60 karine chemla
into the nature of algebraic proofs, on the one hand, and the process of their
emergence, on the other? If a historical link can be established between the
two, what evidence can we fi nd regarding the historical process by which
both kinds of proof were connected? Th is question opens onto another one,
much more general: through what concrete historical processes did alge-
braic proof take shape and develop?
Th e analyses developed in this introduction have brought to light several
elements inherent to that kind of proof as we experience it: textual tech-
niques, refl ections on numbers and problems of generality. What other
elements constitute algebraic proof and how did this cluster crystallize?
What type of historicity is attached to it? Th is book off ers a contribution to
this agenda by identifying elements essential to algebraic proof and hypoth-
esizing a historical scenario regarding the kinds of practice in which these
elements took shape. Clearly, much more remains to be done.
Th ese fi rst results show the benefi ts that broadening of the scope of
sources taken into consideration can produce through the change of
perspective we advocated in the approach to proofs and their history. A
scarcely considered branch of the history of proof thus emerges: namely,
the history of proving the correctness of algorithms. And as it takes shape, it
elucidates parts of the history of proof that still await better understanding.
In correlation with opening new pages in the history of proof, we have been
naturally led to approach the topic of proof more comprehensively. From
this global perspective, we understand more clearly the link between the
devaluation of computation as a mathematical activity, which was and still
is quite widespread, and the exclusive focus on only some proofs, written
in ancient Greece, that has dominated the history of mathematics. Now,
what changes will this outline of the history of proving the correctness of
algorithms bring in the history of proof? How far will these tools of analysis
allow historians to examine anew other proofs, for instance proofs written
in Greek? Th ese remain open questions.
Our exploration of ancient practices of proof has met with another
important issue, which is worth pondering further. As suggested by Lloyd,
Høyrup, Keller and Volkov among others, the interest in proof and, more
specifi cally, in writing proofs down has been stimulated by distinct activi-
ties and social contexts. Among those activities and contexts we have seen,
let us mention the rivalry between competing schools of thought or the
development and promotion of one tradition as opposed to another,
teaching mathematics or interpreting a classic, all activities that need not
have been exclusive of each other. Th e list is by no means exhaustive. Still,
this remark brings to the fore two points that are important much more