214 orna harari
of an object are not physical defi nitions, but are dialectical or empty. His
example of such an empty defi nition is the formal defi nition of anger:
‘anger is a desire for revenge’. Th e adequate defi nition of anger, according
to Philoponus, is ‘anger is boiling of the blood around the heart caused by
a desire for revenge’. 18 Th is defi nition refers to the form, the matter and the
cause. Similarly, in the commentary on the Posterior Analytics , Philoponus
claims that neither the formal nor the material defi nition is a defi nition in
the strict sense; only the combination of these two yields an adequate defi -
nition.^19 Th is conception of defi nition is evidently inapplicable to math-
ematics. Mathematical objects are defi ned without reference to matter or
to their cause, hence formal defi nitions provide an exhaustive account of
these objects.
Th e second consequence of Philoponus’ distinction between physical
and mathematical defi nitions concerns the explanatory or causal rela-
tions in demonstrative proofs. Although in the above-quoted passage
Philoponus contends that the cause is also studied in mathematics when a
relation between a mathematical object and its attributes is proved, it seems
that this cause is diff erent from the one studied in physics. According to
the above passage, physics studies the cause of the realization of form in
matter, but since mathematics does not deal with the matter of its objects,
its explanations do not seem to be based on this type of cause. Furthermore,
Philoponus’ analysis of physical demonstrations in terms of the distinction
between formal and material defi nitions gives rise to a problem that has
no relevance for mathematical demonstrations. Th is interpretation gives
rise to the question of how the material aspect of a physical entity, which is
a composite of form and matter, can be demonstratively derived from the
formal defi nition, given that this defi nition does not exhaust the nature of
the composite entity. Stating this question diff erently, how, in Philoponus’
view, can a proposition regarding a substance taken with matter be
demonstratively derived from a proposition regarding its form, which is
considered in separation from matter? Evidently this question does not
arise in the mathematical context. Mathematical defi nitions do not refer to
matter; hence, they give an exhaustive account of mathematical objects. In
what follows, I show that Philoponus answers this question by appealing to
extra-logical considerations. More specifi cally, I show that the causal role
of the middle term in demonstrations provides Philoponus with the means
of bridging the gap between formal defi nitions and material defi nitions.18 43.28–44.8, Hayduck.
19 365.1–13, Wallies.