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[1092b] [1] and another (treating it as the Equal) as contrary to the Unequal, number must be derived as from contraries.Hence there is something else which persists from which, together with one contrary, number is or has been derived.1

Further, why on earth is it that whereas all other things which are derived from contraries or have contraries perish, even if the contrary is exhausted in producing them,2 number does not perish? Of this no explanation is given; yet whether it is inherent or not, a contrary is destructive; e.g., Strife destroys the mixture.3 It should not, however, do this; because the mixture is not its contrary.

Nor is it in any way defined in which sense numbers are the causes of substances and of Being; whether as bounds,4 e.g. as points are the bounds of spatial magnitudes,5 and as Eurytus6 determined which number belongs to which thing—e.g. this number to man, and this to horse—by using pebbles to copy the shape of natural objects, like those who arrange numbers in the form of geometrical figures, the triangle and the square.7 Or is it because harmony is a ratio of numbers, and so too is man and everything else? But in what sense are attributes—white, and sweet, and hot—numbers?8 And clearly numbers are not the essence of things, nor are they causes of the form; for the ratio9 is the essence, and number10 is matter.E.g. the essence of flesh or bone is number only in the sense that it is three parts of fire and two of earth.11 And the number, [20] whatever it is, is always a number of something; of particles of fire or earth, or of units. But the essence is the proportion of one quantity to another in the mixture; i.e. no longer a number, but a ratio of the mixture of numbers, either of corporeal particles or of any other kind. Thus number is not an efficient cause—neither number in general, nor that which consists of abstract units—nor is it the matter, nor the formula or form of things. Nor again is it a final cause.

The question might also be raised as to what the good is which things derive from numbers because their mixture can be expressed by a number, either one which is easily calculable,12 or an odd number.13 For in point of fact honey-water is no more wholesome if it is mixed in the proportion "three times three"14; it would be more beneficial mixed in no particular proportion, provided that it be diluted, than mixed in an arithmetical proportion, but strong.Again, the ratios of mixtures are expressed by the relation of numbers, and not simply by numbers; e.g., it is 3 : 2, not 3 X 215; for in products of multiplication the units must belong to the same genus. Thus the product of 1 x 2 x 3 must be measurable by 1, and the product of 4 X 5 x 7 by 4. Therefore all products which contain the same factor must be measurable by that factor. Hence the number of fire cannot be 2 X 5 X 3 X 7 if the number of water is 2 x 3.16

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).

4 The theories criticized from this point onwards to Aristot. Met. 14.6.11 are primarily Pythagorean. See Introduction.

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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