The History of Mathematical Proof in Ancient Traditions

28 karine chemla

historiographic ideas were formed, the values which past historians had

deemed central, the agendas they adopted and the critical editions they

produced. We are now brought back to the new agenda we suggested above:

what can be gained by widening our perspective on the practices of proof

while considering a richer collection of sources?

III Broader perspectives on the history of proof

Widening our perspective on Greek texts: epistemological values

and goals attached to proof

Th e biases in the history of proof on which the foregoing analysis

shed light fi rst coloured the treatment of the source material written

in Greek. Historical approaches to proof in ancient Greece have so far

concentrated mainly on a restricted corpus of texts and have limited the

issues addressed. Th e ensuing account was accordingly confi ned in its

scope and left wide ranges of evidence overlooked. Some of the chapters

in this book deal precisely with part of this evidence. To begin with,

Geoff rey Lloyd’s chapter indicates some of the benefi ts that could be

derived from a radical broadening of the corpus of Greek proofs under

consideration. In particular, he discusses some of the new questions that

emerge from this extended context, with regard to the practices of proof

in ancient Greece.

Lloyd fi rst reminds us of the fact that, despite the importance histo-

riography granted to Euclid’s Elements and cognate geometrical texts,

mathematical arguments in ancient Greece were by no means restricted

to proofs of the type that this corpus embodies. As Lloyd illustrates by

means of examples, Greek sources on mathematical sciences provide

ample evidence of other forms of argument as well as discussions on the

relative value of proofs. 38 Enlarging the set of sources under consideration

thus opens a space in which the various practices of proof and the values

attached to them become an object of historical inquiry. Some of these

sources bear witness to the fact that some authors found it important to

use various modes of reasoning. Lloyd recalls the case of Archimedes, who

expounds in Th e Method why it is fruitful to consider a fi gure as composed

of indivisibles and to interpret it in a mechanical way in order to yield the

result sought for. However, as Lloyd insists, although Archimedes deemed

such reasoning essential to the discovery of the result to be proved, in

38 Lloyd has made this point on other domains of inquiry; compare for instance Lloyd 1996b.