was devised with this express purpose in view. But we should now be led to anticipate a different numerical result for some other phenomenon. If this second anticipation were to be disproved by experiment we could invoke some second unverifiable disturbing influence to account for the discrepancy, while leaving the first result unchanged. Let us call this new influence the ether
. We might go on in this way indefinitely.
But it is obvious that our theory of mathematical physics whose object it was to allow us to foresee and to foretell would now be useless. No new phenomenon could be anticipated, since past experience would have shown us that unforeseen influences must constantly be called into play if theory were to be verified by experiment. Under these circumstances we might just as well abandon all attempts to construct a mathematical model of the universe.
Suppose now that by modifying once and for all our initial premises we are led to a theory which allows us to foresee and foretell numerical results that are invariably verified with the utmost precision by experiment, without our having to call to our assistance a number of foreign hypotheses. In this case we may assume that the new theory is correct, since it is fruitful; and that our former theory was incorrect, because it led us nowhere.
The considerations we have outlined have an important bearing on the understanding of the outside world as shared by the vast majority of scientists. If we hold that the simplest of all the mathematical theories which finds itself in accord with experiment constitutes the correct theory, giving us the correct representation of the real world, we shall recognise that it would be a dangerous procedure to saddle ourselves with a number of hypothetical presuppositions at too early a stage of our investigations.
To be sure, we may have to make a certain number of fundamental assumptions, but we must regard these as mere working hypotheses which may have to be abandoned at a later stage if peradventure they lead to too complicated a synthesis of the facts of experiment. We shall see, for instance, that relativity compels us to abandon our traditional understanding of space and time. It is this fact more than any other which has been responsible for the cool reception accorded to the theory by many thinkers. When, however, we become convinced that Einstein’s synthesis is the simplest that can be constructed if due account be taken of the results of ultra-refined experiment, and when we realise that a synthesis based on the classical understanding of separate space and time would be possible only provided we were willing to introduce a host of entirely disconnected hypotheses ad hoc which would offer no means of direct verification, we cannot easily contest the soundness of Einstein’s conclusions.
There are some, however, who argue that we have an a priori intuitional understanding of space and time which is fundamental, and that we should sacrifice simplicity of mathematical co-ordination if it conflicts with these fundamental intuitional notions. Needless to say, no scientist could subscribe to such views. Quite independently of Einstein’s discoveries, mathematicians had exploded these Kantian opinions on space and time many years ago. As Einstein very aptly remarks in his Princeton lectures:
“The only justification for our concepts and system of concepts is that they serve to represent the complex of our experiences; beyond this they have no legitimacy. I am convinced that the philosophers have had a harmful effect upon the progress of scientific thinking in removing certain fundamental concepts from the domain of empiricism where they are under our control to the intangible heights of the a priori. For even if it should appear that the universe of ideas cannot be deduced from experience by logical means but is in a sense a creation of the human mind without which no science is possible, nevertheless this universe of ideas is just as little independent of the nature of our experiences as clothes are of the form of the human body. This is particularly true of our concepts of time and space which physicists have been obliged by the facts to bring down from the Olympus of the a priori in order to adjust them and put them in a serviceable condition.”
In the passage just quoted, Einstein argues from the standpoint of the physicist, but the opinions he expresses will certainly be endorsed by pure mathematicians. They, more than all others, have been led to realise how cautious we must be of the dictates of intuition and so-called common sense. They know that the fact that we can conceive or imagine a certain thing only in a certain way is no criterion of the correctness of our judgment. Examples in mathematics abound. For example, before the discoveries of Weierstrass, Riemann and Darboux the idea that a continuous curve might fail to have a definite slant at every point was considered absurd; and yet we know to-day that the vast majority of curves are of this type. In the same way our intuition would tell us that a line, whether curved or straight, being without width, would be quite unable to cover an area completely; yet once again, as Peano and others have shown, our intuition would have misled us. Many other examples might be given, but they are of too technical a nature and need not detain us. At all events, mathematicians, as a whole, refused to question the soundness of Einstein’s theory on the sole plea that it conflicted with our traditional intuitional concepts of space and time, and we need not be surprised to find Poincaré, one of the greatest mathematicians of the nineteenth century, lending full support to Einstein when the theory was so bitterly assailed in its earlier days.