Mathematical proof: a research programme 53
‘explanation’ which are referenced with specifi c terms. Once the ‘explana-
tion’ is given in the form of such a diagram, it comes to a close. Is it that
the argument is left for the reader to develop or is it that it was developed
orally? It is diffi cult to tell. However, we recognize a feature of proofs that
was frequently mentioned in nineteenth-century accounts of ‘Indian’ math-
ematical reasoning but was subject to divergent assessments, as Charette’s
chapter shows. Seen from another angle, we may note that the written for-
mulation of a proof carried out in relation to a diagram took quite diff erent
forms in history. Further development of a comparative analysis of such
texts arises as a possible venue for future research.
On the other hand, the commentator used the term ‘explanation’
(pratipadita) to refer specifi cally to another component that he intro-
duced: problems solved by means of the algorithm described. In which
ways did the problems contribute to providing an explanation of the
algorithms? Here too, the source material calls for a comparative analysis
of the part allotted by diff erent traditions to problems for establishing
an algorithm.
Th e evidence discussed so far illustrates the variety of contexts that may
have prompted an interest in writing down proofs. Th e sources analysed
by Keller and myself show how commenting upon a canonical text has
been an activity by which proofs were made explicit. In addition Høyrup
suggests the hypothesis that teaching could have motivated an interest in
formulating proofs. In fact, the two explanations are not mutually exclusive,
if we embrace Volkov’s hypothesis that Chinese commentaries were com-
posed within the context of mathematical education. We come back to this
hypothesis below. In addition, the evidence discussed so far also shows the
variety of motivations that led to the formulation of proofs in the ancient
traditions. What they contribute to our historical approach and under-
standing of mathematical proof is an issue to be taken up in the conclusion.
Before we can address our conclusions, however, one more dimension of
our world history is worth considering.
Th e persistence of traditions of proof in Asia
One may be tempted to believe that it is relevant to adopt the perspective
of a world history to deal with mathematical proof in ancient traditions,
but that aft er the seventeenth century, the story to be told is that of the
‘Western’ practices and their adoption worldwide. Th e fi nal two chapters of
the book illustrate two ways in which such a view must be qualifi ed. Th ey
constitute the only incursions of this book into later traditions of proof. Th e