The History of Mathematical Proof in Ancient Traditions

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Tannery has all as alphabetical numerals. Th e most we can say is that, in

the manuscripts, there is an overall tendency to prefer using the same form

within a single phrase, though exceptions to this are found as well. Here we

see Tannery homogenizing, turning numbers into numerals. But we may

also fi nd the opposite, e.g. in i .20, an expression we may translate as

Tannery: Let the two <numbers> be set down as ς 3

τετάχθωσαν οἱ δύο ς 3

Par. Gr. 2485: Let the 2 <numbers> be set down as Numbers, Th ree.

τετάχθωσαν οἱ Β ἀριθμοὶ τρεῖς

Tannery has spelled out the word ‘two’, to signal that it functions here in

a syntactic, not an arithmetical way. But it is neither syntactic nor arith-

metical, it is phonological/orthographic. In the manuscripts, we have the

phonological/orthographic object /duo/ which may be represented, as far

as the scribes are concerned, by either B or δύο: both would do equally well.

Signifi cantly, it is diffi cult to discern a system even in the symbols intro-

duced by Diophantus himself. Consider Par. Gr. 2380, inside ii .10: ς ενος

μοναδων Γ, that is ‘ς one, monads 3’ (I quote this as an elegant example

where both Diophantus’ special symbols, as well as numerals, are inter-

changed with fully spelled out words). Very typical are expressions such

as Par. Gr. 2378, ii .20: Δ υ Δ αριθμους Ε Μ ο , that is ‘Δ υ 4, numbers 5, Μ ο 1’.

Th e ‘numbers’ – alone in the phrase – are spelled out. In general, one can

say that monads appear to be abbreviated more oft en than anything else in

Diophantus’ symbolism: this may be because they are so common there.

But the main fact is not quantitative, but qualitative: one fi nds, in all manu-

scripts, the full range from Diophantine phrases fully spelled out in natural

Greek, through all kinds of combinations of symbols and full words, to fully

abbreviated phrases.

My conclusion is that symbols in Diophantus are allographs : ways of

expressing precisely the same things as their fully spelled out equivalents.

And once this allography is understood, the chaos of the manuscripts

becomes natural. For why should you decide in advance when to use this or

that, when the two are fully equivalent?

One should now understand Tannery’s plight. Th at he systematized his

printed edition is natural: what else should he have done? I am not even

sure we should criticize him for failing to provide a critical apparatus on

the symbols. Th e task is immense and its fruits dubious. In particular, given

Tannery’s goal – of reconstructing, to the best of his ability, Diophantus’

original text – a critical study of the abbreviations seems indeed hopeless.

Th e interrelationships between manuscripts, in terms of their choice of