Th e pluralism of Greek ‘mathematics’ 307
aesthetics that began to be cultivated in the Hellenistic period. 19 I n t h e c a s e
of mathematics, there were occasions when its practitioners delighted in
complexity and puzzlement for their own sakes.
From a comparative perspective what are the important lessons to be
learnt from the material I have thus cursorily surveyed in this discussion?
Th e points made in my last paragraph provide the basis for an argument
that tends to turn a common assumption about Greek mathematics on its
head. While one image of mathematics that many ancients as well as quite
a few modern commentators promoted has it that mathematics is the realm
of the indisputable, it is precisely the disputes about both fi rst-order prac-
tices and second-order analysis that mark out the ancient Greek experience
in this fi eld. Divergent views were entertained not just about what ‘math-
ematics’ covered, but on what its proper aims and methods should be. Th e
very fl uidity and indeterminacy of the boundaries between diff erent intel-
lectual disciplines may be thought to have contributed to the construction
of that image of mathematics as the realm of the incontrovertible – con-
tested as that image was. But we may remark that that idea owed as much to
the ruminations of the philosophers – who used it to propose an ideal of a
‘philosophy’ that could equal and indeed surpass mathematics – as it did to
the actual practices of the mathematicians themselves.
It may once have been assumed that the development of the axiomatic–
deductive mode of demonstration was an essential feature of the develop-
ment of mathematics itself. But as other studies in this volume amply show,
there are plenty of ancient traditions of mathematical inquiry that got on
perfectly well, grew and fl ourished, without any idea of the need to defi ne
their axiomatic foundations. In Greece itself, as we have seen, it is far from
being the case that all those who considered themselves, or were considered
by others, to be mathematicians thought that axiomatics was obligatory.
Th is raises, then, two key questions with important implications for
comparativist studies. First how can we begin to account for the particular
heterogeneity of the Greek mathematical experience and for the way in
which the axiomatic–deductive model became dominant in some quarters?
Second what were the consequences of the hierarchization we fi nd in some
writers on the development and practice of mathematics itself?
In relation to the fi rst question, my argument is that there was a crucial
input from the side of philosophy, in that it was the philosopher Aristotle
who fi rst explicitly defi ned rigorous demonstration in terms of valid
deductive argument from indemonstrable primary premisses – an ideal
19 Netz 2009.