The History of Mathematical Proof in Ancient Traditions

210 orna harari

formulates this question in response to Aristotle’s contention that sentences

whose subject is an attribute, such as ‘the white ( to leukon ) is walking’ or

‘the white is a log’ cannot feature in demonstrations, because they are not

predicative in the strict sense ( Posterior Analytics 83a1–21). Th is conten-

tion jeopardizes, in Philoponus’ view, the status of geometrical proofs. Th e

subject matter of geometry, according to Philoponus, is shapes and their

attributes. Hence, Aristotle’s narrow conception of predication may imply

that proofs that establish that certain attributes belong to shapes are not

demonstrative because they prove that certain attributes, such as having the

sum of the interior angles equal to two right angles, belong to other attrib-

utes, such as triangles (239.11–14). 7 Philoponus dismisses this implication

saying:

Even if these [attributes] belong to shapes accidentally, they are completive [attrib-

utes] of their being ( symplērōtika tēs ousias ) and like diff erentiae that make up the

species they are [the attributes] by which [shapes] are distinguished from other

things. 8 ... Just as ‘being capable of intellect and knowledge’ or ‘mortal’ or any of the

[components] in its defi nition do not belong to ‘man’ as one thing in another, but

[man] is completed from them, so the circle is also contemplated ( theōreitai ) from

all the attributes which are observed in it. Similarly, also the triangle would not be

something for which ‘having three angles equal to two right angles’ or ‘having the

sum of two sides greater than the third’ do not hold, but if one of these [attributes]

should be separated, immediately the being of a triangle would be abolished too. 9

Th is account does not answer Philoponus’ original query; it does not tackle

the question whether proofs that establish predicative relations between

two attributes are demonstrative. Instead, Philoponus focuses here on the

question whether the attributes that geometry proves to hold for shapes

are essential, arguing that mathematical attributes like diff erentiae are

parts of the defi nitions of mathematical entities. However, the analogy

between the diff erentiae of man and mathematical propositions is not as

obvious as Philoponus formulates it. Th e attributes ‘capable of knowledge’

and ‘mortal’ distinguish men from other living creatures; the former dis-

tinguishes human beings from other animals and the latter distinguishes

7 Philoponus presupposes here Aristotle’s categorical scheme, in which terms belonging to the

nine non-substance categories are attributes of terms belonging to the category of substance.

According to Aristotle’s Categories the term ‘shape’ belongs to the category of quality. Hence,

Philoponus claims that geometry studies attributes of attributes.

8 Th e term ‘completive attributes’ ( symplērōtikos ) refers in the neo-Platonic tradition to

attributes without which a certain subject cannot exist. On these attributes and their relation to

diff erentiae , see De Haas 1997 : 201 and Lloyd 1990 : 86–8.

(^9) 239.14–25, Wallies.