30 karine chemla
order to conclusively resolve debates as a social phenomenon. Th e new and
important research programme which he proposes intends to account for
the development of such attempts to adjudicate debates in social terms. In
other words, Lloyd calls for developing a social account of the emergence of
second-order discussions on proof.
Th ese suggestions show how, by concentrating on a set of texts wider than
the usual geometrical writings, one can defi ne new horizons for research on
proof in Greek sources. In recent years, though, new approaches to proofs
in the writings that provided the standard historiography with its basis have
taken shape. Th ese approaches are interesting for us, since they have brought
to the fore epistemological values other than being conducive to truth which,
as far as our sources can tell, may have been attached to proof, thereby side-
lining the issue of certainty that has dominated the discussion on ancient
proofs. To mention but one example, I shall show how, in my view, Ken Saito
has advocated a new way of interpreting proofs in the core corpus.
Saito takes as his starting point the thesis that, when one considers
this collection of texts as a whole , there emerges from the corpus a set of
‘elementary techniques’ that form a ‘tool-box’ on which Greek geometers
relied. 39 Moreover, he argues that the practitioners developed knowledge of
how to combine the elements in the tool-box in standard and locally valid
methods – combinations which he also calls ‘techniques’, or ‘patterns of
argument’. In Saito’s view, the ‘method of exhaustion’, which was named and
discussed as such only in the seventeenth century, constitutes an example
of such a method. His approach not only yields an analysis of the method as
a specifi c sequence of elements taken from the tool-box, but it also embeds
a technique that has been long recognized into a larger set of similar tech-
niques which recur in proofs. What is worth stressing is his remark that, for
reasons yet unknown, these methods do not seem to have been described
at a meta-mathematical level or even named at the time. Nevertheless, the
sources bear witness to patterns of proof which circulate between proofs
and to the stabilization of a form of knowledge about them.
An initial hypothesis can be formulated with respect to these methods:
it is by reading the text of a proof per se and not merely as establishing the39 Th e insight about the ‘tool-box’ was introduced and worked out by Saito from 1994 on (see
Saito and Tassora 1998 and http://www.greekmath.org/diagram/ ). It was further developed in
N1999: 216–35. Th e latter book fi gures prominently among the publications that opened new
perspectives in the approach to deduction in the Greek mathematical texts of what I called the
‘core corpus’. I develop here refl ections on a tiny part of the new ideas that were introduced
in this wider context. Saito’s project on the Greek mathematical tool-box has not yet come to
completion. To present his ideas here, I rely on personal communication and on draft s that he
sent me in 2005 and which contain abstracts of part of his project.