My father saw a young Melbourne comedian named Barry Humphries try out an act as an ordinary Moonee Ponds housewife in a Review at the Phillip Street Theatre in Sydney in about 1955. He and I saw undergraduate mathematician Adam Spencer winning theatre sports improv contests at The Harold Park Hotel in about 1988. As well as being so witty that I would remember his name all this time, he also still had a full head of blonde hair.

## Archive for the 'Matherati' Category Page 3 of 6

Belately, I want to record a play seen at the headquarters of The Royal Society in London last month, Let Newton Be, written by Craig Baxter, but using only Isaac Newton’s own words. The play was interesting although the energy of the play sagged at times, particularly in the first half. The story only barely mentioned Newton’s interest in alchemy, and seemed to overlook his brutal, deadly campaigns against money forgers later in life (or did I nap through that scene?)

The play comprised three actors, two men and a woman, who played Newton at different ages – as a child, as a young-ish Cambridge academic, and as an old man. As a work of drama, the conceit worked well, although it was best when one of the actors was playing another person interacting with Newton (eg, Halley, and later Leibniz, who spoke in an amusing cod-German accent). Perhaps the real Newton was not sufficiently schizoid for three actors to play him, at least not when constrained to only use the man’s written words. As I have remarked before, Newton’s personality was all of a piece: it is only modern westerners who cannot imagine a religious motivation for activities such as scientific research, for example, or who find alchemy and calculus incoherent.

The performance was followed by a panel discussion by the Great and the Good – two historians and two scientists. One of the scientists was the Astronomer Royal, Sir Martin Rees, who has subsequently won this year’s Templeton Prize for Science and Religion. The discussion was interesting, so it is a pity it was not recorded for posterity.

A review of another play about a member of the matherati, Kurt Godel, is here.

Writing in Bertinoro, Italy, I have just learnt of the death earlier this year of J. Martin Harvey (1949-2011), a friend and former colleague, and one of Zimbabwe’s great mathematicians. Martin was the first black student to gain First Class Honours in Mathematics from the University of Rhodesia (as it then was, now the University of Zimbabwe), the first person to gain a doctorate in mathematics from that University, and the first black lecturer appointed to teach mathematics there. (Indeed, his three degree certificates name three different universities – Rhodesia, London, and Zimbabwe – but all were granted by the same physical institution.) He later became an actuary, one of the few of any colour in Zimbabwe, but this was a career that lost value with the declining Zimbabwe dollar: actuarial science is about financial planning under uncertainty, and planning is impossible and pointless in an economy with hyper-inflation. He then became part of the great Zimbabwean diaspora, lecturing at the University of the Western Cape, in South Africa.

Martin was a true child of the sixties, with all the best qualities of that generation – open, generous, tolerant, curious, unpompous, democratic, sincere. He was a category theorist, and like most, a deep thinker. Martin was a superb jazz flautist, and on his travels would seek out jazz musicians to jam with. He wrote and recited poetry, and indeed could talk with knowledge on a thousand topics. I once spent a month traveling the country with him on a market resarch project we did together, and his conversation was endlessly fascinating. Despite our very different childhoods, I recall a long, enjoyable evening with him in a shebeen in rural Zimbabwe talking about the various American and Japanese TV series we had both seen growing up (which I mentioned here). Among many memories, I recall him once arguing that a university in a marxist state should have only two faculties: a Faculty for the Forces of Production, and a Faculty for the Relations of Production. There was great laughter as he insisted that the arts and humanities were essential to effective production, and so belonged in the former faculty; this argument was typical of his wit and erudition.

Our mutual friend, Heneri Dzinotyiweyi, another great Zimbabwean mathematician, has a tribute here. I send my condolences to his wife, Winnie Harvey, and family. Vale, Martin. It has been an honour to have known you.

**Bibliography:**

_{0}-separation in topological categories.

*Quaestiones Mathematicae*, 2 (1-3): 177 – 190. An earlier version apperared in:

*Proceedings of the Second Symposium on Categorical Topology*, University of Cape Town, Cape Town, RSA, 1976.

*Categorical Topology, Proceedings Berlin 1978.*Lecture Notes in Mathematics #719, pp. 102-111. Berlin, West Germany: Springer.

*Proceedings of the American Mathematical Society*, 85 (4): 517-519. Available here.

*Proceedings, Winter Meeting 1982, Durand Mathematical Association*, pp. 35-38.

J. M. Harvey [1983]: Categorical characterization of uniform hyperspaces. *Mathematical Proceedings of the Cambridge Philosophical Society*, 94: 229-233. Available here.

J. M. Harvey [1985]: Reflective subcategories. *Illinois Journal of Mathematics*, 29 (3): 365-369. Available here.

The photo shows the Greek Orthodox Cathedral of All Saints, at the corner of Pratt and Camden Streets in Camden, London. Before becoming an Orthodox Chuch in 1948, the building was an Anglican Church, most recently All Saints Camden. The building was designed by William Inwood and his son Henry Inwood in 1822-24, who had together earlier designed St. Pancras New Church in Euston, London. Both churches borrow from ancient Greek architecture, so it is fitting that one is now filled with Greek icons and text, and used for services in (modern) Greek. All Saints has a low-set but very deep choir balcony, extending from the entrance almost one-third the length of the church; this gives the church a quite intimate feel, despite the height of the main chapel. The current cathedral also has three large, low-hanging white glass chandeliers over the main chapel, which enhances the intimacy. I was reminded of the intimacy of Lloyd Wright’s Unity Temple in Chicago, a building which is similarly deceptive from the outside about the compactness of the space within.

When built, All Saints was called Camden Town Chapel, and its founding pastor was the Rev’d Alexander Charles Louis d’Arblay (1794-1837), son of the author Fanny Burney (1752-1840) and Alexandre-Jean-Baptiste Piochard D’Arblay (1754-1818), emigre French aristocrat and soldier, and adjutant-general to Lafayette. The Reverend d’Arblay was a poet and chess-player, and had been 10th wrangler in the Mathematics Tripos at Cambridge in 1818. He was a friend of fellow-student (but non-wrangler) Charles Babbage and a member of Babbage’s Analytical Society (forerunner of the Cambridge Philosophical Society), and he may have introduced Babbage to recent French mathematics. d’Arblay had been partly educated in France, and was aware of French trends in analysis, which in its rigour and formality was very different to the applied focus of British mathematics. From his time as an undergraduate, Babbage ran a campaign against the troglodytic British mathematics establishment, who were then opposed to rigour, formality and theory, and he sought to introduce modern analysis into mathematics teaching at Cambridge. British pure mathematics, as better mathematicians than I have argued, lost a century of progress as a result of its focus on certain types of applications at the expense of rigour.

Because of his First-Class degree, after his graduation d’Arblay was appointed a Fellow of Christ College Cambridge, which paid him a generous stipend his entire life (presumably while he remained unmarried). He had a remarkable ability to quickly learn and recite from memory long poems, and was obsessed with chess. He once missed an arranged meeting with his father when the latter was returning to France because he was engrossed in a chess game with his uncle, the admiral James Burney. d’Arblay was apparently bilingual, and wrote equally easily in English and French, and translated poetry and literary works from each language to the other. Through his mother, he was friends with the royal family and moved in high society. Ordained Reverend d’Arblay, he served as minister of Camden Town Chapel from 1824-1837, and then briefly at Ely Chapel in High Holborn, London. He died of tuberculosis and was unmarried, although engaged to one Mary Anne Smith. Some of his poetry is on the subject of chess. As the son of Fanny Burney, d’Arblay was the grandson of musician, composer and musicologist Charles Burney FRS (1726-1814), and thus from a remarkable family that included musicians, dancers, novelists, painters, historians, and an admiral.

Alongside d’Arblay, the founding organist at Camden Town Chapel was Samuel Wesley (1766-1837).

An index to posts about the Matherati is here.

**POSTSCRIPT 1 [2011-12-24]:** I have now seen d’Arblay’s poem, “Caissa Rediviva”, published anonymously in 1836. This is a long poem about a chess game. If there were any doubts about d’Arblay’s membership of the Matherati, this publication would allay them: The frontispiece to the poem poses a non-standard chess problem, which only someone with a subtle and agile mathmind could imagine: Given a particular chess board-configuration, find the precise sequence of 59 moves by White, each of which forces a single move by Black, and which ends with Black check-mating White with a particular move.

**POSTSCRIPT 2 [2012-02-18]:** Apparently, the Reverend d’Arblay suffered severely from depression for most of his adult life. Peter Sabor, in a recent talk at a conference in depression in the 18th century argues that d’Arblay’s depression may have arisen from his combination of great (and unrealistic) ambition and great indolence. But, of course, his apparent indolence may have been the result, not the cause, of his depression.

**POSTSCRIPT 3 [2012-02-18]:** d’Arblay was not the last member of the Matherati to become engrossed in intellectual pursuits. The most recent Senior Wrangler at Cambridge, Sean Eberhard (Tripos 2011), is described by his fellow collegians as, *“most likely to neglect children to do crossword”*.

*References:*

An Amateur at Chess [Alexander C. L. d’Arblay] [1836]: *Caissa Rediviva: Or the Muzio Gambit*. London, UK: Sampson Low.

Peter Sabor [2008]: *Frances Burney and Alexander d’Arblay: Creative and Uncreative Gloom*. Invited Lecture at: Conference on Before Depression: 1600 – 1800.

This short biography of Australian philosopher and computer scientist Charles L. Hamblin was initially commissioned by the Australian Computer Museum Society.

Charles Leonard Hamblin (1922-1985) was an Australian philosopher and one of Australia’s first computer scientists. His main early contributions to computing, which date from the mid 1950s, were the development and application of reverse polish notation and the zero-address store. He was also the developer of one of the first computer languages, GEORGE. Since his death, his ideas have become influential in the design of computer interaction protocols, and are expected to shape the next generation of e-commerce and machine-communication systems.

Australian Bureau of Statistics graduate cadets, 1979:

- Phil Aungles
- Gail Bansemer
- Penny Barlow
- Warren Bird
- Wendy Darr
- Ken Henry
- Karen Hyams
- Debra Keillor
- Peter McBurney
- Vivienne Palmer
- Prue Phillips
- Suzanne Sheridan
- Steven Skates
- John Stroud.

John Bennett AO (1921-2010), first professor of computing in Australia and founder of Sydney University’s Basser Department of Computer Science, died last month. The SMH obit, from which the lines below are taken, is here.

Emeritus Professor John Bennett AO was an internationally recognised Australian computing pioneer. Known variously as ”the Prof”, ”JMB” or ”Rusty”, he was a man with a voracious appetite for ideas, renowned for his eclectic interests, intellectual generosity, cosmopolitan hospitality and prodigious general knowledge.

As Australia’s first professor of computer science and foundation president of the Australian Computer Society, Bennett was an innovator, educator and mentor but at the end of his life he wished most to be remembered for his contribution to the construction of one of the world’s first computers, the Electronic Delay Storage Automatic Calculator (EDSAC). In 1947 at Cambridge University, as Maurice Wilkes’s first research student, he was responsible for the design, construction and testing of the main control unit and bootstrap facility for EDSAC and carried out the first structural engineering calculations on a computer as part of his PhD. Bennett’s work was critical to the success of EDSAC and was achieved with soldering irons and war-surplus valves in the old Cambridge anatomy dissecting rooms, still reeking of formalin.

. . .

The importance of his work on EDSAC was recognised by many who followed. In Cambridge he also pioneered the use of digital computers for X-ray crystallography in collaboration with John Kendrew (later a Nobel Prize winner), one of many productive collaborations.

He was recruited from Cambridge by Ferranti Manchester in 1950 to work on the Mark 1*. Colleague G. E. ”Tommy” Thomas recalls that when Ferranti’s promise to provide a computer for the 1951 Festival of Britain could not be fulfilled, ”John suggested … a machine to play the game of Nim against all comers … [It] was a great success. The machine was named Nimrod and is the precursor of the vast electronic games industry we know today.”

In 1952, Bennett married Mary Elkington, a London School of Economics and Political Science graduate in economics, who was working in another section at Ferranti. Moving to Ferranti’s London Computer Laboratory in 1953, Bennett worked in a team led by Bill Elliott, alongside Charles Owen, whose plug-in components enabled design of complete computers by non-engineers. Owen went on to design the IBM 360/30. Bennett remembered of the time, ”Whatever we touched was new; it gives you a great lift. We weren’t fully aware of what we were pioneering. We knew we had the best way but we weren’t doing it to convert people – we were doing it because it was a new tool which should get used. We knew we were ploughing new ground.”

Bennett was proud of being Australian and strongly felt the debt he owed for his education. When Harry Messel’s School of Physics group asked him in 1956 to head operations on SILLIAC (the Sydney version of ILLIAC, the University of Illinois Automatic Computer – faster than any machine then commercially available), he declined a more lucrative offer he had accepted from IBM and moved his family to Sydney.

The University of Sydney acknowledged computer science as a discipline by creating a chair for Bennett, the professor of physics (electronic computing) in 1961. Later the title became professor of computer science and head of the Basser department of computer science. Fostering industry relationships and ensuring a flow of graduates was a cornerstone of his tenure.

Bennett was determined that Australia should be part of the world computing scene and devoted much time and effort to international professional organisations. This was sometimes a trial for his staff. Arthur Sale recalls, ”I quickly learnt that John going away was the precursor to him returning with a big new idea. After a period when we could catch up with our individual work, John would tell us about the new thing that we just had to work on. Once it was the ARPAnet [Advanced Research Projects Agency Network] and nothing would suffice until we started to try to communicate with the Aloha satellite over Hawaii that had run out of gas to establish a link to Los Angeles and ARPAnet and lo and behold, the internet had come to Australia in the 1970s.”

In 1983, Bennett was appointed as an officer of the Order of Australia. After his retirement in 1986, he remained active, attending PhD seminars and lectures to ”stay up to date and offer a little advice” while continuing to earn recognition for his contributions to computer science for more than half a century.

Lest anyone think I’m uniquely deranged for my criticisms of the Cambridge University Mathematics Tripos examination, particularly during the 18th- and 19th-centuries, here is GH Hardy – perhaps Britain’s greatest 20th century pure mathematician – speaking in his Presidential Address to the Mathematics Association in 1926:

My own contribution to the discussion consisted merely in an expression of my feeling that the best thing that could happen to English mathematics, and to Cambridge mathematics in particular, would be that the Mathematical Tripos should be abolished. I stated this on the spur of the moment, but it is my considered opinion, and I propose to defend it at length to-day. And I am particularly anxious that you should understand quite clearly that I mean exactly what I say; that by “abolished” I mean “abolished”, and not “reformed”; that if I were prepared to co-operate, as in fact I have co-operated in the past, in “reforming” the Tripos, it would be because I could see no chance of any more revolutionary change; and that my “reforms” would be directed deliberately towards destroying the traditions of the examination and so preparing the way for its extinction.” [p. 134]

. . .

“I suppose that it would be generally agreed that Cambridge mathematics, during the last hundred years, has been dominated by the Mathematical Tripos in a way in which no first-rate subject in any other first-rate university [page-break] has ever been dominated by an examination. It would be easy for me, were the fact disputed, to justify my assertion by a detailed account of the history of the Tripos, but this is unnecessary, since you can find an excellent account, written by a man who was very much more in sympathy with the Tripos than I am, in Mr. Rouse Ball’s History of Mathematics in Cambridge. I must, however, call your attention to certain rather melancholy reflections which the history of Cambridge mathematics suggests. You will understand that when I speak of mathematics I mean primarily pure mathematics, not that I think that anything which I say about pure mathematics is not to a great extent true of applied mathematics also, but merely because I do not want to criticise where my competence as a critic is doubtful.

Mathematics at Cambridge challenges criticism by the highest standards. England is a first-rate country, and there is no particular reason for supposing that the English have less natural talent for mathematics than any other race; and if there is any first-rate mathematics in England, it is in Cambridge that it may be expected to be found. We are therefore entitled to judge Cambridge mathematics by the standards that would be appropriate in Paris or Gottingen or Berlin. If we apply these standards, what are the results? I will state them, not perhaps exactly as they would have occurred to me spontaneously-though the verdict is one which, in its essentials, I find myself unable to dispute-but as they were stated to me by an outspoken foreign friend.

In the first place, about Newton there is no question; it is granted that he stands with Archimedes or with Gauss. Since Newton, England has produced no mathematician of the very highest rank. There have been English mathematicians, for example Cayley, who stood well in the front rank of the mathematicians of their time, but their number has been quite extraordinarily small; where France or Germany produces twenty or thirty, England produces two or three. There has been no country, of first-rate status and high intellectual tradition, whose standard has been so low; and no first-rate subject, except music, in which England has occupied so consistently humiliating a position. And what have been the peculiar characteristics of such English mathematics as there has been? Occasional flashes of insight, isolated achievements sufficient to show that the ability is really there, but, for the most part, amateurism, ignorance, incompetence, and triviality. It is indeed a rather cruel judgment, but it is one which any competent critic, surveying the evidence dispassionately, will find it uncommonly difficult to dispute.

I hope that you will understand that I do not necessarily endorse my friend’s judgment in every particular. He was a mathematician whose competence nobody could question, and whom nobody could accuse of any prejudice against England, Englishmen, or English mathematicians; but he was also, of course, a man developing a thesis, and he may have exaggerated a little in the enthusiasm of the moment or from curiosity to see how I should reply. Let us assume that it is an exaggerated judgment, or one rhetorically expressed. It is, at any rate, not a ridiculous judgment, and it is serious enough that such a condemnation, from any competent critic, should not be ridiculous. It is inevitable that we should ask whether, if such a judgment can really embody any sort of approximation to the truth, some share of the responsibility must not be laid on the Mathematical Tripos and the grip which it has admittedly exerted on English mathematics.

I am anxious not to fall into exaggeration in my turn and use extravagant language about the damage which the Tripos may have done, and it would no doubt be an extravagance to suggest that the most ruthless of examinations could destroy a whole side of the intellectual life of a nation. On the [page-break] other hand it is really rather difficult to exaggerate the hold which the Tripos has exercised on Cambridge mathematical life, and the most cursory survey of the history of Cambridge mathematics makes one thing quite clear; the reputation of the Tripos, and the reputation of Cambridge mathematics stand in correlation with one another, and the correlation is large and negative. As one has developed, so has the other declined. As, through the early and middle nineteenth century, the traditions of the Tripos strengthened, and its importance in the eyes of the public grew greater and greater, so did the external reputation of Cambridge as a centre of mathematical learning steadily decay. When, in the years perhaps between 1880 and 1890, the Tripos stood, in difficulty, complexity, and notoriety, at the zenith of its reputation, English mathematics was somewhere near its lowest ebb. If, during the last forty years, there has been an obvious revival, the fortunes of the Tripos have experienced an equally obvious decline.” [pp. 135-137]

. . .

“It has often been said that Tripos mathematics was a collection of elaborate futilities, and the accusation is broadly true. My own opinion is that this is the inevitable result, in a mathematical examination, of high standards and traditions. The examiner is not allowed to content himself with testing the competence and the knowledge of the candidates; his instructions are to provide a test of more than that, of initiative, imagination, and even of some sort of originality. And as there is only one test of originality in mathematics, namely the accomplishment of original work, and as it is useless to ask a youth of twenty-two to perform original research under examination conditions, the examination necessarily degenerates into a kind of game, and instruction for it into initiation into a series of stunts and tricks. It was in any case certainly true, at the time of which I am speaking, that an undergraduate might study mathematics diligently throughout the whole of his career, and attain the very highest honours in the examination, without having acquired, and indeed without having encountered, any knowledge at all of any of the ideas which dominate modern mathematical thought. His ignorance of analysis would have been practically complete. About geometry I speak with less confidence, but I am sure that such knowledge as he possessed would have been exceedingly one-sided, and that there would have been whole fields of geometrical knowledge, and those perhaps the most fruitful and fascinating of all, of which he would have known absolutely nothing. A mathematical physicist, I may be told, would on the contrary have received an appropriate and an excellent education. It is possible; it would no doubt be very impertinent for me to deny it. Yet I do remember Mr. Bertrand Russell telling me that he studied electricity at Trinity for three years, and that at the end of them he had never heard of Maxwell’s equations; and I have also been told by friends whom I believe to be competent that Maxwell’s equations are really rather important in physics. And when I think of this I begin to wonder whether the teaching of applied mathematics was really quite so perfect as I have sometimes been led to suppose.” [p. 138]

. . .

“I shall judge the Tripos by its real or apparent influence on English mathematics. I have already told you that in my judgment this influence has in the past been bad, that the Tripos has done negligible good and by no means negligible harm, and that, so far from being the great glory of Cambridge mathematics, it has gone a very long way towards strangling its development.” [p. 141]

* *

*Reference:*

G. H. Hardy [1926/1948]: Presidential Address: The Case against the Mathematical Tripos. *The Mathematical Gazette*, 32 (300): 134-145 (July 1948).

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.

*References: *

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).

I have posted before about the two cultures of pure mathematicians – the theory-builders and the problem-solvers. Thanks to string theorist and SF author Hannu Rajaniemi, I have just seen a fascinating paper by Freeman Dyson, which draws a similar distinction – between the birds (who survey the broad landscape, making links between disparate branches of mathematics) and the frogs (who burrow down in the mud, solving particular problems in specific branches of the discipline). This distinction is analogous to that between a focus on breadth and a focus on depth, respectively, as strategies in search. As Dyson says, pure mathematics as a discipline needs both personality-types if it is to make progress. Yet, a tension often exists between these types: in my experience, frogs are often disdainful of birds for lacking deep technical expertise. I have less often encountered disdain from birds, perhaps because that is where my own sympathies are.

A similar tension exists in computing – a subject which needs both deep technical expertise AND a rich awareness of the breadth of applications to which computing may be put. This need arises because the history of the subject shows an intricate interplay of theory and applications, led almost always by the application. Turing’s abstract cineprojector model of computing arrived a century after Babbage’s calculating machines, for example, and we’ve had programmable devices since at least Jacquard’s loom in 1804, yet only had a mathematical theory of programming since the 1960s. In fact, since computer science is almost entirely a theory of human artefacts (apart from that part – still small – which looks at natural computing), it would be strange indeed were the theory to divorce itself from the artefacts which are its scope of study.

A story which examplifies this division in computing is here.

*Reference:*

Freeman Dyson [2009]: Birds and frogs. *Notices of the American Mathematical Society*, 56 (2): 212-223, February 2009. Available here.