Panini And Euclid Reflections On Indian Geometry

Author(s): Johannes Bronkhorst
Publisher: Johannes Bronkhorst

Page 10

JOHANNES BRONKHORST
PĀŅINI AND EUCLID
they do not derive their validity from proofs. This explains that some incorrect rules have been able to slip into the works of Aryabhata and Bhaskara and remain undiscovered for a long time.
The notion of proof that is claimed to have existed in India and elsewhere by the above-mentioned authors has been examined by G.E.R. Lloyd in the 3rd chapter of his book Demystifying Mentalities (1990), some passages of which are worth quoting. Lloyd begins as follows (p. 74): At the outset we must be clear that "proof' and 'proving' may signify a variety of more or less formal, more or less rigorous, procedures. In some domains, such as law, proving a fact or a point of law will be a matter of what convinces an audience as being beyond reasonable doubt. Again in some contexts, including in mathematics, "proving a result or a procedure will sometimes consist simply in testing and checking that it is correct. Both of these are quite informal operations. But to give a formal proof of a theorem or proposition requires at the very least that the procedure used be exact and of general validity, establishing by way of a general, deductive justification the truth of the theorem or proposition concerned. More strictly still Aristotle was to express the view that demonstration in the fullest sense depended not just on deductive (he thought specifically syllogistic) argument but also on clearly identified premises that themselves had to fulfil rather stringent conditions ... He was the first not just in Greece, but so far as we know anywhere. explicitly to define strict demonstration in that way
Two crucial distinctions have, then, to be observed, (1) between formal proofs and informal ones, and (2) between the practice of proof (of whatever kind) and having an explicit concept corresponding to that practice, a concept that incorporates the conditions that need to be met for a proof to have been given.
Before we jump to the conclusion that these shared features are due to the influence of Paninian grammar on classical Indian geometry, some further facts and arguments have to be taken into consideration. The claimed absence of proofs in classical Indian geometry, in particular, has to be confronted with the conflicting claim that proofs existed already in Indian geometry long before Aryabhata and Bhaskara, and presumably before Panini, viz., in the geometry of the Vedic Sulba Sütras. Frits Staal - whose views about the importance of grammar are central to this essay - maintains this position in a very recent article (1999)," but does not provide arguments to bolster it beyond referring to a number of publications by Seidenberg (1978, 1983; one might add 1962, 1975: 289 ff.) and Van der Waerden (1983: 26 ff.; one might add 1980)." Seidenberg and Van der Waerden - like Staal himself - argue for a common origin of mathematics, or of geometry specifically, as found in various cultures. Staal, for example, pleads for a common origin of Greek and Vedic geometry in Bactria/Margiana; Van der Waerden proposes "a Neolithic Geometry and Algebra, invented somewhere in Central Europe between (say) 3500 and 2500 B.C., in which the 'Theorem of Pythagoras' played a central role" (1980: 29; cp. 1983: 33–35); Seidenberg claims to prove that "(a) common source for the Pythagorean and Vedic mathematics is to be sought either in the Vedic mathematics or in an older mathematics very much like it" (1978: 329; similarly already Schroeder, 1884). More important for our present purposes is that these three authors -as pointed out aboveagree that Vedic mathematics had proofs. Van der Waerden goes further and maintains that already the original mathematicians (whom he calls the pre-Babylonian mathematicians) had them (1980: 8): "I suppose that these mathematicians had proofs, or at least plausible derivations. A pupil who has to solve a mathematical problem can do it just by applying a rule he has leamt, but the man who invented the rule must have had some sort of derivation. I also suppose that our pre-Babylonian mathematicians had a proof of the 'Theorem of Pythagoras'." Seidenberg is hardly less brazen (1978: 332): "The striking thing (in the Apastamba Sulvasutra) is that we have a proof. One will look in vain for such things in Old-Babylonia. The Old-Babylonians, or their predecessors, must have had proofs of their formulae, but one does not find them in Old-Babylonia."
Lloyd points out that the second distinction, in his view, has been ignored or badly underplayed in recent attempts to see the notion of proof as originating long before even the earliest extant Egyptian and Babylonian mathematics. He then turns to Vedic mathematics and argues that the authors of its key texts were not concerned with proving their results at, all, but merely with the concrete problems of altar construction. Vedic mathematics is again dealt with in a "Supplementary note: geometry and 'proof' in Vedic ritual" (pp. 98-104), where Lloyd observes that "the notion that the authors in question had a clear and explicit concept of proof is subject to the general doubt... that to obtain results is one thing, to have that concept as an explicit one is another. ... It also falls foul of one further fundamental difficulty. This is that no clear distinction is drawn in these texts between the rules that are expressed to arrive at what we should call approximations and those that are employed to yield what we should call exact results" (p. 101).
The question could be asked whether the notion of proof is really culture-specific to ancient Greece. Here Lloyd comments (p. 75): The practice of proof, in Greece, antedates by several generations the first explicit formal definition (first given by Aristotle in the fourth century B.C.) and the process whereby such notions us thul of the starting points or axioms came to be clarified wus both hesitant and gradual. That long and complex development, in Greece, belongs to and is a further instance of the gradual heightening of self-consciousness we have