Mathematical proof: a research programme 29
Archimedes’ view, this type of argument could not be conclusive and had
to be followed by another purely geometrical proof. Our explorations into
matters of proof will allow us to come back to this example below, from a
new perspective. Let us stress for now that diff erent kinds of reasoning have
diff erent kinds of value.
Furthermore, Lloyd stresses that in numerous domains of inquiry in
ancient Greece, there were debates about the value of their starting points
or the proper methods to follow, and securing conviction was a key issue.
Keeping too narrow a focus on mathematics in this respect conceals impor-
tant phenomena. Here two points are worth emphasizing.
Firstly, within this extended framework, it appears that proofs carried
out according to an axiomatic–deductive pattern were developed in several
areas and were by no means confi ned to mathematics, although even in
antiquity, geometry came to be perceived as a singular fi eld in this respect.
Th e recurring use of such a practice of proof echoes the variety of terms
used throughout the sources to demand ‘irrefutable’ arguments. One is
hence led to wonder how far, as regards ancient Greece, the history of an
axiomatic–deductive practice can be conducted while remaining within the
history of mathematics, or to what extent the interpretation of this practice
can be based only on mathematical sources. Here too, we encounter the
impact of a form of anachronism. Since this kind of proof is at the present
day deemed to be essential to, and even characteristic of, mathematics,
historiography has approached the question of axiomatic–deductive proof
mainly from within the fi eld of mathematics, disregarding the fact that it
was employed much more widely in antiquity. What greater understanding
of such a practice of proof would a broad historical inquiry of proof more
geometrico yield? Th is is the issue at stake here.
Secondly, such an importance granted to one type of method and organi-
zation of knowledge cannot hide a much wider phenomenon which Lloyd
wants to emphasize: the numerous debates on the correct way of conduct-
ing an inquiry. We seem to have here an idiosyncrasy of ancient Greek writ-
ings, or at least among the writings that have been handed down to us. Th e
unique multiplicity of ‘second-order disputes’ evidenced in ‘most areas of
inquiry’ leads Lloyd to suggest a third expansion.
Lloyd grants that disputes between practitioners of mathematics or
other domains of inquiry are a widespread phenomenon worldwide in the
ancient world. However, his comparison of such debates, in ancient Greece
and elsewhere, leads him to an important observation, namely, that the
modes of settling debates in various collectives appear to diff er. Lloyd thus
invites us to consider engaging in a discussion on the standards of proof in