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Analytics. But mathematics suff ers from a diff erent shortcoming, in his
view, which relates to the ontological status of the subject matter it deals
with. Unlike Plato, who suggested that mathematics studies separate intelli-
gible objects that are intermediate between the Forms and sensible particu-
lars, Aristotle argued that mathematics is concerned with the mathematical
properties of physical objects. 14 While physical objects meet the require-
ments of substance-hood, what mathematics studies belongs rather to the
category of quantity than to that of substance.
While Plato and Aristotle disagreed about the highest mode of phil-
osophizing, ‘dialectic’ in Plato’s case, ‘fi rst philosophy’ in Aristotle’s, they
both considered philosophy to be supreme and mathematics to be sub-
ordinate to it. Yet mathematics obviously delivered demonstrations, and
exemplifi ed the goal of the certainty and incontrovertibility of arguments,
far more eff ectively than metaphysics, let alone than ethics. Once Euclid’s
Elements had shown how virtually the whole of mathematical knowledge
could be represented as a single, comprehensive system, derived from a
limited number of indemonstrable starting points, that model exerted very
considerable infl uence as an ideal, not just within the mathematical disci-
plines, but well beyond them. 15 Euclid’s own Optics , like many treatises in
harmonics, statics and astronomy, proceeded on an axiomatic–deductive
basis, even though the actual axioms Euclid invoked in that work are prob-
lematic. 16 More remarkably Galen sought to turn parts of medicine into an
axiomatic–deductive system just as Proclus did for theology in his Elements
of Th eology.^17 Th e prestige of proof ‘in the geometrical manner’, more
geometrico , made it the ideal for many investigations despite the apparent
diffi culties of implementing it.
Th e chief problem lay not with deductive argument itself, but with its
premisses. Aristotle had shown that strict demonstration must proceed
14 Lear 1982.
15 As noted, the question of whether Hippocrates of Chios had a clear notion of ultimate
starting-points or axioms in his geometrical studies is disputed. In his quadratures of
lunes he takes a starting-point that is itself proved, and so not a primary premiss. Ancient
historians of mathematics mention the contributions of Archytas, Eudoxus, Th eodorus and
Th eaetetus leading up to Euclid’s own Elements , but while the commentators on that work
identify particular results as having been anticipated by those and other mathematicians, the
issue of how systematic their overall presentation of mathematical knowledge was remains
problematic.
16 Th us one of Euclid’s defi nitions in the Optics (def. 3, 2.7–9: cf. Proposition 1, 2.21–4.8) states
that those things are seen on which visual rays fall, while those are not seen on which they do
not. Th at seems to suggest that visual rays are not dense, a conception that confl icts with the
assumption of the infi nite divisibility of the geometrical continuum. See Brownson 1981 ; Smith
1981 ; Jones 1994.
17 Lloyd 2006c.