In another case, however, the same theory seems to arise by perfectly logical processes, although it again leads to a statement that does not seem correct to "common sense." For it shows by a very subtle and ingenious method that all the surface-points of a surface infinitely extended in all directions may be brought to correspond in a reversible single manner to the linear points of a line, however small; so that to every point of the unlimited plane there corresponds a definite point of the line, and vice versa. The same theorem may be extended to three-dimensional space, with the result that we have to reconcile ourselves with the incredible fact that, expressed in popular language, a straight line of however small length exhibits the same potentiality with regard to the number of its points, as all the points in the universe.
For my own part, I must confess that no means suggests itself to me to make this paradox intelligible. But the sacrificium intellectus comes within dangerous proximity. Einstein, who values and marvels at the theory of aggregates as a science, or perhaps more as a work of art built up from the materials of science, gives whole-hearted support to the proof. He refuses to accept the notion of a paradox—that is, he recognizes a contradiction not in our process of reasoning, but only in a habit of thought that is open to correction. I should give much to discover the means of correction!
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A third example arises out of the special theory of relativity. It has a mysterious paradoxical character that vanishes when a clear view of the relationships involved has been obtained.
According to this theory the rate at which events happen alters according to the state of motion of the system under consideration. Let us now consider two twins A and B, that, although born at one place on the earth, are immediately separated, B remaining at rest, whilst A rushes out into space at an enormous rate, describing what, viewed from the earth, is an inconceivably great circle. In this way the rate of happening of all events is reduced very considerably for A in a manner that may be calculated. If A then returns to B, it may happen that the twin who stayed at home is now sixty years old, whereas the wanderer is only fifteen years of age, or is perhaps only an infant still.
The first introduction to this flight of imagination naturally causes profound perplexity. Nevertheless, we are dealing not with a realm of miracles, but with something that is within the range of comprehension.
"In the case of these two twins," Einstein declared, "we have merely a paradox of feeling. It would be a paradox of thought only if no sufficient ground could be suggested for the behaviour of these two creatures. This ground, which accounts for the comparative youth of A, is given, from the point of view of the special theory of relativity, by the fact that the creature in question, and only this creature, has been subject to accelerations. A proper grasp of the reason is furnished only when we adopt the general theory of relativity, which tell us that, from the point of view of A, a centrifugal field exists, whereas it is absent from the point of view of B. This field exerts an influence on the relative rate of happening of the events of life."
It certainly requires a prodigious mechanism to allow the moving twin to gain even only one second of time. If he were to spend a year in a merry-go-round whose circumference were about 19 milliard miles in length, he would have to travel in it at the rate of over 600 miles per second if he is to gain a second on his brother.
This inevitable result that is immediately apparent to a trained scientific mind throws light on the nature of "common sense," the validity of which, as an ultimate criterion, Kant too has refused to recognize, in so far as this "common sense" is incapable of passing beyond the examples offered in its own experience. It circulates, as Einstein says, in the "realms of feeling and analogy." It finds no analogy for a phenomenon like that described above, and since it can apply rules only concretely, many things appear to it paradoxical that, in the light of intensified abstraction, appear logical and necessary.
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