The evolution of scientific thought from Newton to Einstein

by A. D'Abro

Of course, this charge is totally unjustified and issues solely from our philosopher’s faulty understanding of the facts. The heterogeneity was never introduced in the casual way Broad believes. It was introduced by the front door the moment Einstein wrote out his gravitational equations

, as any one familiar with the theory of spaces would have recognised immediately. Far from having been slurred over, this heterogeneity of space-time constitutes the very essence of Einstein’s general theory. Whereas in classical science the sun developed a field of force around itself in a perfectly flat space, in the general theory of relativity the sun develops a heterogeneous space-time curvature. The field of force is then but a manifestation of this curvature. As for the new conception that this heterogeneity forces upon our understanding of the world, it has been subjected to critical enquiry by Weyl and Einstein. Furthermore, it is not so new a conception as Broad appears to believe, for it is one we owe to Riemann following his epochal discoveries in non-Euclidean geometry seventy years ago. As Weyl tells us:

“Riemann rejects the opinion that had prevailed up to his own time, namely, that the metrical structure of space is fixed and inherently independent of the physical phenomena for which it serves as a background, and that the real content takes possession of it as of residential flats. He asserts, on the contrary, that space in itself is nothing more than a three-dimensional manifold devoid of all form; it acquires a definite form only through the advent of the material content piling it and determining its metric relations.” And again, elsewhere:

“It is upon this idea, which it was quite impossible for Riemann in his day to carry through, that Einstein in our own time, independently of Riemann, has raised the imposing edifice of his general theory of relativity.” (We may mention that the reason Einstein was able to carry through Riemann’s ideas is because he applied them to space-time instead of to space alone.)

Of course, it is granted that Riemann’s discoveries and non-Euclidean geometry are not of easy access; yet, on the other hand, the man who ignores at least the implications of non-Euclidean geometry, is in no position to discuss Einstein’s theory or the problem of space even from a purely philosophical point of view. Indeed, it may be said that the philosophical importance of non-Euclidean geometry is even greater than its scientific importance. Such has always been the contention of Lobatchewski, Riemann and other great mathematicians.

So far as the writer is aware, the only philosopher who made any reference to non-Euclidean geometry, prior to Einstein’s discoveries, was Lotze; and Lotze expressed the hope that philosophy would never allow itself to be imposed upon by it. But a perusal of Lotze’s writings on the subject proves that he had a very superficial understanding of what was meant by “curvature.” His opinions seem to have been based on Helmholtz’s more or less successful attempts to popularise the new doctrine by easy illustrations. Unfortunately, the subject is too deep to be explained in any loose way. At any rate, the effect of this negative attitude on the part of philosophers has reacted to their disadvantage in that it has deprived them of a very powerful insight into the problem of space.

Our sole purpose in mentioning these few examples (which we might have multiplied ad infinitum) has been to show how wary we should be of criticising the conclusions of scientists before proceeding to acquire more than a superficial schoolboy knowledge of the physical and mathematical facts which they are endeavouring to co-ordinate. If, as a result of misinterpretation or ignorance on our part, we are acquainted with only a small number of these facts, the conclusions of scientists may well appear strange and unwarranted; but it should be remembered that had not all these additional facts, which we ignore, been known to the scientist, it is quite certain that he never would have been driven to the conclusions that offend our natural views so deeply.

Unfortunately, in many cases, these facts are not of an elementary character; they cannot be explained in an hour or so. More often than not, an appreciation of what they represent would require years of preliminary study, for they are often in the nature of conclusions derived from other facts through the medium of laborious mathematical analysis. These statements, of course, must not be construed as applying solely to theories of mathematical physics. They apply with equal force to the discussion of numerous concepts such as those of infinity, continuity and atomism, dimensionality, number, measurement, rigidity, etc. Discoveries in higher mathematics and in physics have thrown a new light on all these subjects. Students of advanced mathematics know only too well how crude was their understanding of continuity and infinity before they were apprised of the mathematical discoveries of Riemann, Weierstrass, Cantor, Dedekind and du Bois Reymond. Furthermore, no one can have studied advanced mathematics without realising its intimate connection with problems of psychology, for mathematics has brought to light mind-forms of which we were only dimly conscious.

The thorough remodelling of our ideas of the universe which the discoveries of Planck and Einstein appear to be rendering inevitable, makes an understanding of these fundamental points imperative. For example, let us consider the quantum theory. As we shall see, its necessity arises only when we take into consideration a number of empirical discoveries pertaining to the various realms of physical science. Here, for instance, is a heated enclosure. We make a pinhole aperture in its wall and examine the colour and intensity of the light rays streaming out. The experimenter notes that as the temperature increases, the colour of the light rays passes gradually from red to white. He then studies the density of energy of the radiation emitted, and finds it proportional to the fourth power of the absolute temperature of the enclosure, regardless of the material of the enclosure. This empirical discovery is known as Stefan’s law. Then, by splitting up the light through a prism, he finds that the radiation which exhibits the maximum intensity is of a frequency which is proportional to the absolute temperature. This discovery is comprised in Wien’s displacement law. These are the facts of the case; the physicist has accomplished his task and may now retire. What is the significance of these facts? Were science to limit itself to the bare discovery and cataloguing of facts, there would be nothing more to do. But regardless of what philosophers may say to the contrary, science should not and does not limit itself to any such humble rôle. The discoveries of the experimenters are handed over to the theoretical investigators, and it is these who are called upon to interpret their significance.