The History of Mathematical Proof in Ancient Traditions

327

10 Reasoning and symbolism in Diophantus:

preliminary observations

Reviel Netz

In memoriam D. H. F. Fowler

1. Introducing the problem

Th is chapter raises two separate questions, one dealing with the role of rea-
soning in Diophantus, the other with the role of symbolism. 1 Needless to
say, this discussion of symbolism and reasoning in Diophantus is of philo-
sophical interest, as the nature of symbolic reasoning is central to modern
philosophy of mathematics. My main interest, for this philosophical ques-
tion, is to underline our need to consider the demonstrative function of
symbolism cognitively and historically. Th e promise of symbolic reasoning
was oft en seen as a transition into a mode of reasoning where the subjec-
tive mind is excluded, and an impersonal machine-like calculation takes
its place. 2 But in reality, of course, the turn into symbolic proof must have
involved the transition from one kind of subjective operation to another,
from one set of cognitive tools to another. Th e abstract question, concern-
ing the role of formalism as such in mathematics, may blind us to the actual
cognitive functions served by various formal tools in diff erent historical
constellations. Th is chapter, then, may serve as an example for this kind of
cognitive and historical investigation.
Th e specifi c question concerning symbolism and reasoning in Diophantus
is especially diffi cult and interesting. Ever since the work of Nesselmann

1 Th e central idea of this article – that Diophantine symbolism should be primarily understood
against the wider pattern of scribal practices – was fi rst suggested to me in a conversation with
David Fowler. I will forever remember, forever miss, his voice.
2 Th e locus classicus for that is Wittgenstein’s Trac tatu s (Wittgenstein 1922 ) e.g. 6.126: ‘Whether
a proposition belongs to logic can be calculated by calculating the logical properties of the
symbol... ’ (italics in the original); 6.1262: ‘Proof in logic is only a mechanical expedient to
facilitate the recognition of tautology, where it is complicated.’ Probably, though, even the
Wittgenstein of the Trac tatu s would not have denied the possibility of studying the cognitive
and historical conditions under which a certain ‘mechanical expedient’ in fact ‘facilitates
the recognition of tautology’. But the thrust of the philosophy of mathematics suggested by
Wittgenstein’s Trac tatu s was to turn attention away from the proving mind and hand and on to
the proof ’s symbols.