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More About This Textbook
Overview
Winner of the the Susan Elizabeth Abrams Prize in History of Science.
When Isaac Newton published the Principia three centuries ago, only a few scholars were capable of understanding his conceptually demanding work. Yet this esoteric knowledge quickly became accessible in the nineteenth and early twentieth centuries when Britain produced many leading mathematical physicists. In this book, Andrew Warwick shows how the education of these "masters of theory" led them to transform our understanding of everything from the flight of a boomerang to the structure of the universe.
Warwick focuses on Cambridge University, where many of the best physicists trained. He begins by tracing the dramatic changes in undergraduate education there since the eighteenth century, especially the gradual emergence of the private tutor as the most important teacher of mathematics. Next he explores the material culture of mathematics instruction, showing how the humble pen and paper so crucial to this study transformed everything from classroom teaching to final examinations. Balancing their intense intellectual work with strenuous physical exercise, the students themselves—known as the "Wranglers"—helped foster the competitive spirit that drove them in the classroom and informed the Victorian ideal of a manly student. Finally, by investigating several historical "cases," such as the reception of Albert Einstein's special and general theories of relativity, Warwick shows how the production, transmission, and reception of new knowledge was profoundly shaped by the skills taught to Cambridge undergraduates.
Drawing on a wealth of new archival evidence and illustrations, Masters of Theory examines the origins of a cultural tradition within which the complex world of theoretical physics was made commonplace.
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Meet the Author
Andrew Warwick is a senior lecturer in the history of science at Imperial College, London, and coeditor of Teaching the History of Science and Histories of the Electron: The Birth of Microphysics.
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Cambridge and the Rise of Mathematical Physics
By ANDREW WARWICK
The University of Chicago Press
Copyright © 2003 The University of Chicago
All right reserved.
ISBN: 9780226873756
Chapter One
Writing a Pedagogical History of Mathematical Physics
Late to bed and early rising, Ever luxury despising, Ever training, never "sizing," I have suffered with the rest. Yellow cheek and forehead ruddy, Memory confused and muddy, These are the effects of study Of a subject so unblest.
JAMES CLERK MAXWELL, NOVEMBER 1852
1.1. Learning and Knowing
At 9 a.m. on 3 January 1876, J. H. Poynting (3W 1876) and around a hundred of his undergraduate peers concluded their mathematical studies at Cambridge by embarking on nine days of gruelling examination in the University Senate House. They were anxious men. After ten terms of intensive study, they would be ranked by their examiners according to the number of marks they could accrue answering increasingly difficult mathematical problems against the clock. At approximately half past ten on the morning of the fourth day, the most able candidates encountered the first problem set by Lord Rayleigh, the examiner with special responsibility for mathematical physics:
vi. Investigate the equations of equilibrium of a flexible string acted upon by any tangential and normal forces. An uniform steel wire in the form of a circular ring is made to revolve in its own plane about its centre of figure. Show that the greatest possible linear velocity is independent both of the section of the wire and of the radius of the ring, and find roughly this velocity, the breaking strength of the wire being taken as 90,000 lbs per square inch, and the weight of a cubic foot as 490 lbs.
Rayleigh's question was considered difficult for a paper set for the first four days of examination, but Poynting knew he could take no more than eighteen minutes to complete the investigation and solve the problem if he were to remain among the top half dozen men. He would almost certainly have committed a form of the required investigation to memory and begun at once to reproduce the relevant sections as quickly as possible. The next part of the question was more difficult. He had to apply the general theory just derived to solve a problem he had never seen before. What remains of Poynting's efforts that January morning is reproduced below in figures 1.2 and 1.3.
* * *
On 14 December 1883, J. H. Poynting, Professor of Physics at Mason College Birmingham, sent a paper to Lord Rayleigh asking in a covering letter whether he would communicate it to the Royal Society for publication in the prestigious Philosophical Transactions. Poynting's work would turn out to be one of the most important ever published in electromagnetic theory. In the paper he derived a simple mathematical expression relating the flow of energy in an electromagnetic field to the electrical and magnetic forces of which the field was composed. The importance of the expression lay in the fact that it helped to clarify a number of conceptual difficulties that had troubled students of James Clerk Maxwell's new field theory of electricity and magnetism for almost a decade. The "Poynting vector" (as the expression became known) showed, for example, how the energy of a battery was transmitted during conduction around an electrical circuit, even though, as Maxwell's theory seemed to require, nothing flowed along the conducting wires. The final steps in the derivation and Poynting's explanation of the result are shown in figure 1.1.
1.2. Physics in the Learning
At the close of the nineteenth century, Cambridge University had for some seventyfive years been one of Europe's foremost training grounds in mathematical physics. Among those who had learned their craft within its walls were William Hopkins, George Airy, George Stokes, John Couch Adams, P. G. Tait, William Thomson, James Clerk Maxwell, Edward Routh, J. W. Strutt, Osborne Reynolds, W. K. Clifford, G. H. Darwin, Horace Lamb, J. H. Poynting, J. J. Thomson, Joseph Larmor, W. H. Bragg, E. T. Whittaker, and James Jeans. These men, together with a much larger number of only slightly less wellknown figures, had helped to develop what in the twentieth century would be known as "classical physics." From the cosmic sciences of celestial mechanics, thermodynamics, and electromagnetism to the humbler dynamics of the billiard ball, the boomerang, and the bicycle, they considered themselves the mathematical masters of every known phenomenon of the physical universe. Joseph Larmor, shortly to become Lucasian Professor of Mathematics, was about to publish a remarkable treatise, Aether and Matter (1900), in which he would attempt to explain the whole material universe in terms of tiny charged particles moving in a ubiquitous dynamical ether. This was an ambitious project, yet one befitting a man working at the intellectual hub of the largest empire the world had ever known. Cambridge University, situated in a small market town in the East Anglian fens, was home to more expertise in mathematical physics than any other place in Britain or her empireperhaps more than any other place on earth.
This study aims to deepen our understanding of the nature and historical origins of that expertise by exploring it from the perspective of pedagogy or training. The subject of scientific education has received considerable attention from historians over recent years, yet few studies have made any sustained attempt to use the educational process as a means of investigating scientific knowledge. Likewise, little attempt has been made to provide a historiography of the rise of modern mathematical physics in terms of the formation and interaction of communities of trained practitioners. This failure to explore the relationship between learning and knowing is surprising, moreover, as it is now several decades since philosophers of science such as Thomas Kuhn and Michel Foucault drew attention to the importance of training both in the production of knowing individuals and in the formation of the scientific disciplines. Kuhn's most important claim in this regard was that scientific knowledge consisted in a shared collection of craftlike skills leaned through the mastery of canonical problem solutions. Foucault's was that regimes of institutionalized training introduced in the decades around 1800 found new and productive capacities in those subjected to its rigors, and imposed a new pedagogical order on scientific knowledge. But, despite numerous references to their works, Kuhn and Foucault's suggestive comments have received little critical discussion, especially from a historical perspective. Historians of the mathematical sciences have in the main focused on the history of theoretical innovation while simply assuming that their key characters are able to innovate and to communicate with their peers via a takenforgranted collection of shared technical skills and competencies. As an initial point of departure, therefore, I would like to explore the proposition that the history of training provides a new and largely unexplored route to understanding the mathematical physicist's way of knowing.
Consider the second of the episodes in J. H. Poynting's early career outlined above. His 1883 paper on the flow of energy in an electromagnetic field is a classic example of the kind of published source commonly used by historians of mathematical physics to write the discipline's history (see fig. 1.1). What are its key characteristics? First and foremost it presents a neat and novel result in mathematical form. The socalled Poynting vector was recognized in the mid 1880s as a major contribution to electromagnetic field theory and remains to this day an important calculating device in physics and electrical engineering. Second, the paper opens with a clear statement of the origin and nature of the problem to be tackled. According to Poynting, anyone studying Maxwell's field theory would be "naturally led to consider" the problem "how does the energy about an electric current pass from point to point"? Third, the solution to the problem offered by Poynting seems to emerge naturally from Maxwell's theory. Starting with Maxwell's field equations and expression for the total energy stored in an electromagnetic field, Poynting deduces the new vector by the skilful manipulation of secondorder partial differential equations and volume and surface integrals. Finally, the paper concludes with the application of the vector to seven wellknown electrical systems. Poynting shows in each case that the vector makes it possible not only to calculate the magnitude and direction of energy flow at any point in the space surrounding each system but also to trace the macroscopic path of the energy as it moves from one part of the system to another.
Papers of this kind have long appealed to historians of physics, partly because they are easily accessible in research libraries around the world but also because they express the ideals and aspirations of physicists themselves. Poynting's publication was intended to assert his priority in deriving a new and significant result, thereby building his fledgling reputation as an original researcher. This route to establishing one's name in the mathematical sciences has existed since at least the late seventeenth century, the annual volumes of such prestigious journals as the Royal Society's Philosophical Transactions providing an internationally recognized and seemingly unproblematic record of the accretion of new knowledge in mathematical physics. It is important to note, however, that sources of this kind tend to impart a particular and partisan view of the discipline's history. Notice, for example, the stress laid on novelty and priority. Poynting's paper is memorable because it tells us when and by whom a neat and important equation was first published. It tells us nothing of why the equation was found at that particular time nor why it was Poynting rather than another student of Maxwell's work who found the new result. The paper also offers an implicit account of how the vector was derived, and, by broader implication, of how work of this kind in mathematical physics generally proceeds. Taking the published derivation at face value one might imagine that mathematical physicists generally start with a wellrecognized theoretical problem, solve the problem by the application of logical mathematical analysis, and conclude by applying the result to a number of previously troublesome cases. Perhaps most significantly of all, Poynting's narrative makes Maxwell's electromagnetic theory the central player of the piece. We are led to believe that this theory not only prompted his initial investigation but lent meaning to his result. On this showing, the history of mathematical physics is essentially a history of physical theories, mathematics being the mere language used by individuals in making more or less original contributions to the theories' development.
But to what extent does Poynting's paper provide a faithful account either of his motivations for undertaking research in electromagnetic theory or of the route by which he obtained the new result? Contrary to his assertion, one is not led "naturally" to the energyflow problem through the study of Maxwell's work. Only one other student of Maxwell's theory, Oliver Heaviside, attempted to investigate a special case of this problem, and he had reasons for doing so that were not shared by Poynting. Poynting himself almost certainly took little if any research interest in Maxwell's theory until the autumn of 1883 and began to do so only after he had hit upon the new result by a different route. Up until that time his research had concentrated mainly on precision measurement and the theory of sound. Moreover, the neat derivation of the vector published in the Philosophical Transactions was not even entirely his own work. Both of the referees appointed by the Royal Society to review the paper recommended that a number of revisions be made before it appeared in print. Finally, the problem that had actually led Poynting to the new vector was tucked away at the end of the paper as if it were a mere application of the more general expression. In summary, the structure and implicit rationale of the published work is a skilful fabrication bearing little relation to its actual origin and offering an artificial view of how research is undertaken.
My purpose in introducing Poynting's work is not, at least at this stage, to provide a detailed analysis of its origins or content, but rather to highlight the ways in which publications of its kind display some aspects of the mathematical physicist's craft while concealing others. First and foremost it is an exemplary piece of readymade science, carefully tailored for the consumption of the wider expert community. A glance at the page reproduced in figure 1.1 will immediately convince most readers, especially those without a training in advanced mathematics and physics, that this is a specialized form of technical communication aimed at a small and select readership. For most people it will seem a dizzying sequence of incomprehensible jargon and symbols, an unwelcome reminder perhaps of the impenetrability of modern physics or of frustrating hours spent in mathematics classes at school. Poynting clearly assumes that his reader is familiar with the rudiments of Maxwell's electromagnetic field theory and with the operations of higher differential and integral calculus. There are nevertheless limitations to what he expects even his specialist audience to be able to follow. Notice for example that he provides precise page references to equations borrowed from Maxwell's Treatise on Electricity and Magnetism. Poynting does not expect the reader to share his own intimate knowledge of this difficult work. Notice too that he sometimes offers a few words in explanation of a step in the argument and often refers back to equations numbered earlier in the derivation. These remarks and references indicate places where Poynting thought the reader might require additional information or clarification before assenting to the current step in the argument. He effectively anticipates and manages the reader's progress through the technical narrative. Notice lastly the confident and succinct way in which Poynting explains the physical meaning of the final and forbidding equation (7). If the reader agrees with Poynting's derivation and interpretation of this equation, the rest of the paper follows straightforwardly.
What this work reveals, then, is how a brilliant young mathematical physicist chose to present part of his research to a broad audience in the mid 1880s. What it conceals is how he produced the work in the first place, why he chose to present it in the form described above, and how his peers responded to the paper. We might ask, for example, how long it took to write. Was it a few days, a few weeks, or perhaps a few months? Unfortunately the paper offers no clues to the temporality of its own composition. Why did Poynting present his work as a logical succession of problems, solutions, and applications when, as I have already noted, it was produced by a quite different route? Were there conventions regarding the way work of this kind was structured, and, if there were, how did Poynting learn them? How did he recognize the importance of his work to Maxwell's theory so quickly given that he had previously shown no research interest in this area? Did all mathematical physicists of Poynting's generation have an unspoken knowledge of Maxwell's Treatise on Electricity and Magnetism and the tricky theoretical questions it raised but left unanswered?
(Continues...)
Table of Contents
List of Illustrations
Preface and Acknowledgments
Note on Conventions and Sources
1 Writing a Pedagogical History of Mathematical Physics
2 The Reform Coach Teaching Mixed Mathematics in Georgian and
Victorian Cambridge
3 A Mathematical World on Paper The Material Culture and Practice
Ladenness of Mixed Mathematics
4 Exercising the Student Body Mathematics, Manliness, and
Athleticism
5 Routh's Men Coaching, Research, and the Reform of Public Teaching
6 Making Sense of Maxwell's Treatise on Electricity and Magnetism
in MidVictorian Cambridge
7 Joseph Larmor, the Electronic Theory of Matter, and the Principle
of Relativity
8 Transforming the Field The Cambridge Reception of Einstein's Special
Theory of Relativity
9 Through the Convex Looking Glass A.S. Eddington and the Cambridge
Reception of Einstein's General Theory of Relativity
Epilogue: Training, Continuity, and Change
Appendix A: Coaching Success, 18651909
Appendix B: Coaching Lineage, 18651909
Bibliography
Index