Book XIII Chapter 7

First, then, let us inquire if the units are associable or inassociable, and if inassociable, in which of the two ways we distinguished¹. For it is possible that any unity² is inassociable with any, and it is possible that those in the ‘itself’ are inassociable with those in the ‘itself’, and, generally, that those in each ideal number are inassociable with those in other ideal numbers. Now (1) all units are associable and without difference, we get mathematical number-only one kind of number, and the Ideas cannot be the numbers. For what sort of number will man-himself or animal-itself or any other Form be? There is one Idea of each thing e.g. one of man-himself and another one of animal-itself; but the similar and undifferentiated numbers are infinitely⁵ many, so that any particular 3 is no more man-himself than any other 3. But if the Ideas are not numbers, neither can they exist at all. For from what principles will the Ideas come? It is number that comes from the 1 and the indefinite dyad, and the principles or elements are said to be principles and elements of number, and the Ideas cannot be ranked as either prior or posterior to the numbers.

But (2) if the units are inassociable, and inassociable in the sense that any is inassociable with any other, number of this sort cannot be mathematical number; for mathematical number consists of undifferentiated units, and the truths proved of it suit this character. Nor can it be ideal number. For 2 will not proceed immediately from 1 and the indefinite dyad, and be followed by the successive numbers, as they say ‘2,3,4’ for the units in the ideal are generated at the same time, whether, as the first holder⁶ of the theory said, from unequals (coming into being when these were equalized) or in some other way-since, if one unit is to be prior to the other, it will be prior also to 2 the composed of these; for when there is one thing prior and another posterior, the resultant of these will be prior to one and posterior to the other. Again, since the 1-itself is first, and then there is a particular 1 which is first among the others and next after the 1-itself, and again a third which is next after the second and next but one after the first 1,-so the units must be prior to the numbers after which they are named when we count them; e.g. there will be a third unit in 2 before 3 exists, and a fourth and a fifth in 3 before the numbers 4 and 5 exist.-Now none of these thinkers has said the units are inassociable in this way, but according to their principles it is reasonable that they should be so even in this way, though in truth it is impossible. For it is reasonable both that the units should have priority and posteriority if there is a first unit or first 1, and also that the 2’s should if there is a first 2; for after the first it is reasonable and necessary that there should be a second, and if a second, a third, and so with the others successively. (And to say both things at the same time, that a unit is first and another unit is second after the ideal 1, and that a 2 is first after it, is impossible.) But they make a first unit or 1, but not also a second and a third, and a first 2, but not also a second and a third. Clearly, also, it is not possible, if all the units are inassociable, that there should be a 2-itself and a 3-itself; and so with the other numbers. For whether the units are undifferentiated or different each from each, number must be counted by addition, e.g. 2 by adding another 1 to the one, 3 by adding another 1 to the two, and similarly. This being so, numbers cannot be generated as they generate them, from the 2 and the 1; for 2 becomes part of 3 and 3 of 4 and the same happens in the case of the succeeding numbers, but they say 4 came from the first 2 and the indefinite which makes it two 2’s other than the 2-itself; if not, the 2-itself will be a part of 4 and one other 2 will be added. And similarly 2 will consist of the 1-itself and another 1; but if this is so, the other element cannot be an indefinite 2; for it generates one unit, not, as the indefinite 2 does, a definite 2.

Again, besides the 3-itself and the 2-itself how can there be other 3’s and 2’s? And how do they consist of prior and posterior units? All this is absurd and fictitious⁷, and there cannot be a first 2 and then a 3-itself. Yet there must, if the 1 and the indefinite dyad are to be the elements. But if the results are impossible, it is also impossible that these are the generating principles.

If the units, then, are differentiated⁴, each from each, these results and others similar to these follow of necessity. But (3) if those in different numbers are differentiated, but those in the same number are alone undifferentiated from one another, even so the difficulties that follow are no less. E.g. in the 10-itself their are ten units, and the 10 is composed both of them and of two 5’s. But since the 10-itself is not any chance number nor composed of any chance 5’s — or, for that matter, units — the units in this 10 must differ. For if they do not differ, neither will the 5’s of which the 10 consists differ; but since these differ, the units also will differ. But if they differ, will there be no other 5’s in the 10 but only these two, or will there be others? If there are not, this is paradoxical; and if there are, what sort of 10 will consist of them? For there is no other in the 10 but the 10 itself. But it is actually necessary on their view that the 4 should not consist of any chance 2’s; for the indefinite as they say, received the definite 2 and made two 2’s; for its nature was to double what it received.

Again, as to the 2 being an entity⁸ apart from its two units, and the 3 an entity apart from its three units, how is this possible? Either by one’s sharing in the other, as ‘pale man’ is different from ‘pale’ and ‘man’ (for it shares in these), or when one is a differentia of the other, as ‘man’ is different from ‘animal’ and ‘two-footed’.

Again, some things are one by contact, some by intermixture, some by position; none of which can belong to the units of which the 2 or the 3 consists; but as two men are not a unity apart from both, so must it be with the units. And their being indivisible will make no difference to them; for points too are indivisible, but yet a pair of them is nothing apart from the two.

But this consequence also we must not forget, that it follows that there are prior and posterior 2 and similarly with the other numbers. For let the 2’s in the 4 be simultaneous; yet these are prior to those in the 8 and as the 2 generated them, they generated the 4’s in the 8-itself. Therefore if the first 2 is an Idea, these 2’s also will be Ideas of some kind. And the same account applies to the units; for the units in the first 2 generate the four in 4, so that all the units come to be Ideas and an Idea will be composed of Ideas. Clearly therefore those things also of which these happen to be the Ideas will be composite, e.g. one might say that animals are composed of animals, if there are Ideas of them.

In general, to differentiate³ the units in any way is an absurdity⁹ and a fiction; and by a fiction I mean a forced statement made to suit a hypothesis. For neither in quantity nor in quality do we see unit differing from unit, and number must be either equal or unequal-all number but especially that which consists of abstract units-so that if one number is neither greater nor less than another, it is equal to it; but things that are equal and in no wise differentiated we take to be the same when we are speaking of numbers. If not, not even the 2 in the 10-itself will be undifferentiated, though they are equal; for what reason will the man who alleges¹⁰ that they are not differentiated be able to give?

Again, if every unit + another unit makes two, a unit from the 2-itself and one from the 3-itself will make a 2. Now (a) this will consist of differentiated units; and will it be prior to the 3 or posterior? It rather seems that it must be prior; for one of the units is simultaneous with the 3 and the other is simultaneous with the 2. And we, for our part, suppose that in general 1 and 1, whether the things are equal or unequal, is 2, e.g. the good and the bad, or a man and a horse; but those who hold these views say that not even two units are 2.

If the number of the 3-itself is not greater than that of the 2, this is surprising; and if it is greater, clearly there is also a number in it equal to the 2, so that this is not different from the 2-itself. But this is not possible, if there is a first and a second number.

Nor will the Ideas be numbers. For in this particular point they are right who claim that the units must be different, if there are to be Ideas; as has been said before. For the Form is unique; but if the units are not different, the 2’s and the 3’s also will not be different. This is also the reason why they must say that when we count thus-’1,2’-we do not proceed by adding to the given number; for if we do, neither will the numbers be generated from the indefinite dyad, nor can a number be an Idea; for then one Idea will be in another, and all Forms will be parts of one Form. And so with a view to their hypothesis their statements are right, but as a whole they are wrong; for their view is very destructive, since they will admit that this question itself affords some difficulty-whether, when we count and say — 1,2,3-we count by addition or by separate portions. But we do both; and so it is absurd to reason back from this problem to so great a difference of essence.

1 distinguished   

adj.卓越的，杰出的，著名的

参考例句：

• Elephants are distinguished from other animals by their long noses.大象以其长长的鼻子显示出与其他动物的不同。
• A banquet was given in honor of the distinguished guests.宴会是为了向贵宾们致敬而举行的。

2 unity   

n.团结，联合，统一；和睦，协调

参考例句：

• When we speak of unity,we do not mean unprincipled peace.所谓团结，并非一团和气。
• We must strengthen our unity in the face of powerful enemies.大敌当前，我们必须加强团结。

3 differentiate   

vi.(between)区分；vt.区别；使不同

参考例句：

• You can differentiate between the houses by the shape of their chimneys.你可以凭借烟囱形状的不同来区分这两幢房子。
• He never learned to differentiate between good and evil.他从未学会分辨善恶。

4 differentiated   

区分，区别，辨别( differentiate的过去式和过去分词 )； 区别对待； 表明…间的差别，构成…间差别的特征

参考例句：

• The development of mouse kidney tubules requires two kinds of differentiated cells. 小鼠肾小管的发育需要有两种分化的细胞。
• In this enlargement, barley, alfalfa, and sugar beets can be differentiated. 在这张放大的照片上，大麦，苜蓿和甜菜都能被区分开。

5 infinitely   

adv.无限地，无穷地

参考例句：

• There is an infinitely bright future ahead of us.我们有无限光明的前途。
• The universe is infinitely large.宇宙是无限大的。

6 holder   

n.持有者，占有者；(台，架等)支持物

参考例句：

• The holder of the office of chairman is reponsible for arranging meetings.担任主席职位的人负责安排会议。
• That runner is the holder of the world record for the hundred-yard dash.那位运动员是一百码赛跑世界纪录的保持者。

7 fictitious   

adj.虚构的，假设的；空头的

参考例句：

• She invented a fictitious boyfriend to put him off.她虚构出一个男朋友来拒绝他。
• The story my mother told me when I was young is fictitious.小时候妈妈对我讲的那个故事是虚构的。

8 entity   

n.实体，独立存在体，实际存在物

参考例句：

• The country is no longer one political entity.这个国家不再是一个统一的政治实体了。
• As a separate legal entity,the corporation must pay taxes.作为一个独立的法律实体，公司必须纳税。

9 absurdity   

n.荒谬，愚蠢；谬论

参考例句：

• The proposal borders upon the absurdity.这提议近乎荒谬。
• The absurdity of the situation made everyone laugh.情况的荒谬可笑使每个人都笑了。

10 alleges   

断言，宣称，辩解( allege的第三人称单数 )

参考例句：

• The newspaper article alleges that the mayor is corrupt. 报纸上断言该市长腐败。
• Steven was tardy this morning and alleges that his bus was late. 史提芬今天早上迟到的说词是公车误点了。