Wranglers and Physicists: Studies in Cambridge Mathematical Physics in the 19th Century 
edited by P.M. Harman.
Manchester, 261 pp., £27.50, November 1985, 0 7190 1756 4
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Sir Edmund Whittaker’s History of the Theories of Aether and Electricity first appeared in 1910, and is mentioned at the very start of the book under review, though never again. The scope of Whittaker’s book is wide, its material densely organised, but it is a pleasure to read. Whittaker was a stylish writer and a distinguished thinker. Was it really necessary to go over part of this ground again in so much more detail? Unfortunately, there is good reason for doing so, if only because Whittaker did just this when, forty years later at the age of 78, he expanded his original history and added a second volume which was concerned with developments since 1900. The stylishness remains, but the reader’s mind is now alert to the possibility of misrepresentation – the account of the rise of relativity and quantum theory is deeply suspect. There is no suspicion of conscious falsification, rather that the teacher (and he was a great teacher) could not restrain himself from tidying up the arguments. It is with something of a sigh that we retrace the ground in the company of professional historians of science whose prose, it must be regretted, is not always as compelling as Whittaker’s. But their purpose is to tell it as it happened, and even in mathematical physics, whose final expression is so precise, the way truth is attained is by muddle and misconception. One could not believe a tale which unfolded itself too smoothly. I wish, however, that the editor, or one of the authors, had thought to include a critique of Whittaker’s reading of the story they tell, if only as a yardstick for those who will continue to consult him as the only comprehensive historian of a vast and complex phase of scientific thought.

The aim of the contributors is deliberately narrow: ‘the way in which the Mathematical Tripos at Cambridge shaped the physics of men such as William Thomson (later Lord Kelvin) and James Clerk Maxwell’. Of course, nobody believes that even Cambridge in the early years of the last century was so inward-looking that the Tripos and Smith’s Prize examinations were the sole influences on the thought of some of the most imaginative and profound scientists of the time. Nevertheless, they were the instrument by which new ideas might be disseminated, in a way that would be hard to reproduce nowadays. It was perfectly reasonable then for an examiner of Maxwell’s stature to announce a new result by asking the candidates to prove it. After all, only those who aspired to a career in learning, or as coaches for the next generation of wranglers (first-class mathematics graduates), would waste precious time on the arduous pursuit of an otherwise little-valued expertise.

New ways of thought were, indeed, sorely needed, as we can see in retrospect. Newton’s example of a hundred years earlier had been largely unheeded in England, though not entirely in Scotland and Ireland, but had borne fruit on the Continent, especially in France where Lagrange, Laplace, Fourier and many others had developed analytical methods which they applied with great power to problems of natural philosophy. During this period the Royal Society was dominated by Sir Joseph Banks, its president for 40 years, a natural historian who had sailed with Cook and who by temperament was a bigoted anti-mathematician. On the other hand, the empirical study of electricity and magnetism had flourished, and the history of English-speaking investigators from Franklin through Priestley, Michell and Cavendish to Faraday is evidence enough of creative talent applied with uncommon power. Where the Cambridge school of mixed (i.e. applied) mathematics made its mark was in its attempt to marry the qualitative results of the experimenters with the severely deductive mode of French mathematics. Progress was slow but ultimately spectacular, when Maxwell succeeded in formulating the electromagnetic theory which remains the foundation of all subsequent work. Yet the achievement, as frequently happens, left its mark on the victors, who, though still young and capable of great things elsewhere, could not now abandon the conceptual framework which had served them so well. They had to leave the next developments to a new generation of mathematical physicists who were less concerned with logical deduction than with elaborating and testing the consequences of ideas derived from experiment.

This, however, was at variance with the procedures of the French school. Building on Newton’s mechanics and his law of universal gravitation, they had established by 1800 a virtually complete system of celestial mechanics in which the gravitational forces between the Sun and all the planets, and their satellites, could be included. Given the state of the solar system at any instant, its subsequent history could be predicted with as much accuracy as desired, subject only to the patience of the analyst. It was the small discrepancies between prediction and observation that led Adams and Le Verrier to infer the existence of the so far unobserved Neptune, and to calculate with adequate precision where to point the telescope to find it. In this century Pluto was similarly predicted and discovered, and the methods of the French analysts are still very much alive, though now translated into computer terms, in the exploration of outer space. It is not surprising that the mathematicians should have seen their successes as the preliminary to conquering the whole range of physical phenomena.

Yet it could not be overlooked by philosophers that the empire of Newton and Laplace was built on a very dubious foundation. It was all very well to derive everything from a mathematical statement of the law of attraction – the inverse-square law – between massive bodies, but how did this action-at-a-distance really work? It was not just Nature that abhorred a vacuum, but Descartes and every right-thinking man: there must be something (for Descartes, vortices in the plenum) that allowed influences to be transmitted through apparently empty space. It might not be necessary for the mathematicians (for all their genius, mere calculators) to worry about mechanisms, but the natural philosopher must ask himself why the laws take the form they do. To bring rigour to this quest took well over a century. Descartes had not translated his model of vortices into mathematical terms, while Newton had dismissed everything Cartesian from his thoughts. But in the end the nature of the void was a problem that had to be faced, and with the establishment by experiment of the wave-theory of light in the early years of the 19th century the need for an aether was obvious: waves propagating from a luminous source into apparently empty space needed something to work on – something that was at the same time rigid enough to carry them indefinitely far at a very great speed, and pliable enough for the celestial bodies to move through without hindrance. Many were the models put forward, especially as it became clear that the aether had to function as the carrier not only of light but of the electrical and magnetic forces that also act through empty space. The wonderfully useful lines of force that Faraday introduced to describe the latter were to be seen as representations of the state of strain that electric charges and magnets induced in the aether. Long before Maxwell perfected his electromagnetic theory of light it was conjectured that the same fundamental mechanisms were at work, and that one aether alone held the clue to the variety of actions at a distance: but it took half a century to find a satisfying formulation.

In this exploration the Cambridge school were among the leaders, both in developing new mathematical tools – what we nowadays call vector calculus and teach to all physics students – and in devising models of the aether. It is here that the mechanical mode of thought, a legacy of celestial mechanics, shows its strongest influence and its quaintest manifestations. The disembodied equations, free from all mechanical attributes, that Maxwell finally achieved had their origin in an assembly of vortices, all spinning in the same sense and coupled by innumerable idle wheels. One cannot imagine that he ever really believed in their literal existence, however firmly he was convinced that the mechanical aether was a reality. More probably he found such models, as Kelvin certainly did, an encouragement to his imagination, and an equally valuable restraint at the same time. So long as the equations they wrote down had their counterpart in a realisable mechanism, they could be confident that they were not talking nonsense. Wrong they might be, and often were, but what they said was not silly, and anyone pushing into new territory of thought needs some such assurance. In the end, however, Maxwell could divorce the equations he had formulated from the scaffolding of models, writing a purely mathematical description of the results of laboratory experiments in electricity and magnetism, which in addition had solutions in the form of waves travelling at a speed that agreed with the speed of light.

This is not to say that the whole matter was finally resolved. For one thing, Kelvin was never able to accept, in the forty years of life remaining to him, the crucial ‘displacement hypothesis’ that Maxwell had introduced. His own strivings for an acceptable model left him, in this area alone, a hidebound reactionary. Their younger contemporaries were no more ready to abandon vortex models, which continued to proliferate in glorious and self-defeating complexity. It is valuable for a scientist to learn, from studies such as this collection, how hard a struggle a radically new idea has to endure before it triumphs. Max Planck expressed it in a celebrated sentence: ‘A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.’

Maxwell was in his early thirties when he formulated his electromagnetic equations, and died at 48 still wedded to aetherial vortices. To a modern physicist it almost passes belief that the everyday phenomenon of electrical resistance in a current-carrying wire should have been the obstacle at which he baulked. Would he, one wonders, have embraced the resolution of this problem that was initiated soon after his death? It was Helmholtz and Lorentz, among others, who revived and mathematised the old ideas of atomic theory – not so much sweeping the aether away as relegating it to a background against which electrically-charged particles were to play the leading roles. And with Einstein’s Special Theory of Relativity in 1905 the aether was doomed.

Whatever attempts were made after that time, and indeed are still made, if only by cranks, to restore the old ways of thought, something like positivism was now the order of the day. The task of the mathematical physicist was to describe the relationships of observed phenomena through equations of very general validity, but to eschew explanation in mechanical terms. The question ‘why?’ had ceased to be permissible.

Yet model-building cannot be dispensed with. The models we indulge in nowadays, and usually believe in as firmly as Kelvin and Maxwell believed in their aethers, are nevertheless models. One of the most remarkable achievements of ‘positivist’ thought in this century was the Matrix Mechanics that Heisenberg invented in 1926. Apparently without any model in his mind he devised a calculus that yielded the verifiable behaviour of a hydrogen atom and other simple systems. Almost simultaneously Schroedinger arrived at a simpler and mathematically equivalent procedure, Wave Mechanics, which can be visualised in terms of a wave in empty space controlling the behaviour of particles. However strongly one insists that Schroedinger’s wave-function is only a mathematical construct, the fact remains that the imagination is greatly aided if one pretends to oneself that it describes a real wave whose behaviour may often be guessed at without having to solve the equations completely. And it is by no means unusual to hear physicists, in their rare moments of philosophical speculation, referring to ‘the wave-function of the universe’, as though it were a real entity pervading all space – an up-to-date aether, in fact (but they are shy of suggesting that there is any medium carrying the wave).

When his attention is drawn to this aberration the physicist will rarely insist on the reality of the wave-function, but he will not yield so readily to the suggestion that the protons and electrons on which his whole theory of matter is based are equally models. Nevertheless, even though modern physics is inconceivable without them, they must not be regarded as ultimate realities, if only because the properties one is forced by quantum theory to assign to them are as strange and, in everyday terms, as unimaginable as anything the 19th century postulated for the aether. But if it was a wrench to discard the aether, what agonies shall we endure before we are persuaded to abandon our fundamental particles, which provide such a superb explanation for a very much more diverse range of observations? Let us not sneer at the naivety of our predecessors, giants by any standard, because we have outgrown the models by which they laid the foundations for our own. However tedious it may seem to follow the details of studies like those in this book, the exercise is bound to make even the most complacent physicist wonder about the permanence of his world-picture.

Yet there is sound sense behind his complacency, since to most physicists the problems that engage the modern Maxwells and Einsteins are indeed irrelevant. It is all too easy to assign a special importance to the heaven-storming task of constructing a universal theory of the material world, as if this were the primary goal of physics, in whose company earthier activities, like solid state physics, not to mention chemistry and sciences other than physics, are secondary and hardly worth the attention of a really first-rate mind. It must be admitted that the deep questions have attracted a high proportion of really creative mathematical physicists ever since it came to be accepted that the expression of the fundamental laws must be in terms of lapidary simplicity. Yet for all its successes there are severe limitations to a natural philosophy which seeks to build a complete system on the basis of a small number of hypotheses, by the rigorous application of mathematical deduction. In practice, mathematics proves a weak tool for tackling any but the simplest of problems; it is beyond mathematical analysis to work out the structure of any but a few of the lightest atoms from first principles, or any molecule containing carbon, the stuff of all living creatures. In the face of these problems, experiment and theory proceed hand in hand, experiment telling the analyst what sort of solution to look for, and what approximations he may hope to get away with. In no sense, therefore, can the basic laws be taken as the starting-point for a modern Laplace to re-invent the universe. Indeed, only a few in this century have tried, notably Einstein and Eddington in their later years, and Dirac for most of his long life – and all were fated to lose touch with the real universe in their attempts to make it anew.

Since 1945 the majority of those dedicated to fundamental physics have taken a rather less ambitious view. It has been challenging enough to devote huge material and intellectual resources to enlarging by experiment and theory their understanding of the structures underlying the relatively homely electron and proton. Yet only a few astrophysicists among all other scientists have found a use for even this limited advance. Most of us do not understand the new languages of quarks and field theory; we find the old ways good enough – the electron and proton suffice, along with the positron and neutron, more recent, but known since 1932. And for our calculations we use the quantum mechanical principles whose definitive form was reached at the same time.

In all but fundamental physics, the problems of science are generated by the complexity of real materials, for whose understanding mathematical rigour is usually a handicap. When Landau and Lifshitz write in the preface to a distinctly severe text, ‘No attempt has been made at mathematical rigour ... since this is anyhow illusory in theoretical physics,’ they are not rationalising a sense of inferiority compared with real mathematicians. Rather, from the standpoint of highly successful performers they are stating a truth of experience: that the scientist who tests every step for logical soundness will never get anywhere. Guessing the way and travelling hopefully prove to be the best course – time enough when you reach the end to ask if that was where you meant to go and whether there was not perhaps a better way.

The split between mathematician and physicist was not nearly so marked in the last century as it is now; Kelvin could bestride both worlds and dominate the engineers as well. But even then the difficulty was apparent in the limited choice of subject-matter that could be assimilated into the mathematical mode. As one of the authors in this book remarks, with unconscious irony, ‘The progress of that branch of physics which, until the world war of 1939-45, had been least touched by mathematics shows what neglect of advances in physical mathematics entails: one would rather the glorious confusion of [Kelvin’s] mathematical physics than the chaos of contemporary thermodynamics.’ It seems not to occur to him, or to some recent writers who have sought to repair the omission by reducing thermodynamics to a rigorously axiomatic scheme, that distilling a rare essence and packaging it in elegant cut-glass phials puts it out of reach of the ordinary man. To the engineer, the chemist or the only physicists who really care for it, thermodynamics is a tool to be understood and used, not contemplated as a mystery. The mathematician seeks a flawless logical deduction from carefully-stated premises so that no exception, however pathological, shall escape to raise doubts about how many others may lurk undetected. The physicist is perfectly happy with a flimsy structure, once he has satisfied himself that his intuitive grasp of it will warn him in time of its rare failures.

Because both believe they are doing things the right way it is unfortunate that the further development of the Cambridge school of mathematicians and the rise of the Cavendish Laboratory under J. J. Thomson and Rutherford led eventually to an almost complete divorce in Britain between the experimental scientists and the mathematicians. The consequence has been a marked tendency on the part of young mathematicians of talent to neglect the difficult, earthbound problems to which they could contribute, incidentally promoting the kind of science that modern technology needs: instead, they all too often become involved in elaborations of hypotheses that concern no one else. It would be of great interest to carry the story of this book beyond the time of Maxwell to see how the Cambridge schools (and those in other countries) came to diverge. On the evidence before us, the authors of the present study have the ability to take it further. At the same time, one would like to know more about the relationship between pure and applied mathematicians. Cayley, who dominated pure mathematics at Cambridge during the period of this work, gets no more than passing mention and perhaps had only a minor influence on the teaching programme. But as time went on, and especially after the appearance of Hardy and Littlewood, pure mathematics began to take the lion’s share and the applied mathematicians often found themselves respected by neither their pure colleagues nor the physicists. The resulting weakness of British theoretical physics compared to, say, that in the United States and Germany has still not been fully cured.

Here is a topic for research which might well help in a practical way to eliminate a fault in our academic structure, and incidentally give historians of science a standing among practising scientists which at present they lack. The rise and fall of the aether plays no part in the modern physicist’s everyday concerns. It is not like an ancient philosophy or an epoch of political history which still has lessons to teach us: the aether theory simply failed, the technical tricks were incorporated into textbooks, and the research worker can afford to forget it. A study of Cambridge mathematical physics in the 19th century, even one as competent as this, must make its way as an academic exercise in its own right, and this is no joke for a new discipline in times of hardship. Yet it costs little compared with scientific research, and one must hope for a generous appreciation by scientists of the value, in the long run, of ancestor-worship.

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Vol. 8 No. 9 · 22 May 1986

SIR: I enjoyed Brian Pippard’s excellent review (LRB, 20 February). He seems to accept that our scientific theories and well-held tenets are ‘only models’, but in the same breath talks about ‘attaining truth’, and says that it is difficult to accept some quantum consequences as ‘ultimate reality’ because they are ‘so strange’. This suggests that there is an attainable ultimate reality that is true, while admitting that the ultimate criterion of acceptance is aesthetic. Maxwell and the French logicians were pursuing an aesthetic of mathematical elegance that applied Occam’s razor with a vengeance. Its success and appeal are not surprising since this criterion of simplicity has served us very well since Occam’s day and has powered the triumph of Renaissance science over Medieval occultism. But we have reached the end of that road: after seemingly complicating the four elements by creating the hundred or so chemical elements, we then reduced things back to four with the electron, proton, photon and neutron. Then the cycle repeated itself and after the plethora of elementary particles we now suppose a handful of quarks … This seems dangerously like a cul-de-sac, chasing through infinite regress a will-o’-the-wisp. All that is demonstrated is the mathematicians’ imaginative power to construct logical relationships following the ‘severely deductive mode’ of the revolutionary French. Yet the very names of their creations, ‘quarks’ and ‘gluons’ betray the invention, the sheer imagination, that is characteristic of all art. All scientific theories are creations in this sense. Newton didn’t discover gravity, he invented it, and Einstein has distilled it into ‘a rare essence in a cut-glass phial’. Einstein’s picture is more elegant, with the precision of a Canaletto; Newton appeals as does the comfortable impressionistic realism of Constable; the fertility of contemporary abstract theories parallels the abstract art of the 20th century, with a Miro in Heisenberg and a Dali in Schrodinger. It goes without saying that none of these are ‘true’ or ‘truer’, or, strange to say, inconsistent with one another. They are just differing viewpoints, and the sooner physicists explicitly see this, the clearer the nature of what they do will become to them and us.

John Marks
Halton General Hospital, Runcorn

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