The logic of modern physics

by P. W. Bridgman

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

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

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

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

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

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

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

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

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

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