THE FATE OF PHYSICS
THE FATE OF PHYSICSTHE FATE OF PHYSICS
Phenomena and Noumena
To understand the nature of the fate that has befallen physics, it is necessary not only to relate the 'new physics' to its roots in the classical era, but to return to the theoretical roots of classical physics itself. At its very outset, science is confronted with a fundamental dilemma, which it is impossible for it, qua science, satisfactorily to resolve. It is this. In its investigation of the world, science places evidential emphasis overwhelmingly on experience: but not on experience as such. Rather on experience of the world, or the world as experienced. Now, my basic experience of the world, qua world, more especially the core material of physics, the world of interacting bodies in space. is acquired perceptually, via my sense organs. This experience is, of course, part of myself, the perceiver, a consequence of the stimulation of my sense organs by some source, or sources, in the external world. But these sources, ex hypothesi, are external to myself. How, then, can I know what they are? I cannot, in some magical fashion, transcend myself and make a direct comparison of them with the effect which, via my organs of sense, they have upon me. Clearly, if we live in a rational universe, there must be a causal connection between the external source (or noumenon) and the internal effect (or phenomenon). But what is the nature of this connection? R.G. Collingwood defines metaphysics as, " ... the science which deals with the presuppositions underlying ordinary science ..." (An Essay on Metaphysics, p.11); and if it is possible ever satisfactorily to answer the above question then it can only be by a rationally systematic analysis of such presuppositions - that is, by a process of metaphysical reasoning, so defined. But science - ordinary science - has never, as a matter of historical fact, been metaphysically grounded in this sense. In relation to this fundamental dilemma, what position, then, has it taken up?
Commonsense reasoning would suggest that some perceptual attributes (secondary qualities) are entirely the effect of external bodies upon us, whereas others (primary qualities) inhere in the bodies themselves. Thus, common sense would reject as absurd the notion that the pain of a pin prick or the pleasure from a slice of chocolate cake in any sense reside in the pin or the cake. And a little reflection shows it to be only less absurd to believe that colours, sounds, olfactory, gustatory, and tactile sensations, sensations of heat and cold, and so forth, reside in the external bodies which stimulate them in us. On the other hand it seems equally absurd to deny that bodies must endure through time, occupy space, possess solidity, resist attempts to change them, move from place to place, etc. - in effect, since these seem the essential attributes of bodies, to deny that a world of bodies, as such, exists. A further point making for this distinction is that, in the case of secondary qualities, one can discover physical processes conceived in primary terms whose regular variation results in precisely corresponding regular variation of secondary qualities (e.g. the vibration of minute bodies, and the sensation of sound) strongly suggesting that secondary qualities are grounded on primary; that in some way - inevitably an impenetrable mystery for science - the regularities (primary) of the external, physical world are translated into corresponding regularities (secondary) in the internal, mental world. But, that this cannot be done for primary qualities offers a further argument to common sense that they truly belong to the external world.
This commonsense distinction, then, is what science, as embodied in Galileo, its principal founding father, settled for. More especially since this way of regarding the world carries with it a huge methodological bonus: an advantage in no way underestimated by Galileo, who, in common with most of science's great formative figures, was a professional mathematician. This was that the qualities thus selected as primary are also those which are the most amenable to counting and measurement - which serve ideally for that quintessential feature of scientific method, putting numbers into nature. Nor, of course, is it a coincidence that this should be the case. Undoubtedly the primary qualities are more germane than the secondary to the fundamental nature of the physical world, and, equally, owing to the overwhelming repetitiveness of natural processes, counting and measurement cannot but serve as indispensable theoretical tools for their structural elucidation.
However, sound as these arguments undoubtedly seem to the commonsense scientific mind, it is not difficult to see where error - conceivably great error - could have entered in. While it is perfectly reasonable to assume that such properties as 'enduring through time' or 'occupying space' or 'moving from place to place' can legitimately be predicated of the entities of the external world, it would be highly unlikely, in view of the length and complexity of human evolution, if these properties, as inherent in the entities themselves, existed in the same way in which we conceive them. Just, in fact, as the secondary qualities, though differently experienced, are somehow grounded on corresponding primary processes, so here, on a more fundamental level, the apparently primary qualities themselves may well correspond to, and be grounded on, a further set of, truly primary, processes. Both the notorious fact that science finds it impossible to explain how secondary qualities have arisen out of primary, and that these primary qualities, when subjected to systematic, rational analysis, reveal themselves as radically incoherent, strongly imply that this is indeed the case. Now, although, in the nature of things, we cannot become the external physical world, it may well be possible for metaphysics - or, as I prefer to say, ontology - to conceive it more nearly as it really is than ordinary science, which is little better than systematised naive realism, has proved capable of doing.
The Fatal Trap
I have said, above, that the error which the casual commonsense approach adopted by its founding fathers must inevitably have incorporated into the theoretical foundations of science may, conceivably, have been great. And, in fact, this is precisely what occurred. A truly monstrous error was indeed incorporated into the theoretical foundations of classical physics; as the core concept of mechanistic materialism its destructive effect would be hard to overestimate. I once referred to it, truthfully enough, if somewhat sententiously, as " ... perhaps the most vicious - because most humanly destructive - fallacy ever to hold sway over the intellect of man." (Letters, The Listener, 24.10.74). I call this error the Fatal Trap, since, to fall into it is certainly fatal to any chance of understanding either the physical world or ourselves. It bequeaths us two sets of intrinsically insoluble problems: those concerning the origin, nature, and interrelations of the physical fundamentals, and those concerning the causal and substantial connections between these fundamentals and life, mind, and spirit. We shall be discussing it in considerable detail elswhere. Here, we need only define its essence; which is to believe that all physical bodies are composed of ultimate constituents which are, themselves, intrinsically unchanging, but which create the changes without which time would have no meaning, by their motions through an independently existing, quasi-substantial, extensive medium - space. This is the core concept of mechanistic materialism. Whitehead (Process and Reality II II V) castigated it as "sheer error".
The Fatal Trap (a natural outgrowth of that practical 'thing thinking' which conferred upon the higher primates so decisive an advantage over their fellow mammals) first entered systematic thought in the form of Greek atomic theory. It entered science during its formative era through the enthusiastic revival of atomism by Gassendi, Galileo, Descartes, Boyle and others,1 who saw its explanatory potential. And in the consummation of this formative epoch with the publication of Newton's "The Mathematical Principles of Natural Philosophy" of 1686-7, the Fatal Trap became cemented into the theoretical foundations of classical, and hence post-classical, physics. In short, physics had become ineluctably set in a mechanistic mould.
Equations and Processes: Classical Era
The conceptual framework within which classical mechanics conceived the physical world consisted of material bodies - themselves composed of intrinsically unchanging ultimate particles - changing their motions through space solely under the dictates of forces exerted on them by other bodies. To discover the order underlying these motions - that is, to reveal their intricate coordination - precise, systematic, definition of the physical parameters involved, and the precise measurement of their magnitudes as they manifested in particular situations - more especially under those meticulously controlled conditions we call experiments - was an essential prerequisite. To this end unit measures were assigned to all the basic physical attributes - quantity of matter, length, time, velocity, acceleration, force etc. - so that their magnitudes could be expressed as multiples of these units. Moreover, wherever possible, the unit measure of each attribute was defined in terms of mathematical relations obtaining between those of more fundamental attributes, as for example, defining a unit of velocity as a unit of distance (length) divided by a unit of time. In this way it was found that all physical quantities could be expressed ultimately in terms of unit measures of amount of matter (a unit of mass), length (or distance), and time. In practical terms, this meant that all the numbers introduced into physics entered either by counting, or by a length or a time measurement. This held even for mass, since, by Newton's second and third laws, the masses of two interacting bodies are in inverse ratio to their accelerations.
It is clear, then, that the knowledge acquired by the classical physicist, working entirely within the theoretical framework delimited by Newton's laws of motion, was systematised on two intimately related conceptual levels. One level was that of mathematical formalisms, or, as I prefer to say, measure-number equations: equations relating symbols whose magnitudes in any physical situation could be obtained by counting, read off measuring instruments, or calculated from such readings. Clearly, these equations expressed constant correlations among their symbols, since change in magnitude of any one meant that at least one other had to change if the equality was to be maintained. The other level was that of envisaging the coordinated changes of the physical parameters to which the symbols referred: changes of which the equations were the metrical description. Throughout this period, measure-number equations were obtained in great abundance. Some were obtained by simple combination of the defining equations. Others, the most significant, - above all, Newton's equation of universal gravitation - were obtained by a harmonious synthesis of physical reasoning, painstaking observation and experiment, and mathematical logic. And all these were greatly multiplied through the rapid expansion of mathematics, more especially the calculus, by whose means they could be organised into ramifying systems of differential equations of great elegance and power.
For the mechanist, the causes of change in the motions of bodies - which is to say, interactions - fell into two distinct classes: direct and indirect. Transmission of momentum by impact constituted the mechanist's paradigmatic causal mode, since it was obvious that if bodies composed of intrinsically unchanging particles were headed for the same place at the same time, there must ensue some changes in their motions. And these changes were precisely calculable by the law of the conservation of momentum, a direct implication of Newton's laws. But action at a distance was an insuperable barrier to understanding for the mechanist. Inevitably he tried to account for it by postulating some kind of indirect impact. The stance taken up by Newton vis-a-vis this fundamental difficulty is well known: " ... that one body may act upon another at a distance, through a vacuum, without the mediation of anything else through which their action may be conveyed from one to another, is to me so great an absurdity that I believe no man who has in philosophical matters a competent faculty of thinking, can ever fall into it". (Letter to Bentley [1692/3]. Quoted by Collingwood, "The Idea of Nature", p. 144). And Collinwood goes on to say of Newton, "He believed that gravity must be either a peculiar effect of some peculiar kind of impact, which he always regarded as the only possible physical cause of motion, or the effect of some immaterial cause". To make for contact across space, and also to account for the 'subtle fluids' heat, light, electricity, and magnetism, imponderable bodies were postulated. One conception of heat and light comprised minute imponderable corpuscles sent at high speed across space from emitting to receiving body. More widely favoured was an aetheric medium, consisting of particles of imponderable matter, filling all space. The forces of gravitation, electricity, and magnetism were then thought of as tensions and pressures in this medium set up by the ponderable matter embedded in it, and heat and light as some kind of progressive oscillatory disturbance through it.
The two centuries of classical physics saw physical conceptions, and their precise metrical expression, increasingly extended from visible bodies to the realms of the invisibly minute and the imponderable. In the first, the other great physical science, chemistry, made rapid progress within a basic theoretical framework witnessing the happiest of marriages between atomic theory and (another legacy of the Greeks) element theory. By the end of the classical era it was known that there were ninety odd different kinds of atom, and that the immense diversity of material substances arose solely from the different orderly arrangements which these atoms in their different combinations and proportions were capable of taking up in space. It was also realised that the cohesive forces holding the atoms in these arrangements were electrical in nature, and that heat and light originated in rapid, regular, repetitive movements of these minute particles. In the second, the speed of light (c) had been determined, and when it was realised that c = 1/√(κμ), where κ and μ are fundamental constants of electricity and magnetism respectively, it seemed obvious that light must be transmitted through the aether in the form of progressive electromagnetic waves: a theory regarded in the 1880's as confirmed when Hertz succeeded in transmitting electric oscillations through space.
Nor, during these two centuries, did any serious difficulties arise in the task of constructing a sufficiency of measure-number equations relating to the behaviour of these subtle fluids. This was because their effects, cumulative for the most part, on those macroscopic bodies we call measuring instruments could be precisely registered. Moreover, it was found that for this, only one fundamental quantity - an electrical one - needed to be added to mass, length, and time; and that since, like mass, this could be regarded as proportional to acceleration, it, too, required only length and time measurements. But an ever deepening obscurity accompanied all these brilliant advances into the realms of the minute and the imponderable. The envisagement of just what physical processes the measure-number equations were regarded as metrically referring to was inevitably becoming vaguer as the cutting edge of physics moved ever further from the realm of ponderable, macroscopic bodies. But for all this growing separation of the two conceptual levels on which physical theory organised the world, measure-number equations could still legitimately be regarded as metrical descriptions of at least superficially intelligible physical processes.
Equations and Processes: Post-Classical Era
All this was soon to change. The period whose limits were defined broadly by the Michelson-Morley experiment of 1887 and the Davisson-Germer experiment of 1927, saw a flood of anomalous discoveries with which mechanistic theory was quite unable to cope. The mainstream implications drawn from all these experiments were that there was no aether; that matter is essentially electrical in nature; that all bodies move, not in continuous trajectories, but in tiny discrete steps; that all electromagnetic radiation has about it something particulate as well as undulatory; that there is some kind of oscillatory activity intrinsic to matter; that all physical processes are not precisely defined, but probabalistic, in essence; and that space and time are merged indissolubly. So that where, before, despite many obscurities, the mechanistic framework held firm, so many transgressions of its basic tenets now existed as to render it effectively destroyed. The implications of all this for physics were naturally profound: not only for its cognitive content, but also for its methodology, and even its very nature.
This breakdown of mechanism was, of course, inevitable. A fundamentally false theory, despite its ability to provide superficially intelligible descriptions of macroscopic physical processes, it only required physics to attain a certain level of empirical sophistication for its comprehensive inadequacies to become evident to all. The message that should have been read into all this was that any conception of the physical world based upon the notion of intrinsically unchanging entities moving around in a spatial medium was radically false, and hence that what was needed was a systematic physical theory which avoided this Fatal Trap (see above §2). But, although various philosophically inclined scientists and scientifically educated philosophers have come up with some valuable insights, the natural bias of the Fatal Trap has so far proved too strong for any such theory to have emerged. As an inevitable consequence, post-classical physics - the so-called new physics - has been obliged to develop in other directions, all of them radically unsatisfactory. Either it still attempts to say something about the nature of the physical world, in which case it is condemned to a kind of disjointed, self-contradictory mechanism; or it effectually limits its aims to gaining practical control over natural processes; or it claims that its measure-number mathematics is itself an adequate description of the physical world.
As I emphasised earlier, in classical physics measure-number equations could legitimately be regarded as metrical descriptions of intelligible physical processes. Now, although, in order to be part of the metrical science of physics, physical theories necessarily require measure-number equations, the converse is not true: measure-number equations do not necessarily require physical theories. A physicist, let us say, obtains an anomalous experimental result - one that his classical theories can't account for. Clearly, his conception of physical processes must be, in some way, false, since the result suggests physical processes that are incompatible. But no such impasse exists for the measure-number equations which constitute the metrical descriptions of these processes. In the great majority of cases they may be combined or otherwise manipulated to yield a measure-number equation which successfully accommodates the experimental data. Though mechanism is internally inconsistent, the physical world is not; so that the investigation of orderly physical processes is bound to yield coordinated changes of the numbers on the instruments measuring these processes. And given the virtually infinite plasticity of mathematics, and the formidable skills of mathematicians, there usually exists no reason why a satisfactory measure-number equation should not be found.
But it is obvious that a physics whose measure-number equations are no longer metrical descriptions of intelligible physical processes is a physics which has abnegated its role of providing knowledge of an objectively existing physical reality. In the classical era it was taken for granted that this was the primary function of physics. But there exist other conceptions of its nature and purpose.
Physics as Technology
Physics - as, indeed, all science - is Janus-faced, looking towards philosophy on the one hand and technology on the other. Is it primarily a means for understanding the world, or manipulating it? This identity crisis is endemic. The failure of mechanism - the naively realistic approach to investigating the physical world (and, to be frank, the only approach that the majority of physicists seem capable of making) - naturally afforded strong support to those who contended that physics is primarily a discipline for gaining control over physical processes, rather than one which seeks to understand them objectively. Hence, two closely related essentially modern, conceptions of physics have come to the fore. The first is positivism, or phenomenalism, which contends that the essential task of physics is to register constant correlations among physical phenomena. Physical theories are not interdicted but are seen as serving, at most, a merely heuristic purpose. The second, known as pragmatism, operationism, or instrumentalism, sees the facts of physics as a series of operations performed by physicists; and if, from such a series, it can be correctly predicted that such and such a consequence - more especially an instrumental reading - will follow, then this defines what is meant by a truth. So that true and false propositions are no longer defined ontologically, by the criterion of correspondence of their meaning content with the attributes of an objectively existing world, but operationally, according to their success in predicting the outcome of certain precisely defined operational circumstances. These ways of viewing the science of physics degrade it from an activity seeking to make a powerful contribution to the age-old rational aspiration of understanding the universe, to mere formula finding in the interests of its practical applications in the service of humanity - though one might be pardoned for seeing a recipe for disaster in a science brashly confident of its ability to change for the better a world it finds too complex to understand.
No Physics out of Mathematics
But there is another, quite different, way of viewing the role of physics. This is based on the theory, going back at least as far as Pythagoras, that the structural ground of the universe is one of mathematical relationships: that it is by the numerical content, forms, and operations of pure mathematics that the world is organised. So that in organising our physical experience in the form of systems of measure-number equations we are, by that very fact, understanding the universe in the only way in which it is capable of being understood. According to Sir James Jeans, writing in 1930 (The Mysterious Universe, Penguin Ed., p.p.153-4): " And what we are finding, in a whole torrent of surprising new knowledge, is that the way which explains [phenomena] more clearly, more fully and more naturally than any other is the mathematical way, the explanation in terms of mathematical concepts".
But, "It is simply ludicrous to tell us that these numbers which we have just constructed enable us to dispense entirely with the essential constituents of the real world which alone confers on these numbers a significance beyond that of abstract arithmetic." (Alfred O'Rahilly, Electromagnetics. A Discussion of Fundamentals. 1st Pub. Longman's 1938; Dover Ed. 1965, p.846). The simple truth is that pure mathematics tells us nothing more about the universe than that it contains mathematicians. Nor is there any mystery as to why, in general, mathematics should be so powerful a tool for analysing the structure of the physical world. It is, or should be, obvious enough by now that the universe is a process of some kind - a process, moreover, that is enormously repetitive. And where one has collectivities whose individual instances are intrinsically indistinguishable, such as a pile of ball-bearings of a certain size; or, a fortiori, when such instances exist in a spatial or temporal sequence, as in the case of equally spaced marks on a straight rod, or the ticks of a clock - then one can employ cardinal and ordinal numbers in their description. Thus, the equations of physics, relating such numbers, refer to physical objects or structures or processes. And it is the nature of these - matter and its attributes of distance, time, charge, force, etc. - and their interrelationships, not the forms of pure mathematics, that dictate in any physical situation the nature of the available mathematics that can be suitably applied - though, of course, it is the current state of mathematics which determines what mathematics is available.But what if - as a Pythagorean might contend - the physical fundamentals themselves are ultimately mathematical in nature? Now, mathematics, by its very nature, is abstract. Likewise, physical experience is, by definition, concrete. No investigation confined solely to mathematics can ever tell us anything about the concrete reality that mathematics has been abstracted from. The ultimate nature of the physical world can be determined, if at all, only by an ultimate analysis of our experience; and that is a task for ontology, not mathematics. Now, because of the extreme repetitiveness of the physical process, one would certainly expect number and quantity to figure prominently in such an experiential analysis; but to claim that this analysis should involve nothing but mathematics would be to commit the grossest of category mistakes. Moreover, even if it were granted, propter argumentum, that the physical world does consist ultimately of nothing but numbers in relation, it would still hold true that the abstract measure-number mathematics of our physics would be constrained in its physical application by the nature of the concrete 'mathematics' of the ultimate constituents of the physical world.
Certainly the whole history of physics bears witness to the fact that the purely formal relations exhibited by measure-number algebra provide no reliable guide to the quantitative or spatial or temporal relations of the physical entities or situations to which they refer: which is not to deny that occasionally an exact correspondence apparently exists - that between vitreous and resinous electricity and positive and negative numbers being a case in point. In a general way it is easy to see, looking both from the physical and the mathematical ends of the relationship, why no reliable way exists of deducing the nature of physical processes from the formal properties of measure-number mathematics.
Looking first from the physical end: in any physical situation the qualities and relations to which measure-number mathematics is being applied will inevitably only be part of a wider system of causal constraints, with the consequence that not all the numerical values of the measure that are mathematically possible may be physically possible. Obviously, which values make physical sense and which are physical nonsense can in no way be determined from the mathematics. An elementary example of this is afforded by the application of quadratic equations to physical problems, where frequently one of two equally mathematically impeccable solutions has to be rejected on the grounds of its physical impossibility. Another obvious example, obtaining throughout physics, is afforded by the fact that measure-number equation is true only for a certain range of numerical values of its variables. Just what that range is, mathematics is obviously powerless to tell us.
Viewed from its other end, the situation reveals itself as even more treacherous, because of the virtually infinite plasticity of mathematics. This applies even to the changes in form within a single equation, but is immeasurably amplified when substitutions from other equations are involved. I will give a highly significant example from both categories. The equation P = [m/√(1 - v2/c2)] x a, much used in non-Newtonian mechanics, can be transposed to P√(1 - v2/c2) = m x a. It is usually written in the first form and given the physical interpretation that, for a constant impressed force (P), as the velocity (v) of a body tends to the velocity of light (c), its mass [m/√(1 - v2/c2)] tends to infinity. But, when the equation is written in the second form, the natural physical interpretation is that the mass (m) stays constant but, that as v tends to c, the impressed force [P√(1 - v2/c2)] tends to zero. The same equation thus lends itself equally to two physical interpretations utterly at variance. In the latter half of the nineteenth century Maxwell's electromagnetic theory of waves in an aetheric medium displaced the electrodynamic interparticle force theory of Clausius, Weber et alia which incorporated neither waves nor aether. Yet the equations of the two theories, at first glance so different, prove to be largely interconvertible. That Maxwell's field theory gained the allegiance of the majority of physicists was partly because, in their eyes, it offered a superior mechanical explanation, and partly because its mathematics, in the form of differential equations, and lending itself to vector analysis, was more elegant and powerful than that of its rival. Yet, as I hope to show, the latter theory was much closer to the physical truth than Maxwell's. As Burniston Brown, one of the greatest theoretical physicists of our century, wrote in 1981: " ... the mathematics can be right, but the physics fundamentally wrong." (Retarded Action-at-a-Distance, Cortney Publications, p.3).We contend, then, that if the equations of the 'new physics' are not to be taken as mere algorithms of experimental procedures but, when suitably interpreted, as making assertions about the nature of the physical world, the physical theory they are presenting is, at best, no more than a disjointed, or self-contradictory, mechanism, and, at worst, the mere reification or hypostatisation of the abstractions of mathematics. In support of this contention we shall now take a critical glance at certain key conceptions in the two theories which virtually comprise the 'new physics' - quantum theory and relativity.
Quantum Theory
The physical concept which, it is scarcely too much to claim, quantum theory is built around is the wave. As Sir James Jeans wrote in 1930: " ... the tendency of modern physics is to resolve the whole material universe into waves, and nothing but waves." (ibid., p.99). The only waves of which we have direct, perceptual acquaintance, and upon which, therefore, all our wave concepts are ultimately based, are waves in fluids, strings, and membranes, where what are oscillating are material particles - small bodies, in effect. These waves, of course, were intensively studied in the classical era, and numerous measure-number equations derived to provide precise metrical descriptions of their behaviour. Now, the most widely accepted theory explaining the propagation of heat, light, and (later) radio, was, as we have seen, one of electromagnetic waves through the aether. Since no particles of aetheric substance were ever found - let alone found oscillating - why did physicists postulate these waves? Basically, because there existed strong evidence that their emitters were oscillators of some kind, and that the phenomena of interference and diffraction likewise afforded strong evidence for oscillatory processes in the receptors. In addition, as we noted earlier (p.5), the fact that c, the 'speed of propagation through space' was found to be a simple function of fundamental electric and magnetic constants was taken as powerful evidence that these waves really existed, that they were, in fact, progressive oscillations of electric (E) and magnetic (H) field strengths - the concept of field thus acquiring an intimate association with that of wave.
But the clear - and generally accepted - implication of the Michelson-Morley experiment of 1887 was that no aether - and hence no aether waves - existed. Yet the mathematics of the Maxwell (strictly, the Maxwell-Lorentz) theory was deemed so essential to electromagnetism that, in order to retain the notion of electromagnetic waves by giving them some kind of physical basis, the concept of field was proposed as " ... a brilliant compromise ... " (O'Rahilly, ibid., p.658) between the now equally unacceptable void and aether. Now, according to classical notions, every physical body, whatever its spatial location, is being acted upon by forces originating in other physical bodies. So that at any instant. any point in space can be seen as characterised by the resultant force which would act upon a particle at that point if a particle were present. In this way, space can be regarded as a single universal force field. But, of course, since forces are material, not spatial, attributes, such a field possesses no physical reality. It is just a conceptual convenience, and to reify it such that " ... vectors like E and H were made to strut about in their own right" (O'Rahilly, ibid., p.644) as if they were material things, was no more than a crass category mistake.
Incredible as it may seem, this reified measure-number algebra 'constituting' the universal force field (or - since all the basic forces seemed to be oscillating - wave field) of space was actually elevated to the status of the ground of the universe. Thus: "According to our present conceptions the elementary particles of matter are in their essence nothing else than condensations of the electromagnetic field." (Einstein, "Sidelights on Relativity", 1922). Moreover, it is a relatively easy matter mathematically to create a wave group whose members destructively cancel out everywhere except a minute region of space where they constructively reinforce one another to form what is known as a wave packet. Now, the Davisson-Germer experiment of 1927 revealed that electrons, no less than X-rays, produced diffraction effects. And diffraction effects had long been regarded as evidence of wave motion. But all that can strictly be deduced from the diffraction of light (or X-rays) is that some kind of periodic activity is involved. But given the wave orientation of mechanistic thinking, it was assumed that electrons possessed wavelike properties. And, indeed, de Broglie, anticipating this experimental result by more than three years, had already worked out a satisfactory measure-number equation: the famous de Broglie relationship, λ = h/mv. where λ is the wavelength of the 'electron wave', m and v are the mass and the velocity of the electron, respectively, and h is a universal constant of action. It was inevitable that this experimental result should be regarded as confirmation that the electron was a wave packet. And since a wave packet is no more than that minute portion of the universal wave field where a particular group of waves does not destructively cancel out, movement of an electron can only consist, ultimately, of adjustments of phase and frequency of this infinitely extensive wave group.
The wave packet conception was inspired by a well-known phenomenon of real, physical waves: the wave group. In a dispersive medium - a medium where the frequency of a wave differs with its velocity - a group of waves of slightly different frequencies, and travelling, therefore, at slightly different velocities, combine to form what is known as an envelope wave. An important relation holding between this envelope wave (i.e. the group wave) and its component waves, is that the smaller the range of frequencies of the components, the longer is the wavelength of the envelope wave, and conversely. Despite the fact that the universal wave field is a non-dispersive 'medium', Heisenberg drew momentous ontological conclusions, based on this classical property of group waves, from the notion of the electron as a wave packet. The de Broglie relationship, λ = h/mv, stating that the wavelength (λ) of the electronic wave packet is inversely proportional to its velocity (v), taken in conjunction with the inverse proportionality of frequency range and wavelength of a group wave, enabled Heisenberg to deduce, as a virtually direct consequence, his notorious uncertainty principle. This states that it is intrinsically impossible for any material particle, owing to the inverse proportionality of the precision of their respective values, to have both an exact location and an exact velocity. We emphasise that the truth of this principle, from which so many far-reaching consequences concerning man and the universe have been inferred, is wholly dependent upon the notion of ultimate particles as wave packets: in turn dependent upon the reification of measure-number mathematics and the mechanistically grounded conception of radiation as waves.
During the first decades of this century, more sophisticated investigation of the effects of radiation had revealed that light, in addition to its wavelike properties, possessed particulate attributes - that it interacted with matter in ways which were incompatible with the notion of it as a wave spreading through space. But the conception of light - indeed, all radiation - as composed of imponderable oscillating particles (photons) involves such a host of physical absurdities that there is no reason to believe that photons possess any greater physical existence than waves. Both alike are conceptual fictions arising as inevitable consequences from our mechanistic way of conceiving the world.
But, without question, the most important manifestation of the wave concept in the whole of quantum theory is Schrödinger's wave equation. Bohm goes so far as to say: "Practically the entire quantum theory is contained in the wave equation ... " (Quantum Theory, 1951. Dover Ed., p.79). Basically, this equation is no more than the de Broglie relationship, λ = h/mv, combined with a general mathematical property of waves connecting the wave function (y) with its wavelength (λ). We have seen that this relationship for electrons is a metrical description of the diffraction of electron beams. And though this diffraction is interpreted mechanistically as a wave phenomenon, what it is really evidence for is the intrinsic periodicity of matter. This innate periodic nature of the substance of the physical world is the basic ontological fact giving rise to the whole of post-classical physics: since the classical era was, at bottom, no more than that comparatively crude stage in the development of physics when this fundamental attribute of matter did not have to be taken into account. But, still in thrall to the Fatal Trap - the conceptual core of mechanism - contemporary physics has failed to recognise the true nature of its discovery. In default of a new matter theory with which to deal with it directly, it has had no choice but to assimilate it into its outmoded mechanistic conceptual system in the form of a wave. And in the all-important metrical domain it is by means of the Schrödinger equation that this has been achieved. By welding (via λ) the de Broglie relationship with a most general mathematical property of wave motion, this effectively assimilates the intrinsic periodicity of matter into that fictional world of waves to which the mechanistically conditioned physicist is confined. And it is because the periodic nature of matter is the root physical fact out of which the whole quantum world has grown, that practically the entire quantum theory is contained in the Schrödinger wave equation.
Special Relativity
We turn now to relativity, perhaps the most ambitious of all attempts to derive physics out of mathematics. Maxwell's theory, which by the 1880's, had come to assume for electromagnetism almost the importance which Newtonian dynamics possessed for mechanics, postulated, unlike mechanics, an absolute standard of rest, the aether, through which all bodies were thought to be moving. But when, in 1887, the Michelson-Morley experiment failed to detect the presence of an aether, the only way of saving the theory, and, even more important, the differential equations in which it was metrically expressed, seemed to be Lorentz's ad hoc hypothesis that all bodies contracted, and their rhythmical processes (in effect, clocks) slowed by amounts that were exact, mathematically expressible, functions of their velocity through the aether. This would have the consequence of rendering the aether undetectable by any experiment confined to a single body - such as the Earth.
Einstein's special theory followed a quite different line of reasoning, which, if sound, would have achieved the paradoxical result of vindicating Maxwell's equations while discrediting his theory. Einstein's theory proposed that although the velocity of light was absolute, there existed no detectable standard of rest for bodies, whose motions were therefore purely relative. From these postulates, together with an empirical definition of simultaneity, Einstein succeeded in deducing the same equations (known as the Lorentz transformation) which Lorentz, as mentioned above, had advanced as the metrical expression of his aether theory. At first sight this might seem to offer an acceptable solution to the problem. But on further thought a certain difficulty arises - as one might expect from a wholly relativistic theory invented for the purpose of saving equations which were the metrical expression of a theory grounded on absolute velocities.
For Einstein as for Lorentz: a moving clock goes slow. But it is obvious that the expression must mean something very different in the two theories. For Lorentz, 'the moving clock' meant a clock moving relatively to the aether. But for Einstein, 'a moving clock' could only mean a clock moving relatively to some other clock. And since, in his theory, motion was purely relative, this would seem to imply that each of two relatively moving clocks was working more slowly than the other - on the face of it a straight contradiction. But, claims the relativist, this is really only a paradox, or apparent contradiction. As viewed from the coordinate system in which clock k is at rest, the moving clock, k', appears to go slow; and, reciprocally, as viewed from the coordinate system in which clock k' is at rest, the moving clock k appears to go slow. This apparent discrepancy arises because they are using different values - given by the Lorentz transformation - for distances and times.
But this is an unacceptable argument. Apart from his empirical definition of simultaneity, obtained by a process of synchronising clocks - a process which, though never used, is perfectly acceptable in theory - Einstein's argument was conducted throughout on the plane of abstract kinematics. But, unlike mathematics, the science of physics studies physical things in physical situations by means of physical instruments - in this case, clocks. And, unfortunately for Einstein's theory, when the argument is brought down (up?) to this physical level, and the readings of two relatively moving clocks compared, it is found that the seemingly apparent contradiction must, in fact, be an actual one. This must be the case whether clock readings are compared directly by bringing together two previously synchronised clocks (as in Einstein's original, 1905, paper); or indirectly, by means of two relatively moving sets of synchronised clocks - and Einstein and Infeld in "The Evolution of Physics" (1938) state that it is a matter of indifference whether clock comparisons are made using single clocks or synchronised sets. In either case, when the readings of two such relatively moving clocks at the same location are compared (using photographs if necessary), either they will read the same or they will read differently. If the former, then there has been no differential slowing of clocks; so that, howsoever the expression be interpreted, moving clocks do not go slow, the Lorentz transformation is inapposite, and any theory from which it is a necessary inference must be false. If the latter, the slower clock is the moving clock and the hypothesis of universal relative motion is thereby contradicted. In either case, therefore, Einstein's special theory of relativity is shown to be false.
What adds an extra dimension of paradox to the whole extraordinary situation is the relationship obtaining between Einstein's special theory, Maxwell's electromagnetic theory, and the interparticle force theory of Weber et alia. We saw earlier (p.9) that Maxwell's mathematics and that of interparticle force theory are in many important respects equivalent; so that in vindicating Maxwell's mathematics by means of the Lorentz transformation, whilst discrediting his aether wave theory, Einstein was, in effect, vindicating the mathematics of interparticle force theory: and this, postulating no absolute velocities with respect to an aether, and therefore requiring no Lorentz transformation, was, unlike Maxwell's theory, in no way discrepant with Newtonian dynamics. (But it did entail "spooky" action at a distance, which, to Einstein - as to Maxwell - was unacceptable). In effect, therefore, Einstein was a) attempting to vindicate mathematical relationships which, if in a different form, did not require vindicating; b) attempting to reconcile electromagnetism with Newtonian dynamics, despite the fact that, as an interparticle force theory, it needed no such reconciliation; c) condemning both electromagnetic theories. If one then asks the obvious question - What was Einstein's electromagnetic theory? - the answer can only be - unless, that is, one counts the gross ontological blunder of reifying measure-number algebra and calling it a physical theory - that he didn't possess one. As O'Rahilly (ibid., p.421) put it, "Einstein has no physical theory at all; he is simply exploiting an elementary piece of mathematics." - that is, the Lorentz transformation.
The objection that Einstein's theory is vindicated by its successful everyday use in contemporary physics carries no weight at all. What are employed ubiqitously are Maxwell's equations adjusted, for high velocities, by the Lorentz transformation. But since both of these antedated it, relativity can have nothing to do with the matter. Nor, incidentally, contrary to popular belief, has relativity any particular connection with that other piece of algebra built into contemporary physics: E = mc2.
General Relativity
The special theory of relativity confined itself to coordinate systems in uniform relative motion. The general theory extends relativity to accelerating systems. The central idea informing it is that the laws of nature, to be truly expressed, must be expressed in invariant form. The mathematical property of invariance (or covariance) - a property possessed by the Lorentz transformation - seemed to Einstein "to possess an imperious cosmic significance" (O'Rahilly, ibid., p.422). A branch of mathematics called the tensor calculus, perfected by Ricci and Levi-Civita by the end of the nineteenth century, was just what Einstein needed for his enterprise, since its central concern is with the forms which differential expressions must possess if they are to remain invariant under all transformations of coordinates. But, naturally, this calculus is physically empty. It can tell us nothing as to what the laws of the universe actually are, but only, if Einstein's hunch is correct, something about the form in which they must be expressed if they are to be true for all 'observers' - that is, all coordinate systems. Einstein confined his researches entirely to one law of nature: universal gravitation. To find the covariant form of this law, he had to select the tensor which (when equated to zero) most resembled Newton's law expressed as a differential equation, and then, guided by various mathematico-physical assumptions (including, of course, Newton's law), derive his own law of gravitation out of it. We confine ourselves here to pointing out certain fundamental errors involved in these assumptions.
Einstein's theory is confined solely to bodies moving in gravitational fields; but the motion of a planet, in which Einstein was, above all, interested, is determined not only by gravity, but by the force of inertia. To overcome this difficulty, he postulated his Principle of Equivalence: that inertial and gravitational forces are equivalent in the sense that an inertial field could, if one so wished, be regarded as a gravitational field. But, as has so often been pointed out, the vectors of a gravitational field are centripetal, whereas those of an inertial field are parallel. Strictly, therefore, their equivalence - if it holds at all - can hold only at a point.
The tensor calculus, in its comprehensive analysis of the covariance of differential expressions under transformations of coordinates, embraces all varieties of coordinate system, all types of space, and any number of dimensions. Now, ultimately, the entity whose invariance of metrical description, under all these transformations, is being studied, is the shortest distance between two points - that is, the line element, or geodesic - of a space. Thus, in empty, three dimensional space, metricised by three mutually orthogonal axes (x,y,z), the geodesic or line element (strictly, its square, ds22) is given by the Pythagorean relation dx2 + dy2 + dz2, where the 'd' merely signifies 'change in'. Empty space is known as Euclidean space, since Euclidean geometry holds true within it. But if space is not empty, if matter has to be negotiated in moving from one point to another - as, for instance, on the Earth's surface, where the geodesic is an arc of a great circle - the simple Pythagorean relation no longer holds good, the expression for ds2 being more complex. Such a space is said to be non-Euclidean.
According to the special theory, uniformly moving coordinate systems are Lorentz invariant; in that to obtain the same numerical values for an event referred to any one of a uniformly moving set of coordinate systems, the Lorentz transformation equations have to be applied. Now, Einstein's one-time teacher, Minkowski, showed that the expression x2 + y2 + z2 - c2t2 is Lorentz invariant, and interpreted this analytical truism in a sensational way. He claimed that it implied that we do not live in a three-dimensional spatial world where the square, ds2, of the shortest distance between two points is given by dx2 + dy2 + dz2, but, instead, that we inhabit a four-dimensional spatio-temporal continuum, where the square, ds2, of the smallest interval between two events is given by dx2 + dy2 + dz2 - c2dt2. What's more, he showed that by taking four 'orthogonal' axes (x,y,z,ict), the metrical description of an event in one coordinate system can be Lorentz transformed to that in any other moving relatively to it with uniform velocity (v), by rotating the (purely imaginary) ict axis through an angle (likewise purely imaginary) whose tangent is iv.
Now, considered as a physical theory, this is pure nonsense. To begin with, as we saw earlier (§6), it is physics, not mathematics, that determines whether some particular algebraic expression applies to the physical world. Thus, s2 = x2 + y2 + z2 just expresses a relation between four numbers until we discover that it is the metrical expression of the square of the length of a straight line in Cartesian coordinates. But this in no way implies that s2 = w2 + x2 + y2 + z2 has any physical meaning. Space is such that no more than three mutually orthogonal straight lines can intersect in a point. Why this should be so can only be discovered - if at all - by an ontological analysis of the nature of space. And to assume that just because a certain algebraic expression can be given a physical application, an analogous expression can necessarily be given an analogous application is simply a false inference arising from the baseless intuition that the forms of the physical world mirror or parallel those of mathematics. In the case of the Minkowskian metric, the position is doubly absurd since his fourth 'orthogonal axis' cannot even be calibrated in real numbers. There are no such physical attributes as imaginary lengths or imaginary times. And as if all this weren't enough, the Minkowskian 'metric' would be ruled out by the fact that, as our discussion of special relativity disclosed, Lorentz invariance is incompatible with the relativity postulate.
But it was this Minkowskian interpretation of his special theory, concerned only with uniform velocities, that Einstein took as the basis for his general theory enlarged to embrace accelerations. He did this not only because it gave him a mathematical form, compatible with his special theory, eminently suitable for manipulation by the tensor calculus, but also because it seemed to him, reasoning entirely by mathematical analogy, to possess momentous implications for the nature of space, time, force, and motion. The reasoning is broadly as follows: Minkowski's four dimensional, Lorentz invariant, spatio-temporal geodesic for uniform velocities, ds2 = dx2 + dy2 + dz2 - c2dt2, is analogous to the Pythagorean geodesic ds2 = dx2 + dy2 + dz2 for empty, three dimensional space. And since this space is called Euclidean, we may view Minkowski's geodesic for uniform velocities - motions unaffected by forces, since forces produce accelerations - as applying to Euclidean space-time. But the geodesic for any non-Euclidean space - space in which matter must be negotiated - will no longer be a straight line but a curve, and its metric some more complex expression based on the Pythagorean relation. Hence, by analogy, any region of non-Euclidean space-time will be a region in which accelerated motions, due to the presence of matter, occur, and its geodesic will no longer be a straight trajectory, but a curved one, and its metric some more complex expression based upon Minkowski's.
By taking this purely mathematical line of reasoning, Einstein replaced the Newtonian conception of a body following a curved path under the constraints (forces) exerted on it by neighbouring bodies, with the conception that the presence of material bodies changed the geometry of space-time in such way that any body in their vicinity followed a curved trajectory. Now, it is true that Newtonian mechanics has no explanation either of what a force is, or how it acts across space. Such problems are beyond the scope of science as ordinarily understood; though doubtless Einstein thought that by changing the geometrical properties of space he had finally laid the ghost of that "spooky" action at a distance he found so unacceptable. But it is obvious from the above that all his 'changing the geometry of space' really amounts to is assigning to space (or space-time), on no stronger grounds than a fatally flawed mathematical analogy, two properties possessed by bodies - surface curvature, and resistance to penetration by other bodies; as if space were just another body.
So that the tensor calculus (physically empty), Newton's gravitational equation (at least approximately true), the Principle of Equivalence (false), and Minkowski's formulation of his special theory (false), formed Einstein's basic guidelines in deriving his new law of gravitation. This derivation proved a very long and tortuous process, in which appeals, some more plausible than others, to the general nature of the physical world were made, arbitrary constants of integration put to grateful use, and, by illicitly making the velocity of light a dimensionless unit, mass, length, and time melted down into a single all-purpose dimension. The equations that finally emerged differed slightly from Newton's. The observational 'evidence' that was once thought to favour Einstein's equations over Newton's has turned out to be illusory.
In Conclusion
I am contending, then, that contemporary theoretical physics exists in a state of monstrous confusion. This overall confusion resolves itself into three great component confusions. The first concerns confusion as to what physics is actually about. Is it primarily natural philosophy or technology? An activity devoted to understanding the physical world or manipulating it? These two aims are not intrinsically in conflict; such conflict was only latent in the classical era. But today, the obsolescence of the mechanical philosophy and the hijacking of physics by mathematics make it imperative that we decide on the order of our priorities.
The second confusion, even more important, is that resulting from the fundamental misunderstanding of the relation existing between the abstract discipline of mathematics and concrete physical reality. This misunderstanding takes two forms: firstly, the belief that physical structures can be deduced from - instead of occasionally suggested by - mathematical formalisms; secondly, that measure-number equations are themselves the structural elements of the physical world. "The wisest of mankind" got it precisely right nearly four centuries ago (The New Organon, Bk.1 Aph.XCV1): " ... natural philosophy ... is tainted and corrupted ... by mathematics ... which ought only to give definiteness2 to natural philosophy, not to generate or give it birth." And Kant, a century and a half later, echoes him with "... the mathematician, by employing his method in philosophy, can produce only so many houses of cards". To which we might now add: God does not play cards with the universe.
But, of course, it is the third confusion: that between phenomena and noumena - between attributes of the physical world as perceived, and attributes of the physical world as it really exists - which is the most important, since the first two are but consequences of it. By projecting upon the objectively existing world attributes which belong only to the observing subject's perception of it - above all, by incorporating into its foundations those comprising what I call the Fatal Trap (§2) - mechanistic physics ensured its own demise. It only needed experimental techniques to attain a certain level of sophistication for these intrinsic absurdities to reveal their presence by discrepancies between theory and experiment. This, of course, occurred around the end of the last century, but the rise of instrumentalist philosophies and the reification of measure-number algebra tended to obscure for the less philosophical the true nature of the situation.
But to end on a more positive note: for all its shortcomings, physics has succeeded in providing ontological philosophy with an immense body of detailed, systematic knowledge of the physical world. And one should not blame the physicist, who, qua physicist, is no more than a virtuoso of the phenomenal surface, for not organising this knowledge in the form of a rationally coherent world theory. This is a task for the metaphysician - or, as I prefer to say - ontologist. And the first step towards such a theory can only be the total dismantling of that fundamental error syndrome on which physics has hitherto been based - the Fatal Trap. We hope to attempt this in a later paper.
NOTES
1. But not, as is sometimes misleadingly claimed, Bacon; who states (The New Organon, Book 2, Aphorism VIII), "Nor shall we be led to the doctrine of atoms, which implies the hypothesis of a vacuum and that of the unchangeableness of matter (both false assumptions); we shall be led only to real particles, such as really exist.
2. We should now say 'precision'.
