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. We have now to consider in a very brief way certain of the philosophical problems which antedate Einstein’s discoveries, but with which his theory is intimately connected. Long before the advent of Einstein, problems pertaining to the relativity of motion through empty space had occupied the attention of students of nature. There were some who held that empty space, and with it all motion, must [Pg xvii] be relative; that states of absolute motion or absolute rest through empty space were meaningless concepts. According to these thinkers, in order to give significance to motion and rest, it was necessary to refer the successive positions of the body to some other arbitrarily selected body taken as a system of reference. We should thus obtain relative rest or motion of matter, with respect to matter. All these views were in full accord with our visual perceptions and they were expressed by what is known as the visual or kinematic principle of the relativity of motion. Other thinkers preferred to uphold the opposing philosophy. They assumed that space was absolute; that all motion must be absolute; that there was meaning to the statement that a body was in motion or at rest in space, regardless of the presence of other bodies to be used as terms of comparison. The controversy mighty have continued indefinitely, had it not been for the appearance of the scientist with his empirical methods of investigation.