Philoponus and Aristotelian demonstrations 219
Proclus on the conformity between mathematical
proofs and Aristotelian demonstrations
Proclus’ philosophy of geometry is formulated as an alternative to a con-
ception whereby mathematical objects are abstractions from material or
sensible objects. 28 According to Proclus, mathematical objects do not diff er
from sensible objects in their being immaterial, but in their matter. Sensible
objects, in Proclus’ view, are realized in sensible matter, whereas math-
ematical objects are realized in imagined matter. In Proclus’ philosophy of
geometry, then, mathematical objects are analogous to Philoponus’ physi-
cal objects; they are composites of form and matter. Proclus’ philosophy
of mathematics is at variance not only with Philoponus’ views regarding
the ontological status of geometrical objects but also with Philoponus’
views regarding the conformity of Euclid’s proofs to Aristotelian demon-
strations.^29 In his discussion of the fi rst proof of Euclid’s Elements in the
commentary on the fi rst book of Euclid’s Elements , Proclus questions the
conformity of certain mathematical proofs to the Aristotelian model:
We shall fi nd sometimes that what is called ‘proof ’ has the properties of demon-
stration, in proving the sought through defi nitions as middle terms – and this is a
perfect demonstration – but sometimes it attempts to prove from signs. Th is should
not be overlooked. For, although geometrical arguments always have their necessity
through the underlying matter, they do not always draw their conclusions through
demonstrative methods. For when it is proved that the interior angles of a triangle
are equal to two right angles from the fact that the exterior angle of a triangle is
equal to the two opposite interior angles, how can this demonstration be from the
cause? How can the middle term be other than a sign? For the interior angles are
equal to two right angles even if there are no exterior angles, for there is a triangle
even if its side is not extended. 30
In this passage, Proclus claims that Euclid’s proof that the sum of the inte-
rior angles of a triangle is equal to two right angles ( Elements i .32) does
not conform to Aristotle’s model of demonstrative proofs. In so doing,
he focuses on the causal role of the middle term in Aristotelian dem-
onstrations. Proclus argues that Euclid’s proof does not conform to the
Aristotelian model because it grounds the equality of the sum of the inte-
rior angles of a triangle to two right angles in a sign rather than in a cause.
28 In Eucl. 50.16–56.22, Friedlein.
29 A discussion of the relationship between Proclus’ philosophy of geometry and his analysis of
mathematical proofs is beyond the scope of this paper. For this issue, see Harari 2006.
30 206.12–26, Friedlein.