Th e pluralism of Greek ‘mathematics’ 299
in the passage just quoted from Pappus he included both the construction
of models of planetary motion and that of the marvellous gadgets of the
‘wonder-workers’ among ‘the most necessary of the mechanical arts from the
point of view of the needs of life’. Meanwhile the most ambitious claims for
the all-encompassing importance of ‘mathematics’ were made by the neo-
Pythagorean Iamblichus at the turn of the third and fourth centuries ce. He
argued in On the Common Mathematical Science (ch. 32: 93.11–94.21) that
mathematics was the source of understanding in every mode of knowledge,
including in the study of nature and of change.
(3) From among the many examples that illustrate how the question of
the proper method in mathematics was disputed let me select just fi ve.
(3.1) In a famous and infl uential passage in his Life of Marcellus
( ch. 14 , cf. Ta b l e Ta l k 8 2 1, 718ef ) Plutarch interprets Plato as having
banned mechanical methods from geometry on the grounds that these
corrupted and destroyed the pure excellence of that subject, and it is true
that Plato had protested that to treat mathematical objects as subject to
movement was absurd. Th e fi rst to introduce such degenerate methods,
according to Plutarch, were Eudoxus and Archytas. Indeed we know from
a report in Eutocius ( Commentary on Archimedes Sphere and Cylinder 2, 3
84.12–88.2) that Archytas solved the problem of fi nding two mean propor-
tionals on which the duplication of a cube depended by means of a complex
three-dimensional kinematic construction involving the intersection of
three surfaces of revolution, a right cone, a cylinder and a tore. Plutarch
even goes on to suggest that Archimedes himself agreed with the Platonic
view (as Plutarch represents it) that the work of an ‘engineer’ was ignoble
and vulgar. Most scholars are agreed fi rst that that most probably misrepre-
sents Archimedes, and secondly that few practising mathematicians would
have shared Plutarch’s expressed opinion as to the illegitimacy of mechani-
cal methods in geometry.
(3.2) My second example comes from Archimedes himself and concerns
precisely how he endorsed the usefulness of mechanics, as a method of
discovery at least. In his Method (2 428.18–430.18) he sets out what he
describes as his ‘mechanical’ method which depends fi rst on an assump-
tion of indivisibles and then on imagining geometrical fi gures as balanced
against one another about a fulcrum. Th e method is then applied to get
the area of a segment of a parabola, but while Archimedes accepts the
method as a method of discovery, he puts it that the results have thereaft er
to be demonstrated rigorously using the method of exhaustion standard
throughout Greek geometry. At the same time the method is useful ‘even
for the proofs of the theorems themselves’ in a way he explains ( Method