Th e pluralism of Greek ‘mathematics’ 297
device attributed to him, the so-called quadratrix, is too sophisticated for
the fi ft h century.
Although much remains obscure about the precise claims made in diff er-
ent attempts at quadrature, it is abundantly clear fi rst that diff erent inves-
tigators adopted diff erent assumptions about the legitimacy of diff erent
methods, and second that those investigators were a heterogeneous group.
Some were not otherwise engaged in mathematical studies at all, at least to
judge from the evidence available to us. An allusion in Aristophanes ( Birds
1001–5) suggests that the topic of squaring the circle had by the end of the
fi ft h century become a matter of general interest, or at least the possible
subject of anti-intellectual jokes in comedy.
Among those I have mentioned in relation to quadratures several are
generally labelled ‘sophists’, this too a notoriously indeterminate category
and one that evidently cannot be seen as an alternative to ‘mathematician’.
As is well known Plato does not always use the term pejoratively, even
though he certainly has severe criticisms to off er, both intellectual and
moral, of several of the principal fi gures he calls ‘sophists’. Yet Plato himself
provides plenty of evidence of the range of interests, both mathematical
and non-mathematical, of some of those he names as such. As regards the
Hippias he calls a sophist, those interests included astronomy, geometry,
arithmetic, but also, for instance, linguistics: however, whether the music
he also taught related to the mathematical analysis of harmonics or was
a matter of the more general aesthetic evaluation of diff erent modes is
unclear. Again, the fragments that are extant from Antiphon’s treatise Tr uth
deal with questions in cosmology, meteorology, geology and biology. 8
Protagoras, who is said by Plato to have been the fi rst to have taught for a
fee, famously claimed, according to Aristotle Metaphysics 998a2–4, that the
tangent does not touch the circle at a point, a meta-mathematical objection
that he raised against the geometers.
Th us far I have suggested some of the variety within what the Greeks
themselves thought of as encompassed by mathēmatikē together with
some of the heterogeneity of those who were described as engaged in
‘mathematical’ inquiries. But in view of some persistent stereotypes of
Greek mathematics it is important to underline the further fundamental
disagreements (1) about the classifi cation of the mathematical sciences and
the hierarchy within them, (2) about the question of their usefulness, and
(^8) Th e identifi cation of the author of this treatise with the Antiphon whose quadrature is
criticized by Aristotle is less disputed than the question of whether the sophist is identical with
the author called Antiphon whose Tetralogies are extant.