Mathematical proof: a research programme 55
books exerted a strong infl uence. It is in such a Vietnamese source that
Volkov found a model for mathematical examination which he translates
and analyses in his chapter.
Th is piece of evidence leads him to put forward the hypothesis that
the shape taken in China by the mathematical classics and the seventh-
century commentaries may refl ect precisely the requirements of the
teaching institution. On this basis, one can shed light on the connection
between these texts and the examination system from another angle. It is
quite striking, indeed, that the administrative texts analysed by Volkov
describe the tasks to be carried out by students in the seventh-century
Chinese state institutions with technical terms that can be found inter alia
in Th e Nine Chapters and the commentaries on the mathematical classics
that were mentioned above. Th is holds true, as Volkov stresses, for words
like wen ‘problem’, or da ‘answer’, which refer to components of texts like
Th e Nine Chapters. However, most importantly, this also holds true for
terms like yi’ ‘meaning’, which is the second type of meaning given above
for a procedure, a meaning that is intimately connected to the activity of
proving. Such a link between the two types of sources supports Volkov’s
thesis that commentaries played a key part in the training of students,
since terms like yi′ are not to be found in the texts of the mathematical
classics themselves, but only in commentaries. 62 In conclusion, Volkov
suggests a social context for the interest in the proofs of the correctness of
algorithms in ancient China.
Two points are worth emphasizing for our main argument here. Firstly,
let me stress again what was said above: if Volkov’s hypothesis holds true,
we would have at least two cases – East Asia and Babylon – in which the
professional context of teaching was instrumental for composing proofs,
even though the proofs actually written down diff ered in the two contexts.
Secondly, it is worth noting that this piece of evidence confi rms the longev-
ity of practices of proof in East Asia. Th is is but one example which shows
62 One may even go a step further. We mentioned above two commentaries on Th e Nine Chapters :
the one completed by Liu Hui in 263, and the one presented to the throne by Li Chunfeng
in 656. In fact, several scholars have produced clues which indicate that the text of the two
commentaries may have been commingled during the process of transmission (in CG2004:
472–3, I have summarized the current contributions treating this diffi cult issue which awaits
further research). For the question discussed here, it may be relevant to note that many clues
suggest that the concept of yi’ , when used in relation to procedures, may belong to the layer
of commentary from the seventh century. If this is confi rmed, the connection between the
administrative sources and the seventh-century commentary would be even more striking. Th e
correlation between the terms used in both types of documents should invite us, in my view,
to take the occurrence of the terms in the commentaries on the classics into account when
interpreting the administrative prescriptions.