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1 The objection is directed against the Platonist treatment of the principles as contraries (cf. Aristot. Met. 14.4.12), and may be illustrated by Aristot. Met. 12.1.5-2.2. Plurality, as the contrary of unity, is privation, not matter; the Platonists should have derived numbers from unity and some other principle which is truly material.
2 Because it may be regarded as still potentially present.
3 According to Empedocles Fr. 17 (Diels).
5 e.g. the line by 2 points, the triangle (the simplest plane figure) by 3, the tetrahedron (the simplest solid figure) by 4.
6 Disciple of Philolaus; he "flourished" in the early fourth century B.C.
7 cf. Burnet, E.G.P. sect. 47.
8 This is an objection to the view that numbers are causes as bounds.
9 Or "formula."
10 In the sense of a number of material particles.
11 Cf. Empedocles Fr. 96 (Diels).
12 i.e., a simple ratio.
13 It is hard to see exactly what this means. If the terms of a ratio are rational, one of them must be odd. Alexander says a ratio like 1 : 3 is meant. Oddness was associated with goodness (cf. Aristot. Met. 1.5.6).
14 Apparently the Pythagoreans meant by this "three parts of water to three of honey." Aristotle goes on to criticize this way of expressing ratios.
15 Cf. previous note.
16 sc. because if so, a particle of fire would simply equal 35 particles of water.
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