It is a consequence of this view that any correlation is adapted to be an absolutely final element of explanation, and can never be superseded by the discovery of new experimental facts, if the correlation is by definition beyond the reach of further experiment. Such a possibility, for example, is contained in a correlation between the numerical magnitude of the gravitational constant and the total mass of the universe. Something of this sort may be well attempted by those who desire their explanations to take a formally final shape. We shall return to this subject later.

The instinctive demand for a mechanism is fortified by observation of the many important cases in which mechanisms have been discovered or invented. However, the significance of such successful attempts must be subject to most careful scrutiny. The matter has been discussed by Poincaré,[6] who showed that not only is it always possible to find a mechanistic explanation of any phenomenon (Hertz's program was a perfectly possible one), but there are always an infinite number of such explanations. This is very unsatisfactory.

[6]Henri Poincaré. Wissenschaft and Hypothese, Translated into German by F. and L. Lindemann, Teubner, Leipzig, 1906. See especially p. 217.

We want to be able to find the real mechanism. Now an examination of specific proposed mechanisms will show that most mechanisms are more complicated than the simple physical phenomenon which they are invented to explain, in that they have more independently variable attributes than the phenomenon has been yet proved to have. An example is afforded by the mechanical models invented to facilitate the study of the properties of simple inductive electrical circuits. The great number of such models which have been proposed is sufficient indication of their possible infinite number. But if the mechanism has more independently variable attributes than the original phenomenon, it is obvious that the question is without meaning whether the mechanism is the real one or not, for in the mechanism there must be simple motions or combinations of motions which have no counterpart in features of the original phenomenon as yet discovered. Obviously, then, the operations do not exist by which we may set up a one to one correspondence between the properties of the mechanism and the natural phenomenon, and the question of reality has no meaning. If, then, a mechanism is to be taken seriously as actually corresponding to reality, we must demand that it have no more degrees of freedom than the original phenomenon, and we must also be sure that the phenomenon has no undiscovered features. Physical experience shows that such conditions are most difficult to meet, and indeed the probability is that they are impossible.

A mechanism with more independently variable attributes than the phenomenon may prove to be a very useful tool of thought, and therefore worth inventing and studying, but it is to be regarded no more seriously than is a mnemonic device, or any other artifice by which a man forces his mind to give him better service.

There is another possible program of explanation, the converse of that considered above, namely, to explain all familiar facts of ordinary experience in terms of less familiar facts found at a deeper level. The most striking example of this is the recent attempt to give a complete electrical explanation of the universe. The original attempt was to explain electrical effects in mechanical terms; this attempt failed. At about the same time the existence of the electron was experimentally established, so that it was evident that electricity is a very fundamental constituent of matter. The program of explanation was reversed, and an electrical explanation sought for all mechanical phenomena, including in particular mechanical mass. But this attempt has also failed; we recognize that part of mass may be non-electrical in character, we postulate non-electrical forces inside the electron, and further, we usually postulate for electrons and protons the property of impenetrability, a property derived from experience on a higher scale of magnitude.

A program of this general sort is likely to be regarded with considerable sympathy, and indeed the chances of success seem much greater than do those of the converse program, for in our experience large scale phenomena are more often built up from and analyzed into small scale phenomena than the converse. But as a matter of principle we must again recognize that the only appeal is to experiment, and that we have to ask just one question: "Is it true, as a matter of fact, that all large scale phenomena can be built up of elements of small scale phenomena?" It seems to me that the experimental warrant for this conviction has not yet been given. The failure of the attempted electrical explanation of the universe is a case in point. However, the failure to prove a proposition is no guarantee that some time it may not be proved, and many physicists are convinced of the ultimate feasibility of this program. Personally I feel that the large may not always be analyzed into the smaller; the subject will be discussed again.

A conviction of the significance of microscopic analysis has many features in common with the usual conviction of the ultimate simplicity of nature. The thesis of simplicity involves in addition the assumption that the kinds of small scale elements are few in number, but actually this involves no important difference between the two convictions, because we have seen that the elements of which we build our structure become fewer in number as we approach the limit of the experimentally attainable. We may properly grant to convictions of this sort pragmatic value in suggesting new correlations and experiments, but a recognition of the empirical basis of all physics will not allow us to go further.

MODELS AND CONSTRUCTS

In discussing the concept of length, we could find no meaning in questions such as: "Is space on a scale of 10-8 cm. Euclidean?" Nevertheless it will seem to many that they do attach a perfectly definite meaning to a question of this kind. Of course it must be agreed that magnitudes of 10-8 cm. cannot be thought of in terms of immediate sensation. When one thinks of an atom as a thing with any geometrical properties at all, I believe he will find that what he essentially does is to imagine a model, multiplying all the hypothetical dimensions by a factor large enough to bring it to a magnitude of ordinary experience. This large scale model is given properties corresponding to those of the physical thing. For example, the model of the atom which was accepted in the fall of 1925 contains electrons rotating in orbits, and every now and then an electron jumps from one orbit to another, and simultaneously energy is radiated from the atom. Such a model is satisfactory if it offers the counterpart of all the phenomena of the original atom. Now I believe the only meaning that any one can find in his statement that the space of the atom is Euclidean is that he believes that he can construct in Euclidean space a model with all the observed properties of the atom. This possibility may or may not be sufficient to give real physical significance to the statement that the space of the atom is Euclidean. The situation here is very much the same as it was with respect to mechanisms. The model may have many more properties than correspond to measurable properties of the atom, and in particular, the operations by which the space of the model is tested for its Euclidean character may [and as a matter of fact I believe do] not have any counterpart in operations which can be carried out on the atom. Further, we cannot attach any real significance to the statement that the space of the atom is Euclidean unless we can show that no model constructed in non-Euclidean space can reproduce the measurable properties of the atom.