Reasoning and symbolism in Diophantus 355
be lost out of sight, for otherwise the derivation would appear as a con-
jurer’s trick out of which the solution happened to have emerged – precisely
the opposite eff ect of the rational bridge Diophantus aims to construct.
At the same time, the visual component of the bimodal reading serves in
the computation of the expression. Th e eye glances quickly to the correct
spot in the phrase, fi nding the correct value. Even more important, perhaps:
the mind is trained to look for the expressions, so that a visual–spatial
arrangement for the phrase comes to aid the purely verbal computation.
Th is is a speculative statement: I believe it to be true. Let me explain. First
of all, independently of how a particular phrase may be spelled out, through
abbreviations or through fully written-out words, it is certainly read by a
mind that is already acquainted with the fi xed structure of the phrase on
the page, and with its limited arsenal of symbols. Th us the reader would
have triggered in him or her not only the verbal response, but also the visual
response. In other words, it appears to me that, just as the mind involun-
tarily creates a verbal representation of a Diophantine abbreviation, so it
involuntarily creates a visual representation of a Diophantine spelled-out
word. Th us the reader has three resources available: (1) the actual trace of
the page, (2) the verbal representation of the contents, kept by the mind’s
working memory of phonological representations, (3) the visual repre-
sentation of the contents, kept by the mind’s working memory of visual
representations. Resource (1) would then serve to stabilize and keep in
place both resources (2) and (3). It is obvious that the presence of a visual
resource, over and above the verbal resource, helps in the computation of
the expression: I shall return to explain this in more detail below.
What is involved in the computation? Th e reader, above all, verifi es that a
certain relation holds, in the rational bridge, leading from one statement to
the next. In other words, what we need is to have a tool for operating upon
phrases expressing arithmetical values. We need to verify that the product
of an operation on the expression X is indeed the expression Y. So we can
see why the operations themselves do not call for symbolism: they may be
fully spelled out, instead. What we need is symbolism for the arithmetical
values on which the operations operate. We can thus see why Diophantine
symbolism stops at the level of the noun-phrase and does not reach the
level of the sentence.
Th e computation is thus local to the level of the noun-phrase. Indeed, it
is clear that the resources (2) and (3) – the verbal and visual representation
of expressions in the reader’s working memory – are limited in capacity and
duration. In fact, all that they allow is the verifi cation of the relation in a
single stage of the argument – the rational bridge is built one link at a time.
We can now return to i .10 and consider the verifi cation in action: