Philoponus and Aristotelian demonstrations 221
up to their intersection point, is also presented by Proclus in his comments
on propositions i .16 and i .17 of the Elements. In both cases, he regards this
procedure – and not Euclid’s auxiliary construction in which the triangle’s
base is extended – as the true cause of the conclusion. 33 Proclus’ appeal to
this procedure in searching for the true cause of these conclusions indicates
that in attempting to accommodate Euclid’s proofs with Aristotle’s require-
ment that demonstrations should establish essential relations, he grounds
mathematical conclusions in causal relations rather than in logical rela-
tions. Proclus considers the proposition that the sum of the interior angles
of a triangle is equal to two right angles essential not because it is derived
from the defi nition of a triangle, as Aristotle’s theory of demonstration
requires, but because the proposition is derived from the triangle’s mode
of generation. Viewed in light of Philoponus’ interpretation of Aristotle’s
theory of demonstration, Proclus’ attempt to accommodate Euclid’s proof
with Aristotelian demonstrations seems analogous to Philoponus’ account
of physical demonstrations. In both cases, causal considerations are
employed in rendering proofs concerning material objects compatible with
Aristotelian demonstrations.
Th is examination of the presupposition underlying Philoponus’ and
Proclus’ views regarding the conformity of mathematical proofs to
Aristotelian demonstrations has led to the following conclusions.
(1) Th e pre-modern formulation of the question of the conformity of
mathematical proofs to Aristotelian demonstrations concerns the
applicability of the non-formal requirements of the theory of dem-
onstration to mathematical proofs. More specifi cally, this formulation
concerns the questions whether mathematical attributes are proved
to belong essentially to their subjects and whether the middle term in
mathematical proofs serves as the cause of the conclusion.
(2) Th e emergence or non-emergence of the question of the conformity
of mathematical proofs to Aristotelian demonstration is related to
assumptions concerning the ontological status of mathematical objects.
Th is question does not arise in a philosophical context in which math-
ematical objects are conceived of as separated in thought from matter,
whereas it does arise when mathematical objects are conceived of as
realized in matter.
(3) Demonstrations concerning composites of form and matter were
understood in late antiquity as based on causal relations, viewed as
additional to the logical necessitation of conclusions by premises.
33 310.5–8, 315.15, Friedlein.