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...years earlier, astronomer-mathematicians of Kerala, notably Madhava of Sangamagrama and his disciples, had discovered elements of that calculus, the forerunners of modern techniques used in mathematical analysis. Given the existence of a corridor of communication between Kerala and Europe, especially from the sixteenth century onwards, and the crucial importance of calculus in the growth of modern mathematics, one would have expected that the possibility of the transmission of the Kerala mathematics westwards would be high on the agenda for historical investigation. That such an investigation has not yet been carried out may reflect, in our view, the strength and the pervasive nature of Eurocentrism in the history of science.
Introduction
>From early times, Indian mathematics was often bundled together with astronomy and astrology, a fact recognised and commented on by scientists such as Al-Biruni as early as the eleventh century AD. (1) So it is not difficult to understand why the mathematics of the Kerala School (2) was initially developed by astronomers to service astrological prediction and facilitate the construction of a precise calendar. This mathematics included numerical integration methods and infinite series derivations for certain trigonometric functions and was transcribed in palm leaf manuscripts by Kerala astronomers such as Nilakantha Somayaji, Jyesthadeva, Sankara Variyar and others between the fourteenth and sixteenth centuries. (3) These astronomers had elaborated and extended the earlier work of the toaster astronomer and founder of the Kerala School, Madhava of Sangamagrama (c.1340-1425).
It is important to bear in mind that the conceptual and epistemological bases of Madhava's mathematics had little affinity with those of early Greek mathematics. Instead, they were founded on the principles elaborated in 499 AD by the renowned Indian astronomer Aryabhata in his Aryabhatiya. While Aryabhata and his predecessors may have been influenced by the content of classical Greek mathematics they did hot subscribe to the idea of Platonic reality (4) and deductive proof, the essential characteristics of the Greek contribution. Instead, the Indians subscribed to a paradigm in which the truth of mathematical results could be established by both empirical and abstract reasoning (or pramana). No conflict was perceived between pramana, on the one hand, and empirical demonstration or numerical calculation, on the other. Deduction was an integral part of pramana, but it was not imagined, as in the case of Greek mathematics, that the exclusion of the empirical somehow conferred a superior and infallible status on deduction. In addition, the use of irrational numbers, unlike early Greek mathematics, was accepted in Indian mathematics by the use of floating point number approximations. (5)
It is noteworthy that the reluctance of Indian astronomers to accept Greek knowledge unconditionally had its parallels in Europe. The Portuguese and Spanish explorers of the fifteenth and sixteenth centuries cast a critical eye over the immutable laws of antiquity and their insistence on discovery through thought experiment. (6) The Iberian explorers were justified in their scepticism when, in their voyages of discovery, they found certain aspects of the astronomy of Ptolemy and Hipparchus to be incorrect. This is not to say that the Portuguese explorers rejected ancient knowledge; instead they, like the Indians, modified it to suit their own particular ideas. This is most aptly demonstrated by Portugal's greatest medieval scientist Pedro Nunes, who considered Ptolemy's geography to be intrinsically flawed, but nevertheless modified it to develop the concept of the loxodrome as an aid for navigation. (7)
In a similar vein, the Aryabhatan School of Indian mathematics was justified in its epistemological approach when, in approximately the same period as the Iberian explorers, Madhava and his students used procedures involving the passage to infinity and pramana--things alien to the Greek mathematical epistemology--to discover infinite series sums for various functions. The Madhava School was comfortable in dealing with the concept of infinity and, moreover, applied this concept in a sophisticated way to be replicated in the period following the development of general methods of the calculus by Leibniz and Newton. As Rajagopal and Marar point out:
There are two points which emerge from a consideration of the [mathematics of the Kerala School] ... In the first place, it employs relations which would appear not to have been noticed in Europe before modern forerunners and followers of the calculus started investigations ... Our second point is not unconnected with the first. The Hindu mathematicians achieved, without the aid of calculus, results which, for us, are treated best by means of the calculus ... This is not to gainsay the fact that (i) the Hindus' proof of [infinite] series shows their awareness of the principle of integration as we ordinarily use it nowadays (ii) their intuitive perception of small quantities ... is as good as a practical knowledge of differentiation. (8)
This line of argument suggests that the Renaissance mathematicians of Europe, who were schooled in the mathematics of Euclid and who came before 'modern forerunners and followers of the calculus started investigations', may have had some difficulty with the concept of infinity. The controversies over Cavalieri's indivisibles and Newton's fluxions suggest this to be the case. (9) On a prototypical level, we can give an example of this awkwardness in the work of the prominent seventeenth-century English mathematician John Wallis. When Wallis attempted to generalise the result '... 1/5 < 1/4 < 1/3 < 1/2 < 1/1', induction led him to conclude that '1/0 < 1/-1 < 1/-2 < 1/-3 < 1/-4 < ...'. This erroneous conclusion showed a faulty understanding of the transition from positive quantities to negative ones via infinity. Wallis did later formulate a prototypical concept of infinitesimally small quantities, but the awkwardness remained. (10)
This awkwardness in dealing with infinitesimals (infinitely or extremely small quantities) and, by implication, with infinity exhibited by Renaissance mathematicians was most likely due...
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