Let us bear in mind that in aiming at ultimate truths we have no final courts of appeal. Nor are the latter to be assumed even when, in pursuing a theory, we encounter a difficulty, which at first exhibits all the signs of a direct conceptual contradiction. It should rather be realized that a fiction containing an initial but only provisional contradiction serves as a starting-point for just those investigations that are most subtle and that have far-reaching consequences. We should have no Infinitesimal Calculus, no Algebra, no Atomic Theory, no Theory of Gravitation if, to avoid all initial contradictions, we surrender the fiction of differentials, of imaginary quantities, of the atom, of action at a distance. In short, it may, indeed, be said that not only knowledge, but also life, the holding together of people by convention, law, and duty, would become impossible if we did not accept the fiction of free will, which directly contradicts the determinate character of all happening, including actions and motives, which, physically, alone seems recognizable.
Fiction (not to be confused with hypothesis) and anthropomorphism, in spite of their inner inconsistency, are the two poles about which our thoughts and our lives revolve. And no doctrine will ever soar to such heights that it will be able to deny completely its origin from these roots of all thought. The Archimedean thought-centre of the universe, which would enable us to lift the world out of its hinges, is unattainable, because it does not exist at all.
Is this also to apply to the new physics, whose results are to be regarded as the last word in scientific knowledge? Many a hypercritical thinker might be led away by the current of the preceding statement, and feel disposed to answer in the affirmative, were it not that, here too, a contradiction intrudes itself. This is expressed in the fact that not one of the present-day philosophers is in a position to pursue the threads of this theoretical fabric to their hidden ends.
Thus we arrive at a parting of the ways. Whoever aims at becoming thoroughly familiar with Einstein's new world-system finds that the study of the theory claims so much attention that there is scarcely a possibility left of proceeding to an ultimate philosophical analysis. And whoever is absorbed only by the desire of making philosophic investigations soon enough arrives at border-lines of thought, at which his conscience warns him to beware of insufficient scientific knowledge. He will be attacked by doubts as to whether he has properly understood the theory. And he will be confronted with the question whether he is justified in drawing ultimate philosophical conclusions before he has mastered all the mathematical details.
As far as can be judged at present, only one thinker has, so far, had sufficiently wide knowledge to enable him to correlate the physical theory methodically with the theory of knowledge. I mean Professor Moritz Schlick of Rostock, who has set out his ideas systematically in his book Erkenntnislehre, which is extraordinary in itself and in its great scope; it takes us beyond Kant. In Schlick's opinion Einstein's theory furnishes us with the key to new and unexpected chambers of thought; it is a wonderful instrument for opening up new avenues, and would appear more wonderful still if we could use this instrument without having recourse to anthropomorphism. This limitation may lead to a Utopia, or may entail a circulus vitiosus. But we have one philosophy nowadays which applies to what cannot be fulfilled "AS IF" it really is capable of fulfilment. Among the disciples of Vaihinger, the founder of the As-If-doctrine of thought, we, however, notice the tendency to follow anthropomorphic and fictitious paths also in his field of thought.
From numerous utterances of Einstein, I have gathered that he himself does hot give his unqualified approval to all attempts at unravelling the ultimate problems by means of philosophy, that is, by using metaphysics alone. He does not deprecate these endeavours, but even expresses admiration for some of the newer works, as for that of Schlick, yet he sees certain obstacles in the purely philosophical methods, that at least restrain him from taking a systematic interest in them. This reluctant acceptance of, and doubt in, the processes of philosophy, that has never forsaken the exact investigator, this suspicious attitude which scents traces of sophistic and scholastic machinations in all metaphysical arguments, also asserts itself in him in a noticeable form. He feels the absence of rigour and of consistency of direction, which is a guarantee of progress in passing from one result to another, in the method of thought of those who are pure philosophers: and he deplores the spongy and murky appearance of certain expressions of thought, which, it must be admitted, form a poor contrast to the completeness and the crystal clearness of mathematico-physical reasoning. There was an inscription on the portals of the Athenian Academy which stated that entrance was forbidden to all who had had no mathematical training; we may imagine next to it an academy of pure transcendental philosophy, bearing the inscription: No exact research allowed! I believe that this clear-cut distinction would tally with Einstein's view.
In the case of the great Ernst Mach, for whom Einstein has intense admiration, we observe a similar attitude, or we may say that, in the language of allegory, he sang openly the same refrain in another key. He never ceased reiterating that he was properly "no philosopher at all, but only an investigator of Nature." At the beginning of the introduction to one of his works we read his confession: "Without in the slightest degree being a philosopher, or even wishing to be one ..."; and some lines further on he calls himself sarcastically "a mere amateur sportsman" in philosophical regions. Yet, Mach's initial remark is followed by a remarkable result, for the book in question, Knowledge and Error (Erkenntnis und Irrtum), is to be reckoned among the most important works in philosophical literature; and he himself, the amateur sportsman, who did not even desire to be called a philosopher, accepted in 1895 the post of Professor of Philosophy at Vienna University. It was merely his timidity in the face of the philosophical fraternity that had made him emphasize repeatedly the distinction between his own work and that of the philosophers, whereas in his heart he had nourished a passion for Philosophy, the first mother of Science. And in my opinion such a moment may arrive for even the most rigorous investigator when he succumbs to the siren strains from the shores of philosophy.
As far as Einstein himself is concerned, I cannot venture on a prognostication. Even though he belongs to the category and rank of Descartes, Pascal, d'Alembert, and Leibniz, in whom Mathematics and speculative Philosophy are intermingled, he is yet characterized by such a pronounced individuality, that it is quite inadmissible to draw conclusions about him from others. He has no need to experience a day of Damascus, for he carries the gospel of salvation in himself, and it radiates from him. One thing seems possible, in my opinion, namely, that Einstein will occasionally roam into the neighbouring realm merely from æsthetic motives. Although the means of philosophy are nebulous and more indefinite than those of exact science, which are almost glaringly distinct, philosophy itself for this reason is the more closely related to Art. And a theory that applies to the whole universe must assuredly contain many germs that may come to life if subjected to the methods of Art. The connecting link between Kant and Schiller shows in what sense this is to be understood. Even at present there are indications in Art which tend to show that it is ready to establish points of contact with Knowledge. In France symphonic poems were written on the measure relations of the circle, and on logarithms: these are at present only curiosities, but may in future become models. At a much later date, perhaps, the four-dimensional universe may become ripe for treatment by such methods of Art. On the way to this goal there is the treatment with the symbolic, non-rigorous, and semi-poetic means of expression used by Philosophy. Many will use their efforts to achieve this, and perhaps they will come within closer range of success, if Einstein himself lends a helping hand. It will not be possible to arrive at new physical truths by following this path, but those that are actually known will be traced more readily to the great mainstream of philosophy. To fathom the secrets of the world is the work of a recluse, but to make it comprehensible to a wide circle, a preacher is necessary, who uses the beautiful methods of philosophical rhetoric. Cosmos denotes the World and its Ornamentation; its creator, Demiurge, is a master who fashions his forms along the lines of Art.
Thus we have learned what Einstein regards as the sole purpose of Science, namely, the search after Truth. For him, the latter is something absolute in itself, and the possibility of getting nearer to it is as great as the impossibility of deriving results of scientific use from, say, ethical discoveries. For ethics is a field which is haunted by the conceptual ghosts, and the manner of treatment, ordine geometrico, that Spinoza wished to apply to it, is reserved for physics. Einstein leaves the inverse philosophical query: "Is not Truth in itself only something that we have constructed in imagination?" to those who find pleasure in sauntering along paths of thought that are totally unconnected, whereas he himself advances in a straight line with the consciousness that even if the goal is unattainable, he will at least not lose the right direction!
CHAPTER VIII
HIGHWAYS AND BY-WAYS
Practical Aims of Science.—Pure Search for Truth.—Retrospective Considerations.—The Practical Side of Kepler.—A Saying of Kant.—Mathematics as a Criterion of Truth.—Deductive and Inductive Methods.—Conceptual and Perceptual Knowledge.—Happiness and the Pleasures of Theory.—Achievements of Science and Works of Art.—Ethical Results.—Minor Questions.
AGAIN we chanced to refer to the great subject: Can or should theoretical science also pursue practical aims?
It is impossible to overrate the importance of this question. It haunts us daily and often enough looms up threateningly on the horizon of mankind. Observe what form the discussions of educated people take when the finest and most sublime achievements of mind are being debated: one talks of the wonders of research in the remotest corners of astronomy where the structures of world-wide star-systems are being investigated; we hear observations about the theories that aim at tracing the cosmogonic development of universes from the original chaos of countless ages ago. We hear mention of exalted sciences, the Theory of Functions and Numbers, whose founders and representatives are just as remarkable in propounding problems as in solving them, and inevitably the following question obtrudes itself: Of what use is it, ultimately? What can one do with it? Can it be admitted that theoretical science has an object of its own, or have we at least the right to maintain the hope that, sooner or later, it will bring us a real "Utility" expressible in practical terms?
And just as the devotees of pure art have framed the expression, "L'art pour l'art," so Einstein proclaims that science is its own object, "Science for its own sake!" It carries its aims absolutely in itself and must not, through aiming at other purposes, stray from its own highways. "It is my inner conviction," said he, "that the development of science itself seeks in the main to satisfy the longing for pure knowledge, which, psychologically, asserts itself as religious feeling."
"To yourself, Professor, the practical aspect seems comparatively insignificant?"
"I did not say that, and it was not implied in the question. We must not lose sight of our premises. As long as I am interested in working along lines of research—this was the assumption—the practical aspect, that is, every practical result that is found simultaneously or arises out of it later, is a matter of complete indifference to me."
Far be it from me, even in thought, to wish to question this confession of faith, particularly as the fact that it comes from a searcher of the truth gives it the more weight. Yet a certain uneasiness has crept over me because voices have recently made themselves heard that demand for science a totally different tendency. They arise not only from the public at large, but also from academic circles. Just a short time ago I read an exposition by a well-known scientist, W. Wien, in which he indulged in a violent polemic against the view that purely scientific objects are alone valid. Professor Wien addressed himself particularly to German physicists, reproaching them with underestimating technical science, and with regarding it as a "lowering of status" when a physicist enters into practical life.
To this Einstein remarked: "I do not know at whom this reproach is aimed, but I venture to think that my own attitude can never have given rise to an attack of this kind. For I make no divisions of rank, and recognize no higher and no lower status. I affirm only what is the nature of science herself, and the objects according to which she, objectively, has to direct her gaze. Whatever further orientation individual investigators may seek for themselves depends on the determining conditions of life of each, although these conditions do not serve as a means for deducing the main lines of research. The accusation that I am unwarranted in putting forward this view will, I hope, not be levelled at me, for my connexions with practice are manifold enough, and up to the present moment I have often collaborated with practical physicists...."
"As I have regretfully observed when you were obliged to interrupt a conversation with me to give an audience to impatient persons seeking advice in technical matters!"
"My own associations with the world of practice are not, indeed, of recent date. My own parents originally wanted me to become a technical scientist, and I was expected to choose this profession to earn my livelihood. I was not, however, sympathetically inclined to it, for even at an early age these practical aims were to me, on the whole, indifferent and depressing. My idea of human culture did not coincide with the current view, that cultural development is to be measured in terms of technical progress. Doubts, indeed, arose in me as to whether technical improvements and advances would actually contribute to the well-being of mankind. I must add that, later, when I came into actual touch with technical science, my opinion became somewhat modified, for the reason that, here too, pleasures of theory often visited me."
The true position is probably that the technical worker who does not merely think out improvements for machines, but occupies himself with inventions on a higher plane, never ceases to feel himself a theorist, since his achievements are dependent for their inspiration on the fruits of theory. The practical results of to-day are rooted in the theoretical results of decades ago, and what is nowadays regarded as an idea of pure research may in later decades acquire practical value. Whether it actually becomes of value, or not, is of little account in judging the idea. At any rate experience has shown that the beginning of theoretical investigations hardly ever gives us the chance of making prognostications. We spoke of the discoveries of Volta, Ampère, and Faraday. When these were first known, the world might have asked: Why have they been disclosed? To what can they be applied? Of what use are they? Nowadays we know the answers that still lay hidden at that time, and we proudly point to modern dynamos. But does a dynamo really represent the significance of these discoveries? Would the importance and rank of Volta, Ampère, and Faraday be less if the dynamo had not come into existence? Only an out-and-out materialist would affirm this, and, strictly speaking, the question should not even be raised. For it is in a sense equivalent to wishing to judge of the importance and significance of the Polar Star from its usefulness to the navigator on the earth's surface in finding his bearings. We may put the question (although only in the spirit of psychological curiosity, and without expecting much elucidation): Would these discoverers have been particularly happy if they had divined the far-reaching consequences of their work? Did they, indeed, in the course of their abstract researches, have a pre-vision of the future dominated by the dynamo? Einstein refused to answer this in the decisive negative. He left room, if ever so little, for doubts—that is, he considered that, in all probability, these three discoverers had no presentiment of these consequences, and even if they had in a dream caught a glimpse of our present electrical age, their zest for discovery, their "pleasure in theory," could scarcely have been increased; for they were discoverers by nature, who, swept along by their own spirits, did not need to wait to satisfy the desires of practical application.
In Einstein's opinion, the presentiment that a discovery may have practical applications in the future may react on pure research. He quoted bacteriology as a proof of this. In the series of eminent bacteriologists, ranging from Spallanzani to Schwann and Pasteur, there were certainly some whose desire for knowledge was directed primarily towards discovering purely scientific relationships. Pasteur himself started from the theoretical question of the creation of life, that is, from the problem of the origin of organic creatures from inorganic matter without the medium of parent organisms. As a pan-spermist he took up a negative attitude, that is, he tried to prove that it is impossible to discover a bridge between organic and inorganic matter. Yet he doubtless knew that his theoretical efforts stretched out into practical regions, and he may easily have foreseen that they would exert a very important influence on Medicine and Hygiene, although he could not measure its full extent. In this case, then, we cannot fail to recognize that a certain connexion between the desire for pure knowledge and the impulse to apply it practically is possible, serviceable, and justified in itself.
An influence in the opposite direction is also possible, and when, during the course of our conversation, we went in search of examples, we came across one of great interest. It shows us that a question may arise out of ordinary practice that may open up an immense field of pure knowledge, nay, it may lead to a science of very wide scope. As this example is not well known, I shall mention it here; I do so with additional pleasure as the scientist involved is one of those whom Einstein quotes most frequently and for whom he has the greatest admiration, namely, Johannes Kepler. First we have the surprising fact that Kepler, who, even when at the height of his fame, was not free from care, was once the possessor of some money. In the year 1615, his blessed year of fortune, the great astronomer owned a comfortable home in Linz, and even dared to conceive the idea of placing some well-filled casks in his cellar; nay, more, he was in a position to publish a new scientific work at his own expense, and thus appear as his own publisher.
This production of Kepler and his casks of wine are directly connected, as we see clearly from the title: Doliometrie, literally, "The Measurement of Casks." But the title of the work gives not the slightest hint of its importance. For these investigations relating to wine-casks actually became the foundation of a science of sovereign power, the Infinitesimal Calculus.
What was Kepler's aim? It was something entirely practical, and directed to a definite purpose, quite independent of "pleasures of theory," to repeat Einstein's expression. His problem was a question of economy, of using material sparingly and appropriately, in accordance with the requirements of the careful head of a house. How must such a cask be constructed from a minimum of wood to give the greatest cubical content?
His deliberations began by regarding wine as the precious content enclosed by a figure in space, and then conceiving the cask as representing a particular class of "bodies of revolution," that is, of figures in space that may be regarded as produced by the revolution of a curved line about an axis. At this point he at first endeavoured to gain a complete survey of the question. He varied the boards along the sides, the staves, and formed successively ninety-two such bodies of revolution, some of which he named after the fruits which they resembled in shape, as, for example, apple-shaped, lemon-shaped, olive-shaped bodies. He started out by measuring casks, and the final result was that his work, Doliometrie, became the source of all future cubatures or measurements of volume.
Now we come to the deciding point. What conditions has the limiting surface of such a cask-like body of revolution to fulfil, if the body is to have a maximum volume? An epochal discovery here came to light. The practical head of the house soars up into the sublime realms of the theory of magnitudes. Kepler discovered the conception of changes in functions, and their peculiarities at the maximum point. (He did not, of course, use these modern terms.) By this means, long before Newton and Leibniz, he laid the foundations of Infinitesimal Calculus, which later became the heart and soul of mathematics, of astronomy, of theoretical physics, and of technical science, in so far as it is founded in mechanical relations.
On the other hand, Einstein who now, three hundred years later, has set up his differential equations, and, with them, a new world-system, stands before us as a pure discoverer, devoid of practical aims. But in these equations there are elements of analysis that once came to light in a happy idyll. This event did not come out of the grey obscurity of abstraction, but out of a region of earthly happiness, when a ray of light found its way into Kepler's gloomy existence. No poet has yet expressed this curious complex of events in a ballad, telling how Truth, the only object of Science, was pressed out of the grape, and how Practice, inspired by the inquiry of a cooper, found its way to a Theory that stretches to the confines of the Universe.
II
The conversation touched on famous expressions, words carved in stone, in particular a saying of Kant which seeks to fix the foundation and the limits of knowledge. "Every science of Nature," the great philosopher of Königsberg had said, "contains just as much Truth as it contains mathematics." And since, ultimately. Nature includes everything—for a demarcation between physical and mental science no longer seems possible—then, if we follow Kant, we should have to regard mathematics as the sole measure of science.
It is certainly not yet possible to enter into a discussion on this point with historians, medical or legal practitioners. They would be justified in refusing it, since, in their subjects, "truth" is not the sole factor, and because we cannot see at present how the conception of a comprehensive mathematical truth is to find a place in them. But when we question a physicist on this point, who unceasingly uses mathematics as his chief instrument, we should surely expect him to answer with an unconditional affirmative. At least, I should not have been surprised if Einstein had answered in this way, and if he had indeed claimed its validity for every branch of science.
But Einstein considered this quotation to be true only conditionally, in that he accepted it as a principle, but did not regard it as universal. That is, he does not recognize mathematics as the only test of truth.
"The sovereignty of mathematics," said Einstein, "is based on very simple assumptions; it is rooted in the conception of magnitude itself. Its dominant position is due to the fact that it gives us much more delicate means of distinguishing between infinitely varied possibilities than any other method of thought that expresses itself in language and is restricted to the use of words. The greater the field taken into consideration, the clearer does this become; but even in such a narrow range as 1 to 100, an estimate such as 27 is incomparably more exact than can be expressed in words in any other way. If we think of a series of sensations, ranging from pleasure to pain, or from sweet to bitter, we find that words leave us in an uncertain, confused state, and we do not succeed in fixing on a point of the series with the same precision as we above fixed on the 27 out of the 100. But when the theory of magnitude plays a part in the question, as, for example, in a series of tones, whose vibrations exhibit a mathematical sequence, we immediately attain a much higher order of precision by using numbers...."
That is why there is a sort of scientific pleasure in the sequence of tones, so my thoughts ran on. Leibniz remarks that "Music is the pleasure of the human soul, which arises from counting without knowing that it is counting." Here Pythagoras' "Number is the essence of all things" is verified. As soon as we arrive at the stage at which we feel the psychological essence of number, we fall into a sort of ecstasy, because, in our subconscious minds, we experience not only the pleasure of sense but also the underlying truth.
Einstein resumed: "Kant's remark is correct in the sense that it sets up two things in clear contradiction to one another. On the one hand, he has in view the fruits of knowledge of ordinary life, in which our ordinary perceptions and experiences are intermingled and cannot be disentangled by inductive methods and deductive considerations. Opposed to these, and to be regarded of higher rank, are the properly scientific constructions—that is, such in which we find a neat differentiation of connected thoughts that are based on regular foundations and that form the links of a chain of deduction. Whenever our science succeeds in detaching this logically ordered knowledge from its sense-sources, it has a mathematical character, and the amount of truth contained in it will accordingly be determined by Kant's criterion. But Kant demands too much when he asks us to apply this scale to all attainable knowledge of science. It would seem advisable to draw limitations if his remark is to serve as a regulative measure. A great part of biological science will in future still be obliged to make its way independently of purely mathematical considerations."
"Your reflections, Professor, would then also apply to the saying of Galilei: The book of Nature lies open before us, but is written in letters other than those of our alphabet; its characters are composed of triangles, quadrilaterals, circles, and spheres."
"With all due honour to the beauty of this observation, I cannot refrain from doubting its universal validity. If we were to accept it unconditionally we should have to regard the paths of all research as purely mathematical, and this would exclude certain very important possibilities, above all, certain forms of intuition that have shown themselves to be extremely fruitful. Thus, according to Galilei's interpretation, the book of Nature would have been illegible for Goethe, for his spirit was entirely non-mathematical, indeed anti-mathematical. But he possessed a particular form of intuition that expressed itself as a feeling which put him into direct contact with Nature, with the result that he obtained a clearer vision than many an exact investigator."
"Do you then consider intuitive gifts to be separable at all in form and in kind?"
"It would be pedantic to seek to establish a fundamental difference, even if we may regard the non-mathematical intuition of Goethe as a very striking case. Moreover, as I have often emphasized, all great achievements of science start from intuitive knowledge, namely, in axioms, from which deductions are then made. It is possible to arrive at such axioms only if we gain a true survey of thought-complexes that are not yet logically ordered; so that, in general, intuition is the necessary condition for the discovery of such axioms. And it cannot be denied that, in the great majority of minds with a mathematical tendency, this intuition exhibits itself as a characteristic of their creative power."
"From these remarks it would appear that you value deduction considerably higher than induction. Perhaps in using these catchwords I am expressing myself a little vaguely; it seems to me that great things have been achieved, too, by using inductive processes."
"Let us first define what each of these terms means. Deduction is the derivation of the particular from the general, whereas induction is the process of deriving the general from the particular case. Now, quote any example of a brilliant achievement, which you feel illustrates the power of the inductive method. Of whatever kind your example may be, you will soon become aware of the difference in the significance of the two processes."
"For me the most perfect example of induction is given by certain reasoning of Euclid. The question was whether there is a finite or an infinite number of primes (that is, numbers that cannot be divided without leaving a remainder except by unity). Euclid found an elegant proof that the total number is infinite by the following strictly inductive reasoning. If the total number were finite there would have to be a greatest prime. Let us call it n, and then form the product of all primes up to n and including it, finally adding one, thus: 2 x 3 x 5 x 7 x 11 x 13 ... n, plus 1. This new number, say Y, is certainly greater than n, and now there are two possibilities, either n is prime or it is not prime.
"If it is not prime, it must be divisible by some existing prime. But the primes up to and including n cannot divide exactly into Y, as there is always a remainder, namely, 1. Hence Y must be divisible by an existing prime X greater than n. This contradicts the assumption that n is the greatest prime, for X is shown to be greater than n.
"Secondly, if Y is a prime, it immediately follows that n cannot be the greatest prime, for Y is greater than n. Hence, however great may be any prime that we may assume, there will always be one that is greater, and even if we do not succeed in expressing it in figures, we see that it must certainly exist. Thus by studying carefully a particular case—the prime n, which was assumed to be the greatest possible one—we have arrived at a general theorem which states that there is no limit to the number of primes. Is not that, too, a triumph of intuition?"
"Certainly," said Einstein. "But you must not overlook the fact that a theorem of this kind cannot be ranked with a theorem of a fundamentally axiomatic character. The one you have discussed has been derived by a clever process of reasoning, but it does not exhibit the characteristic of a momentous discovery. This theorem of Euclid can be imagined absent from science without the content of truth in science being essentially effected. Compare with it a theorem of axiomatic significance, such as Galilei's Law of Inertia, or Newton's Law of Gravitation. Theorems such as the latter are characterized by being starting-points of knowledge that are inexhaustible in the consequences that may be deduced from them. Your question, earlier, as to whether I consider the deductive method superior to the inductive, was not formulated in correct terms. To this I answered above that the inductive method as a means of discovering general truths usually appears over-estimated. The proper form of the question is: Which truths are of the higher order, those that are found inductively, or those that lead to further deduction? There can scarcely be doubt about the answer."
"No, that is certainly true. If I understand your meaning rightly, the answer may be expressed by an allegory. Intuition of the highest order creates treasure-mines, those of lesser degree individual articles of value that are significant in themselves, although they cannot be compared with the inestimable value of the mines. The fact that the highest intuition is found in minds with a mathematical trend makes it appear possible that Kant's remark may gain more and more credence in the future. It already applies in a measure to subjects to which it seemed inapplicable during Kant's lifetime, for example, in Psychology, in which the relations between stimulus and response have been established mathematically only since the Weber-Fechner Law was set up; and also, since the time of Quetelet, in Moral Science and Sociology, we learn from mathematical methods of statistics and probability that even Man as an active being is subjected to mechanical causality. At any rate it seems manifest that Kant's remark, that in every science there is just as much truth as there is mathematics, has received additional support in recent times."
"That may be admitted," concluded Einstein, "without recognizing his remark as an axiom. It is still far removed from making possible unassailable deductions, and will never quite succeed in doing so; yet it may claim equal significance as a beautifully expressed idea with that of Pythagoras, which asserts number to be the nature of all things."
III
"The lines of demarcation between 'conceptual knowledge' (Erkennen) and 'perceptual knowledge' (Kennen) are being drawn more and more closely nowadays. The former is regarded as being the exclusive possession of the highly developed human mind, and the latter as being characteristic of the lower intelligence of other living creatures. Is this not a pronounced case of anthropomorphism, and does it not mislead us to form opinions that we should at once disown if we succeed in stepping out of our human frames even for a moment?"
"We have to rest satisfied with anthropomorphism once and for all," answered Einstein, "and there is no sense in wishing to escape from it, for the arguments about anthropomorphism are necessarily also diffused with it, itself. We are thus moving in a circle if we imagine we can deduce something outside of human knowledge. As soon as we have argued around the circle, we find ourselves again at the starting-point, and so we are compelled to mark clear lines of division between instinctive knowledge, derived directly by perception, from conceptual knowledge, derived by processes of abstraction and reflection; in this way we award the palm of supremacy to the human mind."
"But what if the following contradiction were to assert itself? Suppose that the logical 'circle' is not a circle at all, but a spiral, so that the final point of the argument lies just a trifle above the initial point. I feel instinctively that such apparently fruitless circuitous arguments might finally lead to a definite piece of knowledge. For example, a certain insect, the ichneumon-fly, although devoid of a knowledge of science in our sense, infallibly plants its sting in a definite point in the rings of a caterpillar, at just the point that serves its purpose of paralysing the caterpillar without killing it. It acts instinctively, and it is open to me to interpret this occurrence in other words. The fly discloses that it 'knows' the anatomy of the foreign creature, although it has no conceptual knowledge of it in our sense. But it immediately follows from this analogy that, from the point of view of the fly, its perceptual intelligence stands higher than our conceptual intelligence—that is, by changing the perspective, I am led to declare the anatomical knowledge of the fly to be of higher rank than the analogous knowledge of the most learned anatomist. In the same way I might persuade myself that the mathematics of a bird of passage stands above the cartographic knowledge of any human explorer. The migratory bird that flies from the interior of Africa in a straight fine to its nest in Mecklenburg must have something in the nature of a co-ordinate system in its organism. The real reason that we assign a higher position to our conceptual knowledge is that we are equally proud of our intelligence as of our science; this is perhaps a deception depending on some compromise, a sort of illicit deal in which the mind draws bills of exchange on science, and, as a return, science meets its obligations by paying in cheques drawn on the mind!"
I must confess that these hazardous suggestions received no welcome from Einstein, and were not even met with the friendly smile with which he usually accompanies his refutations. Nor do I disguise from myself that the question of conceptual or perceptual knowledge can in no way serve as a basis of proof; we may at most base certain conjectures on the difference of these types of knowledge, conjectures that suggest in words what eludes our clear comprehension. Einstein's refusal to allow this possibility certainly rests on much firmer ground than the somewhat Bergsonian views that I tried to present. Perhaps they are of a hair-splitting nature, and deal with things lying on different planes; and are deduced by unjustifiably altering the perspective with a sort of sophistic somersault; perhaps I may be reproached with seeking, like Münchhausen, to reach a higher standpoint without having a support from which to start. Yet how is it that I find it impossible to free myself from this chain of thought? No reason is forthcoming, for it is a purely metaphysical question, and there has never yet been a clear system of metaphysics free from ambiguities and sophism.
Let us rather confine ourselves to the conceptual intelligence characteristic of human beings, with which, according to Einstein, so many pleasures of theory are available. I asked him whether he would recognize differences of degree in these pleasures, dependent on their intensities. Although I rightly felt that he would answer in the affirmative, his answer took a totally different turn from what I had expected. It was, indeed, a great surprise, for in the matter of happiness of spirit he expressed a view, according to which he—a great discoverer!—does not regard Science as the deepest source of happiness!
"Personally," said Einstein, "I experience the greatest degree of pleasure in getting contact with works of Art. They furnish me with happy feelings of an intensity such as I cannot derive from other realms."
"This is indeed a remarkable revelation, Professor!" I exclaimed. "Not that I have ever doubted your receptivity for products of art, for I have often enough observed how you are affected by good music, and with what interest you yourself practise music. But even at such moments when you gave yourself up to the pleasures of the Muses, and were soaring in regions far removed from the earth, I used to say to myself: This is a delightful arabesque in Einstein's existence; but I should never have surmised that you regard this decorative side-issue as the greatest source of happiness. But your confession seems to go further, perhaps even beyond music?"
"At the moment I was thinking particularly of literature."
"Do you mean literature in general? Or had you a definite writer in mind, when you were speaking of the felicitous effect of works of art?"
"I meant it generally, but if you ask in whom I am most interested at present, I must answer: Dostojewski!" He repeated the name several times with increasing emphasis. And, as if to deal a mortal blow at every conceivable objection, he added: "Dostojewski gives me more than any scientist, more than Gauss!"
"If, Professor," said I, after a pause that may easily be accounted for—"if you mention in the same breath the names of two such powerful but essentially different intellects, you open the way to a discussion that cannot be settled by a mere positive assertion. It is possible to admire intensely Dostojewski as one who moulds personalities and who analyses the inner struggles of the soul, and yet to deny him perpetual fame. This depends on individual judgment, and, as for my own, I believe that Dostojewski, in spite of his direct artistic appeal, will not have his name perpetuated through the centuries like that of many another member of Parnassus. It seems to me to be a more important matter whether a common measure can be found for Art and Discovery at all. Perhaps the test of how far a work can be replaced may be regarded as valid for each. When you say that Dostojewski gives you more than Gauss, this probably corresponds with the feeling that without Dostojewski you would have no 'Karamasoffs' and hence would lack a certain life-value that cannot be replaced. But if Gauss had failed to produce one of his fundamental theorems of Algebra, probably some other Gauss would have appeared, who would have achieved this result. According to this, then, our instinct increases the value of a work of art, as we feel that we are dependent on one being alone for its creation."
"But this is only to be admitted conditionally," said Einstein, "for the best that Gauss has given us was likewise an exclusive production. If he had not created his geometry of surfaces, which served Riemann as a basis, it is scarcely conceivable that anyone else would have discovered it. I do not hesitate to confess that to a certain extent a similar pleasure may be found by absorbing ourselves in questions of pure geometry."
"Perhaps we may use a different characteristic as a means of comparison," I suggested, "namely, the permanency of the impression produced on the subject receiving it. For example, a fine piece of music never loses its influence. We can listen to the first movement of Beethoven's Ninth Symphony a hundred times, and, although we know at every beat what will follow, the state of pleasure continues unweakened; indeed, it might rather be said that the expectation of pleasure increases from one hearing to the next."
"This characteristic, too," answered Einstein, "cannot be claimed as the exclusive property of works of art. Its existence cannot be doubted, inasmuch as it belongs to every eminent example of art. Yet we encounter it outside the realm of art, too, in great advances of science, with which we never cease occupying ourselves, and yet the impression continues unweakened."
"Do you include among them the impressions that a discoverer experiences when he reviews in his mind the progress due to his own efforts?"
"Naturally, and these, indeed, quite particularly; and if this question were put to me directly, I should answer unhesitatingly that I find pleasure in reflecting on my own discoveries, and never experience feelings of weariness in passing over them again. So that, to return to our original thesis, we must adopt a new basis of value if we wish to account for the fact that the greatest degree of happiness is to be expected of a work of art. It is the moral impression, the feeling of elevation, that takes hold of me when the work of art is presented. And I was thinking of these ethical factors when I gave preference to Dostojewskis works. There is no need for me to carry out a literary analysis, nor to enter on a search for psychological subtleties, for all investigations of this kind fail to penetrate to the heart of a work such as "The Karamasoffs." This can be grasped only by means of the feelings, that find satisfaction in passing through trying and difficult circumstances, and that become intensified to exultation when the author offers the reader ethical satisfaction. Yes, that is the right expression, 'ethical satisfaction'! I can find no other words for it."
His whole face lit up, and I was deeply touched by his expression. At that moment it seemed to me that he had drawn the last veil from his soul to allow me to share in his ecstasy. Was that the same physicist who interprets the events of the world in terms of mathematics, and whose equations encompass phenomena from electrons to universes? If so, it was a different soul; one which gave utterance, like that of Faust, to the words:
Call it then what thou wilt.
Call it Bliss! Heart! Love! God!
I have no name for it!
Feeling is all in all!
Name is but sound and reek,
A mist round the glow of heaven!"
And, certainly, the book need not have been one of Dostojewski's to excite this feeling in him. He chose the latter to give expression to a mood that may change according to what he reads, but undergoes no fluctuations in its ethical foundation. From other occasions we know how little ethics, that is conducted along systematic lines, signifies to him, and that he does not even include it in the sciences. But at the same time we see now that his inner life is dominated entirely by the ethical principle. His deep love of Art is characterized by it, and receives full satisfaction from the source of ethical joy of which Art is the centre.
IV
During the autumn of 1918 Einstein was feeling indisposed, and, on the advice of his doctor, did not leave his bed. When I entered his room, I saw at once that there was no reason for alarm, for pieces of paper covered with mysterious symbols were lying about, and he was absorbed in making additions to some of them. Nevertheless, I considered it my duty to treat him as a patient under medical care, and did not conceal my intention of leaving him after having inquired about his condition. But he would not accept my visit as a mere call to ascertain his progress towards recovery, and insisted that I should remain with him a while, to converse about amusing little problems as usual.
I pointed out to him that there were two objections to this, the first being that he was unwell, and the second that I was intruding on his work.
"How illogical!" he answered. "If I interrupt my work to chat with you, I am putting aside exactly what the doctor would deny me if I were to allow him. So, let us make a start. You have probably some conundrum weighing on your mind."
"That may not be far wrong. I have been troubled by something in connexion with Kepler's second law. It almost robbed me of my night's sleep. My thoughts kept returning to a certain question, and I should like to know whether there is any sense in the question itself at all."
"Let us hear it!"
"The law in question states that every planet in describing its elliptic path, sweeps out with its radius vector equal sectorial areas in equal intervals of time. But this seems only half a law, for the radius vectors are only considered drawn from the one focus of the ellipse, namely, the gravitational centre. Now, another focus exists, that may be situated in space somewhere, perhaps far away in totally empty regions, if we assume the orbit to be very eccentric. My question is: What form does this law take if the radius vectors are drawn from this second focus and if the corresponding sectorial areas are considered, instead of these quantities being referred to the first focus exclusively?"
"This question is not devoid of sense, but it serves no useful purpose. It may be solved analytically, but would probably lead to very complicated expressions, that would be of no interest for celestial mechanics. For the second focus is only a constructive addition, that has nothing real in space corresponding to it. What else is troubling you?"
"My next difficulty is a little problem that sounds quite simple and yet is sufficiently awkward to make one rack one's brains. It was suggested to me by an engineer who certainly has a keen mind for such things, and yet, as far as I could judge, he did not get a solution for it. It concerns the position of the hands of a clock."
"You surely are not referring to the children's puzzle of how often and when both hands coincide in position?"
"By no means. As I said just now, it is really quite perplexing. Let us assume the position of the hands at twelve o'clock, when both hands coincide. If they are now interchanged, we still have a possible position of the hands, giving an actual time. But, in another case, say, exactly six o'clock, we get a false position of the hands, if we interchange them, for on a normal clock it is impossible for the large hand to be on the six whilst the small hand is on the twelve. The question is now: When and how often are the two hands situated so that when they are interchanged, the new position gives a possible time on the clock?"
"There, you see," said Einstein, "that is just the right kind of distraction for an invalid. It is quite interesting, and not too easy. But I am afraid the pleasure will not be of great duration, for I already see a way to solve it."
Supporting himself on his elbow, he sketched a diagram on a sheet of paper that gave a clear picture of the conditions of the problem. I can no longer recollect how he arrived at the terms of his equation. At any rate, the result soon came to hand in a time not much longer than I had taken to enunciate the problem to him. It was a so-called indeterminate (Diophantic) equation between two unknowns, that was to be satisfied by simple integers only. He showed that the desired position of the hands was possible 143 times in 12 hours, an equal interval separating each successive position; that is, starting from twelve o'clock, the two hands may be interchanged every 5 minutes ²⁄₁₄₃ seconds, and yet give a possible time.