212 orna harari
include the attributes that are present in matter. 12 In this interpretation,
then, the ontological distinction between incomposite and composite
entities accounts for two characteristics of the principles of demonstra-
tion: their indemonstrability and their essentiality. Th e question whether
certain propositions meet Aristotle’s requirements is not answered by an
examination of their logical characteristics, but by the ontological status of
their subjects.
It follows from this discussion that from Philoponus’ viewpoint the
immateriality of the subject of predication is suffi cient to guarantee the
essential relation between a subject and its attributes. 13 Th is assumption
may explain Philoponus’ approach to the issue of the essentiality of math-
ematical propositions. Mathematical objects, according to Philoponus, are
abstractions from matter 14 – that is, they belong to the class of incomposite
objects that serve as the subjects of formal defi nitions. Th us, in light of
Philoponus’ characterization of these defi nitions, it plausible to regard all
attributes of mathematical objects as essential, because the immateriality of
these objects seems to entail, in Philoponus’ view, the essentiality of their
attributes. In what follows, I show that the ontological distinction between
incomposite and composite entities also explains why the causal role of
the middle term is not taken into account in Philoponus’ discussion of the
conformity of Euclid’s proofs to Aristotelian demonstrations.Causal demonstrations
In his commentary on Aristotle’s Physics ii .2, Philoponus examines the
tenability of Aristotle’s criticism of the theory of Forms, which involves,
according to Aristotle, separation from matter of the objects of physics,
although they are less separable than mathematical objects. In so doing,
Philoponus draws a distinction between separability in thought and sepa-
rability in existence, claiming that he agrees with Aristotle that the forms13 Two reasons may explain why Philoponus does not consider the possibility that immaterial
entities have accidental attributes. First, it is commonly held in the ancient tradition that only
individuals have accidental attributes, which belong to their matter. Second, Philoponus’
notion of essential predication is more formal than Aristotle’s. In characterizing essential
predications Philoponus appeals to extensional, rather than intensional, considerations. In his
view, attributes that belong to all members of a species and only to them are essential (e.g., In
An. Post. 63.14–20, Wallies; In DA 29.13–30.1, Hayduck; In Cat. 64.9, Busse).
14 For Philoponus’ conception of mathematical objects, see (e.g.) In Phys. 219.10; In DA , 3.7–11.
For a discussion of this view, see Mueller 1990 : 465–7.12 364.16–18, Wallies.