EINSTEIN: And this flight of thought, which, by the way, has been indulged in repeatedly by others too, has much more sense in it than the former one, because you may make an abstraction which disregards speed altogether. It is only a limiting case of reflection.
M.: I should like to touch on other limiting cases, in particular two that I find it impossible to interpret. Lotze mentions them in his Logic. The first concerns the infinitely long lever whose fulcrum, or turning-point, is at the confines of the universe. According to the Laws of Levers, a mass of magnitude zero will suffice to keep in equilibrium at the end of the other lever-arm any weight, no matter whether it is a million times heavier than the earth. Our imaginations cannot even picture this. Yet I cannot feel satisfied with the mere explanation that it is an exceptional case, an extension of a general law to a case in which it is no longer applicable. The second example is still more perplexing because it does not require a journey into other worlds, but leads us into inconceivable consequences even if we remain on the earth. Lotze considers this second limiting case easier; to me it seems more difficult. It is this: The force that a wedge exerts is inversely proportional to its thickness. If it is infinitely thin, this formula gives an infinitely great result, whereas, actually, the force exerted is nil. This very thin wedge, transformed finally into a geometrical plane, should be able to split in twain any wooden or even steel block. And now, consider a special arrangement of this wedge in which it is resting with its extremely sharp edge vertically downwards, whereas at the top it broadens to a little ledge which supports a weight. We then get the incredible result that this wedge, which can be imagined concretely, should be able to cut through the whole earth with its extremely fine edge, if placed on some base. Where is the fallacy in this case?
EINSTEIN: The mechanical facts have not been taken sufficiently into consideration.—He illustrated his further remarks by drawing a few strokes with his pen, and proved from his diagram that a wedge of this sort would be able to perform what I assumed, only if the base on which it is placed is composed of separate laminae. Otherwise the assumption that the force is infinitely great would be erroneous.
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After this digression to a limiting case on the earth we returned to more general problems, and the question of the finitude or infinitude of the universe. Shortly before, Einstein had given an address to the Berlin Academy on this point, involving difficult calculations, and I hoped to hear from him an easy explanation at least in general terms.
It is one of the ultimate problems. Whoever talks of the limits of the world endeavours also to mark off the bounds of the understanding. The average person, at first sight, almost always decides in favour of an infinite universe, on the ground that a finite world is inconceivable. He argues that, if it were considered finite, we should immediately be confronted with the question: What lies beyond the finite boundary? Something must be present, even if it is only empty space. This brings us into an inevitable conflict with the first of Kant's "antinomies," with the thesis and antithesis, from which there is no escape. What is the meaning of the fact that the apprehensive understanding seeks refuge in "Infinity"? It signifies that he gets entangled in the folds of a negative conception, that furnishes him with no explanation at all, and expresses merely that his first assumption of finitude cannot be thought out to its conclusion.
Besides this, a second disturbing question arises. Is there a finite or infinite number of stellar bodies? If this question refers to an assumed infinite space, even if such space is inconceivable, then there are two possible answers. For it would be possible to imagine a finite number of stars even if no limit could be found for space.
Whereas the general question of space in the universe belongs exclusively to speculative philosophy, the star-question is not purely metaphysical, but is physical, too, and has accordingly been treated by physicists. The great astronomer Herschel imagined he could solve it by means of optical principles, and he arrived at the conclusion that the number of heavenly bodies must be finite, as otherwise the aspect of the starry firmament, from the point of view of illumination, would be entirely different. But this proof did not establish itself among scientists, for the number of stars of the type of the sun might be finite, whilst there was an infinite number of dark stars.
A further question presented itself: Would it be possible for a definite part of the heavens (say, that north of the ecliptic) to contain an infinite number of stars, whilst other parts contained only a finite number? At first this sounds very extraordinary, but it is by no means unreasonable, as a concrete example will show: If, on a scale of temperature, we count the degrees of heat from a certain point, then they stretch apparently to infinity in one direction, whereas they extend only to -273° (Centigrade) in the other direction, that is, to the absolute zero. Thus we can imagine an arrangement which stretches to infinity only in one direction.
To get an insight into the discussion by Einstein which is about to follow, we must first dispose of a certain arbitrariness of language, lying in the customary indiscriminate use of the terms, infinite, immeasurable, and unbounded. Suppose we have a globe about one foot in diameter, the surface of which is inhabited by extremely small, ultramicroscopic creatures that can move about freely and can think. The surface of the sphere constitutes the world of the micro-men, and he has a very good reason for considering it infinite, for, however far and in whatever direction he may move, he never encounters a boundary. But we, who live in our space, look on to this spherical surface, and recognize that his judgment is erroneous. To us his spherical world seems decidedly finite and quite measurable, although it has no determinable beginning and no end, and thus must appear unbounded to the micro-man. In fact, we ourselves may regard it as boundless, if we can succeed in forming an abstraction that leaves out of account its limitations in our own space.