Mathematical proof: a research programme 59
composes a new type of text with which to carry out proofs, one that inte-
grates diff erent agendas. Most importantly, however, if we follow Tian’s
interpretation of it, we can read Li Rui’s discourse and practice as illustrat-
ing the politics of the proof, in that they attempt to embody the ideal of
proving in the ‘Chinese way’, and not in the ‘Western’ way. Some decades
later, the politics of the historiography of mathematical proof would
become by far more visible.
IV Conclusion: a research programme
on mathematical proofs
It is time to gather the various threads that we have followed and conclude,
by considering our fi ndings with respect to ancient mathematics and the
research programme that they open for us.
Let us begin with facts. What we have seen emerging in Section iii is the
outline of a history of proving the correctness of algorithms in the ancient
world. Mesopotamian, Chinese and Indian sources bear witness to the fact
that practitioners have attended to the correctness of the algorithms with
which they have practised mathematics. An analysis of their attempts helps
us identify some of the fundamental operations involved in such proofs.
We have seen that these practitioners have striven to establish how an
algorithm correctly yields the desired magnitude and the value that can be
attached to it. To do so, they have designed devices or dispositifs that have
allowed them to formulate the ‘meaning’ of operations. Th e proofs they
constructed share common features. Th ey also demonstrate specifi cities in
the way in which proof was practised.
Among the specifi cities noted in the way of approaching the correctness
of algorithms, one fact proved of special relevance for a history of proving.
Chinese sources demonstrate the fact that operations – meta-operations,
if one wishes – were sometimes applied to the sequence of operations that
an algorithm constitutes. Th ese meta-operations were used to transform
an algorithm known to be true, qua algorithm, into another algorithm, the
correctness of which was to be established. Moreover, these sources bear
witness to the fact that a connection was established between the validity
of these meta-operations and the numbers with which one worked. I sug-
gested the conclusion that we have here a kind of ‘algebraic proof within an
algorithmic context’.
Th is remark leads to several questions. What kind of understanding can
practices developed specifi cally to prove the correctness of algorithms yield