The logic of modern physics

by P. W. Bridgman

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.