Reasoning and symbolism in Diophantus 357
previous lay and school algebras, but also from the established elite literate
Greek mathematics Diophantus was familiar with. Th is mathematics had
included no such symbolism as Diophantus’. Why would Diophantus intro-
duce such a symbolism, then? In other words, what is the function served
by symbolism, in the case of the problems studied by Diophantus – but
which is not required in the case of the problems studied by previous elite
literate Greek mathematicians?
Th e question can be put precisely: why are Greek geometrical relations
easily computable without symbolism, while Diophantus’ numerical rela-
tions are not? Th e question is cognitive, and so we should look for a cogni-
tive divide between the character of geometrical and numerical relations.
To begin with, then, let us remind ourselves of how Greek geometrical
relations are expressed.
As described in Chapter 4 of N1999, Greek geometrical texts are written
in a system of formulaic expressions, the most important of which is the
family of ratio-expressions, e.g. ‘the ratio of A to B is the same as the ratio of
C to D’ (typically, the slots A, B, C and D are fi lled by spelled-out formulae
for geometrical objects, e.g. ‘the [two letters]’, the standard formulaic rep-
resentation of a line). One may then bring in further information, always
expressed within the same system of formulaic expressions, e.g. that ‘C is
equal to E’, or that ‘the ratio of C to D is the same as the ratio of G to H’.
Extra information of the fi rst kind would license a conclusion such as ‘the
ratio of A to B is the same as the ratio of E to D’, while extra information of
the second kind would license a conclusion such as ‘the ratio of A to B is the
same as the ratio of G to H’.
To repeat, the system is based on formulaic expressions – all within
natural Greek grammar. No special symbolism is involved and the text is
spelled out in ordinary alphabetical writing, so that the mind doubtless fi rst
translates the written traces into verbal representation and then computes
the validity of the argument on the basis of such verbal representations.
Note now that the formulaic expressions of Greek geometry are
characterized by a hierarchical, generative structure. Typically, a formulaic
expression has, as constituent elements subordinate to its own structure,
several smaller formulaic expressions, all ultimately governing the charac-
ters of the alphabet indicating diagrammatic objects. Th us in ‘the ratio of A
is that divisions are not represented as unit fractions but are left ‘in the raw’ – makes such
a verifi cation possible. It is one thing to be told that 6 divided by 8 is 4^3 (where you directly
verify that 8^6 is the same as 4^3 ); another, to be told that 6 divided by 8 is 2′ 4 ′ (where the
verifi cation depends on a relatively complex, separate calculation – usually, much more
complex than in this simple example).