Conversation
I will now use an awful expression. I wanted to talk of a stationary meaning, such as a picture that one has in one's mind, and a dynamic meaning. I was going to say, "No dynamic meaning follows from a stationary meaning." But that is very badly put and had better be forgotten immediately. Another way of putting it is to warn you: Don't think any use collides with a picture, except in a psychological way.Don't imagine a sort of logical collision. But that is also very badly expressed. For one then wants to ask where I got the idea of logical collision from. And one would be perfectly justified in asking. One is tempted to say, "A contradiction not only doesn't work-it can't work." One wants to say, "Can't you see? I can't sit and not sit at the same time." One even uses the phrase "at the same timew-as when one says, "I can't talk and eat at the same time." The temptation is to think that if a man is told to sit and not to sit, he is asked to do something which he quite obviously can't do. Hence we get the idea of the proposition as well as the sentence. The idea is that when I give you an order, there are the words-then something else, the sense of the words-then your action. And so with "Sit and don't sit", it is supposed that besides the words and what he does, there is also the sense of the contradiction-that something which he can't obey. One is inclined to say that the contradiction leaves you no room for action, thinking that one has now explained why the contradiction doesn't work. Suppose that we give the rule that "Do so-and-so and don't do it" always means "Do it". The negation doesn't add anything. So if I say "Sit down and don't sit down", he is to sit down. If I then say, "Here you are, the contradiction has a good sense", you are inclined to think I am cheating you. This is an immensely important point. Am I cheating you? Why does it seem so?
Turing: I should say that we were discussing the law of contradiction in connexion with language as ordinarily used, not in connexion with language modified in some arbitrary way which you like to propose.'
Malcolm: The feeling one has was that we were talking of 'p. -p'as it is now used-to express a contradiction; and you have merely suggested a use in which it would no longer express a contradiction.
Wittgenstein: Yes; you speak of the sentence as expressing a contradiction-as if the contradiction were something other than the sentence and expressed by it.-But doesn't the explanation of this feeling that I have cheated lie perhaps in the fact that I have made a wrong continuation? Now what is it that I have continued wrongly?
Turing: Could one take as an analogy a person having blocks of wood having two squares on them, like dominoes. If I say to you "White-green", you then have to paint one of the squares on the domino which I give you white and the other green. If the point of this procedure is to be able to distinguish the two squares, you will probably hesitate when I say "White-white". -Your suggestion comes to saying that when I say "White-white" you are to paint one of the squares white and the other grey.
Wittgenstein: Yes, exactly. And where does the cheating come in? What is the wrong continuation I have suggested? Why is this continuation in your analogy a wrong continuation? Might it not be the ordinary jargon among painters? The point is: Is it or is it not a case of one continuation being natural for us? Or ought one to say that there is something more to it than that? Ought one to give a reason why one continuation is natural for us? Ought one to say this, for example: "If we learn to use orders of the form 'p', 'q', 'p and q', 'p and not-q'etc, then so long as we give the phrase 'p and not-p'the sense which is
determined by the previous rules of training, it is clear that this cannot be a sensible order and cannot be obeyed. If the rules for obeying these orders-for logical product and negation-are laid down, then if we stick to these rules and don't in some arbitrary way deviate from them, then of course 'p and not-p'can't make sense and we can't obey it." Isn't that the sort of thing you would consider not cheating?
Turing: I should say that it is another kind of cheating. I should say that if one teaches people to carry out orders of the form 'p and not-q'then the most natural thing to do when ordered and not-p'is to be dissatisfied with anything which is done.
Wittgenstein: I entirely agree. But there is Just one point: does "natural" mean "mathematically natural"?
Turing: No.
Wittgenstein: Exactly. "Natural" there is not a mathematical term. It is not mathematically determined what is the natural thing to do. We most naturally compare a contradiction to something which jams. I would say that anything which we give and conceive to be an explanation of why a contradiction does not work is always just another way of saying that we do not want it to work. If you have a tube and a cock which shuts or opens it, your experience may have led you to think that always when the handle is parallel to the tube, the tube is open, and when it is at right angles to it, the tube is closed. But at home I have a cock which works the other way about. And in order to get used to it, I had to think of the handle as lying along the tube and blocking it, so that the tube was closed when the handle was parallel to it. I had to invent a new imagery. Similarly, one needs to change one's imagery in the case of contradictions. One can change one's imagery in such a way that 'p and not-p 'sounds entirely natural, as when we say, "The negative doesn't add anything".
This is most important. We shall constantly get into positions where it is necessary to have a new imagery which will make an absurd thing sound entirely natural. I want to talk about the sense in which we should say that the law of contradiction: - (p. - p) is a true proposition. Should we say that if '-(p. -p)' is a true proposition, it is true in a different sense of the word from the sense in which it is a true proposition that the earth goes round the sun? In logic one deals with tautologies-propositions like '- (p. -p)'. But one might just as well deal with contradictions instead. So that Principia mathematica would not be a collection of tautologies but a collection of contradictions. Should one then say that the contradictions were true? Or would one then say that "true" is being used in a different sense?
Turing: One would certainly say that it was being used in a different way.
Wittgenstein: It is used in a different way because you now say it of things of which you would not say it before. One could put the point this way. One often hears statements about "true" and "false" for example, that there are true mathematical statements which can't be proved in Principia Mathematica, etc. In such cases the thing is to avoid the words "true" and "false" altogether, and to get clear that to say that p is true is simply to assert p; and to say that p is false is simply to deny p or to assert -p. It is not a question of whether p is "true in a different sense". It is a question of whether we assert p. If a man says "It is fine" and I say "It is not fine", I am correcting him and asserting the opposite; and we can then argue about whether it is fine or not, and we may be able to settle the question. But if I am trained in logic, I am trained to assert certain things and not to assert others. This is an entirely different case from being trained to assert that Smith looks sad. I am not trained to assert that he looks sad or that he doesn't look sad. But I am actually trained to assert mathematical propositions that 3 X 6= 18, and not 19-and logical propositions. ''Trained to assert under what conditions? Well, for instance, when I have to pass an exam.-And if, for example, we did logic by means of contradictions, we should be trained to assert contradictions in examinations. It is important in this connexion that there is an inflexion of asserting. We make assertions with a peculiar inflexion of the voice; and there are gestures with this. This is one thing which is very characteristic of assertion. It is also important that assertions in our language have a peculiar jingle; we make them with sentences of a certain form. For instance, "'Twas brillig" is an assertion, although "brillig" is not a normal word. Now suppose that we were trained to use contradictions instead of tautologies in logic. There are circumstances in which we should call it the same logic as our present logic. What are these circumstances? What would be our criterion for saying that this other logic is all absurd, or for saying that it is essentially the same as our present logic?
Malcolm: Wouldn't we say it was the same as our present logic if we used "-" in a different way?
Wittgenstein: Yes, But using "-" in a different way does not here refer to the way in which it is used in the proofs. [In the proofs it] might be just the sameIn ironical statements, a sentence is very often used to mean just the opposite of what it normally means. For instance, one says "He is very kind", meaning that he is not kind. And in these cases the criterion for what is meant is the occasion on which it is used. One might make a deduction and say "He is very kind, therefore we will give him a birthday present "ironically, meaning "He is not kind, therefore we will not give him a birthday present." Thus we could have proofs in our supposed new logic just like the ones in Principia Mathematics, and the assertion sign would appear before contradictions. By the way, this is the way in which a proposition can assert of itself that it is not provable. Besides putting the assertion sign before contradictions I could put it before propositions like 'p--> q'. In the one case I-- p. -p would mean p. -p is refutable and in the other 'I- p--> q' would mean p--> q is not provable.Thus we see that Principia might not only be a collection of tautologies or a collection of contradictions; it might even be a collection of propositions which are neither contradictions nor tautologies.
I may meet Turing in the future, possibly 1939 — Wittgenstein
I think one of the basis of his argument is that mathematics doesn't base itself on the meaning of the symbols or operations but a SAME use. — Wittgenstein
This is a difficulty which arises again and again in philosophy: we use "meaning" in different ways. On the one hand we take as the criterion for meaning, something which passes in our mind when we say it, or something to which we point to explain it. On the other hand, we take as the criterion the use we make of the word or sentence as time goes on. — Wittgenstein
Exactly. "Natural" there is not a mathematical term. It is not mathematically determined what is the natural thing to do. We most naturally compare a contradiction to something which jams. I would say that anything which we give and conceive to be an explanation of why a contradiction does not work is always just another way of saying that we do not want it to work. If you have a tube and a cock which shuts or opens it, your experience may have led you to think that always when the handle is parallel to the tube, the tube is open, and when it is at right angles to it, the tube is closed. But at home I have a cock which works the other way about. And in order to get used to it, I had to think of the handle as lying along the tube and blocking it, so that the tube was closed when the handle was parallel to it. I had to invent a new imagery. Similarly, one needs to change one's imagery in the case of contradictions. One can change one's imagery in such a way that 'p and not-p 'sounds entirely natural, as when we say, "The negative doesn't add anything".
This is most important. We shall constantly get into positions where it is necessary to have a new imagery which will make an absurd thing sound entirely natural. I want to talk about the sense in which we should say that the law of contradiction: - (p. - p) is a true proposition. Should we say that if '-(p. -p)' is a true proposition, it is true in a different sense of the word from the sense in which it is a true proposition that the earth goes round the sun? In logic one deals with tautologies-propositions like '- (p. -p)'. But one might just as well deal with contradictions instead. So that Principia mathematica would not be a collection of tautologies but a collection of contradictions. Should one then say that the contradictions were true? Or would one then say that "true" is being used in a different sense?
But this dialogue is actually from a larger context in which Wittgenstein advocates a finitist viewpoint of mathematics. — Wittgenstein
Algorithmic Decidability vs. Undecidability: If mathematical extensions of all kinds are necessarily finite, then, in principle, all mathematical propositions are algorithmically decidable, from which it follows that an “undecidable mathematical proposition” is a contradiction-in-terms. Moreover, since mathematics is essentially what we have and what we know, Wittgenstein restricts algorithmic decidability to knowing how to decide a proposition with a known decision procedure.
What are the rules, and to what end? Who made the rules and how? It seems to me that if the rules were arbitrary, then they'd be much more malleable. Reality has this tendency of working how it wants to and we are just along for the ride - subjected to the forces that be, and active participants in what the calculus formulas represent, or mean. We all made the the rules as we all live in the same reality that help us determine what the rules should be. It is our common experiences of the same world that make the same rules applicable for others. If we didn't live in the same world, then how is it that the rules work for others and how are they shared?wittgenstein is really making a profound point here. He even has Turing struggling to give a good counterpoint.
I think the central issue is that "rule following" or usage in mathematics is mathematics and not what what meaning we get out of it. If we assume that, it is easy to see why a contradiction isn't a problem if we either assign something to the result obtained after contradiction or leave it there. Read it again perhaps, it will get clearer. — Wittgenstein
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