Mathematical proof: a research programme 63
operations. Moreover, within the context of distinct practices of proof,
these basic components appear to have been composed in various ways
and to have been combined in distinct kinds of technical texts. Among the
kinds of texts and inscriptions we encountered, let me recall a few: texts
for algorithms transparent with respect to the reasons of their correctness;
the material dispositifs by means of which their meaning was made explicit;
symbolic inscriptions of diff erent sorts (including those of Diophantus,
those which Colebrooke fi rst described in the Sanskrit texts, and those of
the Chinese past revivifi ed by Li Rui); and texts composed with formulaic
languages. In addition, it regularly appeared that paradigms in the form of
particular fi gures or mathematical problems were used to formulate general
proofs.
Th is variety of texts developed for proofs merely refl ects the variety of
contexts within which proofs were carried out. Th is means that the design
of texts is, in an important sense, an indicator of the context in which they
were composed. Moreover, the shaping of kinds of texts to carry out proofs
is an aspect of the practice of proof as such which has been little studied so
far. Th is shaping demands study, even if only as a limited component of the
practice of proof. However, there is another equally fundamental reason to
study this range of phenomena.
Th e examples just summarized remind us of the fact that the interpreta-
tion of the text of a proof is a thorny issue, and it is so in relation to the eff ort
involved in the construction of a kind of text adequate for the execution of
proofs of a certain type. In other words, it is because each human collec-
tive which carried out mathematical proofs deliberately designed texts for
this activity that these texts cannot be interpreted straightforwardly. 66 Th is
claim can be illustrated easily with the example of the recently mentioned
transparent algorithms. In order to read a proof in the statement of the
algorithm itself, the historian has to establish the way in which the texts
made sense. Th e interpretation of paradigms as paradigms would constitute
another example.
Th ese remarks explain why the relation between the type of text used
and the kind of proof developed is an essential topic for future research. It
is essential not only because the shaping of texts to carry out proofs is an
aspect of the practice of proof in itself, but also because inquiring into this
issue yields better tools to interpret the texts in question.
66 In this respect, we return to the conclusions that emerged from the collective eff ort published
in Chemla 2004.