Reasoning and symbolism in Diophantus 349
of mathematical contents. Of course, I imagine that he would still prefer a
general proof to a particular one – but only as long as other, no less impor-
tant characteristics of the proof were respected as well. But this, I suggest,
was not the case. I will try to show why in the next section.
Even before that, let us mention the third and most obvious account for
why Diophantus did not present a more general approach. An argument
that comes to mind immediately is that Diophantus did not produce more
general arguments because he did not possess the required symbolism.
Fundamentally, what we then do is to put side by side our symbolism and
that of Diophantus so as to observe the diff erences and then to pronounce
those diff erences as essential for a full-fl edged argument producing a
general algebraical conclusion. Of course, the diff erences are there. In par-
ticular, Diophantus has explicit symbols for a single value in each power: a
single ‘number’ (a single x ), a single ‘dunamis’ (a single x 2 ), a single ‘cube’
(a single x 3 ), etc. Th ere is thus no obvious way of referring even to, say,
two unknowns such as x and y. Th is is a major limitation, and of course it
does curtail Diophantus’ expressive power. Some scholars come close to
suggesting that this, fi nally, is why Diophantus does not produce explicit
general arguments. 23 But by now we can see how weak this argument is, and
this for two reasons.
First, it is perfectly possible to express a general argument without
the typographic symbolism expressing several unknowns, by the simple
method of using natural language (over whose expressive power, aft er all,
typographic symbols have no advantage). Th is is the upshot of text 3 above.
Of course, even though a text such as text 3 does prove a general claim, it
does so in an opaque form that does not display the rationality of the argu-
ment. But this helps to locate the problem more precisely: it is not that, with
Diophantus’ symbolism, it was impossible to prove general claims; rather, it
was impossible to prove general claims in a manner that makes the rational-
ity of the argument transparent.
Second, and crucially, note that it was perfectly possible for Diophantus
to make the rather minimal extensions to his system so as to encompass
multiple variables. Indeed, since the most natural way for him of speaking
of several unknowns was to speak of ‘the fi rst number’, ‘the second number’,
etc., he eff ectively had the symbolism required – all he needed was to make
the choice to put together the less common symbol for ‘number’ together
with the standard abbreviation for numerals: αʹ ʹς would be ‘the fi rst number’,
β´ ς would be ‘the second number’, etc. A bit more cumbersome than
23 See e.g. Heath 1885 : 80–2.