An extraordinary mind – Part 11

Encouraged by his well-wishers, Ramanujan, then 25 years old and had no formal education, wrote a letter to Hardy on January 16, 1913. The letter ran into eleven pages and it was filled with theorems in divergent series. Ramanujan did not send proofs for his theorems. He requested Hardy for his advice and to help getting his results published. Ramanujan wrote : “I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras on a salary of only £ 20 per annum. I have had no university education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at mathematics. I have not trodden through the conventional regular course which is followed in a university course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as “startling“… I would request you to go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published. I have not given the actual investigations nor the expressions that I get but I have indicated the lines on which I proceed. Being inexperienced I would very highly value any advice you give me “. The letter has become an important historical document. In fact, ‘this letter is one of the most important and exciting mathematical letters ever written’. At the first glance Hardy was not impressed with the contents of the letter. So Hardy left it aside and got himself engaged in his daily routine work. But then he could not forget about it. In the evening Hardy again started examining the theorems sent by Ramanujan. He also requested his colleague and a distinguished mathematician, John Edensor Littlewood (1885-1977) to come and examine the theorems. After examining closely they realized the importance of Ramanujan’s work. As C.P. Snow recounted, ‘before mid-night they knew and knew for certain’ that the writer of the manuscripts was a man of genius’. Everyone in Cambridge concerned with mathematics came to know about the letter. Many of them thought `at least another Jacobi in making had been found out’. Bertrand Arthur William Russell (1872-1970) wrote to Lady Ottoline Morell. “I found Hardy and Littlewood in a state of wild excitement because they believe, they have discovered a second Newton, a Hindu Clerk in Madras … He wrote to Hardy telling of some results he has got, which Hardy thinks quite wonderful.”

Fortunately for Ramanujan, Hardy realised that the letter was the work of a genius. In the next three months Ramanujan received another three letters from Hardy. However, in the beginning Hardy responded cautiously. He wrote on 8 February 1913. To quote from the letter. “I was exceedingly interested by your letter and by the theorems which you state. You will however understand that, before I can judge properly of the value of what you have done it is essential that I should see proofs of some of your assertions … I hope very much that you will send me as quickly as possible at any rate a few of your proofs, and follow this more at your leisure by more detailed account of your work on primer and divergent series. It seems to me quite likely that you have done a good deal of work worth publication; and if you can produce satisfactory demonstration I should be very glad to do what I can to secure it” .

In the meantime Hardy started taking steps for bringing Ramanujan to England. He contacted the Indian Office in London to this effect. Ramanujan was awarded the first research scholarship by the Madras University. This was possible by the recommendation of Gilbert Walker, then Head of the Indian Meteorological Department in Simla. Gilbert was not a pure mathematician but he was a former Fellow and mathematical lecturer at Trinity College, Cambridge. Walker, who was prevailed upon by Francis Spring to look through Ramanujan’s notebooks wrote to the Registrar of the Madras University: “The character of the work that I saw impressed me as comparable in originality with that of a Mathematical Fellow in a Cambridge College; it appears to lack, however, as might be expected in the circumstances, the completeness and precision necessary before the universal validity of the results could be accepted. I have not specialised in the branches of pure mathematics at which he worked, and could not therefore form a reliable estimate of his abilities, which might be of an order to bring him a European reputation. But it was perfectly clear to me that the University would be justified in enabling S. Ramanujan for a few years at least to spend the whole of his time on mathematics without any anxiety as to his livelihood.”

Ramanujan was not very eager to travel abroad. In fact he was quite apprehensive. However, many of his well-wishers prevailed upon him and finally Ramanujan left Madras by S.S. Navesa on March 17, 1914. Ramanujan reached Cambridge on April 18, 1914. When Ramanujan reached England he was fully abreast of the recent developments in his field. This was described by J. R. Newman in 1968: “Ramanujan arrived in England abreast and often ahead of contemporary mathematical knowledge. Thus, in a lone mighty sweep, he had succeeded in recreating in his field, through his own unaided powers, a rich half century of European mathematics. One may doubt whether so prodigious a feat had ever been accomplished in the history of thought.”

Today it is simply futile to speculate about what would have happened if Ramanujan had not come in contact with Hardy. It could happen either way. But then Hardy should be given due credit for recognizing Ramanujan’s originality and helping him to carry out his work. Hardy himself was very clear about his role. “Ramanujan was”, Hardy wrote, “my discovery. I did not invent him – like other great men, he invented himself – but I was the first really competent person who had the chance to see some of his work, and I can still remember with satisfaction that I could recognize at once what I treasure I had found.”

It may be noted that before writing to Hardy, Ramanujan had written to two well-known Cambridge mathematicians viz., H.F. Baker and E.W. Hobson. But both of them had expressed their inability to help Ramanujan.

Ramanujan was awarded the B.A. degree in March 1916 for his work on ‘Highly composite Numbers’ which was published as a paper in the Journal of the London Mathematical Society. He was the second Indian to become a Fellow of the Royal Society in 1918 and he became one of the youngest Fellows in the entire history of the Royal Society. He was elected “for his investigation in Elliptic Functions and the Theory of Numbers.” On 13 October 1918 he was the first Indian to be elected a Fellow of Trinity College, Cambridge.

Much of Ramanujan’s mathematics comes under the heading of number theory — a purest realm of mathematics. The number theory is the abstract study of the structure of number systems and properties of positive integers. It includes various theorems about prime numbers (a prime number is an integer greater than one that has not integral factor). Number theory includes analytic number theory, originated byLeonhard Euler (1707-89); geometric theory – which uses such geometrical methods of analysis as Cartesian co-ordinates, vectors and matrices; and probabilistic number theory based on probability theory. What Ramanujan did will be fully understood by a very few. In this connection it is worthwhile to note what Hardy had to say of the work of pure mathematicians: “What we do may be small, but it has certain character of permanence and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something beyond the powers of the vast majority of men.” In spite of abstract nature of his work Ramanujan is widely known.

Ramanujan was a mathematical genius in his own right on the basis of his work alone. He worked hard like any other great mathematician. He had no special, unexplained power. As Hardy, wrote: “I have often been asked whether Ramanujan had any special secret; whether his methods differed in kind from those of other mathematicians; whether there was anything really abnormal in his mode of thought. I cannot answer these questions with any confidence or conviction; but I do not believe it. My belief that all mathematicians think, at bottom, in the same kind of way, and that Ramanujan was no exception.”

Of course, as Hardy observed Ramanujan “combined a power of generalization, a feeling for form and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his peculiar field, without a rival in his day.

Here we do not attempt to describe what Ramanujan achieved. But let us note what Hardy had to say about the importance of Ramanujan’s work. “Opinions may differ as to the importance of Ramanujan’s work, the kind of standard by which it should be judged and the influence which it is likely to have on the mathematics of the future. It has not the simplicity and the inevitableness of the greatest work; it would be greater if it were less strange. One gift it shows which no one will deny—profound and invincible originality.”

The Norwegian mathematician Atle Selberg, one of the great number theorists of this century wrote: “Ramanujan’s recognition of the multiplicative properties of the coefficients of modular forms that we now refer to as cusp forms and his conjectures formulated in this connection and their later generalization, have come to play a more central role in the mathematics of today, serving as a kind of focus for the attention of quite a large group of the best mathematicians of our time. Other discoveries like the mock-theta functions are only in the very early stages of being understood and no one can yet assess their real importance. So the final verdict is certainly not in, and it may not be in for a long time, but the estimates of Ramanujan’s nature in mathematics certainly have been growing over the years. There is doubt no about that.”

Often people tend to speculate what Ramanujan would have achieved if he had not died or if his exceptional qualities were recognised at the very beginning. There are many instances of such untimely death of gifted persons, or rejection of gifted persons by the society or the rigid educational system.

Even after more than 80 years of the death of Ramanujan the situation is not very different as far the rigidity of the education system. Today also a ‘Ramanujan’ is not likely to get a chance to pursue his career. This situation remains very much similar as described by JBS Haldane (1982-1964), a British born geneticist and philosopher who spent last part of his life in India. Haldane said: “Today in India Ramanujan could not get even a lectureship in a rural college because he had no degree. Much less could he get a post through the Union Public Service Commission. This fact is a disgrace to India. I am aware that he was offered a chair in India after becoming a Fellow of the Royal Society. But it is scandalous that India’s great men should have to wait for foreign recognition. If Ramanujan’s work had been recognized in India as early it was in England, he might never have emigrated and might be alive today”.

Ramanujan died of tuberculosis in Kumbakonam on April 26, 1920. He was only 32 years old. “It was always maths… Four days before he died he was scribbling,”  said Janaki, his wife. The untimely death of Ramanujan was most unfortunate particularly so when we take into account the circumstances under which he died.

The story of Ramanujan should be told and retold to our younger people particularly to those who aspire to do something extraordinary but feel dejected under the prevailing circumstances. And in this connection it is worthwhile to remember what Chandrasekhar had to say: “I can recall the gladness I felt at the assurance that one brought up under circumstances similar to my own could have achieved what I could not grasp. The fact that Ramanujan’s early years were spent in a scientifically sterile atmosphere, that his life in India was not without hardships that under circumstances that appeared to most Indians as nothing short of miraculous, he had gone to Cambridge, supported by eminent mathematicians, and had returned to India with very assurance that he would be considered, in time as one of the most original mathematicians of the century – these facts were enough, more than enough, for aspiring young Kashmiri students to break their bands of intellectual confinement and perhaps soar the way what Ramanujan had”.

The author is teaching at Department of Mathematics, University of Kashmir