Mathematical proof: a research programme 41
view, the proofs he reads in the formulation of the procedure intend to
guarantee an understanding of the reasons why the operations should
be carried out. He even suggests they are proofs precisely because they
have this goal. We see how the exclusive focus on the function of proof as
yielding certainty would leave out these sources as irrelevant for a history
of proof. However, these texts demonstrate that one motivating interest
in proofs and their transcription in one way or another may have been
not only – or perhaps not at all – to convince someone of the truth of a
statement but to make one understand the statement. Th is is still a strong
motivation for mathematicians today, as is evidenced, for example, by the
debate analysed in Section ii and it has been so all through the history of
mathematics.
Let us pause for a while to consider the goal of ‘understanding’ within
the context of a practice of proof intended to establish the correctness of
algorithms. Far from being the fi nal point of the analysis, it is in fact only
its beginning. Th e possibility that some proofs aim at providing an ‘under-
standing’ raises an essential question, for which the Babylonian case allows
further inquiry: what techniques or dispositifs were devised to provide an
‘understanding’ of the algorithms in the milieus of scribes? By Høyrup’s
restoration, geometrical diagrams seem to have supported the procedure.
Moreover, these diagrams were made in a way which allowed material
transformations of their shape. Th e specifi c terms which prescribed the
operations designated such material transformations which helped make
sense of the computations. Th e arguments supporting this hypothesis
come from a close analysis of the terms used to write down the algorithm.
However, this conclusion would have remained only speculation, had not
Høyrup discovered some texts from Susa that make explicit the kind of
training required by such a mode of understanding.
Th ese texts are revealing for several reasons. Th e explanations in them
that produce the ‘understanding’ are developed very specifi cally within
the framework of paradigmatic situations similar to those described in
the outline of some geometrical problems. We hold that these explana-
tions reveal how the context of geometrical problems may have provided
situations as well as numerical values with which the understanding of
the eff ect of operations could be grasped. Th e texts from Susa also reveal
how diagrams with highly particular dimensions were used in the same
way. Th is parallel between geometrical fi gures and problems, as well as this
way of using geometrical problems, compellingly evokes the case of some
Chinese mathematical sources, about which two points can be established.
Firstly, the problems were not only a question to be addressed, but, as the