CHAPTER IV. Special Views of Nature

SPECIAL VIEWS OF NATURE

IN this last chapter we propose to discuss certain special hypotheses about the structure of nature, and certain other matters that could best be left until we had examined our fundamental concepts.

We have seen that in setting up the general rules which are to guide us in describing and correlating nature, we have to take extreme care to allow no special hypotheses to creep in, as otherwise we might be restraining possible future experience. Even here there is no hard and fast line of separation of the general from the special, and one might entangle himself in inextricable difficulties if his ideals were too meticulous. How for example is the critic to be answered who says: "Your very endeavor to formulate principles so broad as not to restrict future experience means, when examined in the light of operations, that you are seeking for principles which past experience suggests will not limit the future. It is in the very nature of things impossible to escape all the implications of past experience and therefore to find any completely general principle." I believe that we must admit the critic is right, and that rigorously our goal is impossible of attainment. We may say in partial self-defense that all the discussion of this essay has been subject to one explicit assumption, namely, that the working of our minds is understood, which of course involves the assumption that our minds continue to function in the future in the same way as in the past. Even with this proviso we can not rigorously avoid the implications of the past, but there can be no practical question that we recognize certain assumptions about the behavior of nature to be so special as to limit seriously the physical possibilities, and other assumptions to be less restricting. In the previous discussion we had to make assumptions, but I hope these assumptions will be recognized by all with physical experience to be so broad as not to restrict us seriously. More special assumptions or hypotheses have their very great use, however, when we attempt to push forward the domains of experimental knowledge, because they may suggest new experiments or aid in correlating information already obtained. These special hypotheses may cover a very wide range of generality; some of them are general enough in character to be discussed here.

Among these special hypotheses there is a group which play an important part in the speculation of most physicists, and which have features in common. These are: the hypotheses of the simplicity of nature, of the finiteness of nature in the direction of the very small, and of the determinateness of the future in terms of the present. That these views have points of similarity is obvious if we consider a hypothetical special case. Suppose that no physical structure beyond the electrons and protons can be discovered, or is even suggested by any known phenomenon, so that the entire future behavior of a system can be determined by a specification of the present relations of all its protons and electrons; in this case nature would be both simple and finite and the future determined by the present.

THE SIMPLICITY OF NATURE

Of these hypotheses, perhaps the most important is that of the simplicity of nature, because of its wide spread diffusion and the effect it has had on physical thought. The hypothesis of simplicity assumes several forms; some physicists are convinced that the laws which govern nature are simple, others that the ultimate stuff of which nature is composed is simple (perhaps protons and electrons and energy), or there may be a combination of both views into the belief that ultimately we shall find simple ultimate elements behaving according to simple laws. In one respect it is obvious that nature is not simple, namely numerically--try counting the electrons or atoms or stars!

Consider now the first of these aspects of the thesis of simplicity, which may be expressed as the conviction that the behavior of the entire universe can be comprehended in a few principles of great breadth and simplicity, such as the inverse square law of force, or the second law of thermodynamics, or perhaps still better the equality of the elementary positive and negative charges, which apparently holds to an enormous degree of precision. In explanation of a view like this there is in the first place the mental urge, because we can take a satisfaction almost æsthetic in contemplating such a universe, and there is in the second place a strong suggestion from experience. Practically all the history of physics is a history of the reduction of the complicated to the simpler. For example, the behavior of a large part of the world of immediate experience can be reduced to the simple laws of mechanics. The behavior of another very large group of natural phenomena can be reduced to thermodynamics. The behavior of the heavenly bodies, which at first was described in a rather complicated way in the Ptolemaic system of astronomy, can be reduced to those same laws of mechanics which we find in our immediate neighborhood, with the one addition of the universal law of gravitation, which later refined experiment discloses is really active in our immediate surroundings. Similarly the laws of thermodynamics (except that part dealing with radiation) are reduced to the ordinary laws of mechanics through the additional assumption of the atomic structure of matter. Truly a stupendous accomplishment that may well color our whole future outlook. One may find great justification here for the belief that all nature will ultimately be reduced to a similar simplicity, and, in particular, justification for the attempt to find the explanation of all nature in the action of mechanical laws. Now, of course, as a matter of physical and historical fact, this program could not be carried through, but obdurate physical phenomena were discovered. Electric phenomena, which at first seemed so promising, refused to fit into the scheme, and the converse attempt, to explain mechanical effects in terms of electrical effects, also failed. We still carry our ordinary mechanical notions down into the realm of small electric effects, and still talk, for instance, about non-electrical forces which hold an electron together. Nor are there experiments affording sufficient basis for believing that all the mass of a positive nucleus is electrical in character. We also think of electrical charges as having the property of identifiability, which involves the possession of sharp edges and a change in the law of force at small distances, and this is certainly a property carried over from our large scale experience.

It seems fairly evident then that the laws of nature cannot be reduced to either those of mechanics or of electricity, nor probably, as is suggested by quantum phenomena, to a combination of both. This of course does not preclude the possibility that the laws still may be simple when expressed in other forms. An example of such a broad general law that goes deeper than mechanics or electrodynamics is probably afforded by the second law of thermodynamics when extended to include radiation phenomena.

Examples of attempts to find other such simple laws are Tolman's Principle of Similitude, and Lewis's theory of Ultimate Rational Units, and his recently enunciated principle of Complete Reversibility. The first two of these attempts I do not believe are successful, for reasons I have stated elsewhere, 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, 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 10^4 years, which may be plausibly taken to indicate that every 10^4 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. 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.

We can only conjecture whether the number of fundamental concepts will prove impossible of further increase or not, but present experience seems to give greater probability to the view that as we penetrate deeper the number of fundamental concepts will always tend to become fewer. We have already certainly one example in that the temperature concept disappears when we get to the atomic scale of magnitude, and possibly a second example in the building up of separate concepts for energy and frequency by the combination of great numbers of that one operationally simple thing which characterizes the elementary quantum process in ordinary radiation.

Different sorts of relations between concepts are conceivable in the transition zone where the number changes. We may find that other examples are like that of temperature, which is simply a statistical effect of a great many phenomena which may be described individually in terms of the ordinary concepts of mechanics, so that in this case the number of concepts changes merely by temperature dropping out, leaving the others more or less unaffected. Or all the concepts may be more closely interwoven, so that when the total number of concepts changes it may not be possible to separate out a group of concepts whose defining operations are unchanged. In such a case we must say that the original concepts are not applicable on the new level. The most immediate application of this idea has been already mentioned, namely, to the concepts of space and time. If the operations by which space and time are measured on the ordinary scale of magnitude cannot be carried down as a whole into the region of quantum phenomena, then we must say that the ordinary concepts of space and time are not applicable to these phenomena.

Closely connected with the sharper analysis of the operational structure of our concepts, we may expect in the future also a closer analysis of our inventions. This will take the form of a search for new physical facts which shall give to our inventions the character of physical reality. In case prolonged search fails to disclose such phenomena (as is probably now the case with the field concept of electrodynamics), we must then find some way of embodying explicitly in our thinking the fact that we are dealing with pure inventions and not realities.

INDEX

Absolute, 26 Absolute time, 4 Action at a distance, 46 Analysis of large into small, 51, 220 Arithmetic, 35 Atom, 59

Bell, 84 Birkhoff, 72 Black body, 112 Bohr, 190, 192 Born, 222 Boscovitch, 46 Bothe and Geiger, 116 Bridgman, 201 Brownian movement, 107, 129, 143 Bush, 142

Caloric fluid, 59 Campbell, 117 Carnot engine, 125 Causality, 80 ff Causal train of events, 85 Clifford, 28 Clock, 70 ff, 176 Compton, 116, 188 Continuity, 94 Conservation of charge, 135, 136 Conservative functions, 113 Constructs, 53 Correlation, 37 Cosmic units, 182 Cross word puzzle, 202

Descriptive background, 64 Determinism, 114, 209 ff Discontinuous space, 191 Döppler effect, 166 ds, 72

Eddington, 93 Einstein, vii, 1, 2, 3, 4, 7, 8, 9, 12, 13, 14, 64, 100, 155, 162, 163, 164, 167, 169, 170, 171, 172, 173, 175, 176, 177, 206 Electrical concepts, 131 ff Electrical explanation of universe, 50 Electrical mass, 139 Electric field, 56, 133 Empiricism, 3 Energy, 108 ff, 126 ff Euclidean space, 14, 15, 16, 18, 23, 52, 61, 67 Event, 95, 167 Explanation, 37 Explanatory crisis, 41 Extended time, 77

Faraday, 44, 57, 58, 209 Final explanations, 48 First law of thermodynamics, 126 ff Force, 102 ff Foucault pendulum, 180, 184 Fourth dimension, 74 Future, 222

Gauss, 15, 34 "Go and come" time, 112 Gravitational constant, 91

Haldane, 25 Heat flow, 130 Heisenberg, 222 Hertz, 44 Hoernlé, vi

Identity, 91 ff Isolation, 82

Joule, 124

Kelvin, 45, 110 Kinetic theory of gases, 40

Lagrangean equations, 112 Larmor, 149 La Rosa, 164 Length, 9 ff Lewis, 166, 201 Light, 150 ff Local time, 75 Lorentz, 143, 147, 148, 149

Mach, 183 Mass, 102 ff Mathematics, 60 ff Maxwell, 44, 58, 112, 137, 148 Meaningless questions, 28 ff Measurement approximate, 33 Mechanism, 45 Mercury, 105 Michelson, 15, 26 Michelson and Morley, 66 Models, 45, 52

Newton, 4, 110

Operational character of concepts, 5 Operational thinking, 32 Optical space, 67 Ostwald, 109

Penumbra, 34 Perrin, 107 Physical reality, 59 Planck, 69 Poincaré, 48, 115, 116, 190 Pythagoras, 61

Quantum act, 156 Quantum theory, 40, 47, 186

Radiation and temperature, 123 Relative character of knowledge, 25 Relativity, 150 ff Reynolds, 93 Rotational motion, 178 Russell, 205

Schrödinger, 222 Space, 66 Silberstein, 11 Simplicity of nature, 198 Simultaneity, 7, 8 Spring balance, 103 Statistical methods, 115, 117 Stress, 54 Swann, 204

Table top, 106 Tactual space, 67 Temperature, 118 ff Thermodynamics, 117 Thing traveling, 101, 152, 157, 164 Time, 69 Tolman, 201 Truth, 78 Turbulent motion, 120, 124

Velocity, 97 ff, 213 ff Velocity of light, 100

Whitehead, 167 Wilson, 224