CHAPTER VII

PROBLEMS

Questions of the Future.--Problem of Three Bodies.--Conception of Approximation.--Object of Mechanics.--Simplicity of Description.--Limits of Proof.--Reflections about the Circle.--From the History of Errors.--Causalities.--Relativity on a Physiological Basis.--Physicists as Philosophers.

WE spoke of the objects and problems of science in general, and touched on certain recurrent questions with which reputed men of science are confronted from time to time, so that we may ascertain their opinions about immediate as well as more remote aims, and about worthy objects and those within reach.

"Such stimuli," said Einstein, "may be quite interesting inasmuch as they sharpen the appetite of the public for the works of investigators, and give the latter the opportunity of making wider circles acquainted with their plans. Yet the value of their suggestions must not be overrated, when they are directed at giving trustworthy information about the future lines of development of science. Every scientist, in working out his own research, gravitates to particular points on the boundary which separates the known from the unknown, and becomes inclined to take his particular perspective from these points. It must not, however, be expected that these individual aspects will form a complete picture, and will indicate the only paths along which science can or will advance."

"May I suggest, Professor," I answered, "that we select certain answers that have been given to these recurrent questions for discussion? I have brought along a whole series of them; it would be of value to know what attitude you take up towards some of the statements that have been made about future possibilities."

Einstein acquiesced, and so I read out a number of expressions of opinion, given by eminent authorities, particularly in natural science and mathematics. They came under the heading, "The Future Revolution of Science." At the outset we encountered arguments by Bailhaud, the director of the Paris Observatory; he dealt with the so-called "Problem of Three Bodies," and with "The Finitude or Infinitude of the Universe."

Einstein elucidated these questions as follows. The celebrated Problem of Three Bodies is a special case of the general problem of Many Bodies, the object of which is to discover the exact paths of the heavenly bodies. If we suppose that the planets and the comets are subject only to the attraction of the central body, the sun, then their paths would be exactly those given by Kepler's Laws--that is, they would move about the central body, or, more precisely, about the common centre of gravity in perfectly elliptical orbits. The same result would happen if we regard the orbit of a moon to depend solely on its parent planet. But this assumption is not in agreement with reality, since all the bodies of our system are also subject to their mutual attraction depending on their masses and distances. Consequently we have the so-called disturbances, perturbations, and divergences from the ideal paths; and the problem of ascertaining these disturbances is essentially identical with the Problem of Three Bodies. Regarded from the point of view of pure mechanics, this problem may be considered solved in so far as we are able to write down the equations of motion. But, in addition to this purely mechanical process, there is a mathematical problem which has not been completely solved--that is to say, the integral expressions that occur in it can be calculated only approximately. This makes no difference to the practical calculation, since the degree of approximation, according to the present methods, may be carried as far as we wish. The error may be reduced to any desirable extent, so that it is probably wrong to expect new revelations on this point from future upheavals in physics. We read on and discovered that several of the scientists mentioned did not stop at expecting all advances of the future from pure theory. They had visions of an optimum of happiness, to gain which the increase of knowledge alone did not suffice. Thus the celebrated Swedish astrophysicist Svante Arrhenius had summarized his judgment in a few lines: "After the stupendous progress that has been made in the physical and chemical sciences in recent times, it seems to me that the moment has come for attacking the most important problems of mankind with full success, namely, those of biology, and in particular of the art of healing, with the weapons that are furnished by the arsenal of the exact sciences." And the mathematician, Emile Picard, Membre de l'Académie, expressed himself in still more hopeful terms: "There is no doubt but that the discoveries which the human race is awaiting with impatience are those that are seeking to eliminate sickness and the decrepitude of old age. Injections giving immunity against all diseases, an elixir of life (_une eau de Jouvence_) for persons of advancing age--these are the discoveries that are longed for by every one. There are also sciences that are to be termed 'moral,' from which we are impatiently expecting that guidance which will diminish the hate which seems to be increasing from day to day among the nations. That would be a splendid discovery."

"These are, indeed, noble and inspiring words," said I. "It shows how deeply rooted is the demand for ethical values in human nature, when even a mathematician, whose intellectual interests are directed primarily towards exact results, ranks the discoveries of ethics above all others."

Einstein answered: "We must carefully distinguish between what we wish for in general and what we have to investigate as belonging to the world of knowledge. The question under consideration is not one of wishes and feelings, but was unmistakably aimed at the advances and revolutions in the realm of science. It does not come within the scope of science at all to make moral discoveries! Its one aim is rather the Truth. Ethics is a science about moral values, but not a science to discover moral 'truths.' Ethics, conceived as a science in the usual way, can therefore serve to discover or to promote truth only indirectly. To illustrate my point of view I shall quote an example taken from a totally different field; it is merely to serve as an analogy. Let us consider the game of chess. Its value and its meaning is not to be sought in scientific factors, but in something entirely different, in a struggle which takes place according to definite rules. But even chess, inasmuch as it sharpens the intellect, may exhibit an indirect value for promoting truth. It may, for instance, suggest examples in permutations, which may contain mathematical, that is, purely scientific, truths. I certainly do not deny that there is an ethical factor in all genuine sciences. For being occupied with things for the sake of truth alone emancipates and ennobles the mind."

"This ennobling effect," I interposed, "should surely show itself in a moderation of the passions which were mentioned in the above expression of opinion. With Picard we should expect above all things to see a diminution in the feelings of hate between peoples, the tragic consequences of which we have experienced."

Einstein smiled, and, with a touch of sarcasm, said, "Hate is presumably a privilege of the 'cultured,' who have the time and the energy for it, and who are not the slaves of care." His tone indicated clearly that he used the generic term "cultured" to denote the Philistines of culture, its snobbish satellites, but not those whose intensive work aimed at increasing and deepening the fields of culture. In general he maintained his view that it is an illusion to expect "discoveries" in the realm of ethics, since every real discovery belonged alone to the sphere of truth in which the division only into right and wrong, not that into good and evil, holds good.

This led us to the old question of Pilate: What is Truth? In seeking an answer to this question Einstein first called special attention to the conception of "approximation," which plays a great part in the actual search for truth, inasmuch as every physical truth, expressed in measures and numbers, always leaves some remainder, that marks its distance from the unattainable truth of reality. This conception, which manifests itself so prominently in the relation of Einstein's own researches to the older, so-called classical, mechanics, will be developed here according to his line of thought as far as I can recollect from a number of conversations.

Let us suppose that we overhear two people arguing about the shape of the earth's surface. The one affirms that it is an unlimited plane, whilst the other maintains that it is a sphere. We should not hesitate a moment to say that the first is in error, and that the second gives the true answer. As long as the question was to be decided in favour of a "Plane or a Sphere," the sphere would represent the absolute truth. Yet it would be only relative, for these two statements are contradictory only between themselves, but will no longer be so if a third assertion is made which opposes a new alternative to "sphere."

If this alternative objection is actually raised, the third person would be quite justified in saying that the "sphere" explanation is wrong. For the conception "sphere" requires that all diameters be equal, whereas we know that they are not so, since the distance from pole to pole has been proved to be smaller than that between opposite points on the equator. The earth is an ellipsoid of rotation, and this truth is absolute in the face of the errors which are expressed by the terms, plane and sphere.

It would again have to be added that this absoluteness would stand only as long as this contradiction is regarded as being one between a definite sphere and a definite ellipsoid. If, as in the case of the earth, there are quite different diameters in the equatorial and the diametral planes, then there is complete contradiction between the two statements, and as the supporter of the ellipsoid is right, the one who supported the sphere must now give in, although he previously triumphed over his first opponent. His statement was true compared with the latter, but showed itself to be an error when compared with the statement of the third person.

This does not run counter to the laws of elementary logic. One of these, somewhat inadequately called the Law of Contradiction, states that two directly contrary statements--_e.g._ this figure is a circle, and this figure is not a circle--cannot both be true simultaneously. The truth of the one implies necessarily the falseness of the other. As this cannot be disputed, it follows in our case that we cannot have been confronted with contradictory judgments at all concerning the figure of the earth.

This is to be understood in a geometrical sense. The sphere does not entirely contradict the ellipsoid, since it is a limiting case of the latter: and the plane is likewise a limiting case of the sphere, as well as of the surface of ellipsoids.

But we are not concerned with purely geometrical considerations, for the earth is a definite body, and not a limiting configuration derived from abstraction. We are here dealing with measurable quantities, whose difference can be proved, and hence we must have one of the disputants proclaiming the absolute truth, whilst the other proclaims an absolute error. This, however, again is incompatible with our result that the second person is right in the one case and wrong in the other.

The logical Law of Contradiction overcomes the dilemma in the simplest way. None of these assertions contains the truth, hence none of these judgments allows the falseness of the others to be deduced. Only this may be said, that there is a fraction of truth in each judgment. The true shape of the earth is given by the plane to a first, the sphere to a second, the ellipsoid of rotation to a third, degree of approximation: we reserve the right of further approximations, each of which in succession approaches a higher degree of correctness, but none attains the absolute truth.

This reflection on a particular case may be generalized, and remains when we extend it to our attempts at grasping the states, changes, and occurrences of Nature. Whenever we talk of physical laws, we must bear in mind that we are dealing with human processes of thought, that are subjected to a succession of judgments, courts of appeal, as it were, excluding, however, a final court beyond which no appeal is possible. Each new experience in the course of natural phenomena may render necessary a new trial before a higher court, whose duty is then to give a more definite or different form to the law formulated by us, so as to attain a still higher degree of approximation to the truth.

If we call to mind some of the most valuable statements made by modern investigators about the nature of natural laws, we recognize that they are all connected by a single thread of thought, namely, that even in the most certain law there is left a remainder that has not been accounted for, and that obliges us to consider a greater approximation to the truth as possible, even if a final stage is not attainable.

Mechanics furnishes us with the expression of its laws in equations, whose importance Robert Kirchhoff explained in 1874 by a definition that has been considered conclusive by scientists. According to him, it is the object of mechanics to describe completely (and not to explain) in the simplest manner the motions that occur in Nature.

The postulate of simplicity is derived from the fundamental view of science as an economy of thought. It expresses the will of man's mind to arrive at a maximum of result by using a minimum of effort, and to express the greatest sum of experience by using the smallest number of symbols. Let us consider two simple examples quoted by Mach. No human brain is capable of grasping all the possible circumstances of bodies falling freely, and it may well be doubted whether even a supernatural mind like that imagined by Laplace could succeed in doing so. But if we take note of Galilei's Law for Falling Bodies and the value of the acceleration due to gravity, which is quite an easy matter, we are equipped for all cases, and have a compendious formula, accessible to any ordinary mind, that allows us to picture to ourselves all possible motions of falling bodies. In the same way no memory in the world could retain all the different cases of the refraction of light. Instead of trying to do the impossible task of grasping this infinite abundance, we simply take note of the sine law, and the indices of refraction of the two media in question; this enables us to picture any possible case of refraction, or to complete it, since we are free to relieve our memories entirely by having the constants in a book. Thus we have here natural laws that give us a comprehensive yet abbreviated statement of facts, and satisfy the postulate of simplicity to a high degree.

But these facts are built up on experiences, and it is not impossible that some new unexpected experience will reveal a new fact, which is not sufficiently taken into account in the law. This would compel us to correct the expression for the law, and to seek a closer approximation for the enlarged number of facts.

The Law of Inertia, according to our human standard, seems unsurpassable in simplicity and completeness; it seems to us fundamental. But this law, which prescribes uniform rectilinear motion to a body subject to no external forces, selects only one possibility out of an infinite number as being valid for us. It does not seem evident to a child, and it is easy to imagine a good scholar in some branch of knowledge other than physics, to whom it would likewise not seem evident. For it is by no means necessary a priori that a body will move at all when all forces are absent. If the law were self-evident, it would not need to have been discovered by Galilei in 1638. Nevertheless, it appears to us, now, to be absolutely self-evident, and we can scarcely imagine that it can ever be otherwise. This is simply because we are bound to the current set of ideas that cannot extend beyond the sum of sense-data and experiences that have been inculcated into us by heredity and environment. At a very distant date in the future the average mind may surpass that of Galilei to the same extent as Galilei's surpasses that of a child, or of a Papuan native. And of all the infinite possibilities one may occur to a Galilei of the distant future, which, when formulated as a law, may serve to describe motions of a body subject to no forces better than the law of inertia, proposed in 1638.

These reflections are not mere hallucinations, but have to do with scientific occurrences that we have observed in the twentieth century. Newton's equation that gives the Law of Attraction is beyond doubt a model of simplicity, and it would have occurred to no thinking person of even the last generation to doubt its accuracy. The easily grasped expression k (m.m^1⁄r^2) apparently expresses truth in a law which is valid for all eternity. In this expression, he denotes a gravitational constant, that is, a quantity which is invariable in the whole universe; _m_ and _m_^1 are two masses that act attractively on one another; and _r_ is the distance between them. But Newton has been followed by Einstein, who has proved that this expression represents only an approximate value, that leaves a small remainder as an error that may be detected if the greatest refinement be made in our methods of observation. The equations that have been set up by Einstein represent the approximation that is to be considered final for the present, and that may remain valid for thousands of years. They are certainly very complicated, being included in a system of differential equations of awe-inspiring length, and we may feel tempted to object with the question: how do they agree with Kirchhoff's postulate that the simplest description of the motions must be sought? But this objection falls to the ground if we look carefully into the question. For simplicity consists not merely in being brief or in excluding difficulty from a formula, but rather in asserting the simplest relation to the universe as a whole, which is independent of all systems of reference. When this independence is proved--and in Einstein's case it is so--the complicated aspect of the formula disappears entirely in the light of the higher simplicity and unity of the world-system that presents itself--a world-system that is directed in conformity with the one fundamental law of general relativity as well in the motion of the electrons as in motion of the most distant stars. With regard to the other postulate, that of completeness, _i.e._ absolute accuracy, we have been furnished with proofs that have rightly excited the wonder of the present generation. But are we then to recognize the Principle of Approximation in every direction? Is there then nothing that can be proved rigorously, nothing that is unconditionally valid in the form of knowledge that corresponds exactly to truth?

We are led to think of mathematical theorems, which, when they have once been proved, are evident to the same degree as the axioms from which they have been derived, by virtue of logic which cannot be disputed since a contradiction leads to absurdity. It has been said that mathematics _est scientia eorum_, _qui per se clara sunt_, that is, is the science of what is self-evident.

But here again doubts arise. If we should get to know only a single case, in which the self-evident came to grief, the road to further doubts becomes open. Such a case will now be quoted.

As we know, a tangent is a straight line, which makes contact with a curve at two coincident (or infinitely near) points without actually cutting the curve. The simplest case of this is the perpendicular at the extremity of a radius of a circle. And it agrees fully with what our feeling leads us to expect when it is stated that every curved line that is "continuous," that is, which discloses no break and no sudden bend, has a tangent at every point. Analysis, which treats plane curves as equations in two variables, gives the direction of the tangent in terms of the differential coefficient, and declares accordingly that every continuous function has a differential coefficient, that is, may be differentiated, at every point. The one statement amounts to the same as the other, since there must be an equivalent graphical picture corresponding to every functional expression.

But this apparently rudimentary theorem involves an error, which was not discovered before the year 1875. The theory of curves has been in existence for centuries, but it occurred to no one to doubt the general validity of this theorem of tangents. It was regarded as self-evident, as a mathematical intuition. And certainly neither Newton, nor Leibniz, nor Bernoulli, not to mention the mathematicians of olden times, even dreamed that a continuous curve without a tangent, or a continuous function without a differential coefficient, was possible.

Moreover, a proof of the theorem had been accepted. It appeared in text-books, and was often to be heard in lecture rooms; nor was a shadow of a doubt suggested. For it was not merely a _demonstratio ad oculos_, but it appeared directly to our sense of intuition. And we may safely say that up to the present day no one has ever been able to _imagine_ a continuously curved line which has no tangent; no one has been able to picture even one point of such a curve at which no tangent could be drawn.

Nevertheless, scientists appeared who began to entertain doubts. In the case of Riemann and Schwarz these doubts assumed a concrete form, in that they proved that certain functions are refractory at certain points. But Weierstrass was the first to make a real breach in the old belief that was so firmly rooted. He set up a function that is continuous at every point, but differentiable at no point. The graphical picture would thus have to be a continuous curve having no tangent at all.

What is the appearance of such a configuration? We do not know, nor shall we presumably ever get to know. During a conversation in which this problem of Weierstrass arose, Einstein said that such a curve lay beyond the power of imagination. It must be remarked that, although the mathematical expression of the Weierstrass function is not exactly simple, it is not inordinately complex. Moreover, seeing that one such function (or curve) exists, others will soon be added to it (Poincaré mentions that Darboux actually gave other examples even in the same year that the first was discovered); there will, indeed, be found an infinite number of them. We may go still further, and say that, corresponding to each curve that has tangents, there are an infinite number that have no tangents, so that the former form the exception and not the rule. This is an overwhelming confession that shakes the foundations of our mathematical convictions, yet there is no escape.

How may we apply the principle of "approximation" to these considerations? May we say that the theorem that was believed earlier is an approximation to a mathematical truth?

This is possible only conditionally, in a certain extremely limited sense, namely, if we picture to ourselves that point in the development of science at which the conception and properties of tangents first began to be investigated. Compared with this stage of science, the above theorem denotes a first approximation to the truth, in spite of its incorrectness; for it makes us acquainted with a great abundance of curves that are very important for us and that exhibit tangents at every point. This knowledge brings us a step nearer to the more approximate truth given by Weierstrass's example. In the distant future, the earnest student will learn this theorem only as a curious anecdote, just as we hear of certain astrological and alchemistic fallacies. He will learn, in addition, other theorems that are looked on as proved by us of the present day, although actually they were proved only approximately. For what does it mean when Gauss, for example, repudiated certain proofs of earlier algebraists as being "not sufficiently rigorous," and replaced them by more rigorous proofs? It signifies no more than that, in mathematics, too, what appears to one investigator as flawless, strict, and evident, is found by another to have gaps and weaknesses. Absolute correctness belongs only to identities, tautologies, that are absolutely true in themselves, but cannot bear fruit. Thus at the foundation of every theorem and of every proof there is an incommensurable element of dogma, and in all of them taken together there is the dogma of infallibility that can never be proved nor disproved.

It must appear extremely interesting that, at first sight, this example of the tangent has its equivalent in Nature herself, namely, in molecular motions the investigation of which is again largely due to Einstein.

Jean Perrin, the author of the famous book, _Atoms_, describes, in the introduction, the connexion between this mysterious mathematical fact and results that are visible and may be shown by experiment, to which we have been led by the study of certain milky-looking (colloidal) liquids.

If, for example, we look at one of those white flakes, which we get by mixing soap solution with common salt, we at first see its surface sharply outlined, but the nearer we approach to it, the more indistinct the outline becomes. The eye gradually finds it impossible to draw a tangent to a point of the surface; a straight line which, viewed superficially, seems to run tangentially, is found on closer examination to be oblique or even perpendicular to the surface. No microscope succeeds in dispelling this uncertainty. On the contrary, whenever the magnification is increased, new unevennesses seem to appear, and we never succeed in arriving at a continuous picture. Such a flake furnishes us with a model for the general conception of a function which has no differential coefficient. When, with the help of the microscope, we observe the so-called Brownian movement, which is molecular by nature, we have a parallel to the curve which has no tangent, and the observer is left only with the idea of a function devoid of a differential coefficient.... We find ourselves obliged, ultimately, to give up the hope of discovering homogeneity at all in studying matter. The farther we penetrate into its secrets, the more we see that it, matter, is spongy by nature and infinitely complex; all indications tend to show that closer examination will reveal only more discontinuities.

I have not yet had an opportunity of seeing these Brownian movements under the microscope, but I must mention that Einstein has repeatedly spoken to me of them with great enthusiasm, of an objective kind, as it were, for he betrayed neither by word nor by look that he himself has done research leading to definite laws that have a recognized place in the history of molecular theory.

As soon as we approach the question of molecular irregularities we recognize that, when we earlier spoke of the figure of the earth in discussing the principle of "approximation," we were still very far from the limit that may be imagined. We had set up the three stages: plane--sphere--ellipsoid of revolution, as relative geometrical steps, beyond which there must be still further geometrical approximations. If we imagine all differences of level due to mountains and valleys to be eliminated, for example, and if we suppose the earth's surface to consist entirely of liquid, undisturbed by the slightest breath of wind, even then, the ellipsoid is by no means the final description. For now the discontinuities from molecule to molecule begin, the infinite number of configurations without tangents, the macroscopic parallels of what the white flake soap solution showed as microscopically, and no conceivable geometry would ever be adequate to grasp these phenomena. We arrive at a never-to-be-completed list of functions which can never be described either in words or in symbolic expressions of analysis.

But even if the ultimate geometrical truth is hidden behind the veils of Maya,[6] we are yet left with the consolation that the method of approximation, even when applied to a relatively modest degree, produces remarkable results in the realm of numbers. Let us consider for a moment in the simple figure of a circle the ratio between the circumference and the radius.

[Footnote 6: Maya = appearance.]

As we know, this ratio is constant, and is called in honour of the man who first gave a trustworthy value for it, Ludolf's number, namely, π (pi). Thus it makes no difference whether we consider a circle as small as a wedding-ring, or as large as a circus arena, or even one the radius of which is as great as the distance of Sirius. And it makes just as little difference what happens to the circle whilst it is being measured; the above ratio must remain constant.

But here, too, a contradiction makes itself heard, issuing from one section of modern science. It calls to mind the saying of Dove that when professors are not quite sure about a thing they always preface their remarks with the phrase: "it is well known that" ... We should be well advised in avoiding this method of expression altogether, for even when we feel quite sure, the ghost of the unknown lurks behind what we fain would call well known.

The theorem that all circles without exception are subject to the same measure-relation belongs _a priori_ to the synthetic judgments. But fields of thought have been discovered in which the _a priori_ has lost its power. Mathematics--once a quintessence of synthetic statements _a priori_--is now regarded as being dependent on physical conditions. Physical conditions, however, are empirical and subject to change. Therefore, since the _a priori_ is not subject to change, we encounter a discrepancy. It leads to the question: Is the Euclidean geometry with which we are familiar the only possible geometry? Or, in particular: Is π the only possible measure-relation?

Einstein replies in the negative. He not only shows how another geometry is possible, but he also discloses what once seemed inconceivable, namely, that if we wish to describe the course of the phenomena of Nature exactly by means of the simplest laws, it is not only impossible to do so with the help of Euclidean geometry alone, but that we have to use a different geometry at every point of the world, dependent on the physical condition at that point.

From the comparatively simple example of two systems rotating relatively to one another, Einstein shows that the peripheral measurement of a rotating circle, as viewed from the other system, exhibits a peculiarity which does not accompany the radial measurement. For, according to the theory of relativity, the length of a measuring rod is to be regarded as being dependent on its orientation. In the case quoted, the rod undergoes a relative contraction only when applied along the circumference, so that we count more steps than when we measure the circumference of the same circle at rest, that is, in non-rotation. Since the radius remains constant in each case, we get a relatively greater value for π, which shows that we are no longer using Euclidean geometry.

Yet, formerly, before such considerations could even be conceived in dreams, this π was regarded as absolutely established and immutable; and observers used every possible means of determining its value as accurately as possible.

In Byzantium there lived during the eleventh and twelfth centuries a learned scholar, Michael Psellus, whose fame as the "Foremost of Philosophers" stretched far and wide, and whose mathematical researches were regarded as worthy of great admiration. This grand master had discovered by analytical and synthetical means that a circle is to be regarded as the geometric mean between the circumscribed and the inscribed square, which gives to the above quantity, as may easily be calculated, the value √8, that is, 2.8284271.... In other words, the length of the circumference is not even three times that of the radius.

We have the choice of regarding the result of Psellus as an approximation, or as mere nonsense. Every schoolboy who, in a spirit of fun, measures a circular object, say a top, with a piece of string, arrives at a better result, but the contemporaries of Psellus accepted this entirely wrong figure with credulous reverence, and continued to burn incense at the feet of the famous master. It is all very well for us of the present to call him a donkey. We have just as much right in saying that mathematicians differ, not in their natures, but only in the order of their brain functions. If a man like Psellus missed the mark by so much, it is possible that men like Fermat or Lagrange may also have erred occasionally or even consistently.

No heavenly power will give us a definite assurance to the contrary, and all of us may be just as false in our judgment of accepted celebrities as were the Byzantines eight hundred years ago in their estimate of Psellus.

Whereas the latter had obtained a value "less than 3," there are learned documents of about the same date that have been preserved, according to which the value of π comes out as exactly 4. Compared with this grandiose bungling, even the observations mentioned in the Old Testament are models of refinement. For, as early as three thousand years ago, it is stated of the mighty basin in the temple of Solomon (First Book of Kings,