Learning from one another

Indian research was deeply influenced by the knowledge of foreign works on the subject, and in turn, Indian maths influenced mathematical work in other countries

The skill of doing research, the hard preparation needed for doing new and original work — going beyond the old established knowledge, and indeed the courage to think in novel and daring lines — are all immensely helped by good and exciting teaching. For me, this began at home. My grandfather Kshiti Mohan Sen, who taught at Santiniketan, could excite my interest in Sanskrit studies, including heretical texts in Sanskrit, which still inspire my engagement in that wonderful language, as I pick up a book in Sanskrit today. Sanskrit, we have to remember, is not only the language in which the Hindu and many of the Buddhist texts came; it is also the vehicle, among many other radical thoughts, of comprehensive doubts about the supernatural expressed in the Lokayata texts, and also the medium in which the questioning of class and caste and legitimacy of power would be expressed with spectacular eloquence by Shudraka in his profound play, “Mricchakatikam” (“The Little Clay Cart”). It was great for me to be taught at a very early age the distinction between a great language as a general vehicle of thought and the specific ideas — religious or sceptical — that may be expressed in that language. That distinction remains important today.

I also have to acknowledge my debt to my other teachers — in Santiniketan, at Presidency College, and at Trinity College in Cambridge — in helping me to find my way. I am delighted that the Infosys Foundation has initiated a new scheme for the training of rural teachers of mathematics and science. Since our school education is the basis of all our education — no matter how “high” our higher education maybe — the fruits of investment in good school education can be extraordinarily high. Narayana Murthy, who like me grew up in a family of teachers, knows that with visionary insight.

Wider role of teaching

I also want to say a few things about the wider role of teaching — in linking different nations and different cultures together. Teaching is not just a matter of instruction given by teachers to their individual students. The progress of science and of knowledge depends in general on the learning that one nation, one group of people, derives from what has been achieved by other nations and other groups of people.

For example, the golden age of Indian mathematics, which changed the face of mathematics in the world, was roughly from the fifth to the 12th century, and its beginning was directly inspired by what we Indians were learning from work done in Babylon, Greece and Rome. To be sure, there was an Indian tradition of analytical thinking going back much further, on which the stellar outbursts of mathematical work in India from around the fifth century drew, but we learned a lot about theorems and proofs and rigorous mathematical reasoning from the Greeks and the Romans and the Babylonians. There is no shame in learning from others, and then putting what we have learned to good use, and going on to create new knowledge, new understanding, and thrillingly novel ideas and results.

Indians of course were teaching other Indians. Perhaps the most powerful mathematician of ancient India, Brahmagupta, would not have been able to do such dazzling work without his having been influenced by the ideas of his own teachers, in particular Aryabhata, the pioneering leader of the Indian school of mathematics. Alberuni, the Iranian mathematician, who spent many years in India from the end of the 10th to the early years of the 11th century (and helped to make Arab mathematicians learn even more from Indian mathematics than they were already doing) thought that Brahmagupta was perhaps the finest mathematician and astronomer in India, and possibly in the world, and yet (argued Alberuni), Brahmagupta could be so productive only by standing on the shoulders of the great Aryabhata, who was not only an extraordinary scientist and mathematician, but also a superb teacher. Learning from each other continued over centuries, involving — in addition to Aryabhata and Brahmagupta — Varahamihira and Bhaskara, among many others.

And just as Indian mathematicians learned something from Babylonians, Greeks and Romans, they also taught some brilliantly new ideas to mathematicians elsewhere in the world. For example, Yi Xing [I-Hsing], who lived in China between the seventh and the eighth century, and who was, as Joseph Needham describes him, probably the finest Chinese mathematician of his time, knew all the relevant Indian texts. The Chinese mathematicians as well as the pioneering Arab mathematicians, including Al Khwarazmi (from whose name the term “algorithm” is derived), all knew Sanskrit and the Sanskritic literature in maths. What we are admiring here is not Indian mathematics done in splendid isolation (that rarely occurs anywhere in the world), but mathematics done with a huge role of international and interregional exchange of ideas. Indian research was deeply influenced by the knowledge of foreign works on the subject, and in turn, Indian maths influenced mathematical work even in those countries, including Greece and Rome and Baghdad, from where Indians themselves had learned many things.

Interconnections of traditions

Let me end with an example. The history of the term “sine” in Trigonometry illustrates how we learn from each other. That trigonometric idea was well developed by Aryabhata, who called it jya-ardha, and sometimes shortened it to jya. The Arab mathematicians, using Aryabhata’s idea, called it “jiba,” which is phonetically close. But jiba is a meaningless sound in Arabic, but jaib, which has the same consonants, is a good Arabic word, and since the Arabic script does not specify vowels, the later generation of Arab mathematicians used the term jaib, which means a bay or a cove. Then in 1150, when the Italian mathematician, Gherardo of Cremona, translated the word into Latin, he used the Latin word “sinus,” which means a bay or a cove in Latin. And it is from this — the Latin sinus — that the modern trigonometric terms “sine” is derived. In this one word we see the interconnection of three mathematical traditions — Indian, Arabic and European.

Teaching and learning are activities that link people together. Even as we celebrate science and research, we have to recognise the role of teaching and that of learning from each other — from our teachers, from our colleagues, from our students, from our friends, and from our fellow human beings. There is something extraordinarily great in these interconnections.