Book XIV Chapter 3

As for those, then, who suppose the Ideas to exist and to be numbers, by their assumption in virtue¹ of the method of setting out each term apart from its instances-of the unity² of each general term they try at least to explain somehow why number must exist. Since their reasons, however, are neither conclusive³ nor in themselves possible, one must not, for these reasons at least, assert the existence of number. Again, the Pythagoreans, because they saw many attributes of numbers belonging te sensible bodies, supposed real things to be numbers-not separable numbers, however, but numbers of which real things consist. But why? Because the attributes of numbers are present in a musical scale and in the heavens and in many other things. Those, however, who say that mathematical number alone exists cannot according to their hypotheses say anything of this sort, but it used to be urged that these sensible things could not be the subject of the sciences. But we maintain that they are, as we said before. And it is evident that the objects of mathematics do not exist apart; for if they existed apart their attributes would not have been present in bodies. Now the Pythagoreans in this point are open to no objection; but in that they construct natural bodies out of numbers, things that have lightness and weight out of things that have not weight or lightness, they seem to speak of another heaven and other bodies, not of the sensible. But those who make number separable assume that it both exists and is separable because the axioms would not be true of sensible things, while the statements of mathematics are true and ‘greet the soul’; and similarly with the spatial⁴ magnitudes of mathematics. It is evident, then, both that the rival theory will say the contrary of this, and that the difficulty we raised just now, why if numbers are in no way present in sensible things their attributes are present in sensible things, has to be solved by those who hold these views.

There are some who, because the point is the limit and extreme of the line, the line of the plane, and the plane of the solid, think there must be real things of this sort. We must therefore examine this argument too, and see whether it is not remarkably⁵ weak. For (i) extremes are not substances, but rather all these things are limits. For even walking, and movement in general, has a limit, so that on their theory this will be a ‘this’ and a substance. But that is absurd. Not but what (ii) even if they are substances, they will all be the substances of the sensible things in this world; for it is to these that the argument applied⁶. Why then should they be capable of existing apart?

Again, if we are not too easily satisfied, we may, regarding all number and the objects of mathematics, press this difficulty, that they contribute nothing to one another, the prior to the posterior; for if number did not exist, none the less spatial magnitudes would exist for those who maintain the existence of the objects of mathematics only, and if spatial magnitudes did not exist, soul and sensible bodies would exist. But the observed facts show that nature is not a series of episodes, like a bad tragedy. As for the believers in the Ideas, this difficulty misses them; for they construct spatial magnitudes out of matter and number, lines out of the number planes doubtless out of solids out of or they use other numbers, which makes no difference. But will these magnitudes be Ideas, or what is their manner of existence, and what do they contribute to things? These contribute nothing, as the objects of mathematics contribute nothing. But not even is any theorem true of them, unless we want to change the objects of mathematics and invent doctrines⁷ of our own. But it is not hard to assume any random⁸ hypotheses and spin out a long string of conclusions. These thinkers, then, are wrong in this way, in wanting to unite the objects of mathematics with the Ideas. And those who first posited⁹ two kinds of number, that of the Forms and that which is mathematical, neither have said nor can say how mathematical number is to exist and of what it is to consist. For they place it between ideal and sensible number. If (i) it consists of the great and small, it will be the same as the other-ideal-number (he makes spatial magnitudes out of some other small and great). And if (ii) he names some other element, he will be making his elements rather many. And if the principle of each of the two kinds of number is a 1, unity will be something common to these, and we must inquire how the one is these many things, while at the same time number, according to him, cannot be generated except from one and an indefinite dyad.

All this is absurd, and conflicts both with itself and with the probabilities, and we seem to see in it Simonides ‘long rigmarole’ for the long rigmarole comes into play, like those of slaves, when men have nothing sound to say. And the very elements-the great and the small-seem to cry out against the violence that is done to them; for they cannot in any way generate numbers other than those got from 1 by doubling.

It is strange also to attribute generation to things that are eternal, or rather this is one of the things that are impossible. There need be no doubt whether the Pythagoreans attribute generation to them or not; for they say plainly that when the one had been constructed, whether out of planes or of surface or of seed or of elements which they cannot express, immediately the nearest part of the unlimited¹⁰ began to be constrained¹¹ and limited by the limit. But since they are constructing a world and wish to speak the language of natural science, it is fair to make some examination of their physical theorics, but to let them off from the present inquiry¹²; for we are investigating the principles at work in unchangeable things, so that it is numbers of this kind whose genesis we must study.

1 virtue   

n.德行，美德；贞操；优点；功效，效力

参考例句：

• He was considered to be a paragon of virtue.他被认为是品德尽善尽美的典范。
• You need to decorate your mind with virtue.你应该用德行美化心灵。

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 conclusive   

adj.最后的，结论的；确凿的，消除怀疑的

参考例句：

• They produced some fairly conclusive evidence.他们提供了一些相当确凿的证据。
• Franklin did not believe that the French tests were conclusive.富兰克林不相信这个法国人的实验是结论性的。

4 spatial   

adj.空间的，占据空间的

参考例句：

• This part of brain judges the spatial relationship between objects.大脑的这部分判断物体间的空间关系。
• They said that time is the feeling of spatial displacement.他们说时间是空间位移的感觉。

5 remarkably   

ad.不同寻常地，相当地

参考例句：

• I thought she was remarkably restrained in the circumstances. 我认为她在那种情况下非常克制。
• He made a remarkably swift recovery. 他康复得相当快。

6 applied   

adj.应用的；v.应用，适用

参考例句：

• She plans to take a course in applied linguistics.她打算学习应用语言学课程。
• This cream is best applied to the face at night.这种乳霜最好晚上擦脸用。

7 doctrines   

n.教条( doctrine的名词复数 )；教义；学说；（政府政策的）正式声明

参考例句：

• To modern eyes, such doctrines appear harsh, even cruel. 从现代的角度看，这样的教义显得苛刻，甚至残酷。 来自《简明英汉词典》
• His doctrines have seduced many into error. 他的学说把许多人诱入歧途。 来自《现代汉英综合大词典》

8 random   

adj.随机的；任意的；n.偶然的(或随便的)行动

参考例句：

• The list is arranged in a random order.名单排列不分先后。
• On random inspection the meat was found to be bad.经抽查，发现肉变质了。

9 posited   

v.假定，设想，假设( posit的过去式和过去分词 )

参考例句：

• Several writers have posited the idea of a universal consciousness. 有几个作者都假设存在普遍意识。 来自辞典例句
• All cash receipts should be recorded and de-posited daily. 所有的现金收据应该被每日记录和存放。 来自互联网

10 unlimited   

adj.无限的，不受控制的，无条件的

参考例句：

• They flew over the unlimited reaches of the Arctic.他们飞过了茫茫无边的北极上空。
• There is no safety in unlimited technological hubris.在技术方面自以为是会很危险。

11 constrained   

adj.束缚的,节制的

参考例句：

• The evidence was so compelling that he felt constrained to accept it. 证据是那样的令人折服，他觉得不得不接受。
• I feel constrained to write and ask for your forgiveness. 我不得不写信请你原谅。

12 inquiry   

n.打听，询问，调查，查问

参考例句：

• Many parents have been pressing for an inquiry into the problem.许多家长迫切要求调查这个问题。
• The field of inquiry has narrowed down to five persons.调查的范围已经缩小到只剩5个人了。