Philoponus and Aristotelian demonstrations 223
be essential predicates of their subjects. 36 Th is question, as I showed, was
central in the discussions of the conformity of mathematical proofs to
Aristotelian demonstrations in late antiquity.
Th e non-formal requirements of the theory of demonstration were also
central in the Renaissance debate over the certainty of mathematics. 37
Piccolomini’s objective in his Commentarium de certitudine mathemati-
carum disciplinarum was to refute what he presents as a long-standing
conviction that mathematical proofs conform to the most perfect type
of Aristotelian demonstration, called in the Renaissance demonstratio
potissima. Th e classifi cation of types of demonstrations that underlies
Piccolomini’s argument is based on Aristotle’s distinction between dem-
onstrations of the fact ( hoti ) and explanatory demonstrations or dem-
onstration of the reasoned fact ( dioti ). Th is distinction has been further
elaborated by Aristotle’s medieval commentators and it appears in the
Proemium of Averroes’ commentary on Aristotle’s Physics as a tripartite
classifi cation of demonstrations into demonstratio simpliciter , demon-
stratio propter quid and demonstratio quid est. It is in this context that
Averroes claims that mathematical proofs conform to the perfect type of
demonstration, in his terminology demonstratio simpliciter. 38 According to
this classifi cation, the diff erent types of demonstration diff er in the epis-
temic characteristics of their premises, hence in the epistemic worth of the
knowledge attained through them. Following this tradition, Piccolomini’s
argument for the inconformity of mathematical proofs to Aristotelian
demonstrations focuses on these characteristics. According to Piccolomini
potissima demonstrations are demonstrations in which knowledge of the
cause and of its eff ects is attained simultaneously; the premises of such
demonstrations are prior and better known than the conclusion; their
middle term is a defi nition, it is unique and it serves as the proximate
cause of the conclusion. Mathematical demonstrations, so Piccolomini
and his followers argue, fail to meet these requirements. Th e importance
of the non-formal requirements of the theory of demonstration for the
Renaissance debate over the certainty of mathematics comes to the fore
in the following passage from Pereyra’s De communibus omnium rerum
naturalium principiis et aff ectionibus :
36 Th is question is not utterly ignored in modern interpretations of the Posterior Analytics. See
McKirahan 1992 ; Goldin 1996 ; Harari 2004.
37 For a general discussion of the Quaestio de certitudine mathematicarum , see Jardine 1998. For
the infl uence of this debate on seventeenth-century mathematics, see Mancosu 1992 and 1996.
38 Aristotelis opera cum Averrois commentariis , vol. iv , 4.