EINSTEIN: It follows from the mathematical calculations which I presented in "Cosmological Considerations arising from the General Theory of Relativity," in which the figure I have just quoted is not given. The exact figure is a minor question. What is important is to recognize that the universe may be regarded as a closed continuum as far as distance-measurements are concerned. Another point, too, must not be forgotten. If, in deference to your wish, I used an easy illustration, this must not be regarded otherwise than as an improvised bridge to assist the imagination.
M.: Nevertheless, it will be very welcome to many, who are unable to grasp the difficult Cosmological Considerations. The number that you mention is overwhelming in the extreme. Indeed, it seems to me that a diameter of 100 million light-years suggests an infinitely great distance more than the word "infinity" itself, mentioned per definitionem, which conveys nothing to the ordinary mind. It calls up a regular carnival of numbers, particularly in those to whom the immense number alone gives a certain pleasure. But you were going to give me the number expressing the mass, too?
And then I learned that the weight of the whole universe, expressed in grammes, was 10 multiplied by itself 54 times, that is 1054 (453 grammes = 1 lb., roughly). This seems rather disappointing at first, but assumes a different aspect when we represent to ourselves what this figure signifies. It means that the weight of the universe in kilogrammes is high in the octillions. The earth itself weighs six quadrillion kilogrammes, hence the weight of the Einstein universe bears the same relation to the weight of the whole earth as the latter bears to a kilogramme. Again, the earth's weight to that of the sun is as 1 is to 324,000. Hence we should have to take at least a trillion, that is, a milliard times a milliard, suns to get the weight of the universe. And as far as the linear extent is concerned, let us consider the most distant stars of the Milky Way, which are at an inconceivable distance, expressible only in light-years. If we place 10,000 such Milky Ways end to end we shall arrive at this diameter of the universe, which, accordingly, will have a cubical content a thousand milliard times greater than the region accessible to astronomical observation.
Thus we have a very spacious universe. Yet it is not spacious enough to satisfy all the demands that a mathematician interested in permutations and combinations might make. One of such combinations is exemplified in the so-called Universal Book, that originated in an imaginary experiment of Leibniz. If we picture to ourselves the sum-total of all books that can be printed by making all possible arrangements and successions of our letters, each book differing from any other even if only in one symbol, then, together, they must contain all that can be expressed in sense and nonsense, and everything that is ever realizable actually or in dreams. Hence, among other things, they would include all world-history, all literature, and all science, even from the beginning of the world to the end. If we agree to the convention of operating with 100 different printed signs (letters, figures, stops, spacings, etc.), and of allowing each such book a million paces for signs, so that each book will still be of a handy size, then the number of these books would amount to exactly 10 to the two-millionth power, or, in figures, i.e. 102,000,000.
This fully exhaustive universal library containing all wisdom would consist of so many volumes that it could not be contained in a case of the size of the entire stellar universe. And, unhappily, it must be added that the closed universe, just described by Einstein and having a diameter of a hundred million light-years would be much too small to contain this library.
"Nevertheless," said I, "your universe pictures something inconceivably great; one might call it an infinity expressed in figures. For in your world there still remains one property of infinity, namely, that it imposes no limitations on motion of any kind. On the other hand, the figures proclaim a limited measure in the mathematical sense, however great this measure may be. This calls up the old restlessness of mind, due to the persistent question: What lies beyond? The absolute Nothing? Or is it a something which yet does not occupy space? Descartes and many other great thinkers have never overcome this difficulty, and have always affirmed that a closed world is impossible. How, then, is the average person to reconcile himself with the dimensions you have established?"
Einstein gave an answer which, it seemed to me, offered a last escape to apprehensive minds. "It is possible," so he said, "that other universes exist independently of our own."
That is to say, it will never be possible to trace a connexion between them. Even after an eternity of observation, calculation, and theoretical investigation, no glimpse or knowledge of any of these ultra-worlds will ever enter our consciousness. "Imagine human creatures to be two-dimensional surface-creatures," he added, "and that they five on a plane of indefinite extent. Suppose that they have organs, instruments, and mental attitude adapted strictly to this two-dimensional existence. Then, at most, they would be able to find out all the phenomena and relationships that objectify themselves in this plane. They would then have an absolutely perfect science of two dimensions, the fullest knowledge of their cosmos. Independent of this, there might be another cosmic plane with other phenomena and relationships, that is, a second analogous universe. There would then be no means of constructing a connexion between these two worlds, or even of suspecting such a connexion. We are in just the same position as these plane-inhabitants except that we have one dimension more. It is possible, in fact, to a certain degree probable, that we shall by means of astronomy discover new worlds far beyond the limits of the region so far investigated, but no discovery can ever lead us beyond the continuum described above, just as little as a discoverer of the plane-world would ever succeed in making discoveries beyond his own world. Thus we must reckon with the finitude of our universe, and the question of regions beyond it can be discussed no further, for it leads only to imaginary possibilities for which science has not the slightest use."
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Einstein left me for a while to the tumult of ideas that he had roused up in me. After I had overcome the first shock, I sought to gain a haven in the idea that arose out of the first shadow-argument, in which the spherical bodies occurred that seek to escape towards infinity on the right but reappear, instead, at enormous distances on the left. Has anyone ever had presentiments of this kind of world? Perhaps something of the sort is to be found in earlier books of science? If so, they have escaped my notice. Yet, a passage of a poet occurs to me. It is to be found in a volume by Heinrich von Kleist; it is a volume dealing only with earthly matter and bare of astronomical ideas. Imagine a book the subject of which is a puppet-show, containing, in the middle of it, a section foreshadowing Einstein's universe! Quite by chance Kleist comes to speak of "the intersection of two lines which, after passing through infinity, suddenly appear on the other side, like a picture in a concave mirror, which moves away to infinity and suddenly returns again and is quite close," and, quite in accordance with our new cosmology, he declares: "Paradise is locked and barred, and the cherub is behind us; we must make a voyage round the world, and see whether we cannot discover an exit elsewhere at the other end perhaps."