14 karine chemla
truth of a piece of knowledge and convincing an opponent of the incontro-
vertibility of an assertion seem to be what mathematical proof off ers and
the ideal it embodies.
Clearly, if we adopt this view of proof, we are immediately forced to admit
that starting points (defi nitions, axioms) are mandatory for the activity of
proof, if we are to achieve these goals. Moreover, the validity of these start-
ing points must be agreed upon, regardless of how this agreement is reached.
In his chapter, Geoff rey Lloyd treats at length the variety of terms used to
designate these starting points in ancient Greece and the intensity of interest
in, and debate about, them that this variety refl ects. On this basis, and this is
where requirements such as rigour appear to come in, valid arguments are
required to derive assertions from the starting points in a trustworthy way,
and new assertions depend on the fi rst ones or the starting points, and so on.
In other words, as soon as one has granted the premise that the goal of
mathematical proof is to prove in an indisputable way, then the conclu-
sion follows unavoidably that this aim can be only achieved within the
framework of an axiomatic–deductive system of one sort or another. In the
context of this assumption, Euclid’s Elements is the fi rst known mathemati-
cal writing that contains proofs, and any claim that a given source contains
proofs has to be judged accordingly. And such claims have actually been
judged by that very standard.
Th is is, in my view, the simple device by which Greek geometrical writings
have become so central to the discussion of proof that they cannot possibly
be challenged, and this position lies at the core of the recent rejection of the
claim that Babylonian, Chinese or Indian sources contained proofs by some
part of the community of history and philosophy of science (among others).
Th e reasoning will look simplistic to many. However, I claim that this is pre-
cisely the core of the matter. 21 If I am right, this is the point on which critical
analysis must be exercised for us to open our historical inquiry into proof
wider. Th e feature of mathematical proof just examined is certainly quite
meaningful, and was indeed deemed so outside mathematics. However, on
what basis do we grant ‘incontrovertibility’ as the essential value and goal of
mathematical proof within mathematics itself?21 I formulated the reasoning relying on present-day perception of what yields certainty.
Although certainty, starting points and modes of reasonings based on the latter to secure
the former remained a stable constellation of elements in the history of discussions about
mathematical proof, the meanings and contents attached to them displayed variation in
history. As Orna Harari shows in her chapter in this book, earlier views were quite diff erent
from present-day ones. Compare Mancosu 1996 , especially chapter 1.to this value. Hacking 2000 is a bright analysis of what certainty and its cognate values have
meant for some philosophers.