chapter VII
.): "And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits; and a line of thirty cubits did compass it round about." Thus π here appears as 3, an approximation which no longer satisfied later generations. The wise men of the Talmud went a step further, in saying 3 plus a little more; and this agrees roughly with the actual value.
The view became more and more deeply rooted that this π was a main pillar of mathematical thought and calculation. The more the problem of the quadrature of the circle seized on men's minds, the greater were the efforts made to find the exact value of this "little more" of the Talmud. Since 1770 we know that this is not possible, for π is not rational, that is, it can be represented only as an infinite and irregular (that is, non-repeating) decimal expression. It occupies, further, a special rank as a transcendental quantity; this fact was proved by Lindemann as late as 1882 for the first time. Yet, even nowadays, there are incorrigible devotees of quadrature, who are still hunting a solution because they cannot rid themselves of the hallucination that such a simple figure as the circle must submit ultimately to a constructive process.
The correct way was to carry out an even more accurate determination of the decimal figures. The above-mentioned Ludolf van Ceulen got as far as the 35th place of decimals; at the turn of the eighteenth century the 100th decimal place was reached. Since 1844, thanks to the lightning calculator Dase, we have its value to the 200th decimal place, and this should satisfy even the most extravagant demands. This number, associated with the circle, is a classical example of how an approximation that is expressible in figures of very small value gives an order of accuracy that can be described only by using fantastic illustrations.
If we take a circle of the size of the equator, and also multiply the value of the diameter of the earth by π, we know that the latter result will not be exactly equal to the former, and that there will always be a small remainder. If this discrepancy were less than a metre, the order of exactness would be extraordinarily high, for a metre is practically insignificant compared with a mighty circle of the dimensions of the earth's circumference.
Let us stipulate still greater accuracy. We demand that the error is to be less than the thickness of the thinnest human hair. We find, then, that we must take for π at most 15 places of decimals. Thus, if we use π = 3.14159265358973, we are applying a means of calculation that reduces the possible error in all measurements of circles on the earth to a degree beyond the limits of human perception.
If we pass beyond the world out into celestial space, and consider circles of the dimensions of a planetary orbit, nay, further, if we pass on to the Milky Way or even to the limit of visible stars, to find space for our circle, and if in this case we still reduce the discrepancy so as to be less than any length that is observable under a microscope, then the last given value of π still suffices. Yet we must not forget the proviso: _semper aliquid haeret_, something unsolved still clings to the problem.
Such numerical approximations, however instructive they may be, nevertheless retain a comparatively playful character, and furnish only a superficial analogy to the most important approximations that are contained in our natural laws themselves. It is these, above all, that manifest themselves so clearly in Einstein's life-work, and they bear the same relation to the former as truth bears to correctness. Truth comprises the greatest conceivable circle of ideas and passes far beyond the sphere of correctness, which deals only with measure-relations, and not with the things in themselves. If Einstein, as we learn, emphatically declares truth to be the only object of science, he means the strictly objective truth that is to be derived from Nature, the true relationship of phenomena and occurrences, independently of whether restless philosophy assigns a question mark to this ultimate objectivity. A great discoverer in the realm of Nature cannot and dare not proceed otherwise. For him there is behind the veil of Maya not a phantom that finally vanishes, but something knowable, that becomes ever clearer and more real as he detaches each successive veil in his process of approximation.
During this conversation, when we were talking of the "Future of the Sciences," Einstein gave his ideas free rein, shooting far ahead of the views and prognostications of the above-mentioned scientists:
"Hitherto we have regarded physical laws only from the point of view of _Causality_, inasmuch as we always start from a condition known at a definite cross-section of time, that is, by taking a time-section of phenomena in the universe, as, for example, a section corresponding to the present moment. But, I believe," he added, with earnest emphasis, "that the laws of Nature, the processes of Nature, exhibit a much higher degree of uniformity of connexion than is contained in our time-causality! This possibility suggests itself to me particularly as the result of certain reflections concerning Planck's Quantum Theory. The following may be conceived: What belongs to a definite cross-section of time may in itself be entirely devoid of structure, that is, it might contain everything that is physically conceivable, even such things (so I understood him to say) as, in our ordinary physical thought, we consider impossible of realization, for example, electrons of arbitrary size, and having an arbitrary charge, iron of any specific gravity, etc. By our causality we have adjusted our thought to a lower order of structural limitations than seems realized in Nature. Real Nature is much more limited than our laws imply. To use an allegory, if we regard Nature as a poem, we are like children who discover the rhyme but not the prosody and the rhythm." I interpret this as meaning that children do not suspect the restrictions to which the form of the poem is subject, and just as little do we, with our causality, divine the restrictions which Nature imposes on occurrences and conditions even when we regard them as governed by the natural laws we have found.
Thus a leading problem of science in the future will be to discover the restrictions of Nature as compared with the apparent causality implied in physical laws.
We have in this an example of the transcendental perspectives that are opened up when we accompany Einstein on one of his excursions of thought. In this case it is actually a question of ultimate things, of a region of discovery of which we cannot yet form a conception, and it appears doubtful whether the problems latent in it are to be treated by making investigations into physical nature, or whether they are to be allotted to speculative philosophy.
In the first place, Einstein's remark seems to aim at nothing less than a revision of the conception of causality. However much has been done to purify this conception and to make it clear, we have here, perhaps, a new possibility of refining it by making a synthesis of scientific and abstract philosophical views. We shall just touch very lightly and superficially on the possibility of a synthesis giving us an avenue to truth. Whoever has heard these words of Einstein, feels the need of getting on to firm ground to rescue himself out of the turmoil of ideas into which he has been plunged.
What is Causality? A physiological answer may be given by saying that it is the irrepressible animal instinct, rooted in our brain-cells, that compels us to connect together things that we have experienced and imagined. Poets have defined Hunger and Love as the fundamental elements of our social lifes; we need only add the thirst for causality to this to complete the list of primary instincts. For this mental thirst is not less intense than our bodily hunger, and is even greater in that it never forsakes us for a moment. It is easier for the body to check breathing than for the soul to still the question of the why and wherefore, of the cause and effect, of the antecedent and consequent.
This ceaseless search for a connexion between occurrences has become organized into a fixed and immovable form of thought, which remains mysterious even when we imagine that we have eliminated all the mystery from it. The relations that we seek and that we regard as being of an elementary character are totally foreign to Nature herself. David Hume, the first real, and at the same time the most penetrating, explorer into this form of thought, said that, in the whole of Nature not a single case of connexion is disclosed which we are able to grasp. All happenings appear, in reality, disconnected and separate. One "follows on" another, but we can never detect a connexion between them. They appear "co-joined," but never "connected." And since we can form no idea of what has never presented itself to our outer or inner perception, the necessary conclusion seems to be that we have absolutely no idea of causal connexions or causative forces, and that these expressions are quite devoid of meaning, however much they may be used in philosophical discussions or in ordinary life. This "Inquiry concerning Human Understanding," with its atmosphere of resignation, has been elaborated in manifold ways, particularly by Kant and the Kantians; for it is impossible to take up a philosophic thread without entering on an examination of the fundamental question concerning the existence of a causality which lies outside our instinct for causality. It is also inevitable that, whenever we start out in this direction, we encounter the further question: What is Time? For causality directs itself to the problem of succession, both of sensations and phenomena, consequently the two questions are not only intimately connected, but are really only different expressions of one and the same question. Time, which according to Descartes and Spinoza is a _modus cogitandi_, not an _affectio rerum_, and, according to Kant, is an _a priori_ form of thought, dominates our intelligence with the same sovereign power, as the imagined course of things: what we perceive in the corresponding act of thought is regarded as temporal and causal, and impossible of further analysis.
Now, the conception of time has been entirely revolutionized by Einstein himself; and it may be expected that the conception of causality, too--which, in accordance with custom, we still endow with a separate existence--will also be affected by this revolution.
We thus approach a relativization of causality, and we may advance a step further in this direction, if we call to mind the differences of time-perception that Nature herself leaves open to us. It must be clearly understood that we are not dealing at present with the theoretical time of physics, in the sense of Einstein's theory, but with something physiological that ultimately, however, resolves itself into a relativization of time, and hence also of the causal connexions in time.
To do this, we have to follow the lines of reasoning developed by the celebrated St. Petersburg academician, K. E. von Baer, and we need extend it only very little to get at the heart of causality, if we start from his address of 1860: "Which View of Living Nature is correct?" For the human brain is a part of living nature, and hence the processes of thought may also be conceived as expressions of life.
The starting-point is a figment, the fictitious character of which vanishes as soon as we approach its results. The bridge of thought may be destroyed later; it suffices to carry us temporarily, as long as it lands us in safety on the other side.
The rapidity of perception, of the arbitrary motions, of intellectual life seems in the case of various animals to be proportional approximately to the rapidity of their pulse-beats. Since, for example, the pulse of a rabbit beats four times as quickly as that of a bull, it will, in the same interval of time, also perceive four times as quickly, and will be able to execute four times as many acts of will, and will experience four times as much as the bull. In the same astronomical length of time the inner life and perceptual world, in the case of various animals, including Man, will take place at different specific rates, and it is on these rates that each of these living creatures bases its subjective measure of time. Only when compared with our own measure of time does an organic individual, say, a plant, appear as something permanent in size and shape, at least within a short interval. For we may look at it a hundred times and more in a minute, and yet notice no external change in it. Now, if we suppose the pulse-beat, the rate of perception, the external course of life, and the mental process of Man, very considerably accelerated or retarded, the state of affairs becomes greatly changed, and phenomena then occur, which we, fettered by our physiological structure, should have to reject as being fantastic and supernatural, although, on the supposition of a new structure they would be quite logical and necessary. If we suppose human life from childhood to old age to be compressed into a thousandth part of its present duration, say, into a month, so that the pulse beats a thousand times more quickly than occurs in our own experience, we should be able to follow the course of a discharged bullet very exactly from point to point with our eyes, more easily than we can at present observe the flight of a butterfly. For now the motion of the bullet in a second will be distributed among at least 1000 pulse-beats, and will induce at least 1000 perceptions, and accordingly, in comparison with our everyday perception, it will appear 1000 times slower. If the duration of our life were again to be reduced to a thousandth of its first reduced value, that is, shortened to about forty minutes, then our flowers and herbs would seem just as motionless and immutable as rocks and mountains, in which we only infer the changes without having directly observed them. We would in the course of our lifes see little more of the growth and decay of a bud and a flower in full bloom than we at present see of the geological changes in the earth's crust. The acts of animals would be much too slow to be seen; at most, we could infer them as we do the motions of the stars at present. If life were shortened still further in the same way, light would cease to be an optical occurrence to us. Instead of seeing the things on which light falls, we should become aware of them as being audible, and what we at present call tones and noises would long have ceased to have an effect on the ear.
If, however, we let our fancy roam in the opposite direction, that is, if, instead of compressing the duration of human life, we expand it enormously, what a different picture of the world would present itself! If, for example, the pulse-beat, and hence the rate of perception, were to be made a thousand times slower, so that the average human life would be spread out over, say, 80,000 years, and that we should experience in one whole year only as much as we now experience in a third of a day, then, in every four hours winter or any other season would pass by, vegetation would spring up and as rapidly die. Many a growth would not be perceptible, on account of its relative rapidity compared with the rate of the pulse-beat. For example, a mushroom would suddenly come into existence, like a newly formed spring. Day and night would alternate as a light and a dark minute; and the sun would appear to fly over the heavens like a fiery projectile. If we were again to make the duration of human life a thousand times longer still, and hence the rate of life a thousand times slower still, we should, during the whole of an ordinary year, be able to have only 190 distinct perceptions, so that the difference between day and night would vanish entirely, and the sun's path would be a glowing circular band in the heavens, and all changes of form that seem to us to happen quietly and regularly, and to preserve a certain permanency, would melt together in the wild stream of happening, engulfed in its onward rush.
Are we justified in opposing to this relative perception of time "our own" time, which is something specific and dependent on our constitution as human beings? Should we not rather adopt the view that this specific time, adapted to our particular pulse-beat, gives only a very limited picture of the world, which is conditioned and determined by the limitations of our own definite intelligence? Is it, perhaps, only a distorted picture, a caricature, of actual occurrences?
An intelligence infinitely superior to our own would no longer be dependent on the separate sensations such as are presented to us with the rhythm of the pulse. For such a mind there would be no metronomic foundation in the sequence of occurrences, beyond what represents itself as time to our understanding. He would be situated outside of time in what Thomas Aquinas called the _nunc stans_, in the stationary present, without a retrospect of the past and without expectation of a future. Without the Before and the After, the occurrences of the world would acquire the clearest and simplest meaning, like that given by an equation of identity. What presents itself to us as a "succession" of events would merge together into one whole, just as a succession of numerical calculations become summarized in a rule of calculation, or as a series of logical operations resolves into a logical self-evident truth. If the mind conceived by Laplace actually existed, it would stand above the necessity of introducing time as a quantity into its world-equations, for time is a purely anthropomorphic quantity, produced by our perception, and regulated by our own characteristic pulses. Accordingly, the conception of causality, too, which is indissolubly connected with time, must be regarded as anthropomorphic, as something that we read into, and not out of Nature. We should at least have to recognize that if there is a causality outside ourselves, then we can learn only a minimum about it, and even this only in a world displaced or distorted by the accidental rate of our pulse-beat.
Let us now repeat Einstein's assertion "that the laws of Nature, the processes of Nature, exhibit a much higher degree of uniformity of connexion than is contained in our time-causality! It is possible that what belongs to a definite cross-section of time may in itself be entirely devoid of structure, that is, it might contain everything that is physically conceivable, even such things as, in our ordinary physical thought, we consider impossible of realization, for example, iron of any arbitrary specific gravity." It seems to me that the non-physicist will, perhaps, gain a clearer insight into these highly significant words of Einstein, now that he has received the assistance of these physiological considerations. It must be granted that the philosophic grounds of Einstein are quite different and lie much deeper than those of von Baer, who starts from organic functions and ends by arriving at a mysterious relativity that is yet consistent in itself. Nevertheless, there is one point of contact, inasmuch as in each case possibilities that lie apparently _extra naturam_ are suggested.
Einstein says: "Hitherto we have regarded physical laws only from the point of view of _causality_, inasmuch as we always start from a condition known at a definite cross-section of time, as, for example, a section corresponding to the present moment." At our own risk an easy paraphrase of his words will be attempted:
The time-section of the present contains for us the sum of all previous experiences, out of which the necessary course of our thought sifts out the category of causality.
What is not present in experience cannot appear in our causality. Let us consider for a moment Hume's example of the Indian who has never known ice. Without being told, and if he is dependent only on his own sensations, he would never learn that water freezes in cold climates. The influence of cold on water is not gradual, corresponding to an increase of cold, and not one that may be anticipated in all its consequences, but at the freezing-point water, which a moment before was a very mobile liquid, passes into a very rigid solid. The causality of the Indian cannot account for this. If we tell him of this phenomenon, he has two courses open to him. Either he refuses to believe it--and this would be quite natural, since rigid water is to him as meaningless as is a square circle to us. Or else he believes the story, and then his list of categories incurs a break, passing through the middle of causality. He has then to reconcile himself to the assumption that something that is meaningless to him and that stands outside the connexion of cause and effect is possible of realization. Up to that moment, in his time-section of the present, there was no room for it in his causality. To Torricelli the conception of liquid air, which we have been able to prepare only since 1883, would have appeared impossible and incompatible with his causality.
So there is no room in our causality for the idea of iron with the specific gravity of air, or with one several times that of gold. For, reasoning along the lines of our causality, we should conclude that a substance that is so light or so heavy may, indeed, exhibit chemical relationship with iron, but it would not itself be sufficiently defined by the term _iron_.
Now Einstein also said: "Real Nature is much more limited (or bound) than our laws imply." A sceptic might be disposed to take these statements separately in order to construe a contradiction out of them. For, if there are limiting conditions in Nature, which are foreign to the views expressed in our laws, how would it then be possible for phenomena, which cannot be imagined, to become realized? If Nature can do this, surely she must have more liberty than we seek to impose on her. This apparent contradiction vanishes if we treat the conception of structural design or uniformity as something distinct from the measure of all experience up to the present. This would give us the following interpretation:
Out of the manifold of occurrences that are possible in mechanical Nature, real Nature selects a very closely defined manifold. Thus the true laws imply a much greater degree of limitation than those known to us. For example, the laws known to us at present would not be affected if we should discover electrons of arbitrary size or iron of arbitrary specific weight. But Nature realizes only electrons of a quite definite size and iron of a definite specific weight.
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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!
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