Mathematics In and Out of Indian Universities

We give below a version of V S Varadarajan's lecture Mathematics In and Out of Indian Universities given on 25 March 1982 at the 34th Annual Seminar of the South Asia Regional Studies Department, University of Pennsylvania. A printed version of V S Varadarajan's lecture is at:

V S Varadarajan, Mathematics In and Out of Indian Universities, The Mathematical Intelligencer 5 (1) (1983), 38-42.

Mathematics In and Out of Indian Universities

The subject matter at hand is a very broad one and it is not possible for me to develop more than an outline of it at this time. I have been in the United States for close to twenty years now and so my observations must necessarily be biased in various ways. I have however visited India often during this period and kept in touch with the mathematical community there, so it is reasonable to hope that I am not too far off the mark. I have also consulted many friends, colleagues, and teachers in preparing this talk, and I feel that their comments and insights have been very beneficial. I am especially grateful to Professor K G Ramanathan of the Tata Institute of Fundamental Research, visiting the Institute for Advanced Study at Princeton, New Jersey, and to Professor C Radhakrishna Rao, Jawaharlal Nehru Professor at the Indian Statistical Institute, who is currently at the University of Pittsburgh. Their help was invaluable in preparing this talk. I regard it as a very fortunate circumstance that I was able to get the help of Professors Ramanathan and Rao, because they were important figures in the development of the mathematical sciences in India after the Second World War. Needless to say I alone am responsible for the opinions expressed here and any errors that may have occurred.

There are many aspects to our theme creative, pedagogical and organisational. The creative aspect is of course the dominant one but the others are also important since they determine how favourable the environment is for research work. In any country it is fair to say that most of the important mathematical activity is carried on at the Universities and the Research Institutes, with the Government playing a supporting role. However, it is not just a question of pouring money into new projects or creating new Institutes for advanced study; if this were the case, most countries would have major centres of mathematics. So many things seem to play important roles: tradition, the socio-political and economic environment, the general mood of the people, both at large and in power, and so on. This is perhaps one of the reasons why, if you examined any particular country or region, there are only certain periods that show noticeable peaks of mathematical development.

The Indian situation has some unique features. Although Indian mathematics goes back to ancient days, during most of the colonial period things were very quiet. The first Indian Universities (in the modern sense) were established around the 1850's by the British Government. The general idea was to produce bodies to man the lower echelons of the Administrative and judicial network. Independent thinking was regarded as a needless luxury and was often actively discouraged. So it was a long time before the laws of probability could assert themselves and produce some intellectual stirrings. The entry of India to the modern mathematical stage dates back to Srinivasa Ramanujan (1887-1920). If you are familiar even with the barest outline of the career of Ramanujan you would remember that he was not a "successful" product of the prevailing educational system. How could he be "successful" according to a scheme which was the last word on sterility?

I think Ramanujan's emergence was a watershed for another important reason. It showed (to anyone who was willing to think) that important and even beautiful things could be done in an environment where power was in unsympathetic and uncomprehending hands, and where creative isolation was pervasive. It broke down the psychological barriers and captured the imagination of many. I do not think it is entirely accidental that great scientific personalities like Raman, Bose, Harish-Chandra and Radhakrishna Rao rose within a couple of decades of Ramanujan. However India had to wait for its independence (1947) before excellence in science in general and mathematics in particular became a national goal. I think here she was fortunate to have Jawaharlal Nehru as her first prime minister. For, unmatched among the Indian leaders of that (or any other) era, Nehru had a vision of greatness for Indian Science, the ability to articulate it, and the willingness to take concrete steps to realise it. His efforts led to enormous national support to new institutes as well as to some of the institutes for advanced work that had been established in earlier times, but were still struggling for visibility and recognition. The force provided by his leadership lifted the mathematical sciences in India to the modern epoch.

The key period for us is therefore the post-independence one, starting from 1947. Before going into it let me make some remarks on the general structure of the educational system in India, especially with reference to mathematics. The elementary and secondary schools take up 11 or 12 years and are followed by the undergraduate program for about 4 years. The undergraduate instruction is carried out in the affiliated colleges. The Universities generally confine themselves to the graduate programmes. With a few exceptions, it is only at the Universities that a normal student can expect to be exposed to some serious mathematics. However, the gifted student can go on to the Research Institutes (either immediately after graduation, or, as is more usual, after obtaining a Master's degree in the University). There are also Institutes of Technology where the emphasis is on engineering and applied mathematics. The Institutes are supported by funds from the Government (both Federal and State); supervising this outflow of money are many governmental and quasi-governmental agencies.

As I mentioned earlier, during the Colonial era, instruction in the Universities was based on a very sterile conception of the needs of the society. In mathematics this meant a very watered down version of a small part of the Tripos program, with emphasis on drill and memory. In this barren landscape there were, however, two Institutions that seemed promising. One was the Tata Institute of Fundamental Research (TIFR) which had been established at Bombay, in 1945, as a private Institute supported by the Tata foundation, under the directorship of Professor H J Bhabha, with the aim of doing research in Theoretical Physics in general and cosmic rays in particular. The other was the Indian Statistical Institute (ISI), established at Calcutta in 1931 by Professor P C Mahalanobis, and involved in Statistical Research. The TIFR had, in addition to Professor Bhabha, Professor F W Levi and Professor D D Kosambi in its faculty and had many gifted younger research fellows. The ISI had Professors Mahalanobis, R C Bose, S N Roy, K R Nair as well as younger scientists like C R Rao. When India became independent and Nehru began to emphasise the imperative need to lead India to the mainstream of contemporary science, it was felt that the University system was too archaic and entrenched to allow quick reorganisation. This led to the decision to place the emphasis on development outside the existing educational system, and in particular, to take advantage of the small nucleus of strong faculty and advanced students already in place in TIFR and ISI.

In TIFR a graduate program in mathematics was started by Professor K Chandrasekharan assisted by Professor K G Ramanathan. In ISI there was already a graduate program in statistics; its scope was greatly expanded to include mathematical subjects and Professor C Radhakrishna Rao was in charge of it. The operating procedures in both places were similar. Gifted students who already had a Master's degree were selected and given intensive training so that they could start doing independent work. Leading scientists from other countries were invited to visit for substantial (3-4 months or more) periods of time and give courses of lectures, enabling the students to learn new disciplines relatively quickly. Those young people who did exceptional work were encouraged to go outside India to the great centres of mathematical activity like Paris, Cambridge, Princeton and so on. Many of these (to their huge credit) returned and contributed to the growth of the intellectual community in their institutions. There is no doubt that much of this progress would have been impossible or at least delayed very much, without the active help of many outstanding scientists from outside India. Although they are too numerous to recount I would at least like to mention Laurent Schwartz, Carl Ludwig Siegel, Norbert Wiener, Andrei Nikolayevič Kolmogorov, and J B S Haldane for the inspiration arising out of their visits and their help in bringing attention to the work done at Bombay and Calcutta.

I myself, along with many of my friends, was trained in this milieu, and I can say from first hand knowledge that the faculty and students were responsible for a very exciting and creative intellectual climate that prevailed there during the 1950's and 1960's. Coming out of this period and environment were a number of mathematicians who, by their work in such diverse fields as algebraic geometry, group theory, arithmetic, probability and mathematical physics, have achieved international recognition. I do not believe that similar things have been done in many other places certainly not with such limited resources and in such a short time span (20 years). We should be grateful to Nehru and his associates, especially Bhabha and Mahalanobis, for the farsightedness of their views and their energy to do something about them.

Unfortunately, as the effort to develop the advanced Institutes was not accompanied by a comparable effort to revitalise the instructional program in the Universities, the basic problems have remained unsolved. The mathematics curriculum in the Universities remained old fashioned so that students coming out were hopelessly out of date. This put enormous pressure on the Institutes to conduct background courses and delayed considerably the students' exposure to current ideas; furthermore it made it difficult, from both the human and practical points of view, for the mathematicians coming out of the Institutes to be absorbed by the educational system. New tensions were thus created which had many harmful effects. Some young scientists, convinced that there was no chance of any change in the situation, migrated to the West.

These problems were of course noticed and efforts were made to counteract these disagreeable trends. The TIFR and the ISI started organising summer schools for the faculty at the Universities so that they could be introduced to modern perspectives in mathematics and statistics. The University Grants Commission attempted to encourage modernising trends in the Universities by designating some of them as centres for Advanced Study and channelling financial help to these. New Institutes were created hoping that the success of TIFR and ISI would be repeated. In addition the TIFR started, in collaboration with the Courant Institute of Mathematical Sciences, a programme of developing applications to other sciences.

Varied as these efforts have been, they have not been very successful. The reasons for the failure are many and one cannot go into them in any detail at this time (for some thoughts on mathematical organisation see [4]). In my opinion the foremost are that the basic mathematics curriculum has not been changed except perhaps in a cosmetic fashion and that there is a virtual famine of well-motivated and qualified teachers.

Here I think it is appropriate to ask what the mathematics curriculum should be (see [1], [2], [3]). Briefly, it must focus on the organic unity and historical continuity of mathematics; emphasise general principles in framing and solving naturally arising questions rather than tricky and ingenious solutions; point out the relevance and the (surprising) strength of the mathematical method in the description, interpretation, and prediction of natural phenomena; and above all, convey its aesthetic appeal. With this in mind if we turn to what is still being taught at the Colleges and Universities, we see what we are up against. The graduating student, even with a Master's degree, has almost certainly not heard of homology, differentiable manifolds, quadratic reciprocity or probability; he may have only vague notions about Riemann surfaces, vector spaces and linear transformations, and boundary value problems. In spite of this he may continue downstream and "do a Ph.D." under the supervision of a professor who wants to increase the number of Ph.D.'s he has "produced" so that he may advance in the professional ladder. The dearth of well-motivated and intellectually active teachers may be understood from the fact that the academic system rewards quantity rather than quality of published research.

The prognosis for the future is thus clouded. In my recent visits to India I encountered a certain pessimism that was in sharp contrast to the optimistic, up-beat, view of things we had in the 1950's. The problems are clearly visible and in many cases it is also clear what should be done. However the thrust of the solutions must come from the political and administrative side of the Universities. Being, therefore, essentially human in nature, it will take not only cleverness but wisdom to overcome the obstacles.

Last Update July 2026