Mathematical proof: a research programme 15
To examine this question, let us restrict the discussion to mathemati-
cal proof as such, as carried out within the context of mathematics. Th e
recollection of a simple fact will prove useful here: many mathematical
proofs produced throughout history by duly acknowledged scholars were
not presented within axiomatic–deductive systems. 22 In fact, the periods
during which advanced mathematical writings were predominantly
composed in such a way are much shorter than the periods when they
were not. In tandem with the lack of interest in an axiomatic–deductive
organization of mathematical knowledge, the authors oft en did not place
much emphasis on rigour. Yet they referred to what they wrote as proofs. 23
One may argue that these practitioners of mathematics overlooked
some diffi culties and made errors. But these objections cannot possibly
obliterate the innumerable theories proposed and results obtained with
precisely such types of proof. Th ese remarks have an inescapable conse-
quence: it reveals that for a fair number of practitioners of mathematics
the goals of proof cannot have been only ascertaining incontrovertibility
and assuring certainty through achieving conviction, if such was ever their
goal at all. Nevertheless, they considered it worthwhile to look for proofs,
and their practice of proof was no less productive from a mathematical
point of view.
In my view, this perception of proof still holds true today. Even though,
in their discourse on the contemporary practices of proof, mathemati-
cians may stress the axiomatic–deductive framework within which they
work and emphasize the certainty yielded by proofs as well as the rigour
necessary in their production, 24 the functions they ascribe to proof in their
22 Ironically enough, the proof that lies at the core of Plato’s Meno and that has exerted a huge
infl uence in the history of philosophy (Hacking 2000 ) is not formulated within an axiomatic–
deductive system. Philosophers of the present day such as Lakatos 1970 held ‘a no-foundation
view of mathematics’ (Hacking 2000 : 124). Unfortunately such views have not yet shown any
clear impact on the history of ancient mathematics. Rav 1999 : 15–19 lists several examples of
major domains of mathematics of the present day, for which axioms have not been proposed
and that are nevertheless felt to be rigorous. He further emphasizes the various meanings of
‘axioms’ as used in modern practice.
23 I am not aware of any historical publication which denies that Leibniz, Euler, Poncelet,
Poincaré or others of their ilk wrote down actual proofs and suggests that these men should
be erased from the history of mathematical proof: whatever the evaluation may be, it is
without contest that they contributed to shaping practices of proof. More revealing examples
are discussed in Jaff e and Quinn 1993 : 7–8. Th e fact that Jaff e and Quinn refer to cases of
‘weak standards of proof ’ and suggest that, in some cases, ‘expressions such as “motivation”
or “supporting argument” should replace “proof ”’ in actors’ language indicates that in the
contemporary mathematical literature the label ‘proof ’ refers to a great variety of types of
arguments (Jaff e and Quinn 1993 : 7, 10). Th is topic recurs below.
24 See the very diff erent and lucid account in Th urston 1994 : 10–11. Among other refreshing
insights into the activity of proof, Th urston rejects the ‘hidden assumption that there is