维特根斯坦曾与波普尔对谈半个小时，就有人八卦了一本书；而维特根斯坦和图灵智力交锋了一学期，却很少有人评论，。。。

这是因为人们对“思想”（在与“内容”本身的相对的意义上）比对（人的）“思想形式”（与“内容”的实体相对应）上更不容易看到事物的深层本质。

1939年，维特根斯坦（Wittgenstein，1889－1951）在剑桥大学开讲”数学基础”（Foundations of mathematics），时年五十岁；图灵（Alan Turing，1912－1954）二十七岁，刚在美国普林斯顿的数学家、逻辑学家丘奇（Alonzo Church，1903－1995）的指导下完成博士论文“基于序数的逻辑系统”（Systems of Logic Based on Ordinals），回到英国。图灵在剑桥大学申请讲师未遂，接着当研究员，同年在学校开一门数理逻辑的课程，取名“数学基础”，看了学校的课程表后才知道维特根斯坦也开一门同名课程，于是决定去旁听，有了维特根斯坦和图灵一学期的精彩对话！记载在维特根斯坦的几个学生整理的书中《维特根斯坦剑桥数学基础讲义，1939》。

这里把关于“图灵与维特根斯坦关于矛盾和悖论的对话”翻译出来。

一，译文

维特根斯坦：“说谎者悖论”使人感到困惑，这从某种意义上说很奇怪，。。。。是这样的：如果一个人说“我在说谎”，就得出：我没说谎，所以我说谎；我说谎，所以我没说谎。好吧，那又如何呢？你可以这样继续下去， 直到说得小脸发青。为什么不？ 没关系，。。。

现在假设一个人说“我在撒谎”，而我说“因此你没撒谎，因此你撒谎，因此你没撒谎...” – 哪有问题？ 没有，除了没有用，这只是一个没意义的语言游戏而已，为什么会有人因此而兴奋？

图灵：让大家困惑的是，在一般情况下，有矛盾肯定就有错，但在这个例子中，大家不知道哪出错了。

维特根斯坦：是的，但没什么错啊。

有一种特殊的数学方法，即反证法，也称“避免矛盾”。 在这种方法中，人们暴露出矛盾，然后再指出避免矛盾的出路，但这并不意味着矛盾就是恶魔。

有人可能会说：“矛盾是问题的来源。” 答案是：那么，就不要从矛盾中得出任何结论； 将此作为规则。 您可能会说：当我们得出矛盾时，总是有时间去解决它。当我们得出矛盾时，我们不应该简单地说：“这没有用-我们不会从中得出任何结论”？

图灵：除非有实际情况，否则不会有真正的危险。在实际情况下，桥梁会倒塌或发生类似情况。

桥（第22讲）

维特根斯坦：上次有人提出逻辑或数学中的矛盾在实际情况中的危险，图灵指出桥梁可能倒塌。

说一座桥可能因矛盾而倒塌，听起来有些不对劲。我们对导致桥梁倒塌的想法有错误。

（a）我们掌握的错误的物理法则 – 错误的系数。

（b）计算有误 – 有人乘法做错了。

第一种情况显然与矛盾无关；第二个不太清楚。

图灵：除非你知道其中没有隐藏着矛盾，否则你不会对你的计算充满信心。

维特根斯坦：在我看来那里有严重错误，因为你的演算给出了某些结果，并且你希望桥梁不倒塌。我要说的是，出问题的可能只有两种方式：或者桥梁倒塌，或你在计算中犯了错误 - 例如，你乘法做错了，但是你似乎认为可能有第三种错误：演算是错误的。

图灵：不。我反驳的是桥梁倒塌。

维特根斯坦：但是你怎么知道它倒塌？ 这不是物理问题吗？

图灵：如果有人接受了弗雷格的符号系统，给予某人乘法的技巧，那么通过使用罗素悖论，他可能会得到错误的乘法。

维特根斯坦：这是做一些我们不称做乘法的事情……我要讲的是弗雷格和罗素的逻辑无论如何都不是算术的基础 - 不管矛盾还是不矛盾。 （来自第22讲和第23讲）

二，原文

The Turing/Wittgenstein exchange on contradiction and paradox (Lecture XXI)

Wittgenstein: ‘Think of the case of the Liar. It is very queer in a way that this should have puzzled anyone ... Because the thing works like this: if a man says “I am lying” was say that it follows that he is not lying, from which it follows that he is lying and so on. Well, so what? You can go on like that until you are black in the face. Why not? It doesn’t matter…

Now suppose a man says “I am lying” and I say “Therefore you are not, therefore you are, therefore you are not...” – What is wrong? Nothing. Except that it is no use; it is just a useless language-game, and why should anybody be excited?’

Turing: What puzzles one is that one usually uses a contradiction as a criterion for having done something wrong. But in this case one cannot find anything done wrong.

Wittgenstein: Yes – and more: nothing has been done wrong.

Wittgenstein: There is a particular mathematical method, the method of reduction ad absurdum, which we might call “avoiding the contradiction”. In this method one shows a contradiction and then shows the way from it. But this doesn’t mean that a contradiction is a sort of devil.

One may say, “From a contradiction everything would follow.” The reply to that is: Well then, don’t draw any conclusions from a contradiction; make that the rule. You might put it: There is always time to deal with a contradiction when we get to it. When we get to it, shouldn’t we simply say, “This is no use – and we won’t draw any conclusions from it”?

Turing: The real harm will not come in unless there is an application, in which case a bridge may fall down or something of that sort.

The Bridge (Lecture XXII)

Wittgenstein: It was suggested last time that the danger with a contradiction in logic or mathematics is in the application. Turing suggested that a bridge might collapse.

Now it does not sound quite right to say that a bridge might fall down because of a contradiction. We have an idea of the sort of mistake which would lead to a bridge falling.

(a) We’ve got hold of a wrong natural law – a wrong coefficient.

(b) There has been a mistake in calculation – someone has multiplied

wrongly.

The first case obviously has nothing to do with having a contradiction; and the second is not quite clear.

Turing: You cannot be confident about applying your calculus until you know that there is no hidden contradiction in it.

Wittgenstein: There seems to me to be an enormous mistake there. For your calculus gives certain results, and you want the bridge not to break down. I’d say things can go wrong is only two ways: either the bridge breaks down or you have made a mistake in your calculation – for example, you multiplied wrongly. But you seem to think that there may be a third thing wrong: the calculus is wrong.

Turing: No. What I object to is the bridge falling down.

Wittgenstein: But how do you know that it will fall down? Isn’t that a question of physics?

Turing: If one takes Frege’s symbolism and gives someone the technique of multiplying in it, then by using a Russell paradox he could get a wrong multiplication.

Wittgenstein: This would come to doing something which we would not call multiplying... The point I’m driving at is that Frege and Russell’s logic is not the basis for arithmetic anyway – contradiction or no contradiction. (from Lectures XXII and XXIII)

参考文献：

【1】Turing and Wittgenstein on Logic and Mathematics - The Eighteenth British Wittgenstein Society Lecture, Ray Monk, https://www.britishwittgensteinsociety.org/wp-content/uploads/documents/lectures/Turing-and-Wittgenstein-on-Logic-and-Mathematics.pdf

【2】图灵对掐维特根斯坦：这次维特没有用拨火棍却显出了尊敬，https://www.thepaper.cn/newsDetail_forward_1361179