In a new mini-series, Norman Biggs gives us a brief history of mathematics at LSE. Norman is an Emeritus Professor in the Department of Mathematics at LSE and is also on Twitter: @norman_biggs
When LSE was founded in 1895, Mathematics was not one of the foundation subjects. However, from the very start, Statistics was regarded as an important tool for the social scientist, and A.L. Bowley (right) gave regular lectures on the subject. In 1901 his book Elements of Statistics was published. It went through several editions with only minor changes, but in his preface to the 1920 edition Bowley indicated a major change of emphasis. ‘In the first edition an effort was made to obtain the principal results without the use of the Calculus; but as the subject has developed during the past twenty years, it has become necessary to abandon this attempt.’ In fact, Bowley’s concessions to mathematics were rather limited: he described the normal distribution in terms of an integral, and introduced a small amount of differential calculus. However, from this time forward it is clear that Statistics at the LSE was taught in a more rigorous way. The appointment of mathematically-trained statisticians such as Roy Allen (1928) and Maurice Kendall (1949) was an important factor.
It is worth mentioning that one of the first benefactors of the LSE was Bertrand Russell (left), already becoming famous for his work in logic and the philosophy of mathematics. Indeed, he is reported to have given a course of lectures at the School in 1896, but sadly they were on German Social Democracy.
A more important impetus came, in due course, from Economics. By the beginning of the twentieth century the leading economists, such as Alfred Marshall, were accustomed to using mathematics in their work. In a letter to Bowley, Marshall formulated his famous Six Principles of how to use mathematics in economics:
- Use mathematics as a shorthand language, rather than as an engine of enquiry.
- Keep them till you have done.
- Translate into English.
- Then illustrate by examples that are important in real life.
- Burn the mathematics.
- If you can’t succeed in 4, burn 3. This I do often.
It must be remembered that, to the founders of the LSE, ‘Economics’ was a vague term, covering a broad range of mainly historical enquiries. The prevailing attitude is well-illustrated in a letter from the Director (Hewins) to Sidney Webb, written in 1898: ‘you may be gratified to know … that in Germany you and Mrs Webb are held in the highest estimation of all English writers on Economics. Marshall is nowhere.’ Thus it is no surprise to find little evidence of mathematics being required by LSE economists until the 1920’s. John Hicks, the future Nobel Laureate, came to the LSE as a temporary lecturer in Economics in 1926/7. He had done one year of mathematics at Oxford before switching to PPE, and found that it was ‘sufficient to cope with what anyone (then) used in economics’. But changes were afoot. In 1931 Roy Allen began to lecture on mathematical analysis to LSE students of economics, and his book was published in 1938. He gave a logical development of the subject in the ‘modern’ style, although he referred to Hardy’s Pure Mathematics for the foundations of the real number system. In addition, he covered a wide range of economic applications, based on the work of Marshall, Edgeworth, Hicks, F.P. Ramsey, and others.
In the next twenty years the face of economics changed significantly, and when Allen came to revise his book in 1956 he adopted a different approach. His new book was entitled Mathematical Economics, and its theme was the exposition of economics in mathematical way. He incorporated the mathematics as it was needed, and so we find matrices and vectors discussed as in a prelude to the theory of games and linear programming.
These developments produced a climate of opinion in the LSE (or at least part of it) sympathetic to the application of mathematics in the social sciences, and the Robbins Report on the expansion of higher education in the UK provided the opportunity for action. The Minutes of the Academic Board held on 26 May 1965 contain the following paragraph:
“Over the past few years there has been a great increase in the use of mathematics related to the School’s subjects and there are more students coming up with Advanced level mathematics combined with Arts and Social Science subjects who want to continue the study of mathematics at the School. It is felt that the School should have a group of pure mathematicians to support and expand the work which is already being done by applied mathematicians in social sciences. There is a growing national need for persons qualified in mathematics with reference to the social sciences. The more traditional form of the mathematics degree which contains a large element of applied physics is not as directly relevant to such occupations as operational research, statistics and econometrics. The new degree which is being proposed would remedy this deficiency, and would enable a mathematical specialist to study his subject in relation to social science disciplines rather than those of the physical sciences.”
In due course it was agreed that this plan should be adopted. Cyril Offord (right), a classical analyst, was appointed to a new Chair of Mathematics. He moved from Birkbeck College, where he had become frustrated by the constraints imposed by that institution’s commitment to part-time students. At the LSE, Offord was faced with the task of setting up a programme of mathematics teaching under a different set of constraints. The core degree was the BSc (Econ), which had a rather rigid structure: in particular, three first-year courses (Economics, Government, and History) were compulsory. This left little room for laying the foundations of a mathematics degree. After some delicate negotiations, Offord, with the help of Roy Allen, persuaded the LSE to establish a new BSc degree, distinct from the BSc (Econ).
Part one of the new degree consisted of five courses: (1) Economics, (2) Analysis and Set Theory, (3) Algebra and Methods of Analysis, (4) Further Algebra and Theory of Probability, and (5) either Elementary Statistical Theory or Introduction to Logic. There was also a dispensation that ‘in specially approved cases’ students could take Analysis and Set Theory in the first year of the BSc (Econ). Offord wrote to a number of schools telling them about the new degree. He pointed out that ‘applied mathematics’, which was traditionally represented by Mechanics and Theoretical Physics, would in this case be represented by Economics and Statistics.
While the framework was being sorted out, recruitment of staff continued. Offord was joined by Haya Freedman (below left) in 1967, and by Richard Hornblower and John Bell in 1968. The last of these has written an anecdotal account of the early years, including the part played by the controversial philosopher Imre Lakatos in his appointment. Bell himself was a colourful character – a mathematical child prodigy who went up to Oxford at the age of 15 and became a logician. After twenty years at the LSE he moved to a Chair in Philosophy at the University of Western Ontario.
By the early 1970’s the mathematics group comprised six people with rather diverse mathematical interests. Another complicating factor was that Offord was due to retire, which he duly did in 1973. His successor was Anatole Beck (below centre), a well-known American mathematician with wide interests in classical mathematics. When Beck decided to return to the USA in 1975, the Chair was filled by Ken Binmore (below right), who had joined the staff as a lecturer in 1969. Binmore had made his name as a complex analyst at Imperial College, but he became attracted to the mathematical aspects of game theory, which was to become one of the fundamental paradigms of theoretical economics.
As the years passed, there were minor alterations in the content and structure of the degree. One course was specifically devoted to ‘Mathematical Methods’, and this attracted many students from other disciplines. However, the number of students studying serious mathematics remained small, even after Mathematics and Economics became a ‘special subject’ in the BSc (Econ) degree. The lack of progress was due mainly to the LSE’s traditional approach to internal organisation. This is exemplified by John Bell’s letter of appointment (1968), which did not mention a department, or give any hint as to his duties or to whom he was responsible. Gradually the need for some structure became clear, and the mathematicians became part of a Department of Statistical and Mathematical Sciences (SAMS). In due course SAMS had five ‘sub-departments’, each led by an unfortunate individual with the title of ‘sub-convener’. The BSc (Econ) still played a major part in the life of the School, although several departments had established course-unit degrees, similar to the BSc in Mathematical Sciences.
By 1986-7 changes were afoot. In twenty years the mathematics group had been granted only one new half-post in addition to the original complement of six. Binmore was increasingly moving towards Economics, where he had established an international reputation for his work on Game Theory, and he wished to make his position a fact. John Bell was leaning more towards Philosophy and Haya Freedman was close to retirement. The outcome might have been disastrous for mathematics, but fortunately a Review Group in 1987 recommended that the core group of mathematicians should be maintained. As a result the School decided to appoint a new professor of mathematics, and made a commitment to replace Bell and Freedman. The first step was implemented by the appointment in 1988 of Norman Biggs. In accordance with the School’s traditions, his qualifications did not include any specific knowledge of economics or social science.