18 karine chemla
New perspectives, or: the project of the book
From this vantage point, two conclusions can be discerned.
Firstly, we see how a history of proof limited to inquiry into how practi-
tioners devised the means of establishing a statement in an incontrovertible
way runs the risk of being truncated. Th is, in my view, is what happens
when the Babylonian, Chinese and Indian evidence is left out.
Secondly, and conversely, the outline sketched above suggests another
kind of programme for a history of mathematical proof, one likely to be
more open and allow us to derive benefi ts from the multiplicity of our
sources. We may be interested in understanding the aims pursued by
diff erent collectives of practitioners in the past when they manifested
an interest in the reasons why a statement was true or an algorithm was
correct. We may also wonder how they shaped the practices of proof in
relation to the aims they pursued and how they left written evidence of
these practices. 32
In fact, some of these other functions associated with proof were explic-
itly identifi ed in the past and they were at times perceived as more impor-
tant than assuring certainty. In relation to this, epistemological values
distinct from that of incontrovertibility have been used to assess proofs. In
this respect, one can recall the seventeenth-century debates about how to
secure increased clarity through mathematical proofs, thereby achieving
conviction and understanding. Seen in this light, the versions of Euclid’s
Elements of the past were not much prized, and new kinds of Elements were
composed to fulfi l more adequately the new requirements demanded from
mathematical proof. 33 Th is example illustrates how diff erent types of proof
were created in relation to diff erent agendas for proving.
How would such a programme translate with respect to ancient tradi-
tions? Th is is the inquiry of the present book, as one step towards opening
a wider space for a historical and epistemological investigation into math-
ematical proof.
Th e book is mainly – we shall see why ‘mainly’ shortly – devoted to the
earliest known proofs in mathematics. By the term ‘proof ’, it should be now
clear why we simply mean texts in which the ambition of accounting for the
truth of an assertion or the correctness of an algorithm can be identifi ed as
one of the actors’ intentions. In other words, we do not restrict our corpus32 Th is analysis and this programme develop the suggestion I formulated in Chemla 1997b :
229–31.
33 On this question, see chapter 4 , ‘L’interprétation d’Euclide chez Pascal et Arnauld’, in Gardies
1984 : 85–108.