)”.[154]
These discussions on the form of the universe bring to light another important aspect of the methodology of science. We may notice that the accord which was practically unanimous in the earlier part of the theory now gives way to dissenting opinions. The reason for this is of course obvious. In the earlier part, experimental verifications were soon forthcoming and permitted us to decide matters one way or another. But when we are dealing with the form of the universe, verifications are extremely difficult; they depend on tedious astronomical observations which may require centuries to be completed. Hence, the only method of advance is to explore all mathematical possibilities in the hope that somehow or other we may be led to an experimentum crucis. De Sitter places the emphasis on the mathematical principle of relativity. Einstein adopts the attitude of the physicist and prefers to stress the dynamical aspect of relativity. Weyl and Eddington adhere more closely to the pure physics of the field, following the procedure inaugurated by Maxwell and Faraday.
But in all cases the driving force behind these attempts has been a desire to co-ordinate mathematically a maximum number of experimental facts in the simplest way possible. These facts have been collected from the various realms of physical science, and, indeed, it is owing to their wide variety that the conclusions obtained are compelling. Furthermore, it should be noted that the facts appealed to are the results of highly refined experiments. This again is important, since we have seen that a difference one way or another of a tiny fraction of an angle or a length might be sufficient to overthrow an entire theory.
Summing up these discoveries, we see that a matter-created universe entailing the complete relativity of all motion to the universal masses appears to be mathematically possible. If this solution is the correct one, the cornerstone of Newton’s absolute space is finally destroyed. If, on the other hand, the cylindrical universe should prove untenable, we could scarcely claim that an absolute background to motion had been disposed of. In this measure, at least, Newton’s ideas would stand.
In short, we see that this problem of the relativity of rotation, of the relationships between matter and extension, of the finiteness of the universe, is not one for philosophy and a priori arguments to solve. A mathematical co-ordination of the facts developed by ultra-precise experiment can alone yield us an answer. As Weyl tells us:
“The historical development of the problem of space teaches how difficult it is for us human beings entangled in external reality to reach a definite conclusion. A prolonged phase of mathematical development, the great expansion of geometry dating from Euclid to Riemann, the discovery of the physical facts of nature and their underlying laws from the time of Galileo together with the incessant impulses imparted by new empirical data, finally, the genius of individual great minds, Newton, Gauss, Riemann, Einstein, all these factors were necessary to set us free from the external, accidental, non-essential characteristics which would otherwise have held us captive.”
To be sure, as our knowledge accumulates and our mathematical ability increases, we may be led to revise former conclusions. This is why the conclusions of Einstein differ from those of Newton, just as those of our descendants may differ from ours of to-day. But the main point to grasp is that these variations in our philosophical outlook are brought about by our increase of knowledge both mathematical and physical; and this increase requires centuries: it cannot be obtained overnight. Indeed, had Einstein lived in Newton’s time and ignored the mathematical and physical discoveries of the present day, he could never have improved upon Newton’s solution. Inversely, had Newton lived in our time and been acquainted with the facts known to modern science, it is very possible that he would have created the theory of relativity.
In our analysis of the methodology of scientific theories we have attempted to show that the deductions of the great scientists were always based on experiment and not on wild guesses. But this does not mean that other factors have not also come into play. For instance, we mentioned that a proper choice of facts was of supreme importance. Whereas a certain choice may lead to nothing of interest, some other choice may issue in a great discovery. A case in point was afforded by Einstein’s attitude towards the well-known identity of the two masses.
But even this is not all. In mathematics, as in architecture, certain co-ordinations are beautiful, others top-heavy or unsymmetrical. When, therefore, physical phenomena are translated into mathematical formulæ, the theoretical physicist will always endeavour to obtain beautiful equations rather than awkward ones. Of course, since he must restrict himself to a slavish translation of physical results, his initiative in this respect may be extremely limited. Nevertheless, in certain cases it may be possible to add an unobtrusive term to the equations, though this term may not actually be demanded by experiment. If this additional term renders the equations more beautiful, there will be every incentive to retain it. It was an urge of this sort that guided Maxwell in his discovery of the equations of electromagnetics. The experimental data known in his day could not establish whether a certain additional term was necessary or not. Nevertheless, Maxwell introduced it because it beautified his equations, and was thus led to anticipate the existence of electromagnetic waves. As we know, subsequent experiment has justified Maxwell’s deep vision. It is much the same with the theory of relativity. One of the reasons for its great appeal to mathematicians is the extreme harmony, beauty and simplicity which it permits us to bestow on many of the equations of physics.