Some reflections on the practices of proofs in Sanskrit mathematical texts, with a special emphasis on Śaṅkara Vāriyar’s work on Mādhava’s procedure to approximate the circumference of a circle.
Agathe Keller (Sphere, CNRS / Université Paris Cité)
In his commentary on the Līlāvatī—Bhāksara (b.1114) ’s very popular arithmetical text—Śaṅkara Vāriyar (fl. ca. 1540) launches into a spectacular presentation of the values that Mādhava (14th century) can provide to approximate the ratio of the circumference of a circle to its diameter. He then offers an elaborate proof of one of the highlights of the “Kerala School of Mathematics” attributed to the same Mādhava: a rule to approximate the circumference of a circle which is seen as an equivalent of formulas given later by Gotfried Wilhelm Leibniz (1646-1716) and James Gregory (1638-1675) prefigurating the birth of calculus. In this presentation, I will show how Śaṅkara Vāriyar commentary testifies to new ways of thinking about reasonings and proofs in mathematics, offering many contrasts with the practices of earlier authors writing in Sanskrit. More largely I will describe how authors of mathematical texts in Sanskrit had a great variety of practices of mathematical reasonings. Not all of these practices were about “proving” mathematical truths; reasonings could have many different aims— such as showing that a procedure could be used in different mathematical disciplines, or that a formal computation could be explained by providing each step with a meaning. My aim will be to look at how authors carried out “explanations” (vāsanā) or sought to “establish” a procedure (sadh-, upapad-), and how this questions standard historiographies of proof in Sanskrit mathematical literature on the one hand and of the “Kerala school of mathematics” on the other.