Abstract
Several books have been written on the history of Indian tradition in mathematics.2 In addition, many books on history of mathematics devote a section, sometimes even a chapter, to the discussion of Indian mathematics. Many of the results and algorithms discovered by the Indian mathematicians have been studied in some detail. But, little attention has been paid to the methodology and foundations of Indian mathematics. There is hardly any discussion of the processes by which Indian mathematicians arrive at and justify their results and procedures. And, almost no attention is paid to the philosophical foundations of Indian mathematics, and the Indian understanding of the nature of mathematical objects, and validation of mathematical results and procedures.
This Epilogue is an updated version of the article, M. D. Srinivas, “The Methodology of Indian Mathematics and its Contemporary Relevance”, PPST Bulletin, 12, 1–35, 1987. See also, M. D. Srinivas, ‘Proofs in Indian Mathematics’, in G. G. Emch, R. Sridharan, and M. D. Srinivas, (eds.) Contributions to the History of Indian Mathematics, Hindustan Book Agency, New Delhi 2005. p. 209–248.
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Notes
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, 2001;
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, 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 stark 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 any proof for most of the claims he made, with the realisation 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 Cakravāla algorithm is different and more optimal than the Brouncker-Wallis-Euler-Lagrange algorithm for solving this so-called “Pell’s Equation.”
D. Pingree, Jyotiḥśāstra: Astral and Mathematical Literature, Wiesbaden 1981, p. 118.
K.V. Sarma and B.V. Subbarayappa, Indian Astronomy: A Source Book, Bombay 1985.
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, ‘Pāṇini and Euclid’, Jour. Ind. Phil. 29, 43–80, 2001).
This method seems to be known to Bhāskarācārya I (c.629 AD) who gives a very similar diagram in his Āryabhaṭīyabhāṣya (K.S. Shukla (ed.), Delhi 1976, p. 48). The Chinese mathematician Liu Hui (c 3rd century AD) seems to have proposed similar geometrical proofs of the so-called Pythagoras Theorem. See for instance, D.B. Wagner, ‘A Proof of the Pythagorean Theorem by Liu Hui’, Hist. Math. 12, 71–3, 1985.
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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Sarma, K.V., Ramasubramanian, K., Srinivas, M.D., Sriram, M.S. (2008). Epilogue. In: Gaṇita-Yukti-Bhāṣā (Rationales in Mathematical Astronomy) of Jyeṣṭhadeva. Culture and History of Mathematics. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-36-1_16
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