CHAPTER VI
OF DIFFERENT WORLDS
Imaginary Experiment with "Lumen."--Impossibilities.--A Destroyed Illusion.--Is the World Infinite?--Surface Creatures and Shadow Rambles.--What is the Beyond?--Action at a Distance.--Ideas of Multi-dimensional Regions.--Hypnotism.--Recollections of Zöllner.--Science and Dogma.--The Trial of Galilei.
CONVERSATION held during April 1920 destroyed an illusion which had become dear to me.
It concerned the fantastic figure, "Lumen," conceived as an actual human being, imagined as endowed with an extraordinary power of motion and keenness of sight. Mr. Lumen is supposed to be the invention of the astronomer Flammarion, who produced him in the retort of fancy, as Faust produced Homunculus, to use him to prove the possibility of very remarkable happenings, in particular, the reversal of Time.
Einstein declared outright: "Firstly, Lumen is not due to Flammarion, who has derived him from other sources; and secondly. Lumen can in no way be used as a means of proving things."
MOSZKOWSKI: "It is at least very interesting to operate with him. Lumen is supposed to have a velocity greater than that of light. Let us assume this as given, then the rest follows quite logically. If, for example, he leaves the earth on the day of a great event, such as the battle of Waterloo, and---- May I trace out this example, at the risk of tiring you?"
EINSTEIN: Do repeat it, and act as if you were telling something entirely new. It is clear that the Lumen-story gives you great amusement, so please talk quite freely. But I cannot forgo the privilege of showing later how the whole adventure and its consequences must be demolished.
M.: Well then, the person, Lumen, sets off at the end of the battle of Waterloo to make an excursion into space with a speed of 250,000 miles per second. He thus catches up all the light-rays that left the field of battle and moved in his direction. After an hour he will already have attained a lead of about twenty minutes. This lead will be gradually increased, so that at the end of the second day he will no longer be seeing the end of the battle, but the beginning. What has Lumen been seeing in the meantime? Clearly he has been observing events happening in the reverse direction, as in the case of a cinematograph which is exhibiting pictures backwards. He saw the projectiles leaving the objects they had struck, and returning into the mouths of the cannon. He saw the dead come to life, arise, and arrange themselves into battalion order. He would thus arrive at an exactly opposite view of the passing of time, for what he observes is as much his experience as what we observe is ours. If he had seen all the battles of history and, in fact, all events happening in the reverse order, then in his mind "before" and "after" would be interchanged. That is, he would experience time backwards; what are causes to us would be effects to him, and our effects would be his causes; antecedents and consequents would change places, and he would arrive at a causality diametrically opposite to our own. He would be quite as justified in adopting his view of the happening of things, according to his experiences, and of the causal nexus as it appears to him, as we are justified in adopting ours.
EINSTEIN: And the whole story is mere humbug, absurd, and based on false premises, leading to entirely false conclusions.
M.: But it is only to be taken as an imaginary experiment that plays with fantastic impossibilities to direct our ideas on to the relativity of time by a striking illustration. Did not Henri Poincaré adduce this extreme example to discuss the "reversal" of time?
EINSTEIN: You may rest assured that Poincaré, even if he used this example as an entertaining digression in his lectures, took the same view of Lumen as I do. It is not an imaginary experiment: it is a farce, or, to express it more bluntly, it is a mere swindle! These experiences and topsy-turvy perceptions have just as little to do with the relativity of time, such as it is taught by the new mechanics, as have the personal sensations of a man, to whom time seems long or short according as he experiences pain or pleasure, amusement or boredom. For, in this case, at least the subjective sensation is a reality, whereas Lumen cannot have reality because his existence is based on nonsense. Lumen is to have a speed greater than that of light. This is not only an impossible, but a foolish assumption, because the theory of relativity has shown that the velocity of light cannot be exceeded. However great the accelerating force may be, and for however long it may act, it cannot cause this limit to be transcended. Lumen is supposed to be equipped with the organ of sight, that is, he is supposed to have a corporal existence. But the mass of a body becomes infinitely great when it reaches the velocity of light, so that it is quite absurd to go beyond this stage. It is admissible to operate with impossibilities in imagination, that is, with things that contradict our practical experience, but not with absolute nonsense. That is why the other adventure of Lumen, in which he jumps to the moon, is also an absurdity. In this, he is supposed to leap with a speed greater than light, and, when he reaches the moon, to turn round instantaneously, with the result that he sees himself jumping from the moon to the earth backwards! This jump is logically meaningless; and if we try to make deductions of an optical nature from such a nonsensical assumption, we deceive ourselves.
M.: Nevertheless, I should claim extenuating circumstances for this case on the ground that I am enlisting the help of the conception of impossibility. A journey even at a speed of only 1000 miles per second is impossible for a man or a homunculus.
EINSTEIN: Yes, according to our experience, if we measure it against facts. We cannot state definitely that a journey into the universe at an enormous yet limited velocity is absolutely impossible. Within the indicated bounds every play of thought that is argued correctly is allowable.
M.: Now, suppose that I strip Lumen of all bodily organs and take him as being a pure creature of thought, entirely without substance. A velocity greater than that of light can be imagined, even if it cannot be realized physically. If, for example, we think of a lighthouse with a revolving light, and consider a beam of light about 600 miles long, which rotates 200 times per second. Then we could represent to ourselves that the light at the circumference of this beam travels with a speed of nearly 760,000 miles per second.
EINSTEIN: As for that, I can give you a much better example of the same thing. We need only imagine that the earth is poised in space, motionless, and non-rotating. This is physically admissible. Then the most distant stars, as judged by us, would describe their paths with almost unlimited velocities. But this projects us right out of the world of reality into a pure fiction of thought, which, if followed to its conclusion, leads to the most degenerate form of imagination, namely, to pathological individualism. It is in these realms of thought that such perversities as the reversal of time and causality occur.
M.: Dreams, too, are confined to the individual. Reality constrains all human beings to exist in one and the same world, whereas, in dreams, each one has his own world with a different kind of causality. Nevertheless, dreams are a positive experience, and signify a reality for the dreamer. Even for waking reality it would be easy to construct cases in which the causal relationship is shattered. Suppose a person who has grown up in a confined retreat, such as Kaspar Hauser, looks in a mirror for the first time in his life. As he knows nothing of the phenomena of optical reflexion, he sees in it a new, objective world that gives a shock to, or even subverts, his own idea of causality in so far as it may have become developed in him. Lumen sees himself jump backwards, whereas Kaspar Hauser sees himself performing gestures on the wrong side of his body; should it not be possible to draw a reasonable parallel between these two cases?
EINSTEIN: Quite impossible. However you set about it, your Lumen will inevitably come to grief on the conception of time. Time, denoted in physical expressions by the symbol "t," may, indeed, be given a negative value in these equations, so that an event may be calculated in the reverse direction. But then we are dealing with pure matters of calculation, and in this case we must not allow ourselves to be drawn into the erroneous belief that time itself may travel negatively, that is, retrogressively. This is the root of the misapprehension: that what is allowable and indeed necessary in calculations is confused with what may be thought possible in Reality.[5] Whoever seeks to derive new knowledge from the excursions of a creature like Lumen into space, confuses the time of an experience with the time of the objective event; but the former can have a definite meaning only if it is founded on a proper causal relation of space and time. In the above imaginary experiment the order of the experiences in time is the reverse of that of the events. And as far as causality is concerned, it is a scientific conception that relates only to events ordered in space and time, and not to experiences. In brief, the experiments with Lumen are swindles.
[Footnote 5: Perhaps an analogy will serve to make this clear. Suppose that a certain quantity of some foodstuff is consumed by ⅒ head of population. The false inference would be that a population is possible which has ⅒ heads! In the same way the statistics may be quite correct in arriving at the figure ⅕ suicides, but if we leave the realms of calculation, then the £ suicide loses its meaning entirely.]
M.: I must resign myself to giving up these illusions. I must frankly confess that I do so with a certain sadness, for such bold flights of constructive fancy exert a powerful attraction on me. At one time I was near outdoing Lumen by assuming a Super-Lumen, who was to traverse all worlds at once with infinite velocity. He would then be in a position to take a survey of the whole of universal history at a single glance. From the nearest star, Alpha Centauri, he would see the earth as it was four years ago; from the Pole Star, as it was forty years ago; and from the boundary of the Milky Way, as it was four thousand years ago. At the same moment he could choose a point of observation that would enable him to see the First Crusade, the Siege of Troy, the Flood, and also the events of the present day simultaneously.
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.
Now, it might occur to a particularly intelligent micro-being to undertake a voyage for the purpose of making measurements. He carefully marks his point of departure, walks Straight ahead in a certain direction, describing a circle on his sphere--a circle which he will necessarily regard as a straight line. He continues ever onwards in the firm conviction that he is getting farther and farther away from his starting-point. Suddenly, he discovers that he has reached it again. He discovers, by the mark he made, that he has not been describing a straight line, but a line that merges into itself.
The micro-professor would be compelled to declare: Our world, the only one known to me, is not infinite, although in a certain sense boundless. Moreover, it is not immeasurable, since it can be measured in at least one direction by the number of steps I have walked. From this we may infer that our former geometrical view was either wrong or incomplete, and that, in order to understand our world properly, we must build up a new geometry.
We may assume that the majority of the remaining micro-inhabitants would at first protest strongly against this decision. The idea that a line, which appears to them to be pointing always in the same direction, is curved, seems to them inconceivable and absurd. They would only gradually overcome their scruples of thought by getting an insight into a newly developed geometry that makes clear to them for the first time the conception of a sphere.
In our world of space, which includes all stars, we are the micro-inhabitants. We have been born with, or have inherited, the idea of a straight and ever-advancing path in space, and we become filled with the utmost astonishment if some one asks us to believe that if we undertake a voyage in one direction out into the universe, beyond Sirius and a million times farther, we should finally arrive at our starting-point again, although we had not changed our direction. But the macro-being, who belongs to a universe of higher dimensions and who looks on our world as we looked on the above spherical world one foot in diameter, sees the narrowness of our view. We, too, are in a position to rise above this narrow view by means of a theory founded on our experience, which will lead us to an extended world-geometry, just as the micro-professor used his experience to extend his theory of the circle to include the conception of a sphere.
After these preliminary remarks we shall endeavour to get an insight into Einstein's reasoning, not in the form in which it was originally presented (in the Report of the Proceedings of the Berlin Academy of Science of 8th February 1917), but in a very easy description which was given to me during a conversation. Here, too, I shall try to preserve the sense of Einstein's remarks without binding myself strictly to his words. For although I am indebted to him for his efforts to avoid difficult points, yet the aim of this book is, if possible, to make the explanation still easier. Any lack of accuracy arising from this last simplification is to be debited to me. The new form of representing the argument, which is as important as it is fascinating, is, of course, due to Einstein.
The final result stated by Einstein was: The universe, both as regards extent and mass, has finite limits and can be measured. If anyone asks whether this can be pictured, I shall not deprive him of the hope. All that is required is a power of imagination that is great enough to follow a pictorial description and that can take up the right attitude towards a sort of figurative representation.
Let us again imagine a sphere of modest dimensions with its two-dimensional surface. We are concerned only with the latter, and not with the cubical content. The sphere is to be considered as resting on an absolutely plane white table of unlimited extent in all directions. The sphere touches the table at a single point which we shall call its South Pole; on the top side directly opposite, we have the North Pole. To simplify matters we may make a sketch on paper of a vertical section through the centre of the sphere. This profile-picture will show us the sphere as a circle, and the white table as a straight line; the line joining the two poles is the axis of the globe, and the sectional circle is a meridian.
Let us further suppose a creature (resembling, say, a ladybird in shape) having length and breadth, but no thickness, to crawl along this meridian. Although it has no thickness, we shall imagine it to have one property of a solid body, that of being opaque, so that it can throw a shadow if properly illuminated. We assume the globe itself to be transparent. At the North Pole we suppose a very strong point-source of light, a little electric lamp, that sends out rays freely in all directions.
The insect begins its journey at the South Pole and sets out along the meridian to reach the North Pole. It is illuminated by the lamp all the way, so that it continually throws a shadow on the white table. The shadow moves along the table farther and farther from the South Pole, in proportion as the insect moves up the meridian, with the difference that while the insect is describing an arc of a circle, its shadow moves along a straight line. The position of the shadow can be determined at any moment by drawing the straight line connecting the lamp to the insect, and producing it to meet the white surface of the table; the point of intersection is the projection of the insect on the plane.
At the beginning of the excursion the shadow is exactly as large as the flat insect itself, if we assume that its dimensions are negligible compared with the surface of the sphere, for it will then coincide with its own shadow. But when the insect crawls upwards, its shadow will increase, because of the shortened distance between the insect and the lamp, and because the points of projection on the table separate more and more as their distances from their corresponding points on the sphere become greater. There is thus a twofold increase. The shadows move away more and more rapidly, and at the same time increase in size.
When the insect gets very near the North Pole, its shadow, now of enormous dimensions, has moved to a very great distance; and when finally it reaches the Pole, its shadow becomes infinitely great and thus stretches to infinity.
But let the insect wander on along the meridian, past the North Pole, down towards the South. At the moment when it passes the upper Pole its shadow jumps from the right side to the left. Its shadow now emerges from an infinite distance to the left, and, instead of being infinite size, again becomes finite in dimensions as it approaches. It contracts as it approaches, and, in short, the same process as occurred during the first half of the journey now occurs in the reverse order.
[If we fix on the critical moment of the jump from the right to the left, that is, from plus infinity to minus infinity, we may encounter difficulties. For the surface-creature pursues its way without interruption and continuously, and we experience a wish to ascribe to it a shadow-path that is also unbroken and continuous. This is possible only if we assume the two points at infinity to be connected, that is, if we consider them identical. This assumption will seem more natural if we reason as follows. In the profile-picture the table is represented as a straight line, and it is along this line that the shadow travels. We may regard this line as an infinitely great circle, for an infinitely great circle has zero curvature, just as the straight line, from which it is therefore indistinguishable. The infinitely great circle has, however, only one point situated at an infinite distance, that is, it associates together the two apparent points at infinity of the straight line with which we identify it. Accordingly, we preserve the continuity of the shadow-journey, too. Einstein considers it allowable to say that the right and the left portion each represent a half of the infinite projection, which becomes complete only when the two ends are joined.]
Now we must be prepared for an effort of thought which will need considerable help from our imaginations. Firstly, instead of one surface-creature, we shall suppose several crawling about on different meridians, so that a series of shadows will be moving about along straight lines radiating from the South Pole. Next, let us imagine the whole picture to have its dimensions increased by one, that is, we transform the plane-picture into a space model. The phenomena are to remain the same, except that they are to be strengthened by one dimension, surface conditions becoming space conditions, and surfaces becoming solids.
What we now see are actual insects with round bodies (if we retain our original type of creatures), or, since there is no restriction as to their size--the shadows have assumed all possible sizes--we may assume any solid bodies whatsoever, stars or even star-systems. Their motions take place in exactly the same way as those of the shadows previously thrown by the flat bodies.
This means that, if a stellar body moves, its size increases until it reaches the spherical boundary of space, where it becomes infinitely great, and, at the same moment, passes from plus infinity to minus infinity, that is, it enters the universe from the opposite direction; then, if it continues moving in its original direction (as it has been doing all along), it gradually becomes smaller in size until, finally, it reaches its original position and its original size. If we suppose the body to be endowed with the power of sensation, it would not be able to observe its own changes of size, since all its scale-measures would be altered in the same proportion. This whole complex of phenomena would still be taking place in an infinite world of space, but, according to the General Theory of Relativity, the geometry that is valid in this world would no longer be that of Euclid; it is replaced by a system of laws that arise from physics as a geometric necessity. In this new geometry, a circle described with unit radius is a little smaller than it would be in Euclidean geometry, with the result that the greatest conceivable circle in this world cannot assume an infinite size.
Thus we have to imagine that our solid bodies, say stars, arrive at a point in their travels which we may term only "enormously distant." If we call the directions right and left instead of positive and negative, then the process reduces itself to this: the moving body reaches the point, which is enormously distant on the right, and which is identical with the point enormously distant on the left; this means that the body never moves out of the space continuum of this world, but returns to its initial point of departure even when it moves ever onward in what is apparently a straight line. It moves in a "warped" space.
Einstein has succeeded in finding an approximate value for this non-infinite universe, from the fact that there is a determinable gravitational constant. In the constitution of the universe it denotes the same for the mass-relationships of the earth as the gravitational constant of the earth denotes for us, namely, the quantity from which we can calculate the final velocity attained by a freely falling body during a unit of time. He also assumes a probable average for the density of distribution of matter in the universe, by supposing that it is about the same as that of the Milky Way. On this basis Einstein has arrived at the following result by calculation:
The whole universe has a diameter of 100 million light-years, in round numbers. That amounts to about 700 trillion miles.
M.: Does this follow from the discussion you entered on just now?
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 10^54 (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._ 10^2,000,000.
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."
Perhaps poets of the future will busy themselves with this universe; not lyrical poets, but descendants of Hesiod, Lucretius, or Rückert. They will express in verse that Einstein's world offers a source of consolation to tormented spirits which have sickened of Kant's antinomies. For in this still almost immeasurable world the fateful conception "infinite" has been made bearable for the first time. In a certain way it relieves us from what is quite inconceivable, yet into which we are usually driven, and forms a bridge between the thesis "finite" and the antithesis infinite. We are brought to a common stream, in which both conceptions peacefully flow together. There was no mention of this in our talk, and I had good reason for being cautious about following out the theme along these lines. I must not allow any doubts to arise on this point: Einstein, himself, clings with unerring logic to the strict mathematically defined conception of infinity, and allows no compromise with the non-infinite.
When I, on some previous occasion, sought to lead him on to a compromise, involving a transition-boundary, it availed me nothing that I quoted Helmholtz to support the possibility of such an operation: my effort came to an abrupt end.
* * * * * * * *
In pursuing these considerations about the universe, we arrived at things which, in ordinary language, are usually called "occult." In connexion with this, these remarks ensued: "I am, of course, far from trying to trace out a connexion between the four-dimensionality that you establish, Professor, and the four-dimensionality of certain spiritistic pseudo-philosophers, yet it suggests itself to me that in such occult circles efforts will be made to derive advantage from the fact that the same word is used in both cases. This is more than a conjecture, indeed, for there are no misgivings among the ignorant, and so we actually find the name Einstein quoted in connexion with mediumistic experiments that are flavoured with four-dimensionality."
"It will not be expected of me," said Einstein, "to enter into discussion with ignoramuses and misinterpreted. Discarding them, then, let us confine ourselves to a brief consideration of the conception 'occult,' as this has played a part in serious science. The chief example of this in history is gravitation. Huyghens and Leibniz refused to accept gravitation, for, so they said, according to Newton's view, it is an action at a distance and hence belongs to the realm of the occult. Like everything occult, it contradicts the causal order in Nature. We must not regard Huyghens' and Leibniz's contradiction as being due to lack of perspicacity; rather, they objected on grounds which, as investigators, they had every right to uphold. For, as far as our everyday experience is concerned, every mutual influence of things in Nature occurs only by direct contact, as by pressure or impact, or by chemical action, as when a flame is lit. The fact that sound and light apparently form exceptions is not usually felt as a contradiction to the postulate of contact. The case of a magnet appears much more striking because its effect asserts itself as a direct manifestation of force. I must mention that when I, as a child, made my first acquaintance with a compass--and this was before I had ever seen a magnet--it created a sensation in me, which I consider to have been a dominant factor in my life up to the very present. There is, indeed, a fundamental difference between pressure and impact on the one hand, and what we hear and see on the other, even in everyday experience. In the case of light and sound, something must be 'happening' continually, if the effect is to occur and continue...."
"Yet another difference seems to enter here," I interposed. "Is it possible to give a full explanation of gravitation by using only the conceptions pressure and impact? Perhaps 'pressure at a distance' would not have seemed to contemporaries of Newton as unintelligible as a 'tension or pull at a distance.' It seems to me that it is particularly difficult to imagine a pull or an attraction towards a distant object."
Einstein does not consider this difference considerable, and regards it as possible to overcome it even in a manner which can be directly pictured. "If the force is exerted by a corpuscular transmission," he explained, "we may imagine a 'force-shadow' into which the bombarding corpuscles cannot penetrate. Thus if an obstacle, which produces such a shadow, becomes interposed between a body A and a body B, then there will be a lesser pressure on the side of B facing A, and hence B will experience a greater corpuscular pressure on the other side, with the result that B will be forced in the direction of A, and the observer would gain the impression of a pull from B to A. Nowadays, when the theory of 'fields of force' dominates our physical views, we need trouble just as little about using corpuscular pressures and impacts as about the vortices which Descartes once considered as the ultimate causes of the motions of the heavenly bodies. The efforts of certain reformers to reintroduce these vortices and whirlpools as explanations must be regarded as futile."
"Nevertheless," I answered, "it seems admissible to say that, ultimately, there is always an occult element in every physical explanation, an absolutely final and elementary something which we recognize as a principle, without concealing from ourselves that we have reached the limit of explanation, and our knowledge avails no further. This brings me to another question the discussion of which, as I clearly perceive, leads us on to dangerous ground."
EINSTEIN: Don't hesitate to say what is troubling you. I cannot yet see what you are aiming at.
M.: I am referring to certain phenomena which are also called "occult"--with the object of discrediting them. They may at times degenerate to hocus-pocus and fall into the category of dubious arts. It seems to me, however, that scientists have not always drawn the line with sufficient care, and that they have been disposed to reject as humbug, without examination, everything inexplicable that dares to present itself in the form of open display.
EINSTEIN: In general, they will be in the right, for investigators cannot be expected to occupy themselves with things bolstered up by advertisement, and which are supposed to be connected with some fabulous, occult regions.
M.: Nevertheless, in my opinion even among such displays there sometimes occur phenomena which scientists should not pass over with contempt. I, myself, have experienced such cases, and have said to myself: There are stranger happenings here----
EINSTEIN:--than are dreamt of in your philosophy, you were about to say?
M.: Exactly. These are things that in the guise of sensationalism often hide a physical truth well worthy of study.
EINSTEIN: But you must not overlook the fact that in such cases you have mostly played the part of an onlooker, and hence were exposed to all possible manner of deception. You are baffled on all sides by undiscoverable tricks and by other persons, whose collusion you do not suspect. This renders an objective criticism impossible.
M.: This presumes that the performing artist is not entirely isolated. It is possible to bring about conditions that positively eliminate all tricks from the very outset.
EINSTEIN: If you have experienced any such cases, relate them by all means.
M.: I shall be brief, and shall state only facts....
EINSTEIN: Or, expressed more accurately, only things which seem to have been facts as far as you can trust to memory. Well then, you think that you have grounds for saying that you caught a glimpse of a mysterious world at that time.
M.: It is certainly long ago, more than thirty years. Hansen, the freak, one of the most eminent of his profession, was showing hypnotic and telepathic experiments that were partly identical with experiments that the celebrated scientist Charcot at Paris was performing for purposes of pathology.
EINSTEIN: Well, then, why did you hesitate before? These experiments come under the head of science, and require no occult veil to appear in the open.
M.: This touches the main issue. Hansen did not work in the interests of science, but wished, above all, to earn money. Nevertheless he had in his own way produced marvellous results that were used later for scientific work. Unfortunately in his case, owing to the fact that he cloaked it in occultism at the outset, he was brusquely repudiated by scientists. The result was that Hansen was condemned to a long period of imprisonment in Dresden, thanks to the recommendation of scientists who declared that the experiments were only possible if deception was practised, and hence that Hansen was an impostor who should be made harmless by being incarcerated.
EINSTEIN: And how did you yourself seek to discover whether his experiments were genuine?
M.: Very easily and with absolute certainty. One of my acquaintances, the wealthy race-horse owner, von Oelschläger, had induced him by means of a high fee to experiment at his country house, at some distance from Berlin, in the presence of persons, not one of whom Hansen knew, and in the case of whom there could be no question of secret collaboration. I can assure you that everything succeeded without exception. A single second was sufficient for him to communicate his will to each subject of experiment. He operated like a supernatural being on those present.
EINSTEIN: I should like to hear examples.
M.: Herr von Oelschläger introduced four jockeys, and suggested a race in the great salon. Hansen placed them astride over chairs, hypnotized them on the spot, described the shape of the course, giving distances in kilometres, curves, and even the value of the prizes. He then gave the signal for starting. The jockeys immediately began treating their chairs as race-horses, exhibiting all the signs of extreme strain which accompany the actual ride.
EINSTEIN: This is not yet a positive proof. The subjects of experiment may have become cognizant of the fact that they were to serve some eccentric display. Their acquiescence in a prescribed part need by no means signify that they were subjectively convinced of the genuineness of the affair.
M.: There could be not the slightest doubt on this point. After a few seconds perspiration was streaming over their faces as a result of the exertion, a symptom that exhibits itself only when the participants are convinced of the absolute earnestness of their undertaking. All that gazed on this baffling ride made the acquaintance of a grotesque reality, and were looking into a strange world of dreams, which transformed wooden chairs into living thoroughbreds. In the course of his following experiments in the transference of his will-power, Hansen experimented with an actress who was famous at that time, and with whom he had no more acquaintance than with the others. He again produced deep hypnosis, and gave the order: I shall ask you various questions, all of which you will be able to answer correctly, with one exception: you will have forgotten your name. And so it happened. In her trance the actress gave correct answers, until, when the question, "What is your name?" was asked, her own name, Helene Odilon, had vanished from her memory. And immediately afterwards, she told me herself that, in spite of her state of coma, she had retained full consciousness, had understood everything, and had been possessed of her memory until it came to the critical moment when, in spite of extreme efforts, she could not recollect the words Helene Odilon. But Hansen did not stop at dictating his thoughts to others, he also transformed corporate things. By a single motion of his hand he converted a stable-boy into a rigid block, devoid of sensation. Never would I have thought such an intense state of cramp possible. He placed the boy with his feet and head alone resting on two supports, so that the body itself was poised in space. He then stood on the body with his whole weight, without the rigid body of the boy bending even an inch.
EINSTEIN: How did he, in all these cases, restore the normal state?
M.: Always by a single gesture, which, like everything that he did, worked at lightning speed. I must admit that his display became a little monotonous after a while, and that his programme did not seem capable of much variation. Things were different, however, in the case of a man who, some years previously, had toured the world as an exponent of occult phenomena, and to whom scientists will some time in the future look back with regret. When he appeared, most academicians took only sufficient notice of him to reject him without having given him a trial. It was Henry Slade, the American, who is not to be confused with other Slades who appropriated his name in order to dupe people whose insatiable curiosity was aroused.
EINSTEIN: One might almost suppose that your genuine Henry Slade served as a model for them.
M.: For certain reasons I regard this as out of the question, mainly because the true Slade gave "demonstrations" only occasionally, his chief object being to interest scientists. He, himself, repeatedly asserted that he did not understand his own achievements, and he unceasingly requested the supervision of professional physicists and physiologists, to whom the unusual phases in his nature were to serve as objects of study. The result was that people like Dubois-Reymond, Helmholtz, and Virchow refused to see him, not to mention experiment with him.
EINSTEIN: These men cannot be reproached for acting in this way. Slade was regarded as a representative of a four-dimensional world in the spiritistic sense; serious scientists must avoid all humbug of this sort, since even slight interest in it can easily be misinterpreted by the ignorant public.
M.: Not every one was afraid of compromising himself. After closed doors had greeted Slade in Berlin, he went to Leipzig, where he became an object of study for one important scientist.
EINSTEIN: You are referring to Friedrich Zöllner, who undoubtedly had a reputation as an astrophysicist to preserve. But he would have served his reputation better if he had not entered into this adventure with the American spiritist.
M.: Perhaps there will some day be cause for a revision of opinion on this point. The documents are extant, even if, half forgotten, they are reposing in various libraries. A renewed investigation of Zöllners _Scientific Dissertations_, dating from 1878 to 1891, might lead to the judgment that his ghostly interpretations are to be regarded as occult in the worst sense, and yet one would marvel that a great scientist, such as he was, should have felt himself at a complete loss with his knowledge, so that he was forced to resort to abstruse methods in order to escape from the mental confusion into which Slade had plunged him.
EINSTEIN: That merely shows that Slade, as a cunning practician, surpassed him, and that Zöllner did not succeed in seeing through his machinations.
M.: This would lead one to assume that Slade knew more physics than the Leipzig professor. For in a great number of experiments Zöllner himself had prescribed the conditions, including all contrivances which made deception so much the more unlikely, since Slade himself could not know what Zöllner's intentions were. It was a question of Electricity, Magnetism, Optics including prepared conditions of polarization, involved Mechanics, in short, things that Zöllner as a professional physicist understood thoroughly, and which, moreover, were controlled by others of his profession. Among the latter was the celebrated professor of Electricity, Wilhelm Weber, who, like Zöllner, found himself faced by phenomena that were utterly incomprehensible to him. It would be a profitable undertaking to bring these dissertations to light again, and it would easily be recognized that the things described actually deal with scientific problems and have not the remotest connexion with tricks of magic. For example, there is an account of an incredible anatomical feat. On flour which had been placed carefully in a dish beforehand, there suddenly appeared the imprint of a naked human foot, whilst Slade was present at a certain distance, being fully clothed and subject to careful scrutiny. The footprint showed all the surface-details of the skin, as was confirmed by authorities, just as only a left foot could produce them, but not an artificial copy.
EINSTEIN: And from this Zöllner inferred the intervention of supernatural beings? He would have done better to measure the dimensions of the foot.
M.: So he did--at once. A difference of four centimetres between the length of Slade's foot and the copy was disclosed. This riddle, like so many others, remained unexplained. I must repeat that I am not in the slightest degree disposed to assert that occult phenomena really occur, but am interested only in seeing that they are investigated carefully by qualified persons.
EINSTEIN: Your remarks show that Leipzig scientists did so at that time with no better result than that Zöllners mental confusion became still greater.
M.: The conjecture remains that the Leipzig experiments, abundant as they were, did not suffice. Allow me to ask a direct question, Professor. Supposing another such agent of miracles should appear, would you yourself feel impelled to test him experimentally?
EINSTEIN: Your question is misdirected. I explained above that I share the point of view taken up by Dubois-Reymond and his colleagues.
M.: The following case may be conceived. A certain man, X, might suddenly appear, who has control of a certain natural force that has never before been investigated; like one who knew how to use electricity at a time when people had never experienced any electrical phenomenon. He would be able to give hundreds of demonstrations, all of which we should relegate to the realm of inexplicable magic. We should, for instance, be much astonished if he were to draw sparks from a living person. Now, suppose two professors express an opinion. Professor A declares the whole thing to be a farce, and refuses to look into it at all. Professor B is ready to investigate the achievements of X only if the latter subjects himself from the beginning to all the physical conditions that are to be determined beforehand. And suppose the professor arranges his conditions so that they make impossible the occurrence of electrical phenomena. If, now, all scientists were to behave like A and B, the consequences would be very depressing. For here was an important field of investigation, which is cut off owing to the distrust or obstinacy of scientists, who should have been the first to open it up. It is quite irrelevant whether X had the character of a charlatan or not, for behind his charlatanism there were facts which clamoured for investigation.
EINSTEIN: The most that I can grant is that your imagined case does not lie outside the scope of possibility. Yet the chance that there is such a "natural force" hitherto undiscovered by Man, that is, one that is a "secret force" as far as we are concerned, is so vanishingly small that it may be set down as equal to impossible. I should refuse to take part in any such practices, served up in the form of sensation, for one reason that I should regret the waste of time, as there are better things to do. It is a different matter if the mood takes me to visit a variety entertainment, in order to derive amusement from such mystifications. For example, only yesterday I was in a little theatre, in which, among diverse items, a thought-reading woman was performing. She correctly guessed the numbers 61 and 59 that I had in my mind. But let no one mention this as a case of telepathic actions at a distance or wireless communication between minds, for an intermediate person, the manager, was present, and I had to whisper the numbers to him. The distance to the stage was certainly too great to allow the sound to be conveyed directly to an audible degree. Hence there must have been a different, very cunningly arranged code of signals, which eluded the notice of people in the stalls. The process consists actually in an extraordinary refinement of observation, which does not, however, seem to me any more wonderful than the training of a reckoner who extracts cubic roots mentally, or than the practised muscles of a juggler all working in unison to enable him to perform feats with twelve plates simultaneously.
M.: It gives me enough satisfaction, Professor, that you conceded me before a certain limited chance of finding a last refuge in occultism. And even if you, yourself, as a representative of the most rigorous research of physical reality, refuse to consider it, yet the fact that many others are drawn irresistibly towards mysterious phenomena cannot be denied. Should one feel shame on this account? I believe that, in this matter, we are touching on inner confessions that are quite independent of the standard of the mind in which they are embedded. Newton considered the key of the universe to be a personal God, whereas Laplace proclaimed: _Dieu--je n'avais pas besoin de cette hypothèse_: this contrast allows no inference to be drawn as to their relative keenness of mind. And probably the same may be said of the question whether there are other hidden universes besides the one in which we live. In any case, those who feel enthusiasm for such questions can quote in their support good names from the learned world. Immanuel Kant occupied himself seriously and intensively with the wonders of Swedenborg, Kepler practised Astrology, in which he had a firm belief, Roger Bacon, Cardanus, Agrippa, Nostradamus, van Helmont, Pascal, and, among the modern, Fechner, Wallace, Crookes, are to be counted among the mystics. No matter whether the views they held were theosophical, occult, four-dimensional in the spiritistic sense, or coloured by any other superstition; they proclaimed that things that could be rigorously proved were, alone, insufficient for them. Out of presentiment and conjecture they constructed wings with which to fly into regions _extra naturam_. This is how it happened that, as the common folk could not find a place in science for many extraordinary achievements, they assigned their authors to the realm of magicians, as in the case of Paracelsus, Albertus Magnus, Raimundus Lullus, Sylvester II, who were regarded as sorcerers. And this coin is still current: to Edison, of our times, the term, "sorcerer of Menlo-Park," has become attached. In the minds of the populace discovery and invention, works of genius and supernatural phenomena, become confused and indistinguishable; it may even happen to you. Professor, that your works will become invested with legend. I should not like to conjure up what your fate would have been if your theory of relativity had originated at the time of the Inquisition. For the views put forward by Giordano Bruno are mere child's play compared with your theory of the universe as a quasi-spherical closed space of hyper-Euclidean character. The tribunal of the Inquisition would not have understood your differential equation, gravitational potentials, tensors, and equivalence theory; they would abruptly have declared the whole theory to be a magical formula or a manifestation of the devil, and would have honoured it and you with a funeral pyre.
EINSTEIN: This is clearly a slight exaggeration. Mathematico-physical and astronomical works have never been attacked by the Papal courts, but, on the contrary, have been much encouraged by them down to the present day. This is abundantly clear from the fact that we can set up a whole list of Brothers of Orders, particularly Jesuits, who have made eminent discoveries in natural science. From my personal knowledge of you, I foresee that you will one day sketch a fantastic trial, in which the new world-system will have to defend itself against the _Sanctum Officium_.
M.: This would be a very grateful task, judged from the literary point of view. What a splendid colouring could be obtained by bringing these two worlds of thought into conflict with one another, the Relative against the Absolute, which has been established in tradition and dogma. But we need not even call the historical fancy into action, for, actually, the theory of the structure of the world is even now still at variance with traditional ideas, that act with dogmatic violence. There is no need to deny the fact that every person of education, who makes the acquaintance of Lorentz's, Minkowski's, Einstein's ideas for the first time, feels excited to offer contradictions, and becomes involved in a tumult of pros and cons, and each one experiences in himself the excitement of an inquisitorial tribunal. The triumph of the new theory passes over the corpses of conceptions that lie at the cross-roads of thought and, long after, retain a ghostly existence. Only very few of us are aware of the further inner revolution that awaits us along the line of development of Einsteinian ideas; we have only vague presentiments that whisper to us that the end of forms of thought once considered as irrefragable is drawing nigh. When once the principle of causality has been set on a relative base, and all "properties" have been resolved into occurrence, and all that is three-dimensional has come to be recognized as an abstraction from the four-dimensional world that is alone valid, then the time will have come to arrange for, the death procession of all the philosophies that once served as the main pillars of thought.
A retrospect of the trials of Giordano Bruno and of Galileo Galilei offers certain parallels other than those usually discovered by scholars. And if, to-day, we proclaim Einstein as the Galilei of the twentieth century, it must be added that in character he is fortunately a Bruno and not a Galilei. For it is not true that the latter came out of the persecution as a moral victor with an _eppur si muove_, rather, in spite of the protection of influential prelates and dignitaries, even of the entourage of the Pope, he lacked courage and bowed his head, betraying his science and denying himself as well as Copernicus. Are we to picture how Einstein would have acted under similar circumstances, even if they cannot recur again?
Whoever has even an inkling of his character will entertain no doubts. At that time, three hundred years ago, the materials for a magnificent scene, "one world _versus_ the other," lay ready. Only one condition was wanting, the moral courage of the hero. The lack of this one factor spoilt the final act for the history of that time. The fine ethical feelings of later generations have had to be propitiated by improvising a legend iridescent with beautiful colours.
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