Mathematical proof: a research programme 27
commonplace only in modern times. Th is brings us back to the issue of the
part played in our story by the philosophy of science as it took shape as a
discipline in the nineteenth century.
To make this point, Harari digs into the history of the discussions that
bore on the question of the conformity of mathematical proofs – particu-
larly, those contained in Euclid’s Elements – to Aristotle’s theory of demon-
stration. Her historical inquiry highlights that the present-day discussions
of the issue are at odds with how the question was understood and tackled
from late antiquity until the Renaissance. In contrast to the discussions by
John Philoponus and Proclus which took Aristotle’s theory as their founda-
tion and inquired into whether and how mathematical proofs, and which
mathematical proofs conformed to the Aristotelian theory, the contem-
porary view reversed the perspective. It took Euclid’s Elements as a basis
on which Aristotle’s theory of demonstration had to be interpreted and
understood. Th is repositioning reveals a fundamental shift in the interpre-
tation of Aristotle’s Posterior Analytics. By analysing how Philoponus and
Proclus discussed the issue, she emphasizes that, despite essential diff er-
ences between their approaches, they both understood the key problems to
be whether proofs established mathematical attributes that belong to their
subjects essentially and whether the middle term of a syllogism could serve
as the cause of the conclusion. Th us, for these authors, the problem of the
applicability of Aristotle’s theory of demonstration related to the non-formal
requirements of the theory. Th e same criterion holds true for the discus-
sion until the Renaissance. By contrast, whatever conclusions they reach,
contemporary interpretations of the question only consider the formal
requirements. Th e main point of the discussion has hence become whether
an interpretation of the syllogism could be off ered that could accommodate
what is to be found in, say, Euclid’s Elements. Harari’s contribution thereby
exposes the anachronism underpinning the common, present-day reading
of the relationship of Euclid and Aristotle to each other and highlights how
much stranger they might become – both to us and to each other – if we
attempted to restore them back to the context of the discussions and prob-
lems from which they emerged, so far as this is possible. Can we establish
a correlation between the modern readings of Euclid and Aristotle and the
way in which the critical editions discussed above were carried out? Such
questions are interesting to keep in mind generally when analysing the
various editions of Euclid’s Elements produced throughout history.
Th ese remarks conclude our analysis of past historiographies of proof
and our identifi cation of the factors at play in shaping and maintaining
them. Among these factors, we identifi ed elements of the contexts in which