## Monday, January 31, 2011

### The use and misuse of Srinivasa Ramanujan

**by Hartosh Singh Bal**

Over the past month there have been two separate reasons to return to the story of Srinivasa Ramanujan. The first was the result of an astounding piece of mathematics by Ken Ono and his colleagues on the theory of partitions, bringing to a conclusion some of Ramanujan’s most interesting work in number theory. The second was thanks to Patrick French’s recent book – India, a portrait – which ends with a short two page biography of Ramanujan. The first Ramanujan is of course the Ramanujan who should matter, the mathematician, the second is unfortunately the Ramanujan who has come to occupy public memory, the metaphor.

It is not clear what French’s Ramanujan stands for in a chapter that seeks to explain the specifics of individual, social and organizational behavior on the basis of particular Indian traits such as religion or caste, but given the title of the chapter – Only in India – it does seem that French believes there was something particularly Indian about Ramanujan’s story.

This belief is not unique to French and has only been compounded by Ramanujan’s own description of the Goddess of Namagiri as the source of his inspiration. The result is that Ramanujan has come to embody certain romantic notion of eastern or more specifically Indian thought. Even those who want to allude to Ramanujan the mathematician do so in such terms. Paul Hoffman, in an otherwise entertaining book on the Hungarian mathematician Paul Erdos – The Man who Loved Only Numbers – writes, "While Hardy and Ramanujan’s partnership lasted, the two men stood the world of pure mathematics on its head. It was East meets West, mysticism meets formality, and the combination was unstoppable."

Ramanujan’s otherwise excellent biographer Robert Kanigel devotes the entire first chapter of the book – The Man who Knew Infinity – to Ramanujan’s religious and social upbringing. However important this may have been to Ramanujan the man, the claim that it is central to Ramanujan the mathematician does not stand up to scrutiny. Ramanujan did not learn his mathematics in a temple. By the time he went to school only a few of the traditional Vedic schools still functioned. They had been largely replaced by schools teaching a curriculum based on European science.

In his 1894 book, The History of Education in the Madras Presidency (the region in the South of India where Ramanujan grew up was a separate administrative unit of British India), S. Satthianadhan quotes an 1822 description by a Collector of Bellary, A.D. Campbell, of the mathematical education that used to be handed out at in these traditional schools in the South of India, `` He (a student) then commits to memory an addition table and counts from one to one hundred ; he afterwards writes easy sums in addition and subtraction of money, multiplication and the reduction of money, measure, etc. Here great pains are taken with the scholar in teaching him the fractions of an integer which descend, not by tens as in our decimal fractions, but by fours, and are carried to a great extent. In order that these fractions together with the arithmetical tables in addition, multiplication, and the three fold measures of capacity, weight, may be rendered quite familiar to the minds of the scholars, they are made to stand up twice a day in rows, and repeat the whole after one of the monitors.’’

Taught in this manner if Ramanujan had survived to become a mathematician, he would have had to rediscover all of mathematics. Thankfully he was saved this fate, for one that was only slightly better. After the British Governor General in India, William Bentinck, decided against the traditional Indian school system in 1835, a new school system was instituted in the Madras Presidency in 1854. Sattianadhan’s book describes the ensuing curriculum for the first four standards, the same curriculum that is likely to have been followed in Ramanujan’s times:

I —Notation to thousands, easy addition, and the multiplication table to five times five. English is to be used in all cases.

II —Subtraction, multiplication, and division. The multiplication table to twelve times twelve.

III—Compound rules and reduction, with the ordinary weight, measure and money tables.

IV—Moderately easy practical questions in vulgar fractions and simple proportion.

Clearly what Ramanujan learnt in school did not differ much from what is taught today. The difference lay in the fact that Ramanujan’s obvious mathematical abilities did not come in for the notice they would have attracted at any other time or place. In his biography of Ramanujan Robert Kanigel describes that by the time he was eleven ``his classmates were coming to him for help’’, a year later he was ``challenging his teachers’’ and by the time he was thirteen her had mastered S.L. Loney’s Trignometry, an English text that some select Indian students encounter even today, but only at the age of sixteen or so.

The reason for neglecting such ability was simple. The system schooling Ramanujan was not designed to detect or produce men of outstanding talent. Satthianadhan writes, ``The Despatch of 1854 marks an era in the history of education in the Madras Presidency. It has been of 1854 called the Magna Charta of English education in India. "We have always looked upon the encouragement of education," say the Court of Directors, " as peculiarly important, because calculated not only to promote a higher degree of intellectual fitness, but to raise the moral character of those who partake of its advantages, and so to supply you (Government of India) with servants to whose probity you may with increased confidence commit offices of trust.’’

In Europe at any point after the Renaissance a student of Ramnujan’s genius would have found a mentor. In British India he was allowed to proceed in much the ordinary fashion. His talent actually became a hindrance. Even when he went to Government College, Kumbakonam in 1904 at the age of 17, he had to drop out a year later after failing English Composition. He resumed his degree at Pachaiyappa’s College but failed the Physiology examination and was forced to opt out of college for good in 1907. In 1910 he found a job as a clerk before G.H. Hardy rescued him from oblivion in 1913.

Contrast this with the career of the two great mathematicians he is often compared to, Carl Gustav Jacob Jacobi and Leonhard Euler, both marked by a similar ability to reveal formulas of great depth and beauty. Euler was born in Switzerland in 1706. As a student in the University of Basle he drew the attention of another great mathematician Johannes Bernoulli who persuaded Euler’s father to let him study mathematics. The Bernoulli connection later helped Euler obtain a position in St Petersburg when he was barely 20 and at the age of 26 he took up the leading mathematical position in the Academy. Jacobi born in 1804 was the son of a banker, in 1821 he headed to the University of Berlin. After completing his PhD in 1825 he became a lecturer at the age of 21.* * Euler and Jacobi were no exceptions, if you compare Ramanujan’s background with that of any other great mathematician of the last 500 years, it is clear that none faced similar intellectual and economic handicaps.

Bereft of the knowledge of what was happening in the world of mathematics and cut off from the company of the kind of mathematicians who would have realized his talent Ramanujan was in effect the equivalent of a brilliant (and that is a complete understatement) high school student in his grasp of mathematical rigor. This was compounded by the fact that he was subsequently shaped by E.H. Carr’s *Synopsis of Elementary Results in Pure and Applied Mathematics*. It was a book probably handed down to him by college students lodging in his house. A book that is typical of a system meant to train student for examinations, with no proofs for the results it cited. In Europe a boy of his obvious ability would have been asked at the very least to read Gauss’ *Disquisitiones Arithmeticae,** *and would have had a chance to directly engage with the work of several of the great mathematicians of the 19^{th} century.

It was only the genius of Ramanujan that could transmute the handicaps of colonialism into a triumph. Perhaps an equivalent story is one from the Mahabharata, where a tribal boy Eklavya, brought up in an isolated forest far from the capital where the art of archery was taught by the great teacher Drona, set up a bust of Drona and practiced his art with such talent and avidity that he soon outshone the best of Drona’s pupils.

Egged on by his envious students Drona asked for Eklavya’s thumb as his `fee’ for the instruction in archery. The parallel may not be precise but even so it is not difficult to think of the lack of rigor in Ramanujan’s work as a price extracted for allowing him a glimpse into the world of modern mathematics. It can only be a surmise that born a hundred years later in India Ramanujan may well have been the greatest mathematician of the modern era. But the claim rests not on his being a Tamil Brahmin or an Indian but on his being Ramanujan.

Hardy himself had once noted, `` He would probably have been a greater mathematician if he could have been caught and tamed a little in his youth. On the other hand he would have been less of a Ramanujan, and more of a European professor, and the loss might have been greater than the gain….’’

The qualifier is in keeping with the romanticism that surrounds Ramanujan. It fits in far too comfortably with notions of the mystic East and the rational West, a comparison that has always worked to the advantage of one side. Ramanujan himself would have not chosen the course of life that was inflicted on him, as his attempts to find recognition show. It is no wonder that more than a decade later Hardy was to term his own observation `ridiculous sentimentalism’.

Left to fend for himself at sixteen by Carr, Ramanujan turned his compulsion into a virtue, arriving at mathematical truth through a process of heuristic reasoning all his own. Imagine what he could have done if Gauss, Euler and Jacobi had been his guides.

Posted by Hartosh Singh Bal at 01:00 AM | Permalink