MATHEMATICS appears to have acquired an identity as an independent branch of knowledge early on in human intellectual history. This identity became precise and firmly established thanks to the Greeks in the millennium before Christ. Two characteristics are vital to this identity: abstraction and logical deduction; these are of course present in all scientific enquiries but in mathematics they are defining elements.
Abstraction consists in building mental constructs (often, though not always, born of attempts to understand the concrete). This involves the recognition of common patterns in apparently disparate phenomena and, equally important, the rejection of the irrelevant in an investigation. This is already manifest in the first intimations of mathematical activity: counting. Though almost an involuntary act, what underlies it is profoundly abstract: the human mind recognises that there is an attribute which it can ascribe to a collection of entities – the number of entities in the collection – an attribute which is entirely indifferent to the nature of the individual members of the collection yet can be common to different collections.
What underlies it may be remarkable abstraction, but counting owes its discovery to mundane down to earth compulsions: exchange of goods and barter which required setting values on different commodities. The marketplace was the driving force behind all the arithmetic we learnt at school. And down the ages a great deal of mathematics was born as a response to the needs of diverse human endeavours and has served the cause of these endeavours admirably. This symbiotic relationship is most striking in the case of physics. Attempts to understand motion gave birth to calculus, which has gone on to enhance our understanding not only of motion, but a myriad other natural phenomena in diverse scientific disciplines. The success of mathematics has been so striking that mathematical intervention is what seems to confer the label ‘science’ on an intellectual discipline.
Mathematics is itself of course a science, yet it stands a little apart from other sciences. The title of Newton’s classic, The Mathematical Principles of Natural Philosophy, itself suggests this. Gauss (one of the greatest mathematicians of all time), called mathematics the queen of sciences; indeed, at times, one finds in mathematics the aloofness of the royal personage.
The queen is whimsical – queens are supposed to be – but not quite as arbitrary as the one in Alice’s Wonderland. There is a large body of mathematics that has been created as a result of purely aesthetic impulses internal to the discipline, essentially by the fancy of the mathematician. Mathematicians perceive beauty in purely mathematical constructs and the interrelation among them and probe these simply to savour the intellectual delights they offer. Yet time and again it has turned out that such mathematics has proved itself the right vehicle for apprehending nature. The renowned physicist Wigner has described this pithily as the ‘unreasonable effectiveness of mathematics in the natural sciences’. A striking example of this ‘unreasonable effectiveness’ is the way Descartes’ idea of representing spatial entities by numbers – coordinates – has permeated our thinking in diverse contexts. Descartes’ motivation was the renovation of geometry, as far away as can be from the company chairman pondering the profit curve.
In every creative endeavour there is a certain tension between imagination and discipline. In the sciences other than mathematics, the discipline is essentially external: theories about natural phenomena have to be in tune with observations. The mathematician’s imagination is unfettered by external considerations; the discipline comes from the demands of logical deduction and (the more intangible) aesthetics. In this, mathematics is closer to art than to other sciences.
The glorious days of Greek mathematics ended in the early centuries after Christ and the centre of gravity of mathematical development shifted to the East. From then on right up to the 12th century India had a dominant role. One of the greatest mathematical achievements of all time – the invention of ‘zero’ and the place value system for representing numbers – emanated from this country. It is at once an absolutely brilliant piece of abstract mathematics and simultaneously an unmatched practical device indispensable in practically every sphere of human activity. Arguably, human progress owes more to this one discovery than to any other (mathematical) innovation.
The practical importance of the place value system stems from the ease with which it enables one to handle large numbers though at the time of its discovery there was rarely a serious need to deal with large numbers. Ancient India seems, however, to have been obsessed with large numbers – names had been given to powers of ten way beyond the billion (the ninth power of ten). This suggests that it was the fascination with numbers for their own sake rather than practical considerations that was the motive force behind the discovery – another example of the ‘unreasonable effectiveness’ of mathematics.
Algebra – the mathematics of manipulation of symbols representing variable quantities – also originated from India; and that too is a magnificent intellectual leap. Arab scholars who came into contact with India recognised the strength of Indian mathematics, absorbed and built upon it, contributing many great ideas themselves. Eventually they passed it all onto Europe which in a burst of new energy took the lead.
Great as its contributions were, the East nevertheless had not, in its pursuit of mathematics, enforced the rigorous discipline of the Greeks. Europe restored firmly Euclid’s postulational paradigm as the framework for mathematics. Today we recognise a body of knowledge as mathematical only if it can be fitted into that framework. It means that the body of knowledge is derived from a certain number of postulates (called axioms by Euclid) which are accepted without argument through rigorous reasoning, the methods of deduction themselves being governed by set rules which are also to be regarded as postulates.
An amazingly large part of what we regard as mathematical knowledge does meet even this very rigid criterion with only the axioms needed for the elementary arithmetic of natural numbers (and just a little more) as the foundation. However, this last statement has to be tempered by one fact: many mathematical areas which today can be fitted into this remarkably economical postulational scheme could not meet these stringent demands when they were first apprehended. It is only after Dedekind in the 19th century showed how to extend the number system beyond the rational numbers of elementary arithmetic, with only the axioms needed for elementary arithmetic as the starting point, that calculus could be incroporated into the fold of this scheme. Contemporaries of Leibnitz and Newton (inventors of calculus), however, had no hesitation in hailing the birth of some new and wonderful mathematics; and not merely calculus. Alot of other much less sophisticated mathematics could be derived from the axioms of elementary arithmetic by Euclid’s deductive method but only after Dedekind had had his say. Of course, Newton and his contemporaries as well as other mathematicians did use the deductive principles a la Euclid but based themselves implicitly on a set of much more elaborate unquestioned assumptions than the simple axioms of arithmetic.
The modern history – that of the last 500 years – of mathematics is necessarily Eurocentric: practically all major developments right up to the early 20th century originated in Europe. Descartes and Newton were to be followed by a string of great names: Euler, Gauss, Riemann, and Poincare to name a few. There was a steady and bounteous flow of new ideas through the 18th and 19th centuries, ideas from which emerged whole new areas, and this flow turned into a torrent in the 20th century. Gottingen and Paris were at the heart of this mathematical explosion, an explosion which also fuelled the revolution in physics which was taking place at the same time. The United States of America, not yet a mathematical power, was beginning to display its potential. Czarist Russia was also a participant in the mathematical action in Europe; but with the advent of the Soviet Union, the state came up with a deliberate policy for the promotion of mathematics with resounding effect. By the fifties, Gottingen had declined thanks to the Nazis and was yet to recover from the war while Moscow became a formidable centre rivalling Paris and could boast of a galaxy of extraordinarily creative minds. Sadly, that great school has virtually disintegrated along with the political system: most of the outstanding mathematicians having migrated to the West.
The Americans gave relatively little attention to mathematics in the first half of the 20th century, but the Soviet space programme’s first Sputnik jolted them from their benign indifference into eager support for mathematics. Through the sixties and seventies and even into the eighties, support for mathematics was available on a very generous scale in the US and this had a tremendous effect. It produced an array of brilliant mathematicians and much of the most exciting mathematical developments. Europe (Russia included) has now ceded its pre-eminence to the US.
What then of our country? Intellectual activity had certainly taken a back seat for centuries in our country (there was, however, some exceptional mathematical progress in Kerala anticipating later work in Europe). The first stirrings after the long period of dormancy are to be seen in the Bengal renaissance of the 19th century. In the beginning of the reawakening it was the pursuit of humanities that dominated the scene, but in the early 20th century the twin figures of C.V. Raman and Ramanujan blazed new trails in science.
Raman was an outstanding communicator and his leadership provided immense impetus for the development of physics. Mathematics did not have this advantage: Ramanujan’s brilliant career was tragically cut short in its prime. Nevertheless, his example inspired many people to pursue mathematics. A career in mathematics was of course unattractive in comparison with many others when viewed in terms of the creature comforts that one could command, but in the first half of the 20th century there was compensation in the kind of respect that learning was accorded.
It must be said that both Raman and Ramanujan received reasonable support from the colonial institutions of that period; in the case of Ramanujan, once the people in positions of power were convinced of his extraordinary talent, they acted with an alacrity that today’s bureaucrats would do well to emulate. Of course, Britain was not interested in promoting intellectual activity in this country, but there was some response to sporadic individual achievements. In any event, whatever the rulers thought, Indian society did not have a strong awareness of the importance of science, much less that of mathematics during the colonial days.
With the advent of independence, the national leadership – Jawaharlal Nehru in particular – laid great emphasis on science and propagated the idea of infusing our society with ‘scientific temper’. Nehru’s vision resulted in the creation of many institutions of scientific research and among them a few which actively promoted mathematics. However, even as there exists a general perception of science as an important human activity, this perception is (understandably) based on the concrete and practical role science plays in industrial development. There is much less understanding of the civilizational role of fundamental science in general, of mathematics in particular. There is little appreciation of the fact that a great deal of today’s applicable knowledge was at some period in the past basic science at its frontiers. This applies to mathematics much more than to other sciences.
The glamour attached to physics, thanks to developments in the field of nuclear energy and more recently to biology because of the recent discoveries in genetics, helps attract public attention to these fields. Chemistry too has areas that have captured the public imagination. The Nobel Prize also helped increase public awareness of the importance of these sciences: the prize is a household word and some knowledge of facts about it is practically statutory requirement for school kids taking part in quiz contests. Mathematics does not have the advantage of being able to project such a glamorous image and Alfred Nobel unfortunately did not consider mathematics as worthy of partaking of his huge legacy.
Most people, otherwise well informed, are not aware of the Fields Medal which identifies superior mathematical achievement even as the Nobel does in other fields, perhaps because of the fact that its monetary value is a pittance in comparison with the Nobel. Since its inception about 50 mathematicians (necessarily under the age of 40) have been awarded the medal and if a school kid happens to know the name of one of these recipients, it is a safe bet to assume that his/her family has personal contacts with a mathematician. Even many of our teachers in schools and colleges are unaware of the existence of this medal for mathematics. Very recently, the Norwegian government has instituted a prize similar to the Nobel for mathematics. The mathematical community hopes that it will bring the same kind of visibility to mathematics as Nobel has given to other sciences.
There is a general feeling that unlike physics or other sciences, mathematics is a somewhat other worldly pursuit. Few realize that esoteric problems of cosmology pursued by astrophysicists or the frontier areas of particle physics are as meaningful to the practical everyday world as Fermat’s last theorem. As a mathematician I have come across many people who wonder what there is left in mathematics to discover, an experience that I am sure the physicist does not share.
Despite this general lack of public awareness about mathematics, there is in this country a feeling that we are very good at mathematics and there is a certain pride in its past achievements. We have no doubt major past contributions to mathematics to our credit, as has already been pointed out. The romantic story of Srinivasa Ramanujan no doubt contributes (very justifiably) to this belief. But there are other dubious claims on behalf of Vedic mathematics which are also taken quite seriously and contribute to this confidence in our mathematical proficiency. And there is this feeling that people like Shakuntala Devi who can perform calculating feats represent superior mathematical talent that strengthen this perception.
Since Ramanujan, there have been some substantial contributions to mathematics from Indians (working in India), but by and large the public or even the scientific community outside of mathematics are not aware of them. These have made the international community to sit up and take notice of us. A major mathematical event, the International Congress of Mathematicians (ICM) has been taking place regularly once every four years since 1950 with ever increasing international participation. This is the occasion when the Fields Medals are awarded.
The meet is built around 200 prestigious invited lectures: some 20 ‘plenary’ one-hour lectures address a general mathematical audience and are given by eminent figures who have influenced the evolution of different mathematical areas; and the rest, more specialised talks (of 50-minute duration) by other outstanding contributors to diverse fields within mathematics (the numbers mentioned are from recent congresses).
Some idea of the Indian impact on the world of mathematics may be gleaned from the fact that since 1958, in every Congress except in 1986, there has been at least one invited talk by an Indian mathematician working in India. There have been two plenary talks by Indians but both were non-residents; however, on the whole there have been far fewer talks by non-resident Indians than by those working in India. The Fields Medal has so far eluded us though there are instances of people coming close to winning it. In the developing world, India is certainly in the lead, but China seems poised to overtake us.
That is the ‘feel good’ side of Indian mathematics, but the peaks do not tell the whole story. The general level of mathematics outside a handful of institutions of higher learning is abysmally low. Much of what is passed off as research in our institutions of higher learning is of shamefully poor quality; nor does one find scholarship of note. For example, if one excludes some half a dozen institutions in the country, it will be impossible to find others even distantly familiar with the mathematical areas in which the fifty odd Field Medallists have worked. Much of the peak achievements have in fact come from two institutions: the Tata Institute of Fundamental Research in Mumbai and the Indian Statistical Institute with branches in Kolkata, Delhi and Bangalore.
The present scenario is indeed depressing, and in many ways things are on a further decline. There has been a steady loss of interest in the pursuit of mathematics (in fact, of all basic science) among the young over the last three decades. In the recent past this problem has become even more acute. This is largely due to socio-economic factors. Career options for mathematics graduates are essentially limited to the academic profession, not attractive in terms of emoluments: a fresh management graduate enters his/her career at a salary at which the academic retires. Unattractive economic status is of course not new to the academic profession, but in an earlier era it commanded considerable respect in our society and that is no longer the case.
In the case of talented youngsters even these factors do not thwart them from pursuing mathematics, but many seek to pursue it in Europe or America. In the advanced countries too there is a decline of interest in mathematics despite career opportunities being available outside the academia. These migrants from India and others like them from the Third World are in fact now the mainstay of American graduate schools, safeguarding the future of mathematics there. One reason of course is that in these countries the academic has a much better socio-economic standing in society than in our country. The other is the sorry state of most of our institutions of higher learning – an eager youngster cannot get exposure to truly exciting mathematics except in a limited number of places and the best graduate schools we have, while good, are no match for Harvard or Princeton.
Unfortunately, the policies of the state in regard to higher education do not warrant an optimistic prognosis for the future. The state seems to be reducing, even withdrawing completely such support as it provides now for higher education. This is ostensibly because of the contrast with the meagre resources made available for basic education. There can be no two opinions about the need for enhancing several fold the support given to basic education. But this does not mean that the support given by the state to higher education has to be reduced: it is certainly not excessive by itself. However, it is true that most of the support provided for higher education has ended up as a ‘subsidy’ for business and industry (in the country as well as abroad): the prestigious institutions in the country (run practically entirely with government support) have provided human resources for them.
On the other hand higher studies that do not cater to the needs of commerce get short shrift. State governments have steadily cut back on funds for universities. Many university departments have unfilled positions which cannot be advertised, as the governments are not willing to release the money needed for the purpose. Colleges are run with teachers hired on the basis of contracts on unattractive, humiliating terms. In such a context, it is hardly surprising that a very large number of our college teachers are less than competent. The problem gets more acute with areas like mathematics where the subject has been for several years a last resort option for students and these are the ones with the requisite ‘qualification’ to teach the subject.
In the context of the enormous socio-economic problems facing us, a low priority for the promotion of mathematics may appear entirely justified. But this is definitely short sighted. Mathematics has an ever-increasing role in every human activity. As was pointed out earlier, it played a vital role in the advancement of science and technology. Industry in the advanced countries has benefited immensely from the intervention of mathematics of increasing levels of sophistication. Biology and medicine which seemed at one time to be impervious to mathematics, now use some advanced mathematics. The social sciences too are increasingly turning to mathematics for help. Wall Street employs mathematics PhDs routinely.
All this suggests that the promotion of mathematics is vital to our future well-being. There is another intangible important benefit that society derives from mathematics. Mathematics inculcates habits of thought that help promote scientific temper. There are many initiatives that can be taken to promote mathematics but they will need political will: we need a major overhaul of the educational system and some social engineering as well. It would appear from our track record that there is mathematical talent and undeniably we have an excellent tradition to revive; what is needed is to create the conditions in which that talent can flower.
Despite all the discouraging factors there is a trickle, small though it is, of headstrong youngsters who pursue mathematics and perform at superior levels working in India. This is the one silver lining in the cloud that keeps optimism alive.
