When we consider all these technical difficulties, and realise that solutions of these problems have been obtained thanks only to the most abstract speculations of pure mathematicians, to the invention or discovery of mathematical entities which at first sight would appear to be totally estranged from the world of reality, we must not belittle the practical importance of these seemingly unreal entities.
As a matter of fact, it is little short of a miracle, in view of the insignificance of our mathematical ability, that theoretical physics should have been possible at all. Had our solar system contained two suns such as are present in the double-star systems, or had one of the planets been comparable in size to the sun, it is safe to say that Newton, in spite of his genius, would never have discovered his law of gravitation; for he would have been thrown back on the tremendous problem of the three bodies.
Leaving aside these technical difficulties, the fact remains that in a number of cases mathematics has been found applicable to nature at least in an approximate way, and as a result mathematical laws have been discovered. We shall now proceed to show how these theories of mathematical physics have evolved. To begin with, a large number of empirical facts are first discovered by the experimenter. These facts are then co-ordinated into a consistent whole, general relations or laws are established, and in this way we obtain various groupings of phenomena, such as electricity, magnetism, optics, mechanics, chemistry, biology, etc. In certain cases it may be impossible to get beyond the initial stage; though experimental data are accumulated, the co-ordination of facts and the discovery of general relations and laws may defy analysis. In such cases we can scarcely regard our knowledge as constituting a science.
The next task is to co-ordinate and establish relations between these various realms of science. Confining ourselves to physics, we find that, first of all, relationships were discovered between electric and magnetic phenomena; then between light propagation and the vibrations of elastic solids (Fresnel, MacCullagh); then between electromagnetic phenomena and optical ones (Maxwell); then between heat and molecular motion (Maxwell, Boltzmann). But in spite of these lofty syntheses, one mysterious influence appeared to remain estranged from all the others; this was gravitation. It has been one of the triumphs of the relativity theory to succeed in establishing the connections between gravitation and optics or electricity.
Finally, an amalgamation of all these various realms is sought for in the expression of a gigantic universal law, the principle of action, governing one unique mathematical world-function, the function of action of the universe, which the theories of Weyl and Eddington appear to suggest.
In short, we see that the development of theoretical physics has followed that of pure mathematics in its generalising characteristics. In either case the breaking down of barriers, the discovery of unity in diversity, has been the guiding motive. And yet, in spite of this mathematisation of physics exemplified in the works of the theoretical physicists, physics is not mathematics, and truth in physics is not the same as truth in mathematics.
In the first place, physics progresses by successive approximations and does not attain its goal at one stroke, as is often the case with mathematics. Thus, in the days of Galileo it would have been correct to say: The centre of gravity of a projectile moving in vacuo describes a parabola. Half a century later, Newton recognised that this statement was only approximate. The correct statement then became: Under ideal conditions of isolation, the centre of gravity of the projectile will describe an ellipse, the earth’s centre being situated at one of its foci. But, according to relativity, Newton’s statement in turn is approximate. We must now say that the trajectory lies along an ellipse whose axis is slowly rotating. There is every reason to believe that even Einstein’s mechanics is but an approximation to truth; and Schrödinger’s wave mechanics is already supplanting it in the infinitely small. Indeed, there is every reason to suppose that however far we go, we shall always be dealing with approximations. Here, then, is an essential difference between physics and mathematics. Thus far, it would appear that absolute truth might exist but that our present means of investigation had not allowed us to attain it. But we shall see that more is at stake than a temporary admission of failure.
Consider, for instance, the constants of physics, such as
, the gas constant, or