Capítulo 20. Chapter XX.—Concerning Number
[3.20.151] Since time, it seems, is not found apart from number, it will not be out of place to discuss number briefly.[460] In the customary way we speak undogmatically[461] of numbering a thing and hear number talked of as something which exists; but the extreme methods of the Dogmatists have provoked the attack upon number also.
[3.20.152] Thus, for example, the School of Pythagoras declare that numbers are also elements of the Universe. They assert, in fact, that phenomena are constructed from something, and that the elements must be simple; therefore the elements are non-evident. But of things non-evident, some are corporeal, like atoms and masses, others incorporeal, like figures[462] and forms and numbers. Of these the corporeal are composite, being constructed from length and breadth and depth and solidity, or even weight. The elements, therefore, are not only non- evident but also incorporeal.
[3.20.153] Moreover, each of the incorporeals involves the perception of number, for it is either one or two or more. Thus it is inferred that the non-evident and incorporeal numbers which are involved in all perception are the elements of existing things. Yet not simply ⟨these numbers⟩, but both the Monad also and the Indefinite Dyad which is generated by the expansion of the Monad, and by participation in which the particular dyads become dyads.[463]
[3.20.154] For they say that it is from these that the rest of the numbers are generated—those, that is, which are involved in the perception of numerables—and the Universe is arranged. For the point presents the relation, or character, of the Monad,[464] and the line that of the Dyad (it being regarded as lying between two points), and the surface that of the Triad (for they describe it as a flowing of the line breadth-wise up to another point placed transversely), and the ⟨solid⟩ body that of the Tetrad; for Body is formed by an ascension of the surface up to a point placed above.
[3.20.155] It is in this way that they image forth both the bodies and the whole Universe, which also they declare to be arranged according to harmonic ratios[465]—namely, that of the ‟By-Fours,” which is ‟epitrite,” as is the ratio of 8 to 6; and that of the “ By-Fives,” which is one and a half times, as is the ratio of 9 to 6; and that of the ‟By-Alls,” which is double, as is the ratio of 12 to 6.
[3.20.156] These are the fictions they imagine; and they also make out that number is something else apart from numerables, arguing that if ‟animal” according to its proper definition[466] is (say) one, the plant, since it is not an animal, will not be one; but the plant is one; therefore the animal is not one ⟨qua animal⟩ but in virtue of some other attribute perceived outside itself, whereof each animal partakes and because of which it becomes one. And if number is the numerables, since the numerables are men (say) and oxen and horses, number will be men and oxen and horses—and number will be white and black and bearded, if the objects counted should happen to be such.
[3.20.157] But these things are absurd; therefore number is not the numerables, but it has a reality of its own apart from them whereby it is involved in the perception of the numerables and is an element.
So when they had thus concluded that number is not the numerables, there arose in consequence the difficulty about number. For it is argued that if number exists, number is either the actual numerables or something else apart from them; but number is neither the actual numerables, as the Pythagoreans have proved, nor something else apart from them, as we shall show; number, therefore, is nothing.
[3.20.158] That number is nothing apart from the numerables we shall demonstrate by basing our argument on the Monad, for the sake of lucidity of exposition.[467] If the Monad, by partaking in which each of its participants becomes one, is in itself a real object, this Monad will be either one or as many as are its participants. But if it is one, does each of the things said to partake thereof partake of all of it or of a part of it? For if the one man (say) takes all the Monad, there will no longer exist a monad for the one horse to partake of, or the one dog or any one of all the other things which we declare to be one
[3.20.159] —just as, supposing there are a number of naked men, who possess only one garment amongst them, which one man had put on, all the rest will remain naked and without a garment. And if each thing partakes of a part of it, then, in the first place, the Monad will have a part. and parts, too, infinite in number into which it is divided; which is absurd. And further, just as the part of the Decad, such as the Dyad, is not a Decad, so neither will the part of the Monad be a Monad, and for this reason nothing will partake of the Monad. Hence the Monad whereof the particular objects are said to partake is not one.
[3.20.160] But if the monads, by participation in which each of the particular objects is called one, are equal in number to the numerables to which the term ‟one” is applied, the monads partaken of will be infinite in number. And these either partake of a superior[468] monad or of monads equal in number to themselves, or else they do not so partake but are monads apart from any participation.
[3.20.161] Yet if these can be monads without participation, each of the sensibles also will be able to be one without participation in a monad, and so at once the monad said to be perceived as real in itself is overthrown. Whereas, if those monads are monads by participation, either they all partake of one monad, or each partakes of a monad of its own. And if all partake of one, each will be said to partake either of a part or of the whole, and the original difficulties will still remain;
[3.20.162] but if each partakes of its own monad, we must posit a new monad for each of those monads, and others again for the former, and so on ad infinitum. If then, in order to apprehend that there are certain self-sub-sistent monads by participation in which each existing thing is one, it is necessary to apprehend an infinite infinity of intelligible monads, and to apprehend an infinite infinity of intelligible monads is a thing impossible, then it is impossible to show that there are certain intelligible monads and that each existing thing is one through becoming one by participation in its own monad.
[3.20.163] It is absurd, therefore, to say that the monads are as numerous as the things which partake of them. But if the so-called self-subsistent monad is neither one nor as many as its participants, a self-subsistent monad does not exist at all. So likewise none of the other numbers will be self-subsistent; for one may apply to all the numbers the argument which has now been employed in the typical case of the monad. But if number is neither self-subsistent, as we have shown, nor consists in the actual numerables, as the Pythagoreans have demonstrated, and beyond these there is no other alternative, then we must declare that number does not exist.
[3.20.164] In what way, too, is the Dyad said to be generated from the Monad by those who believe that number is something else apart from the numerables? For when we combine a monad with another monad either something external is added to the monads, or something is subtracted from them, or nothing is either added or subtracted. But if nothing is either added or subtracted, there will not be a Dyad. For neither did the monads, when existing apart from each other, contain the Dyad as involved in the perception of them, according to their own definition, nor has any addition now been made to them from without, just as, by hypothesis, nothing has been subtracted.
[3.20.165] Hence the combination of the monad with the monad will not be a dyad, as no addition or subtraction from without takes place. But if subtraction does take place, not only will there not be a dyad but the monads will even be diminished. And if the dyad is added to them from without, so that a dyad may be generated from the monads, the things which appear to be two will be four; for there exists already a monad and a second monad, and when an outside dyad is added to these the result will be the number four.
[3.20.166] And the same argument applies to all the other numbers which are said to be formed as a result of combination.
If, then, the numbers which are said to be compounded from the superior[469] numbers are formed neither by subtraction nor by addition nor without subtraction and addition, the formation of the number which is said to be independent and apart from the numerables is non-composite. But they themselves make it clear that the numbers formed by combination are not ungenerated by asserting that they are compounded and generated from the superior numbers —from the monad, for example, and the Indefinite Dyad.[470]
[3.20.167] So then number does not subsist of itself. But if number neither is conceived as self-existent, nor subsists in the numerables, then, to judge from the subtleties introduced by the Dogmatists, number is nothing.
Let this, then, suffice as an account in outline of what is called the Physical section of philosophy.