The logic of modern physics

To stop the discussion at this point might leave the impression that this observation of the relative character of knowledge is of only a very tenuous and academic interest, since it appears to be concerned mostly with the character of our descriptive processes, and to say little about external nature. [What this means we leave to the metaphysician to decide.] But I believe there is a deeper significance to all this. It must be remembered that all our argument starts with the concepts as given. Now these concepts involve physical operations; in the discovery of what operations may be usefully employed in describing nature is buried almost all physical experience. In erecting our structure of physical science, we are building on the work of all the ages. There is then this purely physical significance in the statement that all motion is relative, namely that no operations of measuring motion have been found to be useful in describing simply the behavior of nature which are not operations relative to a single observer; in making this statement we are stating something about nature. It takes an enormous amount of real physical experience to discover relations of this sort. The discovery that the number obtained by counting the number of times a stick may be applied to an object can be simply used in describing natural phenomena was one of the most important and fundamental discoveries ever made by man.

Meaningless Questions

Another consequence of the operational character of our concepts, almost a corollary of that considered above, is that it is quite possible, nay even disquietingly easy, to invent expressions or to ask questions that are meaningless. It constitutes a great advance in our critical attitude toward nature to realize that a great many of the questions that we uncritically ask are without meaning. If a specific question has meaning, it must be possible to find operations by which an answer may be given to it. It will be found in many cases that the operations cannot exist, and the question therefore has no meaning. For instance, it means nothing to ask whether a star is at rest or not. Another example is a question proposed by Clifford, namely, whether it is not possible that as the solar system moves from one part of space to another the absolute scale of magnitude may be changing, but in such a way as to affect all things equally, so that the change of scale can never be detected. An examination of the operations by which length is measured in terms of measuring rods shows that the operations do not exist (because of the nature of our definition of length) for answering the question. The question can be given meaning only from the point of view of some imaginary superior being watching from an external point of vantage. But the operations by which such a being measures length are different from the operations of our definition of length, so that the question acquires meaning only by changing the significance of our terms—in the original sense the question means nothing.

To state that a certain question about nature is meaningless is to make a significant statement about nature itself, because the fundamental operations are determined by nature, and to state that nature cannot be described in terms of certain operations is a significant statement.

It must be recognized, however, that there is a sense in which no serious question is entirely without meaning, because doubtless the questioner had in mind some intention in asking the question. But to give meaning in this sense to a question, one must inquire into the meaning of the concepts as used by the questioner, and it will often be found that these concepts can be defined only in terms of fictitious properties, as Newton's absolute time was defined by its properties, so that the meaning to be ascribed to the question in this way has no connection with reality. I believe that it will enable us to make more significant and interesting statements, and therefore will be more useful, to adopt exclusively the operational view, and so admit the possibility of questions entirely without meaning.

This matter of meaningless questions is a very subtle thing which may poison much more of our thought than that dealing with purely physical phenomena. I believe that many of the questions asked about social and philosophical subjects will be found to be meaningless when examined from the point of view of operations. It would doubtless conduce greatly to clarity of thought if the operational mode of thinking were adopted in all fields of inquiry as well as in the physical. Just as in the physical domain, so in other domains, one is making a significant statement about his subject in stating that a certain question is meaningless.

In order to emphasize this matter of meaningless questions, I give here a list of questions, with which the reader may amuse himself by finding whether they have meaning or not.

(1) Was there ever a time when matter did not exist?

(2) May time have a beginning or an end?

(3) Why does time flow?

(4) May space be bounded?

(5) May space or time be discontinuous?

(6) May space have a fourth dimension, not directly detectible, but given indirectly by inference?

(7) Are there parts of nature forever beyond our detection?

(8) Is the sensation which I call blue really the same as that which my neighbor calls blue? Is it possible that a blue object may arouse in him the same sensation that a red object does in me and vice versa?

(9) May there be missing integers in the series of natural numbers as we know them?

(10) Is a universe possible in which 2 + 2 ≠ 4?

(11) Why does negative electricity attract positive?

(12) Why does nature obey laws?

(13) Is a universe possible in which the laws are different?

(14) If one part of our universe could be completely isolated from the rest, would it continue to obey the same laws?

(15) Can we be sure that our logical processes are valid?

GENERAL COMMENTS ON THE OPERATIONAL POINT
OF VIEW

To adopt the operational point of view involves much more than a mere restriction of the sense in which we understand "concept," but means a far-reaching change in all our habits of thought, in that we shall no longer permit ourselves to use as tools in our thinking concepts of which we cannot give an adequate account in terms of operations. In some respects thinking becomes simpler, because certain old generalizations and idealizations become incapable of use; for instance, many of the speculations of the early natural philosophers become simply unreadable. In other respects, however, thinking becomes much more difficult, because the operational implications of a concept are often very involved. For example, it is most difficult to grasp adequately all that is contained in the apparently simple concept of "time," and requires the continual correction of mental tendencies which we have long unquestioningly accepted.

Operational thinking will at first prove to be an unsocial virtue; one will find oneself perpetually unable to understand the simplest conversation of one's friends, and will make oneself universally unpopular by demanding the meaning of apparently the simplest terms of every argument. Possibly after every one has schooled himself to this better way, there will remain a permanent unsocial tendency, because doubtless much of our present conversation will then become unnecessary. The socially optimistic may venture to hope, however, that the ultimate effect will be to release one's energies for more stimulating and interesting interchange of ideas.

Not only will operational thinking reform the social art of conversation, but all our social relations will be liable to reform. Let any one examine in operational terms any popular present-day discussion of religious or moral questions to realize the magnitude of the reformation awaiting us. Wherever we temporize or compromise in applying our theories of conduct to practical life we may suspect a failure of operational thinking.

CHAPTER II

OTHER GENERAL CONSIDERATIONS

It is a general consequence of the approximate character of all measurement that no empirical science can ever make exact statements. This was fairly obvious in the case of mechanics, but it required a Gauss[4] to convince us that the geometry in which we are interested as physicists is an empirical subject, and that one cannot say that actual space is Euclidean, but only that actual space approaches to ideal Euclidean space within a certain degree of approximation. I believe that we are compelled to go still further, and recognize that arithmetic, so far as it purports to deal with actual physical objects, is also affected with the same penumbra of uncertainty as all other empirical science. A typical statement of empirical arithmetic is that 2 objects plus 2 objects makes 4 objects. This statement acquires physical meaning only in terms of certain physical operations, and these operations must be performed in time.

Now the penumbra gets into this situation through the concept of object. If the statement of arithmetic is to be an exact statement in the mathematical sense the "object" must be a definite clean-cut thing, which preserves its identity in time with no penumbra. But this sort of thing is never experienced, and as far as we know does not correspond exactly to anything in experience. It is of course true that in most experience the penumbra is so very thin and snug-fitting that it requires special effort to recognize its presence at all; but scrutiny, I believe, shows that it is always there. If our experience had been restricted to phenomena in a vacuum, and the objects we were trying to count had been spheres of a gas which expand and interpenetrate, it is obvious that the concept of "object" as a thing with identity would have been much more difficult to form. Or, if our objects are tumblers of water, we discover when our observation reaches a certain stage of refinement that the amount of water is continually changing by evaporation and condensation, and we are bothered by the question whether the object is still the same after it has waxed and waned. Coming to solids, we eventually discover that even solids evaporate, or condense gases on them, and we see that an object with identity is an abstraction corresponding exactly to nothing in nature. Of course the penumbra of uncertainty which surrounds our arithmetical statements because of this property of physical objects is so exceedingly tenuous that practically we are not aware of its existence, and expect never to find undiscovered phenomena within the penumbra. But in principle we must recognize its presence, and must further recognize that all empirical science must be of this character.

In most empirical sciences, the penumbra is at first prominent, and becomes less important and thinner as the accuracy of physical measurement is increased. In mechanics, for example, the penumbra is at first like a thick obscuring veil at the stage where we measure forces only by our muscular sensations, and gradually is attenuated, as the precision of measurements increases. But with the arithmetical concept of an individual identifiable object it is the exact reverse; a crude point of view does not suspect the existence of the penumbra at all, and we discover it only by highly refining our methods. Doubtless arithmetic owes its early development to this property.

We may now go still further. Operations themselves are, of course, derived from experience, and would be expected also to have a nebulous edge of uncertainty. We have to ask such questions as whether the operations of arithmetic are clean-cut things. Is the operation of multiplying 2 objects by 2 a definite operation, with no enveloping haze? All our physical experience convinces us that if there is a penumbra about the concept of operations of this sort it is so tenuous as to be negligible, at least for the present; but the question affords an interesting topic for speculation. We also have to ask whether mental operations may similarly be enveloped in a haze.

EXPLANATIONS AND MECHANISMS

Perhaps the climax of our task of interpreting and correlating nature is reached when we are able to find an explanation of phenomena; with the finding of the explanation we are inclined to feel that our understanding of the situation is complete. We now have to ask what is the nature of the explanation which we set as the goal of our efforts. The answer is not easy to give, and there may be difference of opinion about it. We shall get the best answer to this, as to so many other questions, by adopting the operational point of view, and examining what we do in giving an explanation. I believe that examination will show that the essence of an explanation consists in reducing a situation to elements with which we are so familiar that we accept them as a matter of course, so that our curiosity rests.[5] "Reducing a situation to elements" means, from the operational point of view, discovering familiar correlations between the phenomena of which the situation is composed.

There is involved here the thesis that it is possible to analyze nature into correlations, without, however, any assumption whatever as to the character of these correlations. It seems to me that such a thesis is the most general that can be made if nature is to be intelligible at all. This thesis underlies all the considerations of this essay, and we shall not try to find anything more general. We shall, however, recognize that any assumption as to the character of the correlations constitutes a special hypothesis which may restrict the future, and that therefore these special hypotheses are to be subjected to special examination. We return to this matter in more detail in discussing the causality concept, which is closely related to the concept of explanation.

In this view of explanation there is no implication that the "element" is either a smaller or a larger scale thing than the phenomenon being explained; thus we may explain the properties of a gas in terms of its constituent molecules, or perhaps some day we shall become so familiar with the idea of a non-Euclidean space that we shall explain (instead of describe) the gravitational attraction of a stone by the earth in terms of a space-time curvature imposed by all the rest of the matter in the universe.

If this is accepted as the true nature of explanation, we see that an explanation is not an absolute sort of thing, but what is satisfactory for one man will not be for another. The savage is satisfied by explaining the thunderstorm as the capricious act of an angry god. The physicist demands more, and requires that the familiar elements to which we reduce a situation be such that we can intuitively predict their behavior. Thus even if the physicist believed in the existence of the angry god, he would not be satisfied with this explanation of the thunderstorm because he is not so well acquainted with angry gods as to be able to predict when anger is followed by a storm. He would have to know why the god had become angry, and why making a thunderstorm eased his ire. But even with this additional qualification, scientific explanation is obviously still a relative affair—relative to the elements or axioms to which we make reduction and which we accept as ultimate. These elements depend to a certain extent on the purpose in view, and also on the range of our previous physical experience. If we are explaining the action of a machine, we are satisfied to reduce the action to the push and pull of the various members of the machine, it being accepted as an ultimate that these members transmit pushes or pulls. But the physicist who has extended his experimental knowledge further, may want to explain how the members transmit pushes or pulls in terms of the action on each other of the electrons in their orbits in the atoms. The character of our explanatory structure will depend on the character of our experimental knowledge, and will change as this changes.

Formally, there is no limit to the process of explanation, because we can always ask what is the explanation of the elements in terms of which we have given the last explanation. But the point of view of operations shows that this is mere formalism which ends only in meaningless jargon, for we soon arrive at the limit of our experimental knowledge, and beyond this the operations involved in the concepts of our explanations become impossible and the concepts become meaningless.

As we extend experimental knowledge and push our explanations further and further, we see that the explanatory sequence may be terminated in several possible ways. In the first place, we may never push our experiments beyond a stage into which the elements with which we are already familiar do not enter. In this case explanation is very simple: it involves nothing essentially new, but merely the disentanglement of complexities. The kinetic theory of gases, in explaining the thermal properties of a gas in terms of ordinary mechanical properties of the molecules, suggests such a situation. Or, secondly, our experiments may bring us into contact with situations novel to us, in which we can recognize no familiar elements, or at least must recognize that there is something in addition to the familiar elements. Such a situation constitutes an explanatory crisis and explanation has to stop by definition. Or thirdly, we may try to force our explanations into a predetermined mold, by formally erecting or inventing beyond the range of present experiment ultimates more or less like elements already familiar to us, and seek to explain all present experience in terms of these chosen ultimates.

Leaving for the present the third possibility, which is within our control to accept or reject, and is a formal matter, it is merely a question of experimental fact which of the first two possibilities corresponds to the actual state of affairs. The most perfunctory examination of the present state of physics shows that we are now facing the second of these possibilities, and that in the new experimental facts of relativity, and in all quantum phenomena, we are confronted with an explanatory crisis. It has often been emphasized that Einstein's theory of gravitation does not seek at all to give an explanation of gravitational phenomena, but merely describes and correlates these phenomena in comparatively simple mathematical language. No more attempt is made to reduce the gravitational attraction between the earth and the sun to simple terms than was made by Newton. In the realm of quantum phenomena it is of course the merest commonplace that our old ideas of mechanics and electrodynamics have failed, so that it is a matter of the greatest concern to find how many, or indeed whether any, elements of the old situations can be carried over into the new.

An examination of many of the so-called "explanations" of quantum theory constitutes at once a justification of the definition of explanation given above, and of the statement that in quantum phenomena we are at an explanatory crisis. For the endeavor of all these quantum explanations is to find in every new or more complicated situation the same elements which have already been met in simpler situations, and which are therefore relatively more familiar. For example, many quantum phenomena are made to involve the emission of energy when an electron jumps from one orbit to another. But always the elements to which reduction is made are themselves quantum phenomena, and these are still so new and unfamiliar that we feel an instinctive need for explanation in other terms. We seek to understand why the electron emits when it jumps.

The explanatory crisis which now confronts us in relativity and quantum phenomena is but a repetition of what has occurred many time in the past. A similar crisis confronted Prometheus when he discovered fire, and the first man who observed a straw sticking to a piece of rubbed amber, or a suspended lodestone seeking the north star. Every kitten is confronted with such a crisis at the end of nine days. Whenever experience takes us into new and unfamiliar realms, we are to be at least prepared for a new crisis.

Now what are we to do in such a crisis? It seems to me that the only sensible course is to do exactly what the kitten does, namely, to wait until we have amassed so much experience of the new kind that it is perfectly familiar to us, and then to resume the process of explanation with elements from our new experience included in our list of axioms. Not only will observation show that this is what is now actually being done with respect to quantum and gravitational phenomena, but it is in harmony with the entire spirit of our outlook on nature. All our knowledge is in terms of experience; we should not expect or desire to erect an explanatory structure different in character from that of experience. Our experience is finite; on the confines of the experimentally attainable it becomes hazy, and the concepts in terms of which we describe it fuse together and lose independent meaning. Furthermore, at every extension of our experimental range we must be prepared to find, and as a matter of fact we have often found, that we encounter phenomena of an entirely novel character for which previous experience has given us no preparation. The explanatory structure proposed above has all these properties; it is finite, being terminated by the edge of experiment, the final stages of our explanations are hazy in that it becomes more and more difficult to distinguish elements of familiar experience, and every now and then we must admit new elements into our explanations.

The first step in resuming our explanatory progress, after we have been confronted with such a crisis, is to seek for various sorts of correlation between the elements of our new experience, in the confident expectation that these elements will eventually become so familiar to us that they may be used as the ultimates of a new explanation. This is exactly what is now happening in quantum theory.

Diametrically opposed to the views above, there is another ideal of the explanatory process which is held by many physicists, and which has been mentioned above as the third possible way in which the explanatory sequence may be terminated, namely, the endeavor to devise beyond the limit of present experiment a structure built of elements like some of those of our present experience, in the action of which we endeavor to find the explanation of phenomena in the present range. Now a program such as this, as a serious program for the final correlation of nature, is entirely opposed to the spirit of the considerations expounded here. There is no warrant whatever in experience for the conviction that as we penetrate deeper and deeper we shall find the elements of previous experience repeated, although sometimes we do find such repetitions, as in the behavior of gases. Yet this has been the attitude of many eminent physicists, for example, Faraday and Maxwell, in seeking to explain distant electrical action by the propagation through a medium of a mechanical push or pull, or by Hertz, who sought in all phenomena the effect of concealed masses with ordinary mechanical inertia. Although as a general principle this program seems to be absolutely without justification, nevertheless it may be justified if the specific character of the physical facts seems to indicate a repetition at lower levels of elements familiar higher up. Hertz undoubtedly had this justification, as did also Maxwell to a certain extent, in the discovery that the general equations of electrodynamics are of the same form as the generalized Lagrangean equations of mechanics. For Faraday, however, there seems no such justification; the urge to this sort of thing in Faraday came from an uncritical acceptance of his own temperamental reactions.

From a less serious point of view it may, however, be quite justified to make such a working hypothesis as that in the action of electrical forces may be discovered the same elements with which we are familiar in the everyday experiences of mechanics. For such a hypothesis often enables us to make partial correlations which suggest new experimental tests, and thus gives the stimulus to an extension of our experimental horizon. Many physicists recognize the tentative character of such attempted explanations, but others apparently take them more seriously, as for example Lord Kelvin in his continuous life-long attempts to find a mechanical explanation of all physical phenomena. This quotation from Kelvin is illuminating. "I never satisfy myself until I can make a mechanical model of a thing. If I can make a mechanical model, I can understand it. As long as I cannot make a mechanical model all the way through, I cannot understand it.... But I want to understand light as well as I can without introducing things that we understand even less of."

So much for general considerations on the nature of explanation. Coming now to greater detail, many explanations involve what may be described as a mechanism. It is difficult to characterize exactly what we mean by mechanism, but it seems to be associated with an attitude of mind that strives to realize the third of the possibilities mentioned above. As a matter of fact, the mechanism sought for is usually of a particular type, in that the ultimate elements selected are mechanical elements. This point of view is particularly characteristic of the English school of physicists. Although "mechanism" usually implies mechanical elements, we may show by specific examples that we do actually use the word in a broader significance. If, for example, we could devise within the core of an atom a revolving system of electrical charges, acting on each other with the ordinary inverse square forces of electrostatics, such that every now and then the system becomes unstable and breaks up, we should doubtless say that we had found a mechanism for explaining radioactive disintegration.

However, the formulation of a precise definition of mechanism is of secondary concern to us; we are primarily interested in understanding the attitude of mind that feels a mechanism is necessary. A typical example of such an urge to a mechanism is afforded by the gravitational action between distant bodies. To many minds the concept of action at a distance is absolutely abhorrent, not to be tolerated for an instant. Such an intolerable situation is avoided by the invention of a medium filling all space, which transmits a force from one body to the other through the successive action on each other of its contiguous parts. Or the dilemma of action at a distance may be avoided in other ways, as by Boscovitch in the eighteenth century, who, in order to explain gravitation, filled space with a triply infinite horde of infinitesimal projectiles. Now of course it is a matter for experiment to decide whether any physical reality can be ascribed to a medium which makes gravitation possible by the action of its adjacent parts, but I can see no justification whatever for the attitude which refuses on purely a priori grounds to accept action at a distance as a possible axiom or ultimate of explanation. It is difficult to conceive anything more scientifically bigoted than to postulate that all possible experience conforms to the same type as that with which we are already familiar, and therefore to demand that explanation use only elements familiar in everyday experience. Such an attitude bespeaks an unimaginativeness, a mental obtuseness and obstinacy, which might be expected to have exhausted their pragmatic justification at a lower plane of mental activity.

Although it will probably be fairly easy to give intellectual assent to the strictures of the last paragraph, I believe many will discover in themselves a longing for mechanical explanation which has all the tenacity of original sin. The discovery of such a desire need not occasion any particular alarm, because it is easy to see how the demand for this sort of explanation has had its origin in the enormous preponderance of the mechanical in our physical experience. But nevertheless, just as the old monks struggled to subdue the flesh, so must the physicist struggle to subdue this sometimes nearly irresistible, but perfectly unjustifiable desire. One of the large purposes of this exposition will be attained if it carries the conviction that this longing is unjustifiable, and is worth making the effort to subdue.

The situation with respect to action at a distance is typical of the general situation. I believe the essence of the explanatory process is such that we must be prepared to accept as an ultimate for our explanations the mere statement of a correlation between phenomena or situations with which we are sufficiently familiar. Thus, in quantum theory, there is no reason why we should not be willing to accept as an ultimate the fundamental fact that when an electron jumps radiation is emitted, provided always that we can give independent meaning in terms of operations to the jumping of an electron. If there is no experiment suggesting other and intermediate phenomena, we ought to be able to rest intellectually satisfied with this. Of course it is quite a different matter, and entirely justified, to imagine what the assumption of finer details in the process would involve experimentally, and then to seek for these possible new experimental facts.

It is a consequence of this view that any correlation is adapted to be an absolutely final element of explanation, and can never be superseded by the discovery of new experimental facts, if the correlation is by definition beyond the reach of further experiment. Such a possibility, for example, is contained in a correlation between the numerical magnitude of the gravitational constant and the total mass of the universe. Something of this sort may be well attempted by those who desire their explanations to take a formally final shape. We shall return to this subject later.

The instinctive demand for a mechanism is fortified by observation of the many important cases in which mechanisms have been discovered or invented. However, the significance of such successful attempts must be subject to most careful scrutiny. The matter has been discussed by Poincaré,[6] who showed that not only is it always possible to find a mechanistic explanation of any phenomenon (Hertz's program was a perfectly possible one), but there are always an infinite number of such explanations. This is very unsatisfactory.

We want to be able to find the real mechanism. Now an examination of specific proposed mechanisms will show that most mechanisms are more complicated than the simple physical phenomenon which they are invented to explain, in that they have more independently variable attributes than the phenomenon has been yet proved to have. An example is afforded by the mechanical models invented to facilitate the study of the properties of simple inductive electrical circuits. The great number of such models which have been proposed is sufficient indication of their possible infinite number. But if the mechanism has more independently variable attributes than the original phenomenon, it is obvious that the question is without meaning whether the mechanism is the real one or not, for in the mechanism there must be simple motions or combinations of motions which have no counterpart in features of the original phenomenon as yet discovered. Obviously, then, the operations do not exist by which we may set up a one to one correspondence between the properties of the mechanism and the natural phenomenon, and the question of reality has no meaning. If, then, a mechanism is to be taken seriously as actually corresponding to reality, we must demand that it have no more degrees of freedom than the original phenomenon, and we must also be sure that the phenomenon has no undiscovered features. Physical experience shows that such conditions are most difficult to meet, and indeed the probability is that they are impossible.

A mechanism with more independently variable attributes than the phenomenon may prove to be a very useful tool of thought, and therefore worth inventing and studying, but it is to be regarded no more seriously than is a mnemonic device, or any other artifice by which a man forces his mind to give him better service.

There is another possible program of explanation, the converse of that considered above, namely, to explain all familiar facts of ordinary experience in terms of less familiar facts found at a deeper level. The most striking example of this is the recent attempt to give a complete electrical explanation of the universe. The original attempt was to explain electrical effects in mechanical terms; this attempt failed. At about the same time the existence of the electron was experimentally established, so that it was evident that electricity is a very fundamental constituent of matter. The program of explanation was reversed, and an electrical explanation sought for all mechanical phenomena, including in particular mechanical mass. But this attempt has also failed; we recognize that part of mass may be non-electrical in character, we postulate non-electrical forces inside the electron, and further, we usually postulate for electrons and protons the property of impenetrability, a property derived from experience on a higher scale of magnitude.

A program of this general sort is likely to be regarded with considerable sympathy, and indeed the chances of success seem much greater than do those of the converse program, for in our experience large scale phenomena are more often built up from and analyzed into small scale phenomena than the converse. But as a matter of principle we must again recognize that the only appeal is to experiment, and that we have to ask just one question: "Is it true, as a matter of fact, that all large scale phenomena can be built up of elements of small scale phenomena?" It seems to me that the experimental warrant for this conviction has not yet been given. The failure of the attempted electrical explanation of the universe is a case in point. However, the failure to prove a proposition is no guarantee that some time it may not be proved, and many physicists are convinced of the ultimate feasibility of this program. Personally I feel that the large may not always be analyzed into the smaller; the subject will be discussed again.

A conviction of the significance of microscopic analysis has many features in common with the usual conviction of the ultimate simplicity of nature. The thesis of simplicity involves in addition the assumption that the kinds of small scale elements are few in number, but actually this involves no important difference between the two convictions, because we have seen that the elements of which we build our structure become fewer in number as we approach the limit of the experimentally attainable. We may properly grant to convictions of this sort pragmatic value in suggesting new correlations and experiments, but a recognition of the empirical basis of all physics will not allow us to go further.

MODELS AND CONSTRUCTS

In discussing the concept of length, we could find no meaning in questions such as: "Is space on a scale of 10^-8 cm. Euclidean?" Nevertheless it will seem to many that they do attach a perfectly definite meaning to a question of this kind. Of course it must be agreed that magnitudes of 10^-8 cm. cannot be thought of in terms of immediate sensation. When one thinks of an atom as a thing with any geometrical properties at all, I believe he will find that what he essentially does is to imagine a model, multiplying all the hypothetical dimensions by a factor large enough to bring it to a magnitude of ordinary experience. This large scale model is given properties corresponding to those of the physical thing. For example, the model of the atom which was accepted in the fall of 1925 contains electrons rotating in orbits, and every now and then an electron jumps from one orbit to another, and simultaneously energy is radiated from the atom. Such a model is satisfactory if it offers the counterpart of all the phenomena of the original atom. Now I believe the only meaning that any one can find in his statement that the space of the atom is Euclidean is that he believes that he can construct in Euclidean space a model with all the observed properties of the atom. This possibility may or may not be sufficient to give real physical significance to the statement that the space of the atom is Euclidean. The situation here is very much the same as it was with respect to mechanisms. The model may have many more properties than correspond to measurable properties of the atom, and in particular, the operations by which the space of the model is tested for its Euclidean character may [and as a matter of fact I believe do] not have any counterpart in operations which can be carried out on the atom. Further, we cannot attach any real significance to the statement that the space of the atom is Euclidean unless we can show that no model constructed in non-Euclidean space can reproduce the measurable properties of the atom.

In spite of all this, I believe that the model is a useful and indeed unescapable tool of thought, in that it enables us to think about the unfamiliar in terms of the familiar. There are, however, dangers in its use: it is the function of criticism to disclose these dangers, so that the tool may be used with confidence.

Closely related to the mental model are mental constructs, of which physics is full. There are many sorts of constructs: those in which we are interested are made by us to enable us to deal with physical situations which we cannot directly experience through our senses, but with which we have contact indirectly and by inference. Such constructs usually involve the element of invention to a greater or less degree. A construct containing very little of invention is that of the inside of an opaque solid body. We can never experience directly through our senses the inside of such a solid body, because the instant we directly experience it, it ceases by definition to be the inside. We have here a construct, but so natural a one as to be practically unavoidable. An example of a construct involving a greater amount of invention is the stress in an elastic body. A stress is by definition a property of the interior points of a body which is connected mathematically in a simple way with the forces acting across the free surface of the body. A stress is then, by its very nature, forever beyond the reach of direct experience, and it is therefore a construct. The entire structure of a stress corresponds to nothing in direct experience; it is related to force, but is itself a six-fold magnitude, whereas a force is only three-fold.

We have next to ask whether the stress, which we have invented to meet the situation in a body exposed to forces, is a good construct. In the first place, a stress has the same number of degrees of freedom as the observable phenomenon, for it is one of the propositions of the mathematical theory of elasticity that the boundary conditions, which are the experimental variables, uniquely determine the stress in a given body [i.e. a body of given elastic constants]; and of course it is at once obvious, by an inspection of the equations, that conversely a possible stress system uniquely determines the boundary conditions to the significant amount. There is, therefore, a unique one-to-one correspondence between a stress and the physical situation it was made to meet, and so far a stress is a good construct. Up to this point a stress, from the point of view of the operations in terms of which it is defined, is a purely mathematical invention, which is justified because it is convenient in describing the behavior of bodies subjected to the action of force. But we wish now to go farther and ascribe physical reality to a stress, meaning by this that a stress in a solid body shall correspond to some real physical state of the interior points. Let us examine, from the point of view of operations, what the meaning of a statement like this may be. Since we now wish to ascribe an additional physical meaning to a stress beyond that of the mathematical operations in terms of which the stress was determined, there must exist additional physical operations corresponding to this meaning, or else our statement is meaningless. Now of course it is a matter of the most elementary experience that physical phenomena do exist which allow these other independent operations. A body under stress is also in a state of strain, which may be determined from the external deformations, or the strain at internal points may be made more vividly real by those optical effects of double refraction in transparent bodies which are now so extensively used in model experiments, or if the stress is pushed beyond a certain point, we have such new phenomena as permanent set or finally, rupture.

We have, then, every reason to be satisfied with our construct of stress. In the first place, from the formal point of view, it is a good construct because there is a unique correspondence between it and the physical data in terms of which it is defined; and in the second place we have a right to ascribe physical reality to it because it is uniquely connected with other physical phenomena, independent of those which entered its definition. This last requirement, in fact, from the operational point of view, amounts to nothing more than a definition of what we mean by the reality of things not given directly by experience. Since now in addition to satisfying the formal requirements, experience shows that a stress is most useful in correlating phenomena, we are justified in giving to this construct of stress a prominent place among our concepts.

Consider now another construct, one of the most important of physics, that of the electric field. In the first place, an examination of the operations by which we determine the electric field at any point will show that it is a construct, in that it is not a direct datum of experience. To determine the electric field at a point, we place an exploring charge at the point, measure the force on it, and then calculate the ratio of the force to the charge. We then allow the exploring charge to become smaller and smaller, repeating our measurement of force on each smaller charge, and define the limit of the ratio of the force to the charge as the electric field intensity at the point in question, and the limiting direction of the force on a small charge as the direction of the field. We may extend this process to every point of space, and so obtain the concept of a field of force, by which every point of the space surrounding electric charges is tagged with the appropriate number and direction, the exploring charge having completely disappeared. The field is, then, clearly a construct. Next, from the formal point of view of mathematics, it is a good construct, because there is a one to one correspondence between the electric field and the electric charges in terms of which it is defined, the field being uniquely determined by the charges, and conversely there being only one possible set of charges corresponding to a given field. Now nearly every physicist takes the next step, and ascribes physical reality to the electric field, in that he thinks that at every point of the field there is some real physical phenomenon taking place which is connected in a way not yet precisely determined with the number and direction which tag the point. At first this view most naturally involved as a corollary the existence of a medium, but lately it has become the fashion to say that the medium does not exist, and that only the field is real. The reality of the field is self-consciously inculcated in our elementary teaching, often with considerable difficulty for the student. This view is usually credited to Faraday, and is considered the most fundamental concept of all modern electrical theory. Yet in spite of this, I believe that a critical examination will show that the ascription of physical reality to the electric field is entirely without justification. I cannot find a single physical phenomenon or a single physical operation by which evidence of the existence of the field may be obtained independent of the operations which entered the definition. The only physical evidence we ever have of the existence of a field is obtained by going there with an electric charge and observing the action on the charge [when the charges are inside atoms we may have optical phenomena], which is precisely the operation of the definition. It is then either meaningless to say that a field has physical reality, or we are guilty of adopting a definition of reality which is the crassest tautology.

There can be no question whatever of the tremendous importance of the concept of the electric field as a tool in thinking about, describing, correlating, and predicting the properties of electrical systems; electrical science is inconceivable without this or something equivalent. But in addition to this aspect of the field concept, the further tacit implication of physical reality is almost always present, and has had the greatest influence on the character of physical thought and experiment. Yet I do not believe that the additional implication of physical reality has justified itself by bringing to light a single positive result, or can offer more than the pragmatic plea of having stimulated a large number of experiments, all with persistently negative results. It is sufficient to mention the fate of the attempt of Faraday and Maxwell to ascribe a stress like that of ordinary matter to the ether, which failed because, among other reasons, nothing can exist in the ether analogous to the strain of ordinary matter, to indicate the unfruitfulness of the idea of physical reality. It seems to me that any pragmatic justification in postulating reality for the electric field has now been exhausted, and that we have reached a stage where we should attempt to get closer to the actual facts by ridding the field concept of the implications of reality.

Another indispensable and most interesting construct is that of the atom. This is evidently a construct, because no one ever directly experienced an atom, and its existence is entirely inferential. The atom was invented to explain constant combining weights in chemistry. For a long time there was no other experimental evidence of its existence, and it remained a pure invention, without physical reality, useful in discussing a certain group of phenomena. It is one of the most fascinating things in physics to trace the accumulation of independent new physical information all pointing to the atom, until now we are as convinced of its physical reality as of our hands and feet.

A construct which had to be abandoned because it did not turn out to have physical reality, and which furthermore was not sufficiently useful in the light of newly discovered phenomena, was that of a caloric fluid.

The notion of "physical reality" is not of prime importance to this discussion of the character of our constructs; our definition of the meaning of physical reality may not appeal to everyone. The essential point is that our constructs fall into two classes: those to which no physical operations correspond other than those which enter the definition of the construct, and those which admit of other operations, or which could be defined in several alternative ways in terms of physically distinct operations. This difference in the character of constructs may be expected to correspond to essential physical differences, and these physical differences are much too likely to be overlooked in the thinking of physicists. We must always be on our guard not to forget the physical differences between a thing like a stress in an elastic body and an electromagnetic field.

The moral of all this is that constructs are most useful and even unavoidable things, but that they may have great dangers, and that a careful critique may be necessary to avoid reading into them implications for which there is no warrant in experience, and which may most profoundly affect our physical outlook and course of action.

THE RÔLE OF MATHEMATICS IN PHYSICS

Practically all the formulations of theoretical physics are made in mathematical terms; in fact to obtain such formulations is generally felt to be the goal of theoretical physics. It is then evidently pertinent to consider what the nature of the mathematics is to which we assign so prominent a rôle.

We have in the first place to understand why it is possible to express physical relations in mathematical language at all. I am not sure that there is much meaning in this question. It is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention. Furthermore, the mathematics in which the physicist is interested was developed for the explicit purpose of describing the behavior of the external world, so that it is certainly no accident that there is a correspondence between mathematics and nature. The correspondence is not by any means perfect, however, but there is always in mathematics a precise quality to which none of our information about nature ever attains. The theorems of Euclid's geometry illustrate this in a preeminent degree. The statement that there is just one straight line between two points and that this is the shortest possible path between the points is entirely different in character from any information ever given by physical measurement, for all our measurements are subject to error. It is possible, nevertheless, to give a certain real physical meaning to the ideally precise statements of geometry, because it is a result of everyday experience that as we refine the accuracy of our physical measurements the quantitative statements of geometry are verified within an ever decreasing margin of error. From this arises that view of the nature of mathematics which apparently is most commonly held; namely that if we could eliminate the imperfections of our measurements, the relations of mathematics would be exactly verified. Abstract mathematical principles are supposed to be active in nature, controlling natural phenomena, as Pythagoras long ago tried to express with his harmony of the spheres and the mystic relations of numbers.

This idealized view of the connection of mathematics with nature could be maintained only during that historical period when the accuracy of physical measurement was low, and must now be abandoned. For it is no longer true that the precise relations of Euclid's geometry may be indefinitely approximated to by increasing the refinements of the measuring process, but there are essential physical limitations to the very concepts of length, etc., which enter the geometrical formulations, set by the discrete structure of matter and of radiation. This is no academic matter, but touches the essence of the situation. There is no longer any basis for the idealization of mathematics, and for the view that our imperfect knowledge of nature is responsible for failure to find in nature the precise relations of mathematics. It is the mathematics made by us which is imperfect and not our knowledge of nature. [From the operational point of view it is meaningless to attempt to separate "nature" from "knowledge of nature".] The concepts of mathematics are inventions made by us in the attempt to describe nature. Now we shall repeatedly see that it is the most difficult thing in the world to invent concepts which exactly correspond to what we know about nature, and we apparently never achieve success. Mathematics is no exception; we doubtless come closer to the ideal here than anywhere else, but we have seen that even arithmetic does not completely reproduce the physical situation.

Mathematics appears to fail to correspond exactly to the physical situation in at least two respects. In the first place, there is the matter of errors of measurement in the range of ordinary experience. Now mathematics can deal with this situation, although somewhat clumsily, and only approximately, by specifically supplementing its equations by statements about the limit of error, or replacing equations by inequalities—in short, the sort of thing done in every discussion of the propagation of error of measurement. In the second place, and much more important, mathematics does not recognize that as the physical range increases, the fundamental concepts become hazy, and eventually cease entirely to have physical meaning, and therefore must be replaced by other concepts which are operationally quite different. For instance, the equations of motion make no distinction between the motion of a star into our galaxy from external space, and the motion of an electron about the nucleus, although physically the meaning in terms of operations of the quantities in the equations is entirely different in the two cases. The structure of our mathematics is such that we are almost forced, whether we want to or not, to talk about the inside of an electron, although physically we cannot assign any meaning to such statements. As at present constructed, mathematics reminds one of the loquacious and not always coherent orator, who was said to be able to set his mouth going and go off and leave it. What we would like is some development of mathematics by which the equations could be made to cease to have meaning outside the range of numerical magnitude in which the physical concepts themselves have meaning. In other words, the problem is to make our equations correspond more closely to the physical experience back of them; it evidently needs some sort of new invention to accomplish this.

We return later, in discussing Lorentz's equations of electrodynamics, to the disadvantages arising from the present undiscriminating character of mathematics. In the meantime, we must recognize that there are very important advantages here, as well as disadvantages. All experience justifies the expectation that the laws of nature with which we are already familiar hold at least approximately and without violent change in the unexplored regions immediately beyond our present reach. By assuming an unlimited validity for the laws as we now know them, mathematics enables us to penetrate the twilight zone, and make predictions which may be later verified. It is only when we are carried too far afield that we must deprecate this characteristic of our mathematics.

There is another aspect of the use of mathematics in describing nature that is often lost sight of; namely, that any system of equations can contain only a very small part of the actual physical situation; there is behind the equations an enormous descriptive background through which the equations make connection with nature. This background includes a description of all the physical operations by which the data are obtained which enter the equations. For instance, when Einstein formulates the behavior of the universe in terms of the world lines of events, the events as they enter the equations are entirely colorless things, merely 3 space and 1 time coördinate. To make connection with experience there must be a descriptive background giving the physical contents of the events; for example, there may be the statement that some of the events are light signals. This descriptive background is supposed to remain fixed, unaffected by any operations to which the equations themselves are subject. If, for example, the frame of reference of the equations is altered by changing its velocity, the physical significance of the descriptive background is supposed to remain unaltered, or rather no mention is usually made of this question at all. It would seem, however, that this matter needs some discussion. The descriptive background gets its meaning only in terms of certain physical operations. If the descriptive background remains unaltered when the uniform velocity of the frame of reference is changed, for instance, this means that the motion of the frame of reference does not at all affect the possibility of carrying out certain operations. This is pretty close to the restricted principle of relativity itself, which states that the form of natural laws is not affected by uniform velocity. Until a more careful analysis of the situation is made it would seem therefore that there is some ground for the suspicion that the principle of relativity is involved in the possibility of giving to physical phenomena a complete mathematical formulation, understanding "complete" to mean "including the descriptive background."

CHAPTER III

DETAILED CONSIDERATION OF VARIOUS CONCEPTS OF PHYSICS

According to our viewpoint, the concept of time is determined by the operations by which it is measured. We have to distinguish two sorts of time; the time of events taking place near each other in space, or local time, and the time of events taking place at considerably separated points in space, or extended time. As we now know, the concept of extended time is inextricably mixed up with that of space. This is not primarily a statement about nature at all, and might have been made simply by the observation that the operations by which extended time is measured involve those for measuring space. Of course historically the doctrine of relativity was responsible for the critical attitude which led to an examination of the operations of measuring time, but relativity was not necessary for a realization of the spatial implications of time, any more than the discovery of Planck's quantum unit h was necessary for the invention by Planck of his absolute units of measurement, although historically he was inspired to make this invention by discovering h, and in his own mind seems to have thought of the connection as a necessary one.[7]