Mathematical proof: a research programme 21
Klamroth, a historian who specialized in Arabic mathematics. Th e debate
concerned the role ascribed to the editions and translations into Arabic
and Latin carried out between the eighth and the thirteenth centuries – the
so-called ‘indirect tradition’ – in the making of the critical edition of the
Elements. Heiberg’s position was that the Greek manuscripts dating from
the ninth century onwards – the ‘direct tradition’ – were closer to Euclid’s
original text. In contrast, Klamroth argued that the Arabic and Latin wit-
nesses, less complete from a logical point of view, bore testimony to earlier
states of the text, whereas the Greek documents had already been contami-
nated by the various uses to which the text had been put in the centuries
between its composition by Euclid and the transliteration into minuscule
that took place in Byzantium. In brief, Heiberg was committed to the view
that Euclid’s Elements contained a minimum of logical gaps in the math-
ematical composition which it delineated. Th is supposition dictated the
choice of sources on which he based his edition and motivated his rejection
of other documents as derivative. Th is is how his selective treatment of the
written evidence contributed to reshaping Euclid closer to his own vision.
Taking up Klamroth’s thesis, Knorr held the opposite view: for him, the
Arabic and Latin witnesses were closer to the original Euclid, and the addi-
tions of logical steps were carried out by later editors of the Elements. Th e
consequence of the resurgence of the debate was clear: some textual doubts
were thereby raised regarding Euclid’s original formulation of his proofs.
In articulating a critical analysis of this kind regarding the nineteenth-
century edition of the Elements still widely used today for the fi rst time
since the publication of Heiberg’s volumes, Knorr launched a research
programme of tremendous importance to our topic. How much does our
perception of the practice of proof in the Elements depend on the choices
carried out by Heiberg? In other words, how far is his vision of Euclidean
proof, formed at the end of the nineteenth century, conveyed through the
text of his critical edition? Such are the fundamental questions raised. Th e
example illustrates clearly, I believe, a much more general problem, which
can be formulated as follows: how do critical editions aff ect the theses held
by historians of science and the transmission of this inheritance to the next
generations of scholars?
Th is general issue is to be kept in mind with respect to all the sources
mentioned in this volume. However, beyond providing the illustration of a
general diffi culty, the example of the Elements is in itself of specifi c impor-
tance for our topic. In fact, the problem it raises extends beyond the case of
the Elements , since soon aft er the publication of Knorr’s fi rst paper, a dif-
fi culty of the same kind became manifest with respect to Heiberg’s critical