Reasoning and symbolism in Diophantus 351
It is natural that, among the models available to him from elite literate
Greek mathematics, Diophantus would choose that of analysis. While not
the most common form of presenting propositions, it is very markedly
associated with problems rather than with theorems – i.e. with those situa-
tions where one is faced not with a statement, whose truth is to be corrobo-
rated, but with a task which is to be fulfi lled. 24 Th is is of course the nature
of the material Diophantus had available to him. And, since he set out to
produce a systematic, monolithic work, it is natural that he would use the
same form of presentation throughout – resulting in a unique text among
the extant Greek works, consisting of analysis and nothing else.
Th e choice of the analytic form has important consequence for the
nature of the reasoning. Now, it is oft en suggested that analysis is a method
of discovery: that is, it is a way by which Greek mathematicians came to
know how to solve problems. I have written on this question before, in an
article called ‘Why did Greek mathematicians publish their analyses?’ I
shall not repeat in detail what I had to say there, but the title itself suggests
the main argument. 25 Whatever heuristic contribution the analytic move –
of assuming the task fulfi lled – may have had, this cannot account for
writing the analysis down. Th e written-down analysis most certainly is not a
protocol of the discovery of the solution. It must serve some other purpose
in the context of presentation, which is what I was trying to explain in my
article. Like most authors on Greek geometry, I had completely ignored
Diophantus in that previous article of mine, but in fact here is a clear case
for my claim: no doubt, Diophantus in general knew the values solving his
tasks, as part of his tradition. Th e analysis, for him, was not a way of fi nding
those values, but of presenting them.
What is the contribution of analysis in the context of presentation? I have
suggested the following: when producing solutions to problems (unlike the
case where one sets out proofs of theorems) one faces a special burden of
showing the preferability of the off ered solution to other, alternative solu-
tions. Th is, indeed, was a standard arena of polemic in Greek mathematics:
25 Th e selective discussion in that article may be supplemented by my comments on a few
analyses by Archimedes, in Netz 2004 : 207, 217–18.
24 Th is is the main theme of Knorr 1986. In general, for the nature of ancient analysis, the best
starting-point today is the Stanford Encyclopedia of Philosophy entry, with its rich but well-
chosen bibliography: http://plato.stanford.edu/entries/analysis/ , by M. Beaney. Otte and
Panza 1997 are the best starting point in print. I will state immediately my position, that
much of the discussion of ancient analysis is vitiated by paying too much attention to Pappus’
pronouncements on the topic ( Collectio vii .1–2): while Pappus was not an unintelligent reader
of his sources, it is most likely that he presents not so much any earlier theory but rather his
own interpretation, so that his authority on the subject is that of a secondary source.