Panini And Euclid Reflections On Indian Geometry

Author(s): Johannes Bronkhorst
Publisher: Johannes Bronkhorst

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JOHANNES BRONKHORST
PANINI AND EUCLID
62
cxcmplified before, when Necond-order questions came to be raised concerning the nature, status, methods and foundations of different types of inquiry. None of the attendant circumstances surrounding these developments, and none of the steps by which the various interrelated key notions came to be made explicit, can be paralleled in Vedic literature or in the evidence for Vedic society
Does this mean that we should not expect any such notion to have existed in classical India? Is the notion of proof really a Greek invention, determined by the specific social and political situation prevailing in that culture? And does it follow that all cultures that do possess the notion of proof must have borrowed it - directly or indirectly from ancient Greece?
Before trying to answer these questions it seems appropriate to recall that the absence of explicit proofs and of an identifiable notion of proof seems to be a common feature of many cultures that produced geometry. 0. Neugebauer, for example, states about ancient Babylonia (1957: 4546; cited in Seidenberg, 1975: 286): "It must... be underlined that we have not the faintest idea about anything amounting to a "proof conceming the relations between geometrical magnitudes. Richard J. Gillings - in an appendix meant to counter some of the negative criticism addressed at the mathematics of the ancient Egyptians on account of its lack of formal proof - concludes nonetheless (1972: 234); "We have to accept the circumstance that the Egyptians did not think and reason as the Greeks did. If they found some exact method (however they may have discovered it), they did not ask themselves why it worked. They did not seek to establish its universal truth by an a priori symbolic argument that would show clearly and logically their thought processes." Chinese geometry, it appears, did not use proofs either. When early in the seventeenth century Euclid's Elements were translated into Chinese," by Matteo Ricci and Xu Guangqi, the latter of these two wrote in the preface to another work) that only after the translation of Euclid's Elements into Chinese had it become possible to transmit proofs and explanations. In fact, he maintained, the Westem methods of conveying are not essentially different from the methods transmitted in earlier Chinese treatises. What makes Western mathematics more valuable is that it supplies explanations which show why the methods are correct. Joseph Needham (1959: 91) confirms this by stating: "In China there never developed a theoretical geometry independent of quantitative magnitude and relying for its proofs purely on axioms and postulates accepted as the basis of discussion." We learn from Jean-Claude Martzloff's A History of Chinese Mathematics that in the Chinese tradition of geometry "the figures essentially refer not to idealities but to material objects which, when manipulated in
an appropriate manner, effectively or in imagination, may be used to make certain mathematical properties visible". This has various consequences, among them the use of empirical methods: "To show that the side of a regular hexagon inscribed in a circle has the same length as the radius, six small equilateral triangles are assembled and the result is determined de visu. One proof technique for determination of the volume of the sphere involves weighing it. Sometimes the reader is asked to put together jigsaw pieces, to look at a figure or to undertake calculations which themselves constitute the sole justification of the matter in hand". Martzlott concludes (p. 72). "If one has to speak of proofs', it might be said that, from this point of view, the whole of the mathematician's art consists of making visible those mathematical phenomena which are apparent not in Platonic essences but in tangible things". The concern with mathematical objects as parts of objective reality reminds us of the similar concern on the part of Bhaskara, studied above.
Should we then conclude that the notion of proof only belongs to ancient Greece and its inheritors? I think the situation is more complex than this. Recall that India at the time of Aryabhata and Bhaskara did have a clear notion of proof. Such a notion was present in Indian logic, which had been developing since long before Aryabhata and reached some of its peaks in the persons of Dignaga and Dharmakirti precisely during the period that separates Aryabhata from his commentator Bhaskara at any rate, it would appear, had no excuse for not being aware of the notion of proof, and for not providing proofs for his theorems. At first sight the situation of these two mathematicians should not therefore be very different from Euclid in Greece Greek philosophers developed the notion of proof in their logic, and Greek mathematicians did the same in geometry. Indian philosophers developed the notion of proof in their logic, but the Indian mathematicians did not follow suit. Why not? The situation becomes even more enigmatic if we assume that Indian geometry derives from Greek geometry (see note 8, above), and that therefore some Indian mathematicians (presumably well before Aryabhata and Bhāskara) had been familiar with Euclidean procedures.
Here it is important to recall that Indian philosophers of the classical period were engaged in an ongoing debate with each other, in which radically different positions were defended and criticized. This debate went on because all participants were part of what I have called a
tradition of rational inquiry".which translated itself in the social obligation - partly embodied in kings and their courts - to listen to