Philoponus and Aristotelian demonstrations 213
of natural things cannot be separated in existence from matter, but he disa-
grees with Aristotle’s view if it implies that these forms cannot be separated
by reason and in thought. 15 Although Philoponus’ account of the indemon-
strability of the principles of demonstration presupposes the possibility
of separating the defi nitions of both mathematical and physical entities,
the ontological diff erence between these classes of objects is nevertheless
maintained. In his commentary on Aristotle’s De anima , Philoponus draws
a distinction between physical and mathematical defi nitions, arguing that
physical defi nitions should refer to the matter of physical substance, their
form and the cause by virtue of which the form is realized in matter. 16
Mathematical defi nitions, by contrast, refer only to the form:
Th e mathematician gives the defi nitions of abstracted forms in themselves, without
taking matter into account, but he gives these [defi nitions] in themselves. For this
reason he does not mention the cause in the defi nition; for if he defi ned the cause,
clearly he would also have taken the matter into account. Th us, since he does not
discuss the matter he does not mention the cause. For example, what is a triangle?
A shape contained by three lines; what is a circle? A shape contained by one line.
In these [defi nitions] the matter is not mentioned and hence neither is the cause
through which this form is in this matter. Unless perhaps he gives the cause of those
characteristics holding in themselves for shapes, for instance, why a triangle has its
angles equal to two right angles. 17
Philoponus’ distinction between physical and mathematical defi nitions
has two related consequences for the methods employed in physics and
mathematics. First, although both physical and mathematical demonstra-
tions are based on indemonstrable formal defi nitions, these defi nitions
adequately capture the nature of mathematical objects but they fail to
exhaust the nature of physical objects. In the case of physical demonstra-
tions, the formal defi nition captures only one aspect of the object: its
form. Full-fl edged knowledge of physical objects requires reference also
to the matter of this object and the cause of the realization of the form
in matter. Indeed, in both the commentary on Aristotle’s De anima and
the commentary on the Posterior Analytics , Philoponus considers formal
defi nitions of physical objects defi cient. In the commentary on De anima ,
Philoponus argues that defi nitions that do not include all the attributes
15 225.4–11, Vitelli. For the relationship between Philoponus’ discussion of separability in
thought of physical defi nitions and his analysis of demonstrations in the natural sciences, see
De Groot 1991 : 95–111.
16 55.31–56.2, Hayduck.
17 57.35–58.6, Hayduck.