When Hardy said that 317 was a prime because it was so, what did this mean?
哈代说，317就是一个素数，这是什么意思呢？
Did it only mean that people would always agree if they did their sums right?
这否意味着人们总是接受那些他们能算对的东西？
How did they know what were the 'right' rules?
他们怎么知道什么规则是"正确的"？
Wittgenstein's technique was to ask questions which brought words like proof, infinite, number, rule, into sentences about real life, and to show that they might make nonsense.
维特根斯坦的问题，把证明、无限、数学、规则这样的词语引入日常生活中，然后表明它们很可能是讲不通的。
As the only working mathematician in the class, Alan tended to be treated as responsible for everything that mathematicians ever said or did, and he rather nobly did his best to defend the abstract constructions of pure mathematics against Wittgenstein's attack.
作为一个专业数学家，艾伦倾向于支持数学家们的工作，他尽力地维护纯数学的抽象性，抵抗维特根斯坦的这种引入。
In particular, there was an extended argument between them about the whole structure of mathematical logic.
特别是，他们俩进行了一场关于整个数理逻辑体系的争论。
Wittgenstein wanted to argue that the business of creating a watertight, automatic logical system had nothing to do with what was ordinarily meant by truth.
维特根斯坦认为，创造一个严密的、确定的逻辑系统，对于追求真理而言，是没有用的。
He fastened upon the feature of any completely logical system, that a single contradiction, and a self-contradiction in particular, would allow the proof of any proposition:
他紧紧抓住这一点：完备逻辑系统的矛盾特性，会使得任何命题都能被证明。
WITTGENSTEIN: … Think of the case of the Liar.
维特根斯坦：……我们来看说谎者悖论，
It is very queer in a way that this should have puzzled anyone – much more extraordinary than you might think. …
所有的人都在困惑这个问题，这根本就不正常，太不正常了……
Because the thing works like this: if a man says 'I am lying' we say that it follows that he is not lying, from which it follows that he is lying and so on.
一个人说"我在说谎"，我们如果假设他说的是实话，于是他就是在说谎。