The History of Mathematical Proof in Ancient Traditions

222 orna harari

Causal considerations are employed with regard to mathematical dem-

onstrations, when mathematical objects are considered material; they

are not employed when mathematical objects are considered separated

in thought from matter.

Conclusions

In concluding this chapter, I examine the relationship between the modern

formulation of the question of the conformity of mathematical proofs

to Aristotelian demonstrations and its formulation in late antiquity. Th e

modern discussions of the relationship between Aristotle’s theory of dem-

onstration and mathematical proofs focus on Aristotle’s formal requirement

that demonstrations should be syllogistic inferences from two universal

predicative propositions, which relate the subject and predicate of the con-

clusion to a third term, called the ‘middle term’.

Th e disagreement among Aristotle’s modern commentators concerns

whether mathematical proofs can be cast in this logical form. For instance,

Ian Mueller, who says they cannot, argues that in a syllogistic reformulation

of Euclidean proofs the requirement that the inference should have only

three terms is not always met, because the mathematical proofs depend

on the relations between mathematical entities and not on their properties

taken in isolation from other entities. 34 Th e possibility of expressing mathe-

matical relations in syllogistic inferences is also central in modern attempts

to render Aristotle’s theory of demonstration compatible with mathemati-

cal proofs. Henry Mendell, for instance, shows that Aristotle’s theory of

syllogism does have the formal means that make possible syllogistic for-

mulations of mathematical proofs. In so doing, he argues that the relation

of predication, which is formulated by Aristotle as ‘ x belongs to y ’, can be

read fl exibly so that it also accommodates two-place predicates, such as ‘ x

equals y ’, o r ‘ x is parallel to y ’.^35 Mendell’s argument, like Mueller’s, focuses

on the possibility of expressing relations within the formal constraints of

the theory of syllogism. Th e extra-logical consequences of the expansion

of the theory of syllogism to relational terms and their compatibility with

Aristotle’s theory of demonstration are not at the centre of either Mendell’s

or Mueller’s argument. More specifi cally, they do not address the question

of whether relational terms or mathematical properties can be proved to

34 Mueller 1975 : 42.

35 Mendell 1998.