354 reviel netz
in roundabout, extremely subtle, or even paradoxical fashion, so that it is
only by reading them several times over – eff ectively, producing a com-
mentary – that one comes to see their validity. Indeed, such writing is
very typical of the Western philosophical tradition. Diophantus’ world had
also people reading, say, Stoic metaphysics, which is as opaque (and as
precise) as text 3 above. But Stoic metaphysics is the product of a profes-
sional community of specialists who pride themselves in their fl uency in a
complex language. Its subject matter is perceived to have enormous inner
signifi cance. Th us readers prefer the theoretical power of an argument to
its apparent rationality: it is more important to derive a truth than to show
that that truth arises naturally (indeed, there is a premium in a diffi cult-to-
parse argument, in whose production and parsing both author and reader
may take pride). On the other hand, because the author off ers solutions he is
under a special obligation, as argued above, to display the rationality of the
solution as it unfolds, to show that it is not a contrived solution but instead
derives naturally from the terms of the problem. 27
Th is immediately suggests a function for Diophantus’ symbolism.
Obviously, it makes the parsing easier: it abbreviates overall, and it brings
about clear visual signposts with which the text is structured and its entities
identifi ed.
But let us be more precise: just what is being more easily parsed, and
how? To repeat the conclusion of Section 2 above: we see that Diophantus’
symbolism gives rise to a systematic bimodal reading, visual and verbal,
at the level of the noun-phrase. Th is, I argue, directly serves the goal of
constructing a rational bridge leading from the terms of the problem to its
solution.
For what is a rational bridge like? It is a structure where everything is
meaningfully present to the mind, and is also under the mind’s control.
Th e relationships are all calculated and verifi ed, but they are perceived as
meaningful relationships and not as mere symbolic structures lacking in
meaning. In modern terms, we may say that Diophantus needs to have a
semantic derivation; it also ought to be cognitively computable.
Since the derivation must be semantic, a bimodal reading is preferable to
a strictly visual one. For Diophantus, it appears important that the deriva-
tion refers directly to numbers and monads, and does not make use of some
opaque symbols. Th e derivation should be conducted throughout at the
level of the meanings: the signifi ed – and not only the signs – should never27 I follow an explanatory mode comparable to that of Chemla 2003. Considering the closely
analogous case of the use of particular examples in Chinese mathematics, Chemla argues that
these were used because the authors were seeking generality above abstraction. My analogous
argument is that Diophantus sought transparency above generality.