Philoponus and Aristotelian demonstrations 207
certain mathematical proofs to Aristotelian demonstrations is questioned. 2
According to the other approach, found in Philoponus’ commentary on
Aristotle’s Posterior Analytics , the conformity of mathematical proofs
to Aristotelian demonstrations is taken for granted. 3 Nevertheless, these
thinkers did not address the same question that Aristotle’s contemporary
interpreters discuss. Whereas contemporary studies focus on the discrep-
ancy between the formal requirements of Aristotelian demonstrations
and mathematical proofs, the ancient thinkers focused on the non-formal
requirements of the theory of demonstration – namely, the requirements
that demonstrations should establish essential relations and ground their
conclusions in the cause.
In view of this account, I attempt to explain why the question whether
mathematical proofs meet these non-formal requirements does not arise
within the context of Philoponus’ interpretation of Aristotle’s theory of dem-
onstration. Regarding the requirement that demonstrative proofs should
establish essential relations, I show that Philoponus considers it non-
problematic in the case of all immaterial entities including mathematical
objects. I show further that Philoponus’ assumption that mathematical
objects are immaterial renders the requirement that the middle term should
serve as a cause irrelevant for mathematical demonstrations, since accord-
ing to Philoponus causes are required only to explain the realization of
form in matter. Accordingly, the dependence of mathematical proofs on
defi nitions is suffi cient, in Philoponus’ view, to guarantee their conformity
to Aristotelian demonstrations. In substantiating this conclusion, I then
discuss Proclus’ argument to the eff ect that certain mathematical proofs do
not conform to Aristotelian demonstrations. I show that within the context
of Proclus’ philosophy of mathematics, in which geometrical objects are con-
ceived of as realized in matter, consideration of the question whether math-
ematical proofs meet the two non-formal requirements – a question which
Philoponus ignores with regard to mathematical demonstrations – led
Proclus to argue for the non-conformity of certain mathematical proofs to
2 Proclus’ commentary on the fi rst book of Euclid’s Elements was translated into Latin in 1560 by
Barozzi and it played an instrumental role in the debate over the certainty of mathematics. For
the reception of Proclus’ commentary on the Elements in the Renaissance, see Helbing 2000 :
177–93.
(^3) Philoponus’ commentary on the Posterior Analytics has been hardly studied; hence it is
diffi cult to assess its direct or indirect infl uence on the later tradition. Nevertheless, it seems
that the several traits of Philoponus’ interpretation of the Posterior Analytics are found in the
medieval interpretations of Aristotle’s theory of demonstrations, such as the association of
demonstrations of the fact with demonstrations from signs which is found in Averroes (see
n. 38 ) and the identifi cation of the middle term of demonstration with real causes (see n. 27 ).