208 orna harari
Aristotelian demonstrations. As a corollary to this discussion, I conclude my
chapter with an attempt to trace the origins of contemporary discussions of
the conformity of mathematical proofs to Aristotelian demonstrations to the
presuppositions underlying Philoponus’ and Proclus’ accounts of this issue.
I thereby outline a possible explanation for how concerns regarding the
ontological status of mathematical objects and the applicability of Aristotle’s
non-formal requirements to mathematical proofs evolved into concerns
regarding the logical form of mathematical and demonstrative proofs.Philoponus on mathematical demonstrations
In the Posterior Analytics i .9, Aristotle states that if the conclusion of a dem-
onstration ‘All A is C’ is an essential predication, it is necessary that the middle
term B from which the conclusion is derived will belong to the same family
( sungeneia ) as the extreme terms A and C (76a4–9). Th is requirement is
tantamount to the requirement that the two propositions ‘All A is B’ and ‘All
B is C’, from which the conclusion ‘All A is C’ is derived, will also be essential
predications. Th e example that Aristotle presents in this passage for an essen-
tial predication is ‘Th e sum of the interior angles of a triangle is equal to two
right angles’. In his comments on this discussion Philoponus tries to show
that the attribute ‘having the sum of its interior angles equal to two right
angles’ is indeed an essential attribute of triangles. He does so by arguing
that Euclid’s proof meets the requirements of Aristotelian demonstrations:
For having [its angles] equal to two right angles holds for a triangle in itself ( kath’
auto ). And [Euclid] proves this [theorem] not from certain common principles, but
from the proper principles of the knowable subject matter. For instance, he proves
that the three angles of a triangle are equal to two right angles, by producing one
of the sides and showing that the two right angles, the interior one and its adjacent
exterior angle, are equal to the three interior angles, 4 so that such a syllogism is
produced: the three angles of a triangle, given that one of its sides is produced, are
equal to the two adjacent angles. Th e two adjacent angles are equal to two right
angles. Th erefore the angles of a triangle are equal to two right angles. And that
the two adjacent angles are equal to two right angles is proved from the [theorem]
that two adjacent angles are either equal to two right angles or are two right angles.
Whence [do we know] that adjacent angles are either equal to two right angles or4 Th e proof that Philoponus describes is not identical to Euclid’s proof. Philoponus’ reference
to ‘two right angles’ implies that he envisages a right-angled triangle, whose base is extended
so as to create two adjacent right angles. Euclid’s proof refers to an arbitrary triangle. Th is
discrepancy does not aff ect Philoponus’ reasoning, as he states in the sequel that two adjacent
angles are either equal to two right angles or are two right angles.