Thursday, July 02, 2009

The Wranglers

Some time ago I came across the essay What became of the Senior Wranglers?, which describes the history of the Cambridge Mathematical Tripos examination. Interesting excerpts follow below, but I recommend the whole thing.

Difficult exams like the Tripos or present-day international Olympiads in math and science are one of the best ways to identify truly exceptional talent. As discussed below, the tests are able to distinguish between talents at the very far tail of the distribution. But even these exams are only inexact predictors of future success. It's clear that special preparation has an important impact on performance (successful Wranglers typically hired private tutors, see below), and that forcing students to focus on overly technical and narrow exam problems isn't necessarily the best way to measure (or foster) creativity or research ability (see Hardy's criticisms below).

Still, the list of Senior and Second Wranglers is an impressive one!

During the one hundred and fifty seven years (1753-1909) in which the results of the Cambridge Mathematical Tripos were published in order of merit and divided by class of degree into Wranglers (1st Class), Senior Optimes (2nd Class) and Junior Optimes (3rd Class), great prestige attached to those students who had come out in the top two or three places. The securing of the top position as Senior Wrangler was regarded, at the time, as the greatest intellectual achievement attainable in Britain and the Senior Wrangler was feted well beyond Cambridge and accorded pre-eminent status among his peers - indeed years in Cambridge were often remembered in terms of who had been Senior Wrangler in that year. It is curious therefore that no systematic study has ever been made, in so far as the author is aware, of what became of these Senior Wranglers in later years after their triumph. This article may shed a little light on the matter.

Until 1850, Mathematics in Cambridge was dominant over all other University subjects so much so that it was obligatory, astonishing as it now seems, for students who were studying for honours in Classics, first to have taken the Mathematical Tripos.

Because of the prestige attaching to the position of Senior Wrangler and the college from which the Senior Wrangler came, the students, especially the most promising, were subjected, like thoroughbred racehorses, to the most intense training for the Tripos race. The training was in the hands of private ‘coaches’ and not the University professors as often students attended very few lectures and, for example, Charles Babbage gave no lectures in the eleven years, 1828-39, during which he was Lucasian Professor. The best of the coaches, because of their reputation, were able to select the most able students thus perpetuating their reputation for success.

The most famous private tutor was William Hopkins (1793-1866) who himself had been 7th Wrangler in 1827 and was a person of distinction outside his coaching activities being President of the Geological Society 1851-53 and President of the British Association 1853. In 1849 it was said of Hopkins that in the 22 years since his degree he had taught 17 Senior Wranglers, 27 Second or Third Wranglers and 200 Wranglers in total. As William Hopkins continued to turn out Wranglers well after that date his final tally must have been much higher. Hopkins' Wranglers included Clerk Maxwell, Cayley, Thomson (Lord Kelvin), Stokes and Tait. It can be seen with the benefit of hindsight that the greatest of Hopkins' pupils was Clerk Maxwell, but remarkably Hopkins recognised this even when Maxwell was an undergraduate saying "he is unquestionably the most extraordinary man I have met with, in the whole range of my experience".

Galton, who had a nervous breakdown while preparing for the Tripos, analyzed the exam results in his book Hereditary Genius. (See after p.46 here.) In earlier discussions here and here I advanced the claim that modern selection processes are more effective than in the past, with more participants and better access to training. One can quantify this by looking at the scores on, e.g., the International Mathematical Olympiad, which is pretty much as hard an exam as one can devise. Due to the worldwide reservoir of competitors, one finds fairly tight clumping of individuals near the top -- there are often perfect scores, and many nearly perfect ones. (See 2008 scores.) Contrast this with the Tripos score distribution described below, with its outliers and large range of outcomes.

One would be tempted to classify a Senior Wrangler who far outdistanced his competition as a potential Genius, whereas a competitor who falls within the clump of IMO Gold Medalists tends not to stand out very much from his or her peers.

The actual marks were never published but Sir Francis Galton in his book 'Hereditary Genius' refers to having obtained marks in respect of three years (unspecified, but probably around the 1860's). In one of these years, out of a total possible mark of 17,000, the Senior Wrangler obtained 7634 marks, the second Wrangler obtained 4123 marks, the lowest Wrangler obtained around 1500 marks and the lowest candidate receiving an honours degree (Junior Optime) obtained 237 marks. In the second of these years the Senior Wrangler obtained between 5500 and 6000 marks, the Second obtained between 5000 and 5500 and the lowest Junior Optime received 309 marks. In the third of these years when, according to Galton, the Senior Wrangler was conspicuously eminent, he obtained 9422 marks and the Second 5642 marks. Galton makes considerable play of the large discrepancy between the marks obtained by the Senior Wranglerand by the lowest Wrangler.

It can be seen that the Senior Wrangler would typically obtain less than 50% of the marks, the lowest Wrangler less than 10% and the lowest honours candidate less than 2%! This seems to the author a rather curious result and it is not clear what conclusions are to be drawn from it. It suggests that the candidates covered a very wide ability range, that the level of the lowest Wrangler and the lowest honours man was really rather poor by to-day's standards (perhaps university life was more relaxed and the average student did not apply himself very hard?) and that the papers were too long and hard even for the best students.

Curiously, there seem to have been more great physicists among the Wranglers than pure mathematicians!

Among the Wranglers are to be found those who, along with Michael Faraday (1791-1867), William Rowan Hamilton (1805-65) and James Prescott Joule (1818-89), secured for the UK world leadership in physics and mathematical physics in the second half of the 19th century, namely:

James Clerk Maxwell viii (1831-79), 2W 1854.

William Thomson ix(1824-1907), 2W 1845, later Lord Kelvin.

George Stokes (1819-1903), SW 1841, later Sir George Stokes.

John William Strutt (1842-1919 ), SW 1865, later Baron Rayleigh, Nobel Prizefor Physics1904.

John Couch Adams (1819-92), SW 1843, predicted theoretically the existence of the planet
Neptune(also predicted independently by Le Verrier in France).

George Green x (1793-1841),4W 1837, first introduced the concept of potential in a paper of 1828.

Peter Guthrie Tait xi (1831-1901), SW 1852, author with Lord Kelvin of the epoch-making book
'Treatise on Natural Philosophy'.

J.J. Thomson (1856-1940), 2W 1880, later Sir J.J. Thomson, discoverer of the electron in 1897,
Nobel Prizefor Physics, 1906.

University professorships throughout the UK and the British Empire were commonly held by Wranglers in the top two or three places. ...

Given the great attention and prestige attaching to mathematics over the 157 years (1753-1909) we are considering it is curious that the Tripos produced, in contrast to mathematical physics, only a few world class pure mathematicians-only Cayley, Sylvester, Clifford, Hardy and Littlewood. World leadership in pure mathematics in this period remained firmly in France and Germany with each of these countries producing a plethora of world class mathematicians e.g. Gauss, Bessel, Jacobi, Dirichlet, Kummer, Riemann, Dedekind, Kronecker, Weierstrass, Cantor, Klein, Hilbert, Landau, Weyl in Germany and d'Alembert, Lagrange, Laplace, Legendre, Fourier, Poisson, Cauchy, Louiville, Galois, Hermite, Bertrand, Jordan, Poincaré, Hadamard, Cartan, Borel and Lebesguexiii in France.

It was this relative failure of British pure mathematics after the death of Professor Colin Maclaurin in 1748 that so irked G.H. Hardy and he put a large part of the blame on to the Tripos as is evident from his 1926 Address to the Mathematical Association. Hardy's thesis was that the syllabus for the Tripos was out of date and far behind the times since it did not contain any of the important ideas which were dominating contemporary thought in pure mathematics at the time. It was therefore a poor training for a pure mathematician. Furthermore the questions put too much stress on technique rather than ideas and were questions in which professional mathematicians had lost interest many years previously. While accepting these criticisms, it seems curious that those who became professional pure mathematicians apparently found difficulty in shaking off the legacy of the Tripos. After all, the Professors had spent only three years of their active lives on the Tripos during their undergraduate careers and often took little interest in the Tripos thereafter apart from setting some questions for the Smith's prizes. Given their small lecturing load, they had much free time for research, for familiarising themselves withthe latest mathematical ideas and for trying to publish work matching the originality of the papers coming from continental pens. The Cambridge Mathematical Journal had been founded in 1837 by two Scotsmen, A. Smith, SW 1836, and D. F. Gregory, 5W 1837xv. The relative failure of British pure mathematics during this period in comparison with France and Germany remains something of a paradox. A comparative study of the way mathematics was taught and research organised during this period at the Ecole Polytechnique and Ecole Normale Superieure in Paris and at the Universities of Göttingen and Berlin, the centres of European pure mathematics, would be fascinating.


Luke Lea said...

Might it be that the British have always been a pragmatic bunch, and that applied mathematics (including mathematical physics) fits that bill. I once asked a theoretical mathematicisn whether there might be any practical application of the work he was doing. He said he hoped not.

Abhinav said...

Check out this book

Masters of Theory: Cambridge and the Rise of Mathematical Physics

for a much more extensive analysis and history of the Tripos.

Steve Hsu said...

Luke: Hardy was quite proud that there were no applications of his work!

Abhinav: thanks for the recommendation. I read a review of that somewhere and have been meaning to get it out of the library.

Abhinav said...

I would also recommend the delightful article
Old Cambridge Days by Leonard Roth.

bgc said...

Is there any evidence about the first rate UK mathematicians and physicists who were _not_ wranglers (or not Cambridge educated)?

It is conceivable that the wranglers monopolized all the professional positions necessary to a successful career, and thereby prevented non-wranglers from achieving their potential.

This might possibly explain why the UK was not very successful in pure mathematics compared with physics - in the sense that if wranglers were being tested on the wrong abilities necessary for pure math, and if they nonetheless monopolized nearly all the pure math jobs, then wranglers stopped/ inhibited the real pure mathematicians from working in the field.

Steve Hsu said...

BGC: I certainly wouldn't recommend giving out the best positions in the scientific/academic system solely on the basis of a rather specialized exam!

The author of the essay I linked to is clearly interested in your questions -- how did the French and German systems of the same period compare?

It's not entirely clear that the exams were responsible for the physics/math distribution of British research. It's also true that the industrial revolution was centered in England, and lots of things that Lord Kelvin or Maxwell or Faraday got interested in were motivated as much by developments in applied technology as pure science. The British zeitgeist could easily have attracted a lot of the top people away from pure math and into physics.

anon said...

Another example of how the rest of the world values IQ by using tests to determine the quality of a degree, whereas the US uses grades which favors the pushy.

Steve Hsu said...

Abhinav, that essay by Roth was fantastic!

Av said...

One of the chapters in E.Roy Wientraub's book "How Economics became a mathematical science" deals with the influence of the cambridge tripos and a cambridge style of mathematical teaching on the evolution of economics. Alfred marshall was a wrangler for instance...
part of the essay is , if i remember correctly (i read it ages ago), a study of how the cambridge tripos itself evolved over the course of the 19th century.
This link, which is an entry by weintraub in the new palgrave dictionary of economics is the only summary of his argument i could find online:

Robert A. Randall said...

Never mind all the BS-- Comments are intended to IMPROVE customer satisfaction and after 25 years of wearing Wrangler rugged wear I find It time to comment over the lack of consistencly over the past 2-3 years.

curious said...

How do you explain Michael Faraday's genius? The man was reputed to have been unable to understand high school level math and yet he contributed more than physicists who knew more math than he did.

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