298 g e o f f r e y L l o y d
especially (3) on what counts as proper, valid, arguments and methods. Let
me deal briefl y with the fi rst two questions before exemplifying the third a
little more fully.
(1) Already in the late fi ft h and early fourth centuries bce a divergence
of opinion is reported as between Philolaus and Archytas. According to
Plutarch ( Ta b l e Ta l k 8 2 1, 718e) Philolaus insisted that geometry is the
primary mathematical study (its ‘metropolis’). But Archytas privileged
arithmetic under the rubric of logistikē (reckoning, calculation, Fr. 4). Th e
point is not trivial, since how precisely geometry and arithmetic could
be considered to form a unity was problematic. According to the normal
Greek conception, ‘number’ is defi ned as an integer greater than 1. In this
view, arithmetic dealt with discrete entities. But geometry treated of an
infi nitely divisible continuum. Nevertheless both were regularly included
as branches of ‘mathematics’, sister branches, indeed, as Archytas called
them (Fr. 1). Th e question of the status of other studies was more con-
tested. For Aristotle, who had, as we shall see, a distinctive philosophy
of mathematics, such disciplines as optics, harmonics and astronomy
were ‘the more physical of the mathēmata ’ ( Physics 194a7–8). Th e issue of
‘mechanics’ was particularly controversial. According to the view of Hero,
as reported by Pappus ( Collection Book 8 1–2), mechanics had two parts,
the theoretical which consisted of geometry, arithmetic, astronomy and
physics, and the practical that dealt with such matters as the construction
of pulleys, war machines and the like. However, a somewhat diff erent view
was propounded by Proclus ( Commentary on Euclid’s Elements 41.3 – 42.8)
when he included what we should call statics, as well as pneumatics, under
‘mechanics’.
(2) Th at takes me to my next topic, the issue of the usefulness of math-
ematics, howsoever construed. Already in the classical period there was a
clear division between those who sought to argue that mathematics should
be studied for its practical utility, and those who saw it rather as an intel-
lectual, theoretical discipline. In Xenophon’s Memorabilia 4 7 2–5 Socrates
is made to insist that geometry is useful for land measurement, astronomy
for calendar regulation and navigation, and so on, and he there dismissed
the more theoretical or abstract aspects of those subjects. Similarly Isocrates
too distinguished the practical and the theoretical sides of mathematical
studies and in certain circumstances favoured the former (11 22–3, 12 26–8,
15 261–5). Yet Plato of course took precisely the opposite view. It is not for
practical, mundane, reasons that mathematics is worth studying, but rather
as a training for the soul in abstract thought. But even some who empha-
sized practical utility sometimes defi ned that very broadly. It is striking that