Mathematical proof: a research programme 51
wonder whether the algebraic proof in an algorithmic context as demon-
strated in ancient China could not have played a part in the actual emer-
gence of the algebraic proof as we practise it today.
Th is set of issues demonstrates the ways in which the broadening of our
corpus leads to the formulation of new directions of research in the early
history of mathematical proof.
Proving as an element of the interpretation of a classic
Th e Chinese case just examined is not the only historical instance in which
the formulation of mathematical proofs took place within the framework of
commentaries on a classic. Agathe Keller’s chapter is devoted to the earliest
known Indian source in which an interest for mathematical proof can be
identifi ed: it turns out to be the seventh-century commentary by Bhaskara
I on the mathematical chapter of the fi ft h-century astronomical treatise
Aryabhatiya. As in the Chinese case, Keller shows how the development of
arguments to establish the correctness of procedures is part of the activity
of an exegete who comments on a classic. 58
Th e proof is part of Bhaskara’s way of justifying the classic, unless it
justifi es his own interpretation of the classic. A Sanskrit classic is com-
posed of sutras , the interpretation of which requires skills. It is within this
context that, when the classic deals with mathematics, proof – together
with grammar – seems to be a means for a commentator to inquire into
the meaning of the classic and to advance his interpretation. Despite the
fact that commenting on a classic provided the impetus for making proofs
explicit in both Sanskrit and Chinese, the way in which proofs relate to the
interpretation seems to present diff erences between the two contexts.
In the case discussed by Keller, the classic, i.e. the Aryabhatiya , indi-
cates algorithms. Th e commentator Bhaskara states them fully, showing by
means of Paninian grammar how the sutras mean the suggested algorithms
and then accounting for why the suggested algorithms are correct. Bhaskara
manifests his expectation that the classic does not provide explanations.
By contrast, he introduces a set of terms (explaining, verifying, proving)
that indicate how he understands the epistemological status of parts of his
commentary.
Keller provides evidence to support an interpretation of what ‘explain-
ing’, ‘verifying’ and ‘proving’ meant for him, in terms of actual intellectual
58 Srinivas 2005 insists more generally on the fact that in Indian writings proofs occur in
commentaries, and in Appendix A he provides a list of these commentaries.