Mathematical proof: a research programme 31
truth of a proposition that such techniques could be grasped. Th e hypothe-
sis accounts for how the techniques brought to light took shape. It may also
account for one of the motivations at play in making proofs explicit and
writing them down. One can go one step further and speculate about why,
as far as we know, in ancient Greece the methods in question were neither
named, nor analysed in any second-order discussion. Th is point leads me
to a second hypothesis with respect to the text of a proof: were not some of
the proofs written down with the purpose of displaying a given technique
which they put into play? In that case, general techniques would have been
expressed through the proofs of particular propositions and thereby also
motivated the expression of these proofs in writing. In other words, some
proofs were to be read as a kind of paradigm , making a statement of more
general validity than a fi rst reading would indicate. Th e interpretation of the
texts of these proofs would be comparable in that respect to how a problem
and the procedure for solving it made sense in the Babylonian or Chinese
writings.^40 Whatever the case, the essential point here is that the text of a
proof was not read only as establishing a proposition, but also as a possible
source for working techniques. Moreover, the generality and importance of
a textual unit in these books would not lie only in the proposition itself, but
also in the technique brought into play in its proof.
Let us consider these various points one by one to grasp what is more
generally at stake here.
To begin with, the fi rst hypothesis formulated above suggests that readers
were likely to read a proof for itself and not merely for its capacity to estab-
lish the statement proved. Th ere is nothing surprising about this assump-
tion. Th e recent debate on which we commented in Section ii bore witness
to such uses of the text of proofs: some of these mathematicians testifi ed to
the fact that they read proofs, seeking, among other things, techniques and
also concepts. Th is constitutes a challenge for us: how are we, as historians,
to gather evidence in order to take this dimension of the interest in proof
into account more generally and rigorously? Interestingly enough, the
hypothesis on the practice of proof prompted by Saito’s suggestion echoes
with how, as we shall see, proofs of the correctness of algorithms were
conducted in the earliest extant Chinese sources attesting to practices of
proof.^41 In all these contexts, the proofs appear not to have been only means
40 On the latter, a discussion and bibliography can be found in Chemla 2009. Note that I
use paradigm in the sense that grammarians use this word. Also note that the text of a proof
could either state a general technique or document its existence by the fact of bringing it into
play.
41 See below and Chemla 1992 , Chemla 1997b.