The Mathematics PhD in the United Kingdom

 

Notes for the Mathematics Genealogy Project

 

(Don’t be alarmed by discrepancies between what is written below and what is written in some of the MGP entries. The latter are being revised and so the problem is only a temporary one. Some links are to draft entries.)

 

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The UK has an unbroken tradition of 800 years of university mathematics but for nearly all of that time the creation of doctors played no part in the process of preparing the next generation of mathematicians. The PhD is new to the UK, compared to Germany or even to the United States (see NSF US Doctorates). It appeared at the end of the First World War and it was not until after the Second that a PhD became part of the usual preparation of a university mathematician. There may be a chain of teachers back to Bradwardine (d. 1349), to Newton (d. 1727), to Cayley (d. 1895) but there is no chain of doctors.  

 

Today’s entrants into academic mathematics typically have PhDs and they may have some post-doctoral research experience as well. See the advertisements for Academic Jobs on the London Mathematical Society’s website. (The LMS (MacTutor) is the UK’s national association for Mathematics.)

 

The past was very different but, because people and practices last, not so very remote. Coxeter MGP was supervising PhDs until the 1980s. His own PhD, awarded in 1931, was supervised by H. F. Baker MGP who graduated in 1888! A first degree was Baker’s qualification for a distinguished career in pure mathematics. T. M. F. Smith was President of the Royal Statistical Society in 1991-3. His qualification was also a first degree, awarded in 1959. In 1888 the first degree was absolutely standard, in 1959 it was unusual, but not as unusual as Coxeter’s PhD in 1931. The production of PhDs by means of PhDs is a recent phenomenon.

 

When looking up British mathematicians in the Mathematics Genealogy Project database

  • Do not expect to find many PhDs before 1940.
  • Do not be surprised to find supervisors (advisors) with only a first degree.
  • Do not expect long lines of descent. The PhDs before 1940 were almost necessarily ‘first generation’ PhDs but for decades afterwards the system contained many supervisors without PhDs.
  • Do not expect uniformity. The percentage of people entering academic mathematics with PhDs went from 0 in 1920 to nearly 100% today but the movement was not uniform over time, nor uniform across branches of mathematics. The PhD as entry requirement came sooner to pure mathematics than to statistics.

 

The aim of these notes is to save the user of MGP from jumping to too many wrong conclusions. Wrong conclusions cannot be entirely avoided for there are wrinkles enough in the system to confuse anybody. One wrinkle—of historic interest only now—illustrates how qualifications may not be what they seem. Sylvester MGP completed his studies at St. John’s College, Cambridge in 1837. Not being a member of the Church of England (he was a Jew) Sylvester was disqualified from graduating from Cambridge. However, his work at St John’s was recognised by Trinity College Dublin and he was awarded a BA (and an MA) from that institution in 1841. He never studied at Trinity. There are equally surreal PhD stories. In 1929 Wittgenstein MGP was awarded a PhD by Cambridge University. He had been a student of Russell but left in 1913 without a degree. In 1929 Ramsey MGP was designated his supervisor and Wittgenstein presented as his thesis a work written 10 years before, away from Russell, away from Cambridge and while Ramsey, 14 years his junior, was still at school.

 

Notes on the History of the PhD

The image of a PhD system has been present since 1850, there has been a PhD degree since 1920 and a functioning PhD system since around 1950. These notes describe how these changes took place and give some basic information about British degrees and practices. Today there are more than 100 universities in the UK; see here for links to maths departments and here and here for graduate programmes. The universities frame their own degree regulations and these, beyond a basic similarity, are quite varied. I have not written a survey of these institutions, nor a key to how each institution has interpreted the letter D in the course of its history. Further, the notes treat only the UK, although, through the British Empire and Commonwealth, British thought and practice about universities have influenced developments all over the world.  The historian Hank Nelson discusses the PhD in Australia; see also Ian D. Rae (2002) False Start for the PhD in Australia, Historical Records of Australian Science, 14(2) 129-141 abstract. In Canada where the influence of the United States was much stronger the PhD came much earlier: for references see the notes to M. L. Friedland The University of Toronto: A History.

 

General References

  • The basic reference for the origins of the PhD is Renate Simpson How the PhD came to Britain. A Century of Struggle for Postgraduate Education, Society for Research into Higher Education, 1983.
  • Renate Simpson has just published The Development of the PhD Degree in Britain, 1917-1959 and Since: An Evolutionary and Statistical History in Higher Education, Mellen Press. This treats the first fifty years of the PhD system in Britain.
  • The mathematicians of the transition period are very well served by two databases, June Barrow-Green's BritMath and A E L Davis’s Mathematical Women in the British Isles, 1878-1940. BritMath provides detailed information on 750 mathematicians from the period 1860-1940. Mathematical Women gives the qualifications of all the 2500 women who graduated in mathematics between 1878 (when women were first admitted) and 1940. Of the BritMath mathematicians about 150 have PhDs.
  • To illustrate the varieties of personal experience I have made links to the biographies in John O’Connor & Edmund Robertson’s MacTutor History of Mathematics Archive. For additional colour there are links to the title pages of books where authors can be seen displaying their qualifications.
  • 800 years of mathematics in Oxford (the oldest university in the British Isles) are surveyed in Oxford Figures: 800 Years of the Mathematical Sciences edited by John Fauvel, Raymond Flood and Robin Wilson, Oxford University Press 1999. (Amazon.) See also I. W. Busbridge Oxford Mathematics and Mathematicians.
  • For the Scottish university scene see Stuart Wallace “National Identity and the Idea of the University in 19th-Century Scotland” Higher Education Perspectives, 2, 2006.
  • The standard reference work on the universities of the Commonwealth is the Commonwealth Universities Yearbook published by the Association of Commonwealth Universities London.

 

 

19th Century

In 1800, as in 1600, there were two universities in England, four in Scotland (MacTutor) and none in Wales or Northern Ireland—Trinity College Dublin was the only university in Ireland. The national scientific academy, the Royal Society of London (MacTutor) dates from 1662. Election to its fellowship, indicated by the initials FRS, was (and is) a prized distinction. In 1800 the English universities, Oxford and Cambridge, were nothing like modern research universities and almost moribund compared to the two main Scottish universities, Edinburgh and Glasgow. In the course of the 19th century the old universities were reformed and new ones were created. Mathematical Women gives a list of the universities awarding degrees in mathematics in the early 20th century; today there are many more. It also gives the numbers of students graduating (men as well as women) from each institution, which gives some idea of the scale of university mathematics. Information on the number of academics in each institution can be found from BritMath.

 

Reference

  • The two English universities in 1800, Oxford and Cambridge, are placed in a European context in L. W. B. Brockliss “The European University in the Age of Revolution, 1789-1850” ch. 2 of The History of the University of Oxford, volume VI: Nineteenth-Century Oxford (eds. M. R. Brock & M. C. Curthoys), Clarendon Press, Oxford 1997. (Amazon.)

 

 

First degrees, BA and MA

In the 19th century and for many years after the introduction of the PhD in the 20th century the standard degree qualification of the university mathematician was a BA or MA; see e.g. this title page from Todhunter, this from Hardy MGP or this from Fisher MGP.  

 

For most of the 19th century the only mathematics degrees were first degrees—in today’s terms.  At Oxford and Cambridge the Bachelor of Arts (BA) was awarded on the basis of the student’s examination performance. The Master of Arts (MA) involved no further academic work and was essentially a formal recognition that the person actually was a BA! This is still the case in Oxbridge; see here for the current Cambridge practice. The MA was required for full membership of the university and so university teachers were invariably MAs. The existence of two degrees, awarded at different dates for the same course of study, is a source of confusion and explains some of the discrepancies in the dates given by different authorities. In Scotland the first degree was, and is still, called the MA.

 

The University of Cambridge was the centre of mathematical teaching and research in Britain in the 19th century and the first half of the 20th: “The study of Higher Mathematics in the British Empire is now practically concentrated at Cambridge” wrote the Times newspaper in 1906. More than two-thirds of those listed in BritMath were students at Cambridge and Cambridge figures dominate the MacTutor biographies. The Cambridge Mathematical Tripos had great prestige and success in it was widely regarded as the best qualification for an academic career in mathematics. Indeed until the Second World War graduates from other universities would take the Cambridge degree (or part of it) as a form of post-graduate study: Jeffreys MGP, Hodge MGP and Davenport MGP are instances. Jeffreys, for example, had two first degrees; the later ‘higher’ degree is the one given on the MGP page.

 

The new faculties of the old universities and the new universities created in the 19th and 20th centuries used the established terms Bachelor and Master for their degrees, although the new institutions did not follow the ancient universities’ practice of conferring an MA upon a BA. Some universities adopted the name Bachelor of Science (BSc) for their degree in mathematics. The post-graduate Master’s degree is a fairly recent development. It is rare in pure mathematics but since the 1960s it has been widely offered in statistics.

 

 

 

Sources

  • Andrew Warwick Masters of Theory: Cambridge and the Rise of Mathematical Physics, Chicago University Press, 2003. (Amazon.) This contains a very thorough study of the mathematics tripos in the 19th century. (The British mathematical physicists of the 19th century had degrees in mathematics.) There is a useful review by Grattan-Guinness in the SIAM News.

 

 

Doctorates, higher and honorary

The doctorates existing in 1800 were in Divinity, Law, Medicine and Music. Dr Wallis, the Savilian professor of geometry at Oxford 1649-1703, was a doctor of divinity, DD, because he was a theologian before he was a mathematician. These doctorates were not like the modern PhD for the award recognised an established reputation in a field, not the quality of a piece of work done at the beginning of a career. In the course of the 19th century doctorates in other subjects were created on the same pattern.

 

In 1882 Cambridge introduced the ScD, a doctorate open to Cambridge graduates who had produced a significant body of scientific work; a LittD for scholarship in the humanities was also created. Other universities established similar degrees. The British university is not generous with titles, for only a minority of academics have the title “professor”, and perhaps the move was a case of keeping up with titled foreigners. The award of the ScD did not mark the beginning of a career, it recognised a successful career. Writing in the 20th century, Jeffreys described the Cambridge ScD as “more or less equivalent to being proposed for the Royal Society.” Jeffreys himself submitted work for the equivalent degree at the University of Durham and was awarded its DSc. Many distinguished graduates took advantage of this doctoral possibility, but not all; the title page of Whitehead & Russell’s Principia Mathematica shows one author with an ScD, one without. Following the creation of the PhD, the term higher doctorate came to be used for the ScD, or its equivalent elsewhere (most often denoted by DSc). British universities still award higher research degrees but today almost all of those receiving them have a PhD already. See here for what Cambridge expects of an ScD today and here for an application form.

 

Edmond Halley, Wallis’s successor at Oxford, was made a Doctor of Civil Law in 1710. This was not in recognition of any distinction in law but in recognition of Halley’s scholarship in producing an edition of Apollonius’s Conics. Samuel Johnson, the poet and dictionary maker, was a later recipient of an honorary doctorate from Oxford: “they have sent me a degree of Doctor of Laws, with such praise in the Diploma as perhaps ought to make me ashamed …” In the 19th century James Clerk Maxwell (title page) was an honorary doctor of law from Edinburgh University. Another LLD was E. J. Routh; Routh, unlike Maxwell, lived to collect a Cambridge ScD as well (title page). Honorary doctorates are still conferred by British universities; which particular title is chosen will vary with the university but the title is never PhD.  

 

Sources

  • Simpson How the PhD came to Britain chapter 3.
  • The Jeffreys quote is in David Howie (2002) Interpreting Probability: Controversies and Developments in the Early Twentieth Century, New York, Cambridge University Press.
  • The UKCGE has recently surveyed Higher Doctorate Awards in the UK.
  • The story of the award of Johnson’s honorary doctorate in March 1775 is told in Boswell’s Life of Johnson.

 

 

 

Early research degrees, fellowship dissertations, Smith’s Prize essays

In the 19th century British university reformers looked to Germany for a model of the modern university and the higher degree was an element in the German system. Higher degrees to follow on from a bachelor’s degree and serve as a preparation for an academic career started to appear in the 1860s. The DSc introduced by London University in 1860 was a higher degree for advanced study rather than research and was assessed by examination but the Edinburgh DSc had an element of research. In 1885 the London regulations were changed and the DSc became a genuine research degree; Charlotte Scott Math Women MGP was awarded a DSc in 1885, at the end of the examination era. In 1895 all the Scottish universities adopted a five year research degree open to graduates from other universities.

 

In Cambridge there was no feeling that a research degree was necessary for its own graduates but the university created a Bachelor of Arts by Research for graduates from other universities; this could be completed in one year. This was not a popular degree and BritMath lists only one person with it, appropriately the unique Ramanujan. The BA by Research was a curiosity but there were two other research developments that had a lasting influence on mathematical life in Cambridge.

 

Cambridge men had two opportunities to impress, the tripos examinations and the Smith’s Prize, the latter a competition open to the best students. Originally the Smith’s Prize was based on written examinations but from 1885 it was awarded for the best essay. The prize became the focus for post-graduate research as students might spend a year working on their entry. In the early days, at least, there was no system of supervision and so no declaration of ‘parentage.’ Sometimes there was a clear supervisor, e.g. Barnes’s supervision of Littlewood MGP entered the lore of mathematics. Smith’s Prize Winner did not appear with MA after the person’s name but it was a very significant qualification and the list of winners given by Barrow-Green is a who’s who of Cambridge mathematics. In 1936, when Fred Hoyle MGP began his research career, his objective was the prize since “gaining either a Smith’s or a Rayleigh was considered to be almost a guarantee of a post in some university.” The Rayleigh Prize was an additional prize first awarded in 1911. As well as the winners, other creditable performers would be named; in 1935 altogether 9 names appeared. After the Second World War, as the PhD system became established, submitting an essay became a stage in the making of a PhD.

 

The other innovation was the fellowship dissertation. In 1872 Trinity College introduced the dissertation into its fellowship competition and some other colleges followed. See, for example, the MacTutor entries for A. N. Whitehead MGP and H. W. Richmond MGP. Fellowship dissertations continued into the PhD era—see P. Hall MGP—and still exist. Both Cambridge developments are illustrated in the careers of Louis Mordell MGP from the 1910s and Alan Turing from the 1930s. Mordell’s essay for the Smith’s Prize was successful but his fellowship dissertation was not. For Turing one essay was successful both as a Smith’s Prize entry and as a fellowship dissertation. Subsequently Turing went to Princeton for a PhD MGP. Like J. H. C. Whitehead MGP , from Oxford, Turing was attracted to Princeton by the opportunity of working with a particular supervisor, rather than by the PhD qualification which was not required for a career in Britain.

 

It was possible to go abroad for a PhD but there was little incentive to do so. In the nineteenth century it became a tradition for British chemists to go to Germany to obtain a doctorate but in other subjects it was unusual to go abroad. BritMath has two instances of mathematicians with foreign PhDs from the era before the First World War: Grace Chisholm Young Math Women MGP with a PhD from Göttingen and Harry Bateman MGP with a PhD from Johns Hopkins. Both were unusual cases. As a woman, Miss Chisholm had very restricted opportunities in Britain. Bateman already had 60 publications and the most likely explanation for his wanting a PhD is that in America, where he was working, an academic was expected to have one. At Johns Hopkins, thirty years before, Sylvester MGP had been in at the birth of the US doctoral system but his career as a PhD supervisor ended when he returned to England. For Sylvester’s activity in America see Parshall & Rowe “American Mathematics comes of Age: 1875-1900” in AMS History of Mathematics, Volume 3.  

 

Sources

  • For the early research degrees see Simpson chapters 2 & 3.
  • The Cambridge research degree was significant in the history of the Cavendish (physics) laboratory. See ch. 4 of Dong-Won Kim (2002) Leadership and Creativity: A History of the Cavendish Laboratory, 1871-1919, Kluwer. See Google Book Search for an extract.
  • For the Smith’s prize see June Barrow-Green (1999) “A Corrective to the Spirit of too Exclusively Pure Mathematics”: Robert Smith (1689-1768) and his Prizes at Cambridge University, Annals of Science, 56, 271-316. Appendix 1 lists the winners from 1885 to 1940 with the ‘grades’ their essays received. There is no information about unsuccessful entries..
  • Barrow-Green describes the fellowship dissertation system but there does not seem to be any list of who wrote what.

 

 

20th Century

In 1917 British universities resolved to create a PhD degree. However, deciding to create a degree is not the same as deciding that the degree should matter. There was no resolution to create a system in which the PhD would be an essential part of the preparation of the university academic. The system evolved without anybody planning it.

 

The PhD degree

By the beginning of the 20th century the case for a PhD type degree had been made and won—at least outside Cambridge (and Oxford).  In 1917 representatives of the universities met and agreed that they would establish a PhD degree. The key resolution said

 

For the better promotion of research in this country, and for the encouragement of advanced work by “graduate” students from abroad, a degree or title of Doctor should be instituted, attainable after a period of not less than two years whole-time work devoted to advanced study or research …

 

The universities acted together under strong political pressure. Imperial considerations were important. There had long been anxiety that students from the dominions would go to Germany or the United States and be weaned away from the mother country.  

 

Oxford was the first university to institute such a degree, although its choice of title, DPhil, was idiosyncratic. The first Oxford DPhil in mathematics was awarded in 1921. The first Cambridge PhD in mathematics was awarded in 1924 to an Australian Thomas Cherry MGP.

 

A concept of supervision had to evolve. C. R. Rao MGP described his experience as a student in Cambridge in the 1940s and later as a supervisor in India and the US. “I asked Fisher [Rao’s supervisor] to suggest a research problem for my Ph.D. thesis. He said the problem must be mine and that he would only advise me if and when I encountered difficulties. This was good advice. I used to say the same to my Ph.D. students without success. There were only 2 cases (out of 50), where the students chose their own problems.”  The practice in the UK, until recently at least, has been for a student to have a single supervisor.

 

Besides the supervised dissertation, there is another route to the PhD, the staff PhD. A member of staff at a university may obtain a PhD by submitting published work. There is usually no supervisor and the staff PhD is more like a baby ScD than a regular PhD. In 1935 F N David Math Women MGP was appointed as an assistant lecturer at University College London; in 1938 she was awarded a PD. “I took it by sending in 4 papers I had already published.” This route to a PhD still exists; see here for a typical set of regulations.

 

H. E. Daniels RSS president 1974-5  MGP was awarded an external PhD from the University of Edinburgh; an external student does not attend the university. Daniels was ineligible for a staff PhD because he was not working at a university and he was not old enough for an ScD. “Aitken was nominally my supervisor but hadn’t the faintest idea of what I was doing.” Daniels did not really need a supervisor, as he was already a well-established researcher. 

 

On the general issue of the PhD, David recalled, “You did not need [a doctorate] in England, or you did not need it at that time anyway.” She also recalled that Karl Pearson, for whom she had worked, did not think it “a good thing.” Looking back, Daniels did not seem to think it a good thing either. “[Research] is about having ideas. What happens with the run-of-the-mill Ph.D. student is that you lay a trail of clues for him, which he follows, you hope, and in the end he produces his thesis. Well, it seems to me that this is self-defeating.”

 

These reminiscences are from statisticians but the experiences and attitudes they describe have a wider relevance.

 

There has been continuing debate about the character of the PhD and there has been a movement away from the traditional pattern which involved only research to one involving advanced courses as well. The EPSRC (the state science funding body) and its predecessors have been important voices in these debates. The PhD experience is much more variable than the undergraduate experience. Supervision has ranged from “Hello, go away and come back when you are finished!” to very close supervision on a topic worked out in detail by the supervisor. The EPSRC (and its predecessors) have tried to codify and regulate the process. See the guidelines for its current model of how student and supervisor should work together. This document breathes anxiety that the student will not submit the dissertation on time and late completion has been an issue for decades. 

 

 

Sources

  • Simpson chapter 5.
  • The ET Interview: Professor C.R. Rao: Interviewed by Anil K. Bera, Econometric Theory, 19, (2) (2003), 331-340.
  • N. M. Laird, A Conversation with F N David, Statistical Science 4, (3) (1989), 235-246. JSTOR . Project Euclid
  • P. Whittle, A Conversation with Henry Daniels, Statistical Science, 8, (3), (1993) 342-353. JSTOR   Project Euclid

 

 

The PhD system

British universities have never been obliged to recruit only PhDs as lecturers but increasingly they have come to do so. The state played a role in this. The Department of Scientific and Industrial Research (DSIR), the body founded during the First World War to support scientific research, provided scholarships, though only in very small numbers. The number of scholarships was increased after the Second World War. The Engineering and Physical Sciences Research Council (EPSRC), the current version of the DSIR continues to do so.

 

Simpson’s Development charts the first four decades of the PhD system. She finds (p. 321) that 23,505 students were admitted to study for the PhD in all subjects at UK universities in the period 1917-59. Of these admissions 1230 were to mathematics. The top four universities were Cambridge with 568 (46.3% of the total), Imperial College with 195 (15.9%), Oxford with 185 (15.0%) and University College London with 131 (10.7%). Over the decades the numbers grew although the Second World War stalled the growth: the admissions figures by decade for the six largest institutions were 96 (1920s), 207 (1930s), 306 (1940s) and 611 (1950s). Simpson does not report employment destinations.

 

There is information on the growing importance of the PhD as a qualification for an academic job in the research conducted for the inquiry into higher education in Britain which resulted in the Robbins report. One of the volumes reported on Teachers in Higher Education.

  • This found that, in 1961-2, 72% of university teachers in science (the category covering mathematics) had a doctorate of some kind, including a higher doctorate. In the humanities the figure was 29%.
  • It also found that 45% of those recruited in 1959-61 had a doctorate. The authors comment, “the proportion of teachers with higher degrees is lower among those who took their first degree within the last ten years than among older staff, since many of the more recent recruits are of course still working for a higher degree.” The comment reflects the practice of hiring people who had not yet completed their PhD and also the possibility of hiring on condition that the employee starts a PhD. It also reflects the absence of higher doctorates amongst the recruits.

 

In 1989 the Science and Engineering Research Council (successor to the DSIR and predecessor of EPSRC) set up a committee to review mathematics education. By then it had become axiomatic that the university mathematician has a PhD. The Kingman report concluded that the current supply of mathematics PhDs is “dangerously inadequate.” The report contains some interesting data. It gave the number of full time academic staff in university mathematics departments as 1500 with 20 post-doctoral fellows and 120 research assistants. It estimated that in 1990 around 115 PhD students completed their degree. It also estimated that to hold current staff levels constant would require recruitment at the level of 80 posts per year, 30 each in pure and applied mathematics and 20 in statistics and operational research. There is more recent information in the 2004 review of UK research in mathematics which was undertaken by an international team for the EPSRC. 

 

For almost all of its 800 years the university system was a virtually closed system in which the academics were recruited from the ranks of local graduates. De Moivre, the seventeenth century refugee from Catholic France, had no entry into the universities, while refugees from Nazi Germany in the 1930s, especially junior people such as  H. O. Hartley MGP, Bernhard Neumann MGP and Richard Rado MGP often entered the system by taking a second PhD. Recently, however, to meet the shortfall in locally produced PhDs universities have increasingly recruited foreign mathematicians and university mathematics, like the entire university system, has become very open.

 

 

References and further reading

 

 

 

John Aldrich, University of Southampton, Southampton, UK. (home) February 2006. Latest changes December 2009

 

I am grateful to June Barrow-Green, A E L Davis, Anthony Edwards, Karen Parshall, Renate Simpson, Fred Smith and Brian Stewart for information, corrections and suggestions.