62 karine chemla
beginning. Indeed, the main question then raised is to identify how the
way in which the proof is conducted or written down helps practitioners
to reach the goals, achieve the values or implement the qualities they value.
Th is is where the issue of the practices of proof is inextricably linked to the
issue of the expectations actors have with respect to proofs. In relation to
this issue, I introduced the notion of devices or dispositifs that actors have
created in various contexts to carry out key operations with respect to
the proof. We have seen that the dispositifs constructed by Mesopotamian
scribes or Chinese scholars to make explicit the ‘meaning’ of operations in
an algorithm had commonalities as well as specifi c diff erences. Th e diff er-
ences between the two Greek texts dealing with polygonal numbers that
Mueller described can also be approached in these terms: the dispositifs
used by the authors to treat their topic show two distinct attempts at achiev-
ing generality. Seen in this light, axiomatic–deductive systems appear to be
a dispositif designed to yield certainty. Describing these dispositifs appears
to me as a method to attend more closely to diff erences between the various
practices of proof, thereby breaking down what is all too oft en presented
collectively as ‘the mathematical practice’.
Can we spot transformations in the modalities of proof that demonstrate
a change of values or a combination of a larger set of values? Which of these
goals, of these values, of these qualities were held together? Which combi-
nations can we identify and how have these various constraints been held
together? Which of them seem to have been in tension with each other,
because they were diffi cult to fulfi l simultaneously? Archimedes’ practices
of proof off er a case study that can be approached from this perspective.
All the questions that arise in this context now explain, I hope, how an
overly strict focus on the value of certainty would yield an essentially trun-
cated account of mathematical proof. Clearly, such an approach does not
do justice to the variety of agendas that were ascribed to proof and to the
variety of practices that were developed accordingly.
When describing the diverse practices of proof exhibited in ancient
sources, the various chapters of the book collectively bring to the fore
another fact that is, in my view, both important and of general relevance.
Th ey converge on the conclusion that various types of technical texts have
been designed for the conduct of proofs, depending on the context in which
these proofs have been written down and the constraints bearing on them.
Let me gather various hints that support this conclusion.
Th e texts of proofs we have mentioned consist of distinct basic com-
ponents. Among them, one can list equalities, proportions or lists of