no theory seems to have been advanced explaining why the nether regions did not fall, together with Atlas and the earth.

We might consider more scientific examples; but we should find that the case was always the same. is described in terms of , in terms of , and so on. Consider the phenomenon of gravitation. Does any one really imagine that Newton or Einstein has ever attempted to explain gravitation? To say that gravitation is a property of matter or a property of space-time in the neighbourhood of matter is just as much of an explanation as to say that sweetness is a property of sugar; for in the last analysis, what is matter—what is space-time? If we say that matter is an aggregate of molecules, atoms, electrons, protons, what of it? What are electrons? What are protons? We can only confess our complete ignorance and, while attempting to reduce the number of these unknown fundamental entities to a minimum, content ourselves with describing the properties which appear to characterise them and the relationships that appear to connect them. Clearly, those who seek explanations will find no comfort in science. They must turn to metaphysics.

And yet, as a matter of fact, these rather gloomy conclusions are gloomy only because we are expecting too much. If we content ourselves with what we can obtain, we shall find that the descriptions of science are creative and fertile, and not sterile, as descriptions usually are. As we have already explained, a scientific description is creative in that it allows us to foresee and to foretell. A description that could not confer this power on us would be of little use. Thus, we might describe the petals and stamens of a flower, describe its colour, odour and general appearance. A description of this sort would be useful in enabling us to recognise the same plant when we came across it again. But apart from this, it would teach us nothing new. It would not tell us, for instance, whether the root was bulbous or fibrous. On the other hand, if as a result of prolonged botanical study we could establish certain recurrent relationships between the characters of flowers and the other characteristics of the plants, we might be able by a mere examination of the flower to anticipate the nature of the root. In palæontology it is the same. The discovery of a fossilised bone has often been sufficient to enable scientists to reconstruct the entire skeleton of the unknown animal. It is in this respect that a scientific description is creative and differs from the ordinary type of description which we encounter in everyday life.

Now we have already attempted to explain that among all the syntheses of experimental facts known to science some stood out prominently, owing to their extreme simplicity. In these privileged syntheses simple relationships were discovered, and the expression of these permanent relationships constituted the laws of nature. If the experimental facts we are seeking to co-ordinate are few in number, and if the relationships between phenomena, as established by experiment, are more or less crude and uncertain, we may be led to a certain definite synthesis which will appear to be the simplest. But if ultra-refined experiment leads us to more precise relationships at variance with our former crude ones, or, again, if a wider array of facts has become known, it may well happen that our original synthesis will be incapable of co-ordinating all these results unless we appeal to a number of artificial corrective hypotheses. In this case we may be compelled to widen our synthesis or even to reconstruct it entirely. We must realise, therefore, that a synthesis can never be considered final, for we never know what the future may hold in store.

It is the same with our hypotheses and theories. Theories which would appear plausible when a small number of facts are appealed to, often become untenable when a greater number are taken into account. Consider, for instance, Lesage’s theory of gravitation. Lesage had suggested that gravitation might be accounted for by assuming corpuscles to be rushing hither and thither with enormous speeds in all directions through space. Under this hypothesis two bodies, say the sun and earth, would screen each other mutually from the impact of some of the corpuscles. The residual action caused by the impact of the other corpuscles would then be to press the sun and earth together; in this way gravitation would be accounted for mechanically. Needless to say, Lesage’s hypothesis was nothing but a wild guess, but, even so, it was felt that there might still be some truth in it.

The astronomer Darwin attacked the problem and proved that gravitation might be accounted for in this way, not merely qualitatively (for this would have been quite insufficient), but quantitatively as well, provided the corpuscles possessed no elasticity whatever, hence did not rebound after impact. Thus far, then, everything was in order, and the theory was acceptable. But now let us take into consideration a few additional facts. Gravitation, as we know, exerts its action even when bodies are interposed. Hence we must assume that only an insignificant percentage of the corpuscles would be arrested by the earth’s surface. Now, when account is taken of the sizes of the molecules of matter constituting the earth, and of the vacant spaces which separate them, certain limitations are imposed on the size, mass and velocity of the corpuscles. When, furthermore, we realise that the earth moves round the sun without slowing down in any perceptible way, as though its motion encountered no friction, we are compelled to conclude that the density of the gas formed by the corpuscles cannot surpass a certain limit. When all these facts are taken into account, we find that Lesage’s theory cannot be countenanced, for calculation proves that under the impact of the corpuscles, the earth would become white-hot in a very short time. The same difficulty would arise were we to seek to save the hypothesis by substituting for the impacts of material corpuscles the pressure of invisible electromagnetic radiations (possibly of the nature of the Millikan rays). Rather than introduce further hypotheses ad hoc, for the sole purpose of saving Lesage’s theory, Lorentz preferred to abandon it.

We see, then, that it is owing to the incessant accumulation of new facts that scientific theories are constantly submitted to change and revision. Until the advent of Einstein’s theory, however, although theories of mathematical physics had to be subjected to incessant modification, our basic concepts of space, time and matter had remained unaffected. Indeed, there was no reason to deviate from the original synthesis in this respect. Only when ultra-precise experiments were performed was it found impossible to retain the classical spatio-temporal foundation. As a result, Einstein was compelled to construct a totally new synthesis built on an entirely different foundation, that of space-time. Had it not been for our knowledge of ultra-precise experiments, there would of course have been no reason to modify the essential characteristics of the classical synthesis. From this we see that the synthesis which we adopt depends on the extent of our knowledge, or, what is more to the point, on the extent of our ignorance of nature.

There is very little similarity between the erection of a building and the creation of a scientific theory. When a building is erected, we know beforehand the type of structure we propose to erect, and we can lay our foundations accordingly. But in an empirical science such as physics, the structure is never completed. The superstructures are subject to incessant change as our experiments become more precise and as new phenomena are revealed. The result is that we can never lay our foundations once and for all with perfect security. At all times we must be prepared to abandon our theory, remodel the foundations and erect a totally new structure.

It would appear to be unnecessary to stress these points further, for they must appear obvious to those who are at all familiar with the achievements of science. Certain metaphysicians, however, hold that similar reconstructions should never endanger the traditional concepts of space and time. But an attitude of this sort is equivalent to removing the study of space and time from the control of experience; for all concepts which we approach empirically are necessarily subject to change as our experimentation increases in precision. Furthermore, if we adopt this attitude, why not go the limit and assert that the discoveries of the laboratory can be figured out a priori? The entire trend of modern science is hostile to the philosophy of the a priori. Non-Euclidean geometry undermined its foundations a century ago, and to-day the physical theory of relativity has completed the work. To assert in the name of logic or of reason, as so many philosophers have done, that space and time must be separate,[141] or that space cannot manifest variations of curvature from place to place,[142] or that space-time must always be flat,[143] or that matter must be a substance which is conserved, is to impose narrow philosophical doctrines which are likely to cramp the entire synthesis, rendering a simple scheme of natural relations impossible. To quote Weyl:

“Matter was imagined to be a substance involved in every change, and it was thought that every piece of matter could be measured as a quantity, and that its characteristic expression as a ‘substance’ was the Law of Conservation of Matter which asserts that matter remains constant in amount throughout every change. This, which has hitherto represented our knowledge of space and matter, and which was in many quarters claimed by philosophers as a priori knowledge, absolutely general and necessary, stands to-day a tottering structure. First, the physicists, in the persons of Faraday and Maxwell, proposed the ‘electromagnetic field’ in contradistinction to matter, as a reality of a different category. Then, during the last century, the mathematician, following a different line of thought, secretly undermined belief in the evidence of Euclidean geometry. And now, in our time, there has been unloosed a cataclysm which has swept away space, time and matter, hitherto regarded as the firmest pillars of natural science, but only to make place for a view of things of wider scope, and entailing a deeper vision.”

Summarising, we may say that in natural science it is essential to beware of fundamental premises which are claimed to be forced upon us by the requirements of logic or of crude perception.

Yet, on the other hand, it must be admitted that when we have succeeded in constructing some lofty scientific synthesis which takes in the facts of experience, our synthesis must always be such as to permit us to account for the facts of crude observation; for these, to the same extent as the results of ultra-precise experiment, constitute facts which must be co-ordinated. A theory such as Einstein’s must not go counter to the apparent separateness of space and time when phenomena are investigated only in the usual crude way by commonplace observation. In other words, the scientific synthesis must reduce to the primitive synthesis when too great accuracy is not imposed, so that the primitive synthesis must appear in the light of a first approximation. Einstein’s theory satisfies this general condition in every respect. As we have seen, for low velocities or for crude observation the classical separateness of space and time reasserts itself; hence no conflict exists between the relativity theory and commonplace observation. Similar conclusions apply to the classical addition of velocities. This also appears to be verified to a first approximation.

Now it might be contended that since all finality seems to be denied a scientific theory, it would be better to cease theorising and content ourselves with a bare accumulation of facts. But an attitude of this sort would be untenable, for scientific theories, though necessarily inaccurate from a long-range point of view, are nevertheless inevitable if knowledge is to exist. A bare accumulation of disconnected facts would not constitute a science, precisely because, the facts being disconnected, they would afford us no means of foretelling the future sequence of events. Under the circumstances, it is better to be guided by a faulty synthesis which we can improve upon as experience demands, rather than be guided by nothing at all. Furthermore, co-ordinations and theories which have since turned out to be patently inaccurate, have yet been successful in leading to the discovery of new facts.

Thus, Newton’s law, to-day recognised as inaccurate, brought about the discovery of Neptune; it permitted astronomers to assert that it was the same comet which reappeared at definite intervals. Again, it was thanks to Newton’s law that Römer, in 1675, discovered the finite velocity of light, thereby exploding the belief that the existence of an event and our perception of it were simultaneous occurrences. All these discoveries would have been arrived at by other means at some later date; yet it must be agreed that, owing to the law of gravitation, they were considerably precipitated. Indeed, it was not until the nineteenth century that the finite velocity of light was established by direct terrestrial measurement.

In other cases, a scientific theory, whether true or false, directs physicists towards crucial experiments of which they might never have thought. Eddington’s astronomical expedition was prompted by Einstein’s discoveries; had the theory been unknown, it is probable that the bending of light, when finally observed, would have been regarded as an error of observation or as due to some casual disturbing influence. Under the circumstances it might never have become a permanent scientific fact; and it is the same with the Einstein shift-effect. Regardless, then, of whether Einstein’s theory will endure or be scrapped to-morrow, it has led at least, to the discovery of new facts.

Furthermore, we must remember that each successive theory retains certain elements of the one it has supplanted. It rises to a higher plane, but it rises from a rung that has been set in position by its predecessor. It is not in the physical hypotheses that we shall find these elements of permanence, but in the equations themselves; for it is these that constitute the relations of reality. The physical hypotheses, as originally formulated, are but convenient scaffoldings of a more or less temporary nature. This is especially noticeable in the successive theories of electricity and magnetism. From the initial theories of Coulomb and Ampère to the present ones of Larmor and Lorentz, as modified by Einstein’s discoveries, we pass through the intermediary ones of Maxwell, Helmholtz and Hertz. We may follow the equations through these transformations and see how they were supplemented by the adjunction of additional terms or by the refinement of those already present. Thus, Maxwell’s equations differ from Ampère’s by the adjunction of a new term. Lorentz accepts Maxwell’s equation of the free ether without modification; it is only when matter is present that refinements are superadded. Again, Einstein’s gravitational equations degenerate into Newton’s when the field is weak and the velocities are small.

When we realise that all these successive refinements in the theories were brought about by a desire to co-ordinate an ever-increasing number of facts, it would appear that the scientist has very little control over the course his theory will follow, since in every case it is the criterion of simplicity of co-ordination which guides him.

However, it may be of interest to examine whether, in addition to these formal considerations, there may not exist other reasons which prompt scientists towards their solutions. It might be claimed that the various thinkers start from different initial philosophical presuppositions, and that these presuppositions, rather than the number of facts co-ordinated, are responsible for the differences in the nature of the theories and conclusions. Opinions of this sort have been expressed by numerous philosophers who have devoted their attention to what they call the “metaphysics of science”; but we may hasten to say that the opinion appears to be entirely unwarranted. It is solely the criterion of simplicity of co-ordination which decides on the orientation of science. Where the personal idiosyncrasies of the thinker may come into play will be in the directing of his initial attempts of co-ordination along certain lines rather than along others; but if a simple co-ordination is found to be impossible along the lines originally chosen, other attempts in totally different directions will be undertaken in due course. The final result may issue in a co-ordination presenting a philosophical import totally different from the one preferred originally; and this change is due to the facts of the case and not to a change in the philosophic fancies. Of course, it is not easy for the layman to realise this state of affairs, for scientists do not usually write a diary, mentioning all their abandoned attempts. When a theory becomes widely discussed, it is generally presented in its final form, and the numerous hesitations and partial attempts that may have extended over a period of years are lost sight of entirely. And yet, if we revert to the original scientific papers, these hesitations and changes of attitude are apparent.

It would be tedious to give illustrations; they abound everywhere. However, we may quote a few paragraphs of Einstein’s from his paper, “Cosmological Considerations on the General Theory of Relativity.” We shall see that he hesitated a long time between a number of alternatives before even considering the finite universe. We read:

“In the present paragraph I shall conduct the reader over the road that I have myself travelled, rather a rough and winding road, because otherwise I cannot hope he will take much interest in the result at the end of the journey. At this stage, with the kind assistance of the mathematician J. Grommer, I investigated centrally symmetrical gravitational fields, degenerating at infinity in the way mentioned. But here it proved that for the system of the fixed stars no boundary conditions of the kind can come into question at all, as was also rightly emphasised by the astronomer de Sitter recently.... At any rate, our calculations have convinced me that such conditions of degeneration for the ’s in spatial infinity may not be postulated.

“After the failure of this attempt, two possibilities next present themselves.[144] ... From what has been said it will be seen that I have not succeeded in formulating boundary conditions for spatial infinity. Nevertheless, there is still a possible way out, without resigning as suggested under (b). For if it were possible to regard the universe as a continuum which is finite (closed) with respect to its spatial dimensions, we should have no need at all for any such boundary conditions.”

And it is the same in every other case. A number of attempts are made to co-ordinate the facts, and the simplest solution is the one which is accepted by the scientific world, until such time as some newly discovered experimental result conflicts with it.

Of course, all these statements would suffer from an incurable vagueness, were any argument to develop as to what constitutes simplicity. But whatever the reason may be, we find that the agreement among scientists is striking in this respect. There have never been two highly developed theories co-ordinating the same number of facts, in which by common consent one was not recognised as vastly simpler than the other; just as a mathematical expression containing one term is simpler than one containing two.

A case in point would be afforded by a comparison of Einstein’s and Newton’s co-ordination of gravitation and mechanics. If we are concerned solely with the facts of planetary motion and mechanics known to Newton, then Newton’s laws of mechanics and his law of gravitation are by far the simpler. But if we supplement the facts known to him with those more recently discovered, Einstein’s synthesis has the advantage, for Newton’s co-ordination would necessitate a number of additional hypotheses ad hoc. In the same way, so far as our everyday experience is concerned, a non-rotating earth is the simpler solution, whereas, when the planetary motions are taken into consideration, the reverse holds true. It may happen, of course, that the simpler synthesis will conflict with certain philosophical prejudices, but objections of this sort will always give way eventually before the higher criterion of simplicity. Indeed, we know that men finally accepted Galileo’s stand.

If, then, we may assume that the criterion of simplicity presents no further difficulty, the problem reduces to determining whether or not the theoretical physicist introduces presuppositions and assumptions uncritically. We are here concerned solely with the physicist and theoretical physicist, not with the pure mathematician, whose procedure is entirely different. In order to clear up these points, we shall give a rapid survey of some of the basic presuppositions that are common to physicists. Although we fear that it will be impossible to avoid a certain amount of repetition, yet the importance of the subject is such that a mere matter of elegance of presentation will have to be sacrificed.

The purpose of science is, as we have said, to co-ordinate facts and to obtain thereby a common knowledge which will hold for all men. But our co-ordinations, deductions and inductions must be conducted according to rigid rules which all men will recognise. In the absence of such common rules, two men starting from the same premises would arrive at different conclusions; as a result, a common knowledge, and therefore a science, would be impossible. We shall assume that the rules of logical reasoning will serve our purpose in this respect.

Here, of course, it might be argued that we are unwittingly introducing an assumption when we state that beings who neglected to conform to the same rules of reasoning would fail to agree. Of course, if we adopt so hypercritical an attitude, there is nothing left to argue about. The instant we start to talk we are assuming from past experience that sounds will be emitted and that our words will convey at least some meaning to our neighbour. It is the same with the rules of logic. Regardless of what the ontological status of these rules may be, we may assume that in view of past experience it is convenient for men to abide by them if they wish to construct a common science.

Let us consider now what scientists mean by the uniformity of nature. The uniformity of nature may be interpreted in various ways. It may mean that nature is regulated by universal laws. In this form, it is obviously not a presupposition adopted uncritically. The primitive man does not feel the necessity for it; he prefers to believe in miracles. If the modern physicist accepts this uniformity it is because it has been confirmed a posteriori in so large a number of cases that it seems simpler to accept it than to reject it, at least until further notice. Furthermore, it is an assumption which is perfectly well recognised as of limited validity, and scientists are the first to suspect that after a more microscopic survey of nature it may prove untenable. Nature may not be rational and our logic may be unable to cope with it. If this be the case, statistical knowledge may be the only type possible, though even statistical knowledge may fail us.

We can also understand the uniformity of nature in a different sense. We may wish to imply that an experiment attempted on a Monday would yield the same result if repeated on a Tuesday. In other words, the passage of time, though affecting us in a number of ways (aging, etc.), can be disregarded in certain important cases. Suppose, for the sake of argument, that we were to dispute the legitimacy of this assumption; what would be the result? Obviously science would become impossible. A geometrical proof holding in Euclid’s time would no longer hold to-day; knowledge would be intransmissible. There would be no object in performing an experiment to-day, since to-morrow our conclusions would have lost all value. In fact, it would be futile to attempt any experiment at all, since every experiment is spread over a certain time. In a world of this sort, prevision would be quite impossible; a rule of action would be out of the question. Inasmuch as one of the most important functions of science is to serve us with a rule of action, we can realise that science would become unthinkable. This brings out an important difference between science and history, for example. History tells us of events past and gone which, for all we care, may never reoccur; but in science a phenomenon which could never occur again would be of little interest. In short, we must agree that the very existence of a common science proves that our assumption of the uniformity of nature, whether it be justified or not sub specie œternitatis, is justified at least in that limited field of experience which is ours. On no account, then, must our belief be confused with a presupposition adopted uncritically.

As a matter of fact, even were we assured that the uniformity of nature was a mistaken belief, we should continue to be governed by it just the same. We know perfectly well that the uniformity of our stream of terrestrial life does not hold; we know that though we have risen every day, yet at any instant we may die. But unless we wish to exist in a state of complete stagnation, we must act as though our life would continue, and so we plan for the future. Under the circumstances, it appears futile to seek any rational justification for the uniformity of nature, basing our arguments on probability considerations. The fact that things have existed until to-day does not prove they will exist to-morrow; but it suggests that we act as though they would.

Among the principles, special mention must be made of the “principle of sufficient reason.” In physical science, the application of this principle is subject to all manner of difficulties. For instance, when we assert that there is no reason for a body to move here rather than there, we are assuming that we possess at least a general knowledge of all the possible influences which happen to be present.

Obviously, our assumption is apt to lead us to erroneous conclusions. We mentioned a case in point when discussing the problem of space. Assuming space to be perfectly empty, the principle of sufficient reason suggested that it should be homogeneous and isotropic; but as we were by no means certain that physical space was truly empty, the application of the principle was of no great value.

Difficulties of this sort are by no means restricted to problems of physics. Even in pure mathematics it is not always easy to decide in what way the principle should be applied. The mathematical problems in which we are compelled to appeal to sufficient reason are those dealing with probabilities; and the following elementary illustration, discussed by Bertrand, will afford us a sample of the type of difficulties which we may encounter. Suppose, for instance, we are asked: “What is the probability that on tracing the straight line at random, the line-segment contained within a given circle should be greater than the side of the equilateral triangle inscribed in the said circle?”

Now, if we conceive of all possible straight lines intersecting the circle, sufficient reason tells us that any one of these lines is as likely to be traced as any other. Hence, the probability we are seeking will be given by the ratio of the number of different lines that satisfy our stipulated condition, to the total number of different lines that can be drawn to intersect the circle. Clearly, this last number will be given by considering all the lines passing through a given point of the circumference and then conceiving this point as occupying in succession every position on the said circumference. Under these conditions the probability turns out to be ⅓. But we might, with equal justification, have proceeded in another way. We might have considered all the straight lines parallel to a given direction and then conceived of this direction as suffering in succession every possible orientation. In this case, the probability would work out as ½. This example warns us that a large measure of arbitrariness lies concealed even in the most elementary problems where sufficient reason is appealed to.