Th e pluralism of Greek ‘mathematics’ 303
Aristotle was, of course, the fi rst to propose an explicit defi nition of rigor-
ous demonstration, which must proceed by way of valid deductive argu-
ment from premisses that are not just true, but also necessary, primary,
immediate, better known than, prior to and explanatory of the conclusions.
Furthermore Aristotle draws up a more elaborate taxonomy of arguments
than Plato had done, distinguishing demonstrative, dialectical, rhetorical,
sophistic and eristic reasoning according fi rst to the aims of the reasoner
(which might be the truth, or victory, or reputation) and secondly to the
nature of the premisses used (necessary, probable, or indeed contentious).
Yet while the ideal that Aristotle sets for philosophy and for mathematics
is rigorous, axiomatic–deductive, demonstration, he not only allows that
the rhetorician will rely on what he calls rhetorical demonstration, but
concedes that in philosophy itself there may be stricter and looser modes,
appropriate to diff erent subject matter. 12
Th e goal the philosophers set themselves was certainty – where the con-
clusions reached were, supposedly, immune to the types of challenges that
always occurred in the law courts and assemblies. Yet from some points
of view the best area to exemplify this was not philosophy itself (ontology,
epistemology or ethics) but, of course, mathematics. However, the attitudes
of both Plato and Aristotle themselves towards mathematics were distinctly
ambivalent – not that they agreed on the status of that study. For Plato,
the inquiries the mathematician engages in are inferior to dialectic itself:
they are part of the prior training for the philosopher, but do not belong to
philosophy itself. Th e grounds for this that he puts forward in the Republic
are twofold, that the mathematician uses diagrams and that he takes his
‘hypotheses’ for granted, as ‘clear to all’. 13 So although mathematics studies
intelligible objects and so is superior to any study devoted to perceptible
ones, it is inferior to dialectic which is purportedly based ultimately on an
‘unhypothesised starting point’, the idea of the Good.
Aristotle, by contrast, clearly accepts that mathematical arguments can
meet the requirements of the strictest mode of demonstration, since he
privileges mathematical examples to illustrate that mode in the Posterior
12 Lloyd 1996 : ch. 1.
13 Th e interpretation of the expression ‘as clear to all’, hōs panti phanerōn , in the Republic
510d1, is disputed. My own view is that Plato is unlikely not to have been aware that many
of the hypotheses adopted by the mathematicians were contested (including for example the
defi nitions of straight line and point). When Socrates says that the mathematicians give no
account to themselves or anyone else about their starting-points, it would seem that this is
their claim , rather than (as it has generally been taken) their warrant. Burnyeat ( 2000 : 37),
however, has argued that there is no criticism of mathematics in this text, but simply an
observation of an inevitable feature of their methods.