Mathematical proof: a research programme 17
In connection with this issue, and to return to the question whether cer-
tainty is the main motivation for looking for proofs today, it is interesting to
note that many responses to the original paper by Jaff e and Quinn manifest
a concern that too strict a control in order to assure certainty could entail
losses for the discipline. By contrast, the debate also allows one to observe
how many diff erent functions and expectations mathematicians attach to
proof: bringing ‘clarity and reliability’; providing ‘feedback and corrections’,
‘new insights and unexpected new data’ (Jaff e et al. 1993), ‘clues to new and
unexpected phenomena’ (Jaff e et al. 1994), ‘ideas and techniques’ (Atiyah
et al. 1994 ), ‘understanding’,^28 ‘mathematical concepts which are quite inter-
esting in themselves, and lead to further mathematics’; ‘helping support of
certain vision for the structure of ’ a mathematical object (Th urston 1994).^29
Only with this variety of objectives in mind can we account for some oth-
erwise mysterious practices. For instance, how else could we explain why
rewriting a proof for already well-established statements can be fruitful? 30
Restricting ourselves to consideration of proof in the more limited domain
of mathematics brought to light a wealth of reasons which motivated the
writing of proofs for mathematicians. 31 Moreover, it suggests the great loss
for the historical inquiry on mathematical proof if these proofs, the values
attached to them, and the motivations to formulate them and write them
down were not considered.
28 A comment by Martin Davis on the four-colour theorem nicely illustrates this point: the
problem with the computer proof, in his view, is not so much the lack of certainty it entails,
but the fact that it does not put us in a position to understand where the ‘4’ comes from, and
whether it is accidental or not (Martin Davis, 2 October 2007, personal communication).
(^29) As I alluded to it above, rigour is a contested value in these pages (see the contributions by
Mandelbrot, Th om). What is more, it must be stressed that in contemporary mathematics,
as it may have been the case for the Aristotle of the Posterior Analytics , the value attached to
rigour is perhaps linked more to the understanding and additional insights it provides than
to the increased certainty it yields. Hilbert 1900 , for example, testifi es to the idea that rigour
yields fruitfulness and provides a guide to determine the importance of a problem (in the
English translation: Hilbert 1902 : 441). However, as Rav 1999 stresses, even when proofs are
wrong or inadequate, they remain the main source from which new concepts emerge and
new theories are developed. He further suggests that it is in proofs, rather than in theorems,
that mathematicians look for mathematical knowledge and understanding: ‘Conceptual and
methodological innovations are inextricably bound to the search for and the discovery of
proofs, thereby establishing links between theories, systematizing knowledge and spurring
further developments.’ (Rav 1999 : 6).
30 Th is point was stressed in Chemla 1992 , which relies on how Rota 1990 had discussed the
issue.
31 Some historians have attempted to widen the history of proof by suggesting that the
actors of the past used various means to convince their peers of the truth of a statement.
In this vein attention has been paid to the rhetorical means that the actors employed.
Th e preceding remarks show why this does not help frame a wide enough perspective
on the activity of proof.