Philoponus and Aristotelian demonstrations 225
Nevertheless, when the pre-modern discussion of the conformity of
mathematical proofs to Aristotelian demonstrations is viewed in light of its
underlying ontological presuppositions, a conceptual development leading
to the modern formulation of this question may be traced. Discussions of
the conformity of mathematical proofs to Aristotelian demonstrations in
late antiquity were associated with discussions of whether mathematical
objects are immaterial or material; 40 that is, whether they are conceptual or
real entities. Th is ontological distinction is refl ected in diff erent accounts
of the relation of derivation, on which demonstrations are based. Whereas
demonstrations concerning immaterial objects are based on defi nitions
and rules of inference alone, demonstrations concerning material objects
require the introduction of extra-logical considerations, such as the causal
relations between form and matter. Th us, the question of the ontologi-
cal status of mathematical objects refl ects the epistemological question:
whether extra-logical considerations have to be taken into account in
mathematics. When discussions of the conformity of mathematical proofs
to Aristotelian demonstrations in late antiquity are viewed in isolation
from ontological commitments, they seem to be conceptually related to
modern discussions of the nature of mathematical knowledge. Th e need to
take into account extra-logical considerations when mathematical objects
are considered material is equivalent to Kant’s statement that mathematical
propositions are synthetic a priori judgements. Developments in modern
logic led to a reformulation of Kant’s statement in terms of logical forms.
Kant’s contention that mathematical knowledge cannot be based on defi -
nitions and rules of inference alone was regarded by Bertrand Russell as
true for Kant’s time. According to Russell, had Kant known other forms
of logical inference than the syllogistic form, he would not have claimed
that mathematical propositions cannot be deduced from defi nitions and
rules of inference alone. 41 In light of this account, the modern discussions
of the conformity of mathematical proofs to Aristotelian demonstrations,
which focus on whether syllogistic inferences can accommodate relational
terms, may be understood as evolving from the pre-modern discussions
of whether mathematical proofs establish essential relations, and to estab-
lish this conclusion, two conceptual developments have to be traced: the
process by which the question whether mathematical propositions are
40 Th is assumption seems to underlie the Renaissance discussions of this issue as well. In the
eleventh chapter of his treatise Piccolomini attempts to reinstate the status of mathematics
as a science by claiming that mathematical objects are conceptual entities, existing in the
human mind.
41 Russell 1992 : 4–5.