328 reviel netz
( 1842 ), it has been widely recognized that Diophantus’ symbols are not
the same as those of modern algebra: his was a syncopated , not a symbolic
algebra, the so-called symbols being essentially abbreviations (for a fuller
account of what that means, see Section 2 below). Building on this under-
standing, we need to avoid the Scylla and Charybdis of Diophantus studies.
One, which may be called the great-divide-history-of-algebra, stresses that
abbreviations are not symbols: Diophantus is not Vieta, and Diophantus’
symbols have no role in his reasoning. 3 Th e other, which may be called the
algebra-is-algebra-history-of-algebra, stresses that symbols (even when
abbreviations in character) are symbols: Diophantus is a symbolic author
and his writings directly prepare the way for modern algebra (this is
assumed with diff erent degrees of sophistication in many general histo-
ries of mathematics). 4 In this chapter, I shall try to show how Diophantus’
symbols derive from his specifi c historical context, and how they serve
a specifi c function in his own type of reasoning: the symbols are neither
purely ornamental, nor modern.
So I do believe that Diophantus’ use of symbolism has a functional role
in his reasoning. But, even apart from any such function, it is interesting
to consider the two together. Th is combination may serve to characterize
Diophantus’ work. First, the work stands out from its predecessors in the
Greek mathematical tradition, indeed in the Greek literary tradition, by its
foregrounding of a special set of symbols. Th is foregrounding is apparent
not only in that the work in its entirety makes use of the symbols, but also
in that the introduction to the work – uniquely in Greek mathematics – is
almost entirely dedicated to the presentation of the symbolism. 5 S e c o n d ,
the work stands out from its predecessors in the Mediterranean tradition
of numerical problems in its foregrounding of demonstration (in a sense
that we shall try to clarify below). Th e text takes the form of a set of argu-
ments leading to clearly demarcated conclusions, throughout organized(^3) For this, see especially Klein 1934 –6, a monograph that makes this claim to be the starting
point of an entire philosophy of the history of mathematics.
(^4) See e.g. Bourbaki 1991 : 48; Boyer 1989 : 204; besides of course being a theme of Bashmakova
1977). Bourbaki is laconic and straightforward (Bourbaki 1991 : 48): ‘Diophantus uses, for
the fi rst time, a literal symbol to represent an unknown in an equation.’ Boyer is balanced
and careful. Noting Nesselmann’s classifi cation, and stating that Diophantus was ‘syncopated’,
he goes on to add that (Boyer 1989 : 204) ‘with such a notation Diophantus was in a position
to write polynomials in a single unknown almost as concisely as we do today’, however, ‘the
chief diff erence between the Diophantine syncopation and the modern algebraic notation
is in the lack of special symbols for operations and relations, as well as of the exponential
notation’.
(^5) Th e introduction is in Tannery I.2–16, of which i .4.6–12.21 is organized around the
presentation of the symbolism.