I mention this little episode, which is insignificant in itself, merely to give an example of how a great discoverer, too, finds amusement in such distractions. In Einstein's case this tendency to practise his ingenuity on unimportant trifles is so much the more pronounced from the fact that he requires an outlet for his virtuosity in calculation, and gratefully welcomes every suggestion that helps him to relieve his mental tension. Similar characteristics are reported of the great Euler, as well as of Fermat, whereas many another eminent mathematician feels decidedly unhappy if he drifts within reach of the realm of actual numerical calculation. In my mind's eye I still see Ernst Kummer, the splendid savant (who, in his time, conferred distinction on Berlin University by his very presence), suffering agonies whenever ordinary arithmetical tables threatened to appear in the working-out of his formulae. As a matter of fact, these two things, a mastery over mathematics and a talent for ingenious calculation, are to be considered as quite independent, even if we now and then find them present in the same person.
In the case of Einstein this tendency is a symptom of an incredible universality of spirit. It moreover presents itself in the pleasantest forms, and a character-sketch of Einstein would be incomplete if this trait were not mentioned. Every problem which is in any way amusing excites in him a willing interest and enthusiasm. I once directed our conversation to the so-called Scherenschnitte. These are made from long strips of paper or canvas, the ends of which are caused to overlap a little and then pasted together, but instead of being fixed so that a flat wheel results, which rolls on one side of the strip, the strip is twisted one or more times before the ends are fastened together. If now the strip is cut lengthwise right along its centre, various unexpected results occur, depending on the number of twists that have been made before pasting.
Some very complex geometrical difficulties are involved in these problems. This is shown by the fact that learned mathematicians have written extensive disquisitions on these curious constructions (for example, Dr. Dingeldey's book, published by Teubner, Leipzig). Einstein had never taken notice of these wonders of the scissors, but when I began to form these strips, to paste them, and to cut them, he immediately became interested in the underlying problem, and predicted in a flash what puzzling chain constructions would result in each case, with a certainty that would lead one to imagine that he had spent days at it. On another occasion a space-problem dealing with dress came up for discussion: Can a properly dressed man divest himself of his waistcoat without first taking off his coat? One would not have dared to confront Copernicus or Laplace with such a problem. Einstein at once attacked it with enthusiasm, as if it were an exercise in mechanics, the body being the object; he solved it in a trice, practically, with a little energetic manipulation, much to the amazement and joy of the beholder, who asked himself: Is this the same Einstein who developed the work of Copernicus and Newton? A little later, perhaps, the conversation centres around some serious point drawn from politics, political economy, sociology, or jurisprudence. Whatever it may be, he knows how to spin out the suggested thread, to establish contact with his partner in conversation, to open up his own perspectives without ever insisting on his point of view, always stimulating and showing a ready sympathy for the subject of discussion and for all the ideas which it crystallizes, the prototype of the scientist, in the mouth of whom Terence put the words: "I am a human being; nothing that is human is alien to me!"
CHAPTER IX
AN EXPERIMENTAL ANALOGY
Forms of Physical Laws.—Aids to Understanding.—Popular Descriptions.—Optical Signals.—Simultaneity.—Experiments in Similes.
"I WISH to ask you. Professor, to help me over a difficulty and to treat me as the spokesman of a great number who are similarly troubled. In most accounts of your theory of relativity, there is a dearth of definite, concrete, illustrative examples on which we can fix our minds whenever the theorem is to be applied generally without limitation. Let me express this more precisely: Your simplified picture of the structure of the universe is achieved in the theory of relativity by emancipating all observations from fixed co-ordinate systems, and by proclaiming the equivalence of all systems of reference. One of your earliest theorems states that physical laws describing how the states of physical systems alter, remain the same, no matter to which of two co-ordinate systems these states are referred, provided that the co-ordinate systems are moving rectilinearly and uniformly relatively to one another. This theorem entails the following statement. If we—erroneously—adopt a non-relativistic view, we shall come to the conclusion that physical laws depend on the particular system of reference chosen, and will thus assume a different form for each different system. At this point we experience a desire to hear definite examples. What varying forms may a certain given physical law, known under a definite form, assume, and how can we use this law to show that it must adapt itself to the postulate of relativity?"
Einstein explained that such examples cannot be given in special cases, but only in very general terms. If we were to suggest the elliptic orbits of the planets (at which I had hinted in my remarks), we should fall into error, for the law of elliptic orbits is no such law. For, from another point of view, the elliptic paths of the planets might be drawn out into wavy lines, or into spirals, and they would remain ellipses only as long as the lines of motion are referred to the central attracting body. But the constancy of the velocity of light is such a law, as also is the law of inertia, according to which a body that is left to itself moves uniformly in a straight line.
I confessed to him that this limitation to a few very general laws would be a painful matter for many an enthusiast of average attainments, who has great difficulty in distinguishing the laws that are generally valid from those that hold only within circumscribed limits. But if this were not so, we should have to alter our conception of what is conveyed by a popular exposition. For it is called popular, not because it now and then uses the patronizing words "dear reader," but because it anticipates the questions and doubts of the man of average sense, and examines them, proving some to be unjustified and others to be reasonable or unreasonable, as the case may be. "Then there is a further matter that troubles me," I continued. "Let us suppose an ordinary reader of such a popular account to get a first insight into the new conception of Time. He is glad to feel the ideas dawning in him, and, to get a more lasting view of the idea, he repeats the arguments through which he has just threaded his way, and, in doing so, again encounters the phrase 'uniform motion.' At the first reading he imagined that he understood the expression quite well, but the second time he pauses and considers. For now that he knows how much depends on it, he is anxious to find out the exact meaning of a 'uniform motion.' He looks for a definition, and if he cannot find one in the book he is perusing, he endeavours to reason it out for himself. With good luck he arrives at the usual statement: a body moves with 'uniform motion' if it traverses equal distances in equal intervals of time. But equal intervals of time are clearly those during which a body in uniform motion traverses equal distances. In other words, he explains A by means of B, and B by means of A, so that he has involved himself in a vicious circle from which he cannot escape. This is his hour of need, due to the difficulty of 'time.'
"He hopes that further study will remove this obstacle. He meets with the conception of 'simultaneity,' which is defined for him anew, and is disclosed as being 'relative.' He manœuvres further towards the fundamental theorem that every body of reference has its own particular time.
"His popular booklet makes this clear to him by quoting the example of a flying-machine, or, better still, a railway train that is rushing along an embankment at a very great speed, and that carries a passenger. Two strokes of lightning I and II are to take place at two widely distant points on the embankment. The question is then: When are these two flashes of lightning to be considered 'simultaneous'? What conditions must be fulfilled to ensure this? It is found—incontrovertibly—that the light-rays starting out from the two strokes of lightning must meet at the mid-point of the embankment.
"It now follows from a short chain of argument that the observer in the train will see flash II earlier than flash I, if they reach the observer, who is at rest, at the same moment. That is, two events that are simultaneous with respect to the embankment are not simultaneous for a moving system (such as a train or a flying-machine); the converse is, of course, also true.
"Here, again, the eager layman encounters difficulty, for he asks himself: Why should the two events be characterized or defined by lightning-flashes in particular? If acoustic signals were used instead, nothing would be altered in the fundamental determination, for the sound rays (sound-waves) would likewise meet at the mid-point of the line joining the sources of disturbance. What is the reason that the relativity of time arises only when phenomena are regarded optically, and that rays of light play the deciding part in all later developments?
"And this particular query is followed by one which is more general: Why does the popular pamphlet not read this question in my mind? I know that the author of it is more skilled in these matters than I, but just this superiority should help him to divine what is passing in my mind when I make efforts to follow his reasoning."
Einstein had listened to me patiently, and then he explained to me at considerable length why in this case optical signals cannot be replaced by sound signals: light is the only mode of motion that shows itself to be entirely independent of the carrier of the motion, of the transmitting medium. Thus the constancy of velocity is assumed in the above argument, and as this constancy is an exclusive property of light, every other method must be discarded as unallowable for investigating the conception "simultaneity." Furthermore, he showed me how, on the basis of relativity, starting from the embankment-experiment, we may arrive at a perfectly consistent representation of the conception of Time. He certainly did this by applying subtle physical arguments that exceed the scope of the present book.[7] He added, in substance, that it was futile and impossible to discuss in detail all the conceivable objections that might arise in the mind of one reading a popular work of this kind: it was a futile undertaking, because the true purpose was defeated, inasmuch as a clear development of the fundamental thought would be almost impossible under the cross-fire of so many random questions.
[7]In these arguments, arrangements of synchronous clocks occur, which are fixed into the co-ordinate systems, the positions of their hands being compared with one another. The "time" of an event is then defined as the position of the hands of a clock immediately adjacent to the scene of the event.
Thus, in this matter, Einstein takes the same stand as Schopenhauer in the preface of his chief work, in which he says: "To understand this work no better way can be advised than to read it twice (at least), inasmuch as the beginning assumes the end, almost as much as the end assumes the beginning; the smallest part cannot be understood if the whole has not already been understood." Whoever accepts and follows this advice will find that the intermediate objections will gradually balance and cancel one another, and that it is not necessary that they should interrupt the steady and consistent line of development.
The position would be different if a disciple of the new theory should resolve to dispense with strictly scientific reasoning altogether, and should wish to meet the wishes of his readers or hearers by discarding accuracy entirely. Such a programme seems quite feasible.
"This would be merely following the sketchy method of a magazine," Einstein remarked, "but you do not seriously think that it would lead to anything?"
"It would not be a true explanation, which is reserved for technical productions. But I can imagine that it would not be unprofitable to help one who is entirely ignorant on these questions by using makeshifts, in the form of allegories or analogies, which will serve as supports if he should take fright during the course of his earlier studies. These shocks are bound to occur, as, for instance, when he learns that a moving rigid rod undergoes contraction in the direction of motion."
"But this is proved to him!"
"Nevertheless, he does not easily accept it. For the general reader will say to himself: 'A superhuman effort is imposed on my mind. A rigid rod is the most constant of all things, and never before has one been compelled to regard something that is constant as variable.'"
"If he does not grasp it, no analogy will teach him."
"But perhaps it is possible. The analogy is to show him that the effort is not superhuman, and that thinking Man has already had occasion to become familiar with such transformations from constant to variable factors."
"I am afraid your analogy will prove a failure."
"From the scientific point of view this is probably true, inasmuch as all comparisons are imperfect, but the analogy may yet be of service as a last resort. For example, I should say to my general reader: 'Picture to yourself a savant of the Middle Ages who reflects on the constitution of animals and plants. One fact seems to him to be irrevocably true, namely, that the species are unchangeable! A palm tree is a palm tree, a horse is a horse, a worm a worm, and what is once a reptile remains a reptile. A species in itself denotes something absolutely invariant.'"
"The expression is wrong when taken in this connexion; you mean invariable."
"A little inaccuracy more or less does not affect the analogy. For the sake of my picture I should like to retain the conception-couple, variable and invariant. Well, then, the species give our savant the impression of invariance, as in the view that was held by Linné and Cuvier. This view necessarily has its counterpart in his thought. He argues that every species has its own original root, and that, in this sense, there is very extensive variation. The fundamental roots are extremely manifold; Nature has produced innumerable variations in her individual acts of creation. But now the Theory of Descent of Lamarck, Goethe, Oken, Geoffroy St. Hilaire, enters the field and produces a complete inversion of these two elements; the two parts of the earlier point of view change places. Our savant has to revise his whole world of thought. Now all organisms are to be traced back to a single original root: the latter, which was variable before, becomes an invariable unicellular primitive organism, but the apparently unchangeable species now becomes variable, in the widest possible sense. And even if this savant should exclaim: 'How am I to reconcile myself to this view?' his descendants later find no difficulty in accepting the idea that the organic roots are uniform, and that it is the species that are subject to all manner of variation as a compensating feature."
Einstein expressed himself very little pleased with this attempt at an analogy, and found that it was so far fetched that it could not be considered admissible.
"Then I must ask your permission to continue my attempt; perhaps something useful may yet result from it. I now picture to myself a human being who lived in classical times and who, following Ovid and the great majority of his contemporaries, regards the earth as a disc. On this disc, each inhabitant of the earth has his own particular position, for the disc has a centre with reference to which the position of a person can be specified if his distance and his angular displacement from a given initial radius is specified. Thus, there is a variation of position if various persons are considered. On the other hand, the Above and the Below is absolutely invariable for all persons, for the lines running between Above and Below are all parallel for them, since they all have uniformly the same disc under their feet and the same heaven above their heads. Ovid would therefore have refused to entertain for a moment the suggestion that Above-Below is a variable. But his distant descendants accepted the view that the earth is spherical and that there are antipodes as self-evident, and they found not the slightest difficulty in considering the line Above-Below to vary with their own position, making all possible angles with an initial line extending to direct oppositeness. Referred to the centre of the sphere, all people have now an 'invariant' position, whereas, in compensation, the Above-Below is subject to every conceivable variation. And now I again address myself to the average reader, and say that the meaning of these analogies is that every doctrine that leads to a great uniformity converts what was formerly invariant into a variable quantity, and vice versa. The theory of relativity makes all considerations about the physical world independent of all co-ordinate systems; it establishes completely invariable uniformity, removed from all changes due to varying points of view. Hence what was previously invariable—such as a rigid measuring-rod—will now become variable. It is not surprising that this requires a new method of thought, a revision of our mode of reasoning, for the above analogies show that these radical adjustments are characteristically necessary in the case of comprehensive theories, and that such theories are able to overcome apparently firmly established ideas. The parallels that I drew above will at least inspire the average reader with a certain confidence, for they show him how results of reasoning that were once considered incredible were regarded as self-evident by later generations."
I have already emphasized sufficiently that Einstein regards as inadequate these auxiliary pictures that have presented themselves to me. Yet in the course of the conversation I gained the impression that his judgment grew somewhat milder, and that, with certain reservations, he was disposed to let them pass as tolerably useful helps—and they are not intended to be more than this. I think, therefore, that I am not acting counter to his wishes in citing these allegorical examples here, particularly as they arose in the course of our talks.
Since then, I have had many opportunities of testing these examples on certain persons, and may mention that they proved quite useful. Analogies of this kind may offer a friendly help in moments when the uninitiated feel themselves in peril, and encounter a difficulty which they imagine to be insurmountable. They do not remove the difficulty, but they impart a certain power of expansion to the intellect and encourage a continuation of effort, which would probably otherwise be relaxed at the first sign of something which is imagined to be inconceivable. There is thus no room in textbooks for such helps, but they may justifiably find a place in a book that departs from the methodical route, and hopes to discover in by-ways things that are suggestive and instructive.
CHAPTER X
DISCONNECTED SUGGESTIONS
Conditionality and Unconditionality of Physical Laws.—Conception of Temperature.—Grain of Sand and Universe.—Are Laws unalterable?—Paradoxes of Science.—Rejuvenation by Motion.—Gain of a Second.—Deformed Worlds.—Atomic Model.—Researches of Rutherford and Niels Bohr.—Microcosmos and Macrocosmos.—Brief Statement of the Principle of Relativity.—Science with reduced Sense-Organs.—Eternal Repetition.—Higher Types of Culture.
IN all branches of reasoning, no word and no conception has played a more important part than that of law. Physical laws denote the barrier that separates strictly chance and arbitrariness from necessity, and it seems to us that the region of the latter must ever extend so that finally nothing will be left of the former, which will have become amalgamated with necessity. We shall be constrained to believe more and more in a supreme law that will be a complete expression of all the partial laws which science presents to us as more or less permanent results of individual researches.
Our conversation was centred about these individual laws, such as those that are taught in the theory of gases, optics, etc., and that are associated with the names, Boyle, Gay-Lussac, Dalton, Marriotte, Huyghens, Fresnel, Kirchhoff, Boltzmann, and others. In connexion with these I asked Einstein whether he regarded the laws as things unconditioned in themselves, and capable of proof under every set of circumstances; and whether absolutely valid laws existed or could exist.
Einstein's answer was essentially in the negative. "A law cannot be final, if only for the reason that the conceptions, which we use to formulate it, show themselves to be imperfect or insufficient as science progresses. Let us consider, for example, an elementary law such as Newton's Law of Force. From our more recent point of view we find the conception of direct action at a distance to be inexact in Nature. For it has been shown that action at a distance is not an ultimate factor, but must be resolved into a multiplicity of actions between immediately neighbouring points (The Theory of Action by Contact or Contiguous Action). Another example is provided by the conception Temperature. This conception becomes meaningless if we endeavour to apply it to molecules: it leads to no result if we try to impose it on the smallest parts of matter as such. The reason is that the state, the velocity, and the inner energy of the individual molecules fluctuates between very wide limits. The conception 'temperature' is applicable only to a configuration composed of many molecules, and even then it is not applicable quite generally. For let us picture to ourselves an extremely rarefied gas contained in a closed receiver. Two opposite walls are to be at different temperatures, the one being cold and the other being hot In a gas at such very low pressure the molecules come into collision so seldom that, practically, we have to take into account only the collisions of the molecules with the confining walls. The molecules that rebound from the hot wall have greater velocities than those coming from the colder wall, and hence the conception of temperature becomes untenable for this gas."
"Would the temperature-scale on the thermometer then denote nothing?" I asked. "The greater or lesser degree of warmth of a body, in this case of the mass of gas, depends on the more rapid or less rapid motion of its smallest parts. The motions are in any case present, so what would a thermometer indicate?"
"It would betray only that it had nothing to indicate. If a thermometer that is blackened on one side were inserted into the vessel containing the gas, then different temperatures would be recorded if the thermometer were gradually turned about its own axis; and this signifies that the conception of temperature has become meaningless for this configuration of molecules. And passing beyond the quoted examples, I should maintain that all our conceptions, however subtly they may have been thought out, are shown in the course of progressive knowledge to be too rough hewn, that is, too little differentiated."
We spoke of the "Properties of Things," and of the degree to which these properties could be investigated. As an extreme thought, the following question was proposed:
Supposing it were possible to discover all the properties of a grain of sand, would we then have gained a complete knowledge of the whole universe? Would there then remain no unsolved component of our comprehension of the universe?
Einstein declared that this question was to be answered with an unconditional affirmative. "For if we had completely and in a scientific sense learned the processes in the grain of sand, this would have been possible only on the basis of an exact knowledge of the laws of mechanical events in time and space. These laws, differential equations, would be the most general laws of the universe, from which the quintessence of all other events would have to be deducible."
[This thought may be spun out in yet another direction. Every piece of research, however specialized it may appear and of whatever minor importance it may be, retains a relationship with researches into the universe, and may prove to be valuable for this latter task. If we accept the view that science is capable of realizing perfection, then every contribution to knowledge, even the most insignificant, is essentially indispensable for attaining this goal.]
Can a physical law alter with time? In more precise language, can time, as such, enter explicitly into laws, so that, for example, an experiment that is carried out at different times leads to different results? This question has been treated several times, among others, by Poincaré, who answered it with an emphatic "No!" but also by others to whom the invariability of physical laws did not seem to hold for all eternity. If my memory does not play me false, Helmholtz once expressed faint doubts about the constancy of laws.
Einstein answered this question with a decided negative. "For a law of physical nature is, by definition, a rule to which events conform wherever and whenever they take place. Thus, if we were to be compelled as a result of experience to make a law dependent on time, it would be a necessary step to seek a law independent of time, which would include in itself the law dependent on the time as a special case. The latter would be excluded from the category of physical laws, and would henceforward play the part only of a result deduced from the law which is independent of the time."
What attitude should we adopt if, in studying a scientific doctrine, we encounter paradoxical results even though the inferences have been drawn correctly—that is, if we meet with a deduction to which our reasoning powers object, although no fallacy is discoverable in the argument?
Before we deal with cases which seem to me, personally, to be interesting, let us hear what is Einstein's attitude in general. "As soon as a paradox presents itself, we may, as a rule, infer that inaccurate reasoning is the cause, and should thus examine in each particular case whether an error of logic is discoverable, or whether the paradoxical result denotes only a violent contrast with our present views."
Let us first take examples from an entirely modern science, from the Theory of Aggregates founded by Georg Cantor of Halle. We shall follow the argument by the only possible method for this book, namely, by rough indications that will serve our purpose and do not claim to be accurate in expression or in sense.
If we take an aggregate of three objects, for example, an apple, a pear, and a plum, we may, by definition, form six partial aggregates, namely:
the apple
the pear
the plum
the apple and the pear
the apple and the plum
the pear and the plum.
The aggregate of the partial aggregates, which contains six elements, is thus greater than (actually twice as great as) the original aggregate, in which only three elements occur.
If the original aggregate contains an additional element, for example, a nut, the following partial aggregates may be formed:
the apple
the pear
the plum
the nut
the apple and the pear
the apple and the plum
the apple and the nut
the pear and the plum
the pear and the nut
the plum and the nut
the apple, the pear, and the plum
the apple, the pear, and the nut
the apple, the plum, and the nut
the pear, the plum, and the nut.
Thus, in this case, the aggregate of the partial aggregates is already considerably greater than the original aggregate. This numerical excess increases rapidly with each successive increase in the original aggregate, so that if we apply the same reasoning to an infinite aggregate, the aggregate of partial aggregates becomes an infinity of a higher order. This is expressed by saying that the infinite aggregate of partial aggregates has a greater potentiality than the infinity of the elements of the original aggregate.
So we see that the one infinity is, in popular language, much more comprehensive, more powerful than the other. Our minds do not find it impossible to grasp this. But in a definite imaginary experiment it is found that this theorem of progression not only fails in its application, but leads to flagrant contradiction.
For if we start from the primary aggregate of "all conceivable things," its infinity can certainly not be transcended by any other infinity. But according to the above theorem the "aggregate of all partial aggregates" would have a greater potentiality, although it itself cannot extend further than to the conception of the maximum of all conceivable things. We thus arrive at an insoluble paradox, a typical example of how, in the system of conceptions involved, something is insufficient or not in conformity with logical thought. And this sceptical view receives support from various remarks of Descartes, Locke, Leibniz, and particularly Gauss, who, long before the advent of the Theory of Aggregates, raised a protest against inexact definitions of infinity.
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!
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.
Let us speculate on the following question. If all things in the universe should increase or decrease enormously in dimensions, and if, at the same time, in a manner totally concealed from us, certain physical conditions should become changed, we should lack all means of discovering the difference between things before and after the change. For since all measuring-rods, including those furnished by our senses, would have become changed in the same proportion, the two conditions could not be differentiated from one another. It may easily be shown that this would necessarily occur, if an extramundane power were non-uniformly to displace, deform, compress, or bend all things in the universe, provided that our instruments and senses participated in this transformation. Accordingly it is permissible also to regard the universe known to us as one that is deformed, and one that is derived from another, the original form of which will ever remain a secret to us.
Is there any connexion between this grotesque speculation and the theory of relativity?
We can establish only one that is negative and that arises e contrario. "These deformations," said Einstein, "are in themselves abstractions that are physically meaningless. Only relations between bodies have a physical meaning, for example, the relation between measuring-rods and the objects they measure. Therefore, it is reasonable to talk of deformations only when we are dealing with the deformations of two or more bodies with respect to one another, whereas the conception of deformation has no sense, unless a real object is specified, to which it is referred. The philosophical merit of the general theory of relativity, as compared with previous views of physics, consists in the fact that the former avoids entirely these meaningless abstractions with respect to space and time."
[According to this, it is not purposeless to enter on these grotesque trains of thought, even if they are untenable physically. For since the new physics teaches us to avoid these false tracks, it seems of value to know what it is that is to be avoided. Just as we must study scholastic thought if we wish to grasp thoroughly the philosophy which sprang up after the scholastic fetters were burst. Moreover, these reflections on concealed universes are not without a certain attraction, reminiscent of the sorcerer's wand, if they pursued any other goal than that of making universes distorted. It is true that they hold out latent temptations that may in some cases lead us on to dangerous ground, in encouraging us to venture on analogies beyond the scope of geometry and physics. Would it be possible to enter suddenly into a world that is distorted and deformed with respect to its ethics, its culture, and its reasoning intellects, without our observing the difference? Are we ourselves perhaps living under such deranged conditions, of which we cannot become aware, because our perceptual organs have likewise become deformed? I must frankly confess that I do not regard it as quite inconceivable that this argument of deformation may be spun out in this direction, but I must add that Einstein rejects absolutely all such extensions, since, as he emphasizes, they lead to regions that are merely fields for the exhibition of "verbal gymnastics."]
The question whether Nature makes leaps or not is very old. In the theory of descent it forms the foundation of the difference between revolutionists and the evolutionists, who uphold the axiom natura non facit saltus, with all its consequences. Recently attempts have been made, particularly by psychologists, to propound and justify a natural principle of discontinuity. They assert that our own perceptions and sensations are discontinuous in themselves, and that the mechanism of every perception is akin to that of a cinematograph with its extremely rapid interruptions. If this should actually be the case, we should scarcely have a means of solving definitely the question whether continuity reigns, or not, in Nature.
Einstein does not recognize the possibility of this alternative for a moment. If a doubt had ever arisen, the researches of Maxwell would in themselves have been sufficient to dispel it. Our universe that is to be described in terms of differential equations is absolutely continuous.
"But," I interjected, "does not modern physics offer a certain support to the assumption of a discontinuity? Does not the Quantum Theory point to an atomistic structure of energy, and hence also of events that are to be imagined as happening in jerks and as involving relations expressible in whole numbers?"
Einstein gave an answer of epigrammatic brevity and flavour. "The fact that these phenomena are expressible in whole numbers must not be construed into an argument against continuous happening. Just imagine to yourself for a moment that beer is sold only in whole litres; would you then infer that beer, as such, is discontinuous?"
What achievements are to be expected of astronomy in the present era?
This question would have a special meaning if it were assumed that the astronomer who works in observatories is surrounded by solved problems, and can no longer hope to solve problems having the universal significance of those of Copernicus or Kepler. This assumption, however, would not be in agreement with the actual state of affairs.
Einstein indicated to me a number of fundamental problems that present themselves to modern astronomy, and the solution of which he expected of future times.
Above all, the geometrical and physical constitution of the stellar systems will, in the main, become revealed.
At present we do not yet know whether Newton's Law of Attraction holds, at least approximately, for configurations of the type of the Milky Way and of the spherical clusters of stars—that is, in extents of space in which the influence of space-curvature would become appreciable. The rapid progress of recent astronomy justifies our great hopes that the solution of this universal problem will be found within the coming decades.
In distant connexion with this we also touched on the question of the habitability of other worlds. This theme of Fontenelle, "la pluralité des mondes habités," which has again become a centre of public interest, owing to investigations of Mars, has evoked a storm of discussion. We hear the noisy war-cries of geocentric scientists who wish to regain for the earth her shattered supremacy in astronomy, and who claim the existence of organic forms as the sole prerogative of our planet. It is scarcely necessary to mention that Einstein rejects the motives of these human and all-too-human individuals as small-minded and short-sighted. Creatures in distant worlds are derived from, and are subject to, conditions of organic nature, of which we can form no idea by deductions from the world which we inhabit. But to deny their existence on numberless constellations, or to demand an ocular proof of their presence, is no better than to assume the point of view of an infusoria to whom there is no life other than that in a dirty drop of ditch-water.
The idea of the atom as the ultimate structural element involves a philological as well as a conceptual contradiction. For atomos signifies the indivisible, the no-further-divisible, whereas the idea of a body, however small, an element of structure differing from zero, demands, at least geometrically, further divisibility. Even the original founders of the theory of atoms, Leukippus, Epicurus, and Democritus, assigned definite forms to the ultimate components, and we may read in the splendid work of Lucretius how he infers from the nature of substance that the ultimate particles are smooth, round, or rough, or have the shapes of hooks and eyes. The further analysis pressed forward, the more the simplicity of the original idea vanished. Microcosms came to be regarded as copies of macrocosms, and the atoms of present-day science actually exact from us that we should regard them as worlds in themselves.
Einstein acceded to my request that he might give a sketch of the latest achievements of science sufficient to provide an approximate idea of the atomic model. According to the researches of Rutherford and Niels Bohr, we are to picture it as a planetary system.
The central body of this system is represented by a positively charged nucleus, which constitutes almost the whole mass of the atom, surrounded by a certain number of electrons, negative charges, that move in uniform circular or elliptic orbits about the nucleus. There is thus a certain analogy that allows us to regard the nucleus as the sun, and the electrons as the planets of this system.
The number of these electrons varies between the limits 1 and 92, according to the chemical constitution of the element. The smallest number occurs in the case of helium (in which there are two), and of the hydrogen atom, in which only one electron-planet describes its circular path about the nucleus. In other atoms there are probably more complicated orbits, although they are more or less approximately circular. According to this still very new theory, which is supported by very convincing facts, the electrons are to be imagined as arranged in concentric shells (like the layers of an onion), among which the innermost shell plays a distinctive part inasmuch as the number of the electrons arranged in it decides the chemical character of the atom in question. It sometimes occurs that electrons spring, under external influence, from one orbit to another; when the electron jumps back to the original orbit, light is emitted. An essential fact is to be noted: Whereas any arbitrary orbits of any arbitrary radius may occur in a planetary system of the celestial regions, the manifold of these orbits in the case of the electrons is restricted, in that only certain orbits are possible, namely, those that are determined mathematically by the quantum condition.
"Perhaps," I interrupted, "the whole analogy may be inverted. If the atom is considered analogous to a planetary system in the model, it should be admissible to regard our true planetary system as a cosmic atom. And then, long after we have become accustomed to regard our earth as playing the part of a grain of sand, the sovereignty of the sun, too, would be past. The whole majesty of the solar system as far as the orbit of Neptune would then shrink to a configuration compared with which the world of a grain of sand would be infinitely complex."
"This fantastic inversion is permissible up to a certain extent," said Einstein, "but we must not lose sight of the fact that there is a cardinal difference. If we disregard the enormous disparity in dimensions, the analogy is far from exact owing to the circumstance that the atom is only an element of structure, whereas the true planetary system is an extraordinarily complex structure in itself. Thus the difference between a simple thing and one that is very highly complex still remains."
"But, Professor, may not a similar complexity yet be discovered in the atom? It may be merely a difference of philosophical view from the primary idea to that of regarding the electrons as circulating like planets. May we not conjecture that in each successive step we are merely carrying out a true regressus in infinitum?"
"That seems highly improbable," he replied, "although, of course, structural investigations can never cease. At first they are directed at the more remote object of finding out why certain atoms are radioactive, that is, exhibit a tendency to disintegrate. It has already been established that this tendency is a property of the positive nucleus, of which little is as yet known. This means that the nucleus is not simple, yet it does not open up the possibility of an unending regression. Our aim must be to get a clear insight into the constitution of the nucleus, as regards the positive and negative charges, and it is my opinion," he concluded, "that beyond this there will be no further subdivision of matter."
When Goethe writes of the immovable pole in the flux of phenomena, we recognize that his beautiful remark pronounces an elegy to the possibility of attaining ultimate simplicity. Einstein's utterance, if I understand him aright, converts this elegy into a song of hope. If the subdivision of matter actually has an end somewhere, then we are now on the threshold of ultimate things, we are near the immovable pole, which we are capable of reaching.