ROTATIONAL MOTION AND RELATIVITY

Physically there is a great difference between the behavior of systems in uniform relative rectilinear motion and those in uniform relative rotation. The special theory of relativity states that there is a triply infinite number of systems with all possible uniform rectilinear velocities with respect to each other, in all of which physical phenomena have exactly the same mutual relations, that is, natural laws are the same. Now the mere formulation of the principle suggests the sense in which "system" is here used. It is obvious that "system" refers only to a part of the universe; we are not making a hopelessly academic statement about what would happen if we had an infinite number of universes to experiment with, but are talking about operations that may be approximately realized in our own universe. The "system" of the formulation we may think of as a completely equipped laboratory, out in empty space, so far from the heavenly bodies that they can have no effect. The different systems of the formulation are different laboratories, all built to exactly the same architectural blue prints. The phenomenon to which the postulates of relativity apply are phenomena which pertain entirely only to one or another of these laboratories. The meaning of this restriction is not completely definite and has, in any special case, to be judged partly by the context. Obviously, to see from the window of a laboratory another laboratory passing with a certain velocity cannot be counted as one of the allowed phenomena. Still less is it one of the allowed phenomena to observe that the center of gravity of the entire stellar universe has a certain velocity of translation with respect to the laboratory. The special principle of relativity contains by implication therefore the statement that certain very large and important classes of physical phenomena may be isolated and treated as taking place unaffected by the rest of the universe. Granted now the possibility of isolation, we have a second statement, which is usually treated as if it were the entire statement of the restricted principle, namely, that there is a triply infinite set of systems in which these phenomena run in the same way independent of the relative motion of the systems with respect to each other. When once the significance of the observation is grasped that absolute motion has no meaning in terms of operations, we see that this last statement takes immediately a most simple and satisfying aspect, in fact, so simple and inevitable that we are inclined to see in this the complete essence of the situation and regard the meaninglessness of absolute motion as affording peremptory proof of the restricted principle.

With this bias we now turn to examine the facts of rotary motion, and are disconcerted to find them quite different. No meaning in terms of measuring operations can be given to absolute rotary motion any more than to absolute translation, but nevertheless phenomena are obviously entirely different in different systems in relative rotary motion (phenomena of rupture, for example), so that apparently there are physical phenomena by which the concept of absolute rotary motion might be given a certain physical significance. Given two worlds like our own in empty space, but surrounded by impenetrable clouds, and each provided with a Foucault pendulum, then we believe that it is physically possible that we may find on one of these worlds the plane of rotation of the pendulum gradually changing in direction, while on the other it remains stationary. This difference we regard as possible without other accompanying physical phenomena which are causally related to the rotation of the pendulum (of course we have to make the two worlds of infinitely rigid material and eliminate other phenomena which we regard as purely incidental), so that we apparently have here a contradiction of our cardinal physical principle of essential connectivity. We are certainly not inclined to give up our principle, and we believe that as a physical fact, if the clouds could be evaporated, an observer in one world would find that he was rotating with respect to the system of the fixed stars, whereas the corresponding observer on the other world would find that he was stationary. Our principle of essential connectivity is therefore maintained, in that the rotation of the plane of the pendulum is connected with a rotation with respect to the rest of the universe of the entire world in which the pendulum is mounted. As far as I am aware, no other way of maintaining our principle has ever been suggested. But this demands that we give up our physical hypothesis of the possibility of isolating a system. There is here no question of limiting behavior; we believe that no matter how far our rotating world gets from the rest of the universe the Foucault pendulum would always behave in the same way; the system can never be isolated, but such local phenomena as the invariance of the plane of the pendulum are always essentially determined by the rest of the universe.

If now our system cannot be isolated, we must return to the phenomena of translational motion. In principle the act of isolation cannot be performed, the rest of the universe cannot be disregarded, and we should expect that different states of translational motion as well as different states of rotational motion with respect to the rest of the universe would have an effect on phenomena. We set ourselves the problem of understanding this apparent enormous difference between phenomena of translation and rotation. We remark that what apparently is a difference in principle may, in virtue of the approximate character of all measurement, be only a difference in magnitude, and that translational effects may exist too small to detect. A physical basis for such a difference may be found in the enormously different numerical values of translational and rotational velocities with respect to the rest of the universe attainable in practice. In describing phenomena of cosmic magnitude, we may plausibly measure the phenomena in units commensurable with the scale of the phenomena. Thus in measuring linear distances, we may perhaps choose as the unit of length the diameter of the stellar universe, and in measuring rotation, a complete reversal of direction with respect to the entire universe. This last means a change of angular orientation of 2 π, the first means a length of the order of 10⁶ light years. Measured in such cosmic units the angular velocities attainable in practice are incomparably greater than linear velocities. We now see that it is possible that the real state of affairs is as follows: namely, phenomena in any system are affected by motion with respect to the entire universe, whether that motion is of translation or of rotation, and the magnitude of the effect is connected with the velocity of the motion by a factor which is of the general order of unity when velocity is measured in cosmic units. This last is merely an application of the argument so often made in physics as to the order of magnitude of unknown numerical factors, and will be found expanded on page 88 of my book on Dimensional Analysis. The linear velocities attainable in practice are now so exceedingly low that their effect has not yet been detected experimentally, but angular velocities are high, and the effect is easily demonstrable. In this light the special principle of relativity is no different in character from any other physical law; it is only approximate, and some day our measurements may become refined enough to detect its limitations.

We have made a hypothesis here, which we may call the hypothesis of the immanence of the entire universe, namely, that isolation is impossible, or that the rest of the universe, no matter how distant, always has a local effect on at least some phenomena. This is essentially the hypothesis of Mach,[31] and leads to a situation which can, I think, be contemplated with logical equanimity, although it has always seemed to many physicists most highly antiphysical in character.

It must certainly be admitted that most physical experience justifies us in thinking that effects may be made as small as we please by getting far enough away from the cause of the effect. But if we accept the considerations of the preceding pages, we must be prepared to admit that as phenomena change in range their character may change, and that in these new realms we must, at first at least, be satisfied with a mere statement of correlations. Certainly we have very strong physical evidence of a formal correlation between the Foucault pendulum and the rest of the universe. But a correlation of this sort may be without significance because of its very breadth; we never can prove the significance of the correlation by performing an experiment with the rest of the universe absent. Have we really done anything more than merely get things into such a formal situation that they cannot be assailed, a possibility which the mere laws of our thinking seem always to leave open, as has been suggested, or is there any physical content to what we have done? We have seen that if our correlation is also suggested by other phenomena, then we may accept it as having physical content. Now there is just a glimmer of a suggestion that our hypothesis of the immanence of the universe may be needed in other ways. The gravitational constant and the velocity of light are always treated as arbitrary magnitudes thrust on the universe from outside with no connection with other phenomena. Nevertheless, I suppose that no one regards this situation as ultimately satisfactory and does not entertain the hope that some day we may be able to give some sort of account of the numerical magnitude of these constants. We have not hitherto succeeded in finding any connection between these constants and small scale phenomena such as the charge on the electron, its mass, etc., so that there is some plausibility in expecting that a connection may be sometime found with cosmic things; indeed general relativity theory already prepares us for exactly this possibility. Now the velocity of light and the gravitational constant control small scale experiments, for of course these two constants can be measured by local experiments, so that if the cosmic connection is found, we should have a control of local behavior by cosmic things, and therefore another example of the immanence of the entire universe. There is no need for me to waste time in apologizing for the highly speculative character of all this. It is worth while to emphasize, however, that our general considerations on the meaning of "explanation" have prepared us to admit as reasonable just the sort of explanation contained in the hypothesis of the immanence of the universe, and therefore to reserve a place in our physical thinking for possibilities of this sort, in spite of the fact that such considerations are not usually entertained, and may seem to many opposed to the spirit of physics.

QUANTUM CONCEPTS[32]

The history of quantum theory up to the present is a repetition in many respects of that of the early theories of electricity, in that all our thinking has been in mechanical terms. As far as we now know, quantum phenomena are always associated with atoms. We make for the atom a mental model with all the properties of the mechanisms of the ordinary scale of magnitude and with a few impressed properties in addition which represent the new quantum relations. As we now think of it, the atom has a massive core about which electrons revolve under an inverse square law, the connection between the mass of the electron, its acceleration, and the force acting on it being that usual in Newtonian mechanics.

The space in which the electron circulates is thought of as Euclidean, and the motion is described in time, which may be measured with clocks in the usual way. The general equations of electrodynamics do not apply; there are no propagation effects inside the atom, the motion of the electrons does not produce a magnetic field, and there is no radiation when the electron is in one of its possible stable states, in spite of the acceleration. We may, if we please, in working out the character of the motion, entirely neglect the electrical origin of the inverse square law, and treat this merely as an impressed force without further implications. Superposed on the ordinary spatial, temporal, and mechanical characteristics of the model are additional quantum properties, one which determines the particular orbit in which the electron moves [∫ pdq = nh], and another which determines the frequency of the radiation emitted when the electron passes from one allowed orbit to another. No mechanism is suggested to account for these quantum conditions, although the conditions are formulated in mechanical terms.

We now have to ask what is the meaning in terms of operations of our usual concepts of space-time and mechanics when applied to phenomena of this order. It is of course evident, as has already been emphasized, that the concepts have entirely changed in character, because we do not measure an electron orbit, for example, by stepping off the diameter with meter sticks, or by measuring the time required for light to travel across the diameter. The particular feature of immediate interest in this changed situation is the change in number of our concepts on the atomic level. I shall not attempt to find by an exact analysis the number of independent concepts at this level; probably such an analysis is not possible. We may, however, make an approximate suggestion. Apparently the most important concept in describing relations inside a quantum system corresponds to that of energy on the ordinary scale. Changes of energy determine the frequency of emitted radiation, as well as the relations during collisions of atoms and electrons; these collisional relations make direct connection with experiment through the voltages applied to electrons in collision experiments. The analogue of the momentum concept also seems to have independent significance, as shown by the Compton effect. The frequency of emitted radiation is also something with independent experimental significance. I believe that these three things are all that have direct significance for quantum experiments made up to the present time. In any event, it is perfectly evident that on the quantum level the concepts which at present have operational significance are considerably fewer than on the level of ordinary experience.

Apart from the question of convenience, there may be justification in continuing to use our old mechanical forms of thought if new experimental relations are thereby suggested. That a very large number of such as yet undiscovered relations may be suggested in some such way is at once evident. Thus we have no present knowledge of any phenomenon associated with what the electron does when passing from one energy level to another. How long does it take to make the passage? What is its path during passage? Is it subject to the ordinary laws of electrodynamics during passage? When and where is the radiation emitted that corresponds to passage? When the electron leaves one stable orbit is the orbit on which it will eventually land already determined? Does the radiation train emitted during a change from one energy level to another have a definite length in space, or may it have a variable length and correspondingly something that corresponds to variable amplitude? What happens to the radiation when the electron passages are interfered with before the emission of a quantum has been completed? What is the mechanism by which the quantum conditions are imposed? Is it not possible that part of the clew to the riddle of the manner of transition from purely quantum behavior to the behavior of classical mechanics may be found in the behavior of the electron during passage from one energy level to another? Certainly we have a tendency to the classical behavior under those conditions, such as at high temperature or in strongly condensed systems, in which the time occupied in passage might be expected to become a more important part of the total time.

Corresponding to these questions there should be many as yet undiscovered phenomena, and the mechanical point of view therefore has its value in suggesting experiments to detect such effects. It is of course too early to see what the final result will be here; we cannot tell whether eventually enough new experimental kinds of behavior will be found to restore the number of independent concepts to that of the level of ordinary experience or not, or whether indeed it will turn out that a greater number of concepts is required. It is contrary to our instincts to expect a greater number, and a smaller number now seems to us not unnatural, but the considerations of this essay should prepare us for either possibility.

It is often said that quantum phenomena are inconsistent with ordinary mechanics, and proofs of this assertion are often offered. I believe that no such proof, in the spirit in which the attempt is usually made, can be correct, for it seems to me that the remark of Poincaré applies, namely, that any sort of behavior can be imitated by a mechanical system, provided it is only complicated enough. A peremptory proof of this can be given to any one who is not a believer in vitalism. If a sentient being can be regarded as a mechanical system, we merely have to station inside each atom a Maxwell demon, with instructions to make the atom react according to quantum rules. Opposed to the spirit of this sort of reduction of quantum phenomena to mechanical terms, we have to remember that it makes sense to talk about the character of our conceptual structure only when the number of concepts is reduced to the number that have independent operational significance, that is, to the minimum number.

In the meantime let us examine what may be the significance in the light of present experiment of statements like those ascribed to Bohr that our usual concepts of space and time may be inapplicable in dealing with quantum phenomena. This idea is often given the more explicit form that space and time may be essentially discontinuous at the quantum level. From the operational point of view, it is most difficult to see exactly what this more explicit statement means, at least in terms of those operations by which length and time were originally defined. Thus if space were discontinuous, it might mean that a point exists which may be reached by laying off a meter stick fourteen times, for example, and another point by laying off sixteen times, but that no point can be found with fifteen applications. Such a state of affairs seems to be inconsistent with our definition of the counting operation and to have no concern with any properties of space; for what shall we mean by laying off a meter stick sixteen times if it cannot be laid off fifteen times? It is conceivable that space might end, in the sense that beyond a certain limit there might be some irremovable physical hindrance to the continued laying off of distances with a meter stick (although I think that we should be inclined to describe such a state of affairs in terms of matter enclosing empty space rather than as the end of space), but to say that space may be discontinuous seems to be meaningless. In the same way, I believe it meaningless to speak of discontinuous time. We may have phenomena discontinuous in space and time, but not discontinuous space or time.

It seems then that we must give up the idea that in the quantum domain the usual concepts of space and time may fail, in the specific sense that they may become discontinuous. What may we understand by the failure of these concepts in a more general sense? No one of course would expect that even eventually the concepts will have the same operational significance for the inside of an atom that they have on the ordinary scale; it must be a modified sort of concept with which we are concerned, such as we have already seen is given by the field equations of electrodynamics. If now the number of operationally independent concepts on the quantum level turns out to be the same as on the level of ordinary experience, and if there is also the possibility of continuous transition from the operations of the quantum domain to those of ordinary experience, then it seems to me that we should say that our usual concepts of space and time still apply in the quantum domain. But if the number of operationally independent concepts is either greater or less than on the ordinary level, then I believe we must say that the ordinary concepts of space and time cannot apply. One might still look for the possibility of separating out from the complex of concepts on the quantum level a group which might change continuously to those of space and time on the ordinary level, but I think that such a possibility is very remote when one considers that the total number of concepts changes, and that in the zone where the number changes the definitions are not unique by which one extrapolates a concept from one domain to another.

If Bohr's idea is true that space and time cannot be used in describing ultimate quantum phenomena, one of the most immediate implications in terms of experiment might be that phenomena corresponding to intermediate positions of the electron between stable orbits do not exist.

Finally, we must comment on the general tactics of the quantum situation. It would seem that there have already been a sufficient number of unsuccessful attempts to formulate quantum behavior in terms of ordinary mechanics to justify the expectation that ultimately something quite different must evolve. The difficulties of an unmodified carrying over of ordinary mechanical notions to quantum phenomena may be illustrated by a simple example. Consider a particle of mass m rotating in a frictionless circular track of radius r. Then according to quantum conditions it can move stably on this track only with certain definite velocities, such that ∫ pdq = mv 2πr = nh. Suppose now the particle rotating with one of the allowed velocities, and a tangential force applied. If the usual mechanical notions of force are still valid, the particle must respond by moving in its track with continually increasing velocity. After the velocity has been increased by a small amount, we remove the force. The motion is now no longer one of the allowed ones, and the particle must in some way change its velocity; it must either slow down or speed up. In the first case it must either radiate energy, which a system of the simple mechanical properties we have supposed is not capable of doing, or else the law of conservation of energy fails, and also Newton's first law of motion during the process of acquiring the steady condition. If, on the other hand, the particle speeds up, it must increase its energy from nowhere, and again ordinary mechanics does not apply.

It seems then a mistake to attempt to formulate the quantum conditions in terms of the notions of ordinary mechanics (momentum, and position coördinates in either the ordinary or the generalized Lagrangean sense). It would seem, on the other hand, plausible to expect that mechanics is not a fundamental thing, but is in some way an effect produced by the aggregate action of a great many elementary quantum processes. Amplitude of radiational vibration, for example, may be such a statistical aspect of a great many processes, in some such way as on the ordinary level of experience temperature is a statistical aspect of the average kinetic energy of the atoms. One possibility of this kind has already been more explicitly indicated; in the elementary process of emission of radiation, frequency and energy are not two independently assignable variables, but are connected [E = hν]. That is, on the quantum level radiation has only a single property, which is properly neither energy or frequency. [We are now neglecting the polarization aspect of radiation.] On a higher level, that of ordinary radiation, the single elementary property has expanded itself into two (energy and frequency) through the additional variable of the number of elementary quantum processes in the complex radiation.

The program of the immediate future should be an extension of something of this sort, namely, to invent new concepts corresponding to the experimentally independent things on the quantum level (such perhaps as the resultant of the fusion of the energy and frequency concepts for radiation), and then to show how the ordinary concepts of mechanics (and very likely those also of space and time) are generated by statistical effects in aggregates of great numbers. Perhaps it is yet too early for an attempt of this sort, because it may seem that there are still too many possibilities of new experimental discoveries which might upset the results of elaborate theoretical speculation. If this should really be felt to be the case, I believe that physics ought for the present to hold in partial abeyance its theoretical activities in this field, and devote itself to acquiring as rapidly as possible the necessary experimental facts. We may emphasize again that the possibility of carrying out this plausible program can be proved only by experiment; it may be that more concepts will be required on the quantum level than for ordinary experience.

The invention of new concepts is certainly not an easy thing, and is something which physics has always deliberately, and perhaps justifiably, shirked, as shown by the persistent attempts to carry the notions of mechanics down into the finest structure of things. This shirking has not had bad results, but on the contrary good results, as long as physics has been primarily concerned with phenomena near the range of ordinary experience, but I believe that as we get farther and farther away from ordinary experience, the invention of new concepts will become an increasing necessity.

CHAPTER IV

SPECIAL VIEWS OF NATURE

Examples of attempts to find other such simple laws are Tolman's Principle of Similitude,[33] and Lewis's theory of Ultimate Rational Units,[34] and his recently enunciated principle of Complete Reversibility.[35] The first two of these attempts I do not believe are successful, for reasons I have stated elsewhere,[36] the third also seems somewhat doubtful.

With regard to the general question of simple laws, there are at least two attitudes; one is that there are probably simple general laws still undiscovered, the other is that nature has a predilection for simple laws. I do not see how there can be any quarrel with the first of these attitudes. Let us examine the second. We have in the first place to notice that "simple" means simple to us, when stated in terms of our concepts. This is in itself sufficient to raise a presumption against this general attitude. It is evident that our thinking must follow those lines imposed by the nature of our thinking mechanism: does it seem likely that all nature accepts these same limitations? If this were the case, our conceptions ought to stand in certain simple and definite relations to nature. Now if our discussion has brought out any one thing, it is that our concepts are not well defined things, but they are hazy and do not fit nature exactly, and many of them fit even approximately only within restricted range. The task of finding concepts which shall adequately describe nature and at the same time be easily handled by us, that is, be simple, is the most important and difficult of physics, and we never achieve more than approximate and temporary success. Consider the example of time. The original concept of local time, which for long seemed satisfactory, turns out to be inadequate, and has to be replaced by extended time, which is so complicated that it is questionable whether we shall ever be able to grasp it with the confidence that we must demand in a useful concept (by "grasp" I mean intuitive command of all the implications of the operations which are involved). The concept has not yet been found which describes simply the temporal relations of the universe.

Not only are concepts hazy around the edges and so incapable of fitting nature exactly, but there is always the chance that there are concepts other than those which we have adopted which would fit our present phenomena. Finding concepts to fit nature is much like solving a cross-word puzzle. In the puzzle there may be some parts of the pattern which we fill completely and easily, but sometimes we find parts in which we can fill in everything except one or two obstinate definitions, so that we are sure we are on the right track, and rack our brains for the missing words, when with a flash of inspiration we see that the obstinate words can be fitted in by a complete change in those which we had already accepted. It may be that we are soon to witness a similar change in our concept of the nature of light. An important difference between the cross-word puzzle and nature is that we can never tell when we have filled in all the squares in any of the parts of nature's puzzle; there is always the possibility of new phenomena which our present scheme does not touch.

Considering, then, the nature of our conceptual material, it seems to me that the overwhelming presumption is against the laws of nature having any predisposition to simplicity as formulated in terms of our concepts (which is of course all that simplicity means), and the wonder is that there are apparently so many simple laws. There is this observation to be made about all the simple laws of nature that have hitherto been formulated; they apply only over a certain range. We have not extended the laws of gravitation to small bodies, nor have we found that our electrical laws will work on a cosmic scale. It does not seem so very surprising that over a limited domain, in which the most important phenomena are of a restricted type, the conduct of nature should follow comparatively simple rules.

A tempting question is whether there may not be some laws of nature that are really simple, without relation to our mode of formulation, such as the law of the inverse square. I leave it to the reader to decide whether this question has meaning. In this connection it is possibly significant that the average physicist is strangely reluctant to tamper with the inverse square law. I find in myself a lack of sympathy, which I cannot justify by any of the considerations of this essay, with attempts like the recent one of Swann,[37] for example, to explain a wide variety of hitherto obstinate effects by the assumption of slightly unequal departures from the inverse square law by the electrons and protons. Of course I hope that this feeling will turn out not to be prejudice, but will perhaps be justified by some such general observation as that a departure from the inverse square law so slight as by definition to be forever beyond detection by direct experiment is meaningless; but of this I am not at all sure.

We are now ready to consider the second respect in which nature may be simple, namely, because the material of which it is built may reduce to a few sorts of elements. In this discussion it will be convenient to consider also at the same time the more inclusive simplicity arising from simple laws acting on simple elements. The immediate question for us here is one of fact: does nature seem to be getting intrinsically simpler as we get toward small scale phenomena? There is much room for difference of opinion here; personally I feel that this expected simplicity is not in evidence, at least to the extent that we could desire. For instance, the fact that the electrons must have both electrical and mechanical properties is a straw in this direction.

It must also be remembered that a certain simulation of simplicity is inevitable as we approach the limits of experimental knowledge, whatever the actual structure of nature, for the mere reason that near the limit our possible experimental operations become fewer in number, and our concepts fewer also. The question which we are trying to answer has, therefore, its real meaning only in terms of the possible future. Do we believe that if we drive in our stakes at a certain point on our present frontiers, this point will gradually, as physics advances, become possessed of a continually richer experience, so that nature at this point will appear increasingly complicated? Or do we expect a termination of this process of expansion fairly soon? It seems to me that as a matter of experimental fact there is no doubt that the universe at any definite level is on the average becoming increasingly complicated, and that the region of apparent simplicity continually recedes. This, however, is not the opinion of all observers. Thus Bertrand Russell, in "What I Believe", page 10, writes, "Physical Science is then approaching the stage where it will be complete, and therefore uninteresting."

This is perhaps a particularly favorable epoch in the history of physics to urge the essential complexity of nature, because all our new quantum phenomena indicate a vast wealth of hitherto unsuspected relations on the very edge of the attainable. There is one aspect of quantum relations, as also of our ideas of the nature of the structure of the nucleus of the atom, which is particularly significant in this respect, namely, that we have to describe phenomena by statistical methods. Now a statistical method is used either to conceal a vast amount of actual ignorance, or else to smooth out the details of a vast amount of actual physical complication, most of which is unessential for our purposes. There can be no doubt of the amount of ignorance that the statistical method conceals when applied to these phenomena, but there are also strong indications, particularly when applied to the nucleus, that it covers a vast amount of actual physical complications. The nucleus of a radium atom becomes unstable on the average every 10⁴ years, which may be plausibly taken to indicate that every 10⁴ years the radium nucleus gets itself into some particular configuration. Considering the time scale on which we suppose events in the atom to take place, and also considering the fact that radioactive disintegration seems unaffected by outside agencies, this would indicate a perfectly appalling amount of structure. We are similarly driven to statistical methods in quantum theory, as for example, in Einstein's analysis of the details of equilibrium between emitting and absorbing atoms and radiation.

In general, we cannot admit for a minute that a statistical method, unless used to smooth out irrelevant details, can ever mark more than a temporary stage in our progress, because the assumption of events taking place according to pure chance constitutes the complete negation of our fundamental assumption of connectivity; such statistical methods always indicate the presence of physical complications which it must be our aim to disentangle eventually.

It appears then that present experimental evidence makes very probable structures beyond the electron and the quantum; we may go even further and say that there is no experimental evidence that the sequence of phenomena in nature as we go to ever smaller scales is a terminated sequence, or that a drop of water is not in itself essentially infinite. (This statement contains by implications the meaning that we attach to infinite.) All the more, then, there is no evidence that nature reduces to simplicity as we burrow down into the small scale.

Whatever may be one's opinion as to the simplicity of either the laws or the material structure of nature, there can be no question that the possessors of some such conviction have a real advantage in the race for physical discovery. Doubtless there are many simple connections still to be discovered, and he who has a strong conviction of the existence of these connections is much more likely to find them than he who is not at all sure they are there, and is merely hunting for anything that may turn up. It is largely a matter of psychology. Everyone knows that the mere suggestion that a problem has a solution, or the knowledge that someone has already solved it, is often sufficient to suggest a relation that otherwise might not have been noticed. The chances are, therefore, that the relations between phenomena will be found by those who are previously convinced that the relations exist. The observation that most of the discoveries are made by men with particular sorts of conviction naturally strengthens the belief that their convictions are true. But this picture has an obverse side. The man who is convinced that there is a relation where none exists may waste all his time in vain seeking for it. Granted that nature has no particular predisposition to simple relations, the conviction that there are such relations is, from the point of view of any one individual, as likely to be a hindrance as a help. From the point of view of physical society, on the other hand, it is desirable that there be such convictions, for in such a society there will be more discoveries than in a society without such convictions. We have here again the old conflict between the individual and society. As in all other similar conflicts, society will not be able to demand permanently from the individual the acceptance of any conviction or creed which is not true, no matter what the gain in other ways to society. If nature is not simple, physicists will not continue to believe that it is, even if such a conviction does increase the total number of discoveries. It is an impossible attitude to expect that one can maintain. Does this then mean that physics is to face a drab future, becoming continually more prosaic, with new discoveries ever rarer, made by a continually decreasing number of misguided but fortunate enthusiasts? There may be such a danger, but the greatest part of the danger is avoided if its nature is clearly recognized. One of the problems of the future is the self-conscious development of a more powerful technique for the discovery of new relations without the necessity for preconceived opinions on the part of the observer.

There is an aspect here of our physical research that is often lost sight of, namely, the small proportion of successful discoveries compared with the number of investigators. Certainly the number of unsuccessful attempts, even in the case of those fortunate individuals who make the great discoveries, is very much greater than the number of their successful attempts. (Faraday's reputed satisfaction with a ⅒% return comes to mind.) This must always be taken into account in estimating the probable chances of correctness of any new theory. With so many physicists working to devise new theories, the chances are high that many false theories will be found, in which a number of phenomena may apparently fit together into a new relation, but which eventually prove to be inconsistent with other phenomena, so that the proposed theory has to be abandoned. As physics advances and the number of investigators and the amount of physical material increases, one has to be more and more exacting in one's requirements of a new theory. One must be particularly on guard against numerical coincidences. An interesting chapter might be written on numerical relations which have been hopefully published, but later had to be abandoned as without significance.

DETERMINISM

If we are right in supposing that physical evidence gives no warrant for the idea that nature is finite downward, we have not only repudiated the thesis of simplicity but we have also made a very important observation on the other general thesis mentioned at the beginning of this chapter, namely, the thesis of physical determinism. By determinism we understand the belief that the future of the whole universe, or of an isolated part of it, is determined in terms of a complete description of its present condition. [What we mean by present condition will be discussed later.] It is popularly assumed that every physicist subscribes to some such thesis as this. But now if there is infinite structure even in a small isolated part of the universe, a complete description of it is impossible, and the doctrine as stated must be abandoned. It seems to me that all present physical evidence prepares us to admit this possibility. I suppose, however, that most physicists would subscribe to some modification of the original thesis, perhaps along the following lines. Given a description of an isolated part of the physical universe in the most complete terms that have physical meaning, that is, down to the smallest elements of which our physical operations give us cognizance, then the future history of the system is determined within a certain penumbra of uncertainty, this penumbra growing broader as we penetrate to finer details of the structure of the system or as times goes on, until eventually all but certain very general properties of the original system, such as its total energy, are forever lost in the haze, and we have a system which was unpredictable. I suppose that it is a further conviction of at least many physicists that by sufficiently refining our measurements, the amount of haze at any fixed point in the future may be made indefinitely small, and many might even go further and hope by studying the haze (perhaps statistically) to obtain some inferential evidence of structure beyond that yet experienced. In fact it may be that this last contains the germs of the ultimate method of investigation, if we ever reach a stage when we can no longer refine our methods of measurement.

Determinism to the physicist is simply a way of stating certain implications of his conviction of the connectivity of nature. We have seen that the broadest possible statement of the thesis of connectivity is: Given two isolated systems with identical past histories up to a certain epoch, then the future histories will also be identical. The thesis of the determinism of the future by the present constitutes a specialization of this general thesis in that we suppose that identity of all past history is not necessary for identity of future behavior, but only identity of present condition. The general and the special thesis are not equivalent by any means: if past histories are identical then present conditions are also identical, but the converse does not necessarily hold at all.

Now I believe that the general thesis (which I suppose all physicists will admit, but whose truth is nevertheless subject to the verification of experience) gets turned into the special thesis by a feeling of somewhat metaphysical content, which we may perhaps state by saying that we can see no way by which the past can affect the future except through the present. We do not like to think of the effect of a cause distant in the past jumping over the present and affecting the future without touching the present at all. It is the analogue of that attitude of mind to which action at a distance in space is inconceivable; just as it is difficult to conceive of a body here affecting a body there without in some way an action propagated through intermediate space, so we do not like to think of a past cause jumping over time and producing a future effect without some sort of continuity in the causal chain through all intermediate time.

So far our discussion has been purposely loose: it is evident that what we mean by "present state" is crying for definition. What is meant by this may depend somewhat on the specific hypothesis that one adopts about the structure of nature. Historically the conviction of future determinism has been most intimately associated with a mechanical picture of the structure of the universe, so that it may be well to begin from this point of view. Suppose the simplest possible system composed of point masses without structure, as in the kinetic theory of gases. What sort of specifications do we believe necessary to fix the present state of such a system? The mechanical view of nature gives a definite answer. By present state we mean the positions and velocities of all the masses. This is sufficient for the complete determination of any purely mechanical system, in which the forces between the elements are known functions of only their relative positions. By a sort of extension of these ideas valid for mechanical systems, it seems to be often thought that the present state of any system is determined by a complete specification of the positions and velocities of all the ultimate elements of the system (provided always of course that this number is finite). This principle, however, does not appear to bear the check of experiment when applied to electrical systems with radiation. The theorems of the retarded potential show that such systems are determined by the present position and velocities of the charges in the immediate vicinity, and by the corresponding data at remote points given for proper epochs in the past; in this case, therefore, past and present history are necessary to determine the future. But if we consider the electrical field as part of the system, we may fix the future in terms of the present positions of the charges, their velocities, and the values of the field vectors all over space, thus returning to a certain formal resemblance to mechanical systems, and suggesting a reason for ascribing physical reality to the electric field. This analogy with a mechanical system is, however, loose; complete analogy would allow the instantaneous values of the time derivatives of the field to be given also, and this is not possible.

How is it that velocity can strictly be regarded as characteristic of the present state of the system? Certainly the usual operations for measuring velocity demand that we know the configuration of the system at two different times, and calculate the velocity from certain differences of the system at these two times. The velocity is defined as a limiting result, but even in the limit the essential physical fact does not disappear that we must know the positions of the system at two times. We may now go further; if the velocity is properly included in the present attributes of the system, we can see no reason for not including a specification of all the higher time derivatives also. In the case of the simple gaseous system under present consideration we can answer this question by examining the operations by which we actually go to work to determine the future of such a system. The problem of determining the future condition of such a system reduces to the problem of writing the differential equations of motion of all its parts. If the system is a mechanical system, as in this case, these equations are of the second order in the time derivatives of the position coordinates, and also involve the forces, which we suppose are known in terms of the relative positions of the parts of the system. Given, then, the positions and the way in which the forces depend on the relative positions of the parts, the equations of motion can be written down for any configuration of the system, and these equations may be integrated (at least approximately) in terms of the proper initial conditions. Now the only boundary conditions on a second order equation are the initial positions and velocities. This is the reason that velocities have to be specified in giving the present condition of the system, and that it is not necessary to give the higher derivatives. Apparently the reason why we instinctively include velocity among the present properties of the system is not because velocity is by its nature strictly a present property of the elements of the system, but rather because our wide experience with mechanical systems has shown that as a matter of fact velocity is necessary in such systems to determine future motion.

But now if the equations of motion of the parts of the system are not those of mechanics, they will in general be much more complicated in appearance and will involve higher derivatives of the time than the second. Suppose for the moment that the equations contain only derivatives and the mutual positions of the parts of the system. Then to integrate the equations and determine the motion we have to know the initial positions and the initial values of all the derivatives up to an order one lower than the highest which occurs in the equations. The equations of motion of an electron are even more complicated than this, in that the positions of distant parts of the system have to be given throughout an interval of time instead of merely an instant. It would seem that the feeling that the present state of a system may be determined in terms of positions and velocities does not as a matter of fact apply to all the systems of our experience.

The discussion up to this point has been subject to the fundamental assumption that the behavior of the system is entirely determined if we can give the position of each part as a function of time. This assumption is implicitly contained in Einstein's formulation of the general principle of relativity, namely, that there is nothing more to a physical system than a set of space-time coincidences, and that the system is fixed in terms of the space time coördinates of all its parts. Already in discussing the assumption of relativity we have indicated reasons for dissatisfaction with this as a means of reproducing all experience, because in giving only the space-time coordinates of events we have entirely omitted the descriptive background of the equations, which gives physical color to the system in question. This discussion also assumes that a specification of the positions, velocities, and higher derivatives (if necessary) of the elements of the system is possible, which amounts essentially to the assumption that the system contains only a finite number of elements. Now in view of the experimental fact that there is no reason for supposing that the structure of the universe is finite, this conclusion must be modified, but I do not believe that the necessary modification affects the essential argument. In view of the possible infinite structure it would seem that we cannot expect more than that the future is determined by the present within a certain penumbra of uncertainty, and this penumbra may be made less important by digging down deeper into the structure when specifying the present condition.

We have also slurred over the ambiguities in "present" condition when the system is spread over space. Probably a unique ascription of meaning to "present" is not possible for an extended system, but at least one possibility is indicated by relativity theory. Imagine a staff of assistants distributed throughout space, each equipped with clocks synchronized and set with the master clock by light signals in the conventional manner, and each fully equipped with the necessary measuring instruments. Then what we mean at this point of the argument by "present" state of the system is the aggregate of all the information about the positions and velocities of the ultimate elements which I determine in my immediate vicinity at my origin of time plus the reports of similar observations made by all the assistants, each local observation being made at the time origin of each local clock.

Going back now to the main argument, we have shown that the feeling that the present condition of the universe may be specified in terms of positions and velocities arose from experience with purely mechanical systems, and that the more general formulation, in which we add to the velocities the higher time derivatives, applies only to systems in which the ultimate elements move according to differential equations of higher order than the second. Furthermore, our analysis seems to have shown that systems in which there is radiation do not allow a determination of the future in terms of a present condition specified in terms such as these. It seems, however, that the general principle of the determinism of the future by the present may be saved by a change in the definition of what we mean by the present condition of the system, ridding it of its mechanical and other special implications, and making more immediate connection with direct experiment. Let us understand by present condition of a system the aggregate of all information that can be obtained by any physical means whatever, with any sort of physical instrument, not attempting to get out of this analysis information about hypothetical ultimate physical elements, with the proviso that the measurements are to be made now, extending the concept of "now" to points distant in space in the way intimated above. With such a general definition of the meaning of "present" we can now deal with systems in which there is radiation, noticing that our assistant observers must be stationed throughout apparently empty space as well as in the neighborhood of matter. That this does adequately cover the case of radiation is suggested by considering again the two systems of dark lanterns with screens and distant mirrors which we have previously considered, in one system a light signal having been despatched 0.5 second ago and in the other 1.5 seconds ago. Our thesis demands that there be some present difference in these two systems, because their future history is different, in one of them a light signal arriving after the lapse of 1.5 seconds, and in the other after only 0.5 second. Now there is a present difference as reported by our assistants, for the assistant stationed half way between lantern and mirror reports in one system a flash of light on the side of a screen which is turned toward the lantern, and in the other system on the side of the screen turned toward the mirror.

This more general point of view answers the question whether velocity may be regarded as a present attribute of the system, for the parts of a system which are in motion have momentum, and momentum may be detected by placing against such parts comparatively rigid members which will receive a minute deformation, so that velocity has a meaning in terms of physical measurements made at a single instant of time.

There is a subtle and difficult question here, namely, whether in talking about operations of measurement we can ever get rid of temporal implications, and therefore, whether a condition of the system in which temporal implications remain can properly be described as "present." I shall not attempt to answer this question: there must be some practically satisfying answer, involving perhaps the physical analogue of differentials of different orders in mathematics, short of carrying the analysis to such a degree of refinement that the concept of present becomes meaningless, as we can see might easily happen.

With this enlarged understanding of what we mean by present state of the system, it seems to me that physical evidence is now rather favorable to the view that the present determines the future, subject to qualification about the penumbra, at least as far as large scale phenomena are concerned. It appears much more doubtful when we come to small scale phenomena, and in particular it is doubtful whether the principle can be applied to the details of the quantum process, and in fact it is not certain that it has meaning. It is certain that if it is true an enormous amount of structure beyond any that has yet been detected is implied.

ON THE POSSIBILITY OF DESCRIBING NATURE
COMPLETELY IN TERMS OF ANALYSIS

There is a certain thesis that is loosely related to the view that nature is finite downward, namely, that an explanation of the universe is possible in which we start with small scale things, and explain large scale phenomena in terms of their small scale constituents, the thesis, in other words, that all the properties of the large are contained in the properties of the small and that the large may be constructed out of the small. Some such thesis as this seems implied in the general attitude of many physicists. Let us examine the physical basis for this. To maintain this thesis would demand that aggregates of things never acquire properties in virtue of their numbers which they do not already possess as individuals. Is this true? Consider, for example, the two-dimensional geometry on the surface of a sphere. This is non-Euclidean. Is the geometry of the individual elements of the surface of the sphere non-Euclidean, or do they acquire this property in changing scale? Is the kinetic energy of a number of electrons all moving together in such a way as to constitute an electric current the sum of the kinetic energies of the individual electrons, or is there an additional term? Is the mass of an electron the sum of the masses of its elements?

A mathematical consideration is suggestive here. Those properties of a system which can be described in terms of linear differential equations have the property of additivity; the effect of a number of elements is the sum of the effects separately, and no new properties appear in the aggregate which were not present in the individual elements. But if there are combination terms (as in the electrical energy, which contains the square of the field), then the sum is more than (or different from) its parts, and new effects may appear in the aggregate. Now of course the linear equation is of enormous importance in describing nature, but many examples of systems with other types of equation can be found, as that above for electromagnetic mass. In expecting to find in nature such non-additive effects, we need not commit ourselves at all to the view that nature is governed by differential equations, but by analogy may expect similar effects if difference equations, for instance, should prove to be fundamental, or even something beyond present mathematical formulation.

It is certainly very much easier to handle a system physically if the total action can be built up from that of its parts, because the analysis which establishes the connection between the elements is easier to perform. It is obviously easier to show that an explanation in such terms is correct, because we have seen that explanation involves making experiments with representative elements absent or altered, and it is easier to vary the small things than the large things. Those explanations which involve working from the small up will therefore be made first, and will appear to be of disproportionate importance. Places where I look for an explanation from the large to the small are perhaps in accounting for the values of the gravitational constant and the velocity of light and in those phenomena which general relativity theory indicates may depend on all the matter in the universe, as the Foucault pendulum experiment. We must, of course, also be prepared for such non-linear effects in the domain of unexplored quantum phenomena.

A GLIMPSE AHEAD

Some of the general considerations of this essay may, with considerable plausibility, be expected to play a part in the future of both speculative and experimental physics. The most important effect may be expected from the clearer recognition of the operational character of our physical concepts. Indeed during the writing of this essay there has been a very marked increase in emphasis on the necessity of understanding in terms of physical operations such fundamental concepts as that of the electron, by the new quantum mechanics [the mechanics of Heisenberg-Born and Schrödinger of 1925-26].

We are to expect then in the first place a more self-conscious and detailed analysis of the operational structure of all our physical concepts. [It has been beyond the scope of this essay even to begin to attempt a systematic and thoroughgoing analysis of this character.] This future analysis will show precisely how, as we extend the range of experience, the physical character of the operations changes by which we define our concepts, as, for example, in mechanics the notion of force disappears at high velocity and is replaced perhaps by the notion of momentum. In the region of change in the nature of our concepts, special study will be made of the accuracy of our physical measurements, and new experiments devised of greater accuracy, in order that we may know precisely to what extent the new concepts are equivalent to the old. Past experience suggests that we may perhaps expect to find new phenomena especially in those regions where the difficulty of carrying out the usual operation forces us to change the operational character of our concepts. There will be questions of a more or less formal nature to answer, as for example the best way of extending concepts when there are several possible courses open to us.

We may expect more interesting results, however, when we get so far beyond ordinary experience that the character of the possible physical operations has become so restricted as to result in an apparent decrease in the number of independent concepts. It seems plausible to expect that the structure of nature is more fundamentally connected with the number of independent concepts necessary for a complete description than with the precise details of the structure of the individual concepts, such, for example, as whether space is measured optically or tactually. In those regions where the number of concepts decreases, we must make the most thoroughgoing experimental examination to discover if possible new sorts of operations by which the number of concepts may be brought back to normal. In searching for such new experimental operations it seems to me that by far the greatest promise for the immediate future is offered by improvements in our powers of dealing with individual atomic and electronic processes, such as we now have to a limited extent in the various spinthariscope methods of counting radioactive disintegrations, or Wilson's β-track experiments.[38] In this self-conscious search for phenomena which increase the number of operationally independent concepts, we may expect to find a powerful systematic method directing the discovery of new and essentially important physical facts.