218 orna harari
demonstrative derivation rests on two relations: the transitivity of the pre-
dicative relation that the premises state and the causal relation between the
middle term and the conclusion. Th is distinction is applicable to physical
demonstrations, for which the cause of the realization of form in matter is
sought. Th e demonstrative derivation in these demonstrations is based not
only on logical relations but also on causal relations. Mathematical enti-
ties, by contrast, have only one facet: the form. Accordingly, Philoponus’
account of the conformity of mathematical demonstrations to Aristotelian
demonstrations focuses only on the formal requirements of the theory
of demonstration. Th e conformity of mathematical demonstrations to
Aristotelian demonstrations is guaranteed if the conclusions can be shown
to depend on the defi nitions of mathematical entities. Since mathematical
objects have no matter, mathematical demonstrations can be based only
on logical derivation; the question whether the middle term is the cause of
the conclusion does not arise in this context, as the separation from matter
renders superfl uous questions concerning causes. 27
Th e analysis of Philoponus’ interpretation of Aristotle’s theory of dem-
onstration reveals the importance of the ontological distinction between
simple and composite entities for his account of conformity of mathemati-
cal proofs to Aristotelian demonstrations. Th e assumption that mathemati-
cal objects are analogous to simple entities by being separated in thought
from matter does not give rise to two questions that may undermine the
conformity of mathematical proofs to Aristotelian demonstrations. Th e
fi rst question is whether mathematical predications are essential; the
second is whether the middle term in mathematical proofs is the cause of
the conclusion. Th e fi rst question does not arise because the separation
from matter implies that only the essential attributes of entities are taken
into consideration. Th e second does not arise because causal considerations
are relevant only with regard to composite entities, as it is only in their case
that the cause of the realization of form in matter can be sought. Hence,
given the assumption that mathematical entities are separated in thought
from matter, the question whether mathematical proofs conform to the
non-formal requirements of Aristotle’s theory of demonstration does not
arise. Th is conclusion gains further support from Proclus’ discussion of the
conformity of mathematical proofs to Aristotelian demonstrations.27 Th is conclusion may explain Proclus’ otherwise curious remark that the view in which
geometry does not investigate causes is originated in Aristotle ( In Eucl. 202.11, Friedlein). If
this explanation is correct, Philoponus’ conception of mathematical demonstrations seems to
refl ect a widespread view in late antiquity.