First, then, we must inquire whether the limits are addible or inaddible; and if inaddible, in which of the two ways which we have distinguished. Footnote 1Aristot. Met. 13.6.2, 3. For it is possible either (a) that any one unit is inaddible to any other, or (b) that the units in the Ideal 2 are inaddible to those in the Ideal 3, and thus that the units in each Ideal number are inaddible to those in the other Ideal numbers.
Now if all units are addible and do not differ in kind, we get one type of number only, the mathematical, and the Ideas cannot be the numbers thus produced;
for how can we regard the Idea of Man or Animal, or any other Form, as a number? There is one Idea of each kind of thing: e.g. one of Humanity and another one of Animality; but the numbers which are similar and do not differ in kind are infinitely many, so that this is no more the Idea of Man than any other 3 is. But if the Ideas are not numbers, they cannot exist at all;
for from what principles can the Ideas be derived? Number is derived from Unity and the indeterminate dyad, and the principles and elements are said to be the principles and elements of number, and the Ideas cannot be placed either as prior or as posterior to numbers. Footnote 2Since the only principles which Plato recognizes are Unity and the Dyad, which are numerical (Aristotle insists on regarding them as a kind of 1 and 2), and therefore clearly principles of number; and the Ideas can only be derived from these principles if they (the Ideas) are (a) numbers (which has been proved impossible) or (b) prior or posterior to numbers (i.e., causes or effects of numbers, which they cannot be if they are composed of a different kind of units); then the Ideas are not derived from any principle at all, and therefore do not exist.
But if the units are inaddible in the sense that any one unit is inaddible to any other, the number so composed can be neither mathematical number (since mathematical number consists of units which do not differ, and the facts demonstrated of it fit in with this character) nor Ideal number. For on this view 2 will not be the first number generated from Unity and the indeterminate dyad, and then the other numbers in succession, as they Footnote 3The Platonists. say 2, 3, because the units in the primary 2 are generated at the same time, Footnote 4This was the orthodox Platonist view of the generation of ideal numbers; or at least Aristotle is intending to describe the orthodox view. Plato should not have regarded the Ideal numbers as composed of units at all, and there is no real reason to suppose that he did (see Introduction). But Aristotle infers from the fact that the Ideal 2 is the first number generated (and then the other Ideal numbers in the natural order) that the units of the Ideal 2 are generated simultaneously, and then goes on to show that this is incompatible with the theory of inaddible units. whether, as the originator of the theory held, from unequals Footnote 5i.e., the Great-and-Small, which Aristotle wrongly understands as two unequal things. It is practically certain that Plato used the term (as he did that of Indeterminate Dyad
) to describe indeterminate quantity. See Introduction. (coming into being when these were equalized), or otherwise—
since if we regard the one unit as prior to the other, Footnote 6This is a necessary implication of the theory of inaddible units (cf. Aristot. Met. 13.6.1, 2). it will be prior also to the 2 which is composed of them; because whenever one thing is prior and another posterior, their compound will be prior to the latter and posterior to the former. Footnote 7So the order of generation will be: (i) Unity (ungenerated); (2) first unit in 2; (3) second unit in 2; and the Ideal 2 will come between (2) and (3).
Further, since the Ideal 1 is first, and then comes a particular 1 which is first of the other 1’s but second after the Ideal 1, and then a third 1 which is next
after the second but third after the first 1, it follows that the units will be prior to the numbers after which they are called; e.g., there will be a third unit in 2 before 3 exists, and a fourth and fifth in 3 before these numbers exist. Footnote 8This is a corollary to the previous argument, and depends upon an identification of ones
(including the Ideal One or Unity) with units.
It is true that nobody has represented the units of numbers as inaddible in this way; but according to the principles held by these thinkers even this view is quite reasonable, although in actual fact it is untenable.
For assuming that there is a first unit or first 1, Footnote 9i.e., the Ideal One. it is reasonable that the units should be prior and posterior; and similarly in the case of 2’s, if there is a first 2. For it is reasonable and indeed necessary that after the first there should be a second; and if a second, a third; and so on with the rest in sequence.
But the two statements, that there is after 1 a first and a second unit, and that there is a first 2, are incompatible. These thinkers, however, recognize a first unit and first 1, but not a second and third; and they recognize a first 2, but not a second and third.
It is also evident that if all units are inaddible, there cannot be an Ideal 2 and 3, and similarly with the other numbers;
for whether the units are indistinguishable or each is different in kind from every other, numbers must be produced by addition; e.g. 2 by adding 1 to another 1, and 3 by adding another 1 to the 2, and 4 similarly. Footnote 10This is of course not true of the natural numbers.
This being so, numbers cannot be generated as these thinkers try to generate them, from Unity and the dyad; because 2 becomes a part of 3, Footnote 11i.e., 3 is produced by adding 1 to 2. and 3 of 4, and the same applies to the following numbers.
But according to them 4 was generated from the first 2 and the indeterminate dyad, thus consisting of two 2’s apart from the Ideal 2. Footnote 12Cf. sect. 18. Otherwise 4 will consist of the Ideal 2 and another 2 added to it, and the Ideal 2 will consist of the Ideal 1 and another 1; and if this is so the other element cannot be the indeterminate dyad, because it produces one unit and not a definite 2. Footnote 13The general argument is: Numbers are produced by addition; but this is incompatible with the belief in the Indeterminate Dyad as a generative principle, because, being duplicative, it cannot produce single units.
Again, how can there be other 3’s and 2’s besides the Ideal numbers 3 and 2, and in what way can they be composed of prior and posterior units? All these theories are absurd and fictitious, and there can be no primary 2 and Ideal 3. Yet there must be, if we are to regard Unity and the indeterminate dyad as elements. Footnote 14i.e., if numbers are not generated by addition, there must be Ideal (or natural) numbers.
But if the consequences are impossible, the principles cannot be of this nature.
If, then, any one unit differs in kind from any other, these and other similar consequences necessarily follow. If, on the other hand, while the units in different numbers are different, those which are in the same number are alone indistinguishable from one another, even so the consequences which follow are no less difficult.
For example, in the Ideal number 10 there are ten units, and 10 is composed both of these and of two 5’s. Now since the Ideal 10 is not a chance number, Footnote 15I think Ross’s interpretation of this passage must be right. The Ideal 10 is a unique number, and the numbers contained in it must be ideal and unique; therefore the two 5’s must be specifically different, and so must their units—which contradicts the view under discussion. and is not composed of chance 5’s, any more than of chance units, the units in this number 10 must be different;
for if they are not different, the 5’s of which the 10 is composed will not be different; but since these are different, the units must be different too. Now if the units are different, will there or will there not be other 5’s in this 10, and not only the two? If there are not, the thing is absurd Footnote 16i.e., it is only reasonable to suppose that other 5’s might be made up out of different combinations of the units. ; whereas if there are, what sort of 10 will be composed of them? for there is no other 10 in 10 besides the 10 itself:
Again, it must also be true that 4 is not composed of chance 2’s. For according to them the indeterminate dyad, receiving the determinate dyad, made two dyads; for it was capable of duplicating that which it received. Footnote 17Cf. Introduction.
Again, how is it possible that 2 can be a definite entity existing besides the two units, and 3 besides the three units? Either by participation of the one in the other, as white man
exists besides white
and man,
because it partakes of these concepts; or when the one is a differentia of the other, as man
exists besides animal
and two-footed.
Again, some things are one by contact, others by mixture, and others by position; but none of these alternatives can possibly apply to the units of which 2 and 3 consist. Just as two men do not constitute any one thing distinct from both of them, so it must be with the units.
The fact that the units are indivisible will make no difference; because points are indivisible also, but nevertheless a pair of points is not anything distinct from the two single points.
Moreover we must not fail to realize this: that on this theory it follows that 2’s are prior and posterior, and the other numbers similarly.
Let it be granted that the 2’s in 4 are contemporaneous; yet they are prior to those in 8, and just as the 〈determinate〉 2 produced the 2’s in 4, so Footnote 18In each case the other factor is the indeterminate dyad (cf. sect. 18). they produced the 4’s in 8. Hence if the original 2 is an Idea, these 2’s will also be Ideas of a sort.
And the same argument applies to the units, because the units in the original 2 produce the four units in 4; and so all the units become Ideas, and an Idea will be composed of Ideas. Hence clearly those things also of which these things are Ideas will be composite; e.g., one might say that animals are composed of animals, if there are Ideas of animals.
In general, to regard units as different in any way whatsoever is absurd and fictitious (by fictitious
I mean dragged in to support a hypothesis
). For we can see that one unit differs from another neither in quantity nor in quality; and a number must be either equal or unequal—this applies to all numbers, but especially to numbers consisting of abstract units.
Thus if a number is neither more nor less, it is equal; and things which are equal and entirely without difference we assume, in the sphere of number, to be identical. Otherwise even the 2’s in the Ideal 10 will be different, although they are equal; for if anyone maintains that they are not different, what reason will he be able to allege?
Again, if every unit plus another unit makes 2, a unit from the Ideal 2 plus one from the Ideal 3 will make 2—a 2 composed of different units Footnote 19Which conflicts with the view under discussion. ; will this be prior or posterior to 3? It rather seems that it must be prior, because one of the units is contemporaneous with 3, and the other with 2. Footnote 20The implication seems to be, as Ross says, that the Platonists will refuse to admit that there is a number between 2 and 3.
We assume that in general 1 and 1, whether the things are equal or unequal, make 2; e.g. good and bad, or man and horse; but the supporters of this theory say that not even two units make 2.
If the number of the Ideal 3 is not greater than that of the Ideal 2, it is strange; and if it is greater, then clearly there is a number in it equal to the 2, so that this number is not different from the Ideal 2.
But this is impossible, if there is a first and second number. Footnote 21i.e., if numbers are specifically different. Cf. Aristot. Met. 13.6.1. Nor will the Ideas be numbers. For on this particular point they are right who claim that the units must be different if there are to be Ideas, as has been already stated. Footnote 22sect. 2-4 above. For the form is unique; but if the units are undifferentiated, the 2’s and 3’s will be undifferentiated.
Hence they have to say that when we count like this, l, 2, we do not add to the already existing number; for if we do, (a) number will not be generated from the indeterminate dyad, and (b) a number cannot be an Idea; because one Idea will pre-exist in another, and all the Forms will be parts of one Form. Footnote 23i.e., the biggest number.
Thus in relation to their hypothesis they are right, but absolutely they are wrong, for their view is very destructive, inasmuch as they will say that this point presents a difficulty: whether, when we count and say 1, 2, 3,
we count by addition or by enumerating distinct portions. Footnote 24This is Apelt’s interpretation of κατὰ μερίδας. For this sense of the word he quotes Plut. Mor. 644c. The meaning then is: If you count by addition, you regard number as exhibited only in concrete instances; if you treat each number as a distinct portion
(i.e. generated separately), you admit another kind of number besides the mathematical. Aristotle says that number can be regarded in both ways. But we do both; and therefore it is ridiculous to refer this point to so great a difference in essence.