Mathematical proof: a research programme 49
in the classic. Further analysis reveals that, beyond the similarity which we
suggested above with the Babylonian case as interpreted by Høyrup, the
understanding made explicit in the Chinese commentaries was not only
provided by ‘geometrical’ interpretations, but could also be achieved, more
generally, by recourse to the situation described in the statement of any
kind of problem. 57 In this sense, the way of generating a semantic analysis
of operations diff ered. A landscape of similarities and diff erences starts
emerging in our world history of mathematical proof in ancient traditions.
Secondly, the fact that the commentators made explicit the reasons
underlying the correctness of the algorithms in such cases is one aspect of
a much more general phenomenon. In eff ect, the commentaries system-
atically established the correctness of the algorithms contained in Th e Nine
Chapters , thereby bearing witness to a considered practice of proof for such
kinds of statements.
My own chapter focuses on one dimension of this practice, which, as far
as I know, appears to be specifi c to ancient China. Th is dimension, which
reveals another fundamental operation used to establish the correctness of
procedures, sheds light on why the texts of algorithms are not all transpar-
ent about the reasons for their correctness.
As I show, in some cases, to establish that an algorithm correctly ful-
fi lled the task for which it was given, the commentators, on the one hand,
established another algorithm fulfi lling the same task and, on the other
hand, carried out operations on the text of this algorithm to transform it
into the proper algorithm, the correctness of which was to be established.
Moreover, in such cases the commentary usually made explicit the reasons
they adduced for explaining why, although the former algorithm was trans-
parent, the classic substituted the latter algorithm for it.
My chapter mainly focuses on the section of such a proof in which the
algorithm is reworked by means of transformations carried out on the list of
operations directly. My claim is that, within a context in which mathematics
was worked out on the basis of algorithms, this section of the proof repre-
sents a practice of algebraic proof.
By algebraic proof, I mean, in this context, a proof that starts from a
statement of equality, fi rst established in a given way that is not of interest
here and then transforms this original equality as such and in a valid way
into other equalities, until the desired equality is obtained. Th e fi rst part
of my claim is thus that the commentaries record proofs of precisely this
kind, with the only diff erence being that algorithms, and not equalities, are
57 Compare chapter A in CG2004, Chemla 2009.