This conference organised by the Asiatic Society and Council '"for Philosophical Research was planned to be a Foundational and Methodological issues in Indian Astronomy and Mathematics between Pandits and Modern Scientists: In fact the following questions (mostly pertaining to foundations of mathematics) were circulated prior to the conference as the issues for the dialogue session.

1. What is the nature of mathematical knowledge? How does it differ from, say, knowledge in Ayurveda on the one hand and from Linguistics and Astronomy on the other?

2. What is the nature of those objects whose knowledge is sought in the science of mathematics? Are they, like other spatio-temporal objects whose knowledge is sought in the various sciences? If not how are these objects encountered in experience to become objects of knowledge?

3. How Ms mathematical knowledge validated? Is this process of validation essentially different from that which obtains in the other Sastras?

4. What is the relation between different branches of mathematics? Are these different branches concerned with varieties of the same object or of different objects? If the latter is the case, what is the point of grouping them under one discipline?

5. Is Astronomy the only science which has to use mathematics as a necessary instrument for its study? If so, what are the special characteristics of the object of astronomical knowledge which necessitate this?

The conference was attended by over 50 scientists and by only about half a dozen pandits. For this and for various other organizational reasons (which are perhaps inevitable in a pioneering effort like this) the intended dialogue did not, so to say, take off. In fact the first two days of the conference were mainly devoted to the pre-sentation of papers on the Indian tradition in astronomy and mathematics. While these papers did bring out lot of interesting perhaps novel -features of the Indian tradition, in astronomy mathematics, hey were mostly confined to the standard pattern of research in history of Indian Science which is one of enumeration of particular results or achievements of Indian astronomers and mathematicians.

It was pointed out right at the outset by Prof. K.V.Sharma (Adyar Library and Research Centre that for an effective dialogue between pandits and modern scientists there should be a serious effort on the part of the latter to understand the methodology, and the world views of the Indian tradition in science. He pointed that there were Indeed quite a few already published texts, which provided an idea of what he called "the mental makeup” of the traditional Indian scientists.

There was an interesting paper of by Prof. Amalendu Bandhopadhyay (Positional Astronomy Centre, Calcutta) on the astronomical work of, Chandrasekhara Samanta, the 19th century astronomer from Orlssa. His discovery of all the major corrections to lunar motion (namely eviction, variation and annual equation) and his corrections to solar parallax were highlighted by the speaker. He also mentioned of a plan to publish a new edition of Chandrasekhara Samanta's work 'Siddanta Darpan' with an English translation. Another interesting paper on Indian astronomy was that of Shri S.Hariharan (Bangalore) which described the method of the Kerala astronomer Madhava- (AD 1342-1455) to find the declination of the moon and the planets while not moving on the ecliptic. He also pointed out how Madhava's correction was later rediscovered by Tycho Brahe in Europe. Prof.S.N.Sen (Asiatic Society) presented a new analysis of the planetary theory of the Indian astronomers. He argued that the procedure of first correcting the planetary positions by half the so called "sighra and manda corrections ensures that the theoretical approximation made in computing these corrections become in some sense optimal.

Pandit Bibhutl Bhushan Bhattacharya (Banaras) presented a very interesting paper which brought out how some of the methods and procedures of Indian mathematics have been misunderstood by Europe-in mathematicians both at the time of European Renaissance (when Indian mathematical results, especially in Arithmetic and . Algebra were; introduced in a big way into Europe) and even later. His comments mainly concerned the Trairasika or the rule of three, the Citi Ganita or Combinatirics and Sredhi Ganita or theory of progressions and series. Pandit Bhattacharya argued that Indian Sredhi Ganita incorporates most of the methods for which modern mathematics resorts to infinitesimal calculus. He also drew attention to the unfortunate fact that many pandits at the Banaras Sanskrit College in the last century (even illustrious ones like Pandit Bapudev Sastri and Pandit Sudhakara Dwivedi) were persuaded to adopt the obscure European interpretations, which had left a lot of confusion in the Uabi and text books of various Sanskrit Colleges, which continues even today.

Amongst other interesting papers on Indian mathematics was the paper by Navjyoti Singh (N1STADS, New Delhi) on the mathematics of asankhyata or unnameable finite numbers, developed in Oainai tradition, and the paper by Dr.R.C.Gupta (University of Ranch!), on the derivation of the formula for volume of the sphere In Indian, Chinese and European traditions.

Perhaps the most Interesting sessions of the conference were those held on the last day to discuss the various questions that were circulated (listed above). A few key-note papers were presented to initiate a dialogue on these issues. The speakers were Dr.V.Shekhawat (Rajasthan Untversity), Pt. Bibhuti Bhattacharya, Sri Navjyoti Singh, Dr.M.D.Srinivas (University of Madras) and Prof.D.K.SInha (Calcutta University). Many important foundational Issues were brought out during the course of these presentations and the discussions that followed.

The first issue that was deal with was the nature of mathematical objects as understood in the Indian tradition, especially 1n the Sankhya and Nyaya - Vaiseshika Darshanas. This led to various issues of contemporary interest in Foundations of Mathematics. The unresolved issues In (the contemporary debates on Foundations of mathematics between the Platonists, Formalists and Inst1tut1onists were highlighted. So also was the .currently existing gap between the philosophy of mathematics and the working philosophy of a mathematician. The fact that there 1 does not exist fan adequate philosophical account of the nature of mathematics and even more so of the nature of creativity 1n mathematics (in other words,1 of the mathematical activity Itself) was Hicr.jecpH at some length.

All this led to the question of what the Indian, views on the Foundations of Mathematics are. It became clear soon, that so far very little work has been done 1n recent times to explicate the foundational methodology of, the Indian tradition of astronomy and mathematics, The already (printed source materials are perhaps adequate to show that the Indian scientists did place very great emphasis on methodological and foundational issues in mathematics and astronomy. It was pointed out that the already published commentaries of some of the major works (such as the commentaries of Ganesa Daivajna and Krishna Daivajana on the works of Bhaskaracharya) show that the Indian scientists provide upapattis (demonstrations' or almost every result or procedure discovered by them. An these upapattis should throw much light on the fecundation methodologies of Indian mathematics and astronomy. It was pointed that the Indian notion of upapattis was perhaps very different from the, 'Greek or the modern Western notions of "proof" where the attempt was to prove the "absolute truth” of a mathematical statement proofs) for analysis of by deriving 1t from a set of postulates. The purpose of upapatti was more to convince or make clear given result or a procedure. While systematic logical- argumentation was involved in providing an % upapattl, there seems to be no stress on starting from a set of postulates considered as given, once and for all. Another important feature of the Indian methodology of mathematics is that the nature of mathematical objects as understood in Indian tradition does seems to play a crucial role in the kinds of upapattis or demonstrations provided in it. Even more significant seems to be fact that the method of indirect proof (or reduction ad absurdum") is acceptable only for providing the non-existence of an entity. The Indian mathematicians will not accept any proof of existence solely based upon the method of indirect proof. And in this sense they display what Is today termed, a constructivist approach to the question of mathematical existence.

In the course of discussions it became clear that a study of the methodology of Indian mathematics and astronomy would throw up a whole lot of interesting problems and even many novel directions of research of contemporary interest. A reference was made to current work in theoretical computer science showing the optimality of the Indian procedures for solving indeterminate equations It was emphasized that at least now we should make a departure from the conventional approach to the study of history of science in India, which was so far exclusively oriented to recounting some of the results and achievements of Indian science to the utter neglect of the methodologies, and philosophy of science in India. The conference therefore adopted the following resolutions:
Serious research work should be initiated on the methodology, and foundations of mathematical sciences in India (1) with a view to placing the Indian tradition in mathematical sciences in the proper perspective and -(2) with a view to foster creative use of insights from the Indian tradition in mathematical sciences in current research. In order to undertake such research it is very essential that the vast source materials (mostly in manuscript form) on Indian mathematical sciences should be made accessible to our scholars in * microfilm or, preferably in published form, on a priority basis".


INDIAN ASTRONOMY: A SOURCE BOOK By B. V. Subbarayappa and K. V. Sarma (Nehru Center, Bombay, 1985)

This compilation of about 3,000 verses (mostly in Sanscrit) from various Indian astronomical texts has been published by the Nehru Centre and was released during the General Assembly of the International Astronomical Union held in New Delhi in 19CJ6. The source verses, together with English translation, have been compiled under 5 major divisions. The section on "General Ideas and Concepts deals with Jyothisastra and its place in Indian literature, the qualities of an astronomer, the basic cosmological ideas on the universe, the sun, moon, stars, planets and the earth, Indian number system and the Indian measures of time. The second section on Astronomical Instruments deals with the armillary sphere, descriptions of some of the instruments in the various observatories and descriptions of other instruments. The third section on Computations' includes an explanation of the five constituents (Panchanga) of the Indian Calender, enumeration of stars, detailed procedures for calculating longititudes of planets, the effects of precession of equinoxes and gnomonic calculations. The section on Occulatiori deals with the computations of eclipses, phases of the moon, heliacal raising and setting of planets and stars, and conjunctions of stars and planets. The last section on "Innovative Trends deals with some novel innovations of the Indian astronomers and some of the rationales provided in the Indian astronomical texts.

In their introduction, the authors briefly deal with the development of Indian astronomy. Most of the issues dealt with pertain to the earlier periods, (say prior to 12th century) and very little is said on the recent period (say 16-I9th century). However in their own "Select Bibliography of (source works in) Indian Astronomy', appended to the text, the authors cite about 50 works written prior to 12th century, about 75 works written between 12-15th centuries and about 165 works written between 16-19th centuries. Most of these later works are yet to be studied in any depth and detail the Source Book, abounds in highly valuable and interesting quotations from our great astronomers. We shall here give two quotations, just to give a flavour of their writings and their thinking. The following is a statement from the 16th century Kerala Text Orikkarana (In Maloyalam) which gives a brief chronology of the series of Improve-ment? that the Kerala astronomers made in their theory, on the basis of astronomical observations.

The Source Book, abounds in highly valuable and interesting quotations from our great astronomers. We shall here give two quotations, just to give a flavour of their writings and their thinking. The following is a statement from the 16th century Kerala Text Orikkarana (In Maloyalam) which gives a brief chronology of the series of Improve-ment? that the Kerala astronomers made in their theory, on the basis of astronomical observations.

Now, I shall set out in brief what the early astronomers enunciated. Before Kali 3000 (100 BC), the eclipses and other observed phenomena did not tally with the astronomical manuals or the Siddhantas. Then, in the Kali year jnanatunga* (3600=A.D.499) an astronomer by name, Aryabhata was born in this world. In the Kali year glritunga (3623=A.D. 522) was his work Aryabhatiya composed and therein he enunciated the revolutions (of the planets). He had adjusted these revolutions by reduction and addition in such a way that there was non zero-correction at the beginning of Kali. In course of time, deviations were observed in (the results arrived at by) this computation. Then, in the Kali year Mandasthala (3785=A.D 684) equivalent to Saka tanuta (606), several astronomers gathered together and devised, through observation, (a system) wherein (the correct mean longitudes were to be found) by multiplying the current Kali vear glritunga (Kali 3623, viz., the Aryabhatan epoch) as directed by the Vagbhaya, i.e. bhatabda or sakabda, correction enunciated by Haridatta and applying the correction.

This system was termed Parahita and many followed it, assuring them of its accuracy. When a long time had lapsed, there occurred substantial deviations. Then (Paramesvara), a nobel brahmana residing on the coast of the western ocean, revised it (i.e. the Parahita system) by means of (astronomical) observation, Tn tFe Kali year rangasobhanu (4532=A.D. 1431)... The work Tantrasangraha (by Tlilakanta), (with revised constants) is Tor twelve years later. The revolutions given therein (i.e. in Tantrasangraha) too, becoming imperfect (in course of time), observations were continued by the astronomers on the west coast for thirty years, from the Kali year jasustaya (4678=A.D. 1577) to the Kali year janaseva nu (4708=A.D. 1607) and,.by observation, the astronomical tradition was revised accurately. Hencefore, too (the deviations) that would occur should be carefully observed (and revisions effected).

Here is another quotation from the 16th century Kerala astronomer Neelakantha Somayajin, ,which very clearly sets forth the Indian epistemological position concerning theories in natural sciences.

Commentator on the-Hanasa (viz. Laghumanasa of Munjala) has lamented: Indeed, the Tiddhantas, like Paitamaha, differ from one another (in giving the astronomical constants). Timings are different as the Siddhantas differ (i.e. measures of time at a particular moment differs as computed by the different Siddhantas). When the computed timings differ, Vedic and domestic rituals, which have (correct) timings as a component (of their, performance) go astray. When rituals go astray, worldly life gets disrupted. Alas we have been precipitated into a big calamity'.

Here; it needs to be stated: "0 faint-hearted, there is nothing to be despaired of. Wherefore does anything remain beyond the ken of that intent on serving at the feet of the teachers (and thus gain knowledge)? One has to realise that the five Siddhantas had been correct (only) at a particular time. Therefore, one should search for a Siddhanta that does not show discord with actual observations has tb be ascertained by (astronomical) observers during times of eclipses etc, When Siddhantas show discord, observations should be made with the use of instruments and the correct number of revolutions etc., (which would give results which accord with actual observation) found, and a new Siddhanta enunciated.

In his preface to the book, Dr. Raja Ramanna, General Secretary of Nehru Centre, and then Chairman of Atomic Energy Commission, states: A fact of great importance is that India has produced a vast 1iterature" on different aspects of astronomy. According to the American scholar, David,Pingree, who has surveyed extensively the Jyotisa literature of India, it would appear that more than 1,00,000 manuscripts on Jyotisa (including astronomy and astrology) still survive in the public and private repositories in India and outside, of which)a very substantial part, running to several thousands, relates to astronomy... The scientific manuscript wealth of India is indeed enormous. An authentic discovery of India's scientific heritage demands uncritical evaluation of the original sources and a rational presentation of the scientific heritage, through original sources. Guided by this view, Nehru Centre, as part of its Discovery of India Project, took a step in this direction two years ago leading to the present publication.

While the source book is indeed an admirable effort and a very useful compendium, the material contained therein is mostly that which has already been printed and translated. It does not take us far in the process of authentic discovery of India's scientific heritage, which would involve a complete compilation publication and critical study of' the enormous "scientific manuscripts wealth of India.

In fact, Indian scholars are yet to make even a list of what are indeed the manuscripts of Jyotish which was estimated by Pingree to be of over 1,00,000 in extent." Let us hope that the "Discovery of India Project' of the Nehru Centre or some of our national research centers will indeed initiate the collection of these manuscripts at one or several centers in India and their study and analysis. There is also an urgent need for a systematic exposition of Indian astronomy and mathematics in modern Indian languages and English, not in the form of translation of texts, but nevertheless employing only the Indian technical terms and concepts. This would enable our young students' in schools and colleges to understand the methodology - (the simplicity as well as sophistication of the concepts and procedures) of Indian mathematics and astronomy.

Author:M.D. Srinivas


* See for instance: C.O. Selenius: Rationale of the Chakrayala Process of Jayadeva and Bhaskara II, Historia Mathematica 2, 167-184 (1984); Subhash Kak: Computational Aspects of the Aryabhata." Algorithm, Indian Journal of History of Science," 62-71 (1986).

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