Th e pluralism of Greek ‘mathematics’ 305
from premisses that are themselves indemonstrable – to avoid the twin
fl aws of circular argument and an infi nite regress. If the premisses could
be proved, then they should be, and that in turn meant that they could
not be considered ultimate, or primary, premisses. Th e latter had to be
self-evident, autopista , or ex heautōn pista. Yet the actual premisses we fi nd
used in diff erent investigations are very varied. To start with, the kinds
or categories of starting points needed were the subject of considerable
terminological instability. Aristotle distinguished three types, defi nitions,
hypotheses and axioms, the latter being subdivided into those specifi c to
a particular study, such as the equality axiom, and general principles that
had to be presupposed for intelligible communication, such as the laws of
non-contradiction and excluded middle. Euclid’s triad consisted of defi ni-
tions, common opinions (including the equality axiom) and postulates.
Archimedes in turn begins his inquiries into statics and hydrostatics by
setting out, for example, the postulates, aitēmata , and the propositions that
are to be granted, lambanomena , and elsewhere the primary premisses are
just called starting points or principles, archai.
As regards the actual principles that fi gure in diff erent investigations, they
were far from confi ned to what Aristotle or Euclid would have accepted as
axioms. In Aristarchus’ exploration of the heliocentric hypothesis, he set out
among his premisses that the fi xed stars and the sun remain unmoved and
that the earth is borne round the sun on a circle, where that circle bears the
same proportion to the distance of the fi xed stars as the centre of a sphere to
its surface. Archimedes, who reports those hypotheses in the Sand-Reckoner
2 218.7–31, remarks that strictly speaking that would place the fi xed stars
at infi nite distance. Th e assumption involves, then, what we would call an
idealization, where the error introduced can be discounted. But in his only
extant treatise, On the Sizes and Distances of the Sun and Moon , Aristarchus’
assumptions include a value for the angular diameter of the moon as 2°, a
fi gure that is far more likely, in my view, to have been hypothetical in the
sense of adopted purely for the sake of argument, than axiomatic in the
sense of accepted as true. Meanwhile outside mathematics, we fi nd Galen,
for example, taking the principles that nature does nothing in vain, and
that nothing happens without a cause, as indemonstrable starting points
for certain deductions in medicine. In Proclus, the physical principles that
natural motion is from, to, or around the centre, are similarly treated as
indemonstrable truths on which natural philosophy can be based.
Th e disputable character of many of the principles adopted as axiomatic
is clear. Euclid’s own parallel postulate was attacked on the grounds that
it should be a theorem proved within the system, not a postulate at all,