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 thinker¹ 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.² 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.³ 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 mixture⁴; (b) the result of mixture is something different; and unity will not be separable,⁵ 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.⁶ Is it derived as from seed? But nothing can be emitted from that which is indivisible.⁷ 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 thinker⁸ regards unity as contrary to plurality, 1092band 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.⁹ 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,¹⁰ 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.¹¹ 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,¹² e.g. as points are the bounds of spatial magnitudes,¹³ and as Eurytus¹⁴ 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.¹⁵ 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?¹⁶ And clearly numbers are not the essence of things, nor are they causes of the form; for the ratio¹⁷ is the essence, and number¹⁸ 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.¹⁹ 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.

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,²⁰ or an odd number.²¹ For in point of fact honey-water is no more wholesome if it is mixed in the proportion “three times three”²²; 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 2²³; 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 3.²⁴ 1093aIf all things must share in number, it must follow that many things are the same; i.e., that the same number belongs both to this thing and to something else. Is number, then, a cause; i.e., is it because of number that the object exists? Or is this not conclusive? E.g., there is a certain number of the sun’s motions, and again of the moon’s,²⁵ and indeed of the life and maturity of every animate thing. What reason, then, is there why some of these numbers should not be squares and others cubes, some equal and others double? There is no reason; all things must fall within this range of numbers if, as was assumed, all things share in number, and different things may fall under the same number. Hence if certain things happened to have the same number, on the Pythagorean view they would be the same as one another, because they would have the same form of number; e.g., sun and moon would be the same.²⁶ But why are these numbers causes? There are seven vowels,²⁷ seven strings to the scale,²⁸ seven Pleiads; most animals (though not all²⁹) lose their teeth in the seventh year; and there were seven heroes who attacked Thebes. Is it, then, because the number 7 is such as it is that there were seven heroes, or that the Pleiads consist of seven stars? Surely there were seven heroes because of the seven gates, or for some other reason, and the Pleiads are seven because we count them so; just as we count the Bear as 12, whereas others count more stars in both. Indeed, they assert also that Ξ, Ψ and Ζ are concords,³⁰ and that because there are three concords, there are three double consonants. They ignore the fact that there might be thousands of double consonants—because there might be one symbol for ΓΡ. But if they say that each of these letters is double any of the others, whereas no other is,³¹ and that the reason is that there are three regions³² of the mouth, and that one consonant is combined with σ in each region, it is for this reason that there are only three double consonants, and not because there are three concords—because there are really more than three; but there cannot be more than three double consonants. Thus these thinkers are like the ancient Homeric scholars, who see minor similarities but overlook important ones. Some say that there are many correspondences of this kind; e.g., the middle notes³³ of the octave are respectively 8 and 9, and the epic hexameter has seventeen syllables, which equals the sum of these two; 1093band the line scans in the first half with nine syllables, and in the second with eight.³⁴ And they point out that the interval from α to ω in the alphabet is equal to that from the lowest note of a flute to the highest, whose number is equal to that of the whole system of the universe.³⁵ We must realize that no one would find any difficulty either in discovering or in stating such correspondences as these in the realm of eternal things, since they occur even among perishable things. As for the celebrated characteristics of number, and their contraries, and in general the mathematical properties, in the sense that some describe them and make them out to be causes of the natural world, it would seem that if we examine them along these lines, they disappear; for not one of them is a cause in any of the senses which we distinguished with until respect to the first Principles.³⁶ There is a sense, however, in which these thinkers make it clear that goodness is predicable of numbers, and that the odd, the straight, the equal-by-equal,³⁷ and the powers³⁸ of certain numbers, belong to the series of the Beautiful.³⁹ For the seasons are connected with a certain kind of number⁴⁰; and the other examples which they adduce from mathematical theorems all have the same force. Hence they would seem to be mere coincidences, for they are accidental; but all the examples are appropriate to each other, and they are one by analogy. For there is analogy between all the categories of Being—as “straight” is in length, so is “level” in breadth, perhaps “odd” in number, and “white” in color. Again, it is not the Ideal numbers that are the causes of harmonic relations, etc. (for Ideal numbers, even when they are equal, differ in kind, since their units also differ in kind)⁴¹; so on this ground at least we need not posit Forms. Such, then, are the consequences of the theory, and even more might be adduced. But the mere fact that the Platonists find so much trouble with regard to the generation of Ideal numbers, and can in no way build up a system, would seem to be a proof that the objects of mathematics are not separable from sensible things, as some maintain, and that the first principles are not those which these thinkers assume.