American Scientist

Masters of Theory: Cambridge and the Rise of Mathematical Physics.
Andrew Warwick.
xiv + 572 pp. University of Chicago Press, 2003. $89 cloth, $29 paper.

By the end of the 19th century, such well-known British physicists as William Thomson, James Clerk Maxwell, J. J. Thomson, W. H. Bragg and James Jeans had trained for and survived the Cambridge University Mathematical Tripos, an analytical written examination for the bachelor's degree, lasting several days and demanding an exacting virtuoso performance in the mathematical techniques of a classical physics based on principles rooted in Newtonian mechanics. Masters of Theory, Andrew Warwick's long-awaited study of this test, is a major contribution to the historical scholarship on science teaching and learning and on mathematical physics. The central element of Warwick's imaginative and richly documented story is the emergence of a training style that required students to prepare for the examination outside the classroom with special tutors known as "coaches," the most well-known of whom was Edward Routh. That the Tripos had such a profound effect on the intellectual orientation and research style of so many physicists is all the more remarkable when one considers that it was the apotheosis of Cambridge undergraduate education, in which research was scarcely the goal of instruction.

Based on the author's doctoral dissertation, Masters of Theory is a part of the growing literature on science pedagogy that has drawn its theoretical inspiration from Michel Foucault, Thomas Kuhn, Jerome Ravetz and others, who early on made provocative statements about scientific training but lacked sufficient data from empirical studies to substantiate them. Warwick provides an extremely important case study.

A part of the liberal undergraduate education at Cambridge, where men were for the most part trained for the liberal professions rather than for scholarship, the Mathematical Tripos had its origins in the 18th century in the university's Senate House Examination, an oral disputation originally conducted to test a student's knowledge broadly. Merely passing was not the point. In 1747 university officials began to publish the rank order of grades on the examination, listing the highest scorers (known as "wranglers") at the top and the low-scoring hoi polloi at the bottom; only those with the highest ranking could compete for scholarships. By the 1760s, though, the examination had changed markedly: The emphasis on mathematics had increased, written answers were required for honors status, and the competition had intensified. When an examination for classical studies (the Classical Tripos) was founded in 1824, the Senate House Examination became known as the Mathematical Tripos.

Through Warwick's masterful treatment, the Tripos becomes a revealing artifact of Victorian culture. Just as Frederic Lawrence Holmes's studies of laboratory notebooks have done, Masters of Theory diverts attention from a scientist's published papers to knowledge as it is being made, demonstrating once again how published scientific literature masks the route to discovery and novelty. But unlike the notebooks Holmes analyzed (intensely private records, which seemed to have no contact with life outside the laboratory), Warwick's Tripos examinations were public events saturated with palpable stress, surrounded by elaborate ritual (both before and after the exam) and imbued with social significance (especially status)—elements that make them richer sources of cultural history than laboratory notebooks could ever be. Warwick's own historical virtuosity is especially evident in chapters devoted to these cultural dimensions of the Tripos: its linkages to the history of body and gender during the Victorian era, its cultivation of a system of morals and values that defined the Victorian gentleman, and its dependency on forms of sociability in instruction that underscored the central role of face-to-face encounters and human bonding in small-group learning under special tutors. These chapters in particular should provoke historians to reexamine their own assumptions about the historical meaning of technical knowledge in context, just as they will challenge educators and textbook publishers to rethink the rush toward online instruction.

Yet although copies of the Tripos examinations themselves are abundantly available, the answers—the real evidence for knowledge-in-the-making—are less so. Warwick successfully reconstructs the content and more so the experience of working through those answers by triangulating from sources that touch on the learning experience: diaries, correspondence, autobiographies and textbooks. Likewise, although Warwick has only two sets of coaching notes (Stephen Parkinson's from the 1850s and Robert Webb's from the 1890s), his wide-ranging knowledge of Cambridge culture permits him to extrapolate general observations about the coaching and training experience as a whole from his sources. (Interestingly, coaching declined in the 1880s.)

Masters of Theory ends with four chapters describing how Cambridge practitioners reacted to new developments, including Maxwell's Treatise on Electricity and Magnetism (1873) and Albert Einstein's general theory of relativity (1916). Of note is Warwick's sustained and convincing argument that Maxwell's Treatise was not a response to study of electrical metrology by the British Association for the Advancement of Science, but rather the outcome of unresolved pedagogical issues after electromagnetic theory was reintroduced into the Tripos in 1868. These final chapters, laced with the intellectual ruminations of Cambridge-trained mathematical physicists wrestling with new theories, demonstrate how difficult it was to transcend the intellectually conservative mindset shaped by the pedagogical regime of the Tripos.

Warwick strongly and repeatedly emphasizes that the broader international significance of his work is found in the material culture of the "pen, ink and paper" that constituted the tools of the Cambridge mathematical physicist, who worked at his desk, not in a laboratory. He also suggests that examinations were central to the development of mathematical physics elsewhere. For instance, at one point he interprets the Prussian system of state teaching examinations as being shaped in the Cambridge mold, and he draws strained conclusions. The Cambridge Tripos—which tested students for analytical virtuosity—was for undergraduates, was not in service of the professions and was strongly tied to a local culture. By contrast the Prussian state examinations—which were designed to assess only competency—were for graduate students, were aimed at professional certification (for example, teaching mathematics or science at secondary schools) and were administered by academics for the entire state, as had been similar examinations in forestry, architecture and civil engineering in the 18th century. So to the degree that individuals such as the theoretical physicist Franz Neumann complained about the absence of questions on physics (or other subjects) on the Prussian teaching examination, they were expressing a desire for greater career opportunities for their own pupils, not for a material culture of pen and paper, which in any event was already well established, as 18th-century German textbooks for mathematically technical subjects indicate. For this reason, and others, the more significant implications of Warwick's work lie, I believe, beyond both the material culture of pen and paper and the examination system that relied upon it.

If one measure of the value of a book is the number of new avenues of research it suggests, Warwick's succeeds remarkably: He tantalizingly hints at several implications of his study, all of which warrant further investigation. The most pressing concerns the actual percentage of Tripos examinees who went on to become professional physicists or teachers of physics (or mathematics) at any level, a statistic that would also be useful for understanding why a research school in mathematical physics emerged at Cambridge in the 1870s but not earlier. Another issue concerns the relationship between the techniques of the Tripos and experimental results. Warwick calls his Tripos examinees "masters of theory," but because he does not delve into their conception of physical reality, they seem more like "masters of technique." The establishment of the Cavendish Laboratory in 1874 and the appearance of Maxwell's Treatise on Electricity and Magnetism in 1873 are important starting points for understanding how empirical results, especially ones based on precision measurement, meshed with the intellectual perspectives cultivated in the Tripos and its attendant lectures, coaching sessions and textbooks.

Finally, given the profound and lasting influence of the Mathematical Tripos on the development of mathematical physics in Great Britain, it is important to learn why, precisely, a technical subject came to dominate Cambridge undergraduate studies by the end of the 18th century. The link between Newton's work and Anglicanism, detailed by Warwick, is certainly part of the reason. Warwick's book, like Joan Richards's Mathematical Visions: The Pursuit of Geometry in Victorian Britain (1988), motivates us to take a broader look at the cultural role of mathematics in building character, in transforming merit into a social phenomenon, and in achieving social mobility and status in a world that was rapidly changing—economically, politically and socially.

The most important conclusions of Warwick's study, however, concern everyday practices in science, especially those that are involved in the regeneration of the discipline and the reproduction of its personnel. Pace Thomas Kuhn (and even Ludwik Fleck), "normal science" issues not from a canonical textbook, but rather from specific pedagogical practices repeated over time that may eventually become the foundation of a textbook, although not necessarily. Those pedagogical practices—which undergird the successful transmission of technical skills—are for Warwick a form of life-sustaining sociability necessary for disciplinary survival (an observation also made by Sharon Traweek in her 1988 book Beamtimes and Lifetimes: The World of High Energy Physicists).

Thus the broader significance of Warwick's book lies in its articulation of the complex social systems required to keep science operating. Science does not simply exist as a body of knowledge perched on a library shelf and poised to survive no matter what. It requires forms of sociability to exist and persist: a cadre of good teachers, attention to the introductory stages of learning in any subject, the coordination of secondary and higher education, a system of values that holds scientific and mathematical education in high regard, and finally, the assignation of social prestige not only to a scientific career but also (and perhaps more importantly) to science teaching. In this, Masters of Theory is a sober admonition to policy makers that the neglect of science education can only spell the demise of science itself.