The Kerala School, as Joseph calls them, marks a continuity of scholarship in mathematics in India, which was believed to have declined after Bhaskara II in the 12th century. Joseph, who holds honorary appointments at University of Manchester, UK and University of Toronto, Canada, argues that the knowledge of the Kerala School travelled from India to Europe via Jesuit scholars and influenced European mathematics. In an interview with Amrith Lal, Joseph talks about the pluralistic origins of mathematics and how this science was practised in India, especially in Kerala. Excerpts:

Q.The Crest of the Peacock looked at knowledge traditions across the world. In A Passage to Infinity, you have focused on what you call the Kerala School of mathematics. What prompted you to focus on this school?

There is a widespread view that while the non-European roots were quite important in the early history of mathematics, these were primarily ‘elementary’ and had contributed little to the development of modern mathematics. The work of the Kerala School should make one rethink this perception.

An important strand in the genesis and conceptualisation of calculus is the development and application of infinite series techniques, originating from the medieval Indian mathematics through the Kerala conduit. In particular, the expression of certain trigonometric functions as infinite series of the relevant variables is found first in the Kerala work.

Q.You have argued the achievements of the Kerala school disputes the conventional wisdom that mathematics in India stopped growing after Bhaskara II in the 12th century AD. What explains the flourishing of mathematics in Kerala three centuries after Bhaskara?

Despite CM Whish’s notification in 1834 and the work of Rajagopal and his collaborators highlighting the contributions of Kerala mathematics, little of this information percolated into the standard histories of mathematics until recently. The view that still prevailed was that Indian mathematics after Bhaskara II had made ‘only spotty progress until modern times’. The reasons for the emergence of Kerala mathematics three centuries after Bhaskara are difficult to untangle since there are both mathematical and social influences to be considered. At the root of Kerala’s achievements was the mathematical motivation provided by Aryabhata.

In Aryabhatiya, there is his implicit estimate of p as 3.1416, which he describes as ‘approximate’. There is also a throwaway remark in the Aryabhatiya that by dividing the circle into finer segments, the computed value of its circumference could be made increasingly accurate. These hints were taken up by Madhava, the founder of the Kerala School, applying them successfully in his search for an accurate value of p and the sine (cosine).

The other strand, the social context of the Kerala School’s achievements, is found in the history of Kerala between the 7th and 14th centuries AD. Around that period, groups of Brahmins began to migrate from the North (mainly from Maharashtra and the Konkan coast of Goa). This continued for the next three or four centuries, bringing with them not only their rituals but their Sanskritic learning.

The impact of this wholesale importation of knowledge and skills cannot be over-estimated. Not only astronomy and mathematics, but architecture, literature and even the very language, Malayalam, were affected beyond recognition. From this group emerged the Nambuthiri Brahmins, wealthy and highly influential in the courts of the rulers in Kerala. Almost all the members of the Kerala School (with two notable exceptions) were from this group.

It is quite likely that these members of the Kerala School constituted a leisured class as the younger sons, who because of the peculiar primogeniture system prevalent, had little or no family responsibilities and could concentrate on scholarship and study, including mathematics and astronomy, if they were so inclined.

Q. How was this knowledge produced?

It was the practice among the students and teachers of the Kerala School to live and work in illams (Nambuthiri homes). Membership was limited to a leisured class of younger sons of land-owning families whose economic needs were provided for by their families and who custom dictated could not marry. In the illams, which specialised in mathematical astronomy, a small, self-selected and highly motivated group of students were set to memorise verses summarising results from the past, which then became the major texts of the Kerala School. It is from the works of these scholars that we can piece together the Kerala episode of Indian mathematics.

And for the most part, the writings of these scholars rarely circulated much beyond their locality. Contrast this with the wider campus within which the proliferation of mathematical discoveries occurred in Europe during the 17th and 18th century and perhaps we have some clue as to the different course of mathematical development in the two cultures.

However, an overemphasis on the role of the Nambuthiri Brahmins should not be at the expense of the contribution of the other communities in Kerala. It should be noted that there were significant groups of Buddhist, Jain populations, located in the areas where the Kerala School thrived, who had access to Sanskritic knowledge.

It is even argued that certain Ezhavas (other backward castes) were Buddhists who maintained the tradition of Ayurveda and Sanskrit learning. So, a closer study of the non-Brahmin roots of mathematics may well be fruitful.

Q. What are the major achievements of the Kerala School? How is it important to the history of Indian mathematics?

Apart from the work on the infinite series I’ve mentioned, familiar techniques of calculus — such as integration by parts, multiple integration, derivation of the power series for pi, sine and cosine — make their first appearance in the Kerala work. And, so do the remarkable approximations of trigonometric functions including the well-known Taylor series approximations for the sine and cosine functions.

These results had apparently been obtained without the use of infinitesimal calculus and only using the geometry of the circle as well as a sophisticated arithmetic of fractions and negative numbers within a well-developed positional decimal number system. The Kerala episode has helped to restore the unbroken continuity in the history of Indian math from very early times to the time before the emergence of European presence on the Indian subcontinent.

Q. You have indicated that some of the findings of Indian mathematicians, including those of the Kerala School, predate similar findings in Europe. Was there a transmission of knowledge from India to Europe?

It is important to recognise that while Kerala was very much on the periphery of the politics and culture of the Indian subcontinent, its trade linked it with the wider world. Contacts have existed since time immemorial with the African coast and West Asia and for a time with the Roman Empire. And with the coming of Vasco da Gama, windows were thrown open beyond the Iberian Peninsula. Transmission of knowledge from India to Europe through the Arab conduit is well known and included Indian numerals and computations, ideas of Indian trigonometry and algebra and Indian astronomy.

However, while one can show circumstantial evidence of an European search for better navigation and calendrical techniques in Kerala and elsewhere, and even identify possible conduits — such as the Jesuits, Jewish and Arab traders — through whom mathematical ideas from Kerala could have been transmitted, the fact remains that after a three-year research project undertaken by myself and other associates searching various European archives, we came to the conclusion that there is no direct documentary evidence of the conjectured transmission through the Jesuits.

However, one should remember that some of the archival material relevant to this project no longer exists, having been destroyed either by earthquake (Lisbon 1755), or burnt (by the Dutch of the Jesuit library in Cochin), or even disappeared into private libraries in Italy and other parts of Europe.

Q. How much do we know about the possibility of transfer of knowledge through craftsmen belonging to different countries, say ship builders on the Malabar Coast to ship builders of European countries?

This is an area in which there is need for more empirical work. As far as I know, apart from general works such as DF Lach’s Asia in the Making of Europe little has been done.

Q. You mention Jyeshtadeva and his work Yuktibhasha, a mathematical treatise written in the 16th century in Malayalam. Did we have a tradition of preparing scientific treatises in Indian languages other than Sanskrit?

Transmissions from Sanskrit to regional languages and vice versa remain a largely unexplored area of research. However, SR Sarma has made a start in his paper, Mathematical Literature in the Regional languages of India. What is interesting about Yuktibhasa, apart from the fact that it is in Malayalam, is that unlike many other texts in Indian mathematics, which seems in most cases to be a compilation of results with little indication of how the results were derived, Jyeshtadeva placed great importance on yukti (or reasoned justifications).

Yuktibhasa contains proofs of virtually every result given, and in particular the infinite series derivations mentioned earlier. It remains the one key text that should be read by anyone who wants to comprehend the scale of the Kerala work. Professor KV Sarma has now translated the book into English with the title Ganitayuktibhasa, with insightful commentaries by three scholars.

Q. In The Crest..., while discussing Srinivas Ramanujan, you raised three themes for discussion. One, how far did cultural influences determine Ramanujan’s choice of subjects or his methods; two, is it possible to identify any features in his own culture which were conducive to creative work in Mathematics and; three, is there a need to conform to a particular method of presentation before something is recognisable as Mathematics? Two decades have passed after you raised these questions. In the light of new research, how do you approach these questions?

Q. So, how did Ramanujan produce such remarkable mathematics with his limited formal education?

Mathematics and numbers have always had a special significance within the Brahmanical tradition as extra-rational instruments for controlling fate and nature. And there’s an enduring aspect of Indian mathematics over a long time: a fascination with numbers, a positive delight in calculations, and interest in recreational mathematics. Also, the numerate character of the traditional pursuit of astrology, carpentry, architecture should be recognised.

Ramanujan’s mother, the dominant figure in the family and who was to have a decisive influence on Ramnujan throughout his life, was well-known locally for her interest in astrology and numerology. Traditional astrologers in South India are known for their agility in mental arithmetic. It is possible that Ramanujan’s extraordinary intuition for, and ability to manipulate numbers, owed a lot to his mother. There is now greater latitude both in schools and higher institutions in the way that mathematics is presented and in the approach to solutions of problems.

The Kerala approach has been tried out in introducing calculus to college students. It has been found to have certain pedagogical strengths absent in the conventional approach to calculus.