358 reviel netz
to B is the same as the ratio of C to D’ one can detect three levels: the level
of the proportion statement, which is in turn a structure of two ratio state-
ments, each of which in turn is a structure of two object descriptions (which,
in the Greek original form, would refer through characters of the alphabet
indicating diagrammatic objects). Th e structure is hierarchical in that its
constituents are related to each other in relations of syntactic subordination;
it is generative in that such constituents can be added and substituted at will.
Th is substitution is in fact one of the two bases of the computation of
the validity of the geometrical argument in Greek mathematics – the other
being the diagram, which we may ignore here. It is feasible precisely because
the formulaic expression is hierarchical and generative. Mathematical
computation here is parasitic upon syntactic computation. Th e mind is
equipped with a tool for computing substitutions on hierarchic, generative
syntactic structures. It is thus a matter of immediate inspection that, from
the two expressions ‘the ratio of A to B is the same as the ratio of C to D’ and
‘C is equal to E’, the expression ‘the ratio of A to B is the same as the ratio of
E to D’: one unfailingly knows where to affi x the correct substitution, based
on one’s structural grasp of the expression ‘the ratio of A to B is the same as
the ratio of C to D’. Since natural language syntax is the mental tool brought
to bear when computing the validity of such arguments, it is only natural
that they are represented verbally and not visually.
We see then that, to the extent that expressions possess a hierarchic
structure, they may be eff ectively computed through natural language tools.
And it is important to notice that Greek geometrical formulae are indeed
characterized by such hierarchic structures, with proportion as the central
operation in this type of mathematics.
Not all expressions in natural language, however, have this hierarchic
structure based on subordination. Alongside subordinate structure, natural
language uses another structural principle, that of paratactic arrangement,
i.e. the concatenation of phrases to create larger phrases without introduc-
ing an internal structure of dependency. Th is is the diff erence between
expressions of the type ‘Th e A of the B of the C’ and expressions such as
‘A and B and C’. Expressions of the fi rst kind contain, in their syntactic
representation, internal structure, which the mind can use in manipulating
them. Expressions of the second kind are syntactically represented as mere
concatenation lacking internal structure, so that there is nothing syntactic
computation can latch onto.
My suggestion, then, is obvious: the central Diophantine expression –
the phrase representing the sums of, e.g., dunamis, number and monads –
is paratactic and not subordinate in structure. It thus essentially diff ers