# 图灵与维特根斯坦关于矛盾和悖论的对话 - 第21讲（1939年）

1939年，维特根斯坦（Wittgenstein18891951）在剑桥大学开讲数学基础Foundations of mathematics），时年五十岁；图灵（Alan Turing19121954）二十七岁，刚在美国普林斯顿的数学家、逻辑学家丘奇（Alonzo Church19031995）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》。

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。

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)

1Turing 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

https://blog.sciencenet.cn/blog-2322490-1248627.html

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