Mathematical proof: a research programme 47
application to the result. On that count, her conclusion is that the overall
structure of the text makes a statement regarding the fact that the algorithm
for computing reciprocals is its own reverse algorithm. Similar tablets can
be found for square-root extractions, displaying that squaring and square-
root extraction are in the same way the reverse of one another. A similar
interest in algorithms that are the reverse of one another – where one algo-
rithm cancels the eff ect of the other – emerges as central to a type of proof
to which Chinese early mathematical sources bear witness. 52 It is to this
type of proof that my own chapter is devoted.
Algebraic proofs in an algorithmic context
Like some of the Babylonian tablets analysed above, the earliest Chinese
writings attesting to mathematical activity stricto sensu are composed of
problems and algorithms solving them. Th e practice of proof to which they
bear witness also aims at establishing the correctness of algorithms.
Among these writings, those that were handed down through the
written tradition are of a type quite diff erent from that of the Babylonian
tablets just examined. 53 Th e most important one for our purpose, Th e Nine
Chapters on Mathematical Procedures ( Jiuzhang suanshu ), was probably
completed in the fi rst century ce and considered a ‘classic’ soon thereaft er.
In correlation with this adoption, commentaries on it were composed,
some of which were felt to be so essential to the reading of Th e Nine
Chapters that they were handed down with it. Th ese are the commentary
composed by Liu Hui and completed in 263 as well as the one written
under the supervision of Li Chunfeng and presented to the throne in 656.
Two key facts regarding the commentaries prove essential for us in relation
to mathematical proof.
First of all, the commentaries attest to how ancient readers approached
the classic as such. Th is highlights why, as historians, when we interpret
Th e Nine Chapters , we are in quite a diff erent situation from that confront-
ing historians who deal with sources for which no ancient commentary
53 In addition to the source material handed down through the written tradition, we now
have recourse to writings that archaeologists excavated from tombs. Th e most important
of them, the Book of Mathematical Procedures ( Suanshushu ), found in a tomb sealed in c.
186 bce , is useful for, but not central to, our purpose. Such sources can be compared to the
Babylonian tablets with respect to the way in which they were found and the conditions in
which we can interpret them. However, it is not yet clear within which milieus and how they
were used.
52 Chemla 1997 –8.