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the complete review - history / mathematics

Masters of Theory

by
Andrew Warwick

general information | review summaries | our review | links | about the author

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Our Assessment:

A- : fascinating, but overextended

See our review for fuller assessment.

  From the Reviews:

• "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. (...) 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." - Kathryn M. Olesko, American Scientist
  
  
• "(T)he motivated reader will discover a rich, deep and altogether fascinating historical reconstruction of a critical period. The book also benefits from the continuity and thematic control that can be achieved by a single author -- Andrew Warwick -- of great knowledge and skill." - Daniel Siegel, Physics World

Please note that these ratings solely represent the complete review's biased interpretation and subjective opinion of the actual reviews and do not claim to accurately reflect or represent the views of the reviewers. Similarly the illustrative quotes chosen here are merely those the complete review subjectively believes represent the tenor and judgment of the review as a whole. We acknowledge (and remind and warn you) that they may, in fact, be entirely unrepresentative of the actual reviews by any other measure.

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The complete review's Review:

       Masters of Theory looks at the evolution of the study and practise of mathematics (and mathematical physics) at Cambridge University in the 19th and early 20th century. Cambridge was the major centre of mathematical study in Britain, and the university attended by many of the men who would go on to become leading British scientists, and Warwick shows how the tradition there -- and specifically the training oriented towards success in the so-called Tripos (the exams by which students were ultimately judged and ranked) -- led to it becoming such a leading centre, as well as some of the limitations of the system.
       Warwick follows the fascinating and unusual evolution of mathematical study in England and specifically at Cambridge, first hampered by a reliance on Newtonian notation (his fluxional calculus, which differs from the Leibnizian notation that has become the standard one) that cut English mathematicians off from much of the work that was being done on the continent, and then by a very specific Cambridge approach to mathematics.
       The Tripos were what students strived to succeed at at Cambridge. Originally the maths required was fairly rudimentary -- exams were oral, which limited what could be demanded of students, memorization trumping actual problem-solving. The shift to written exams had a particularly great impact on maths, as it immediately allowed for much greater complexity in the questions that could be posed (as opposed to other subjects, where essay-answers were required: the difference between what could be present orally or in written form differed nowhere near as much).
       A presence throughout Masters of Theory are the private tutors -- the coaches -- that helped students prepare for their exams. As it turned out, the course-offerings at Cambridge didn't adequately prepare students for exam-taking, and from early on private tutors thrived (and had such an influence that there were frequently efforts to ban them). Much like American SAT prep courses, or British and Japanese crammers, much of the basic, broad knowledge that students were supposed to know -- and test preparation in the form of problems sets and the like -- was only provided outside the university itself. It's a fascinating history, and Warwick profiles many of the leading coaches and their methods (changing over the decades, as does the focus of the Tripos) and successes.
       The Tripos were significant: to come out top -- as so-called "senior wrangler" -- was a great achievement and honour, and a great career boost (especially for those from poorer families). Several British greats were senior wranglers -- notably William Thomson (Lord Kelvin) and A.S.Eddington --, with the annual results published in newspapers and the top finishers fêted.
       Over the decades the nature of the (mathematical) Tripos changed radically, maths moving from a minor part of what was expected of students to (in 1824) a separate set of exams all by itself to a highly specialised exam, with students able to choose to answer questions only dealing with specific areas. Noteworthy also is the continuing focus on physics and real-world applications, training that played a significant role in making Cambridge such an important centre for physics.
       (Sadly, also, much of the study seems to have been only for the sake of the Tripos, with many who had great success in the exam becoming coaches. While some coaches did also make significant contributions to the mathematics of the day, there clearly seems to have been quite a bit of a waste of talent here.)
       Warwick provides many case-histories, of both coaches and pupils, a fascinating look at how academia changed in the 19th century. The Tripos became incredibly demanding, and the pressure to succeed enormous. Warwick describes the mental and physical breakdowns -- and also devotes a chapter to efforts by some students to balance the mental rigours with physical exercise, showing the rise of interest in both competitive sport and a more widespread belief in mens sana in corpore sano.
       Of particular interest is also how the Tripos defined -- and limited -- mathematics (and physics) at Cambridge, enough to allow for incredible success, but also limiting influence and, ultimately, openness to advances from elsewhere. Warwick neatly shows how Cambridge maths -- and the nature of Tripos problem-solving -- essentially became a language of its own: the closed circle there all approached maths and physics in the same way, familiar with a certain way of presenting and then tackling problems. The demands of the exams required quick responses, jumps and shorthand and techniques familiar to all Cambridge but hard to understand for those who had not been trained in the system. Similarly, those in Cambridge did not focus nearly as much attention on what was being done elsewhere. Essentially, Warwick argues, the Tripos cut off much of Cambridge mathematics from the rest of the world. The great success of much of it obscured this fact to some extent, but eventually it caught up with them: in some fascinating chapters on the Einstein-reception in the early 20th century, Warwick shows how (and why) Cambridge mathematicians and physicists were unwilling and unable to deal with this paradigm-shift.

       Masters of Theory is an incredible book, packed with information. Warwick's detailed study, ranging from personal portraits to the changes caused by a shift from slate to pen(cil) and paper to Tripos-problem examples, is absolutely fascinating. It is also simply too much -- at least too much to be a neatly argued, elegantly presented book for easy, casual perusal. It is simply bursting with information and minutiae, footnotes galore looking temptingly elsewhere, making for a very unwieldy tome.
       The information provided (and the generally very clear analysis) is wonderful -- but also over-abundant. There's so much here -- and so much that is simply fascinating -- that attention is pulled in all sorts of different directions, making reading it utterly exhausting. It makes for a fascinating experience, but it is more a book to be studied than read -- not studied page by page, but perhaps section by section, with frequent excursions to the many fascinating secondary sources cited. (We could live for a year on a selection of the books listed in the mouth-watering bibliography alone.)
       Heady stuff, but an absolute must for anyone interested in the history of science or the history of education.

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Links:

• University of Chicago Press publicity page

• Physics Today

• Homepage for the Natural Sciences Tripos

• See Index of books dealing with Mathematics
• See Index of books on Philosophy
• David Lindley's biography of senior wrangler William Thomson, Degrees Kelvin

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About the Author:

       Andrew Warwick teaches at Imperial College, London.

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© 2004-2021 the complete review

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