I
When Archbishop Davidson, in the early days of relativity theory, asked Einstein what effect his theory would have on religion, Einstein answered: "None. Relativity is a purely scientific theory and has nothing to do with religion." This answer seems to give short shrift to any attempt at aligning with a mystical view of the universe the revolution in scientific thought which Einstein brought about. But Eddington suggests that Einstein's remark must be under- stood in the context of the times in which it was made. In those days, Eddington, explains, one had to become expert in dodging persons who were persuaded that Einstein's four- dimensional continuum was what spiritualistic seances were supposed to reveal: Einstein's hasty evasion was therefore not surprising. But, according to Eddington, the compartments into which human thought is divided are not water- tight: fundamental progress in one cannot be a matter of indifference to the rest. He caustically offers an analogy:
"Natural selection is a purely scientific theory. If in the early days of Darwinism the then Archbishop had asked what effect the theory of natural selection would have on religion, ought the answer to have been 'None. The Darwinian theory is a purely scientific theory, and has nothing to do with religion'?"
Is Eddington's interpretation of Einstein's remark correct? Before we pass judgment we must note, by the way, that Eddington's excuse for Einstein does not seem quite pertinent. Einstein may have wished to dissociate his theory from the claims based on table-rapping and the ouija board. But Archbishop Davidson could scarcely have appeared to him their champion. The more serious-minded among the religious interpreters of relativity theory believed that Einstein confirmed an attitude which was usually considered favour- able to religion, the attitude of subjectivism. When, for
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instance, Einstein declared that space and time are not absolute but relative and that measurements of them depend on the state of the observer, he appeared to make the observing mind the determinant of space and time. Physics prior to Einstein was supposed to present us with a world in which the observing mind made no difference to what was observed; but if space and time vary with the state of the observer, does not the world become permeated with subjective values and does not the materialistic world-view based on 4he old physics collapse, giving precedence to the power of the mind?
No doubt, Einstein himself employs the word "subjective", yet to connect his theory with religion via subjectivism in the common acceptation of the term is to misconstrue him. The word has a special connotation in physics, and what Einstein calls subjective has, in the universe of discourse to which it belongs, no psychological content in any determinant sense: it does not mean that the differences in measurement arise from the state of the observer's mind and occur because he is making use of his consciousness. On this point there is a consensus of scientific authorities. Sullivan, in his article The Physical Nature of the Universe in An Outline of Modern Knowledge (page 99), writes: "It is hardly necessary to say that by referring to an 'observer' we do not imply that there is anything 'subjective' or 'psychological' about this theory of relativity. Instead of 'observer' we could substitute the phrase 'automatic measuring apparatus' without affecting the validity of any of our conclusions." Whitehead, on page 142 of Science and the Modern World, has the same thing to affirm: "There has been a tendency to give an extreme subjectivist interpretation to this new doctrine. It is perfectly legitimate to bring in the observer, if he facilitates explanations. But it is the observer's body that we want, and not his mind. Even his body is only useful as an example of a very familiar form of apparatus." Jeans makes a similar statement on page 65 of Physics and Philosophy: "It is the body of the observer we want and not his mind; a laboratory equipped
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with cameras and various instruments of measurement would serve our purpose just as well." Eddington also, on page 183 of Space, Time and Gravitation, speaks of relativity employing "the different possible impersonal points of view..., those for which the observer can be regarded as a mechanical automaton and can be replaced by scientific measuring appliances." Bertrand Russell explains on page 219 of The ABC of Relativity: "People have been misled by the way in which writers on relativity speak of 'the observer'. It is natural to suppose that the observer is a human being, or at least a mind; but he is just as likely to be a photographic plate or a clock. That is to say, the odd results as to the difference between one 'point of view' and another are concerned with 'point of view' in a sense applicable to physical instruments just as much as to people with perceptions. The 'subjectivity' concerned in the theory of relativity is a physical subjectivity which would exist equally if there were no such things as minds or senses in the world."
All these pronouncements have, of course, to be taken in reference to a particular limited issue and not to the general philosophical problem whether anything can exist independently of consciousness or else, existing, be to consciousness anything other than what the constitution of consciousness makes it like. They should also not be taken in reference to the truism either that even physical subjectivity can have no meaning and no place in physics in the absence of conscious- ness or that it is, for the purposes of science, always a part of the plan and procedure which emanates from and depends on consciousness. The question fronting us is nothing more than the following: Is the observer's consciousness directly and immediately necessary for the "odd results" of relativity physics? To return a true answer let us pause a moment on the phrase "physical subjectivity". It is worth while bringing the meaning of it to sharp focus by marking it off from other species of subjectivity similarly leading to disagreement between observer and observer. Subjectivity by which, within our sphere of discussion, differing statements can be made are of three types.
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There is psychological subjectivity. If I recite Sri Aurobindo's Savitri to an audience, half of whom know English and half do not, those who know it will understand him, while, for the rest, wonderful lines like
The superconscient realms of motionless peace
Where judgment ceases and the word is mute
And the Unconceived lies pathless and alone
will be no more than a series of rhythmically arranged sounds. Nor will all those who know English find in the lines the same wonderfulness. Lovers of mystical poetry will be enchanted and exalted, lovers of modernist poetry will not respond so whole-heartedly; lovers of the so-called matter- of-fact will be quite out of tune with Sri Aurobindo's profound and spellbinding vision of superconscient peace and will perhaps feel because of it only a desire to stretch their legs and have a quiet nap in a comer. All these different impressions are instances of psychological subjectivity. Then there is physiological subjectivity. My audience may be composed of those who hear well, those who are hard of hearing and those who are stone-deaf. So, some will catch every word, some will miss a word or two here and there, some will only see my lips moving inaudibly. The different impressions result not from states of mind but from the body's normality or defectiveness. Then there is physical subjectivity. Fart of the audience may be near me, part far from me. Or else some hearers may be standing in one place, others moving about. What I read will be received differently by the near, the far, the standing and the moving. The differences will depend neither on the mind's condition nor on the state of the body's organs but only on the circumstances of situation and motion. The minds may be all akin, the bodies may be utterly similar, and yet these differences will come to pass. For, they are purely physical and can be reproduced exactly by putting, in place of the people, recording instruments all alike. They arise from "observation"
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in a sense in which observation can go on without minds or bodies being present, since all that is required is recording instruments. And if the human observer himself acts as such an instrument the character of the subjectivity involved is not changed: it remains physical and, within our sphere of discussion, has no bearing in the least on any problem connected with psychological subjectivity.
Physical subjectivity is what all physics, classical or Einstein speaks of. According to Einstein, if the observer is moving at such and such a speed relative to an object under observation, a particular kind of effect will be observed in the object: the object will have a certain behaviour. Change the speed and the behaviour of the observed object will be changed. Obviously what affects the observation is the state neither of the observer's mind nor of his bodily organs but the motion of his body. This implies that the point about the difference either of mental or bodily condition does not arise:
were the mental and bodily condition exactly the same in all observers but the rate of motion dissimilar, the variations observed would still be there. The essential factor is not psychological, not even physiological but totally physical. And if it is totally physical we can break up the observation into two parts: an instrument's recording an effect and the observer's reading off what is thus recorded. It is with this break-up in view that Philip Franck, in Between Physics and Philosophy, makes what is the final elucidation in brief of the whole issue. He writes: "It is only essential in relativity that in accordance with the motion of the measuring instruments the results of the measurements will be different. But in this there is nothing psychological, at any rate not more than in classical physics. The role of the observer is in both cases entirely the same: he merely substantiates the fact that in a certain instrument a pointer coincides with a division mark on a scale. For this purpose the state of motion of the observer is of no account." To sum up in our own words: a moving laboratory fitted with recording apparatus can be a substitute for both the mind and the body of the observer
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and, though the observer's mind-body accompanies the moving laboratory in order to substantiate its readings, the mind-body need not move at all for a particular behaviour to be registered of an object. The observer plays no determinant part in any immediate sense raising a special subjectivist issue.
So, relativity theory, in this context, is to be understood simply as changing our old ideas about what happens when one frame of reference or co-ordinate system or physical standpoint is in relation of motion to another: the term "observer" can be dropped from a formulation of it with as much or as little impunity as from a formulation of classical physics. And, unless we choose general philosophical grounds having nothing to do in particular with any physics, we cannot here subscribe to subjectivism in the common acceptation of the term. In all physics the "subjective" does duty only for the "variant", and the "objective" for the "invariant." The variant is the different characteristics an object or event has in its relation to diverse physical stand- points outside itself, carrying measuring instruments: the invariant is what characteristics must be possessed by it or be attributed to it in order to correlate and unify the variant characteristics. The variant is the pointer-readings unique to a particular place: the invariant is the common factor found in or suggested by the unique pointer-readings from all possible places. The variant is the local or relative feature of an object or event: the invariant is the feature that is universal or absolute.
If Einstein, as Eddington believes, was discouraging popular confusions when he refused to see any religious significance in his theory, the mixing up of the scientific variant with the psychologically subjective was more probably in his mind than seances. But it is doubtful whether any excuse that could be found for him has force. Einstein's remark is really of a piece with all his other pronouncements on science and religion. Religion, in his view, is of two sorts:
either it considers God to be personal, a Being other than the
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universe and interfering with natural events, or it feels that a mighty intelligence is within the universe, imbuing it with a rationality, an ordered regularity, which nothing can break and which is discovered progressively by the human mind when this mind functions as scientist. Science, according to Einstein, is in conflict with the first sort, while it is actually based on the second. "What deep faith," he exclaims, "in the rationality of the structure of the world, what a longing to understand even a small glimpse of the reason revealed in the world, there must have been in Kepler and Newton!" But, beyond the derivation of science from a "cosmic religious feeling", there is for Einstein an utter divergence between science and religion. Science, he believes, deals with what is, religion with what should be: the one with truth, the other with value. Science is impotent to provide principles necessary for judgment and action, it is not even able to justify its very basis - the value of the search for truth. Religion is equally impotent to give any knowledge of the world-process, it is not able to tell us what principles operate in the working of cosmic nature. "Science without religion," says Einstein, "is lame; religion without science is blind." When the two, by being complementary, are entirely different in field and function, how, asks Einstein, can science have any bearing on religion and how can we talk of any religious significance in the theory of relativity?
Einstein's conception of science and religion is open to criticism on many heads. We, however, do not need to go into a detailed philosophical discussion. Suffice it to say that his cutting asunder of science as the realm of truth, from religion as the realm of value, is arbitrary. What science gives is a certain type of truth: surely we cannot restrict the discovery of truth to the scientific method. The human consciousness has many modes of operation and each comes into contact with reality in a different aspect: we cannot dogmatically deny that the artistic imagination or the mystical intuition is incapable of finding truth. The truth they find may be of another type than the scientific, but truth it can
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remain no less. The knowledge of what is cannot be confined to science; even the realm of value is concerned with the discovery of what is, for unless we know that our ethical aspirations are supported, however secretly, by the nature of reality we shall have no genuine sanction against selfishness and cruelty and deception. As Einstein realises, the pursuit of scientific truth itself has no justification without our being convinced that reality is such as to make it worth while for the researcher in pure science to tear himself away from immediate practical life and devote himself to the terrific exertions without which pioneer creation in scientific thought can never come into being. It is not only a "cosmic religious feeling", a sense of an ordering mind within the universe, that inspires a Kepler and a Newton and an Einstein, but also an admitted or unadmitted reliance on the discovery by intuition that reality supports by its nature the ideal of truth. Take away from the non-scientific domain the quest for knowledge of what is and you make the disinterested passion of the pure researcher in science a mighty foolishness. Religion without science is not blind: it is blind merely to scientific truth while being open-eyed to truth of a different class.
And this different class of truth comprises not only value but also, in a subtle sense, structures, as should be evident even from what "cosmic religious feeling" implies. If there is an ordering mind within the universe, then that mind has an order within itself, owing to which the universe is ordered in such and such a way. The mind that is expressed in the structure of phenomena must have, as a support for its expression, its own structure. Religion is concerned with - among other things - the experience of this structure and has made pronouncements about it. For, religion tells us not merely that God is good and blissful and beautiful in a perfect superhuman manner or that He is omniscient and omnipotent but also that He has certain purely existential characters - namely, that, while being omnipresent and manifested in space and time. He is beyond them, too. If this
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is so, it would be an entirely legitimate question whether one scientific theory is more favourable than another to religion. Eddington is justified in holding that human thought is not divided into watertight compartments. His citing the example of Darwin's theory of natural selection is most apt. Samuel Butler revolted against Darwinism by shouting: "It banishes mind from the universe." Strictly speaking, he is not right. Darwinism merely stated that evolution proceeds not by an urge in the organism towards a certain way of living but by a series of accidental variations out of which some are accidentally preserved by the fact that they happen to fit in with the environment while the others do not. How can such a theory banish mind from the universe? It just banishes the operation of mind in the evolution of species. To banish mind from the universe can mean either of two things: there is no consciousness present in the world or, if there is, it is totally explicable in physical terms. Darwin points to neither conclusion. But his theory does definitely lessen the importance of consciousness in the world-process. If religion involves stress on the play of consciousness, as it certainly must, the Darwinian theory is anti-religious. If there is an intelligence at the back of the world, we surely do not deny it by saying that it chooses natural selection as the means of evolution; but we cannot overlook the extreme oddness in this intelligence's manifesting itself in a mode which so little stresses the role of consciousness. The unimportance of the role of consciousness strongly suggests, though it does not prove, that there is no intelligence at the back of the world and that soul and freewill and miracle are non-existent. Similarly, a theory in physics may favour materialism or lean towards a mystical world-view: all depends on the implications of the mathematical structure it regards as final. The terms employed are not abstract symbols as in pure mathematics: they are a symbolic language interrelating, at the end of a long or short process of deduction, actualities of observation and experience. In physics, unless the contrary is proved, every formula of
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structure can be taken to correspond to a world-reality, and the nature of that reality to be suggested by the manner in which the terms of the formula are combined. We commit no "howler" in inquiring whether relativity theory sheds on world-reality a light in the direction of mysticism.
2
To come to the correct conclusion it is best to make a short survey of the rise and development of Einsteinian physics. Einstein versus the ether! That is the form in which the first battle was fought between the new physics and the old. For, the ether was vitally bound up with the problem with which he was occupied: Newton's absolute motion, absolute space and absolute time.
To observe absolute motion we should have a frame of reference absolutely at rest. Otherwise motion would be merely relative - that is, a body' to which reference is made when calculating another body's motion may itself be moving but is taken to be at rest for only convenience's sake and so the rate of motion it yields for the other body is not absolute. Thus the sea is moving relatively to the earth which seems to be at rest, but the earth is itself moving round the sun and the sun too is moving relatively to the so-called fixed stars and they in turn are moving relatively to one another. Newton, however, declared that though all bodies are in relative motion the frame of reference that is absolutely at rest is space and that such a frame is necessary for the purposes of physics. He further declared that there is one time flowing uniformly so that at any chosen moment we can say that events are happening everywhere in space simultaneously with it. Indeed, as motion is measured as a certain number of space-units traversed in a certain number of time- units, time as well as space has to be absolute if we are to
' A "body" in physics does not mean a human body: it is a brief way of designating a frame of reference or a co-ordinate system.
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have absolute motion. But is any experiment possible by which these absolutes can be verified?
Space can hardly prove a frame of reference absolutely at rest if we regard it as empty. Luckily, to explain the phenomenon of light the undulatory theory was invented. Light was found to behave as if it were a wave. To be a wave was thought to imply something in which the wave could form. A universal substance called ether was postulated to fill all space to permit light's vibratory motion. Having postulated the luminiferous ether, physicists naturally attempted to discover its other properties than that of being vibrated. Certain experiments seemed to show it to be dragged with celestial bodies, like a super-atmosphere, as they moved; many more seemed to show it to be not dragged with such bodies but fixed in space, though capable of internal movements such as light.
Taking it to be some sort of subtle material stuff, composed of particles like all matter, Michelson devised an experiment which has often been repeated. Our earth's atmosphere is dragged with the earth, but for moving objects on the earth it is fixed as a whole, though capable of internal movements. That is why we find an air-drift when we speed through air in a plane. When we are stationary, there is no air-drift of the same kind. Similarly, if the ether is dragged with the earth, an appropriate stationary apparatus will record no ether-drift. But if it is not dragged and is fixed in space the apparatus will record the drift. By measuring the speed of the ether-drift we should know our own speed through a fixed ether and thus know the earth's motion in reference to something not only at absolute rest but also practically playing the role of space by being all-pervasive. Here was a chance to get at absolute motion and absolute space. Further, by knowing one piece of absolute motion we can translate all known motions into absolute terms. Thus, we can know the absolute speed of light and by making allowance for the time-lag between the moment when light leaves a source and the moment when it reaches us we can
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know what moment anywhere in space could be considered as simultaneous with the moment at which we receive the light-rays. So absolute time also can be found if we can measure the ether-drift.
Of course, the ether is too subtle to tackle directly. But if light is an ether-phenomenon we can get at the drift by marking light's movement in a particular fashion. Michelson's experiment was meant precisely to do this. In principle it consisted in sending two beams of light from the same source in two directions, one in the direction in which according to astronomical observations the earth was judged to be moving and the other at right angles to this, and then getting them reflected by mirrors fixed at equal distances from the source. The reflection of the beam sent in the direction of the earth's movement would naturally take less time to reach the source than the other reflection, for the earth would be moving forward to meet it and there would be less distance for the beam to travel. The difference in the two times would indicate the speed with which the ether- drift was felt by the moving earth and therefore the rate at which the earth was moving in reference to a fixed ether. The experiment was a masterpiece of delicacy and could have detected even one-hundredth of the extremely minute difference expected; but it failed totally. There was no difference at all. Light coming towards us as we moved towards its source travelled with the same speed as from other sources!
Could it be there was no ether? Michelson, rather than face an etherless physics, concluded that the ether was dragged with the earth and thus counteracted the difference in speeds. But the majority of physicists, relying on astronomical data, would not hear of any dragging. Fitzgerald opined that somehow the rod with which the distances travelled by the two beams on their return journey were measured had contracted when put in the direction of the earth's movement. Lorentz went further and deduced from the then-current laws of electrodynamics that the electrons composing the rod would so readjust themselves in the
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direction of the rod's movement with the earth that the rod would get short by exactly the amount that would be needed to make the two distances travelled by the returning beams appear equal. According to Lorentz, there was no wonder that the times taken were the same and that light behaved so paradoxically: the null result of the Michelson experiment was due not to absence of the stationary ether but simply to rod-contraction.
Einstein was the only thinker unsatisfied with Lorentz's idea. He brought three arguments against it. In the first place, if rod-contraction would always exactly hide the speed of anything in reference to a stationary ether, then, whatever other functions the supposed ether might serve, the function of being a concrete form of Newton's absolute space would never be served by it. A subtly material stationary ether is as good as non-existent for physics, since every measurable quantity it might have yielded is precisely compensated for by a contraction in the measuring rod. Such an ether is a useless hypothesis. In the second place, if we take for reference a body which is moving relatively to another body but which for convenience's sake we regard as being at rest relatively to our own motion, our rod will show exactly the same contraction as we attribute to it in reference to a hypothetically stationary ether. So there is no reason to believe that the rod-contraction conceals from us a stationary ether, an absolute frame of reference. By means of the rod- contraction there is no possibility of distinguishing between an absolute and a relative frame. In the third place, since in all relativities of motion between two bodies the mathematical terms remain the same whether the first body be accounted as moving and the second as at rest or the second as moving and the first as at rest, the rod on either body must be thought of as showing the same contraction. The unchanging mathematical terms imply that from the standpoint of the one body the rod going with it would contract while from the standpoint of the other body the contraction would occur in its own rod. The contraction is a common and mutual
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feature of relative motion and gives not the least indication as to which of the two bodies is moving in reference to a stationary ether. In the face of this ambiguity, what sense can there be in talking of it as compensating for and hiding any particular quantity of motion which reference to such an ether might yield?
Why then cling to a subtly material ether which must be considered as capable of being a frame of reference at absolute rest? Why even hypothetise that it is dragged and therefore inaccessible as such a frame? Do we require it as a medium for light's vibrations? Clerk Maxwell proved light to be a species of electromagnetism. For several years physicists tried to figure out an electromagnetic wave in terms of waves of air or water or a jelly-like solid; but all attempts failed. Hertz, to some extent, and Lorentz, fully, made it clear that light could not be explained as a vibratory movement carried on from particle to particle as in the case of matter. If light was a wave, it was a wave sui generis and could not be understood in terms of oscillating particles, like all other waves. The medium postulated for light's transmission was left sufficiently immaterial by its being not composed of particles. And if a subtly material ether was unnecessary for even the mathematical description of motion, why not eliminate it?
Einstein eliminated the ether composed of fine particles which had stood for Newton's absolute space. Absolute space, he said, does not exist for physics. If absolute space is non-existent for physics, no absolute motion can be measured. And if absolute motion cannot be measured, how shall we measure absolute time? To know what time it is at a distant place when the clock here shows a certain hour, we must have a message from that place, a signal by light or radio: every message, be it ever so fast, travels at a finite speed and, if we never know the absolute value of any speed, how to allow correctly for the time-lag between the starting and the arrival of the message? Hence physics has no means of judging absolute time: a time flowing uniformly in
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the whole universe cannot figure in our equations. With the impossibility of the ether's serving as a frame of reference at absolute rest, the entire construction of Newtonian physics topples down.
Einstein began a new construction founded on the fact that light's speed is constant, whether we move towards it, away from it or stay where we are. Speed being measured, as we have already said, in terms of certain space-units travel- led in certain time-units, Einstein argued that not only rods shorten because of movement but also clocks slow down. This conclusion at once illumined the equation Lorentz had based on rod-shortening, for in it there had figured a term which could not be identified: now Einstein identified it as a sign of change in time-measurement owing to the slowing down of clocks. He even indicated a method by which the slowing down could be experimentally confirmed in a direct fashion by studying rhythmically vibrating atoms. In 1936, H. Ives of the Bell Telephone Laboratories, New York City, carried out the experiment with positive results. So Einstein has sound experimental backing. According to him, both the shortening of rods and the slowing down of clocks are proportional to the rate of movement of one body in relation to another which is itself moving. Reading of space and time made on a moving body are shared by another only if the latter has the same speed. When the speeds differ, the readings also must differ. As a consequence of relative motion, there is a relativity of space and time.
We have nothing non-materialistic, nothing mystical, so far. But to get to the heart of relativity theory we must fix our attention on one fact pointed out by Einstein and already noted by us: the reversibility of relation when motion is considered relatively. Just as our rod is shortened and our clock slowed down when we are in a certain relation of movement to a body which is itself moving, a rod and a clock on that body would in relation to the body which is our frame of reference undergo exactly the same changes which we observe in our instruments. If the mathematical terms
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denoting relative speed remain the same, whether we regard one body as at rest and the other as moving or vice versa, the question crops up: on which of the two bodies can really the rod be said to shorten and the clock to slow down? Well, if we consider both the bodies together as standpoints, the judgment of Daniel would be that the change really occurs in rod and clock on both. If we adopt one of the two stand- points, the change is real on one of them. If we adopt neither standpoint but some entirely other, a different reality will be registered by our measuring instruments. They give us variants according as we adopt one standpoint or another. To say this is to say that measuring instruments like clocks and rods can never give a reading that would be invariant from all standpoints.
But physics always aims at invariants. The laws of nature must be so formulated that they hold for all standpoints. It is not sufficient to find a "transformation" rule by which we may make the requisite adjustments in calculation as we pass from standpoint to standpoint. We must find a rule for the same reading from every standpoint: then alone can we give a description that is universal and absolute, a calculation of the fundamental quantity that different standpoints differently evaluate. But how are we to get beyond the relativity Which Einstein disclosed of all space-measurements and time-measurements?
The mathematician Minkowski showed the way. When the time-measurements are multiplied by themselves - that is, squared and then substracted from the square of the space-measurements we get a quantity which is the square of what is universal, absolute, invariant. Distance of space and distance of time alter with the rate of motion, but as soon as we follow Minkowski's rule we strike upon a distance or interval between two events which is found to be unaltered no matter what the rate at which we move. This rule is somewhat analogous to the one for calculating the distance or interval between two points in three-dimensional space. The latter rule is: Take the three co-ordinates of both the
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points, x and xi, y and yi, z and 2,1, and deduct the lesser co- ordinates from the greater and, squaring the result, add up all the squares: the sum gives the square of the distance or interval. Considering x, y and z to be the lesser co-ordinates, we write the equation:
D2=(x1-x)2 +(y1-y)2+(z1-z)2
Minkowski's equation introduced a fourth co-ordinate, as it were, which was time, and had a minus sign unlike the others:
D2=(x1-x)2 +(y1-y)2+(z1-z)2-(t1-t)2
Mathematicians, however, cannot be completely at ease with this equation. In the first place, the minus sign is not quite to their liking: it is an irregularity. Minkowski comes to their rescue by saying: "Multiply the minus-signed time- measurements by the square root of minus one and, as every schoolboy knowing mathematics will understand, we reach immediately a plus quantity like all the other dimensional quantities, and the equation becomes:
D2 (xi-x)2+(yi-y)2+ (zi-z)2+(ti-t)2.
.
Everything is now symmetrical and there is no technical distinction between time and the other variables. It is as if we had, instead of a continuum of three space-dimensions, a continuum of four space-dimensions completely isotropic - that is, similar in all directions - for all measurements; no direction can be picked out in it as fundamentally distinct from any other." But one step more is required to systematise everything for mathematical purposes. In the new equation, as in the old, time-measurements are left in time- units. How can units like seconds be added to or subtracted from units like inches? We can multiply or divide time-units and space-units by each other: for example, we divide the number of inches a moving object traverses by the number of
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seconds elapsed and we get the velocity of that object. But every schoolboy knows that it is mathematically inadmissible to subtract seconds from inches or add them. Minkowski again comes to the rescue. He says: "Luckily, in all the measurements concerned in relativity theory the speed of light remains constant. So we can use it as a common denominator. Thus, we can consider one second as equal to the 186,000 miles which light travels during a second. So we substitute for the time-measurements the miles which light would travel. Then we have complete symmetry, and the whole equation is completely as of a space of four indistinguishable dimensions. Further, the new statement of the equation facilitates the employment of the equation and any development that may be possible."
The procedure adopted by Minkowski in the interest of systematisation is often looked upon as vital to the conception of the four-dimensional continuum. This is a capital mistake and is responsible for the notion that the four- dimensional continuum is created by artificialities. It is argued: "What can be the justification of the square root of minus one and how can the substituting of miles for seconds give us a time-dimension really like the space-dimension?" Well, if Minkowski's systematisation did create the concept, we can look on his two steps as acts of analytic insight discerning and supplying what was missing in the steps by which a necessary concept was to be created. This way of looking is open as an alternative to the view of his steps as being artificialities, though that view is likely to be more stressed. But the alternative is not even called for. There is a fact which modifies the entire complexion of the controversy. The fact is: when Minkowski found that the time-measurement had to be subtracted from the space-measurements in a certain manner in order to get the invariant without which physics ceases to be physics, he found space-time to be the unavoidable invariant without needing to flourish in the face of the world his square root of minus one and the miles equivalent of a second. In Minkowski's original formula
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which can be accused of waving no such stage-conjuror's wand as the square root of minus one and the miles-equivalent of a second might seem, we have space-time no less than in the new formula, since it actually subtracts time terms from space-terms and therefore implies terms that are neither space nor time or are both together and are best described as neutral. Of course, we have not shown why, if space and time are indistinguishable, there should be a subtraction sign: we shall touch on this point later. It is not crucial here: here we are concerned not with the de jure indistinguishableness but the de facto indistinguishableness which is involved by subtracting one quantity from another. To such indistinguishableness Minkowski's seeming artificialities make no odds. We can drop them without jeopardising anything essential. When we realise this, we learn to see them as neither artificialities nor creative acts of analytic insight, they lead not to the creation of the concept but to the schematisation of it so as to make it most amenable to mathematical employment and possible development.
All the same, it must be admitted that the concept would never have emerged clearly without the schematisation. For, thus alone the invariant wanted by Einstein's physics was cast into a proper mathematical mould of four indistinguishable dimensions and brought to a focus. But, when we avoid the impression that it was not implicit in Minkowski's original equation, we must also avoid the impression that Minkowski gave us a four-dimensional space. Time is indeed spatialised by the form he put forth as mathematically the best for the four-dimensional continuum, and the process begun by physics of reckoning time in space-terms by means of a clock or any other clock-like space-mechanism reaches its apex in a manner undreamt of by the old physicists. But it would be a mistake to think that the four-dimensional continuum is conceived with a space-bias. Of course, a clock stands for time in physics, but after multiplying time-terms by the square root of minus one in order to get a plus sign we can perform the next operation in just the opposite direction:
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we can reduce space-terms to time-terms by considering 186,000 miles as equivalent to one second. The temporalisation of space instead of the spatialisation of time is equally possible. The point is that somehow space and time should be made indistinguishable dimensions. The four-dimensional continuum, therefore, is best designated as space-time or time-space rather than space or time. The dimensions, being equally designable as four of space or four of time, cannot be reckoned in terms either of time or space. The reading made for any event must be taken to be in units which are neutral. Also, the interval between any two events must be read in neutral units.
The neutral character can be realised, too, from another angle. It was found, mathematically, that to get the invariant interval between two events we had to attend to three conditions. If the space-distance between the events is such that an object can travel from one to the other before light from them can reach an observing standpoint, the interval between them for all standpoints is just what a clock on that object would record as the time taken by the object during its travel. The interval is then to be called "timelike." But if an object cannot travel between two events before light from them reaches a standpoint, the interval is just what a rod on that object would record as the space-distance travelled by the object. The interval is then to be called "spacelike". If the two events are the leaving of a ray of light from a source and the reaching of it at any standpoint the interval is such that both a rod and a clock would record it as zero. For, at the progressive rate at which, during motion, a rod shortens and a clock slows down, the rod would be shortened to nothing and a clock stop completely if they were put on a ray of light which travels 186,000 miles per second. The progression-rate can be understood if, for instance, we look at the increase of mass due to motion: here there is a progressive increase instead of decrease but the essence of the rate is the same. At half the velocity of light the mass of an electron or any object is increased by one-seventh. At nine-tenths the velocity the
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mass is nearly two and a half times greater, while at ninety- nine-hundredths of the velocity the mass has seven times its value at relative rest. At higher speeds the mass increases with such leaps that at light's velocity it must become, mathematically, infinite. From this we can say that no object can travel as fast as light: light becomes a limiting velocity. We can also say that like the mass-increase the decrease in the size of a rod and in the rhythm of a clock would be, mathematically, infinite. The interval, therefore, in terms of an object carrying a rod and a clock and travelling with light from the event which is light's leaving a source to the event which is light's reaching a measuring instrument is nil, if the interval is to be invariant from all standpoints. To sum up: the interval is in certain cases reckonable as timelike, in others as spacelike and in yet others as no time and no space! Obviously, it must be a neutral unit and we get clean beyond space-terms and time-terms to terms of space-time or time- space in which the interval cannot be legitimately deemed either space or time. If it can be either in different cases and neither in particular cases, it is something sui generis: we can also regard it as a fusion of space and time, in which both are indistinguishable and become a tertium quid, a "third some- thing".
The indistinguishableness of space and time is most commonly underlined by also pointing out that from different co-ordinate systems in relative motion at different rates the interval between any two events will be differently split up into time and space. Suppose we take the famous eruption of Krakatoa and the outburst on the star Nova Persei. The interval between these two events may be measured from a coordinate system on the earth as so many years and so many millions of miles. But a system on the Nova will measure it as a different number of years and miles. A third system, neither on the earth nor on the Nova, will have still different readings. And what figures as miles in one measurement will figure as years in another!
We must not conclude, however, from the indistinguishableness and from the fusion,
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that Minkowski meant to deny the difference between space and time in common human experience by any given individual. In this experience they are indeed inseparable - at least as far as measurements are concerned, no place having been measured except at an instant and no instant having been measured except at a place. But they are felt to be different in spite of their inseparableness. Space has three dimensions, while time has only one, since we can move only from past to present to future as in a straight line. There is also a difference psychologically in the very texture, so to speak, of extension which is space and duration which is time. Even in physics the experimental modes are dissimilar: a clock is indeed a measurement in space-terms, yet it is not at all a mode like the measurement in space-terms which we call a rod. Moreover, when a physicist measures his own movement in space-coordinates and a time-coordinate, the two are never interchanged. Relativity leaves all these unlikenesses what they are in common human experience by any given individual of his own history and what they were in the old physics. What is new is, in the first place, the discovery of the way in which with relative motion both time-terms and space-terms vary in measurement. A variation of a kind had been acknowledged in space-terms in even the old physics. Thus, if a stone falling from a tower to the ground were measured from different standpoints moving at different rates, the space-coordinates would be different. But the difference did not take into account the rod-shortening and it was fitted into the context of absolute space. Also, it did not go hand in hand with any difference in time-measurements. Time was thought to be unvarying and every moving stand- point was thought to give the same measurements of time. Now that both time-terms and space-terms are declared to be radically variant with standpoints, a novelty is introduced, which, when we search for the goal of all physics - the invariant, the uniform, the absolute from all standpoints, the universal reading - necessitates the concept of fused space
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and time. Therefore what is new is, in the second place, the concept of a four-dimensional continuum in which the dimensions are indistinguishable.
What is the precise import of the concept? And is it merely a mathematical convenience or does it represent a state of reality of which we have no cognisance in common human experience and the old physics had no idea? Is it an utterly revolutionary concept with serious supra-physical consequences?
3
Einstein, in several places, has made pronouncements tending to dissipate the air of mystery which comes with the idea of four-dimensionality. Thus, in collaboration with Infeld, he writes on page 219 of The Evolution of Physics: "Four numbers must be used to describe events in nature. Our physical space as conceived through objects and their motion has three dimensions, and positions are characterised by three numbers. The instant of an event is the fourth number. Four definite numbers correspond to every event; a definite event corresponds to any four numbers. Therefore: the world of events forms a four-dimensional continuum. There is nothing mysterious about this, and the last sentence is equally true for classical physics and the relativity theory." On pages 54 and 55 of his book. Relativity, the Special and the General Theory, Einstein informs us: "The non-mathematician is seized by a mysterious shuddering when he hears of 'four- dimensional' things, by a feeling not unlike that awakened by thoughts of the occult. And yet there is no more common- place statement than that the world in which we live is a four-dimensional continuum.... That we have not been accustomed to regard the world as a four-dimensional continuum is due to the fact that in physics before the advent of the theory of relativity, time played a different and more independent role, as compared with the space-coordinates." Again, on page 30 of The Meaning of Relativity, Einstein
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declares: "The conception of something happening was always that of a four-dimensional continuum; but the recognition of this was obscured by the absolute character of the pre-relativity time."
Is not Einstein forgetting that the new role played by time in his theory has converted the old inseparableness of time and space into indistinguishableness? Is he not ignoring the essence of the situation by labelling the old inseparableness as four-dimensionality? To count a continuum's dimensions just by the descriptive numbers required for an event is a loose manner of specification. Space is a three-dimensionality strictly and precisely because its dimensions are indistinguishable in basic character and composed analogously to one another and form one methodical block. If pre- relativity time which is the time of common calculation can only accompany but never fall in step, so to speak, with this methodical block and increase a dimension systematically instead of by a mere tackling-on, can it legitimately be held to constitute, together with space, a four-dimensional continuum? And would the mere loss by it of its absolute pre- relativity character bring any completely satisfying difference? Let us make a brief inquiry into the meaning of the terms involved in this discussion and ascertain how dimensions must be conceived if they are to build up a continuum.
A continuum, of any number of dimensions, is something that is continuous, admitting of no gap anywhere. Mathematically, this is expressed by saying that between any two specified components of it there can be an infinite number of arbitrarily small steps. If the number were finite and the steps not arbitrarily small, there would be no continuity: each step would be distinct and disparate and, instead of a continuity, we should have a mere aggregate. A line is a one- dimensional continuum and it is made up of an infinite number of successive points. A surface is a two-dimensional continuum and it is made up of an infinite number of successive lines. A volume is a three-dimensional continuum and it is made up of an infinite number of successive
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surfaces. Thus each continuum is formed by a continuous succession of components of the next lower number of dimensions. And a four-dimensional entity would consist of an infinite number of successive volumes. Suppose we take a brick which is a three-dimensional object, and ask how we are to conceive it as constituting by continuous succession an entity of four dimensions. Just a row of bricks will not do: it will not give us a four-dimensional entity. Moreover, a row is not a continuity such as we want: between any two bricks we cannot put an infinite number of bricks. Also, no fourth dimension of space is available: the only dimension other than the available three of space is time, the continuum of moments or instants. But if we take time to be the fourth dimension to make a four-dimensional entity, the continuous succession of a three-dimensional entity in it must be properly understood. Time must be a genuinely new dimension which was not there for any of the entities of the other dimensions. As a line is strictly one-dimensional, a surface strictly two-dimensional, a volume strictly three-dimensional and none of them has any other dimension than those already specified, an entity having time as its fourth dimension must be something that extends in time in a sense in which none of these entities do. But time, as ordinarily understood, is already there for a line, a surface and a volume: all of them need it in order to be themselves. Simply to continue in time as normally things do is not to have the genuinely new fourth dimension we require. To revert to our brick: a brick existing in time will not thereby become a four- dimensional entity any more than an infinite number of bricks in a row in three-dimensions will do so. It will have to exist in the time-dimension in an entirely new way. What the new way must be can at once be grasped by analogy. A line has a co-existence of continuously consecutive points, a surface a co-existence of continuously consecutive lines, a volume a co-existence of continuously consecutive surfaces: similarly, our four-dimensional entity must have volumes coexisting along the time-dimension in a continuous
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consecutiveness. In other words, all the moments of time in which a brick continues must be co-existent! This is the sole valid sense we can attribute to time as a continuum adding a fourth dimension not already present for entities made up by the other three.
It is in this sense that time operates in Minkowski's continuum. Evidently, here is much more than is meant by the necessity felt even by the old physics to regard space and time as inseparable though not indistinguishable and to use four numbers in describing an event. Nor have we here what Einstein appears on occasion to suggest in The Evolution of Physics: merely a four-dimensional form of the representation known to us in a geometrical graph. The stone falling from a tower, which we have mentioned before, can be geometrically plotted, after it has fallen, as if the time-dimension were also stretched out like a space-dimension and all the moments during the fall were co-existent. There would be a time-axis perpendicular to a one-dimensional space-axis showing the one direction "down", and lines joining different moments on the time-axis to different positions on the space-axis would be time-coordinates and space-coordinates and a line drawn through the joinings of the coordinates would indicate the path of the stone's fall downward through space and time. Indeed, what is done here with two dimensions is implied with four-dimensions in Einstein's concept. But there are two dissimilarities beyond the fact that a four-dimensional representation is not picturable and can be expressed in nothing save mathematical terms, a mathematical and not a graphical geometry. One is that the time- dimension is not concerned with only what has already occurred: it is also concerned with what is occurring and is going to occur, all the moments are co-existent in it. The other is that because of the co-existence of all the moments we cannot equate the graph with things as we ordinarily observe them. The line showing the stone's fall answers to what was observed: the "world-line", as Minkowski called the path of events in his continuum, crosses the present and
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the future no less than the past and we can answer with it to observed things only by choosing not to conceive it in its total significance. The four-dimensional concept is more than a plotting out °f Nature, it plots out that which seems to transcend or underlie Nature, and of which Nature seems a projection within a certain framework. This means that even the loss by time of its absolute pre-relativity character would not give us the co-existence of continuously consecutive volumes we want. The loss would indeed point to something more than inseparableness of time and space: it would be an index to indistinguishableness and make the time and space we know a truer limited projection of what is beyond Nature, but it could never cover the full character of the Wonder that is the authentic being of the world we live in. So, all matters considered, Einstein misses the mark when he deprecates the mysterious shuddering and the thoughts of the occult which he finds in the non-mathematician on hearing what he calls the commonplace statement that the world in which we live is a four-dimensional space-time continuum. There is really nothing commonplace in the statement if the reference is to Minkowski's concept which Einstein accepts as integral to his own theory; and the mathematician, deluded by the ease with which he can abstractly tackle any number of dimensions through his symbols, is likely to overlook the definitely supraphysical suggestion here. To be precise, Einstein's continuum carries the suggestion of what philosophers have conceived as Totum Simul, the All-at-once, a state of existence in which the whole past and present and future are a grand simultaneity and all that is in space is not only existent together but each thing is existent in its reality at all moments past, present and future!
This state of existence is deterministic in one sense, for all is already there and cannot be changed, but the determinism is not of the ordinary kind, since in ordinary determinism the present is dictated by the past and the future by the present whereas here there is no sign of any direction and we can with the same justification say that the future dictates the
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present and the present the past or that the present dictates both the past and the future! In the last alternative we have room for an utter freewill; in the first alternative we have room for an utter fixity of fate; in the middle alternative we have room for - God knows what! Living as we ordinarily do in the present, with the past vanished and the future unrealised, we are permitted by Einstein's concept as much to believe in freewill as to be determinists.
The concept of freewill is a most difficult one to state, for in common statements it looks like asking for something which is unconnected with the past to the degree to which the freedom is granted. Especially scientific thought feels foreign to such a lack of connection, since in it the convention has been to regard the past as leading to the future: most of experimental physics is concerned with expecting results which, however unforeseeable at times, are supposed to follow from antecedents and all theoretical physics is concerned with forming mathematical equations of rigorous interconnectivity. No doubt, indeterminism is ascribed by many to quantum phenomena, but Einstein is not one of these many: he is an out-and-out determinist, hoping for a "unified field theory" which would account for all quantum phenomena without the assumption of indeterminism. And he is unable to conceive of freewill. "Honestly I cannot understand," he remarks, "what people mean when they talk of the freedom of the will. I feel that I will to light my pipe and I do it, but how can I connect this up with the idea of freedom? What is behind the act of willing to light the pipe? Another act of willing? Schopenhauer once said: Man can do what he wills, but he cannot will what he wills." Evidently, Einstein implies that if we allowed the statement "I will that I will to light my pipe", we should have to explain the new willing and say "I will that I will that I will": we should have to go on like this without end and that seems meaningless as well as contradictory of our experience. The reasonable thing, in Einstein's opinion, is to postulate, behind every act of willing, a number of events we are not
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aware of whenever we have the feeling that we are free to will something. In other words, our willing is determined by other events that are themselves determined and we can never significantly be thought free. Einstein's idea is that all happenings hang together without any break and with complete continuity, the whole universe thus hangs together at every instant and its hanging together every instant hangs together with all that precedes and follows every instant. The idea is in consonance with the four-dimensional continuum and the geometrical mode of representing events in all space and all time. But, as we have seen, determinism which makes the future an effect of the present and the present an effect of the past is only one of several conclusions from it, and, philosophically speaking, Einstein's continuum does not negate freewill. All it negates is discontinuity such as quantum physics seems to demand, and Einstein is a determinist essentially in the sense that he is all for continuity: the view that negates freewill and makes the past determinative of everything else is merely a conventional interpretation of the kind of continuity involved in a continuum of four dimensions - a kind which, if established over even quantum phenomena, would not philosophically discomfit freewillists.
Perhaps our knowing only the present and having the past and the future clipped off is a clue to the alternative we should regard as the best of the three offered by the continuum. Instead of saying the future dictates everything or the past dictates everything we may say the present dictates both the past and the future and holds them actually co-existent in itself. Then the Totum Simul would be also a Nunc Stans, an ever-standing Now. But it is impossible to equate the ever- standing Now with any space-terms or time-terms or both terms merely combined. A neutral factor is to be posited and this can only be called what traditional language has called Eternity which is also Infinity. Not, of course, an Infinity- Eternity negating space and time: it holds in itself their essence, as it were, and that is why it allows itself to be
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materialist? He may protest that there is no security for the non-materialist either. But that is surely to confess that fundamental issues go into the melting-pot as soon as we warm up to relativity theory. And when a large look is taken at the riddle of the universe, even the most rabid materialist must grant that the Totum Simul is more in tune with the concept of God than with the concept of a universe having no consciousness at its back and bearing a soulless insignificant humanity on its blind breast. Mystical experience gives the closest description possible of a God, who, besides being many other wonders, is a Totum Simul. And if the Totum Simul is a reality and no convenient device for calculation, mystical experience seems more to be trusted than anything else. The poor materialist is in for a severe headache once he concedes the reality of Einstein's continuum.
Has he any ground for not conceding it? We have now to find an answer. Let us ask what would be meant by calling this continuum a mere convenient device. We have seen that no charge of artificiality can be levelled against the concept of it as put forth by Minkowski. Perhaps it will be urged: "Time in physics is measured by a clock or some clocklike mechanism which gives space-quantities and no genuine time at all. If such artificialised time is shown to be fused with space, we have only a convenient device." But, Mr. Convenient Device, are you not forgetting that time, even in the old physics, was measured in space-quantities by a clock? Nobody ever maintained that in the old physics space and time helped materialism just because time was measured in space-quantities. To think spatially of time through a clock was more than irrelevant to the issue of materialism, for everybody was saying that though time was measured spatially by a clock it could never be fused with space and any suggestion of a fusion would have occasioned a doubt about materialism. So, if the fusion has to take place, the clock-measurement of time cannot logically be pressed against the fusion having a significance which is non- materialistic.
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Is there any other argument left? Well, the very idea of looking for an argument becomes ridiculous if we but analyse the phrase "convenient device". There are many convenient devices in physics. Have they any resemblance to the fusion of space and time? If the fusion is a convenient device, it is one on which hangs the whole status of physics as a science. If physics cannot reach the invariant, uniform, absolute, universal description of phenomena, it cannot satisfactorily move forward. There is an unavoidable and basic necessity here. To compare the fusion of space and time to any mathematical quantities created for convenience is to fail to mark this necessity; none resembles it in being unavoidable for the very basis of physics. They are also dissimilar in never involving the literal fusion of any two terms combined. Take the concept of "light-year". Two entirely different ideas are joined to render easy the indication of astronomical distances. Instead of running into inordinately long series of integers to indicate how distant a star or nebula is, we adopt the device of employing as a unit the number of miles traversed in a year by light travelling at the rate of 186,000 miles per second. The light-year is no necessity: it is an arbitrary combination, we can do without it altogether and nothing in physics will suffer: it may be called also a figurative fusion and not a literal one, since in no sense are light and a year to be taken as indistinguishable. The light-year is not in the least comparable to space-time: it falls into another category.
Now look at a quantity like momentum or horse-power. We multiply mass and velocity to give momentum, divide energy by time to give horse-power. Is space-time like these quantities? Hardly. They may not be dispensable conveniences like the light-year; they may be necessary to physics but even they are necessary only for getting variants. Like mass and velocity themselves, momentum is always a variant; like energy and time themselves, horse-power is always a variant. They change as the standpoint changes. Space-time is an invariant. The two necessities are not on the same footing.
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Besides, when we construct momentum or horse-power, there is no implication that mass and velocity are indistinguishable, or energy and time are indistinguishable. The implication would come only if one is added to or subtracted from the other. It is only addition or subtraction that, according to mathematics, shows the essential sameness of terms.
Perhaps the sole concept that gets nearest to the fusion of space and time is the interchange ability of mass and energy a conclusion drawn by Einstein himself from his own theory of relativity. We shall not explain this concept at the moment, but it has two characteristics relevant to the discussion in hand. First, the interchange ability implies a genuine one- ness, so much so that we can actually convert mass into energy and energy into mass. There is no question of a device here. Second, the genuine oneness is still on the level of the variant. Space-time is on the level of the invariant. It is, thereby, a deeper necessity for physics and, in any case, it is not shown to be a device. So to call it a mere device adopted for convenience is to institute everywhere in congruent comparisons. It cannot be likened to any other combination or fusion effected by physics of two different kinds of terms. In short, no meaning attaches to the labelling of it as a convenient device!
It is possible that a difficulty may be sought to be conjured up by thrusting under our noses some oddities connected with space-time. Thus, there are what are called "lumber" equations. Relativity mathematics grinds out equations that seem to have no equivalents in perception, nothing we can verify by experimental observation. If we regard space-time as a reality, how are we to act Sherlock Holmes to their missing perceptual equivalents? Perfectly easy, my dear Watson! We have only to fall into the arms of Eddington and agree with him to regard the "lumber" equations as the mathematical symbols of unperceived properties of some- thing objective. To Eddington, even perceived properties of the world are subjective in the Kantian sense that they are
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imposed on an unknown objective reality by the constitution of one's mind. But the lightness or wrongness of this point is not of importance here: what is of importance is that all the mathematical symbols ground out by relativity correspond, in the ordinary sense, to objective properties: some of these properties are perceived, some unperceived. The unperceived are shadowed in the "lumber" equations.
It may be argued by the Convenient Device: "Aren't you bringing in our old friend, the 'unobservable', under the disguise of 'lumber' equations and giving him a lodging in physics?" The first answer would be the cocky one: "What if we are? How does that affect the controversy we are engaged in - namely, whether space-time is a convenient device or a reality? Even as a convenient device, space-time has to avoid being like the other device: absolute space and absolute time. That is to say, in its own manner it would have to avoid being what would otherwise be called an 'unobservable'. And the 'lumber' equations, as accompaniments of a convenient device, would still raise the query: Aren't they like absolute space and absolute time, that other and rejected device? If they are suspect in our scheme, they would in your scheme be equally suspect. So no advantage accrues to you on suggesting that they are our old friend or, rather, our old enemy the 'unobservable'. Surely, you do not wish relativity theory to be jettisoned because of the 'lumber' equations?"
The truth is that the only person who has a right to challenge with a hopeful gleam in his eyes our acceptance of the "lumber" equations is the anti-relativist. And to his jibe that they introduce by the back-door such entities as Einstein threw out by the front one, our answer is: "Einstein threw out what insisted on being regarded as the basis of the observable when it was really playing no such indispensable part. The 'lumber' equations do not pretend to be basic in the least. They are not at all in the same case as the 'unobservable' rejected by Einstein: they do not introduce by any back-door entities like the latter. If they were not there,
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nobody would urge that what was there became foundation- less, lost all rationale. In fact, most of us would be happy to find them nowhere on our horizon. Discussion of the legitimacy of our theory would be involved as little by their absence as by their presence and at least the bother of pointing out their differences from the 'unobservable' would be spared us."
The "lumber" equations constitute really no difficulty on any ground. And once we interpret them as corresponding to unperceived objective properties we get a new "slant" on the fact that no experimental observation has found anything to contradict them. They may not be confirmed, but why are they not contradicted? Some equations of Einstein's are marvellously confirmed, some are not, but none are contradicted. Is this not curious? One single contradiction would disprove his theory. Einstein, being no epistemological physicist like Eddington, does not dare to say that no contradiction will ever appear. But it is a tremendous tribute to his theory that the last forty years and more have not disclosed anything to throw doubt on its essential correctness within the domain of macro physics. With all objections out of the way, this tribute gains the colour of an argument that "caps, crowns and clinches all."
To sum up: nothing disproves the actuality of the four- dimensional continuum whose concept seems so much a mathematical formulation of the mystic's vision.
4
When we go from the special or restricted theory of relativity propounded in 1905 to the general theory developed in 1915 after years of intense concentration on several possibilities, the implication of a supra-physical reality becomes acuter.
The special theory proved Newton's invariants to be no invariants for physics, but it did not cover all the problems Newton had dealt with. The chief problem it kept aside was
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"acceleration". Newton had divided motion into two parts:
one was uniform, the other was accelerated. Accelerated motion meant change in the rate of speed or in the direction of speed. There was one factor which was" thought by Newton to induce on the widest and most general scale acceleration both in rate and direction. This factor he called gravitation and enunciated a law for it. The law extended to a vast range of phenomena, but at certain points it broke down. Its failure as well as the invalidity shown by Einstein of Newtonian invariants in physics made it impossible of acceptance. An alternative was badly required and to search for it became the master-passion of Einstein, particularly as, in the first place, gravitation involved both types of accelerated motion which still remained outside the relativistic scheme and, in the second place, gravitation involved peculiarities marking it out from any other force - namely, that heating or cooling a body, difference of chemical constitution and the interposing of a screen have no effect in the least on gravitational attraction, and the attraction is across a distance without any medium seeming to convey it.
Broadly speaking, the problem was to explain how the planets of the solar system remain moving in their elliptical paths round the sun and how at certain arcs of their paths they move faster. Newton said that all objects have an attractive force and an enormous object like the sun must draw towards itself smaller ones like the planets. The sun, according to Newton, would bring all the planets crashing into it, were it not that they were in motion and this motion acted so as to make them fly away from the sun but was not strong enough to free them from the sun's gravitation and, as the result of balance of forces, could only set them moving round it in elliptical paths. On analysis, we see that Newton's picture is systematic only if we grant one thing: an object in motion tends to follow a straight path in space unless disturbed by another object, either through impact or through gravitation. If we do not grant this, there is no compulsion to believe that the planets are held in elliptical
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paths by the sun's gravitation, thus being prevented from moving straight away into outer space. Gravitation as a force directly acting on objects is not a necessary concept without the concept of the straight path as the most natural for motion. The change in speed, on the other hand, can be ascribed to gravitation only if we grant that an object tends to move not only straight but also at a uniform rate. Einstein, therefore, - when faced with the question: How are the facts ascribed to gravitation to be accounted for, so as to need none of Newton's absolutes nor his law, which had been found faulty, of a force of gravitation? - decided to throw doubt on the concept that straight uniform motion is the natural one for bodies.
In this he was helped by Minkowski's equation of a continuum of four symmetrical and isotropic dimensions. For, Minkowski had formulated a geometry of the continuum symbolised by his equation. Geometry, we may remark, is essentially an abstract Science and the mathematician does not bother what meaning in terms of common human experience is to be attached to the dimensions he symbolises. On the analogy of a geometry of three dimensions such as worked out by Euclid the mathematical geometrician can build up many self-consistent systems. Whether a system applies to the conditions of common human experience is an issue to be tested by instrumental observation. In the nineteenth century, mathematical geometricians like Gauss, Lobatschewsky, Bolayi and Riemann built up strange systems different from Euclid's. They admitted that if Euclid's initial axioms and postulates were right all his propositions logically followed. But they refused to admit that his axioms and postulates were self-evident as truths. Neither would they admit that Euclid had been completely borne out by instrumental observation. Riemann emphasised in particular the fact that a triangle drawn on a curved surface does not have the sum of its angles equal to two right angles, nor the ratio of the circumference of a circle to the diameter equal to n, the well-known Ludolph number 3.14159265... .
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And just as Euclid had extended to three-dimensional space the geometry of a flat surface, Riemann dared to extend the geometry of a curved surface to three dimensional space. He posed the query: "Why should geometrical figures in space not exhibit properties as do geometrical figures on the curved surface of a sphere?"
His query remained academic because nothing was found to contradict Euclid directly. There was the indirect contra- diction in the faultiness of Newton's law of gravitation in small isolated cases; for, the law assumed space to be Euclidean, with the straight line as the shortest and therefore most natural path for an object in motion to pursue. But everything else was overwhelmingly on Euclid's side. When the special theory of relativity dethroned Newton's absolutes and, with them, his law of gravitation, it became possible to think of space non-Euclideanly. But not till Minkowski's genius came to the aid of Einstein's was the possibility taken advantage of. Minkowski rendered it easy to tackle the mathematics of Einstein's continuum and thus tackle also the riddle whether the world of common human experience, the world of space and time which this continuum was meant to correlate by means of invariants, was Euclidean or no.
Minkowski's own answer, in effect, was: "When you have the geometry of a continuum which is as much time as space, Euclidean geometry which is exclusively of space cannot be wholly valid. Take a triangle ABC. In terms of space, if you measure with a scale from A to B and B to C, the sum of your readings will be greater than the reading obtained from A to C. Two sides of a triangle are greater than the third. But if you take three events A, B and C and measure with a clock the time which would be taken in moving from A to B and B to C, there crops up a condition which is unique. To measure with a scale from A to B, the scale must lie so as to touch A with one end and B with the other: the scale has to be present at both A and B. Similarly, the clock has to be present at A as well as B. This means that when the events A and B happen, the clock must run with
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such speed as to be present at both the events. Now, speed affects the rhythm of a clock: the faster a clock moves across space the slower its rhythm. So, if the distances from A to B and B to C are very great, while that from A to C is small, the clock in running along the two greater distances will go much slower than in running along the one shorter distance. The sum of the readings of the two sides of a triangle formed by three events will be less in this instance than the reading of the third side. Euclid's geometrical rules will not universally .hold. The geometry of space-time is not quite Euclidean: it is semi-Euclidean./'
The suggestion that a geometry other than Euclid's could be actually applicable to the invariant reality whose variants are observed by means of scientific apparatus fell like a most fruitful seed into Einstein's mind. Could gravitation be accounted for in terms of the geometrical structure of space- time? In answering that it might be thus accountable Einstein was aided considerably by his realising more and more that gravitational effects could not be distinguished from other phenomena of acceleration. For instance, when a lift starts to rise, the occupants feel all the effects of a sudden though temporary increase of weight. Indeed a mass hung from a spring-balance would weigh heavier till the upward speed of the lift becomes uniform. Further, it is not logical to say that when an object is falling freely through space it gives rise to the phenomenon of weight which we attribute to gravitation:
only when it is prevented from falling by a weighing machine placed under it the phenomenon of weight is shown. Weight therefore may be regarded as due to upward acceleration impressed on the object by the bombardment of the molecules of the piece of the weighing machine upon which it drops. Again, a motorcyclist riding in a circle and trying to keep at a uniform speed will feel that the constant bending of his movement, the constant change of direction he has to maintain, acts as if he were drawn towards the centre of the circle. This draw would make him fall inside the circle unless, to avoid the slant induced, he inclined his machine to the vertical.
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Finally, the mass of a body as measured by the amount of resistance to another body colliding with it or dragging it and thus being liable to produce non-gravitational acceleration is exactly equal to the mass as measured by the amount of attraction between them according to Newton's law. In all these instances are such effects as are attributed to gravitation.
Instead of stipulating a gravitational force acting directly from body upon body across a distance, why not formulate laws embracing all phenomena of motion both uniform and accelerated? The only conceivable laws which would not exclude gravitational phenomena would be of some change worked upon whatever is between bodies, some change guiding the less massive body into the accelerated motion which imitates or is equivalent to the assumed effects of a direct gravitational force. In other words, what is between the two bodies should be so affected around the massive one as to induce the acceleration of the less massive. Gravitational force as such will not be denied, but it will not be a force of the Newtonian, kind: a body will involve a certain structuring of what is between bodies and this structuring will make the less massive behave towards the more massive as if pulled in its direction at a constantly increasing rate.
Now, what is between bodies is, in ordinary computation, either empty space or some medium filling space. There is no air in the outer expanses of space where gravitation still acts. The luminiferous ether, even if it exists, has never been found competent to explain gravitation. But electromagnetism itself, of which light is a phenomenon, is proved impossible to interpret in terms of an ether composed of particles - that is, a subtle material ether which can serve also as a static frame of reference for absolute motion. Einstein opened Lorentz's eyes to the mathematical superfluity of postulating a static frame. So a subtle material ether cannot be thought even to exist. Empty space is all that remains - unless we introduce a new concept.
But before we introduce a new concept we must recollect
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that there can be no concept without a background of space- time. Acceleration, like all quantities, is relative: from different coordinate systems different readings would be obtained. The laws of motion both uniform and accelerated which would cover gravitational phenomena as well as others and which would operate through a structure of what is between bodies must arise from a four-dimensional continuum of fused space and time. The symmetrical and isotropic equation of Minkowski, involving a semi-Euclidean geometry, turned Einstein's attention to the many systems of non-Euclidean geometry of symmetrical and isotropic dimensions built in the near past. Riemann's extension of the geometry of curved surfaces to three dimensions struck him as the most promising. He extended the geometry to four dimensions and took the simplest formula for what would be an unobstructed body's natural path in them. The natural path of an unobstructed body on a flat surface is the straight line between two points: it is the shortest path. On a curved surface it is the shortest curve. The shortest distance or interval is called the geodesic. Einstein found the formula for the geodesic in a four-dimensional continuum and, translating it into terms of separate space and time, compared the result with gravitational observations. Eureka! The problem was solved. As Whetham puts it on page 255 of The Recent Development of Physical Science, Einstein's geodesic of space- time is found to bend in space towards a mass of matter and, in time, to move faster the nearer it passes to the mass - precisely like the path of a planet swinging round the sun.
Einstein then connected the amount of mass with the character of the geodesic. Geodesies are different according as the amounts of mass present. If the masses are not disproportionate the geodesic describes in space the natural motion of a body as Newton conceived it. If they are disproportionate, the natural motion in space would not be a straight line but a curve. The curve is not due to a pull of gravitation directly from body upon body: it is due simply to the structure of space when disproportionate masses are
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present in space-time: space is as if non-Euclidean or Riemannian in the neighbourhood of bodies - not flat but curved.
With the help of his Riemannian geometry Einstein found he could explain all the facts of gravitation Newton had explained as well as one important fact which the Newtonian theory had not explained - the erratic behaviour of the planet Mercury in certain sections of its elliptical path. On top of this, he offered beforehand the calculations which would be obtained if two crucial experiments were carried out. One concerned the passing of the rays of stars through the sun's neighbourhood and the other the rate of vibration of atoms in the sun. The study of the star-rays was made by several astronomical expeditions during two eclipses of the sun when alone the rays could be distinctly measured. Einstein had predicted the curved continuum would deflect the rays to such and such an amount: his prediction was as good as confirmed as against that which the Newtonian theory allowed. The measuring of the vibration-rate of the sun's atoms proved very difficult but the results were regarded as a satisfactory approximation to what Einstein had foretold. Hence the curved four-dimensional continuum was accepted as scientifically proved.
Not only gravitational phenomena but all other motions become natural deductions from the formula connecting the character of geodesies in space-time with the amount of mass present. An immense simplification is achieved: a vast correlation is made. But we seem to be confronted with a puzzle in the idea of curved space and also of the curved space-time that results in space-curvature. The puzzle, however, is verbal. We mean by curved space nothing more puzzling than what we mean by flat space. How do we conceive space, which is not a surface, to be flat like a surface? What is the sense in calling space Euclidean? All we can mean is simply that, just as on a flat surface, the shortest line is the straight line between two points, the shortest line between two points in space is straight. On the basis of this
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we deduce a whole geometry of how bodies behave in space:
a triangle in space would have the sum of its three angles equal to two right angles. Similarly, by non-Euclidean or Riemannian or curved space we simply mean that the shortest line is a curve. And in the geometry of curved space a triangle would not be found to be as on a flat surface. In no other sense is space-curvature to be understood. That is to say, it must not be understood literally any more than space- flatness. When forms existing in three dimensions are measured, they exhibit certain geometrical characteristics. It is these geometrical characteristics that we meet with our instruments when we meet curvature.
In manifesting these characteristics Einstein's theory of gravitation specifies mathematically the phenomena of gravitation. But this is not the end of the story. The characteristics come about because of something happening between bodies. When we think of something happening between bodies so that they exhibit non-Euclidean characteristics we bring in again the notion of some kind of force. The characteristics describe a "potential" or "stress" or "strain" in what is between bodies. If what is between is empty space, there can be no stress manifesting itself in the Riemannian behaviour of bodies: empty space cannot get structured so as to guide bodies into Riemannian behaviour. On Einstein's theory of gravitation space becomes "substantial" without being composed of particles or having any qualities which would lead us to deem it subtle matter like the old ether or would make it serve as a frame of reference for motion. Inasmuch as it is "substantial" and not void, it is legitimate to bring in the term "ether" again: only, this ether does not fill space but is itself identical with space!
In the sense that it is not empty Einstein calls this ether- space physical. Physicality, however, is here Pickwickian: it is devoid of all that can properly be called physical unless we can speak of physical emptiness. Emptiness is itself a disconcerting concept, but becomes physicalised when we fill it with an all-pervading ether which is subtle matter. Take
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away such an ether, and what is left? Surely not something which we can clearly identify as "vital" or "mental" or "spiritual", yet something so non-material, non-physical and still substantial as to look like the most natural emanation, as it were, of a conscious omnipresent Being in terms of a stretching out of itself for the holding together of objects that are physical and material.
So much for Einstein's space. What about his time? If ultimately time and space are fused, it is impossible to regard time as an emptiness either. No empty space, no empty time. A time-ether, non-physical, non-material and still substantial, has to be conceived - the most natural emanation, as it were, of a conscious omnipresent Being in terms of a stretching out of itself for the deployment of a movement carrying physical and material objects.
When we take the space-ether and the time-ether in fusion, we have as a result of the curvature-concept a space- time explicitly substantialised. All the more it becomes no fiction, no convenient device but a reality existing in its own right. And there is now yet another helpful feature which emerges on our asking: Are the material masses, which lie at the centre of the curvature-pattern and whose amounts bear a fixed ratio to the pattern, the cause of this pattern or themselves a peculiar manifestation of it? On the mere strength of the general theory of relativity we cannot give an entirely decisive answer, but important indications are against their being the cause. The fundamental quantity termed interval of space-time yields a number of mathematical expressions which call for comparison with mathematical expressions concerning what physics names matter. Matter, for instance, is conserved. It is curious that precisely an expression for conservation is derived also from the quantity named interval. But what is here derived refers to some property of space-time - a specific kind of curvature. We may, therefore, submit that where there is this curvature there is, in another language of mathematical equations, conserved matter. Several other observations we associate
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with matter are similarly matched. The physical quantities we know as density, velocity, internal stresses etc. obey certain mathematical relations. Now, some equations got by analysis from the interval happen to have exactly the same number of components as matter, and these components are put together in exactly the same way. The query, as stated by Sullivan, inevitably occurs: "May we not affirm that these components which express features of the space-time continuum are identical with density, stress and the like?" That is to say, what we usually name matter may be what space-time holds as curvature of a certain sort. The curved four- dimensional continuum appears to be the original reality and matter its manifestation.
What relation matter has with space and time as we normally know them we need not here consider. It is sufficient for our purpose to have shown how the general theory or relativity drives home more vividly than the special theory a reality which, to say the least, renders materialism utterly inadequate and, to say the most, suggests a spiritual substance of Totum Simul variously manifesting itself.
A question, however, which we must tackle is raised by the concept of curvature vis-a-vis our description of the four- dimensional continuum as Eternity-Infinity. An ever-new endlessness of time is not doubted by science, but there is the phrase current in Einsteinian physics: "boundless yet finite space". Some years ago, the astronomer Edwin Hubble calculated that on the average the distribution of masses in the universe known to the telescope is .0000000000000000000- 00000000001 gramme per cubic centimetre. According to Einstein's equations of the relation between masses and curvature, Bubble's figure involves a small uniform curvature of all space over and above the non-uniform curvatures involved by the different masses. Thus space becomes a hypersphere, finite yet unbounded in three dimensions in a way analogous to that in which the two-dimensional surface of a sphere is limited in area yet allows endless repetitive movement over it. Hence curved space-time, it may be
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argued, cannot be an infinity, and infinity is negated by science and a barrier set up against mysticism which talks of an infinite Being.
Two answers may easily spring to the mind. We may argue that all concepts of curvature call for room in which the curving can take place: beyond the hypersphere there must be space to accommodate it, just as the curving of the two- dimensional surface is accommodated by a third dimension. Or we may argue: "Let us be clear about the terms we use. Curved space means that there is no straight line except as a short-distance illusion and consequently the universe is 're- entrant' in space. If one could ferry oneself across space and survive for an enormous number of years and always continue along what one would believe to be a straight line, one would at length arrive somewhere near one's starting-point. Nothing can escape the 're-entrance'. But surely here is just the fact that there are geometrical limits to our exploration of space. Boundless yet finite space implies this fact: it does not imply the negation of space-infinity by science."
Unfortunately, neither argument is cogent. The first is built on a double error. To begin with: a two-dimensional surface has room to curve in because it curves in space which has three dimensions; but a hypersphere has itself three , dimensions and if it curves in anything it would be in a fourth dimension, but a fourth dimension of - what? The hypersphere, being three-dimensional, exhausts all the dimensions of what we call space; so, what it might curve in cannot be space! By the analogy of a two-dimensional surface we do not get the space-infinity that is denied. Furthermore, in geometrical physics the term "curvature" has no reference to room to curve in. Surface-geometry in physics, though having a background of actual flatness or curvature, is essentially concerned only with the behaviour of measuring- rods and the properties deducible from various arrangements of them: it abstracts the rods, as it were, from the surface on which they are laid and omits reference to the surface's actual shape. A reference to it would stop all
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extension of geometry from a surface to space which is evidently not a surface; and the reference is technically avoided by the device of calling a surface itself a space. Thus a curved surface is called a two-dimensional space with a curvature measured in terms of arrangements constituted by rods. The question of curved shape within room to accommodate it is ignored and rendered irrelevant from the beginning: the physical presence of the measuring rods is the sole connecting link between two dimensions and three:
unless this were so, the concepts of surface-geometry could never be adapted to three dimensions or more. If there is known to be room to curve in, we accept it but scientifically no such room is considered in our concepts and when we speak of a hypersphere we confine ourselves to the behaviour of measuring rods and never bring in a hypothetical room in which it can be hyper spherical. We have to hold on to relations which exist within the space we speak of and drop reference to anything external to it.
The answer to the second argument - and this could apply also to the first - is quite short: "Since in very principle we can never observe what may lie beyond our 're-entrant' space and since under no conceivable circumstances can anything beyond it figure in our equations any theory assuming such a beyond is superfluous in science and science can supply no basis to any philosophy erected on such a theory."
We have to look for other arguments if we are to talk of infinity of extension on scientific grounds. Only one argument is possible. It is admitted by all that, according to its very nature, Einstein's finite though boundless hypersphere cannot be stable: it must either contract or expand and the current astronomical interpretation of the red shift in the spectrum of nebulae as a sign of their recession tends to show that the hypersphere is expanding. Eddington computes that the present circumference of the universe is between 6,000 and 60,000 million light-years but that the size of the universe is doubled about every 1,300 million years.
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A curious point is that it is not the masses of matter that are expanding but the space between them: if these masses expanded together with the space there would be no means of measuring any expansion. But an expanding universe, however "re-entrant", involves the concept of more and more space not merely in the sense of boundlessness through which we may move over and over again as on the surface of a sphere: it is analogous to a growing larger of the very surface! And though at each stage of the expansion the amount of space remains finite, it is a greater finite each time and what we have is extra and new space. The extra and new space definitely involves a beyond of space to each amount of finiteness. The hypersphere could not have this extra and new space if none were available beyond the geometrical limits of our exploration at each stage. And once we admit this availability constantly coming into our observation we break through the concept of the finite and though we do not directly have the infinite we have it indirectly in the constantly realised possibility of the hypersphere's expansion. Hence physics with its curvature does not shut the door against the infinity of the four-dimensional continuum but points in its direction, and the boundlessly finite space that is expanding can be taken to correspond to a certain diminishing delimitation within the limitless Spirit - a selective play, as it were, of the original reality so that a particular range of possibilities is actualised with a wider and wider scope. There is nothing here against the infinite Being of whom mysticism talks.
Even if there were anything, we should do well to remember that while non-uniform curvatures connected with different masses are accepted by science as proved, Hubble's estimate has no finality and the small uniform bending of all space is only a rather plausible speculation. Convincing proof is wanting - and it can come only if the light of a distant object in the sky, sent out in all directions, reaches us not only from the front of it but also from behind it by getting curved in the long run and arriving on earth by the
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opposite route! General space-curvature should enable us to see a remote nebula twice — in a front-view and a back-view in parts of the sky exactly opposite to each other. Unless the new 200-inch telescope recently set up at Mount Palomar gives us the two views the hypersphere offered us will not pass from plausibility to certainty and infinite space will not be disproved. Astronomers have little hope of getting these views. Of course, the failure may be due to the radius of overall general curvature being too great; but it may just as well be due to utter lack of such curvature.
5
Two topics remain now to be dealt with in order that we may have a complete picture of the mystical implications of relativity. One is the equivalence of mass (or what is commonly called matter) and energy, about which we have already spoken en passant.
Before Einstein came on the scene of physics, the atom of matter which had been supposed to be the ultimate constituent of things had already been broken up and found to consist of electrons and protons. Today we know of many other particles - neutron, positron, meson. But to approach Einstein's concept of equivalence of mass and energy we need consider only the electron, the particle of negative electricity. The electron in very rapid motion had been observed to increase in mass while ordinary matter in all the motions that had been observed never disclosed any in- crease. Of course, one could have said that ordinary matter had never been observed moving so fast as the electron. But here it was pointed out that Maxwell had established certain equations of electromagnetism, which described the behaviour of electric energy. From these equations we could deduce that if the electron was a concentration of electric energy it would show exactly the increase of mass that it did. Since the atom of matter was electrically neutral, ordinary matter was supposed not to increase in mass with motion.
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Einstein broke down this distinction. There are several ways of indicating how he did it. We may choose a few simple ones. As change of velocity of any kind brings about a change in our measurements of the space-quantity of length and the time-quantity of duration, whatever is associated with velocity must undergo a change as measured by our instruments: mass, therefore, of all bodies and not simply that of an electron can never remain constant. Again, motion being relative, we can reverse the relation of movement and quite legitimately regard our frame of reference as moving and the electron as at rest, instead of doing the opposite as at present. So, not the electron only but our own co-ordinate system from which we observe it can be said to increase in mass: from the electron's point of view it is our co-ordinate system that is whizzing past at 100,000 miles per second and getting its mass increased! Electrical properties have thus no monopoly of conducing to increase of mass. Finally, when we consider the rate at which with increase of motion rods shorten and clocks slow down we observe that the progression of shortening and slowing down is such that at light's speed - represented in physics by the letter c - rods would shorten to absolute nothingness and clocks stop dead. This shows that nothing can exceed the speed of light. With this conclusion before us, we can reason in the following manner:
"When motion increases, momentum increases with it. Momentum is mass multiplied by velocity, but if a body were to move at c, the momentum would not increase by an increase in velocity, since c cannot be exceeded. So what would be affected and change is the mass. The extra momentum would be as if a body with more mass were moving at velocity c. This means that, since there would always be an impossibility for the motion of a body to increase so as to reach c, there would be with every increase of momentum a certain increase of mass resulting from the thwarted development in motion." The increase mathematically calculated by Einstein from several angles happened to be exactly the same as had been experimentally observed in
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the electron before the relativity theory was formulated. It was proved all-round that the electron's increase in mass is not due to any electrical properties and that it was observed merely because its speed is sufficiently large to make the increase perceptible and that all bodies whatever have the
mass-increase though their small speeds prevent it from being perceived.
Nor was the universality of mass-increase with motion the sole revolutionary concept introduced by Einstein in this context. Still more revolutionary was the new concept of energy in general, which implied that we could consider matter itself and all particles constituting matter as a state of energy. How the concept was arrived at can be grasped if we examine the rate at which the mass of a body increases with motion. The rate, as we have already noticed, is such that if motion reached light's speed the mass would become infinite. This, meaning as it does that motion can never reach c, means too that as motion increases we find it more and more difficult to increase it further. According to Newtonian physics, the speed of matter makes no difference to the amount of force wanted to increase the speed: a certain amount of matter needs the same force to increase speed from 10 miles per hour to 11 miles as to increase speed from 100 miles to 101, and this force depends exclusively on the amount of matter. Now, if motion, the faster it becomes, is found to resist increase more and more, it acquires the property usually attributed to matter - resistance to whatever acts upon it - the property which makes a greater amount of matter resist more than a smaller one does. This property is inertia, involving mass and weight. Now, motion is a form of energy: as a body moves, it acquires what is named kinetic energy. Kinetic energy, therefore, behaves like matter. Nor is this all. The increase of mass the material body gets by motion is exactly the increase of mass the kinetic energy exhibits. The two masses are one and the same - we can look upon the extra mass as either the kinetic energy's or the material body's. Hence the kinetic energy and the material
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body must be the same kind of entity. But kinetic energy is only one form of energy: it can be converted into other forms - chemical, electric, radiant. It is the conversion of one sort of energy into another that leads to the law of conservation of energy: the amount of energy remains constant throughout the conversions. So all energy must be deemed the same kind of entity as a material body. And if it is the same kind, all energy and all matter must be inter convertible. In that case, how can the law of conservation of energy be kept apart from the law of conservation of matter which tells us that matter remains constant in amount throughout its conversions? The two laws get merged into one law based on the inter convertibleness of matter and energy. And through calculation of the amount of energy which leads to the increase of mass the law can be made to tell us that a very small amount of matter represents a very great amount of energy - in fact, the amount of energy into which matter can be converted is 34,596,000,000 times the amount of matter.'
Here the question becomes pertinent: If matter and energy are inter convertible, is matter proved to be energy or energy proved to be matter? In physics, energy used to be defined as matter's capacity for doing work. The capacity was considered a property of matter and matter the more fundamental reality. Now it is shown that the capacity itself possesses the essential property of matter: inertia, mass, weight. If energy is a property of matter, it cannot itself possess a materiality of its own and bring extra inertia, mass, weight to matter whose property it is! This argument is irrefutable and final. Energy, therefore, cannot any more be regarded as a property of matter. Can we say that it has itself become a state of matter because it exhibits inertia, mass, weight? How can we? It exhibits something else, too - namely, work-capacity: otherwise it would not be energy. Can we then say that matter has become a state of energy?
' It is Einstein's new law of conservation and his calculation of the energy- amount to which matter is convertible that formed the basis of the research whose result was the atom bomb.
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Well, matter must be thought of as exhibiting inertia, mass, weight differently than energy does. The difference lies evidently in the work-capacity exhibited with inertia, mass, weight by energy. In matter there is no work-capacity shown. But, since matter and energy are inter convertible because of the common property of inertia, mass, weight, the work-capacity must be looked upon as what is hidden in matter and brought out in energy. To be more accurate:
when a certain quantity of inertia, mass, weight is in the phenomenon called matter, it hides the work-capacity which is brought out when the same quantity is in the phenomenon called energy. With the passing of the one phenomenon into the other and the conversion of matter into energy, this quantity does not disappear, as commonly supposed: it remains in existence but is part of a phenomenon not found before. Thus, in the sun's radiation that is the energy into which the sun's matter is converted, there is the precise inertia, mass, weight of the converted matter - the energy radiated every second is computed to weigh 4,200,000 tons - but there is with these tons something else not openly carrying them prior to the conversion. Hence the concept of energy is fuller than that of matter, and we can regard matter as concealed energy. Conversely, we can regard energy as matter revealed in its completeness. But this just means that there is an incompleteness in matter as such and energy is the more comprehensive and fundamental phenomenon whose checked and bound state is matter. It is because of being checked and bound that a very great amount of energy is represented by a very small amount of material substance.
When energy is no longer a property of matter but a more comprehensive and fundamental phenomenon which is work-capacity with inertia, mass, weight of its own, it cannot be considered purely physical, though we cannot explicitly designate it as Life or Mind or Spirit. Or, rather, it becomes something in which the physical is subsumed under a mysterious more-than-physical reality - quite unlike the old energy which was never independent of matter and was
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subsumable under a reality quite physical. An indirect testimony to the metamorphosis is the new definition found necessary in the fourteenth edition of The Encyclopaedia Britannica: "That by which work is done and which diminishes in proportion to the work done." This is so vague that it is merely an effort to define the physically indefinable.
The mysterious more-than-physical energy to which, if we analyse Einstein's ideas, the entire universe is reduced joins up philosophically in the most natural manner with the curved space-time, the conscious divine Totum Simul suggested by relativity theory, because this energy renders modern physics open to a non-materialistic interpretation: the world as a Will at work. On the scientific plane itself, a connection between it and the curved continuum has been attempted. As we saw, what we ordinarily call matter seems to be what space-time holds as curvature of a certain sort. And, if matter is concealed energy, energy would be this curvature interpreted in terms of space and time instead of in terms of space-time, the Totum Simul. It appears to be the World-Will of the Infinite and the Eternal.
6
The second topic is the philosophy implicit in the scientific method established by Einstein. This method falls into two parts. To begin with, there is the principle of rejection of the unobservable. Every statement must be made with reference to what can be observed. Of course, this brings in always the observer, but, as we have already shown, there is no subjectivism here. By observation we mean in physics the procedure of reading off the results produced on scientific measuring instruments by nature's phenomena either as they are or as adapted to particular ends in the laboratory. The observer in Einstein's physics plays the same role as in classical physics. To quote Sullivan: "we must not interpret the word 'observable' too narrowly. It would be more correct to substitute for 'observable' 'definable in terms of physical
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processes'. If an entity is to be considered as a scientific entity we must be able to say what physical processes would enable us to detect it. This is the basis of Einstein's objection to Newton's absolute space and absolute time." That is to say, we know of no physical operations, no experimental techniques, no manipulation of scientific apparatus and instruments by which absolute space and time can be measured or even their existence detected.
By "observable", however, is not meant something of which we" must have direct experimental evidence. Consider the interior of the earth. There are no experiments by which we can observe it. But the absence of observation is due to practical difficulties. We disregard practical difficulties and, on indirect evidence, assume that the earth has an interior. Einstein has no quarrel with an unobservable of this kind. Nor are such quantities as the electron's mass rejected. We do not directly observe an electron's mass, but there are observations from which we infer or deduce this quantity. It is the imperfection of scientific apparatus that keeps the electron's mass away from observation. The unobservable that came under Einstein's censure is not due to imperfection of scientific apparatus or to practical difficulties.
It is due to a special factor which may be called compensation. When a quantity investigated is always and automatically and exactly compensated for by an equal and opposite one, it can never be observed. There seems to be a conspiracy on the part of nature's processes to keep certain quantities for ever beyond observation. If, when an effect x is supposed to be produced on phenomena, we find that there is also produced a countervailing effect -x, all processes of nature appear to be in a league against the observer. How are we to interpret such a perfectly organised conspiracy of compensation? Are we to go on saying that the quantity under inquiry still exists although unobserved? If we do, what utility are we to ascribe to it? Since physics will never come across it, it is for physics an utterly useless and gratuitous supposition and as good as non-existent. It will
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never figure in the observations we make by experiments: so, we must build our equation as if it were not there at all. It may be in itself logically conceivable, but it is not logically admissible in physics; it may be in itself philosophically necessary, but the philosophy of physics can have no place for it.
Not that physics should be confined to the observable actual or potential. What it has to do is, in the first place, no1 to allow a quantity that is in utile by compensation to enter the equations built on the observable, and, in the second place, not to allow such a gratuitous quantity to enter fundamental theory. Fundamental theory is the postulate or set of postulates by which we seek an explanation of observed phenomena: it correlates and unifies them. If compensated quantity enters it, the postulates will never admit real verification: there will always remain in th6 alleged verification a hypothetical and superfluous component. After rejecting the unobservable, the method O1 physics a la Einstein is, therefore, concerned with finding the correct type of fundamental theory beyond observation actual or potential.
Here we strike upon an extremely significant characteristic of Einsteinian science. Although Einstein acts th8 "observer" in essentially the same manner as Newton or Galileo or even Archimedes and imports no special subjectivism into physics, yet when it comes to correlating the data of observation and reaching fundamental theory he worK5 with a radical dissimilarity to the manner of physics in the past: his mind so proceeds as to give consciousness a'"1 entirely new value and to convince us that the path t0 ultimate truth in physics lies not in an effort to arrive at a mere generalisation from the observed world but in a creative flight breaking away from observation. Doubtless, observation cannot be dispensed with: we have to start from it for reaching fundamental theory and we have to return to it in order to test the theory, but our theory is no longer at the mercy of what is observed. The mind is made to act with a
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certain degree of independence of all observation of the world, and in this independence is a hint not only that truth in physics is to be found subjectively but also that the reality at the back of phenomena is of the nature of consciousness.
Perhaps the most concise approach to this hint is in some passages of Einstein's The World as I See It. Writes Einstein on page 180: "The theory of relativity admirably exemplifies the fundamental character of the modern development of theoretical science. The hypotheses with which it starts become steadily more abstract and remote from experience. On the other hand it gets nearer to the grand aim of all science, which is to cover the greatest possible number of empirical facts by logical deduction from the smallest possible number of hypotheses or axioms. Meanwhile the train of thought leading from the axioms to the empirical facts or verifiable consequences gets steadily longer and more subtle. The theoretical scientist is compelled in an increasing degree to be guided by purely mathematical, formal considerations in his search for a theory, because the physical experience of the experimenter cannot lift him into the regions of highest abstraction. The predominantly inductive methods appropriate in the youth of science are giving place to tentative deduction. Such a theoretical structure needs to be very thoroughly elaborated before it can lead to conclusions which can be compared with experience. Here, too, the observed fact is undoubtedly the supreme arbiter; but it cannot pronounce sentence until the wide chasm separating the axioms from the verifiable consequences has been bridged by much intense hard thinking. The theorist has to set about this Herculean task in the clear consciousness that his efforts may only be destined to deal the death-blow to his own theory. The theorist who undertakes such a labour should not be carped at as 'fanciful'; on the contrary, he should be encouraged to give free rein to his fancy, for there is no other way to the goal. His is no idle day-dreaming, but a search for the logically simplest possibilities and their consequences."
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What have we here? An underlining of "purely mathematical, formal considerations" rather than "the predominantly inductive methods appropriate in the youth of science" and a clear realisation that the final formulas are "abstract and remote from experience" and an open admission that the theorist has "to give free rein to his fancy". Surely this is no denial of the principle which rejects the unobservable: theory is not to assume quantities unobservable through compensation but, provided it does not assume them, it can be any kind of mathematical construct, no matter how unfamiliar and unvisualisable in its terms. What makes direct contact with the world known to experiment is not the theory but only the consequences logically deduced from it: the theory itself remains akin to pure mathematics - that is, to structures raised with no immediate practical aim but as mere expressions of imaginable possibilities, it is what Einstein, on pages 135 and 136 of his book, calls a free fiction or free invention or free creation of the mind. Its only difference from the various other structures that can be freely created is that it is not only self-consistent but also makes the fewest possible assumptions from which consequences are to be logically derived for verification by means of scientific apparatus.
Apropos this difference Einstein makes on page 136 a pronouncement which is the most significant in the methodology of modern physics. "If the axiomatic basis of theoretical physics," he says, "cannot be extracted from experience but must be freely invented, can we ever hope to find the right way? Nay more, has this right way an existence outside our illusions? Can we hope to be guided in the right way by experience when there exist theories (such as classical mechanics) which to a large extent do justice to experience, without getting to the root of the matter7 I answer without hesitation that there is, in my opinion, a right way and that we are capable of finding it. Our experience hitherto justifies us in believing that nature is the realisation of the simplest
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conceivable mathematical ideas.' I am convinced that we can discover by means of purely mathematical constructions the concepts and the laws connecting them with each other, which furnish the key to the understanding of natural phenomena. Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it. Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients believed. '
Free creations thus fall into two classes - those that correspond to reality and those that do not. What is the precise significance of the former for a philosophy of physics? First we must note the status which, among ideas, Einstein accords to free creations. Can they be put on a par with what Kant calls a priori ideas? The so-called a priori ideas are those that some other philosophers label as logical generalisations from experience: Kant considers them forms of thought inherent in the mind and imposed by it on the stuff of sensation. When Einstein declares the fundamental concepts of physics to be no generalisations from experience, he sets them outside the class of ideas taken to be a priori. The fundamental concepts, according to Einstein, are not dictated by any necessity arising either from experience or from the mind's inherent forms of thought. Of course, they must have contact both with experience and whatever inherent forms of thought there may be in the mind, but they are still free and found by a creative activity of the mathematical conscious- ness, akin to the activity of the artist.
Being artistic in quality they are reached by a sort of inner vision, a sort of intuition. They are divined. In Einstein's view, the mathematician's mind has a capacity of sheer
' "Simplest" does not mean for Einstein the easiest to conceive or memorise:
as already mentioned in the preceding paragraphs, it means the ideas which, however difficult or complex, are the minimum required to correlate the greatest number of observations.
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insight into reality. The capacity, as far as Einstein knows, is not quite like what the ancients believed - a royal and plenary penetrativeness; it does not carry an absolute self- certainty and must get its final seal from experiment after logical deductions from the axioms found by it have been made, but something it still retains of the self-guidance, the direct grasp, the interior apprehension attributed to pure thought by the ancients. Such a capacity implies, however pin-pointedly, that somehow the mind is able to be one with reality and know it from the inside, as it were, by getting identified with it. The oneness, the inside knowledge by identification, argues for reality itself having a secret nature analogous to the mind.
No wonder Einstein believes in a world-intelligence and regards physics as a search for truth which has at its source a unison between one's mind and the world-intelligence, a pre-established harmony without which neither the search for truth nor the divining of truth can have sufficient explanation. Einstein is a Spinozist, affirming with Spinoza that the ultimate reality is a universal Substance with the dual aspect of mind and matter. He also calls himself a pantheist, with - in his own words - "the firm belief, which is bound up with deep feeling, in a superior mind revealing itself in the world of experience."
Here we may hark back for a moment to Einstein's contention, noticed at the beginning of our essay, that only science can find truth and that what religion finds cannot be given the same name. "'Religious truth,'" he has said, 'conveys nothing clear to me at all." But once intuition is granted a place in knowledge, we should be arbitrary if we ruled out of court various other kinds of intuition than the one which the mathematician practises in physics. Mystical insight is an intense type of intuition and there is no point in denying off-hand that it can both arrive at fundamentals of the universe and perceive their unfoldment in a wider banner than mathematical intuition plus mathematical logic can. What mystical insight would lay bare may not be
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mathematical structure; but that does not invalidate it as knowledge, as discovery of truth, as the formulation of what is.
Spinoza himself does not seem to have depreciated mystical insight as a gateway to knowledge. Einstein's Spinozism is therefore somewhat faulty and narrow. Nor is Spinozism an entirely satisfying philosophy in itself. For one thing, it does not realise that if universal mind and universal matter are dual aspects of one Substance they need not be merely parallel attributes but must be capable of interaction. Again, it puts the two universal aspects on an equality in spite of universal matter seeming to be expressive of universal mind's scheme and purpose and therefore to be its manifestation in a new form. Further, this logical primacy of universal mind argues the one Substance to be an ultimate Existence that is Consciousness. Lastly, Spinozism leans towards an impersonal divinity and does not account for the individual human soul and its supporting truth in a Super- person who is more than the universal existence He has emanated. Yes, the metaphysics of Spinoza, for all its sweep and grandeur, does not go far or high enough. But the general theoretical method of Einstein's physics unequivocally suggests that if Spinozism is to be criticised the criticism must come from above it and not below. A cosmic consciousness into which we have a pin-point entry through mathematical divination is the irreducible minimum this method implies. To make that minimum yield a Spinozistic philosophy is to express a particular type of intellectual and emotional disposition. A deeper and richer Weltanschauung may be extracted from it. So we need not take Einstein's Spinozism as the only possible scientific philosophy to which the free mathematical creativity exemplified by his theorising from the data of relativity is a pointer. But, while something more than Spinozism may be approached, nothing less than it will serve. This means that the general theoretical method of Einstein's physics turns away from materialism in a subjectivist manner foreign to classical physics. Not that
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classical physics knew of no divination, no intuitive leap of the mind, by which a gap in physical knowledge was filled - a leap which necessity could justify neither from experimental evidence nor from inherent forms of thought. Clerk Maxwell made such a leap when in stating his famous equations for electromagnetism he postulated a certain term which was not necessitated by anything at the time and which was found correct by experiment many years later. Without that term wireless and radio and radar would have been impossible. But the Maxwellian leap figured in the method of classical physics as an astonishing freak and had no pervasive significance. Einsteinian theoretical physics, making it the common rule instead of an astonishing freak, acquires an utterly new orientation. Although the attempt to escape materialism by putting a subjectivist interpretation on the role of the observer is misguided, the moment Einstein's physics tries to correlate and unify facts found by observation it stresses a creative and intuitive activity of the mind, by which, from subjective depths within us, a glimmer is brought of the vast subjectivism of a Supreme Spirit who is the single secret self of human observers and of the whole universe and whose consciousness not only pervades but seems to have become all things.
7
We may now briefly take stock of our conclusions from Einstein's relativity physics. By three independent routes we arrive at an undeniable implication of the supra-physical, the mystical: 1) the Einsteinian "field" whose four-dimensional continuum of indistinguishable space and time is revealed by the special theory of relativity as a mathematical approximation of the mystic's Infinity-Eternity and by the general theory of relativity as an utterly non-material space-time ether rendering the approximated Infinity-Eternity all the more real and even originative of matter; 2) the Einsteinian "energy" which, by positing something indefinable by any
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scientific concept, points beyond materialism to a World- Will; 3) the Einsteinian theoretical method with its "free creation", involving the discovery of scientific truth by our mind "insighting" a World-Intelligence that seems all-formative. The independence we have given to each of the three routes results in a threefold strength to the suggestion of the supra-physical and the mystical.
Einstein himself does not appear to be always aware of the direction in which his theory leads. This is because of many reasons. He lacks a full intimate grasp of the relation between science and religion. There is missing also the reading of the true philosophical significance of four-dimensionality. Again, little philosophical endeavour is made to identify the sense of his new concept of energy. Only in connection with his theoretical method he seems to discern a direct liaison between science and the religious frame of mind which, he confesses, can never be absent in the true scientific pioneer. Einstein is content in general to affirm an indirect liaison - a liaison merely of an original stimulus and initiative to scientific research by a pantheistic feeling and outlook. But the fact that he has not himself sounded all the philosophical depths of his own theory is no argument against the existence of those depths. Neither is it an argument against the supremacy of scientific genius that is Einstein's. We should be extremely thankful to this supremacy for providing us, independently of mysticism, with mathematical formulas and processes which we can interpret best in terms of mystical experience.
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