1, La rencontre entre Turing et Wittgenstein, Cambridge 1939 [1]
En 1939, Wittgenstein (1889-1951) a donné un cours de philosophie sur « Les fondements des mathématiques » à l'Université de Cambridge.
Turing (1912-1954), rentré depuis quelques mois de Princeton USA où il a fini sa thèse sous la direction d’Alonzo Church (1903-1995), donne lui-même un cours sur la question des fondements des mathématiques. Il initie ses étudiants au formalisme d’Hilbert et aux principaux résultats obtenus dans son fameux travail de 1936, Théorie des nombres calculables, suivie d’une application au problème de la décision (Girard, 1995), dans lequel il a mis au point la notion de « machine de Turing ».
Après avoir être au courant du cours du même titre que le sien, Turing a décidé d’aller à assister le cours de Wittgenstein. Leurs discussions pendant un semestre ont été enregistrées dans le livre "Wittgenstein Cambridge Mathematics Fundamentals,1939 » (Mauvezin, Editions T.E.R.).
Voici je partage leur discussion sur la contradiction et le paradoxe.
2, La discussion sur la contradiction et le paradox [2]
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)
Référence :
[1] Turing et Wittgenstein, Cambridge 1939, Patrick Goutefangea
https://hal.archives-ouvertes.fr/hal-01648506/document
[2] 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
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