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KUMARI L. A. MEERA(1961–1985)

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Kumari L. A. Meera, the youngest daughter of Mr. and Mrs.L. K. Ananthakrishnan, was born on June 1st, 1961, in New Delhi. Her academic career was uniformly brilliant with a distinction at the higher secondary level and a University rank in the B.Sc. (Honours) examination in Physics, from St. Stephens College, Delhi. After again obtaining a rank in her M.Sc., which was at the Indian Institute of Technology in Madras, she took up a Masters programme in Computer Science at the same institution with a view to preparing herself for the emerging applications of this area to physics. She was strongly oriented towards research, having attended summer schools at the Indian Institute of Science and the Tata Institute of Fundamental Research. Her teachers and colleagues recognised her promise and motivation and confidently expected an outstanding career in research. Her personality was warm and friendly with a strong streak of hard work, determination, and helping others. These characteristics endeared her to those who knew her.

She joined the University of Pennsylvania in Philadelphia in September 1985 for doctoral studies in physics. But in November of the same year, her bright career was tragically cut short by her untimely death.

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M. S. RAGHUNATHAN

M. S. Raghunathan is a mathematician working in the area of Lie Theory to which he has made important contributions. He is currently DAE Homi Bhabha Professor at the Tata Institute of Fundamental Research from which institution he retired as Professor of Eminence in 2006. Raghunathan is a Fellow of the three national science academies in the country as well as of the Royal Society of London. He is a recipient of the Bhatnagar Prize of CSIR and of the Third World Academy Prize for mathematics. Raghunathan has also been engaged in promotional activities for mathematics: he headed the National Board for Higher Mathematics for 1987–2006 and currently continues as a member. He was a member of the Executive Committee of the International Mathematical Union during 1998–2006. Currently he is the Chair of the Governing Council of the Harish-Chandra Research Institute in Allahabad and the Steering Committee of the Kerala School of Mathematics in Calicut.

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Eighteenth Kumari L. A. Meera Memorial Lecture

The Queen of Sciences: Her Realm, Her Influence and Her Health

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M. S. Raghunathan

What is mathematics? That is evidently a difficult question to answer. Nevertheless mathematics has had an independent identity as an intellectual discipline since antiquity. In this talk, I will first discuss the main characteristics that contribute to giving mathe-matics its identity challenging in the process some common negative perceptions about mathematics, viz., that mathematics is a dry unimaginative subject, that its rigorous discipline is forbidding and that its cold logic destroys all sensitivity in its practitioners. I will then briefly dwell on the importance of mathematics to our society and finally say something about the state of mathematics research and teaching in our country

This talk is about mathematics. Carl Friederich Gauss, who along with Archimedes and Newton is considered one of the three greatest mathematicians of all time, described mathematics as “the queen of all sciences”; and so the title. The “queen” of sciences does seem to have many of the qualities associated with that royal personage. There is a certain aloofness about royalty and that can be seen in mathematics in relation to other sciences: the very title of Newton’s magnum opus “The mathematical principles of natural philosophy” suggests that mathematics stands a little apart from other sciences, even while it is its patronage that decisively confers the title “science” to any body of knowledge. The queen in history has largely been a decoration, an ornament, not a centre of power; and such power as she wielded was largely indirect. The king is the power centre. His aides make front page news while those of the queen have to scurry to page three. Even in the nursery rhyme the queen is self-

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indulgent and inconsequential: “she was in the parlour eating bread and honey” while the king was “counting out his money”. In all this too the apellation “queen” fits mathematics very well. Perhaps that explains why Gauss's claim for mathematics is seldom disputed. The queen of sciences is whimsical – queens are supposed to be. The way mathematics develops is mostly determined by an internal dynamic, the imagination of the mathematician. It does every now and than draw upon natural sciences for inspiration, but even when that happens, the new mathematics that is born, takes on a life of its own and often charts a path that has little relevance to its origins: the mathematician's imagination takes over. Whimsical she is, yes, but the queen of sciences is not quite as arbitrary as the queen in Alice's wonderland. There is an overriding constraint in the way mathematics evolves: it has to be beautiful, and beauty of course is the foremost quality that the popular mind associates with the queen. Mathematics is indeed beautiful. The first intimations of mathematical activity are no doubt to be found in counting. Counting is almost an involuntary act, but behind it is profound abstraction, a great leap in imagination. The human mind recognises an attribute that is common to a plethora of collections, an attribute entirely indifferent to the nature of the individuals in the collections, namely, the number of objects in the collections – a box of chalks and a box of cheese cubes can indeed have that common attribute or for that matter, share it with a collection of wedding gifts, no two of which may have anything in common with each other beyond belonging to that collection. It is a first step in introducing some order in the chaos that a simultaneous contemplation of disparate collections entails; and seeking order in the midst of chaos is surely a search for beauty. Perhaps the anaesthetic of familiarity – to borrow a phrase from Richard Dawkin's book “Unweaving the rainbow” – inures us to this beauty. But the evident pleasure some children take in counting is perhaps evidence of that beauty. Abstraction is at the core of all mathematics: it consists on the one hand, of the rejection of the irrelevant in an investigation and on the other hand of recognition of commonalities in apparently disparate phenomena or situations. Counting, as I pointed out, is an instance of both aspects of abstraction at play. It took Galileo's genius to treat friction as irrelevant in the first instance in studying

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motion – the crucial step for turning mechanics into a mathematical discipline. Mathematics, especially pure mathematics is practically all imagination. The concept of a number in the abstract is, as I pointed out, the result of a feat of imagination. In geometry, one has idealisations such as points and lines – one imagines objects of smaller and smaller dimensions and end up in the idea of a point, an object of zero dimensions. The word concept itself means something which has an existence only in our imagination. And mathematics is all about concepts and their inter-relations, inter-relations that can be deduced by application of rigourous logical reasoning. By and large a concept is considered mathematical if it can be related to concepts that developed from the integers. All of geometry can be fitted into this paradigm – that was essentially achieved by Descartes through his analytic geometry. Analytic geometry essentially converts all problems of geometry into problems in algebra. Some of the concepts like the ones I mentioned just now have their inspiration in the world around us, but there are others that are the result of imagination applied to other mathematical concepts themselves. The realm of mathe-matics is thus imagination. Whatever its origin, once a concept is introduced into mathematics, mathematicians find it interesting in itself irrespective of any meaning it may have outside of mathematics; and a host of new problems about it, which are the result again of imagination, become their preoccupation. Gauss, when he declared that “Mathematics is the queen of all sciences”, went on to add that “Arithmetic (that is Number Theory) is the queen of mathematics” – and Number Theory is the area of mathematics that has almost no contact with the world outside mathematics. The problems in number theory are mostly generated by internal considerations. And aesthetics is the guide that dictates the choice of the problem to pursue from among a myriad possibilities. Let me give you an example of such a problem and its resolution by Euclid which requires only a minimal back-ground in mathematics. Most – if not all – mathematicians consider it a beautiful piece of mathematics; if you too find it beautiful, your aesthetic sensibilities will find a ready resonance in the mathematical mind. Elementary arithmetic of whole numbers was no doubt born in the market place. Primitive barter of goods required setting relative

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values to commodities such as equating three goats to a cow and as the markets grew more sophisticated, one needed addition subtraction, multiplication an so on to transact business. It is the market place that triggered the development of all the arithmetic we learn at school. But already in the second half of the millennium before Christ, that arithmetic set one foot in the ivory tower in Greece. The notion of a prime crops up as soon as division is introduced: let me recall that a prime is a positive integer p other than 1 such that the only numbers that divide it exactly – that is without leaving a remainder – are 1 and itself. That concept however leaves the market place cold, but the ivory tower is fascinated. The first few primes are 2, 3, 5, 7, 11, 13.... Euclid (whose geometry is familiar to us) raised the following question about prime numbers (in 4th century BCE):

Is the collection of all prime numbers a finite collection?

He answered the question in the negative: there are infinitely many prime numbers. I will now proceed to give you his elegant proof of that assertion. Suppose that the collection P of all primes is finite. Let p1.p2.p3...pn–1, pn be an enumeration of all the primes (in P). Let N be the integer p1.p2.p3...pn–1.pn+1. Then N cannot be a prime since N is greater than all the primes. This means that N is (exactly) divisible by some integer r with 1 < r < N. Let d be the smallest of such integers r. Now if d' is a divisor of d, d' will also divide N (exactly). Since d is the smallest of all the integers, not equal to 1, dividing N, d' = d or 1. In other words d is a prime. It must therefore be none of the pi in P. But (unlike d), none of the pi divide N exactly. In fact all of them leave a remainder 1 when dividing N. Thus our assumption that P is finite thus leads us to a contradiction. So P cannot be a finite collection. Since Euclid mathematicians have raised and are still raising (and answering) any number of questions about primes, questions that have nothing to do with the world outside mathematics. Let me give you two more examples of major mathematical developments that resulted from an internal dynamic rather than any external stimulus. Pierre de Fermat was a French mathe-matician of the 17th century. He was an amateur – by profession, he was a judge in the provincial town of Toulouse in France.

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Mathematics was a hobby, but nevertheless a consuming passion and his work places him among the all-time greats in the history of mathematics. He possessed a copy of the book “Diophantus” – an account of the work of the Greek mathematician of that name – which he studied avidly making brief notes in the margins. One such note discovered posthumously was in a page in that book devoted to Pythagorean triplets, that is triples of positive integers (a, b, c) such that a2 + b2 = c2. They are called by that name for obvious reasons – they are the two sides and the hypotenuse of a right angled triangle. Pythagorean triplets (of integers) abound: in fact if p, q are any pair of integers (p2 – q2, 2.p.q, p2 + q2) is a pythagorean triplet. Fermat's note read “On the contrary, it is impossible to separate a cube into two cubes, a fourth power into two fourth powers, or, generally any power above the second into two powers of the same degree. I have a truly marvellous demonstration (of this) which this margin is too narrow to contain”. In other words Fermat claimed that if an + bn = cn with a, b, c non-negative integers and n an integer greater than 2, then a or b is zero. That note was written some time in 1637 but Fermat did not write down a proof anywhere except for the special case n = 4. Sometime after Fermat's death, his son brought the marginal note to the attention of some mathematicians. Professional and amateur alike became fascinated with the question and generations of mathematicians tried without success, their hand at proving Fermat's claim. It defied their ingenuity for some three and a half centuries till in 1994, Andrew Wiles, a British mathematician working in Princeton, produced a proof. And that proof used the vast machinery developed by several leading figures in mathe-matics over the intervening period, a lot of it in efforts at solving the Fermat problem or other problems that arose as off-shoots of attempts at solving it. An entire area known as algebraic number theory was developed during that period. In the light of this history, mathematicians now generally believe that Fermat mistakenly believed that he had a proof when he recorded that note in his copy of “Diophantus”. Fermat's last theorem is a question that arose out of pure curiosity about whole numbers. Pythagorean triplets – which seem to have some nebulous practical connection – were the trigger for Fermat's imagination in posing the question. But there

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the connection with the outside world stops and fascination with numbers take over. Here is a second episode in the history of mathematics which too illustrates the role of internal dynamics in the development of mathematics. We are all familiar with the quadratic equation. Mathematicians of ancient India knew how to solve the quadratic equation – the Bakshali manuscript discovered in 1881 near Peshawar in Pakistan dating back to 4th century CE or before, bears witness to it. The roots of the quadratic equation ax2 + bx + c = 0, as we learnt in school, can be expressed in terms of the coefficients of the equation by performing arithmetical operations on them and taking square roots as well. The solutions of the equation ax2 + bx + c = 0 are:

 {–b + (b2 – 4ac)1/2}/2 and {–b – (b2 – 4ac)1/2}/2.

The quadratic equation turns up in diverse practical situations, but higher degree equations seldom crop up. Nevertheless mathe-maticians got curious about the cubic equation which yielded its secrets with some struggle as did the next – the fourth degree equation. Then began a quest to solve the fifth degree equation and that went on for some three centuries without any result, with good reason as it turned out. A young Norwegian Hendrik Abel, in his early twenties, had the daring imagination to think very differently: he proved that one cannot express the roots of a general fifth degree equations in terms of the coefficients using only the arithmetical operations and extractions of roots of all orders. The cubic and the biquadratic were already of no great interest to people outside mathematics, the quintic even less. But mathe-maticians became obsessed with them. They had to know, to understand, the quintic. Nor did it stop with the quintic. One wanted to understand equations of any degree. And along came a young man named Evariste Galois from France to pick up that challenge. Galois developed a brilliant and comprehensive theory, now known as Galois Theory that could tell us when a given equation can be solved in the manner in which equations up to the fourth degree were solved. The general theory was the result of a purely mathematical quest. Abel and Galois both died young: Abel at 26 and Galois when he was 21. You cannot script a more poignant tragedy than the stories of their lives.

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 Those are some illustrations to highlight mathematical develop-ments that are the result of internal concerns with aesthetics as the guide. On the other hand Newton's development of Calculus is the spreme example of mathematics that was inspired and guided by the world external to it – it was motivated by the urge to understand motion, planetary movements, in particular. And Calculus and its off-shoot, Differential Equations have had a symbiotic relationship with the natural sciences. Even in these cases however, mathematical developments have followed parallelly a path dictated by aesthetics. Many of the leading figures of 19th century mathe-matics spent time perfecting the Calculus, putting it on rigorous foundations following the Euclidean paradigm. It would appear that great mathematicians tend to set greater store by mathematics that is concerned with its own constructs rather than mathematics that enlists itself into the service of other disciplines. Gauss's partiality to Number Theory certainly indicates that. When admonished by Joseph Fourier, a major figure of eighteenth century mathematics for pursuing useless mathematics, his greater contemporary Carl Gustav Jacobi responded with “A savant like Fourier ought to know that the sole end of science is the glory of the human mind and under that title, a question about numbers is worth as much as a question about the system of the world”. All that illustrates the central role imagination plays in mathe-matics. Yet, in public perception, mathematics is a “dry” subject! Many people find the discipline demanded by mathematics forbidding. Euclid's geometry, when first encountered evokes a sharp reaction: either one of immense pleasure or one bordering on despair. I would suggest that the former is the natural reaction while the latter is often the result of the inadequacies of the teacher. The discipline demanded really amounts to asking one to think, think logically and once one learns to do that learning mathematics becomes a pleasurable experience – the discipline becomes instinctive. In every creative endeavour, there is a tension between imagi-nation on the one hand and discipline on the other. In the natural sciences or in mathematics applied to them, the discipline is imposed from outside. A theory seeking to explain a natural phenomenon has to be in tune with the observations relating to the phenomenon and that reins in the imagination. But with mathe-

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matics, the constraints on imagination are internal: There is of course the discipline imposed by the exacting standards of rigour in reasoning (which I talked about just now). Less widely appreciated is the other factor, which too I have already talked about: aesthetics. Not all problems that arise in the study of mathematical constructs are pursued avidly: one makes a deliberate choice in favour of those that are seen to be pretty. In this – in that the discipline is entirely internal – mathematics is closer to the arts than the sciences. As I have already said some mathematics was developed to meet the needs of other endeavours. But even the mathematical ideas born of the purely aesthetic impulse, have time and again proved to be the right tools for apprehending nature, a phenomenon described pithily by the physicist Wigner as “the unreasonable effectiveness” of mathematics in the natural sciences. Decarte's idea of representing points on the plane by a pair of numbers was a device he introduced to renovate geometry; it was not motivated by any practical consideration. But graphs today are tools that shed light on a myriad things – from Physics to Biology to Economics, to Commerce.... . A lot of sophisticated number theory goes into modern cryptography, which is vital for information security; and all that number theory came out of fascination with numbers for their own sake. Major developments in mathematical analysis, much of it the result again of curiosity about mathematical concepts, have been applied to communication problems with tremendous success. Group theory which was developed largely in the context of Galois theory, has had a fundamental role in quantum physics. Riemann – one of the all-time greats in mathematics in – invented what we call Riemannian geometry, motivated largely by aesthetics but fifty years after Riemann, it was to provide the right frame work for Einstein's general relativity. Some of the very sophisticated Algebraic Geometry developed relatively recently for purely aesthetic reasons is already being used by physicists. Mathematics has been playing an ever increasing role in every field of human activity. I mentioned some of its intervention in other areas that effected remarkable progress in them. Biology which some hundred years ago seemed impervious to mathematics seems to be using more and more of it, none of which was developed with Biology in mind. Technology these days uses mathematics of a very high level of sophistication, mathematics

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which was created not so long ago for purely aesthetic reasons. In recent years business and finance have benefitted immensely from the use of sophisticated mathematical ideas of Probability theory. The work of many of the Nobel Laureates in Economics is mathematical. There is no gainsaying the all pervasive influence that mathematics wields on practically ever sphere of human activity. There is another less tangible benefit that mathematics confers on us. Pursuit of mathematics even at the elementary level helps develop logical thinking – an asset in any pursuit whatever. The scientific temper that we want to instill in our people is perhaps best achieved through teaching mathematics. In any attempt to build a knowledge society, considerable attention needs to be paid to mathematics. And of course that means that the first step is to strengthen our school system to ensure that children are taught mathematics properly. But before I go on to talk about education let me address one other issue: there is a perception that the “cold logic” of mathe-matics is the very anti-thesis of sensitivity and so there is some-thing unfeminine about the pursuit of mathematics. The poet Keats in one of his poems asks the rhetorical question:

 “Do not all charms fly At the mere touch of cold philosophy?”

and later in the same poem, he says:

 “Philosophy will clip an angel's wings, Conquer all mysteries by rule and line”

Philosophy means natural philosophy by which term science was known in Keats's days. As with most sweeping statement of this kind, it cannot be sustained by any serious evidence. Bhaskara, one of ancient India's mathematical stars, in his book “Leelavati” poses mathematical problems in verses full of interesting imagery. It is perhaps not very well known that Omar Khayyam was a leading mathematical figure of his times – he did not use poetry in writing mathematics though. Lewis Carol was the pen-name of the mathe-matician Rev Dodgson, a don at Oxford. Carol's superb humour is difficult to match, but a sense of humour is by no means rare to

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come by among mathematicians. I may also add that much of Lewis Carol's sense of humour has its basis in “cold” logic. Sonja Kowalevski, another big name in mathematics, was a novelist of some standing. Henri Poincare , the great French mathematician, wrote beautifully about mathematics. Hardy's book “A mathe-matician's apology” is a sensitive, yet unsentimental portrayal of a mathematician's perceptions of his profession. The role played by many leading men of mathematics of the 19th century in the fight against gender prejudice to get Sonja Kowalevski the recognition she richly deserved makes a heart warming story. And again in the 20th century the great mathematician Hilbert and his colleagues fought a long, eventually successful, battle against gender prejudice that kept Emmie Noether, one of the great mathematicians of all time out of academia for a long time. In public perception, the mathematician is an absent minded creature entirely absorbed in the world of mathematics and generally devoid of practical commonsense: Professor Calculus of the Tintin comics is the prototype; better that, I suppose than Professor Moriarty, Shelock Holme's criminal adversary. There are no doubt some mathematicians who will fit the Professor Calculus mould, but not many. On the other hand there is among mathe-maticians a reasonable smattering of a wide variety of interesting people as in any other group of professionals. There are flamboyant characters who attract attention, but there are others as austere as they come, yet no less interesting. There is this book, “Men of mathematics” by E T Bell which gives short biographical accounts, with a little mathematics thrown in, of great mathematicians from antiquity down to the end of the 19th century. It makes delightful reading and you will find these people very interesting indeed. Bell has been rightly criticised for historical inaccuracies and over-romanticisation; nevertheless, I believe that he captures the personalities of his subjects very well. Here for you is a gallery of interesting men of mathematics of the 20th century. Hermann Weyl, Carl Ludwig Siegel, Andre Weil, Renee Thom, Alexander Grothendieck. There are many more of course, but these have an Indian connection and I happen to have had the good fortune of having met all of them except Weyl. Weyl was a renaissance intellectual, as much a physicist as a mathematician. He has a little book titled “Symmetry” which is a

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kind of public outreach for mathematics. Among other things, the book explores the mathematical basis of aesthetics in visual arts – Keats would not have approved! Weyl gifted his entire collection of the volumes of the journal Mathematische Annalen to TIFR – he was persuaded to do so by K. Chandrasekharan, the man responsible for building the School of Mathematics at TIFR. Minakshisundaram another big name in Indian mathematics worked on problems closely related to some of Weyl's work. Carl Ludwig Siegel is considered one of 20th centuries greatest mathematicians. He successfully pretended to be mentally unstable to escape military service in Nazi Germany and left the country as he could not stomach antisemitism. He returned home after the war to Gottingen and perhaps thought of himself as a successor to Gauss. He was baptised Karl with a K which he changed to a C – apparently after Gauss – which caused considerable difficulty to archivists looking for a record of his birth in his home village! Siegel has visited TIFR several times for extended periods of stay. He has had considerable influence on Indian mathematics. Andre Weil is another major figure of the twentieth century. He spent two years (1930–32) at Aligarh University as Professor and Head of the department of mathematics. He had a deep interest in India and was particularly fascinated by Indian philosophical thought. He refused to do military service during the second world war – he cited the Bhagawad Gita whose ostensible purpose was to get Arjuna to fight, for justifying his stand: his “dharma” was the pursuit of mathematics, not soldiering, however just the cause! Weil's mathematics has also been a big influence on Indian mathematics. When he visited India some thirty years after his first sojourn, his hosts in India included the then president Zakir Hussain and Dharam Vira, governor of West Bengal, both personal friends from the Aligarh days! Rene Thom was one of the most original topologists of the 20th century. He embarked on providing a mathematical model for morphogenesis in biology. For a while his catastrophe theory made waves and the French media went overboard announcing the arrival of a French Newton. But his work in topology gives him a lasting place in the history of mathematics. Though French, his mathematical style was very different from the predominant Bourbaki style. Shortly before his retirement he went into philosophy and on one occasion talking to me he said that he went

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into philosophy because mathematics had become too difficult from him! Thom too has visited India professionally more than once. Alexander Grothendieck is an extraordinary phenomenon. He began working in functional analysis with Laurent Schwarz, scarcely meeting his adviser but producing a thesis which was published as a book and is the last word , a bible on the subject. He then switched to algebraic geometry where too he effected a revolution and the agenda he set for the subject 50 years ago is still the directive force. That agenda was in fact announced at a confereve in Mumbai in 1968. And then, all of a sudden in 1970, he decided to quit mathematics which he declared an unimportant pursuit, faced as humanity was with serious survival problems. He took to farming failed miserably and kept coming back to acdemia to earn money with lectures to sustain his farm. He started a journal called Survive – survival in English – which folded up in a year! He had a formidable reputation for hard work and when asked by a colleague if it was true that he worked sixteen hours a day his response was “No, not every day!”. So much about the nature of mathematics. Now let me briefly indicate where our country stands in the world of mathematics. We have a long and glorious mathematical tradition. The Pythagoras theorem was known to our ancestors – it is to be found in the Baudhayana Sulva Sutra dating back to 7th century before Christ, some two centuries before Pythagoras. The idea of conferring the status of a number to zero emanated from India, perhaps as far back as the 2nd century before Christ. Our present way of representing numbers by the place value system using the zero is also of Indian origin and that is arguably the single most important mathematical discovery in the history of all science – it is at once a brilliant piece of abstract mathematics and at the same time a practical tool of the greatest value. This discovery may appear to be an excellent example of the old adage that necessity is the mother of invention. But it may well be the result of an aesthetic quest. The real need for an efficient and economical way of representing numbers really arises when you have to deal with large numbers. The Greeks at the time of Alexander and the Persians before them must have felt that need if only to manage their huge armies; but they did not arrive at the system. On the other hand ancient India had a fascination for large numbers – names had been given to powers of ten up to the eighteenth and beyond – numbers that

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would not crop up in any practical context. So it may well have been the aesthetic drive to find a way to write down larger and larger numbers that may have resulted in this discovery. Algebra the manipulation of variable numbers seems also to have originated in this country. There was no question that through the middle ages that India was in the lead in the international scene. Arya Bhata and Brahma Gupta are formidable mathematical intellects to reckon with. So were the two Bhaskaras. Then there was a school in Kerala during the 14th and 15th centuries whose leading figure Madhava had essentially discovered the calculus, 300 years before Newton. And in Ramanuajan we had one of the greats of the twentieth century. All that is good reason for us to believe that our country has considerable mathematical strength and people at large do seem to think that. And in fact some of the post Ramanujan work in this country stands up to the best done anywhere in the world, though the public at large or even many members of the scientific community have no awareness of that. On the other hand spurious claims about so called “Vedic” mathematics unfortunately seem to have impressed many people. Despite this, the negative perceptions about mathematics I mentioned are also very prevalent. Such perceptions are global, though and not confined to our country. But one important contributory factor for the unfavourable perception is the poor quality of mathematics teaching in our schools. The principal reasons for this is that, over several decades now, a large number of our mathematics teachers have been less than competent at their job. This is of course not a problem which is exclusive to mathematics. In the rest of this talk, what I have to say will apply to our education system as a whole. The ignorance of many of our teachers, of the subject that they are supposed to teach, is of, what I call the second order. Not only do they not know, they do not know that they do not know. With some others it may not be ignorance but a total lack of enthusiasm for teaching that is the reason behind their not delivering. The poor quality of the teaching profession in the country is simply due to the fact that the pool of applicants for teaching jobs itself has been of indifferent quality. This has come to pass because, over many years, the teaching profession has not been an attractive career option for most bright young people. The surprising thing is that one

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does find more good teachers than you can expect in such a context. The poor emoluments of the teacher is of course an important factor in rendering the teaching profession unattractive. Other professions which at one time were behind that of teaching have since overtaken it, to say nothing of new ones that have cropped up in the wake of liberalisation of the economy. One sometimes hears shrill voices about how teachers in government schools are irresponsible despite good salaries and are not held accountable. There is no denying that there is a problem there. But I do not quite understand what is meant by “good salary”. The teacher's emoluments do not compare favourably with that in many less demanding professions. A Bollywood star or a cricketer makes more money out of a two-minute commercial brainwashing our kids into drinking a cola they do not need, than a teacher charged with the responsibility of shaping their minds, would in ten lifetimes! Our society's values and demands for accountability are indeed grossly skewed. Salary is by no means the only reason for the profession being unattractive. Working conditions in our schools leave a lot to be desired. The unwieldy sizes of the classes makes even the most committed teachers loose heart. Even passing on information effectively to large classes is a daunting task, but in a subject like mathematics, where you have to convey concepts, the task becomes close to impossible. The workload in terms of the number of hours of teaching is also in general much too heavy. Teachers need time to prepare their lessons – even those who have experience of years. Every batch of students is different and a good teacher would want to device strategies suitable for them and that needs time. Also it will be the endeavour of any intelligent person to constantly up-grade one's own knowledge and that too needs spare time. The teacher has also other duties apart from teaching – setting and correcting exams, interacting with parents etc. They are, in government supported schools, often forced into tasks not related to their professional responsibilities such as election duties, family planning work etc. The general infrastructure in our schools is of very poor quality. The physical environment in which they work is often discouraging in the extreme.

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 Way back in the past, the school teacher found some compensation to some of the unattractive features of the profession in the respect accorded to him or her. Our social environment has come a long way since and that respect has all but vanished. Respect, in our milieu, seems predicated exclusively on economic status. The buzzword these days is “knowledge society”. Evidently the first step towards that is the renovation of our education system and there can be no two opinions on the need for that. If our education system is to deliver, the crucial thing is to ensure that the work-force in the system is of good quality. That can be achieved only by making the teaching profession attractive and sought after by bright young people. I outlined the various reasons that make youngsters shun the teaching profession and these problems have to be dealt with. Emoluments need to be much better than they are – one needs to think in terms of doubling or tripling them. The student teacher ratio must come down drastically – classes of not more than thirty children should be the target. Work-load has to be cut drastically: anything more than 3 hours of teaching in a day is not conducive to maintaining good standards. Infrastructure has of course to be improved by leaps and bounds. These suggestions for making teaching an attractive career option are of course not easy to implement. The financial outlay needed will perhaps be several times what we spend on education now. But if it achieves what it sets out, it is well worth the money. There will of course be many practical difficulties in achieving the goal even if the money and the political will are available but we can find solutions to these practical problems. The school teacher is rendering greater service to us than most other professionals in our midst and our society should ensure that he or she enjoys a socio-economic status commensurate with that. Urgent steps are needed – the problem of quality in education is fast reaching crisis proportions. Unfortunately even while we talk of building a “knowledge society”, we do not seem to be paying attention to the foundations of that edifice. All talk about educational reform is about curriculum, examination system etc. issues which are important, but far less so than the problem with human resources. Well I am reluctant to end on that grim note. So I will tell you a nice story about Gauss which involves a school teacher.

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 Gauss as a child displayed unusual intellectual precocity. At age three he once corrected his father's calculation, when the latter, a farmer, was distributing wages to his peasants. Gauss was all of ten when he made his first exciting discovery. It is at that age that he was formally admitted to a class in arithmetic. Buttner, the teacher was the kind of lazy sadistic character who inspires the shrill voices I talked about. When he did not feel like teaching he set the class an arithmetical problem that was sufficiently tedious for the kids to be at it no end while he could relax in his chair. On one such occasion, he asked the children to add up all the whole numbers between two numbers which were far apart. All the kids except Carl Friederick laboured hard adding numbers on their slates. Carl wrote down a number on his slate in two minutes flat and put it on the master's table and sat back while the rest toiled for the hour before they too piled up the slates on the table. Carl's was the only correct answer. He had discovered on the spot the well known method for summing up arithmetic progressions! Buttner was stunned, promptly redeemed himself by becoming a humane teacher at least to the star pupil. Paying out of his own pocket he acquired the best text book in arithmetic that was available and presented it to Gauss (who breezed through it no time). Moral of the story: Bad teachers are good for spotting incipient Gausses. New York jewish families, I am told, know better. When their kid turns ten, the parents set him or her the problem of summing up an arithmetic progression and wait with bated breath for two whole minutes!

Thank you for your attention

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KUMARI L. A. MEERA MEMORIAL TRUST(Reg. No. 239 of 28.8.1989)

The Kumari L. A. Meera Memorial Trust was established on 28.8.1989 by Mr. L. K. Ananthakrishnan in Palghat, Kerala State in memory of his daughter, Kumari L. A. Meera. The Trust is dedicated to the service of mankind, with a wide range of activities in the fields of Physics, Mathematics and Computer Science besides Anthropology, and welfare and charitable activities. The Trust is managed by a Board of Trustees, the Managing Trustee being Mr. Ananthakrishnan until his demise in March 1998. The present Managing Trustee is Prof. V. Balakrishnan. The main focus of the Trust is to foster scientific interaction and activity in the fields of Physics, Mathematics and Computer Science in various scientific institutions, and encourage scientific education and advanced scientific research in these and interface areas by providing scholarships/stipends/awards for meritorious and deserving students. The Trust is committed to the development of excellence in Physics, Mathematics and Computer Science. It has instituted awards and scholarships in memory of Meera in several schools and colleges and has provided financial grants for upgradation of library, laboratory and computer facilities in a number of institutions in India. In addition it assists in the promotion of Sanskrit and traditional cultural values. The Trust has instituted awards and prizes for the encouragement of excellence in Physics in the Indian Institute of Technology, Madras, St. Stephen’s College, Delhi, the Indian Institute of Science, Bangalore and other educational institutions. The Trust also organises the Kumari L. A. Meera Memorial Meeting on Frontier areas in Physics. The first meeting in the series, on the topic “Geometry and Topology in Physics”, was held at Dhvanyaloka, Mysore from February 8 to 14, 1996. The second meeting, on “Chaos, Complexity and Information” was held from February 1 to 7, 1997; the third on “Optics: Modern Trends” from January 31 to February 5, 1998; the fourth on “Probability and Physics” from January 25 to January 30, 1999; the fifth on “Soft Condensed Matter” from January 27 to February 2, 2000; the sixth on “Physics of Biological Systems” from February 1 to 8, 2001; and the seventh on “Quantum Information and Quantum Computation” from January 30 to February 5, 2002; all at the same venue. The eighth meeting on “Astro-physical, Geophysical and Atmospheric Fluid Dynamics” was held at the Centre for Learning, Bangalore from January 3 to January 7, 2003. The ninth meeting on “Some Aspects of Quantum Mechanics” at the same venue from 2 to 6, 2005; and the tenth meeting on “Topics in Optics, A Classical Selection” was held at the Centre for Learning, Bangalore December 25–31, 2008.

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The Kumari L. A. Meera Memorial Lectures

N. Mukunda – ‘The Task of Picturing Reality’ (1990)

V. Radhakrishnan – ‘Flying Slowly’ (1993)

G. Venkataraman – ‘Parallel Computers’ (1994)

V. S. Ramamurthy – ‘Molecules to Materials –The novel transition domain ofClusters and Nano Structures’ (1995)

J. V. Narlikar – ‘Myths, Beliefs and Facts inAstronomy’ (1996)

D. Balasubramanian – ‘Expanding Visions of theNew Biology’ (1997)

S. Ramanan – ‘The Role of Groups in Artsand Sciences’ (1998)

R. Rajaraman – ‘Can Relativity and QuantumMechanics Co-exist?’ (1999)

S. R. Shetye – ‘The Indian Summer Monsoonand the Waters around India’ (2000)

S. M. Chitre – ‘Windows on the Sun’s Interiorand Exterior’ (2001)

Vijayalakshmi Ravindranath – ‘Life and the Brain’ (2002)

P. K. Kaw – ‘Hymn to Agni the God of Fire’ (2003)

Sunil Mukhi – ‘The Dual World of Science’ (2004)

Sushanta Dattagupta – ‘The Myth about Einstein’ (2005)

J. N. Goswami – ‘Origin of the Solar System: Our Present Understanding’ (2006)

M. S. Ananth – ‘The Changing Environment of Higher Education and Some India –  Centric Concerns’ (2007)

P. Balaram – ‘Chemical Analysis in the Age of Biology’ (2008)

Special Kumari L. A. Meera Memorial Lectures

R. Chidambaram – ‘Nuclear Energy and Safety’ (1996)

D. D. Bhawalkar – ‘Laser Applications in Medicine’ (1998)

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Sri L. K. Ananthakrishnan Memorial Lectures

V. Rajaraman – ‘Video on Demand’ (1999)

G. Srinivasan – ‘The Present Revolution inAstronomy’ (2001)

A. K. Raychaudhuri – ‘Wonderful world of oxides:From art to modern Technologyand frontier science’ (2001)

A. K. Sood – ‘Carbon Nanotubes:

Fun unlimited’ (2002)

N. Panchapakesan – ‘Symmetry Breaking and Our Presence in the Universe’ (2003)

Deepak Mathur – ‘Matter under extreme stress: from molecules to cells’ (2004)

Rohini M. Godbole – ‘The Heart of Matter’ (2005)

Alladi Sitaram – ‘Harish-Chandra: A Mathematician's Mathematician’ (2008)

Vidyanand Nanjundiah – ‘Origin of Species after 150 years’ (2009)

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Eighteenth Kumari L. A. Meera Memorial Lecture

The Queen of Sciences: Her Realm, Her Influence and Her Health

by

M. S. RaghunathanTata Institute of Fundamental Research

Mumbai

December 3, 2009

Kumari L. A. Meera Memorial TrustPalghat, Kerala

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