Mathematical proof: a research programme 13
Th ey worked independently of each other and the proofs they discussed
were quite diff erent in nature. Moreover, their interpretation of the facts
confronting them was not uniform. However, they brought forward exten-
sive evidence, partly new, partly old, which challenged the received view of
the early history of mathematical proof. It is interesting to note that, in a way,
they were partly returning to a past historiography.
A puzzling fact is that, beyond the strict circle of specialists in the same
domain, these results were at best ignored, but, more frequently, were
rejected outright. Clearly, these publications have so far not managed to
bring about any change in the view of the early history of mathematical
proof held by historians and philosophers of science at large, or the wider
population.
Th is sustained failed reception needed to be analysed. Th us, this book is
not only devoted to the history but also contains a section on the histori-
ography of mathematical proof. Needless to say, much more remains to be
done in this domain. Th ese circumstances also explain why I chose to begin
this introduction with historiographical remarks. Some further factors are
at play in how mathematical proof is approached in our societies at large,
and we need to recognize these factors in order to restore some freedom to
the discussion and come to grips with the new evidence.
On the basis of the analysis outlined above, we see two types of obstacles
which could hinder the development of the discussion. Firstly, the whole
question of mathematical proof is entangled with extrascholarly uses in
which it plays an important part – among these uses are those of the issues
addressed earlier which are related to claims of identity. 19 Additionally, and
in relation to this point, an image of what a mathematical proof endeavours
has crystallized and blurs the analysis. My claim is that this image is biased
and that dealing with the new evidence mentioned above presents an
opportunity for us to locate this distortion and to think about mathematical
proof anew.
We have reached the crux of the argument. Let me explain in greater
detail. Th e essential value usually attached to mathematical proof – topmost
for its wide cultivation and esteem outside the sphere of mathematics – is
that, as the word ‘proof ’ itself indicates, it yields certainty: the conclusion
which has been proved can (hopefully) be accepted as true. 20 Securing the
19 How social groups construct identity through science or history of science is more generally a
key issue, on which much more research ought to be done.
20 Grabiner 1988 argues that certainty and applicability were the two features through which
mathematics was most infl uential to ‘Western thought’. Certainly, these two features occupy
a prominent position in Xu Guangqi’s preface to the Chinese translation of Euclid’s Elements
(Engelfriet 1998 : 291–7). Grabiner’s analysis of how the certainty yielded by proof was
infl uential, especially in theology, reveals dimensions of the importance regularly attached