• Wittgenstein
    442
    Please read this all and l believe it will be really enjoyable to get through it.

    This is from Wittgenstein's lectures on the foundation of mathematics.


    Explanation
    What would go wrong, if anything, if we didn't recognize the law of contradiction-or any other proposition in Russell's logic? We treated the question of double negation as parallel to that: If some people used double negation to mean affirmation, and others used double negation to mean negation, should we say then that they were using negation-or double negation-with "different meanings"? We discussed whether a particular meaning of negation made a certain usage correct, or whether that meaning consists in using negation in that way. 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.First of all, to put the matter badly and in a way which must be corrected later, it is clear that we judge what a person means in these two ways. One can say that we judge what a person means by a word from the way he uses it. And the way he uses it is something which goes on in time. On the other hand, we also say that the meaning of a word is defined by the thing it stands
    for; it is something in our minds or at which we can point.

    The connexion between these two criteria is that the picture in our minds is connected, in an overwhelming number of cases-for the overwhelming majority of human beings-with a particular use. For instance: you say to someone "This is red" (pointing); then you tell him "Fetch me a red bookm-and he will behave in a particular way. This is an immensely important fact about us human beings. And it goes together with all sorts of other facts of equal importance, like the fact that in all the languages we know, the meanings of words don't change with the days of the week. Another such fact is that pointing is used and understood in a particular way-that people react to it in a particular way.If you have learned a technique of language, and I point to this coat and say to you, the tailors now call this colour 'Boo' ", then you will buy me a coat of this colour, fetch one, etc.The point is that one only has to point to something and say,"This is so-and-so", and everyone who has been through a certain preliminary training will react in the same way. We could imagine this not to happen.

    If I just say, "This is called 'Boo' " you might not know what I mean; but in fact you would all of you automatically follow certain rules. Ought we to say that you would follow the right rules?-that you would know themeaning of "boo"? No, clearly not. For which meaning? Are there not 10,000 meanings which "boo" might now have?-It sounds as if your learning how to use it were different from your knowing its meaning. But the point is that we all make the SAME use of it. To know its meaning is to use it in the SAME way as other people do. "In the right way" means nothing. You might say, "Isn't there something else, too? Something besides the agreement? Isn't there a more natural and a less natural way of behaving? Or even a right and a wrong meaning?"-Suppose the word "colour" used as it is now in English. "Boo" is a new word. But then we are told, "This colour is called 'boo' and then everyone uses it for a shape. Could I then say, "That's not the straight way of using it"? I should certainly say they behaved unnaturally. This hangs together with the question of how to continue the series of cardinal numbers. Is there a criterion for the continuation-for a right and a wrong way-except that we do in fact continue them in that way, apart from a few cranks who can be neglected? We do indeed give a general rule for continuing the series; but this general rule might be reinterpreted by a second rule, and this second rule by a third rule, and so on.


    One might say, "But are you saying, Wittgenstein, that all this is arbitrary?"-I don't know. Certainly as children we are punished if we don't do it in the right way. Suppose someone said, "Surely the use I make of the rule for continuing the series depends on the interpretation 1 make of the rule or the meaning I give it." But is one's criterion for meaning a certain thing by the rule the using of the rule in a certain way, or is it a picture or another rule or something of the sort? In that case, it is still a symbol-which can be reinterpreted in any way whatsoever. This has often been said before. And it has often been put in the form of an assertion that the truths of logic are determined by a consensus of opinions. Is this what I am saying? No. There is no opinion at all; it is not a question of opinion. 'They are determined by a consensus of act a consensus of doing the same thing, reacting in the same way. There is a consensus but it is not a consensus of opinion. We all act the same way, walk the same way, count the same way. In counting we do not express opinions at all. There is no opinion that 25 follows 24-nor intuition. We express opinions by means of counting. People say, "If negation means one thing, then double negation equals affirmation; but if it means another thing, double negation equals negation." But I want to say its use is its meaning. 'There are various criteria for negation.-Think of the ways in which a child is taught negation: it may be explained by a sort of ostensive definition. You take something away from him and say "No". A child is trained in a certain technique of applying negation long before the question of double negation arises. If a child is taught the use of negation apart from all this, and then goes on to use double negation as equivalent to negation, would you say he is necessarily using negation now to mean something different? If you say, "It must have a different meaning now-this says nothing, unless you mean that a different picture will be associated with it. Let us go back to the law of contradiction. We saw last time that there is a great temptation to regard the truth of the law of contradiction as something which follows from the meaning of negation and of logical product and so on. Here the same point arises again.


    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.
  • Harry Hindu
    5.1k
    I dont get it. Is Witt using logic to render logic meaningless?

    I dont see how you can't use logic and expect to make any sense and communicate. Communication requires logic. Maybe that is why it is so difficult to understand Witt's scribbles, and why there are different interpretations of his scribbles.
  • Wittgenstein
    442


    I apologize for my terrible copy paste above but l can tell, 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.
  • fdrake
    6.5k


    Please cite where you got this from? Also, you should edit the giant copypaste so it's more readable.
  • Wittgenstein
    442

    It's difficult to edit it as it's a giant copy paste. I will try to see what l can do, l can't see if anyone would be bothered to read it though.
  • Streetlight
    9.1k
    Lectures on The Foundations of Mathematics is the source, no?
  • Wittgenstein
    442
    I hope it's a bit neat now.
  • god must be atheist
    5.1k


    I never knew you were so smart, @Wittgenstein. You are absolutely amazing.

    But this other member of this site, @turing, is elusive, I can't see any posts by him.

    When did you two have this conversation? And on what forum platform?
  • Wittgenstein
    442

    In Cambridge, around 2005. He was delivering lectures on the foundation of mathematics and he decided to pay me a visit as l was teaching a similar course. Sadly, he left after a while. I have to go to Norway this year and build myself a hut. I may meet Turing in the future, possibly 1939.

    :wink: :wink:
  • god must be atheist
    5.1k
    I may meet Turing in the future, possibly 1939Wittgenstein

    Please look out for those Nazi stards when you meet. They are a mean bunch, and getting meaner and meaner. They may not like that to you, very sensibly, the dasein is not conceptual ambient defractionism, but rather a postmodernistically mapped contradiction in Darwinian speedtime. They are touchy about that.
  • Wittgenstein
    442

    By Nazi, l take that you mean Donald Trump. :smile:
  • Wittgenstein
    442

    You have ignored my best friend Malcolm. That's not nice.
  • god must be atheist
    5.1k


    With all due respect to Donald Trump, I imagine that when he's in Germany, he would be reaching for Immanuel, thinking he is one of those he likes to grab.
  • god must be atheist
    5.1k
    (-: I am sure the people you listed won't notice there are no women present in the or**. That would be gender categorization, anyway, which is SO not PC.

    So let them be, close the door slowly and silently, and tip-toe away from the room.
  • Wittgenstein
    442

    I heard you will be providing some special services :wink:
  • softwhere
    111
    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

    The issue is even bigger than math. 'Meaning' is external. Sure, we have various intuitions, but intelligibility is primarily a social phenomenon. This short video is worth a look.

    @https://www.youtube.com/watch?v=x86hLtOkou8

    Here's a crucial piece of what you quoted above.

    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

    As the video suggests (and more reading on the subject consolidates), the supposed interior of the mind is only accessible via public (arbitrary, conventional) signs. Indeed, 'interior' is a sign. Note the metaphorical bleed. I don't deny that we can investigate our intuitions of words, but these intuitions aim are of decontextualized essences, as if language was simply a nomenclature, neglecting the 'context effects' in which it actually lives.

    I like the Heideggerian approach to Wittgenstein. A person learns to speak a language as they learn a form of life. This is knowing what one does in different situations, including knowing how words are used. It's only after the mostly unconscious assimilation of such conventions that one can think of oneself as a relatively free individual mind. 'History is a nightmare from which I am trying to awake.' The culture that I might want to improve or rebel against is what gave me the ability to speak and think. We wake up on a galloping horse, neck-deep in conventions we did not choose.
  • fresco
    577
    What appears to be missing in much of the above is an overview of communication called 'discourse analysis', which transcends individual units we might call 'sentences' or 'statements'. As real life social interactions progress, 'meanings' are negotiated such that what appear to be 'contradictions' when taken as static juxtapositions are resolved dynamically. Clearly 'classical logic' fails as a sufficient basis for semantics precisely because it relies on static (rigid) 'set membership'.
  • Wittgenstein
    442

    I believe Wittgenstein thinks that a system in which there are contradictions can still be meaningful and even helpful in certain cases as inconsistency robustness has proved itself. I have selected the following passages from on online source that elucidates how allowing contradictions makes sense in our daily life. It is as if we impose on ourselves some limitations.

    Inconsistency robustness is information system performance in the face of continually pervasive inconsistencies---a shift from the previously dominant paradigms of inconsistency denial and inconsistency elimination attempting to sweep them under the rug.

    Inconsistency robustness differs from previous paradigms based on belief revision, probability, and uncertainty as follows:

    Belief revision: Large information systems are continually, pervasively inconsistent and there is no way to revise them to attain consistency.
    Probability and fuzzy logic: In large information systems, there are typically several ways to calculate probability. Often the result is that the probability is both close to 0% and close to 100%
    Uncertainty: Resolving uncertainty to determine truth is not a realistic goal in large information systems.

    Besides this is Wittgenstein's take on it,
    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?
  • Wittgenstein
    442

    This can certainly be included into the topic of discourse analysis as Wittgenstein thinks meaning is use and particularly for mathematics, he considers it to be simply rule following and what he wants to address is that we can change our conventions in mathematics like we do in any other human activities. But this dialogue is actually from a larger context in which Wittgenstein advocates a finitist viewpoint of mathematics.
  • softwhere
    111


    What occurs to me is that, when we discovered that 2 had no square root , we just invented a new kind of number, extended the concept. To me it looks like we have conventions that we sometimes are inspired to modify.
  • softwhere
    111
    But this dialogue is actually from a larger context in which Wittgenstein advocates a finitist viewpoint of mathematics.Wittgenstein

    I wonder how relevant the finitism still is. Note that we apply math with finite computers, and we use a finite subset of rational numbers to do so. Or a finite set of symbols in computer algebra systems. Technologically math has always been finite. And proofs have always been finite. So Wittgenstein can only really being chiding people, it seems, for their philosophical projections on these finite procedures. Or perhaps he could endorse intuitionistic logic, but I'm not aware of him doing serious work on that.

    Perhaps at the time there was more fervor about set theory being metaphysical and not just formal.

    I have studied some of Wittgenstein's philosophy of math, and it is fascinating. But it's also quite eccentric. Wittgenstein sometimes seems (after his youth mysticism) allergic to profundity.
  • Wittgenstein
    442

    I don't think it will be wrong to say that he was influenced by Brouwer and they did meet and discussed intuitionist logic. In the beginning of of his lectures on the philosophy of mathematics, he mentions that philosophers cannot believe that their activities will prove or discover something new that isn't available to the mathematicians. Infact, he thinks that is a wrong approach. Based on his concept of math being a human invention (later) and a repeated substitution of symbols (early view) , he develops a sort of hybrid philosophy of mathematics. It has elements of being based on logical grounds while allowing greater flexibility in adopting different rule following in maths. He doesn't show what will we get if we take that path but that there isn't a problem if we do.

    He was a constructionist and he didn't believe that statements that are undecided fall in mathematics. Consider his controversial remarks around fermet's last theorem. He didn't regard it as a mathematical statement. He probably thought it could never be proved or disproved and sadly, he was wrong on this point. This is a crucial aspect of his mathematical philosophy as he diverges from what Brouwer advocated, to quote Stanford Encyclopedia of Philosophy,

    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.

    Consider all the propositions for which l think we do not have any algorithm decidability, like goldbach conjecture.( possibly ) I think it is wrong to characterize them according to our present knowledge. The absence of a convergent algorithm at the present moment does not indicate that it will always be that way. A lot of the time mathematicians can feel the breakthrough coming. In that case, if a statement that is not decidable gets proved. It will become a mathematical statement. This conversation from being non mathematical to mathematical is really awkward.
  • softwhere
    111


    It is indeed a complicated issue. I have been quite attracted to intuitionism at times. I haven't studied it closely because my institution was not only mainstream in this regard but never even discussed its position as a choice. I did study computability in Sipser's excellent book. I very much sympathize with wanting a math that is concrete and computable as possible. It's more beautiful and real that way.

    One problem that we haven't touched on is the increasing complexity of math as one looks at more complicated theorems. At some points proofs become so big that one can no longer hold them all in one's mind. One can check the links in the chain, of course, but some of the beauty is lost as one is forced to trust the machinery of logic. Unfortunately this is necessary, at least if one wants to ground certain technical practices in pure math's proofs.

    From what I remember about intutionism (and in response to your specific point), the strangest thing was the state of a proposition being neither true nor false before a proof establishes which. This yanks math out of eternity, often violating intuitions that I find trustworthy. Intuitively a given Turing machine operating on a given input will or not halt, even if we know which while it's still running. What does your intuition say about that?
  • Harry Hindu
    5.1k
    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
    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?

    It seems to me that the end is to ultimately predict future events - like landing a spacecraft on Mars in 6 months after it has just left Earth. Since meaning is the relationship between cause and effect mathematics has meaning as the relationship between what caused the formulas, and variables in some formula, to exist (our shared experience of the world) and how they are applied to create some effect (like landing on Mars).
  • TheMadFool
    13.8k
    Wittgenstein's argument:

    1. Meanings are of two types:
    a. Meaning as use
    b. Meaning through mental imagery and pointing out

    2. Meaning type b involves to a large extent meaning type a

    3. Change a contradiction of form (p & ~p) to simply p, ignoring ~p.

    4. 3 above is simply a play on words

    5. Isn't this actually a case of wrong continuation? If given all the definitions and rules then (p & ~p) is impossible.

    6. yes in that if 5 is true it would be "natural" to be dissatisfied with any course of action.

    7. There is nothing "natural" in math

    8. Therefore, it's just a matter of switching gears and agree to make (p & ~p) "natural"

    9. Does 8 make contradictions true or is "true" being used differently?

    10. "True" is being used differently

    11. Forget about true and false since "p is true" = p and "p is false" = ~p. Only assert and don't declare truth values.

    12. When is a math based on 11 similar to standard or accepted math?

    13. When "~" is being used differently

    14. There are ironic statements where the literal meaning has an opposite truth value to the real meaning.

    15. We can mark ironic statements (14) and get for example "|-- (p & ~p)" to mean contradictions are refutable or false

    The argument doesn't work for me because at 15 he rejects what he tries to introduce viz. the contradiction (p & ~p) into math, after all "refutable" implies that something is false.

    Also, notice how Wittgenstein avoids the claim that contradictions are true in 9 by giving us the option of "true" being misused.
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