Reading proofs in Chinese commentaries 425
were driving the inquiry into mathematics in ancient China. For instance,
they reveal that generality was a key theoretical value and that fi nding out
the most general operations was an aim pursued by the practitioners of
mathematics.^6 However, a crucially important fact for us lies elsewhere:
aft er the description of virtually every algorithm presented in Th e Nine
Chapters , or between the sentences prescribing its successive operations,
the commentators set out to prove its correctness. Th ese texts thus provide
the earliest evidence available today regarding the practice of mathematical
proof in ancient China, and this is the reason why, in this chapter, we shall
concentrate on them.
In contrast to what can be found in ancient Greek geometrical sources,
where statements are proved to be true, the Chinese commentators system-
atically strove to establish the correctness of algorithms. 7 It can hence be
assumed that the commentaries bear witness to a practice of mathematical
proof that, as a practice, developed independently from what early Greek
sources demonstrate. However, we shall not dwell on this issue here. Instead,
and as a prerequisite to tackling this question in the future, we shall aim at
better understanding this practice of proof. Th ereby, we may hope to cast
light more generally on some of the fundamental operations required when
proving the correctness of algorithms – a section of the history of mathemat-
ical proof that, to my knowledge, has been so far almost entirely neglected.
Even though it constitutes an oversimplifi cation to be refi ned later, let
us say, for the present, that an algorithm consists of a list of operations that
can be applied to some data in order to yield a desired magnitude. In this
context, proving that such an algorithm is correct involves establishing that
the obtained result corresponds to the desired magnitude. It can be shown
that, when fulfi lling this task, the commentators systematically made use of
some key operations. Moreover, they employed specialized terms to refer
to concepts related to these operations. 8 Th ese facts disclose that, far from
being ad-hoc developments, these proofs complied with norms familiar to
the actors, since they devised technical terms related to them. Th e way in
6 Chemla 2003 establishes these points. Below, we shall fi nd additional evidence supporting
these theses.
7 It can be shown that this is how the commentators themselves conceive of the aim of their
reasonings. See Chapter A in CG2004: 26–8. I do not come back to this point here. Note that
the commentators leave some of the most basic algorithms without proof. Guo 1992 : 301–20
stressed this fact, emphasizing that this feature meant that the commentators were shaping an
architecture of algorithms, the proofs of which depended on algorithms proved previously.
From another angle, one can argue that reduction to fundamental algorithms, and not to
simple problems, is also a key point at stake in the proofs carried out by the commentators.
8 Chapter A of CG2004: 26–39 sketches these points.