Abstract
Contrary to the widespread belief that Indian mathematicians did not present any proofs for their results, it is indeed the case that there is a large body of source-works in the form of commentaries which present detailed demonstrations (referred to as upapatti-s or yukti-s) for the various results enunciated in the major texts of Indian Mathematics and Astronomy. Amongst the published works, the earliest exposition of upapatti-s are to be found in the commentaries of Govindasvāmin (c.800) and Caturveda Pṛthūdakasvāmin (c.860). Then we find very detailed exposition of upapatti-s in the works of Bhäskaräcärya II (c.1150). In the medieval period we have the commentaries of Śaṅkara Vāriyar (c.1535), Gaṅeśa Daivajña (c.1545), Kṛṣṅa Daivajña (c.1600) and the famous Malayalam work Yuktibhāṣā of Jyṣṭesthadeva (c.1530), which present detailed upapatti-s. By presenting a few selected examples of upapatti-s, we shall highlight the logical rigour which is characteristic of all the work in Indian Mathematics. We also discuss how the notion of upapatti is perhaps best understood in the larger epistemological perspective provided by Nyāyaśāstra, the Indian School of Logic. This could be of help in explicating some of the important differences between the notion of upapatti and the notion of “proof developed in the Greco-European tradition of Mathematics.
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We may cite the following standard works: B.B. Datta and A.N. Singh, History of Hindu Mathematics, 2 parts, Lahore 1935, 1938, Reprint, Delhi 1962
C.N. Srinivasa Iyengar, History of Indian Mathematics, Calcutta 1967
A.K. Bag, Mathematics in Ancient and Medieval India, Varanasi 1979
T.A. Saraswati Amma, Geometry in Ancient and Medieval India, Varanasi 1979
G.C. Joseph, The Crest of the Peacock: The Non-European Roots of Mathematics, 2nd Ed., Princeton 2000.
C.B. Boyer, The History of Calculus and its Conceptual development, New York 1949, p.61–62. As we shall see in the course of this article, Boyer’s assessment — that the Indian mathematicians did not reach anywhere near the development of calculus or mathematical analysis, because they lacked the sophisticated methodology developed by the Greeks — seems to be thoroughly misconceived. In fact, in marked contrast to the development of mathematics in the Greco-European tradition, the methodology of Indian mathematical tradition seems to have ensured continued and significant progress in all branches of mathematics till barely two hundred year ago; it also lead to major discoveries in calculus or mathematical analysis, without in anyway abandoning or even diluting its standards of logical rigour, so that these results, and the methods by which they were obtained, seem as much valid today as at the time of their discovery.
Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford 1972, p.190.
Andre Weil, Number Theory: An Approach through History from Hammurapi to Legendre, Boston 1984, p.24. It is indeed ironical that Prof. Weil has credited Fermat, who is notorious for not presenting proofs for most of the claims he made, with the realization that mathematical results need to be justified by proofs. While the rest of this article is purported to show that the Indian mathematicians presented logically rigorous proofs for most of the results and processes that they discovered, it must be admitted that the particular example that Prof. Weil is referring to, the effectiveness of the cakravāla algorithm (known to the Indian mathematicians at least from the time of Jayadeva, prior to the eleventh century) for solving quadratic indeterminate equations of the form x2 — Ny2 = 1, does not seem to have been demonstrated in the available source-works.
In fact, the first proof of this result was given by Krishnaswamy Ayyangar barely seventy-five years ago (A.A. Krishnaswamy Ayyangar, “New Light on Bhāskara’s Cakravāla or Cyclic Method of solving Indeterminate Equations of the Second Degree in Two Variables’, Jour. Ind. Math. Soc. 18, 228–248, 1929–30). Krishnaswamy Ayyangar also showed that the cakravala algorithm is different and more optimal than the Brouncker-Wallis-Euler-Lagrange algorithm for solving this so-called “Pell’s Equation.”
D. Pingree, Jyotihśāstra: Astral and Mathematical Literature, Wiesbaden 1981, p.118.
K.V. Sarma and B.V. Subbarayappa, Indian Astronomy: A Source Book, Bombay 1985.
Buddhivilāsinī of Ganesa Daivajna, V.G. Apte (ed.), Vol I, Pune 1937, p.3.
Bījaganita of BhDZskaracarya, Muralidhara Jha (ed.), Varanasi 1927, p.69.
Ignoring all these classical works on upapatti-s, one scholar has recently claimed that the tradition of upapatti in India “dates from the 16th and 17th centuries” (J. Bronkhorst, ‘Panini and Euclid’, Jour. Ind. Phil. 29, 43–80, 2001).
We may, however, mention the following works of C.T.Rajagopal and his collaborators which discuss some of the upapatti-s presented in the Malayalam work Yuktibhäsä of Jyesthadeva (c.1530) for various results in geometry, trigonometry and those concerning infinite series for π and the trigonometric functions: K. Mukunda Marar, ‘Proof of Gregory’s Series’, Teacher’s Magazine 15, 28–34, 1940
K. Mukunda Marar and C.T. Rajagopal, ‘On the Hindu Quadrature of the Circle’, J.B.B.R.A.S. 20, 65–82, 1944
K. Mukunda Marar and C.T. Rajagopal, ‘Gregory’s Series in the Mathematical Literature of Kerala’, Math Student 13, 92–98, 1945
A. Venkataraman, ‘Some Interesting Proofs from Yuktibhāsā’, Math Student 16, 1–7, 1948
C.T. Rajagopal ‘A Neglected Chapter of Hindu Mathematics’, Scr. Math. 15, 201–209, 1949
C.T. Rajgopal and A. Venkataraman, ‘The Sine and Cosine Power Series in Hindu Mathematics’, J.R.A.S.B. 15, 1–13, 1949
C.T. Rajagopal and T.V.V. Aiyar, ‘On the Hindu Proof of Gregory’s Series’, Scr. Math. 17, 65–74, 1951
C.T. Rajagopal and T.V.V. Aiyar, ‘A Hindu Approximation to Pi’, Scr. Math. 18, 25–30, 1952.
C.T. Rajagopal and M.S. Rangachari, ‘On an Untapped Source of Medieval Keralese Mathematics’, Arch, for Hist, of Ex. Sc. 18, 89–101, 1978
C.T. Rajagopal and M.S. Rangachari, ‘On Medieval Kerala Mathematics’, Arch, for Hist, of Ex. Sc. 35(2), 91–99, 1986.
Bījapallava of Krsna Daivajna, T.V. Radhakrishna Sastri (ed.), Tanjore, 1958, p.54.
A translation of the upapatti may be found in M.D. Srinivas, ‘Methodology of Indian Mathematics and its Contemporary Relevance’, PPST Bulletin, 12, 1–35, 1987.
Āryabhatīyabhāsya of Nīlakantha, Ganitapāda, K. Sambasiva Sastri (ed.), Trivandrum 1931, p.142–143.
Yuktibhāsā of Jyeṣṭhadeva, K. Chandrasekharan (ed.), Madras 1953. Ganitādhyāya alone was edited along with notes in Malayalam by Ramavarma Thampuran and A.R.Akhileswara Aiyer, Trichur 1948. The entire work has been edited, along with an ancient Sanskrit version, Ganitayuktibhāsā and English translation, by K.V.Sarma, with explanatory mathematical notes by K.Ramasubramanian, M.D.Srinivas and M.S.Sriram (in press).
Proclus: A Commentary on the First Book of Euclid’s Elements, Tr.G.R. Morrow, Princeton, 1970, p.3, 10.
Both quotations cited in Ruben Hersh, ‘Some Proposals for Reviving the Philosophy of Mathematics’, Adv. Math. 31, 31–50, 1979.
For the approach adopted by Indian philosophers to tarka or the method of indirect proof see for instance, M.D. Srinivas, “The Indian Approach to Formal Logic and the Methodology of Theory Construction: A Preliminary View”, PPST Bulletin 9, 32–59, 1986.
For a discussion of some of these features, see J.N. Mohanty: Reason and Tradition in Indian Thought, Oxford, 1992.
Sibajiban Bhattacharya, ‘The Concept of Proof in Indian Mathematics and Logic’, in Doubt, Belief and Knowledge, Delhi, 1987, p.193, 196.
N. Bourbaki, Elements of Mathematics: Theory of Sets, Springer 1968, p. 13
see also N. Bourbaki, Elements of History of Mathematics, Springer 1994, p. 1–45.
I. Lakatos, Proofs and Refutations: The Logic of Mathematical Discovery, Cambridge 1976.
Philip J. Davis and Reuben Hersh, The Mathematical Experience, Boston, 1981, p.354–5.
C.H. Edwards, History of Calculus, New York 1979, p.79.
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Srinivas, M.D. (2005). Proofs in Indian Mathematics. In: Emch, G.G., Sridharan, R., Srinivas, M.D. (eds) Contributions to the History of Indian Mathematics. Culture and History of Mathematics, vol 3. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-25-5_9
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