Th e pluralism of Greek ‘mathematics’ 301
heavenly bodies, but in that of events on earth on their basis – astrology,
in other words, on our terms – that study is downgraded precisely on the
grounds that it cannot deliver demonstration. It is conjectural, though he
would claim that it is based on tried and tested assumptions.
(3.5) Fifthly and finally there are Pappus’ critical remarks, in the opening
chapters (1–23) of Book 3 of his Collection , on certain procedures based on
approximations that had been used in tackling the problem of fi nding two
mean proportionals in order to solve the Delian problem, of doubling the
cube.^10 Although certain stepwise approximations can yield a result that is
correct, they fall short, in Pappus’ view, in rigour. Pappus himself distin-
guishes between planar, solid and linear problems in geometry and insists that
each has its own procedures appropriate for the subject matter in question.
What we fi nd in all of the cases I have taken is a sensitivity not just to the
correctness of results or the truth of conclusions, but to the appropriateness
or otherwise of the methods used to obtain them. It is not enough just to
know the truth of a theorem: nor is it enough to have some means of justify-
ing the claim to such knowledge. No: what is required is that the method of
justifi cation be the correct one for the fi eld of inquiry concerned according
to the particular standards of correctness of the author in question. Th at is
the recurrent demand: yet it is clearly not the case that all Greek investiga-
tors who would have considered themselves mathēmatikoi agreed on what
is appropriate in each type of case or had uniform views on what counts as
a demonstration.
Similar second-order disputes recur in most other areas of inquiry that
the Greeks engaged in, and this too is worth illustrating since it suggests
that the phenomenon we have described in mathematics is symptomatic of
more general tendencies in Greek thought. Sometimes we fi nd such disa-
greements within what is broadly the same discipline, sometimes across dif-
ferent disciplines. In medicine the Hippocratic treatise On Ancient Medicine
provides examples of both kinds. Th e author fi rst castigates other doctors
who try to base medical practice on what he calls ‘hypotheses’, arbitrary
postulates such as ‘the hot’, ‘the cold’, ‘the wet’, ‘the dry’ and anything else
they fancy ( CMG 1 1, 36.2–21). In this author’s view, that is wrong-headed
since medicine is and has long been based on experience. Th e investigation
of what happens under the earth or in the sky may be forced to rely on such
postulates, but they are a disaster in medicine, where they have the result of
narrowing down the causal principles of diseases. While that drives a wedge
between medicine and ‘meteorology’, he goes on in chapter 20 (51.6–18)
10 I may refer to the detailed analysis in Cuomo 2000 : ch. 4.