The logic of modern physics

by P. W. Bridgman

Now it appears to me most questionable whether the analysis of nature into events is possible or sufficient. With regard to the coincidence point of view, it seems perfectly obvious that the world of our immediate sensation cannot be described in terms of coincidences; how, for example, shall we describe in terms of space-time coincidence the photometric comparison of the intensity of two sources of illumination, or the comparison of the pitch of two sounds, or the location of a sound by the binaural effect?

[29]A. N. Whitehead, An Enquiry Concerning the Principles of Natural Knowledge, Cambridge University Press, 1919, Chap. V.

To justify the coincidence point of view we apparently have to analyze down to the colorless elements beyond our sense perception. It does not seem unreasonable, perhaps, to expect that the universe is completely determined in terms of the positions as a function of time of all the positive and negative electrons; but to introduce such a thesis now certainly goes beyond present experimental warrant, and is contrary to the general spirit of relativity, which nowhere else involves any reference to the small scale structure of things. Even if we were willing to overlook all these objections, we would still have the fact that the difference between a positive and negative electron is not contained in any specification of the mere coordinates.

A further very important doubt in principle as to the possibility of the analysis of nature into events is afforded by the character of the concept of event itself. We have seen that the idea of event involves the existence of discontinuities, and that this can correspond only approximately to the physical fact, because discontinuities apparently lose their abruptness as we make our measurements more refined. The thesis that nature can be described in terms of discontinuities of a very small scale seems much too special to be made a fundamental part of a theory of the general pretensions of that of relativity. In fact this, as well as a consideration to be mentioned later, suggests that the argument and result of general relativity may be intrinsically restricted to large scale phenomena.

We now pass from these somewhat special questions to ask why it is that Einstein was able in the general theory of relativity to obtain new and physically correct results from general reasoning of an apparently purely mathematical character. We are convinced that purely mathematical reasoning never can yield physical results—that if anything physical comes out of mathematics it must have been put in another form. Our problem is to find where the physics got into the general theory.

There are two questions to be disentangled here: we have to consider in the first place the significance of the fact that Einstein has been able to describe relations in nature in mathematical form, and in the second place of the fact that he was able to arrive at the mathematical formulation of these physical relations by reasoning of apparently a purely mathematical character, from postulates of merely formal mathematical content (invariance of natural laws in generalized coordinates). Now the theory of relativity does not seem to differ in the first respect from any other branch of mathematical physics, such as the classical mathematical theory of electricity and magnetism, for instance, and this matter has already been touched in an earlier chapter. It is a fact that the behavior of nature can in many cases be expressed to a high degree of precision in mathematical language, and relativity is not unique in this respect. In any event, we must not allow this possibility of mathematical formulation to obscure the essential fact that all physical knowledge is by its nature only approximate, so that we may expect at any time to find, when we have carried our measurements to a higher degree of precision, that our mathematical expression of the laws was not quite exact, as seems now to be the case with Newton's law of gravitation, for example. I do not suppose that Einstein would claim that the statements of relativity differ in this respect from any of our other statements about nature, although apparently some of his followers see something more here. (From the operational viewpoint the meaning to be attached to "something more" is somewhat obscure.)

With respect to the second question, we may stop to notice that the special theory stands in quite a different position from the general theory. The special theory is much more physical throughout; its postulates are physical in character, and it is obvious that the physics got into the results through the postulates. It seems to me without question that Einstein showed the intuitive insight of a great genius in recognizing that there are mutual relations between physical phenomena which can be described in very much simplified language in terms of concepts slightly modified from those already in common use. In view of the remarks made on the nature of light, it is legitimate to wonder, however, whether the formulations of even the special theory will always stand. It seems to be true that all the facts of nature, even in the absence of a gravitational field, cannot be connected by the simple formulations of the special theory; that the physical relations are simple only in a sub-group; and that if we wish to deal with all optical phenomena, we have carried our simplifications too far, for the emission of a light signal is not a simple event, and light is not in nature like a thing travelling. Just the sorts of physical thing which are ignored in treating light as the special theory does are coming to be more and more important in the minds of physicists, and this is a reason for wondering whether ultimately Einstein's special theory may not be regarded merely as a very convenient way of tying together a large group of important physical phenomena, but not as being by any means a full or complete statement of natural relations.

With respect to the general theory, however, I believe the situation is quite different. The fundamental postulate that the laws of nature are of invariant form in all coördinate systems is highly mathematical, and of an entirely man-made character. Of what concern of nature's is it how man may choose to describe her phenomena, and how can we expect the limitations of our descriptive process to limit the thing described? Furthermore, Einstein's method of connecting his mathematical formulation and nature by way of coincidences of 4-events (three space, one time coordinates) seems to be very far removed from reality, since it entirely leaves out the descriptive background in terms only of which the 4-event takes on physical significance. Nevertheless, three definite conclusions about the physical universe have been taken out of the hat by the conjuror Einstein (shift of the perihelion of Mercury, displacement of apparent position of stars at the edge of the sun's disk, and the shift toward the infra-red of spectrum lines from a source in a gravitational field), and the problem for us as physicists is to discover by what process these results were obtained.

An examination of what Einstein actually did in deriving his results will show, I believe, that the situation is really different from that suggested above. In the first place, the requirement that the laws of nature be of invariant form actually places no restriction, as any one can see by setting himself the task of expressing, for example, an inverse cube law for gravitation in terms of generalized coordinates. The work of expressing such a law can be attacked in a perfectly routine way. (The essential difference between the invariability requirements of the special and general theories is to be noted; the special theory requires that the velocity of light, for instance, have the same numerical value in all allowed systems: the general theory merely that all laws have the same literal form, but with variable numerical coefficients.) But, as Einstein says, if any one actually attempts to carry through the work of expressing an inverse cube law in generalized coordinates, he will find the task prohibitively complicated, and will seek for some simpler formulation. What Einstein actually did, therefore, was to require that the laws of nature be simple in generalized form. Now we know that the law of gravitation as formerly expressed in ordinary coordinates as an inverse square law was approximately exact, and was also simple. Any deviations from this law are small, and all experience leads us to expect that to the first order of small quantities the deviations can be taken care of mathematically in the form of small correction terms to this law. This by itself gives nothing, however, because a small correction term can be added to our equations in an infinite number of ways. If, however, we know that the equation must be of a certain type after the correction terms have been added, the possibilities are so much restricted that the form of the correction term may be determined. In arguing as to the probable type of the equation, Einstein advanced the considerations by which physics gets into the situation.

In the first place, the special theory had prepared us for the possibility of finding that our measuring instruments might be modified in a gravitational field, analogously to the shortening of a meter stick when set into motion. In fact, special theory had indicated that in an accelerated system the modifications might be too complicated to be treated by that theory. In the absence, then, of specific information we must be prepared for the most general possible alteration in space-time in a gravitational field. In describing space-time we must therefore use coördinates adapted to handling the most general possible relations, and these are the generalized coördinates of Riemann, which had been already discussed by mathematicians. Going back now to Einstein's criterion that the equations are to be simple, we have the demand that the equations be simple in generalized coördinates, and of course they must also reduce to the ordinary equations (that is, the equations of special relativity) in space where there is no gravitational field. In deciding the further question as to what the type of equation probably is, we are influenced by considerations of convenience as well as by physical considerations. Practically the only type of equation that can be handled mathematically is linear, so that we shall certainly try first whether this type of equation may not continue to hold. Now the Newtonian law of the inverse square may be expressed in terms of a linear differential equation of the second order in the old Cartesian coördinates (Poisson's equation), so that our most immediate suggestion is that the equations remain linear and of the second order in generalized coördinates. As a matter of fact, this requirement turns out to be sufficient to determine the small correction term by which the ordinary equations can be generalized; Einstein's papers must be studied to see how this works out in detail.