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Thread: How a wrong logic could affect mathematics?

  1. Top | #21
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    That is all I have to say. You are stuck in classical logic which today has little use outside of maybe lawyers. Logic and reasoning is now centered in computer science. AI for example.

    Get a book on math proofs, do some problems, then reopen your assertion mathematicians have it all wrong.

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    Quote Originally Posted by Speakpigeon View Post
    I believe mathematical logic is wrong. I mean, really, really wrong. I mean actually all wrong. However, I'm only really worrying, and more generally, as to the possible consequences for mathematics of using a formal logic that would be wrong.

    This question is in fact quite difficult to assess. Nearly all mathematicians use in fact their logical intuition to prove theorems. Thus, they don't have to rely on any method of formal proof and thus it doesn't seem to matter that mathematical logic should be wrong. At the same time, most mathematicians probably receive a comprehensive training in formal logic, and I can indeed routinely spot problematic statements, presented as "obviously" true, being made by mathematicians when they discuss formal logic questions, suggesting that their logical sense may be wrongly affected by their formal logic training. Yet, I'm not sure whether that actually affects the proof mathematicians produce in their personal work.

    It seems to me it's inevitable that it does. I know of specific proofs that are wrong in the sense that it's not something humans would normally accept. Mathematicians who accept them are obviously affected by their training in formal logic. However, these are proofs of logical formulas, not of mathematical theorems and these are much more difficult to assess in this respect.

    Yet, even if it is the case that actual proofs done by mathematicians using their intuition are wrongly affected by their training in formal logic, I'm still not clear what could be the consequences of that in practical term.

    One possible method to assess the possible consequences would be to compare proofs obtained using different methods of mathematical logic, such as relevance logics, intuitionistic logics, paraconsistent logics etc. However, I can't find examples of mathematical theorems proved using these methods. Further, all these methods are weaker than standard, "classical", mathematical logic, meaning that they deem valid a smaller number of logical implications and therefore, presumably, would end up with a smaller subset of the theorems currently accepted by mathematicians. Which may be good or bad but how do we know which?
    EB

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    Speakpigeon,

    I'd rather try to see whether we will have a civil conversation in the other thread, and whether there is any progress over there, before I risk participating in two threads that might go very wrong. But just to give you a suggestion: how a wrong logic would affect mathematics depends on how the logic is wrong.

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    Quote Originally Posted by Angra Mainyu View Post
    Speakpigeon,

    I'd rather try to see whether we will have a civil conversation in the other thread, and whether there is any progress over there, before I risk participating in two threads that might go very wrong.
    Do as you see fit.

    Quote Originally Posted by Angra Mainyu View Post
    But just to give you a suggestion: how a wrong logic would affect mathematics depends on how the logic is wrong.
    I'm absolutely certain it is very seriously wrong. However, I think that even a very seriously wrong logic may not have any apparent deleterious impact depending on how it is used.

    However, it seems also inevitable that the mathematicians' failure to understand logic and the general acceptance of the flawed mathematical logic has a direct impact on our capacity to even consider complex problems.

    Basically, human science has become myopic, and precisely at a time when we're trying to address ever more complex problems.

    Can you give the names of the three best logicians active today?
    EB

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    Okay, so for now, the other thread seems to be civil. So, I will try to address this:
    Quote Originally Posted by Speakpigeon
    I'm absolutely certain it is very seriously wrong.
    I'm absolutely certain it is not. But I wasn't trying to raise that issue. What I meant is that it's not the same if the correct logic is what you believe it is, or the correct logic is another logic different from what is most usually accepted in mathematics (e.g., intuitionistic logic). That would result in somewhat different outcomes. Still, if you want me to address your question under the hypothesis that you are correct that classical mathematical logic (CML) is wrong and Aristotelian logic (AL) is correct (correct or wrong with regard to matching human logic), I would say the answer is somewhat different depending on the answer to the following question:

    1. Are mathematical statements always either true or false? Or are there mathematical statements that are neither true nor false?

    I will for the time being assume all mathematical statements are either true or false. Let me know if you think otherwise, so I address the matter under that, different assumption.

    So, mathematicians got it wrong. Here are some of the consequences:

    1. All things that follow from some set of premises by AL, also follow by CML. Moreover, mathematicians can use all of the tools of AL to derive conclusions - and more tools as well. So, there is no mathematical truth that could be discovered with the correct AL that could not be discovered with CML, with arguments no longer than those used in AL.
    2. As it happens, mathematicians do derive many things from their premises that cannot be derived under AL.
    3. Moreover, very probably, there are things that can be derived on AL but it's much more difficult than deriving them on CML.

    As a result of 1.,2., 3., mathematicians are proving - using CML - many results that they would not have proven if they had been using AL.

    Now, an interesting point is that under CML, an argument is valid if it is impossible for the premises to be true but the conclusion to be false. So, as long as the premises are true, CML guarantees that so is the conclusion. Given this and the above, it turns out that one consequence of having the wrong logic is that mathematicians are able to find many truths that they would never be able to find if they had the correct logic.

    Of course, if the premises are false, also CML yields more results, and likely, more false ones. But then again, in mathematics, papers are checked and re-checked many times. False statements (that result from, say, misapplication of CML to true ones) are rooted out reliably, so unless some of the most basic intuitive statements about math (e.g., the Peano postulates) are false (which would make math pretty much doomed even with AL), it seems a consequence of having the wrong logic is being able to find many more truths than they would otherwise have been able to find.

    In short, it turns out that, as a tool for finding mathematical truth, the wrong logic CML is superior to the right logic AL.

    On the negative side, mathematicians believe that their theorems follow from their premises, but in many cases, they do not: they are merely guaranteed to be true by the premises and the correct application of CML, but - by the assumption that AL is correct and CML is not correct -, they do not follow.
    What's the solution?
    Assuming it can be shown that the assumption I'm making here is correct and CML is the wrong logic but AL is the right logic, then mathematicians who see the argument should stop believing that their theorems follow from their premises, and instead only believe that they follow by CML, but not by AL - and thus, not by human logic. But still, they should keep using CML: it is the wrong logic in the sense that it fails to match human logic, but it is a superior tool for reliably finding mathematical truths. So, a good idea is to stop claiming it is human logic, but still use the best tool we have for finding truths - namely, CML.

    That aside, if the answer to 1. is negative and some mathematical statements are neither true nor false, a somewhat different argument is required. If you believe that that is the case, please let me know and I will address it. On the other hand, if you think that the answer to 1. is affirmative, then my answer is as given above, so we can discuss the matter.

    I would like to address another point you made:

    Quote Originally Posted by Speakpigeon
    This question is in fact quite difficult to assess. Nearly all mathematicians use in fact their logical intuition to prove theorems. Thus, they don't have to rely on any method of formal proof and thus it doesn't seem to matter that mathematical logic should be wrong.
    Actually, mathematicians prove plenty of things that cannot be proven under AL. In the other thread, Bomb#20 gave a much simpler example than complex mathematics. In fact, CML is very intuitive to most mathematicians. So, it seems mathematicians are managing to develop intuitions to use a logic that is superior to human logic, at least in the context of mathematics (whether it's also superior in other contexts is a matter that requires a different discussion, but it definitely is in mathematics, in the sense it is better for finding truths).

    Mathematics would not look as it does now under AL. It would look radically different, as even very simple things could not be proven.

  6. Top | #26
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    When confronted with a proof that needs to be done unless you are omniscient you do not know if the proof is possible.

    A proof takes intuition based on experience as how to start along with trail and error until you get what appears to be a proof.

    Then the proof has to withstand peer review.

    The rules of logic apply, but there are no rules on how to apply them to an arbitrary problem.

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    Quote Originally Posted by Angra Mainyu View Post
    Okay, so for now, the other thread seems to be civil. So, I will try to address this:
    Quote Originally Posted by Speakpigeon
    I'm absolutely certain it is very seriously wrong.
    I'm absolutely certain it is not. But I wasn't trying to raise that issue. What I meant is that it's not the same if the correct logic is what you believe it is, or the correct logic is another logic different from what is most usually accepted in mathematics (e.g., intuitionistic logic). That would result in somewhat different outcomes. Still, if you want me to address your question under the hypothesis that you are correct that classical mathematical logic (CML) is wrong and Aristotelian logic (AL) is correct (correct or wrong with regard to matching human logic), I would say the answer is somewhat different depending on the answer to the following question:

    1. Are mathematical statements always either true or false? Or are there mathematical statements that are neither true nor false?
    I think all statements are either true or false.

    Quote Originally Posted by Angra Mainyu View Post
    I will for the time being assume all mathematical statements are either true or false. Let me know if you think otherwise, so I address the matter under that, different assumption.

    So, mathematicians got it wrong. Here are some of the consequences:

    1. All things that follow from some set of premises by AL, also follow by CML.
    No. Some implications (in effect an infinity of them) are valid according to Aristotelian logic and not valid according to mathematical logic. And you also have the opposite situation.

    So, no, not all things that follow in Aristotelian logic also follow in mathematical logic. As I said from the start, mathematical logic is seriously wrong.

    Quote Originally Posted by Angra Mainyu View Post
    Moreover, mathematicians can use all of the tools of AL to derive conclusions - and more tools as well. So, there is no mathematical truth that could be discovered with the correct AL that could not be discovered with CML, with arguments no longer than those used in AL
    Again, that's not true. Mathematical logic produces two types of error. First, it declares valid inferences that are not valid. Second, it is myopic in that it cannot prove valid some inferences that are valid. And again, when I say some, there is in fact an infinity of these inferences.
    EB

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    Quote Originally Posted by Angra Mainyu View Post
    2. As it happens, mathematicians do derive many things from their premises that cannot be derived under AL.
    And then what? The derivations you're talking about are just wrong, as is the case with EFQ and ACQ for example.

    Quote Originally Posted by Angra Mainyu View Post
    3. Moreover, very probably, there are things that can be derived on AL but it's much more difficult than deriving them on CML.
    There is as of today no formal method that does Aristotelian logic. You seem to be under the misconception that there is one. You're not the only one in that. Aristotle wasn't able to offer a method of logic doing Aristotelian logic.

    Quote Originally Posted by Angra Mainyu View Post
    As a result of 1.,2., 3., mathematicians are proving - using CML - many results that they would not have proven if they had been using AL.
    Proofs that have zero value.

    Quote Originally Posted by Angra Mainyu View Post
    Now, an interesting point is that under CML, an argument is valid if it is impossible for the premises to be true but the conclusion to be false. So, as long as the premises are true, CML guarantees that so is the conclusion. Given this and the above, it turns out that one consequence of having the wrong logic is that mathematicians are able to find many truths that they would never be able to find if they had the correct logic.
    What you call truths here are in fact invalid conclusions. I'm not sure there's anything interesting in doing that.

    Quote Originally Posted by Angra Mainyu View Post
    Of course, if the premises are false, also CML yields more results, and likely, more false ones. But then again, in mathematics, papers are checked and re-checked many times. False statements (that result from, say, misapplication of CML to true ones) are rooted out reliably, so unless some of the most basic intuitive statements about math (e.g., the Peano postulates) are false (which would make math pretty much doomed even with AL), it seems a consequence of having the wrong logic is being able to find many more truths than they would otherwise have been able to find.
    Truths that are not truths

    Quote Originally Posted by Angra Mainyu View Post
    In short, it turns out that, as a tool for finding mathematical truth, the wrong logic CML is superior to the right logic AL.
    Mistaking invalid conclusions for valid ones is not superior to anything. It's zany.

    Quote Originally Posted by Angra Mainyu View Post
    On the negative side, mathematicians believe that their theorems follow from their premises, but in many cases, they do not: they are merely guaranteed to be true by the premises and the correct application of CML, but - by the assumption that AL is correct and CML is not correct -, they do not follow.
    What's the solution?
    Assuming it can be shown that the assumption I'm making here is correct and CML is the wrong logic but AL is the right logic, then mathematicians who see the argument should stop believing that their theorems follow from their premises, and instead only believe that they follow by CML, but not by AL - and thus, not by human logic. But still, they should keep using CML: it is the wrong logic in the sense that it fails to match human logic, but it is a superior tool for reliably finding mathematical truths. So, a good idea is to stop claiming it is human logic, but still use the best tool we have for finding truths - namely, CML.
    A "mathematical truth" which is not a truth is not a mathematical truth.

    Quote Originally Posted by Angra Mainyu View Post
    That aside, if the answer to 1. is negative and some mathematical statements are neither true nor false, a somewhat different argument is required. If you believe that that is the case, please let me know and I will address it. On the other hand, if you think that the answer to 1. is affirmative, then my answer is as given above, so we can discuss the matter.
    I'm not sure there's much to discuss. Mathematicians have come to believe their own fantasies. Their methods of logic show they don't understand logic at all. Again, funny they should very aptly use their own logical sense to prove theorems, including theorems that are validly inferred from premises but that are false because the premises are false. A classical situation in logic but somewhat ironical given it's mathematicians.
    EB

  9. Top | #29
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    Quote Originally Posted by Angra Mainyu View Post
    Okay, so for now, the other thread seems to be civil. So, I will try to address this:
    I would like to address another point you made:

    Quote Originally Posted by Speakpigeon
    This question is in fact quite difficult to assess. Nearly all mathematicians use in fact their logical intuition to prove theorems. Thus, they don't have to rely on any method of formal proof and thus it doesn't seem to matter that mathematical logic should be wrong.
    Actually, mathematicians prove plenty of things that cannot be proven under AL. In the other thread, Bomb#20 gave a much simpler example than complex mathematics. In fact, CML is very intuitive to most mathematicians. So, it seems mathematicians are managing to develop intuitions to use a logic that is superior to human logic, at least in the context of mathematics (whether it's also superior in other contexts is a matter that requires a different discussion, but it definitely is in mathematics, in the sense it is better for finding truths).

    Mathematics would not look as it does now under AL. It would look radically different, as even very simple things could not be proven.
    Yeah, I looked at Bomb#20's example but it's wrong. I'm not going to explain why but it's wrong, and to be honest, it's rather easy to see what's wrong with it.

    OK, thanks for giving me a run-down of your arguments but not one of them is conclusive simply because your assumptions, and therefore premises, are systematically false.

    You also seem to have a rather strange conception whereby there would be "mathematical truths" proven with the wrong logic... I don't buy that. There's just one unique deductive logic and it's probably a universal (Aliens would have the same, inevitably). So, if it is not true, it's not a mathematical truth, irrespective of whether you think you have a mathematical proof of it and I've seen mathematical proofs that are effectively wrong.

    Anyway, thanks again for going into the details of what you think. This is very helpful.
    EB

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    Quote Originally Posted by Speakpigeon
    I think all statements are either true or false.
    Alright, so let us stipulate from now on, and for the sake of the argument, that that is correct (it makes my case easier, though I could make my case as well it if you had said otherwise). Now consider the definition of validity in classical mathematical logic (CML for short). A proof is valid in CML if and only if it takes a form that makes it impossible for the premises to be true but the conclusion to be false.

    It follows that CML-valid proofs with true premises have true conclusions, always. So, CML leads from truths to truths.

    Now, Aristotelian-valid proofs with true premises also have true conclusions. But there are proofs that are CML-valid but not Aristotelian-valid. In fact, for any proposed system of logic PL in which all valid proofs with true premises have true conclusions (whether the Aristotelian-system, or intuitionism, or any other system), any proof that is PL-valid is also CML-valid. Why? Because, by definition, a proof is valid in CML if and only if it takes a form that makes it impossible for the premises to be true but the conclusion to be false.

    The conclusion?

    CML-validity is the strongest form of truth-preserving validity. Any mathematical truth that can be discovered by any other truth-preserving proposed system of logic (whether intuitionistic, Aristotelian, or any other), can also be discovered (and with no greater difficulty) from the same premises with CML-logic. On the other hand, for any proposed truth-preserving system of logic PL, there are arguments with true premises that are CML-valid but not PL-valid. As a result, there are mathematical truths that can be discovered by means of CML, but not PL. Moreover, in the case of systems like Aristotelian logic, some proofs might be extremely difficult to obtain even if valid, whereas CML provides more tools for finding truths.

    Note that all of this is independent of whether CML matches human logic, or not.

    Now, it is also true that if the premises are false, CML will likely lead to more errors than weaker systems, like the Aristotelian system or Intuitionistic Logic. However, in mathematics, that is not likely problematic, for two reasons:

    1. The basic, self-evident statements taken as axioms in different fields are clearly true (and if some of them weren't, then human mathematical intuition would seem hopelessly lost, so that a weaker system of logic would be of no help).
    2. Sometimes, proofs begin with previous results, which already are in no way self-evident. But those more complicated results have been proven from earlier results, etc., until we get to something really basic. And the proofs have been checked by other mathematicians. Errors are usually soon discovered, and when they rarely persist for a while, they're still discovered.

    So, in short, CML is the best way of finding mathematical truths. Now, there are mathematical truths that we will never find. But we will find a lot more with CML than with any other proposed system PL. Granting now for the sake of the argument that CML is the wrong logic in the sense that it does not match human logic, and further granting that your arguments on the matter are persuasive, then I would say we should still do mathematics using the best truth-finding method we have - namely, CML
    Again, in this case, the wrong logic is superior to the right logic, as a tool for finding mathematical truths.


    Quote Originally Posted by Speakpigeon
    No. Some implications (in effect an infinity of them) are valid according to Aristotelian logic and not valid according to mathematical logic.
    That is false (of course, we are talking about deductions). There are no implications that are Aristotelian-valid but not CML-valid. Remember, if a proof is CML-invalid, then by definition, it takes a form such that it is possible for the premises to be true but the conclusion false. That would imply that Aristotelian logic fails to be truth-preserving, which would be a devastating blow for it, regardless of whether it matches human logic.
    Fortunately for Aristotelian logic, every proof that is Aristotelian-valid is also CML-valid.

    Quote Originally Posted by Speakpigeon
    And you also have the opposite situation.
    Indeed, you do have that. There are infinitely many proofs that are CML-valid, but not Aristotelian-valid. But it's not merely that there are infinitely many proofs like that. More importantly, examples among those infinitely many are all over the place. As I mentioned, a very simple example was provided by Bomb#20 here.

    Quote Originally Posted by Speakpigeon
    So, no, not all things that follow in Aristotelian logic also follow in mathematical logic. As I said from the start, mathematical logic is seriously wrong.
    Granting for the sake of the argument that mathematical logic is seriously wrong as you believe (which I do not believe at all, but that's not the point here), the fact remains that all proofs that are valid in Aristotelian logic are also valid in CML. Moreover, if there were Arisotelian-valid proofs that are not CML-valid, then Aristotelian logic would not be truth-preserving.


    Quote Originally Posted by Speakpigeon
    Again, that's not true. Mathematical logic produces two types of error.
    It produces neither. But I'm willing to assume in this thread and for the sake of the argument that it is the wrong logic, in the sense that it fails to match human logic. Under that assumption, it produces the first but not the second type of error. But it's an error easily fixed if we assume there is a persuasive argument to the conclusion that CML fails to match human logic.

    Quote Originally Posted by Speakpigeon
    First, it declares valid inferences that are not valid.
    Yes, by assumption in this thread (not in reality). Adding the further assumption that there is a good argument for the conclusion that some inferences are CML-valid but not human-logic-valid, the solution is to refrain from claiming that all CML-valid inferences are also human-logic-valid, or valid in the colloquial sense of the words.

    However, CML remains superior to human logic as a logic for mathematics, as it can be used to obtain all of the mathematical truths that can be obtained with human logic, and more truths as well (assuming human logic is Aristotelian logic or some other truth-preserving system; on the other hand, if human logic is not truth-preserving, the superiority of CML is even greater).

    Quote Originally Posted by Speakpigeon
    Second, it is myopic in that it cannot prove valid some inferences that are valid. And again, when I say some, there is in fact an infinity of these inferences.
    And again, that is false. Fortunately so, because any inference that is not CML-valid is of a form that is not truth-preserving. If there were a single inference that is valid but not CML-valid, then validity (unlike CML-validity) would fail to be truth-preserving.

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