Reasoning and symbolism in Diophantus 359
from expressions such as ‘the ratio of A to B is the same as the ratio of C to
D’. For this reason, purely verbal representations of the Diophantine phrase
are of limited value, and Diophantus naturally was led to look for further
tools for easing computation, in the principle of allography present in his
scribal culture.
I fi nd it striking that the same seems to be true of numerical expres-
sions in natural language as a whole. It seems that numerical expressions
tend to be paratactic, rather than subordinate: this may be because they
are essentially open-ended in character, ‘A and B and C and D’. Th us they
always off er incentives for non-verbal representation in which their com-
putation is aided by more than natural language syntax. Number symbol-
ism itself is the primary example. For aft er all the earliest and most central
case of symbolic argument is precisely that – the algorithm, manipulating
number-symbolism via a translation of numbers from natural language
into a visual code. 29
4. Summary
Th e suggestion of this article can now be put forward as follows. Involved
in the deuteronomic project of fi tting in previously available texts within
established forms, Diophantus set himself the task of presenting lay and
school algebra within the format – and expectations – of Greek geometrical
analysis. Th is entailed the task of constructing a rational bridge leading
from the setting of the problems to their solutions. Since the expressions
involved were numerical in character (rather than standing for qualitative
relations), their structure was not subordinate, but paratactic. As a con-
sequence, the syntax of natural language no longer helped in their com-
putation and could not support the task of constructing a rational bridge.
Instead, Diophantus reached for the tool available to him in his culture –
allography – to construct expressions whose visual structure could support
the same task. Th ese two features of Diophantus’ context – deuteronomy
and allography – both may have to do, ultimately, with the material history
of writing in late antiquity. And so, the relationship between reasoning and
symbolism in Diophantus is found to be dependent upon the very specifi c
historical conditions of late antiquity.
Th e complex, many-dimensional nature of the account sketched here is
in itself signifi cant. Why does Diophantus use his particular symbolism?
29 Allard 1992.