Mathematical proof: a research programme 43
derive a result. Even though he discards this method as inappropriate for a
proof, as did a tradition of scholars who developed comparable proofs, the
question remains open for us to understand what this kind of interpretation
actually achieved for him.^49
Getting back to our Mesopotamian documents, I am aware that some
historians may question whether such modes of establishing the correct-
ness of algorithms ought to be considered proofs. Th e Babylonian source
material allows us to shed light on the diffi culty that this division would
entail in the history of mathematics. In fact, the techniques that the scribes
used to provide an ‘understanding’ of the type discussed above could be,
and appear to have been later on, taken up in other practices of proof –
where the qualifi cation as ‘proof ’ is less disputed. As Høyrup has suggested
in previous publications, there is a strong historical continuity between
the modes of argumentation alluded to above, which appear to have
been developed in Babylonian scribal milieus on the one hand and what
are explicitly recorded as proofs in Arabic algebraic texts from the ninth
century onwards on the other hand.^50 If only for this reason, these tech-
niques of ‘understanding’ do belong, in my view, to the history of mathe-
matical proof. Th e continuity evoked is of the same kind as that mentioned
above with respect to the textual techniques devised by Diophantus to
develop his arguments.
As a provisional conclusion, one may suggest that the text of a proof is
a technical text, the shaping of which may have benefi ted from all kinds of
resources available. Conversely, in some cases, the formation of a techni-
cal text for working out a kind of proof led to developing techniques that
could be brought to bear in other mathematical activities. In the case of
Babylonian tablets, not only the operations used in a procedure, but – as
is clearly shown by the Susa texts – also the transformations of algorithms,
49 In the same way, Krob 1991 has developed a proof of a combinatorial theorem based on an
interpretation involving a plate, beads and pebbles. Such features are unusual in mathematical
publications. Th ey occur more frequently in some fi elds, like combinatorics, than in others.
Th e reasons why it is so are worth exploring, since they shed light on social aspects of proving.
It is clear that precisely because of these features, not all mathematicians of the present day will
accept the proof Krob 1991 gives as a proof. Th is approach to the question, however, leaves
unanswered the questions which I fi nd quite fascinating: what does the interpretation do for
the reasoning? And why do practitioners fi nd it appealing to make use of such devices or
dispositifs of interpretation within proofs? Approaching the problem through the controversies
among mathematicians would yield interesting results.
50 See, for instance, Høyrup 1986. In his recent edition and translation into French of
al-Khwarizmi’s book on algebra, Rashed 2007 puts forward a diff erent hypothesis for the
history of these proofs, interpreting them rather as composed within the framework of
Euclidean geometry.