Four Outstanding Errors

The problem to be faced here, as I have said, is that of explaining how it has come about that physical scientists, almost to a man, have for so long allowed themselves to accept a theory that demands of a clock such an obvious impossibility as that it shall work steadily both faster and slower than an exactly similar one. The problem is anything but simple, in either sense of the word; i.e. it is both complex and difficult - though the difficulty, I believe, lies only in the discarding of false notions that have been automatically accepted as true, and not in grasping the actually true ones. It is therefore largely psychological, and, being no psychologist, I can only record what commonsense indicates concerning the various attitudes which physicists have adopted towards criticisms of the theory. Just as I cannot explain why physiologists of distinction rejected Harvey's demonstration, which to us seems so convincing, of the circulation of the blood because it conflicted with what Galen drought in the second century, so I cannot explain why physicists think that calculations which they perform on measurements connected with cosmic rays, which Einstein had never heard of, can answer my question why he felt entitled to conclude from his theory that an equatorial clock worked slower, and not faster, than a polar one. I can only record it as a fact, and do my best to analyse the course of events in physics that led to the fact.

Before I come to the historical events, however, I think it will be helpful to pinpoint four basic misunderstandings, an awareness of which will make it easier to understand why special relativity has been accepted for so long in spite of its clear untenability. They are concerned with first, the relation between mathematics and physics; second the confusion of different meanings of the word 'time': thirds the misinterpretation of 'co-ordinate systems' as 'observers'; and fourth, what I can best describe briefly as the literal interpretation of metaphors - the acceptance, as direct observations, of what are actually remote implications of possibly erroneous theories.

I begin, then, with the most dangerous of all, that pervades not only this subject but the whole of modern physics - the false conception of the relation between mathematics and physics (or, more generally, experience; but the experiences, or observations, dealt with in physics are those which specially concern us here). For, though our problem has two distinct aspects first, why do the 'mathematicians' tell the 'experimenters' to believe such absurdities? second, why do the 'experimenters' believe them? - the first is the basic one.

There is a vitally important distinction between mathematics, which belongs wholly to the realm of pure thought, and physics, which belongs wholly to the realm of experience; and these two elements of our general awareness are (at least at the level with which science is concerned; I make no assertion at all concerning philosophies which may attempt to relate them fundamentally) wholly independent of one another. We can, however, use mathematics in the service of science because our aim in physics is to relate together experiences which at first seem unrelated, and by limiting our considerations to the experiences revealed by measurements, as we do in physics (though science in general, of course, goes beyond measurement), we may express them in terms of numbers, or symbols representing numbers, and then apply our mathematical knowledge to those symbols, regarded now as elements with which our mathematical theorems deal. Let me try to elaborate this a little.

Mathematics in itself, as I say, is independent of experience. It begins with the free choice of symbols, to which are freely assigned properties, and it then proceeds to deduce the necessary rational implications of those properties. Thus, the symbols may be straight lines, circles, etc., having properties such as those described by Euclid, or other similar ones, and their consequent relations form the mathematical discipline of geometry, euclidean or otherwise. If the symbols are numbers, their relations, according to assigned rules of addition, multiplication, etc., constitute arithmetic. If the symbols are letters, with rather more extended properties, we have the discipline of algebra, and so on. There is nothing of experience in this: the symbols and properties may be chosen arbitrarily, and all that is required of or implied by the resulting corpus of theorems is that it conforms faithfully to them.

Science is the attempt to find relations between experiences. Some such relations are obvious; e.g. that the fruits of an apple-tree are like each other and unlike those of a pear-tree. Others, such as the relation between the rate of fall of an apple and the motion of the Moon, are discovered only by careful research. It is here that mathematics is of such service to science. If we try in general to find a relation between the apple and the Moon, we shall fail; they seem wholly independent of one another. But by making measurements related to the two things, we arrive at numbers, and these are things with which mathematics deals. We can accordingly apply the relations between numbers discovered by mathematical research, to the measurements related to the apple and the Moon, and discover a parallelism which enables us to say that our experiences of the motions of the apple and the Moon are related to one another in a way not at all predictable or discoverable without the use of mathematics.

By extending this over as wide a range of measurements as possible, we reach the vast body of related experiences that constitutes modern physics. Those experiences whose properties are paralleled by the properties of me symbols in a particular existing mathematical discipline are naturally selected for special study, since we can at once see that they will be related in accordance with me theorems of that discipline. Thus, having ascertained mat the experience of counting certain objects is a process, which can be represented by numbers; we can apply the abstract truths of arithmetic to the corresponding operations with those objects. We can conclude, for example, that if, to a box of apples, we add three and then subtract two, we shall be left with the same number as if we had first subtracted two and then added three; we shall not need to perform the actual operations to assure ourselves of this. Similarly, if we add 3,521 apples to 765, we shall not need to count the total to know that it is 4,286; we can determine that in a few seconds with pencil and paper. The whole relation between mathematics and physics is of this character. Mathematical structures of thought are built up by pure reason applied to arbitrary axioms, and men, having found a realm of experience mat is paralleled by one such structure] we can use the parallelism to extend our knowledge of physical relations over the whole field of experience to which it applies.

But now certain questions arise. How do we know what branch of mathematics applies to a particular realm of experience? How do we know that a parallelism which we have discovered in a limited range of experiences of a certain kind will be valid over the whole range of the mathematical structure on the one hand and the whole range of experiences of me same kind on the other? How do we know that a particular branch of mathematics will have any corresponding possibilities in experience? The answer to all these questions is that we can know these things only by experience itself, i.e. by trial and error. Since, as I shall maintain and have adumbrated in Chapter 1, it is the oversight of this fact, and the illegitimate assumption that there is some necessity for whatever is true in mathematics to impose its inevitability on experience, that is primarily responsible for the error of special relativity, I should like to quote a passage from Einstein which shows that, as a general proposition at least, he was well aware of this contingent element in the relation between mathematics and experience. In a letter to the late Viscount Samuel, published with his consent in the latter's book, Essay in Physics, in the German original and in English translation, he wrote this:

For example, Euclidean geometry, considered as a mathematical system, is a mere play with empty concepts (straight lines, planes, points, etc., are mere 'fancies'). If, however, one adds that the straight line be replaced by a rigid rod, geometry is transformed into a physical theory. A theorem, like that of Pythagoras, then gains a reference to reality. On the other hand, the simple correlation of Euclidean geometry is being lost, if one notices that the rods, which are empirically at our disposal, are not 'rigid'. But does this fact reveal Euclidean geometry to be a mere fancy? No, a rather complicated sort of co-ordination exists between geometrical theorems and rods (or, generally speaking, the external world) which takes into account elasticity, thermic expansion, etc. Thereby geometry regains physical significance. Geometry may be true or false, according to its ability to establish correct and verifiable relations between our experiences.1

What I believe to be the basic misconception of modern mathematical physicists evident, as I say, not only in this problem but conspicuously so throughout the welter of wild speculations concerning cosmology and other departments of physical science is the idea that everything that is mathematically true must have a physical counterpart; and not only so, but must have the particular physical counterpart that happens to accord with the theory that the mathematician wishes to advocate. I have already (Chapter 1) given some examples of this, associated with the greater generality of mathematics as compared with physics; here I wish to show some other aspects of the same fundamental misconception. It is seen easily enough in some very simple examples where it is so obvious that no one could possibly make the error in question; nevertheless, that error is made almost automatically when we get into realms unfamiliar in ordinary experience.

Take, for example, the simple mathematical equation, 1+1=2. Over a wide range this holds good in experience as well as in mathematics, where it is always true. If we have one penny, and someone adds another to our wealth, we have two pennies. If we have one apple and add to it another and count the total, we find that it is two apples. And so on. We may therefore be inclined to generalise, and say that if we add one anything to another of the same thing, we have two of those things; in other words, x+x=2x, whatever x may be. But this is far from the truth. If we add one water drop to one water drop we get not two water drops but one larger drop. If we add one rabbit to one rabbit we may get a continent of rabbits. Even believers in special relativity will assert that if we add a velocity of 1 foot a second to a velocity of 1 foot a second we get a velocity slightly less than 2 feet a second. If we add one idea to one idea we may get a philosophy. If we add one commandment to one commandment we get all the law and the prophets. Browning wrote that it was only in music that one note added to two notes made a star, but in fact experience abounds in that kind of addition. Similarly, although adding three apples and subtracting two gives the same result as subtracting two and adding three, it is certainly not true that adding cocaine to a gum and subtracting a tooth gives the same result as subtracting the tooth and adding the cocaine.

So it is with other operations of mathematics. In algebra, if a = b, then 2a=2b. This was applied in the Middle Ages to prove the immortality of the soul. To be half dead was the same as to be half alive: double both, and it follows that to be dead is to be alive. This particular argument would carry little weight now, but equally naive applications to experience of mathematical truths do flourish. Not long ago the mathematical fact that log 1 = 0 was applied to prove that there was no difference between something and nothing. The late Professor E. A. Milne proposed a theory called kinematical relativity2, according to which it was equally legitimate to represent the measurement of time by a certain symbol and by its logarithm. It was a short step from this to the conclusion that the question whether a distant nebula was moving rapidly away from us or remaining at the same distance was a 'no-question'; the two processes, were the same, since the only difference lay in our free choice of the way of measuring time, and we could equally well measure it directly or logarithmically. When it was pointed out that, if this were true, the principle could be applied equally well to a stone that is thrown at you, so that whether you would experience the impact or not would depend on what kind of watch you carried, Milne refused to consider the physical application of the mathematics. On one scale of time the stone hit you a few seconds after being thrown; on the other an infinite time would elapse; this was mathematically certain, and therefore the two cases were equivalent.

I attach no weight whatever to the verbal descriptions I have occasionally attempted [he wrote]: the core of the matter lies in the mathematics.... It is no use objecting to the results themselves; the critics should find flaws in the trains of mathematical deduction.... Until then, I have nothing to add to my constructive papers.'

This theory receives little attention now, though its obituary notices speak of it with great respect4, but its neglect in application appears to be due only to the fact that other theories, equally unrelated to experience, have superseded it - notably the so-called 'steady state' theory of the universe, according to which, because a certain tensor (a mathematical quantity) can give values for another mathematical quantity which changes from o to 1 when a third called t increases, matter is continually being created out of nothing. There is not the slightest physical evidence for this, or for anything like it; there is only the fact that, in another connection, other tensors can be associated in a reasonable way with other physical quantities which can be observed. In other words, the argument has the same validity as this - that because two water drops when brought together coalesce into one, so two apples similarly treated will coalesce into one - except that we can so treat the apples, and the processes postulated in the steady state theory are wholly imaginary.

Examples could be multiplied ad lib, but I think these are enough to show how general and how dangerous is the prevailing illusion that all that is necessary to entitle a physical theory, however absurd, to respect is to discover some mathematical process whose symbols can be arbitrarily correlated with the physical entities of the theory, without regard to evidence or probability or commonsense. We shall see in due course that the supposed justification of special relativity by the 'mathematicians', to whom the 'experimenters' entrust it, lies wholly in the impeccability of its mathematical structure: the impossibility of the application to experience of that structure, in the manner postulated by the theory, is left out of consideration altogether, just as the fact that the stone hits you within five seconds is left out of account in assessing the credentials of kinematical relativity: the mathematics shows that if you measure time differently the moment of impact is infinitely far ahead, so you have no cause for alarm. Similarly, in special relativity mathematics proves that one clock goes both faster and slower than another; therefore it must do so.

I cannot leave this subject without bringing to attention an aspect of it, which has very serious general implications. I think it is impossible for anyone who reflects on the few examples I have given, and realises that they are not exceptional in their general character but typical of most mathematical physics of the present day, to doubt that, as a general rule, the practice of mathematical physics goes hand in hand with lack of elementary reasoning power and of that normal form of human wisdom, somewhat misleadingly called commonsensc, that provides its own corrective of premature judgement and never allows the requirements of reason and experience to be overcome by the seductions of attractive speculations. I repeat that I am no psychologist, and it is with diffidence that I admit an unwillingness to conclude that this is an inescapable psychological necessity; it is more comforting to hope that it denotes a failure of our educational system to recognise an ever-present danger and to take precautions against it. It is usually taken for granted that the processes of mathematics are identical with the processes of reasoning, whereas they are quite different. The mathematician is more akin to a spider than to a civil engineer, to a chess player than to one endowed with exceptional critical power. The faculty by which a chess expert intuitively sees the possibilities that lie in a particular configuration of pieces on the board is paralleled by that which shows the mathematician the much more general possibilities latent in an array of symbols. He proceeds automatically and faultlessly to bring them to light, but his subsequent correlation of his symbols with facts of experience, which has nothing to do with his special gift, is anything but faultless, and is only too often of the same nature as Lewis Carroll's correlation of his pieces with the Red Knight and the White Queen - with the difference that whereas Dodgson recognised the products of his imagination to be wholly fanciful, the modern mathematician imagines, and persuades others, that he is discovering the secrets of nature.

The processes of mathematics are to be contrasted rather than identified with the process of rational drought. Professor A. N. Whitehead, himself an accomplished mathematician, long ago recognised this truth before the unawareness of it became the ominous danger that it now is.

By the aid of symbolism [he wrote] we can make transitions in reasoning almost mechanically by the eye, which otherwise would call into play the higher faculties of the brain. It is a profoundly erroneous truism, repeated by all copybooks and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilisation advances by extending the number of important operations, which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle - they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.'

If this is a true diagnosis and I think facts available since Whitehead wrote these words unmistakably confirm it mathematical ability and ability to conduct operations of thought are distinct faculties, and although I know of no reason why they should not co-exist in the same person, it is only too clear that at the present time, except in a rare instance, they do not. The danger that the cultivation of the former should cripple the latter is thus so real that positive steps should be taken to counteract it. In fact, however, our methods of education augment it. In the application of mathematics to physics the results of this are shown in such examples as those which I have given, and in many others.

Of more general concern is the possibility that the movement to introduce scientists (in particular, mathematical physicists) into the government of the country may receive support based on ignorance of the realities of the matter. What Whitehead did not foresee (he wrote before the first world war) was that civilisation would soon be concerned less with advancing administratively than with ensuring its own survival, so that operations of thought would need to be the rule rather than the exception. Government now demands, above all things after moral rectitude, intelligent thought, and it must be recognised that mathematical physicists are, of all our citizens, the least fitted to provide it. To the scientist, writes Professor (now Sir) Fred Hoyle, war starts because human behaviour is representable in terms of mathematical equations possessing discontinuous solutions.6 This must not be dismissed as a humorous wisecrack: Hoyle, and others of his type, really believe that this is so. They were not necessarily born with a deficiency of commonsense: they have exceptional mathematical ability, which has been mistaken for exceptional intelligence, and have been so trained that their normal intelligence has expired through desuetude; much mathematics hath made them - what they are. They are now no more able to perceive the advantages of intelligence than the blind men in H. G. Wells's story could perceive the advantages of sight.

Let us, however, return to our present concern. The circumstances in which special relativity came to birth and acquired its present almost impregnable position in physics will be recounted later, but it will be appropriate here to give, in the barest outline, an account of the way in which the ground was gradually prepared for such an event to occur. As we have seen, Galileo - who, more than any other single person, can be regarded as the originator of modern physical science - claimed that the book of nature was written in the mathematical language. He meant exactly what he said: mathematics was a language, a means of expressing something, not the thing that it was important to express. Both Galileo and Newton took experiments or observations as their starling-point, and used mathematics only as a tool to extract the maximum amount of knowledge from the experiments and as a means of expressing that knowledge. In the later developments of their work throughout the eighteenth and nineteenth centuries in the field of mechanics, this relation between experience and mathematics was maintained. Although Adams and Leverrier 'discovered' Neptune by mathematical calculations, it was Galle's observation of the planet that was rightly regarded as its discovery: had he looked and found nothing, the work of Adams and Leverrier would have been forgotten. In chemistry also it was observations that determined the ideas held. When, in order to explain certain observations, it was suggested that phlogiston had a negative weight, that was not because the symbol w in mathematics could be given a negative as well as a positive sign; it was because, in terms of the phlogiston theory, the observations required it. (This may be contrasted with a recent cosmological theory in which it has been proposed that the 'pressure' in the universe -a conception to which it is impossible to assign any physical meaning other than a positive one - is negative, solely because the mathematical symbol p can be given a positive or a negative sign). In electricity and magnetism, of the two great pioneers in the first half of the nineteenth century, Faraday was almost completely innocent of any mathematical knowledge, while Ampere, equally proficient in experiment and mathematics, invariably used mathematics to interpret the results of his experiments and never to dictate them.

Of course, there are numerous instances in which mathematics has suggested physical possibilities which have later been realised and might not have been thought of without the suggestion; its service in this respect can hardly be over-estimated. An outstanding instance is the re-naming of 'vitreous' and 'resinous' electricity as 'positive' and 'negative', and the subsequent deduction of their interactions on the supposition that they accord with those of the mathematical signs bearing those names - although it may be that we are in danger of forgetting that the correlation is ultimately empirical and may well break down in certain extreme cases. But, so far as I have been able to discover, the first serious example of the mastery, instead of the servitude, of mathematics in relation to physics came with Maxwell's theory of the electromagnetic field7, and that, as would be expected, only in a very tentative way and not without resistance. In brief. Maxwell showed that Ampere's law in electromagnetism, expressed mathematically - which, of course, as I have said, was a mathematical expression of results found by experiment - did not satisfy the equation of continuity but could be made to do so by a purely mathematical modification. Accordingly he assumed that this modified form was the actual physical law. But he was too conscious of the true relation between mathematics and physics not to be aware that this was quite unjustifiable unless an actual physical relation existed which was represented by his proposed equation; and since he knew of none, he made the assumption that what he called a 'displacement current' existed in a dielectric. The physical feature that distinguishes a dielectric from a conductor is that the latter, but not the former, can convey an electric current, so this was quite inadmissible on observational grounds, but Maxwell assumed that a 'displacement' of electricity could occur in a dielectric, which had the same physical effects, so far as these were required by the equations, as a current in a conductor. If that were so he could proceed to build up a general 'dynamical theory of the electromagnetic field' which gave an electromagnetic interpretation of light and was so beautiful and comprehensive that he could not refrain from postulating the actual physical reality of his 'displacement current' as a justification for his mathematics.

At that time, when the true relation of mathematics to physics was still a prevalent influence in science, this highly artificial conception naturally aroused strong opposition, though what is most significant from our point of view is the clear indication it gives of the need Maxwell felt to provide some physical basis for his mathematics. Among the foremost opponents of this way of doing it - i.e. of the invention of unobserved 'phenomena' to suit the mathematics instead of adapting the mathematics to what was observed was Lord Kelvin, who said bluntly, 'I want to understand light as well as I can without introducing things that we understand less of.'8 Other physicists had similar feelings, and even Hertz who was instrumental, through his experimental discovery of electromagnetic waves, in establishing Maxwell's theory, notwithstanding its mystical character, as the fundamental truth about electromagnetism - wrote:8 'Many a man has thrown himself with zeal, into the study of Maxwell's work, and even when he has not stumbled upon unwonted mathematical difficulties, has nevertheless been compelled to abandon the hope of forming for himself an altogether consistent conception of Maxwell's ideas. I have fared no better myself.' He proceeded to describe three representations of Maxwell's theory, and went on: ' I shall thus have an opportunity of stating wherein lies, in my opinion, the especial difficulty of Maxwell's own representation. I cannot agree with the oft-stated opinion that this difficulty is of a mathematical nature.' He sums up: 'To the question, "what is Maxwell's theory?" I know of no shorter or more definite answer than the following: - Maxwell's theory is Maxwell's system of equations.'

Nothing could more clearly express the change that had come over physics. Experiments more and more confirmed the deductions that were made from the theory when the symbols in the equations were given certain physical meanings, while the justification for giving the symbols those meanings continued to elude everyone. Lorentz generalised Maxwell's theory to make it apply to moving as well as static systems - we shall come to this later - and, all unconsciously, a state of mind was generated in physicists by which, while still formally adhering to the principle that observation was basic and mathematics a useful tool, they were ready to accept mathematical requirements as an adequate substitute for a genuine theory, even though they could see nothing intelligible that corresponded to it physically. It was a short step from acceptance of the physically unintelligible to the physically absurd, but the description of this must be postponed until we come to the origin of the special relativity theory itself. In the meantime I hope it has been made clear how the atmosphere of the time had become propitious for the advent of a theory that in earlier days would have been dismissed without a second thought. Its survival, thus made possible, was rendered almost inevitable by the actual sequence of historical events, but of this anon.

(This will be a convenientplace to interpolate a note on a general misunderstanding concerning the relation of Maxwell's to Faraday's ideas for which Maxwell himself is ultimately responsible but which would have been noticed and corrected long ago if writers on physics had cultivated the habit of reading original papers instead of relying solely on second-hand accounts of them. We shall, alas, meet with other misunderstandings of the same kind. Maxwell begins his 1865 paper7, in which his theory of the electromagnetic field is set forth, by establishing the existence of the ether on which the whole of what follows is to be based.

The theory I propose [he writes] assumes that in that space [space in the neighbourhood of the electric or magnetic bodies] there is matter in motion by which the observed electromagnetic phenomena are produced... We may therefore receive, as a datum from a branch of science independent of that with which we have to deal, the existence of a pervading medium, of small but real density, capable of being set in motion, and of transmitting motion from one part to another with great, but not infinite, velocity.

Later in the paper he writes:

The conception of the propagation of transverse magnetic disturbances to the exclusion of normal ones is distinctly set forth by Professor Faraday (Phil. Mag., May 1846) in his 'Thoughts on Ray Vibrations'. The electromagnetic theory of light as proposed by him is the same in substance as that which I have begun to develop in this paper, except that in 1846 there were no data to calculate the velocity of propagation.

A reference, however, to Faraday's 'Thoughts on Ray Vibrations' shows that his idea was quite different. He proposed, in fact, 'to dismiss the ether' which was the basis of Maxwell's theory, and to endow each elementary source from which light was emitted with a system of rays, extending indefinitely in all directions, the vibrations on which constitute light. The difference may be of fundamental importance, for Einstein's special relativity theory, designed to save Maxwell's equations, could do so only by sacrificing the ether which was the basis of Maxwell's theory. Had Einstein first sought to bring Maxwell's equations into conformity with Faraday's idea which amounted to a separate 'ether' for every atom instead of a single universal ether which could serve as an absolute standard of rest he might have produced a theory not subject to the fatal defects of the one he did produce. This is a task, which might well repay the effort it would demand).

I pass now to the second of the general considerations previously mentioned, that help us to understand the confusion and misconceptions that surround the word 'relativity': it is the multiplicity of meanings associated with the word 'time'. If it is the misconception of the relation between mathematics and physics that is chiefly responsible for the belief that special relativity is physically true, this second confusion is chiefly responsible for the widespread conviction that it is an esoteric and difficult subject and indeed anything but a rather simple physical theory of the well-understood traditional kind. In dealing with words I must, of course, restrict my comments to one of the numerous languages used in the literature of relativity. That must perforce be English, though reference to many German papers shows that what I have to say about 'time' applies also to the German word 'Zeit', though not every detail is necessarily applicable. The basic confusion, however, exists independently of the language in which it is expressed. Consider a journey. We may say of it the following three things:


(a) The journey occurred in time.

(b) The time of starting was 1 o'clock.

(c) The time occupied by the journey was 2 hours.

The same word, time, is used here in three quite different senses, as may be seen by considering the corresponding statements about space:


(a) The journey occurred in space.

(b) The place of starting was London.

(c) The length of (or distance covered by) the journey was 60 miles.

Here we use three different words - space, place, length (or distance), none of that could be substituted for either of the others without depriving the sentence of meaning. The same distinctions, thus brought to light, exist in the set I, but they are obscured by the use of the same word, 'time', for three quite different ideas.

To distinguish the three meanings of 'time' I will re-express the set I in the following not unnatural ways:


(a) The journey occurred in eternity.

(b) The instant of starting was 1 o'clock.

(c) The duration of the journey was 2 hours.

(In using the word 'eternity' I wish to imply nothing concerning the philosophical problems associated with everlastingness, eternal recurrence, etc., which the word often suggests. I mean by it only what everyone understands by 'time' in the well-known lines

Time, like an ever-rolling stream,

Bears all its sons away).

Now what will probably surprise many readers is that Einstein's special relativity theory, as he expounded it in his 1905 paper, has nothing at all to do with time in the sense of 'eternity'; it is concerned only with instants and durations (as intervals between instants). This fact - for it is an unmistakably verifiable fact - has an importance that can hardly be exaggerated, because one of the chief factors - probably the chief factor - in creating the illusion that relativity is unintelligible, or even difficult, is the notion that it has something to say, and something quite unimaginable to say, about the nature of 'time', of the continuum that St. Augustine and Kant and other philosophers have puzzled themselves about. In fact, time, the ever-rolling stream, has no more to do with the existence of clocks than with that of sausages, while time, in Einstein's theory as in physics generally, means only clock-readings. It is because of this confusion that the 'experimenters' have left relativity to the 'mathematicians'. Their concern has been only with what can be the subject of observations, and 'eternity' cannot be observed; it can only be thought about, and the 'experimenters' leave that kind of thought to mathematicians and philosophers. These draw deductions about 'eternity', and pass them on to the 'experimenters' as relating to 'instants' and 'durations'. They are accepted as such, without understanding but with blind trust. The reader may foresee what will ensue, if this process is allowed to continue.

The achievements of physics - the establishment of relations between measurements of various kinds are summed up in a number of equations, in which the symbol t occurs with great frequency, but never with the meaning of 'eternity'; it always means an 'instant' (i.e. directly or indirectly a clock-reading) and t2 - t1 means a 'duration'. Whenever a general physical formula has to be applied to any particular case, t means the instant of occurrence of some event, and, so far as physics is concerned, the idea of time as a featureless continuum, infinitely or finitely extended, has no more significance than the idea of a featureless continuum of mass or electric charge or any other physical quantity represented by a symbol in an equation. This was recognised perfectly clearly by Einstein (or, rather, was so obvious to him that he was not consciously aware that he was taking it for granted) when he formulated his special theory in 1905.

The theory to be developed [he wrote] is based - like all electrodynamics - on the kinematics of the rigid body, since the assertions of any such theory have to do with the relationships between rigid bodies (systems of co-ordinates), clocks, and electromagnetic processes,

and nowhere in the development of the theory does any interpretation of the word 'time', other than that of instant or duration, appear at all. It was Minkowski who later took the fatal step of introducing 'eternity' into the theory, as we shall see in due course.

When once the distinction between eternity, instant and duration is recognised, the general literature of the subject of relativity is seen to be in utter confusion. The writer, quite unaware that the word 'time' has different meanings, unconsciously oscillates between them, and the reader, equally unconsciously, becomes the victim of one non sequitur after another, in which he can see no failure of reasoning but yet no possibility of making sense of the conclusion: thus is generated the illusion that relativity is incomprehensible to the ordinary mind.

Take, for example, Eddington's standard work, The Mathematical Theory of Relativity 10 In the first chapter, after some general remarks about 'eternity' (called 'time', of course), he remarks that 'in the mass of experimental knowledge which has accumulated, the words time and space refer to one of the "fictitious" times and spaces' i.e. he is asserting that experimental knowledge refers to 'eternity'. But almost immediately he goes on to say that our experimental knowledge is concerned only with 'durations', not with 'eternity' or even with 'instants'. Expressing himself now in terms of space, though it is clear that he intends his remarks to apply to time also, he says:

in our common outlook the idea of position or location seems to be fundamental. From it we derive distance or extension as a subsidiary notion... The view which we are going to adopt reverses this. Extension (distance, interval) is now fundamental... Any idea contained in the concept location which is not expressible by reference to distances from other objects, must be dismissed from our minds... Accordingly our fundamental hypothesis is that - Everything connected with location which enters into observational knowledge - everything we can know about the configuration of events - is contained in a relation of extension between pairs of events.

So, within a page, the content of our 'experimental knowledge' or 'observational knowledge' has switched from 'eternity' and 'space' to 'durations' and 'distances' without a word of apology. But it does not last. Time, in the sense of 'eternity', returns and disappears apparently without rhyme or reason, until on p. 166 we read this:

Events before t = - ? may produce consequences in the neighbourhood of the observer and he might even see them happening through a powerful telescope.

So presumably something that 'enters into observational knowledge' may be assigned to an instant before 'eternity' begins, and so is neither in time nor in a relation of extension between pairs of events. Such is the confusion that abounds in relativity literature, and arises without limit from the use of the word 'time' to denote quite different things.

An example very relevant to the main purpose of this book is afforded by Synge's reply to my criticism (pp. 76-7). He agrees that special relativity is incompatible with my concept of a regularlyrunning clock and that one of them must be abandoned. He chooses to abandon my concept of a regularly-running clock because it is 'equivalent to Newton's concept of absolute time'. But, like Einstein's special relativity theory, it is altogether independent of any concept of absolute time, whether Newton's or another's. What Synge here calls 'Newton's concept of absolute time' can refer only to a concept of what I have called 'eternity'; my concept (or anybody else's for that matter) of a regularly running clock is a concept of an instrument that marks 'instants' and measures 'durations', and these imply no particular concept at all of 'eternity'.

To sum up, then, the whole of Einstein's special theory, as set out in his paper of 1905 which is still generally acknowledged to be its canonical expression, is concerned with concrete, observable things - clocks, instants, durations, distances, events; it is totally independent of all conceptions of the nature of space and 'eternity'. It treats of the relations between observable things in different 'coordinate systems'; i.e., apart from trivial differences, it deals with the values, which those things take when the observable physical system under consideration is regarded as having different states of uniform motion. That is a problem which had been considered for centuries and regarded as solved until an ambiguity arose when it was found that the relations accepted with the events treated in mechanics were incompatible with those which seemed to be demanded with the events treated in electromagnetism. Einstein's theory was designed to provide a relation that held for both kinds of events. It was wholly physical, and concerned wholly with a problem of the traditional kind, involving only traditional concepts. We shall see later how, through the delayed action of Minkowski's metaphysical interpretation of his own mathematics, it came to be enveloped in a metaphysical cloak that had nothing whatever to do with its essence.

The third of the four most prominent sources of confusion that have led to the general illusion concerning special relativity is the substitution of 'observers' for 'co-ordinate systems'. In the literature of relativity there is almost invariably a great deal about 'the observer', and statements about what different observers, in different states of motion, will observe; and the impression is given that this is an essential, if not the essential, feature of the theory. Indeed, the late Professor Milne based his whole conception of relativity on a comparison of the experiences of such observers; he declared that relativity was 'a complete denial of the solipsist position'2 - a position with which it has no more to do than with bimetallism. That, however, is an extreme case, but writers generally have been prone to elevate 'the observer' from a convenient accessory in the task of explaining the real essence of the theory, into part of the essence itself. This distortion has misled not only the general reader, but many specialists in the subject as well.

The fact is that the observer is concerned in relativity no more and no less than in any other department of science - or perhaps it would be truer to say that he is concerned less in relativity than in most other departments. For example, if we are calculating the circumstances of an eclipse of the Sun, what the observer will see will depend very much on where he happens to be, and it is generally not difficult to choose one's station so as to observe the particular aspect of the eclipse in which one is interested. But in special relativity theory, the observers whom it is generally considered worth while to compare are those whose relative motion is very great indeed - far greater than anyone has yet managed to make possible. Apart, therefore, from the needs of science fiction, we can leave the observer out of our account of the theory altogether.

Indeed, a moment's thought will show that this must be so. All science is based on observation, and whatever we say about the world studied in science must justify itself ultimately in terms of what we actually observe, not of what we infer that hypothetical observers would experience in circumstances impossible yet to attain. Now effectively, in all matters with which special relativity is concerned, there is only one observer - a terrestrial one - for the relative motions possible to terrestial observers are so small as to be negligible in this connection. Hence the theory must be wholly expressible in terms of the experiences of that one observer alone. Why, then, does the observer figure so prominently in expositions of the theory? It is simply because he has been falsely identified with a co-ordinate system. Now a co-ordinate system apart from characteristics, which are trivial here and may be ignored - is simply a state of motion. A person in a train travelling at 60 miles an hour through a station is said to be at rest in a co-ordinate system which is moving at 60 miles an hour with respect to a coordinate system in which a person standing on the station platform is said to be at rest; and, conversely, the latter person is said to be at rest in a co-ordinate system moving at 60 miles an hour (but in the opposite direction) with respect to the co-ordinate system in which the former is at rest. Their observations of the surrounding landscape will, of course, be very different, in ways with which we are quite familiar, but this has nothing to do with the special relativity theory: so far as that theory is concerned there are no differences at all in their observations, for it is an essential feature of the theory that either of the two observers has the same right as the other to say that he is at rest and the other moving. In other words, no observations are possible that would entitle either person to claim a state of rest (or, indeed, any particular state of uniform motion at all) for himself, that are not available for the other person to make the same claim for himself.

It follows that all phenomena generally associated with relativity - relative contraction of rods, relative slowing down of clocks, etc. are not matters of observation but are wholly concerned with co-ordinate systems, and the essential difference between an observer and a co-ordinate system is that the same observer (and we have seen that there is effectively only one observer In the universe on whose observations all the science we have yet achieved can be based) can choose any of an infinite number of co-ordinate systems that he pleases (provided that in special, though the limitation does not exist in general, relativity their relative motion is uniform). The observer on the station is not bound to suppose himself to be at rest; he can suppose the train to be at rest and himself (with, of course, the surrounding landscape) to be moving at 60 miles an hour. The observer in the train has an equal right to suppose whatever motion he pleases for himself. Obviously, nothing whatever that either will observe will be changed if he changes his mind about his state of motion - i.e. if he changes his co-ordinate system. It is, of course, usually convenient for both observers in this case to make the same choice that of the co-ordinate system in which the station is at rest and the train moving (though I remember a cartoon, in the days when relativity was a popular sensation, in which a passenger calls out 'Hi, guard, does Manchester stop at this train?'), but that is incidental. In other circumstances we freely change our co-ordinate system as we change the problem under consideration. In laboratory experiments we usually choose a system in which the Earth is at rest. In dynamical astronomy we choose one in which the Sun is at rest and the Earth moving at 18 1/2 miles a second. In stellar astronomy we choose one in which the Earth is moving round the Galaxy at a few hundred miles a second. And so on. Clearly, nothing whatever that we observe is changed by our change of mind.

In Einstein's basic paper on the theory 'the observer' is not mentioned after the first two short sections right up to the end of the description of the theory. In those two sections Einstein is clearly preparing the ground for the serious business of the theory, for he was well aware that, at the time of writing, he was introducing ideas at variance with what had up to then been taken for granted, and something in the nature of a picturesque account was necessary. But what he regards as the theory proper starts at section 3, which is entitled 'Theory of the Transformation of Co-ordinates and Times from a Stationary System to another System in Uniform Motion of Translation Relatively to the Former'. In everything that follows, down to the conclusion: 'We have now deduced the requisite laws of the theory of kinematics corresponding to our two principles, and we proceed to show their application to electrodynamics', there is no reference at all to the observer; it is all concerned with the change of values which a single physicist must make in the coordinates he assigns to objective events when he decides to change his co-ordinate system. What he observes will change no more than what we observe will change when we stop thinking of ourselves as resting in bed and reflect that we are moving round the Sun. The changes in the co-ordinates, however, will be different from those believed to be necessary before the theory was devised, and it is on these differences alone that the theory must be judged.

The last of the errors mentioned as permeating the literature and the general appraisement of relativity has been, I think, the most effective in persuading the 'experimenters' that the theory must be right, notwithstanding their inability to make sense of it. I have described it as the literal interpretation of metaphors, and I can best illustrate it by a particular example. I take the earliest of all the supposed experimental verifications of special relativity -that which is described as the increase of mass of a body with velocity - not only because it is perhaps the simplest example of what is at best an extremely complex matter, but also because it serves the additional purpose of exemplifying, quite indubitably, the general oversight of the fact that all the supposed experimental verifications of special relativity can with exactly the same justification be advanced as verifications of Lorentz's earlier and quite different theory which is described in Chapter 8. This is so because both theories have the same mathematical structure and give indistinguishable physical interpretations to the symbols involved so far as the experiments so far performed are concerned, though there are quite irreconcilable differences of interpretation between which it is not yet possible to decide or at least between which no existing experimental knowledge can decide, though I think the necessary knowledge would be obtained readily enough if there were a sufficient appreciation of the fact that special relativity is still possibly open to doubt. The compatibility of the mass/velocity relation with Lorentz's theory is indubitable because Lorentz himself cited it, and shown to agree with observations already made, before Einstein's theory was published.

What do we mean when we speak of the mass of a body - a lump of lead, for example? We mean that if we place it in one pan of a balance, and the pointer rests in the central position when a weight of 10 lb. is placed in the other pan, the mass of the lump of lead is lolb. But what do we mean when we speak of the mass of an electron? We certainly do not put an electron in a balance pan and compare it with weights in the other pan. We could not do so because not only can we not capture an electron but also we do not know what it is. A hundred years ago the word denoted a rather vaguely conceived unit of electricity of unknown character. By the end of the nineteenth century it seemed to have been definitely revealed as a particle of negative electricity with measurable properties of the kind familiar in ordinary matter, but thirty years later it was found to possess undeniably wave-like characteristics. The idea then arose that it was a sort of mist of electricity, and Eddington probably gave it the most candid description as 'something unknown doing we don't know what'. We are no wiser today; nevertheless, we speak of the mass of an electron as though it was equivalent to the mass of a lump of lead.

Not only so, but we give it a fairly precise value about which there is no disagreement: how on earth do we reach that value? To explain that completely (i.e. to state first, in their bare essence, the actual observations made, and then the full course of the reasoning by which it is inferred from them that the mass of the electron is such and such) I should have to write a large text-book of physics. This, of course, is out of the question here, but two things can be said about it with absolute certainty, and no one would dream of disputing them: first, that nowhere, in the whole description, would there appear any comparison of the electron with standard weights in a balance; second, that whatever choice might be made of the basic operations and the various ways of reaching the conclusion from them, it would be impossible to avoid steps depending indispensably on the Maxwell-Lorentz electromagnetic theory.

What, then, can we mean when we say that special relativity receives confirmation from the verification of its prediction that the mass of a body increases with its velocity? I need hardly say that the "velocity" of the electron in the supposed verification resembles Roger Bannister's velocity of a mile in four minutes no more closely than the 'mass' of the electron resembles that of the lump of lead, in order to make it clear that what we confirm by the experiments (i.e. by the observations and our inferences from them) is that the whole complex of conceptions that yields the highly metaphorical 'mass' and 'velocity' hangs together if we include special relativity (or Lorentz's theory) as a part of it. This would indeed argue in favour of one of those theories if that theory were independent of the previously existing complex of conceptions, for our object in physics is to relate apparently independent phenomena in a single system, but when the theory (Lorentz's or Einstein's) is conceived for the purpose of justifying an essential part of that complex - to wit, the Maxwell-Lorentz theory it proves nothing at all. It is like claiming, as a proof that a man always speaks the truth, the fact that he says he does.

We shall see that this is precisely the case with this (and indeed every other) supposed confirmation of special relativity involving hypothetical particles. Einstein, as he said (see pp. 159-60), designed his theory to conform to the Maxwell-Lorentz electromagnetic theory which he accepted as equivalent to 'certain'. All that the supposed confirmations support is therefore the fact that special relativity was well designed for its purpose. They tell us nothing whatever about the truth of either electromagnetic theory or the special relativity (or Lorentz's) theory itself. An example of the illusion that they do that we have already met is that advanced by Sir Lawrence Bragg concerning cosmic rays (p. in) and expressed in the usual jargon in the editorial in Nature (sec Appendix) in the words, 'short-lived mesons in the cosmic rays appear to observers on the surface of the Earth to last long enough to reach the ground'. It needs not saying that the duration and distance of their fall are not measured by a stop-watch and measuring-tape but are first inferred from a course of reasoning that includes the original Maxwell-Lorentz theory, and is then 'corrected' by the special relativity theory designed for the purpose of correcting it. Is it surprising that the answer comes out right?

It is impossible to believe that men with the intelligence to achieve the near miracles of modern technology could be so stupid as to fall into his elementary error had they not, through long familiarity with the words, unconsciously come to believe that mass, time, distance, and such terms mean the same for hypothetical particles as for the world of the senses. Physicists have forgotten that their world is metaphorical, and interpret the language literally. I do not think Einstein would for one moment have regarded these cosmic ray observations as evidence for his theory, but only as an application of it. His theory in itself was wholly kinematical: it corrects electromagnetic theory because it created a new kinematics for mat end; it can therefore be tested only by straightforward kinematics with sensible bodies, and by reasoning in which the words used have their literal, and not their metaphorical, meanings.

These four matters - the relation between mathematics and physics, the confusion of meanings of the word time, the mistaken identification of co-ordinate systems with observers, and me literal interpretation of metaphors are, I believe, the chief sources of the misunderstanding of the theory and, above all, of the illusion that it is in any way more esoteric or mystical or generally unintelligible than any other department of physics. It is, on the contrary, a rather simple theory far simpler than Maxwell's theory of electromagnetism, or thermodynamics, for example. It is no more difficult than the first principles of Newton's kinematics; indeed, the two systems are on a par with regard to practically every feature they are alternative systems of kinematics, i.e. of the fundamental relations between motion and the readings of measuring rods and clocks. Both had initial prejudices to face - those of Newton (though Galileo had done much to smooth the path for him) were a mixture of mediaeval ideas based on the principles of Aristotle and the metaphysical ideas of Descartes, while those of Einstein were rooted in the conviction that Newton's kinematics was unquestionable. Both were framed with an ultimate purpose in view that of Newton was to provide a basis for a theory of gravitation, that of Einstein to provide a basis for justifying the electromagnetic equations of Maxwell and Lorentz. They are both attempts to provide an impregnable basis for all physical science - fundamental principles on which all future theories can be built with safety and must be built if they are to survive. They can therefore appeal to nothing more fundamental, but each must justify itself on grounds of pure reason allied with experiences so simple as to be unquestionable.

It would seem that only one system of kinematics could possibly satisfy this condition, and when once stated must be self-evidently true. How, then, is it possible that two different systems have succeeded in convincing scientists, over periods of many years, that they are the necessary foundations of science? We know from Einstein's critique what he regarded as the defect in Newton's system: it was that Newton had assumed, on inadequate grounds, that the time by a clock of an event at a distance from that clock had a unique value, and had omitted to state how that" value could be determined. This is certainly true: nevertheless, since Newton's theory of gravitation could not be applied to distant bodies without assigning to events on them times according to the same clock as I that used for terrestrial times, there must have been implicit in Newton's work an assumption concerning what those times were, and what that assumption amounted to was that a clock was not I affected by uniform motion. Indeed, this almost followed from Newton's first law of motion because, since all clocks in uniform motion relative to a standard clock at rest were, like that resting clock, unacted upon by a force, it was only reasonable to suppose that there was nothing to change their rate. In effect, therefore, Newton's kinematics assumed that the time of a distant event was that shown by a clock at the place of that event, that had been synchronized with a terrestrial clock when adjacent to it and then moved at a uniform speed to that place.

Lorentz, as we shall see, was the first to challenge that assumption, by postulating that motion of the clock through the ether changed its rate; but Einstein, discarding the ether, fell back on the fact that it was an assumption that a distant event had any unique instant of occurrence at all (or, to put it in another way, if one spoke of the time (instant) of a distant event, it was necessary to give the word a meaning, and one was free to choose the meaning). In the absence of any self-evident, necessary way of determining such an instant, Einstein claimed the right to define it in such a way as to save the electromagnetic theory without violating the principle of relativity of motion. Furthermore, he succeeded in discovering such a definition. It was a veritable stroke of genius, but it is most important to notice this. Einstein had not disproved Newton's implied requirement that the rate of a clock was not affected by uniform motion; he had only shown that it was not a necessary requirement, and that, in the absence of evidence to the contrary, any other self-consistent assumption about the effect of motion on the rate of a clock was permissible. It is because the assumption which he made has been believed to be self-consistent - and, still more effectually, because, if it is, it does save the electromagnetic equations and make them accord with numerous electromagnetic observations that have been made since (including, for example, the cosmic ray phenomena cited by Sir Lawrence Bragg), that Einstein's theory has succeeded in displacing Newton's. The criticism of Einstein's theory made here is that his assumption is, after all, not selfconsistent because it requires each of two clocks to work steadily faster than the other, which is clearly impossible.

The way is now clear for a description of Einstein's theory and an account of the circumstances that have led to the remarkable oversight of what is, in fact, a very simple defect. This I now propose to undertake. The one thing necessary - and it is absolutely essential is the abandonment of the now almost instinctive conviction that there is anything mysterious in the whole thing, and the recovery of trust in elementary reasoning and commonsense. If that can be achieved the rest is simple.

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