Impure mathematics at Cambridge

I have remarked before that the Mathematics Tripos at Cambridge, with its impure emphasis on the calculations needed for mathematical physics to the great detriment of pure mathematical thinking, understanding and rigor, had deleterious consequences across the globe more than a century later.  Even as late as the 1980s, there were few Australian university mathematics degree programs that did not require students to waste at least one year on the prehensile, brain-dead calculations needed for what is wrongly called Applied Mathematics.    I am still angered by this waste of effort.    Marx called traditions nothing more than the collected errors of past generations, and never was this statement more true.  What pure mathematician or statistician or computer scientist with integrity could stomach such nonsense?
I am not alone in my views. One of the earliest people who opposed Cambridge’s focus on impure, bottom-up, unprincipled mathematics – those three adjectives are each precisely judged – was Charles Babbage, later a computer pioneer and industrial organizer.  I mentioned his Analytical Society here, created while he was still an undergraduate.     Now, I have just seen an article by Harvey Becher [1995] which places Babbage’s campaign for Cambridge University to teach modern pure mathematics within its full radical political and nonconformist religious context.   A couple of nice excerpts from Becher’s article:

As the revolution and then Napoleon swept across Europe, French research mathematicians such as J. L. Lagrange and S. P. Laplace, and French textbook writers such as S. F. Lacroix, made it obvious that British mathematicians who adhered to the geometrically oriented fluxional mathematics and dot notation of Newton had become anachronisms.  The more powerful abstract and generalized analysis developed on the Continent had become the focus of mathematicians and the language of the physical sciences. This mathematical transmutation fused with social revolution.  ‘Lagrange’s treatises on the calculus were written in response to the educational needs of the Revolution’, recounts Ivor Grattan-Guinness, and Lagrange, Laplace and Lacroix were intimately involved with the educational and scientific reorganizations of the earlier revolutionaries and Napoleon.   Thus, French mathematics became associated with revolutionary France.
This confluence of social and mathematical revolution washed into the heart of Cambridge University because the main purpose of the Cambridge mathematics curriculum, as the core of a liberal education, Cambridge’s raison d’etre, was to produce [page-break] educated gentlemen for careers in the Church, the law and academe. With a student clientele such as this, few were disturbed that the Cambridge curriculum stuck to emphasizing Euclidean geometry, geometric optics and Newtonian fluxions, mechanics and astronomy. However, it was not the landed sons (who constituted the largest segment of the undergraduates), but the middle class and professional sons who, though a minority of the student body as a whole, made up the majority of the wranglers.   For them, especially those who might have an interest in mathematics as an end in itself rather than as merely a means to a comfortable career, the currency of the mathematics in the curriculum might be of concern.
Even though a Cambridge liberal education catered to a social/political elite, most nineteenth-century British mathematicians and mathematical physicists graduated from Cambridge University as wranglers. The Cambridge curriculum, therefore, contoured British mathematics, mathematical physics and other scientific fields. Early in the century, the mathematics curriculum underwent an ‘analytical revolution’ aimed at ending the isolation of Cambridge mathematics from continental mathematics by installing continental analytics in place of the traditional curriculum. Although the revolution began before the creation of the undergraduate constituted ‘Analytical Society’ in 1811, and though the revolution continued after the demise of that Society around 1817, the Analytical Society, its leaders – Charles Babbage, John Herschel and George Peacock – and their opponents, set the parameters within which the remodelling of the curriculum would take place.  This essay is an appraisal of their activities within the mathematical/social/political/religious environment of Cambridge.  The purpose is to reveal why the curriculum took the form it did, a form conducive to the education of a liberally educated elite and mathematical physicists, but not necessarily to the education of pure mathematicians.” [pages 405-406]

And later:

As Babbage and Herschel were radicals religiously and socially, they were radicals mathematically. They did not want to reform Cambridge mathematics; rather, they wanted [page-break] to reconstruct it. As young men, they had no interest in mixed mathematics, the focal point of Cambridge mathematics. In mixed mathematics, mathematics was creatively employed to achieve results for isolated, particular, sometimes trivial, physical problems. The mathematics created for a specific problem was intuitively derived from and applied to the problem, and its only mathematical relevance was that the ingenious techniques developed to solve one problem might be applicable to another. The test of mathematical rigour was to check results empirically. Correspondingly, mathematics was taught from ‘the bottom up’ by particular examples of applications.
Babbage’s and Herschel’s concerns lay not in mixed mathematics, but rather, as they put it in the introduction to the Memoirs, ‘exclusively with pure analytics’. In the Memoirs and other of their publications as young men, they devoted themselves to developing mathematics by means of the mechanical manipulation of symbols, a means purely abstract and general with no heuristic intuitive, physical, or geometric content. This Lagrangian formalism was what they conceived mathematics should be, and how it should be taught.  Indeed, they believed that Cambridge mathematicians could not read the more sophisticated French works because they had been taught analysis by means of its applications to the exclusion of general abstract operations. To overcome this, they wanted first to inculcate in the students general operations free of applications to get them to think in the abstract rather than intuitively.  On the theoretical level, they urged that the calculus ought not to be taught from an intuitive limit concept, to wit, as the derivative being generated by the vanishing sides of a triangle defined by two points on a curve approaching indefinitely close to one another; or by instantaneous velocity represented by the limit of time over distance as the quantities of time and distance vanished; or by force defined as the ultimate ratio of velocity to time. Rather, they urged that students start with derived functions of Lagrange, that is, successive coefficients of the expansion of a function in a Taylor Series being defined as the successive derivatives of the function. This was algebra, free of all limiting intuitive or physical encumbrances. It would condition the student to think in the abstract without intuitive crutches. And on the practical level, pure calculus, so defined, should be taught prior to any of its applications. To achieve this would have inverted the traditional Cambridge approach and revolutionized the curriculum, both intellectually and socially, for only a handful of abstract thinkers, pure mathematicians like Babbage and Herschel, could have successfully tackled it.   The established liberal education would have been a thing of the past.” [pages 411-412]

POSTSCRIPT (Added 2010-11-03):
I have just seen the short paper by David Forfar [1996], reporting on the subsequent careers of the Cambridge Tripos Wranglers.    The paper has two flaws.  First, he includes in his Tripos alumni Charles Babbage, someone who refused to sit the Tripos, and who actively and bravely campaigned for its reform.  Forfar does, it is true, mention Babbage’s non-sitting, but only a page later after first listing him, and then without reference to his principled opposition.  Second, Forfar presents overwhelming evidence for the failure of British pure mathematics in the 19th- and early 20th-centuries, listing just Cayley, Sylvester, Clifford, Hardy and Littlewood as world-class British pure mathematicians – I would add Babbage, Boole and De Morgan – against 14 world-class German and 17 world-class French mathematicians that he identifies.   But then, despite this pellucid evidence, Forfar can’t bring himself to admit the obvious cause of the phenomenon – the Tripos exam.  He concludes:  “The relative failure of British pure mathematics during this period in comparison with France and Germany remains something of a paradox.” No, Mr Forfar,  there is no paradox here; there is not even any mystery.    (En passant, I can’t imagine any pure mathematician using the word “paradox” in the way Forfar does here.)
Forfar says:  “While accepting these criticisms [of GH Hardy], it seems curious that those who became professional pure mathematicians apparently found difficulty in shaking off the legacy of the Tripos.” The years which Tripos students spent on the exam were those years generally judged most  productive for pure mathematicians – their late teens and early twenties.  To spend those years practising mindless tricks like some performing seal, instead of gaining a deep understanding of analysis or geometry, is why British pure mathematics was in the doldrums during the whole of the Georgian, Victorian and Edwardian eras, the whole of the long nineteenth century, from 1750 to 1914.
Harvey W. Becher [1995]:  Radicals, Whigs and conservatives:  the middle and lower classes in the analytical revolution at Cambridge in the age of aristocracy.   British Journal for the History of Science, 28:  405-426.
David O. Forfar [1996]:  What  became of the Senior Wranglers?  Mathematical Spectrum, 29 (1).

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