Technology Rig Some days you wish had never started, then along comes one you hope will never end. Anything can happen in a day.

Turing attended Wittgenstein's lectures on the philosophy of mathematics in Cambridge in 1939 and disagreed strongly with a line of argument that Wittgenstein was pursuing which wanted to allow contradictions to exist in mathematical systems. Wittgenstein argues that he can see why people don't like contradictions outside of mathematics but cannot see what harm they do inside mathematics.

Wittgenstein: Think of the case of the Liar: It is very queer in a way that this should have puzzled anyone — much more extraordinary than you might think... 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. Well, so what? You can go on like that until you are black in the face. Why not? It doesn't matter. ...it is just a useless language-game, and why should anyone 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, ... where will the harm come? 

Turing: The real harm will not come in unless there is an application, in which a bridge may fall down or something of that sort…. You cannot be confident about applying your calculus until you know that there are no hidden contradictions in it. 

Wittgenstein: There seems to me an enormous mistake there. ... Suppose I convince [someone] of the paradox of the Liar, and he says, 'I lie, therefore I do not lie, therefore I lie and I do not lie, therefore we have a contradiction, therefore 2x2 = 369.' Well, we should not call this 'multiplication,' that is all... 

Turing: Although you do not know that the bridge will fall if there are no contradictions, yet it is almost certain that if there are contradictions it will go wrong somewhere. 

Wittgenstein: But nothing has ever gone wrong that way yet...

Turing is exasperated and points out that such contradictions inside mathematics will lead to disasters outside mathematics: bridges will fall down. Only if there are no applications will the consequences of contradictions be innocuous. Turing eventually gave up attending these lectures. His despair is understandable. The inclusion of just one contradiction (like 0 = 1) in an axiomatic system allows any statement about the objects in the system to be proved true (and also proved false).

When Bertrand Russel pointed this out in a lecture he was once challenged by a heckler demanding that he show how the questioner could be proved to be the Pope if 2 + 2 = 5. Russel replied immediately that 'if twice 2 is 5, then 4 is 5, subtract 3; then 1 = 2. But you and the Pope are 2; therefore you and the Pope are 1'! A contradictory statement is the ultimate Trojan horse.

Therefore classical logic is not always applicable to real-world situations, no matter how well the parts we've used so far seem to have worked. And we have no theory of when it will be applicable and when it will fail (at least, we didn't in Wittgenstein's time; some might argue that relevance logics give us such a theory now).