Although we may be repeating ourselves unnecessarily, we must again draw attention to the fact that this knowledge of molecules, which is obtained through a co-ordination of experimental results, is in all respects (other than in degree of certainty) of the same type as our knowledge of the spherical shape of the sun or of our knowledge of space and of the table. On the other hand, it is essentially different from our awareness of feeling hot or tired.
Finally, let us consider a last example, namely, our belief that light is atomic. In this case our knowledge is still more uncertain. The justification for a belief in quanta was arrived at by Planck (as we have explained elsewhere), and arose from the peculiar phenomenon of black-body radiation. The mathematical treatment of problems of this sort is based on the calculus of probabilities. When calculations were conducted on the assumption of the continuous emission of light, we obtainedRayleigh’s law of radiation, which was refuted by facts. Planck noticed that by substituting discontinuous probabilities for continuous ones, the results of observation would be anticipated with great precision. When, a few years later, Nernst and then Einstein showed that the extension of these same discontinuities to the energy of molecular motions would account for the curious anomalies and variations of the specific heats of gases and solids, and when in addition Einstein succeeded in accounting with high precision for the photo-electric effect, Planck’s ideas appeared to gain in probability. The belief was enhanced still further when Bohr presented science with his model of the quantum-emitting atom.
On the other hand, there were a number of phenomena which the quantum theory was unable to explain. Light intensities inferior to a quantum had been observed, and the interference of light waves seemed to suggest continuity rather than discontinuity of emission. What is required is a more general synthesis, combining these two conflicting forms; and until a synthesis of this description has been formulated we can scarcely maintain that the existence of quanta has been established beyond dispute.[134]
The purpose of these different illustrations has been to show how slight and gradual is the transition from our commonplace knowledge of space and of the table to the loftiest forms of scientific knowledge. The methodology is ever the same, consisting in the formulation of a mental construct capable of co-ordinating in a rational and simple manner the sum total of our sense impressions. With the child and hisknowledge of space and of the table, this co-ordination is so simple that he obtains it without perceptible mental effort; whereas, in the more sophisticated cases of scientific knowledge, the synthesis is the result of abstruse mathematical speculations. Apart from this difference in degree, however, there exists no essential change.
And now let us examine the significance of these co-ordinations and investigate what they have revealed. We have seen that what we call reality reduces to the simplest co-ordination of the facts of observation, and that these facts, in the last analysis, represent co-ordinations of sense impressions. Hence we may say that it is these mind co-ordinations of sense impressions which yield us our knowledge of the so-called external universe. This scientific interpretation of reality may offend those who believe in the existence of some more absolute type of reality, the reality of the world in itself. To be sure, it appears natural enough for us human beings to assume that there must exist some sort of real world representing a reality of a more concrete category than the mental construct we have been discussing; a world which subsists in the absence of all human observers and in which the causes of our sense impressions originate. But what the nature of this real world of things in themselves may be, whether it could ever be described in human language, whether it is identical with the idea we have formed of it, are questions which are probably meaningless, and in any case are insoluble. As soon as we start arguing on these problems, it is a case of every man for himself; no one seems to be able to convince any one else. Fortunately, discussions of this kind are of no interest to science. In this chapter, the word reality will therefore be held to connote scientific reality, which means the simplest co-ordination of scientific facts, hence, ultimately, of sense impressions. That so restricted an interpretation of reality is ample for the needs of science issues from the following considerations:
The mere fact that men appear to understand one another when discussing the external world is sufficient proof that they have reached some common conception as to its nature. Hence, even if we assume that this so-called external universe, which each one of us believes he has discovered, represents but an individual dream which exists only in our respective minds, the fact remains that we must conceive of this dream as one shared in common by all men. As such, it manifests itself to all intents and purposes as an objective reality which we may regard as pre-existing to the observer who discovers it bit by bit.
It follows that the distinction between idealism and realism is purely academic in science, for our rule of action will be the same whichever of the two opposing philosophies we may prefer.[135]Thus the physicist who studies the properties of matter will proceed in precisely the same way, co-ordinating his results with the maximum of simplicity, regardless of whether he believes in matter as a metaphysical reality or as a mere mind construct. We have thus arrived at an understandingof what is meant by the objective world of science.
Now it is obvious that had it been impossible to discover or create a common objective world the same for all men, of which the various observers would obtain private perspectives, science would have been quite impossible; for science deals with the general, not with the particular or the individual. Were it not for the fact that the objective world of John is also that of Peter, John and Peter would never agree on the most elementary subjects. An exchange of commonplace knowledge, and, to a still greater degree, of scientific knowledge, would be quite impossible. We must recognise, therefore, that the very existence of science proves that the co-ordinations of the sensations of John possess the same structure as those of Peter. This is all we can say. We have no means of discovering whether what one man sees as red the other might not see as blue were our observers to exchange eyes and brains while retaining their memory of past sensations. On the other hand, we can assert from experience that, for normal human beings, two objects which appear to be of the same colour to one observer will appear to be the same colour to the other.
When the importance to science of one same common objective world is realised, the necessity of ridding it of all purely individualistic appearances becomes obvious; and we are thereby led to the problem of illusions or hallucinations. By an illusory object we mean one that has only a private existence, yet whose appearance can be accounted for scientifically, in terms of other phenomena, to which we may concede a common objective existence. Thus, in the case of a mirage, if we insist upon saying that water lies before us in the desert, our objective world will differ from that of our friend who, standing where we see the water, will claim that no water is present. The unfortunate consequences of our error might be very great, as in the case of the dog who dropped the bone for its image.
Hence, either we must abandon a belief in a common objective universe and relinquish all attempts at framing a science, or else we must succeed in interpreting the differences in our opinions by appealing to some phenomenon, such as refraction, which all men will accept. In the case of the mirage, for instance, the introduction of refraction saves the objective universe, hence saves science. Finally, we may say that the bright band in the desert is a reality from the point of view of one particular individual, but that it is an illusion from the point of view of the generality; whereas refraction is a reality from the point of view of all.
In the case of hallucinations, we are dealing with delusions of another kind. Consider the fever-stricken man who sees snakes around him. Of course, the snakes, no more than the water in the desert, can have a place in the common objective world; so that in this respect the hallucination and the illusion appear analogous. Nevertheless, there exists an important difference between the two. Thus, in the case of the mirage, all we had to do was to stand at a given point to beholdthe bright band in the desert as easily as any one else. A certain community in the illusion would still subsist. The hallucination, on the other hand, is essentially private. We cannot step into the patient’s place and suffer from his hallucination. Something more deep-seated is at stake than a mere change in our post of observation in a common objective world.
While it is true that the physiologist and brain specialist may succeed in accounting for a hallucination in terms of objective changes in our organic condition (blood pressure, etc.), yet even so we should be dealing with problems far more complex than those which the modern physicist is required to investigate. For all these reasons, while it is the aim of the physicist to account for illusions in terms of the common objective world and the laws of physics, the study of hallucinations belongs to another field of scientific research. In many cases the distinction we have established between a hallucination and an illusion lacks definiteness. For instance, the stars that we see when punched on the nose would not be considered the result of a hallucination, and yet, according to the division established above, the phenomenon would be more in the nature of a hallucination than of an illusion. Similar considerations would apply to dreams.
There exists a more popular interpretation of illusions. It results from a confusion between ignorance and illusion. Suppose, for instance, that we were to receive visual sensations which led us to assert that a table was present, but that on approaching the table we were to find that we could walk through it. Ifwe, andwealone among men, could see the table, we should be justified in assuming, in view of past experience, that we were suffering from some hallucination, since any other alternative would deny us the possibility of conceiving of an objective world the same for all men. But if all who came into the room perceived the same table and failed, as we had done, to derive any tactual impressions from its presence, it would be taking too much for granted to assert that we had all of us been the victims of error. The mystery might be cleared up without difficulty were we to find it possible to account for this appearance in terms of phenomena already known; but if this procedure failed, the wiser course would be to agree that some new mysterious manifestation had been discovered. At all events, it would be wrong to maintain that the appearance must necessarily be the result of a collective illusion on the ground that its reality would conflict with the laws of matter. Our knowledge of natural laws is obtained only by generalising from experience; and where experience is incomplete, as it must always be, the laws can lay claim to no measure of certainty. For example, a man knowing nothing of atmospheric pressure might well assume that a balloon could never rise in the air without coming into conflict with the law of gravitation, or that a firefly could not emit light without getting burnt. The reason why the theoretical scientist is compelled to approach problems of a seemingly mysterious nature with great open-mindedness is because onso many previous occasions he has been confronted with discoveries which yield nothing in strangeness to the one we have mentioned; and strangeness is not necessarily the product of illusion. More often than not, it is the result of ignorance and prejudice. In this respect, we have only to conceive of the surprise we should experience, had we never heard of an electromagnet, on discovering that distant objects flew towards it without being pulled by strings.
And now let us return to our objective world. We will assume that illusions have been barred therefrom; and the world we have thus obtained can be conceived of as existing in precisely the same way for all observers. If we assume that there is a real world of things in themselves, constituting the underlying cause of our sense experience, the point we wish to ascertain is whether scientific procedure can throw any light on its nature.
In the first place, when discussing the outside world as an existent reality, we must differentiate between its substance and its structure, or form. If, with science, we consider that our knowledge of the external universe can be arrived at only by a rational synthesis of facts of experience, we must recognise that substance escapes us completely; all we can hope to approach is structure, or relationships. Of course, a view of this sort presupposes that we are justified in asserting that knowledge can be acquired only by means of rational co-ordinations. If we dispute the legitimacy of this point the entire argument collapses. Hence the scientific argument can constitute no refutation of the views of the mystic. A man who “knows†of things through intuition or faith, or a man who tells us that he knows that in his previous incarnation he was Julius Cæsar, is a type of opponent with whom the scientist cannot even argue; for whatever can be neither proved nor disproved by observation or reason constitutes a form of knowledge which is meaningless, or at least completely irrelevant to science. If we did not place some kind of limitation on what we were to regard as knowledge, there would be no reason to prefer the opinions of a Newton or an Einstein to the ravings of an ignoramus or a lunatic; and human knowledge would become so conflicting as to lose all significance. It is not denied that intuition may often lead to discovery. Even in mathematics examples are numerous; but the fact remains that intuition alone has so often led men astray that unless its disclosures can be submitted to some kind of test, no reliance can be placed in them.
Furthermore, it is not asserted that intelligence is everything. We know that with animals and, still more, with insects, instinct plays a major rôle. But it is to be noted, first of all, that instinct is scarcely apparent in human beings. And in the second place, with bees, for example, which have selected the hexagonal shape for their combs, a choice which mathematical calculation proves to have been the best they could have selected, instinct has yielded exactly the same results that intelligence would have done; so that aside from the lack of consciousness which accompanies instinct, no difference is detected in the results to which it leads. Hence, it can scarcely be deemedto yield a new form of knowledge. If these points be granted, we may return to our subject and enquire why it is that substance escapes us completely when by “knowledge†we refer to a rational co-ordination of facts and sense impressions.
Rational co-ordinations have their most perfect prototype in mathematical co-ordinations—in those of mathematical physics, for example. Not all co-ordinations can be constructed mathematically; chemistry, and of course, to a still greater degree, biology, afford us illustrations where mathematics is comparatively useless. Nevertheless, as the methods of co-ordination are essentially the same in all cases, the mathematical ordinations, in view of their greater clarity, can be studied to advantage as typical instances in this respect.
Now mathematical equations are nothing but relations, and from initial relations all we can deduce are other relations. In other words, our equations can never yield us more than we originally put into them. It follows that were all relationships in nature to be preserved and the substances changed, no observable difference could be detected; and we should never be able to differentiate between a whole class of worlds identical in structure but differing in substance. If, then, we discard the procedure of the mystic, or of the metaphysician who claims a knowledge that cannot be submitted to the control of experiment, we must recognise that substance escapes us completely and that our knowledge of the real world can at best reduce to a skeleton or structure.
An illustration given by Eddington presents the same problem in a more concrete manner. He assumes that in some future age the game of chess may be dead and forgotten; and he compares our position in respect to the real world with that of archaeologists who will have unearthed curious records of the game written in the usual obscure symbolism. Eventually they would succeed in reconstructing the moves of the different pieces, the two-dimensional ordering relation of the partitions, and the rules of the game. To this extent they could claim to have understood the game of chess. Yet certain aspects would escape them completely. The shape of the partitions, whether square, round or oblong, the aspect of the pieces, the nature of the board, whether of wood or of stone, would remain unknown. But it is to be noted that these elements of knowledge that escaped them would be irrelevant. The board might be of wood or stone, the partitions of diamonds or squares, and yet the game could be played as before. If, on the other hand, the ordering relation of the partitions were changed, if the permissible moves of the pieces were modified, the written records would make no sense. It is the same with the real world. The substance may change in nature, yet no difference will be perceived; but if the relationships are modified, a new world will divulge itself to our observation. To this extent, therefore, we are justified in saying that the most our observations can reveal reduces to the structure of the real world. As the French mathematician Bertrand once said, the age of thecaptain, the number of the crew, the height of the mast can yield us no information about the position of the ship.
Structure, as outlined above, resolves itself into those mathematical relationships or equations which appear to account for the sequence of phenomena. But structure in the more usual sense would apply to the hidden mechanisms themselves, to the deep-seated, underlying causes. When, however, we wish to attain this more concrete knowledge of structure, we encounter a number of difficulties. As Poincaré points out, every time conservation of energy and the principle of least action are found to be satisfied (as is the case in electromagnetics, for example), an indefinite number of different mechanical models are possible; so that our knowledge of the structure of the world is vitiated by the fact that the substructure remains indefinite. Clifford expresses the same idea when he says:
“Whatever can be explained by the motion of a fluid can be equally well explained either by the attraction of particles or by the strains of a solid substance; the very same mathematical calculations result from the three distinct hypotheses; and science, though completely independent of all three, may yet choose one of them as serving to link together different trains of physical enquiry.â€
In a certain measure, difficulties of this sort may prove temporary. It may happen that further experiment will enable us to determine which of the various hypotheses corresponds to concrete reality. Such, indeed, was the hope of the scientists of the eighteenth and early nineteenth centuries. And in our days the practical isolation of electrons and molecules is a proof that these hopes were not always unfounded. Yet, even so, we should have advanced but a step, and should again be faced with the problem of determining the structure of these electrons, and so onad infinitum. It would be of no avail to say that our aim had been realised when the ultimate constituent elements were isolated. For either these supposedly ultimate elements would possess no structure, in which case we could never know anything further about them, or else they would present a structure; but then they would not constitute ultimate elements, since this structure would imply relationships between their various parts, and we should not be at the end of our journey.
With the discoveries of modern science, still other difficulties bar the route to success. It would now seem that as we wended our way into the microscopic, exact laws might vanish. If this be the case, the microscopic structures must ever elude us. Yet, even if we waive these latter difficulties, it is apparent that the scientific method can yield no knowledge of substance; the structure of reality is all we can approach.
But this reality, once again, is nothing but the expression of the simplest co-ordination of scientific facts; no genuine metaphysical reality is implied. If we discard the criterion of simplicity, we may conceive of our world in as many different ways as we please. Consider, for example, Einstein’s theory. One of the principal discoveries ofthe theory has been to disclose unsuspected relationships between space and time, leading to the four-dimensional space-time structure of the universe. And yet, if we wished, we might account for all of Einstein’s previsions while retaining our traditional belief in a three-dimensional structure of space and in an unrelated time. Thus, Michelson’s experiment would be accounted for by Ritz’s or Stokes’ hypothesis; Bucherer’s experiment, by the dragging along of a certain mass of ether by the moving electron just as a ship drags along part of the water surrounding it; Fizeau’s experiment, by a modification of the ether in transparent bodies (and this was, in fact, Fresnel’s original explanation). The double bending of a ray of light, and Mercury’s motion, would be accounted for by a modification of the Newtonian law of gravitation; the Einstein shift-effect, by a modification in the intra-atomic structure of the solar atom, and so on. It is a pure waste of time to consider the various hypotheses that might be suggested, for we can invent as many different ones as we choose. Needless to say, all the simple beauty and unity of nature revealed by Einstein’s theory would be lacking; and prevision would be impossible, seeing that new postulatesad hocwould be needed at all further stages. But when all is said and done, what have simplicity, beauty, unity and prevision to do with the reality of the metaphysician?
Thus the reality of structure which the metaphysician would defend (quite aside from the reality of substance) presupposes hisa prioribelief in the inevitable simplicity of nature. But simplicity is, after all, but an expression of human appreciation. Furthermore, even if this view were contested, there would seem to be no reason to suppose that nature should prefer simplicity to complexity in the first place. As a matter of fact, what little we know of nature proves that the reverse is true when we cease to view phenomena in an approximate macroscopic way and adopt a more microscopic standpoint. To illustrate: There is no more simple law in physics than that of perfect gases; and yet we know that this apparent simplicity is due to our macroscopic observations and that it conceals the most bewildering chaos and uncertainty. It is the same when we consider numerous other phenomena, such as intra-atomic changes.
From this it follows that it is at least a questionable procedure to identify reality with simplicity of co-ordination. And if we discard this tentative identification, reality (even reality of structure) escapes us completely. Of course, the type of reality which we have been branding as elusive is the absolute reality, the “true being†of the metaphysician. The scientist also appeals to the word “reality,†but he employs it in a different sense. For him, reality is identified with simplicity of co-ordination, and he states his views explicitly, realising full well that a reality of this type is far from being absolute and that it is essentially pragmatic. It is for reasons such as these that the vast majority of scientists are agnostics at heart, not on account of anya prioripredilection, but becausea 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 firstodd numbers is equal tohowever greatmay 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 ofmathematical 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 thatifthe sum of the first ()odd numbers is (), then the sum of the firstodd numbers must necessarily be.We assume that this law will hold however greatmay 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 theleit-motifof mathematics.[136]A few specificexamples 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, justas 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 linealwaysintersects the second-order curve in two true points, whether real or imaginary. Further generalisations require the introduction ofideal points,points at infinity,imaginary linesandangles.
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 thecircular points at infinity. When we take these two imaginary points into consideration we are able to state thatallsecond-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-Euclideanand Euclidean) can be treated in terms of projective geometry. We see that in all cases mathematicians attempt to replace the wordsometimesby the wordalways, 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 callthe value of the function and represent the variable by,thenis a mathematical function (andbeing assigned constants). For every value ofa 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 thevery simplest type of functions, we may mention those given by the unending serieswhere 3! stands forfor.These functions were known, respectively as, “sine†(or sin), “cosine†(or cos)and theexponential function.
It was then found that the functions sinand cospresented very rigid relationships as though they belonged to the same family. The namecircular functionswas given to the family; the relation we have in view being given byOn 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;wherestands forandstands 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 wasIn this formula we have only to make,and we obtainThus, not only has the introduction of imaginary numbers revealed an unsuspected relationship between what were widely different types of functions, but the two numbersand,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 theelliptical functionsdiscovered by Abel and Jacobi, then the modular functions of Hermite, and finally theautomorphous functionsof 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 anumber 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 ofscience 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.â€
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 unityin 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 movingin vacuodescribes 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 asor.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 capriciousway.[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.â€
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 showsus 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.â€
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 thematrix 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 theneural 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 thereal shapeof 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 tothis 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 fromhererather than the shape as seen fromthere. 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.