Philoponus and Aristotelian demonstrations 211
them from divine entities, which are also capable of knowledge but are not
mortal. By contrast, the geometrical attributes that Philoponus mentions
in this passage do not distinguish triangles or circles from other shapes.
Admittedly, the attribute ‘having the sum of the interior angles equal to
two right angles’ holds only for triangles, yet, unlike ‘having three sides’, it
is not the feature that distinguishes triangles from other shapes. It seems,
then, that in accounting for the essentiality of mathematical attributes,
Philoponus expands the notion of diff erentia , so as to include all the attrib-
utes of mathematical entities. He does not distinguish between attributes
that enter into the defi nition of an entity and necessary attributes; he con-
cludes from the statement that a triangle will cease to be a triangle if one
of its attributes were separated from it that these attributes are essential.
Th us, rather than explaining why mathematical attributes are essential in
Philoponus’ view, this passage refl ects his assumption that the essentiality
of mathematical attributes is evident. Th is assumption, I surmise, can be
understood in light of Philoponus’ interpretation of the principles of
demonstration.
In his comments on the Posterior Analytics ii .2,^10 Philoponus accounts
for the distinction between indemonstrable premises and demonstrable
conclusions in terms of the distinction between composite and incomposite
entities. Incomposite entities, according to this discussion, are simple or
intelligible substances such as the intellect or the soul, which are considered
( theōroumenon ) without matter. 11 In the case of such entities, Philoponus
argues, the defi ning attribute is not diff erent from the defi nable object
and therefore propositions concerning such entities are indemonstrable
or immediate. Another characterization of indemonstrable premises is
found in Philoponus’ interpretation of Aristotle’s discussion of the relation-
ship between defi nitions and demonstrations in the Posterior Analytics
ii .2–10. In addressing the question whether it is possible to demonstrate a
defi nition, Philoponus draws a distinction between two types of defi nition:
formal and material. Formal defi nitions are the indemonstrable principles
of demonstration that defi ne incomposite entities; they include, accord-
ing to Philoponus, the essential attributes ( ousiodōs ) of the defi ned object.
Material defi nitions, by contrast, serve as demonstrative conclusions and
10 Th e editor of Philoponus’ commentary on the Posterior Analytics , M. Wallies, doubted the
attribution of the commentary on the second book of the Posterior Analytics to Philoponus
(v–vi). Th e authenticity of the commentary on the second book does not aff ect my argument,
because all the references I make here to the commentary on the second book accord with
views expressed in Philoponus’ other commentaries.
11 339. 6–7, Wallies.