Reasoning and symbolism in Diophantus 329
by such connectors as ‘since’, ‘therefore’, etc. Since the text is at the inter-
section of the Greek mathematical tradition with the Mediterranean
tradition of numerical problems, it follows that these two characteristics
- foregrounding symbolism and foregrounding reasoning – may be taken
to defi ne it.
Th is chapter follows on some of my past work in the cognitive and
semiotic practices of Greek mathematics. I bring to bear, in particular,
three strands of research. I extend the theoretical concepts of deuteronomy
(Netz 2004 ) and analysis as a tool of presentation (Netz 2000 ), arguing that
Diophantus was primarily a deuteronomic author – intent on rearranging,
homogenizing and extending past results – employing the format of analy-
sis as a tool of presentation that highlights certain aspects of his practice. I
further contrast Diophantus’ use of symbolism with the geometrical prac-
tice of formulaic expressions (N1999, ch. 4 ), arguing that Diophantus’ use
of symbolism is designed to display the rationality of transitions inside the
proof and that this display is better supported, in the case of Diophantus’
structures, by symbols as opposed to verbal formulae. In short: because
Diophantus is deuteronomic, he uses analysis; because he uses analysis, he
needs to display the rationality of transitions; because he needs to display
the rationality of transitions, he uses symbols. 6
Further, Diophantus needs to display rationality in a precise way: both
allowing quick calculation of the relationship between symbols, as well
as allowing a synoptic – as well as semantic – grasp of the contents of
the terms involved. To do this, he uses symbols in a precise way, which
I call bimodal. Th e symbols are simultaneously verbal and visual, and in
this way they provide both quick calculation and a semantic grasp. What
fi nally makes Diophantus’ symbols have this property? Th is, I argue,
derives from the nature of the symbolism as used in scribal practice in
pre-print Greek civilizations. Th is involves the one main piece of empirical
research underlying this chapter. I have studied systematically a group of
Diophantine manuscripts, and consulted others, to show a result which is
mainly negative: it must be assumed that, in the manuscript tradition, the
decision whether to employ a full word or its abbreviation was left to the
(^6) By ‘Diophantus’ I mean – as we typically do – ‘the author of the Arithmetica ’. I have no fi rm
views on the authorship of On Polygonal Numbers , a work closer to the mainstream of Greek
geometrical style. If indeed the two works had the same author (as the manuscripts suggest) we
will fi nd that, for diff erent purposes, Diophantus could deploy diff erent genres – not a trivial
result – but neither one to change our understanding of the genre of the Arithmetica. But we
are not in a position to make even this modest statement so that it is best to concentrate on the
Arithmetica alone.