Section 1

With regard to this kind of substance, 1 then, let the foregoing account suffice. All thinkers make the first principles contraries; as in the realm of natural objects, so too in respect of the unchangeable substances.

Now if nothing can be prior to the first principle of all things, that first principle cannot be first principle if it is an attribute of something else. This would be as absurd as to say that white is the first principle, not qua anything else but qua white, and yet that it is predicable of a subject, and is white because it is an attribute of something else; because the latter will be prior to it.

Moreover, all things are generated from contraries as from a substrate, and therefore contraries must most certainly have a substrate. Therefore all contraries are predicated of a subject, and none of them exists separately. But there is no contrary to substance; not only is this apparent, but it is borne out by reasoned consideration. 2 Thus none of the contraries is strictly a first principle; the first principle is something different.

But the Platonists treat one of the contraries as matter, some opposing the unequal to Unity (on the ground that the former is of the nature of plurality) and others plurality.

For according to some, 3 numbers are generated from the unequal dyad of the Great and Small; and according to another, 4 from plurality; but in both cases they are generated by the essence of unity. For he who speaks of the unequal and Unity as elements, and describes the unequal as a dyad composed of Great and Small, speaks of the unequal, i.e. the Great and Small, as being one; and does not draw the distinction that they are one in formula but not in number. 5

Again, they state the first principles, which they call elements, badly; some say that the Great and the Small, together with Unity (making 3 6 in all), are the elements of numbers; the two former as matter, and Unity as form. Others speak of the Many and Few, because the Great and the Small are in their nature more suited to be the principles of magnitude; and others use the more general term which covers these—the exceeding and the exceeded.

But none of these variations makes any appreciable difference with respect to some of the consequences of the theory; they only affect the abstract difficulties, which these thinkers escape because the proofs which they themselves employ are abstract.

There is, however, this exception: if the exceeding and the exceeded are the first principles, and not the Great and the Small, on the same principle number should be derived from the elements before 2 is derived; for as the exceeding and the exceeded is more universal than the Great and Small, so number is more universal than 2. But in point of fact they assert the one and not the other.

Others oppose the different or other to Unity; and others contrast Plurality and Unity.

Now if, as they maintain, existing things are derived from contraries, and if there is either no contrary to unity, or if there is to be any contrary it is plurality; and if the unequal is contrary to the equal, and the different to the same, and the other to the thing itself then those who oppose unity to plurality have the best claim to credibility—but even their theory is inadequate, because then unity will be few. For plurality is opposed to paucity, and many to few.

That unity denotes a measure 7 is obvious. And in every case there is something else which underlies it; e.g., in the scale there is the quarter-tone; in spatial magnitude the inch or foot or some similar thing; and in rhythms the foot or syllable. Similarly in the case of gravity there is some definite weight. Unity is predicated of all things in the same way; of qualities as a quality, and of quantities as a quantity.

(The measure is indivisible, in the former case in kind, and in the latter to our senses.) This shows that unity is not any independent substance. And this is reasonable; because unity denotes a measure of some plurality, and number denotes a measured plurality and a plurality of measures. (Hence too it stands to reason that unity is not a number; for the measure is not measures, but the measure and unity are starting-points.)

The measure must always be something which applies to all alike; e.g., if the things are horses, the measure is a horse; if they are men, the measure is a man; and if they are man, horse and god, the measure will presumably be an animate being, and the number of them animate beings.

If the things are man, white and walking, there will scarcely be a number of them, because they all belong to a subject which is one and the same in number; however, their number will be a number of genera, or some other such appellation.

Those 8 who regard the unequal as a unity, and the dyad as an indeterminate compound of great and small, hold theories which are very far from being probable or possible. For these terms represent affections and attributes, rather than substrates, of numbers and magnitudes—many and few applying to number, and great and small to magnitude— just as odd and even, smooth and rough, straight and crooked, are attributes.

Further, in addition to this error, great and small and all other such terms must be relative. And the relative is of all the categories in the least degree a definite entity or substance; it is posterior to quality and quantity. The relative is an affection of quantity, as we have said, and not its matter; since there is something else distinct which is the matter both of the relative in general and of its parts and kinds.

There is nothing great or small, many or few, or in general relative, which is many or few, great or small, or relative to something else without having a distinct nature of its own. That the relative is in the lowest degree a substance and a real thing is shown by the fact that of it alone 9 there is neither generation nor destruction nor change in the sense that in respect of quantity there is increase and decrease, in respect of quality, alteration, in respect of place, locomotion, and in respect of substance, absolute generation and destruction.

There is no real change in respect of the relative; for without any change in itself, one term will be now greater, now smaller or equal, as the other term undergoes quantitative change. Moreover, the matter of every thing, and therefore of substance, must be that which is potentially of that nature; but the relative is neither potentially substance nor actually.

It is absurd, then, or rather impossible, to represent non-substance as an element of substance and prior to it; for all the other categories are posterior to substance. And further, the elements are not predicated of those things of which they are elements; yet many and few are predicated, both separately and together, of number; and long and short are predicated of the line, and the Plane is both broad and narrow.

If, then, there is a plurality of which one term, viz. few, is always predicable, e.g. 2 (for if 2 is many, 1 will be few 10 ), then there will be an absolute many; e.g., 10 will be many (if there is nothing more than 10 11 ), or 10,000. How, then, in this light, can number be derived from Few and Many? Either both ought to be predicated of it, or neither; but according to this view only one or the other is predicated.