Mathematical proof: a research programme 61
generally. On the one hand, proving is an activity that takes place in specifi c
social and professional groups which have specifi c agendas. On the other
hand, as we saw, the practices of proof betray a variety of modalities which
one can attempt to correlate to the social groups which sustain them. Th is
leaves us with two tasks: fi nding the means to describe the practices in their
variety and identifying the social and professional contexts that are relevant
to account for their formation and relative stability.
Such a research programme is quite meaningful to inquire into the
history of proof in the ancient world. Indeed, only along these lines can
we hope to bring to light and accommodate the variety of practices in a
way more satisfactory than the old model of competing civilizations which
has been pre-eminent from the nineteenth century onwards. However,
the research programme is laden with diffi culties. Th e evidence available
with respect to ancient time periods is in general so scanty that rigorously
reconstructing the social environment in which proofs were actually com-
posed is an ideal for the most part out of reach. One can only put forward
hypotheses. In that context, concentrating on the description of the varying
practices appears to be an initial means of overcoming the diffi culties and
perhaps discerning from mathematical sources diff erent social groups that
carried out the practice of proof.
Th is is the project on which we focus in the book and what our explo-
rations into matters of proof open to refl ections of wider relevance. Th e
conclusions which we propose bring forth some suggestions for the task of
describing practices of proof whose value appears to me to exceed the scope
of the ancient world to which we have restricted ourselves. Let me comment
on some of these suggestions by way of conclusion.
Among the various sets of sources which they treat, the chapters in this
book identify diff erent goals ascribed to proof, diff erent values attached to
proving and diff erent qualities required from a proof. In this Introduction, I
have outlined some of them. We have seen that some proofs seem to be con-
ducted in order to understand the statement proved or the text which states
it. In other cases, proofs have appeared to have had as one of their goals the
identifi cation of fundamental operations or the display of a technique. We
have also seen that in some contexts, proofs were expected to be general or
to comply with an ideal of generality. In others, they should bring clarity,
yield fruitfulness or manifest simplicity. Much more remains to be done in
identifying goals and values practitioners have attached – and still attach
today – to proof and the constraints they imposed on themselves.
What is important is that in each of these cases the identifi cation of
these elements, far from being the end of the inquiry, constitutes only its