If, then, it is impossible both not to include the Good among the first principles, and to include it in this way, it is clear that the first principles are not being rightly represented, nor are the primary substances. Nor is a certain thinker Footnote 1Evidently Speusippus; cf. Aristot. Met. 14.4.3. right in his assumption when he likens the principles of the universe to that of animals and plants, on the ground that the more perfect forms are always produced from those which are indeterminate and imperfect, and is led by this to assert that this is true also of the ultimate principles; so that not even unity itself is a real thing. Footnote 2Speusippus argued that since all things are originally imperfect, unity, which is the first principle, must be imperfect, and therefore distinct from the good. Aristotle objects that the imperfect does not really exist, and so Speusippus deprives his first principle of reality.
He is wrong; for even in the natural world the principles from which these things are derived are perfect and complete—for it is man that begets man; the seed does not come first. Footnote 3Cf. Aristot. Met. 9.8.5. It is absurd also to generate space simultaneously with the mathematical solids (for space is peculiar to particular things, which is why they are separable in space, whereas the objects of mathematics have no position) and to say that they must be somewhere, and yet not explain what their spatial position is.
Those who assert that reality is derived from elements, and that numbers are the primary realities, ought to have first distinguished the senses in which one thing is derived from another, and then explained in what way number is derived from the first principles. Is it by mixture? But (a) not everything admits of mixture Footnote 4e.g. to admit of mixture a thing must first have a separate existence, and the Great-and-Small, which is an affection or quality of number (Aristot. Met. 14.1.14) cannot exist separately. ; (b) the result of mixture is something different; and unity will not be separable, Footnote 5sc. when it has once been mixed. Cf. Aristot. De Gen. et Corr. 327b 21-26. nor will it be a distinct entity, as they intend it to be.
Is it by composition, as we hold of the syllable? But (a) this necessarily implies position; (b) in thinking of unity and plurality we shall think of them separately. This, then, is what number will be—a unit plus plurality, or unity plus the Unequal.
And since a thing is derived from elements either as inherent or as not inherent in it, in which way is number so derived? Derivation from inherent elements is only possible for things which admit of generation. Footnote 6And numbers are supposed to be eternal. Cf. Aristot. Met. 14.2.1-3. Is it derived as from seed?
But nothing can be emitted from that which is indivisible. Footnote 7i.e., unity, being indivisible, cannot contribute the formal principle of generation in the way that the male parent contributes it. Is it derived from a contrary which does not persist? But all things which derive their being in this way derive it also from something else which does persist. Since, therefore, one thinker Footnote 8Speusippus: Plato. Cf. Aristot. Met. 14.1.5. regards unity as contrary to plurality, 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. Footnote 9The 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.
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, Footnote 10Because it may be regarded as still potentially present. 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. Footnote 11According to Empedocles Fr. 17 (Diels). 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, Footnote 12The theories criticized from this point onwards to Aristot. Met. 14.6.11 are primarily Pythagorean. See Introduction. e.g. as points are the bounds of spatial magnitudes, Footnote 13e.g. the line by 2 points, the triangle (the simplest plane figure) by 3, the tetrahedron (the simplest solid figure) by 4. and as Eurytus Footnote 14Disciple of Philolaus; he flourished
in the early fourth century B.C. 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. Footnote 15cf. Burnet, E.G.P. sect. 47.
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? Footnote 16This is an objection to the view that numbers are causes as bounds. And clearly numbers are not the essence of things, nor are they causes of the form; for the ratio Footnote 17Or formula.
is the essence, and number Footnote 18In the sense of a number of material particles. 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. Footnote 19Cf. Empedocles Fr. 96 (Diels). And the number, 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.