The History of Mathematical Proof in Ancient Traditions

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4 John Philoponus and the conformity

of mathematical proofs to Aristotelian

demonstrations

Orna Harari

One of the central issues in contemporary studies of Aristotle’s Posterior

Analytics is the conformity of mathematical proofs to Aristotle’s theory of

demonstration. Th e question, it seems, immediately arises when one com-

pares Aristotle’s demonstrative proofs with the proofs in Euclid’s Elements.

According to Aristotle, demonstrative proofs are syllogistic inferences of the

form ‘All A is B, all B is C, therefore all A is C’, whereas Euclid’s mathemati-

cal proofs do not have this logical form. Although the discrepancy between

mathematical proofs and Aristotelian demonstrations seems evident, it is

only during the Renaissance that the conformity of mathematical proofs

to Aristotelian demonstrations emerges as a controversial issue. 1 Th e

absence of explicit discussions of the conformity of mathematical proofs

to Aristotelian demonstrations in the earlier tradition seems puzzling from

the perspective of contemporary studies of Aristotle’s theory of demonstra-

tion. Th e formal discrepancies between Aristotelian demonstrations and

mathematical proofs seem so obvious to us that it is diffi cult to understand

how the conformity between mathematical proofs and Aristotelian dem-

onstrations was ever taken for granted. In this chapter I attempt to bring to

light the presuppositions that led ancient thinkers to regard the conformity

of mathematical proofs to Aristotelian demonstrations as self-evident.

Neither an outright rejection nor an explicit approval of the conform-

ity of mathematical proofs to Aristotelian demonstrations is found in

the extant sources from late antiquity; however, two approaches to this

issue can be detected. According to one approach, found in Proclus’

commentary on the fi rst book of Euclid’s Elements , the conformity of

1 Th e fi rst Renaissance thinker to reject the conformity of mathematical proofs to Aristotelian

demonstrations is Alessandro Piccolomini. His treatise Commentarium de certitudine

mathematicarum disciplinarum , published in 1547, initiated the debate known as the Quaestio

de certitudine mathematicarum , in which other Renaissance thinkers, such as Catena and

Pereyra, sided with Piccolomini in stressing the incompatibility between mathematical proofs

and Aristotelian demonstrations, whereas other thinkers, such as Barozzi, Biancani, and

Tomitano, attempted to reinstate mathematics in the Aristotelian model. I discuss this debate

and its ancient origins in the conclusions.