Writing about the significance of the Hindu numeral system, Keith Devlin of Stanford University and a Fellow of the American Mathematical Society, points out that it is much easier to read symbolic expressions and know what a number means, than to read a description in words. He quotes a study conducted by experimental psychologists, where brain lesions were found to have destroyed number and language capacities. “This demonstrated that our brains store numbers along with — and arguably through — symbols that represent them. Our sense of numbers depends on symbols. The modern symbolic notation for numbers is the world’s only truly universal language,” he says. And it was India that gave the world this universal language.
A look at Indian contributions to mathematics shows the richness of the scientific tradition in India. Even the word ‘sine’ has its etymological origin in India. Indian mathematicians used the term caapa or dhanus for the arc, and jyaa or jeevaa, for the chord. When the Arabs picked up these concepts, jeevaa became jiba. Europeans mistook jiba for jaib, an Arab word which means an opening in a garment at the chest. Europeans translated jaib to the Latin “sinus,” which means chest. Sinus became sine!
In 2016, Donald Knuth, in a speech at Stanford University, while talking of the Knight’s tour problem, drew attention to the works of Rudrata, Ratnakara, Bhoja, and to Someswara’s Manasollasa and Vedanta Desika’s Paduka Sahasram. Knuth said that his “major failing as a teacher” was that he was not able to get a single one of his 28 Ph.D. students to realise “what a thrill it was to work on source material.”
And if not for manuscript libraries how can one get source material? Dr. M.D. Srinivas, retired professor of Theoretical Physics, Madras University, is Vice-President of the K.V. Sarma Research Foundation, which has an excellent collection of manuscripts. In an interview, Srinivas threw light on the ancient Indian mathematical tradition. Excerpts:
Can you give an approximate date when a written symbol for zero originated?
The concept of zero is present in the notion of lopa in Panini’s Ashtadhyayi (prior to 5th century BCE). Pingala (prior to 3rd century BCE) mentions the symbol for zero — ‘rupe shunyam.’
What would be the earliest date for the Indian place value system?
You find decimal place value nomenclature for numbers in the Vedas, which speak a number as a place value. Aryabhatiya (499 C.E.) gives all the rules for calculations in a decimal place value system. In 30-odd verses, Aryabhatta covers most of high school maths taught today and more. Epigraphs show the use of decimal place value notation from the 6th century onwards in India and South East Asia.
One criticism against Indian mathematics is that there are no proofs. Did we use reductio ad absurdum ?
Much of modern scholarship on Indian mathematics dealt with only the original works, but not the commentaries, and this led to the misconception that we had no proofs. Bhaskaracharya I, Bhaskaracharya II, Nilakantha Somasutvan, Ganesa Daivajna, Muniswara and Kamalakara wrote commentaries for their own works, and in the commentaries, we find upapattis (demonstrations) of the results discussed in the original text. Govindaswamin (800C.E.) gives upapattis in his work. Bhaskaracharya II in his Siddhantasiromani , says without upapattis or proofs, a mathematician will not be respected in an assembly of scholars. Nor will he be free from doubts. Ganesha Daivajna says upapattis are needed for buddhi vriddhi - elevation of the intellect.
Indian logicians did not consider reductio ad absurdum , known as tarka , to be an independent means of valid knowledge. They occasionally used it to prove the non-existence of something. Krishna Daivajna used it to show that a negative number has no square root among the known numbers. But Indians never used this method for proving the existence of an entity, whose existence could not be established by any other direct method. They followed a constructivist approach to the existence of mathematical objects.
How many source texts in maths and astronomy have been translated/ analysed? How many source texts remain to be translated from manuscripts?
American scholar David Pingree, who published several volumes of Census of Exact Sciences in Sanskrit, estimated jyotish sastra manuscripts alone to be around 100, 000. Of these at least 30,000 pertain to mathematics/astronomy, and these in turn are associated with roughly 9,000 source works on astronomy and mathematics in Sanskrit. During the last two and a half centuries, 450 source works on mathematics/astronomy have been edited. Of these only 95 source works have been translated and analysed for their technical content.
At least 190 well-known source works are yet to be edited and published. Looking at the bibliographies of David Pingree and K.V. Sarma, we know that manuscripts for these are available. But this is merely indicative of the huge corpus of source works on mathematics/astronomy which are available, but are yet to be studied.
Tribute to scholarship
Krishna Venkateswara Sarma was born in Chenagnnur, Kerala in December 1919. Graduating in Physics and Chemistry, he did his Masters in Sanskrit. He worked for some years in the Manuscripts section of the Kerala University Oriental Research Institute. Later he worked in Madras University’s Sanskrit department, Visveshvarananand Institute of Indological studies, Hoshiarpur, and Adyar library. In 1997, he established the Sree Sarada Education Society and Research centre. He published a survey of Sanskrit manuscripts on sciences found in repositories in Kerala and Tamil Nadu, where he listed over 450 Sanskrit works on Mathematics and Astronomy from Kerala. Sarma’s grandson — Dr. S.A.S. Sarma is continuing the work on surveying manuscript repositories.
Sarma copied out manuscripts in the possession of various families in Kerala. His monumental efforts resulted in a collection of 900 manuscripts, of which about 400 are on astronomy. “Some of his early publications were Grahacaranibandhana of Haridatta, Siddhantadarpana of Nilakantha, Venvaaroha of Madhava, Goladipika and Grahanashtaka of Parameswara. While at Hoshiarpur, he published more than 50 books, mostly on Kerala astronomy, like Parameswara’s Digganita , Golasara of Nilakantha, Tantrasangraha of Nilakantha with the commentaries Yuktidipika and Laghuvivritti of Sankara,” says Dr. M.S. Sriram, retired Professor of Physics, Madras University, and President of K.V. Sarma Research Foundation. Sarma published more than 20 important source works of Kerala mathematicians and astronomers. He brought out 104 books and 450 research papers.
K.V. Sarma Research Foundation was established in 2010, in memory of Sarma’s scholarship, and it houses his manuscripts collection. Prof. Siniruddha Dash, former Head of the department of Sanskrit, Madras University, is the director of the Foundation. Dash has edited two volumes of articles by K.V. Sarma, who had translated Jyesthadeva’s mathematical and astronomical treatise in Malayalam — Ganita Yuktibahsa (1530 CE). Sarma wanted the translation to be supplemented by explanations, diagrams and notations. So, he roped in Dr. M.S. Sriram. Dr. K. Ramasubramanian of IIT Bombay and Dr. M.D. Srinivas. The book, published in 2008, has a foreword by mathematician Dr. C. S. Seshadri and was released at the Chennai Mathematical Institute, and Fields medallist Dr. Mumford was present during the release.
Dr. Mamata, secretary and managing trustee of the Foundation, is working on Laghumanasa , a commentary by Suryadevayajvan on Munjala’s Laghumanasa . Dr. Rama Kalyani is doing an English translation of Buddhivilasini , a commentary on Lilavati . Dash is guiding Prabha Rajagopal, with a Survey of Lexicons on different subjects available in Pali, Prakrit and Sanskrit. All three projects are funded by the TATA Trust. Awaiting publication are some translations and explanatory books — Lagnaprakarana of Madhava, Panchabodha with Sankara Variar’s commentary, Drkkarana,Selected Works of Parameswara , and Selected Works of Puthumana Somayaji . Aditya Kolachana, K.Mahesh, Venkateswara Pai, Veena Bhat and Dinesh Mohan Joshi, have collaborated with Sriram and Ramasubramanian on these texts.
Stressing the importance of preservation of manuscripts, Dash says, “The more manuscripts of a work a scholar refers to, the more authentic is his research. Bhandarkar Oriental Research Institute in Pune collected 4,500 manuscripts of the Mahabharata from all over India. Of these, they used 734 to come up with a critical edition. The TV serial ‘Mahabharata’ was based on this edition, and may therefore said to be an authentic version of the Itihasa.”