Professor P P Divakaran is an eminent mathematician who retired from the Tata Institute of Fundamental Research, Mumbai. He is the author of the book, “The Mathematics of India, Concept, Methods, Connections, which was published in 2018.
In an interview, Prof. Divakaran talked about Madhava of Sangamagrama, the Indian mathematician who developed the concept of calculus much before Newton did. Prof. Divakaran also sheds light on the history of the mathematical traditions of India and its outstanding contribution to the world.
Prof. Divakaran was never a mathematician, but a mathematical physicist by profession. Sharing an anecdote on how he developed an interest in the subject, Prof. Divakaran credits distinguished historian Dr. K V Sharma, whom he met in Chennai.
“He was, at that time, working on the translation into English of the ‘Yukti-Bhasa’ which had not been translated before,” he says.
“He asked me, do you want to read this book and I said yes, I would like to. Then he said, why don’t you try this in Malayalam? I did it for over a year. That is how I got interested in this subject,” Prof. Divakaran adds.
He also developed an interest in the subject as Yukti Bhasa was in his native Malayalam language.
Madhava and Aryabhata
Prof. Divakaran has always been fascinated with the works of Madhava of Sangamagrama, the Indian astronomer-mathematician, known for the discovery of power series. In fact, according to Prof. Divakaran, the works of Madhava and Aryabhata are at par with each other and he considers them, the two main pillars of Indian mathematics.
“The two of them, Aryabhata and Madhava have a distinction which other Indian mathematicians don’t have. They created entirely new disciplines of mathematical inquiry,” he says.
The Nīla School in Kerala
Prof. Divakaran stresses that not many people know there existed two Kerala Schools of Mathematics. In the 9th century, there was a school for mathematicians and astronomers in Kodungallur (then known as Mahodayapuram). Two students from that school were prominent- Govinda Swami and his son, Shankaranarayan.
According to Prof. Divakaran, Shankaranarayana wrote a book called ‘Shankaranarayaniyam’. This was the first Kerala School of Mathematics.
Prof. Divakaran likes to call the second Kerala school ‘The Nila School of Mathematics’, partly due to his bias, and partly due to his special knowledge about the place as a Malayali. He says that the Nila school was a special part of Kerala and India.
Prof. says that during the 16th and 17th centuries, all the great works in Kerala’s culture, be it arts, Kathakali, or Ayurveda, all were influenced and flourished from here. But, he says what made the Nila school special was that Madhava was the first from the second school.
Nilakantha, another mathematician, who in Prof’s view was only second to Madhava in a different way, and his disciple, Jaishree Deva also belonged to this school.
“Why is it special? Because of the continuity, (it is) very unusual to find in the ancient time a line of succession of outstandingly good people working in any special area whether it is Ayurveda in India or mathematics and astronomy in India or mathematics or astronomy in other sciences in later Europe,” he adds.
Madhava- the articulator of calculus
Madhava was born around 1350 while Newton, around 1650. Madhava preceded Newton in time, however, talking of calculus, many things that are proved in Yukti Bhasa are claimed to have been first formulated by Madhava.
However, there is one respect in which Newton and Leibniz are ahead of Madhava, not in time, but in technical privilege. Calculus was applied in Kerala by Madhava and his disciples to sort out problems of a specific kind in the study of trigonometry functions. Contrary to this, Newton and Leibniz from the beginning recognized and unknowingly applied the same methods to a much larger class of mathematicians, objects, functions.
“But of course, these are not the reason for neglecting Madhava’s work in calculus, because subsequently if you see the applications and uses of calculus today in mathematics, Newton and Leibniz would have been astonished. They would not have recognized their own ideas what is now called calculus,” says Prof. Divakaran on Madhava’s contribution to calculus.
Historical evidence to prove Madhava’s contribution
Stressing on the fact that Madhava himself wrote very little, Prof. Divakaran shares that Madhava, what now is believed, scattered his wisdom to whoever was willing to listen to him. It was his followers who put them in well-constructed works of technical quality. Yukti Bhasa is the real source of our information about his work, however, the trouble is that it is in Malayalam.
“He only wrote one or two little books and these are manuals, astronomical manuals,” the Prof adds.
Madhava and members of his schools applied the geometrical methods to obtain their findings which is different from the modern methods of calculus
Prof. Divakaran quashes the idea. He says that in India, geometry and other disciplines of mathematics like algebra, were never separated and had special names. “They would use algebraic techniques very freely in their geometry and their geometry to prove algebraic expressions,” he adds.
Reasons for delay in Madhava’s recognition
The fundamental reason, according to Prof. Divakaran, was that the Nila school was largely self-sufficient and had no close contact with other parts of India. The other reason, Madhava never went outside Kerala.
Prof. Madhava attributes another reason to Madhava’s delay in recognition in other parts of the country. He says that during those days most of the books were written in Sanskrit, which was supposedly known by all scholars in India.
Yukti Bhasa, the most sophisticated account of Madhava’s ideas and techniques, is in Malayalam. This had a tremendous impact and he remained unknown for a very long time. The other reason is that Madhava’s work remained neglected by the westerners. “They were colonizers, why would they be interested in us,” he asks.
Hindrance to Kerala School of Mathematics
People were writing books on astronomy using mathematics and subsequently to write down the rules to be followed by astrologers, which were widely popular among the people. On the contrary, Madhava’s work was purely mathematical and had very little application in astronomy. This could not appeal to people.
Prof. also adds that the beginning of scientific activity in modern Europe is a product of the enlightenment that happened about the same time colonization started in India by the Europeans.
International acceptance to Madhava’s work
It is now that the international community has fully accepted and recognized the contributions made by Madhava, not only in calculus but also in other works which are very modern in spirit. Many of his works are now getting recognized. “It took a long time but finally…,” Prof says.
Indian Mathematicians and their western counterpart
Speaking on whether Indian mathematicians receive less acknowledgment compared to their western counterparts, Prof believes it is only natural. He says that the rediscovery of the greek heritage played a deep role in the enlightenment of the renaissance of intellectual and aesthetic activities in Europe. Even after the first texts were published from India, it took the Europeans a very long time to get involved in the works of Indians.
However, there are still some who find it difficult to believe and acknowledge Madhava’s work.
Prof. says the best way to not feel that Indians are any less than their western counterparts, is to do good research about our history.
Need of in-depth studies on Nila School of Mathematics
There are thousands of manuscripts lying around, almost all of them in Sanskrit. A person who does not know the language has to depend on its translation, only if available in one.
Only 10% of the works in Nila School have a translation. A simple translation would be beneficial for all, especially, those not from India too can follow the detailed works.
Furthermore, on rediscovering the lost manuscripts in modern times, Prof. Divakaran says that we have only lost the manuscripts, not the knowledge. We can try to find the continuity of where an idea came from.
Rightful place of ideas that emerged from India
To bring the works of Indian mathematics to the forefront, Prof Divakaran says it would happen when we have a community of scholars who are people of integrity, objective historians of the highest quality and impeachable objectivity in the way they look at doing things.
Moreover, this will make us better historians, help in the understanding of the reality in which we live, is good for the young people, and gives an idea of India’s achievements. However, this would take time, he adds.
Possibilities of new areas of technology and innovation
The application of mathematics through technology and innovation sciences happens in a very natural way. The mathematical concepts of the 18th and 19th centuries are very useful, for instance, weather forecasting is still done using mathematical concepts, which were first discovered probably in the 19th century.
However, mathematical ideas from the very ancient times are rarely used for modern technology, subject to certain exceptions.
Panini’s work in the classification of objects into certain sets is defined in a very modern way. This went on to be used in modern computers. Thus, the application of mathematics through technology and innovation is already happening.