Reading proofs in Chinese commentaries 481
We saw how proofs relied on algorithms, which had already been estab-
lished as correct, and how proofs articulated these algorithms as a basis
for establishing the correctness of other procedures. Most importantly,
the algorithms, together with the situations in relation to which they were
introduced, provided means for determining the ‘meaning’ of an operation
or a sequence of operations. Th is appears to be a key act for proving the cor-
rectness of algorithms, and it is noteworthy that a term ( yi ‘meaning’) seems
to have been specialized to designate it in ancient China.
Furthermore, as was stressed above on several occasions, the evidence
provided by the commentaries seems to manifest a link – perhaps specifi c
to ancient China – between the way in which the proof of the correctness
of algorithms was conducted and a systematic interest in the dimension of
generality of the situations and concepts encountered. 78 Th e fact that proofs
oft en relate to each other, as we emphasized several times, can be correlated
to this specifi city. However, it will be only when historical studies of such
proofs develop that we will be in a reasonable position to conclude whether
this feature is characteristic of Chinese sources or intrinsic to proving the
correctness of algorithms in general.
Finally, the second key operation in the activity of proving the correct-
ness of algorithms that is documented in ancient China, and on which we
focused in this chapter, was what I called the ‘algebraic proof in an algorith-
mic context’. So far, I can locate it only in ancient Chinese source material,
as far as ancient mathematical traditions are concerned. But again this
conclusion may have to be revised in the future. Again, whatever the case
may be, what can we learn from this occurrence regarding algebraic proof
in general?
If we recapitulate our analysis in this respect, we have seen that several
technical terms were introduced in relation to this dimension of proof: fu
‘restoring’, huan ‘backtracking’,^79 baochu ‘dividing in return’, 80 fan ‘inverting’,
78 I have dealt with this issue on several occasions, from Chemla 1991 onwards. However, given
the complexity of this link, I cannot fully discuss it within the framework of this chapter. I plan
to revisit the issue in another publication that would be entirely devoted to it. Note, however,
that, here again, the commentators introduced a technical term in relation to this facet of the
problem. In my glossary, I transcribed it as yi’ ‘meaning, signifi cation’, to distinguish it from yi ,
and the reader will fi nd in these two entries partial discussion of the problem. Yi’ designates a
‘meaning’ that captures the fundamental procedures that proofs disclose to be at stake within
each algorithm dealt with.
79 A variant for this operation is huan yuan ‘return to the origin’. On all these terms, the reader is
referred to my glossary in CG2004.
80 A variant for this concept is the pair of terms ru ‘enter’/ chu ‘go out’. See the glossary in
CG2004.