The History of Mathematical Proof in Ancient Traditions

90 bernard vitrac

or ‘ἤ καὶ οὕτως... ’ (‘Or, also thus ...’). 56 In the same way, in the Arabo-

Latin translation of Gerard of Cremona, the great majority of the second

proofs are explicitly presented as such, thanks to indications of the type ‘ in

alio libro ... invenitur ’ (‘in another book is found ...’). On the other hand

the identifi cation of proofs as distinct is much more delicate when it is a

question of comparing two solitary proofs appearing in diff erent versions –

for example, when one compares a proof from a Greek manuscript and its

corresponding proof in the Arabic translation, or one from Adelard of Bath

and the other from Gerard of Cremona. Th e intricacies of the manuscript

transmission prevent two proofs which have only minimal variations from

being considered as truly diff erent. If this were not so, there would be as

many proofs of a Proposition as there are versions or, even, manuscripts!

Th is is why it has proven necessary to introduce the division between

local and global. Ideally, it ought to be possible to identify the ‘core argu-

ment’ which characterizes a proof and to distinguish it from the type

of ‘packaging’ which is stylistically or didactically relevant but which is

neither mathematically nor logically essential. Th e expression ‘substitution

of proof ’ (global modifi cation) will be reserved for those cases where there

is a replacement of one core argument by another. Th e distinction between

‘core’ and ‘packaging’ is not always easy to establish, but it may be thought

that the distinction will be better understood if the diff erent methods of

‘packaging’ have been previously delineated. In other words, in order that

Figure 1.2 Euclid’s Elements. Typology of deliberate structural alterations.

Addition/Suppression of Material Modification of Presentation

DELIBERATE ALTERATIONS

Local

Logical

Interventions

Abridged Construction

or Shortened Proof

Global

Stylistic

Interventions

Substitution of Proof

Addition/Suppression

of Cases

Double Proofs (Existence of Alternative Proof)

Alteration of Proofs

Change in Order Change of Status

Fusion of 2 Propositions into 1

Division of 1 Proposition into 2

Different

Formulations

(^56) Nonetheless, there are confusions. Th us, the addition at vi. 27 is introduced as if it were an
alternative proof (ἄλλως). See EHS: ii 231.2.