How a wrong logic could affect mathematics?

[steve_bank]

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.

 

[Speakpigeon]

Reminder

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

 

[Angra Mainyu]

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.

 

[Speakpigeon]

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.

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

 

[Angra Mainyu]

Okay, so for now, the other thread seems to be civil. So, I will try to address this:

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:

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.

 

[steve_bank]

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.

 

[Speakpigeon]

Okay, so for now, the other thread seems to be civil. So, I will try to address this:

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.

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.

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

 

[Speakpigeon]

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.

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.

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.

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.

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

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.

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.

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

 

[Speakpigeon]

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:

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

 

[Angra Mainyu]

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.

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.

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.

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.

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.

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).

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.

 

[Angra Mainyu]

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.

If you mean they are wrong in the sense that they are not valid in the usual, colloquial sense of the English term 'valid', yes, that is so (by assumption in this thread; in reality, it is false, but that's another matter).
However, those derivations are derivations of true conclusions. Why? Because the premises are true, CML is truth-preserving. So, the solution to the (assumed) problem is to recognize that they are not valid, and then continue to use CML, not as a match to human logic, but as a superior method for finding mathematical truths.

Proofs that have zero value.

Zero value to whom?
Different people value different things. I value finding mathematical truths by a reliable method. If that method is human logic, great. But if we have a more effective truth-preserving method and we can find mathematical truths that human logic would not have allowed us to find, then I value both the proofs and the fact that we have such superior method - again, by assumption, etc.; I do not think CML is a departure from human logic, but if there is, it's a positive departure: that is what I've been arguing.

What you call truths here are in fact invalid conclusions. I'm not sure there's anything interesting in doing that.

What I call truths are truths. CML is truth-preserving. We start with true premises, we make a CML-valid argument, and we get a true conclusion. That may not be interesting to you because we are obtaining truths by means of invalid arguments. But then again, I do find it interesting that we have a truth-preserving method (namely, CML) that is superior to human logic (at least in the context of mathematics), and which allows us to find mathematical truths that would have otherwise never been found (well, I would find it interesting if I thought that some CML-valid inferences are not valid in the usual sense). I also find the mathematical truths that we find in that manner (some of them, anwyay) interesting. If you do not, well, different people have different interests. But the question of the thread is about the consequences of having the wrong logic. And a big consequence is that we can reliably find mathematical truths that we could never find if we had the right logic, without getting falsehoods as a side-effect. Having this particular sort of wrong logic has pretty good consequences for the practice of mathematics.

Truths that are not truths

Yes, they are truths. When we start with true premises and make CML-valid arguments, we get true conclusions.

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.

As I said, it is a superior tool for finding mathematical truths. However, the colloquial meaning of the word 'valid' is (by assumption, not in reality) such that some arguments are not valid but are CML-valid, and of course believing they're valid is not good. So, assuming that there is a persuasive argument showing that some arguments are not valid but are CML-valid, then mathematicians who read the persuasive argument in question should ditch the false claim that all CML-valid arguments are valid. And then, they should reckon that they have a truth for finding mathematical truth that is superior to human logic, and then keep using it.

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.

Indeed. But my points remain. For the reasons I have given, under the assumption that CML is the wrong logic in the sense it does not match human logic, we should reckon it is a superior method of finding mathematical truths - not "truths" that aren't truths, but truths.

 

[Angra Mainyu]

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.

If you mean that the problem is the use of "livestock" instead of "farm animals", that was used as a synonym by Bomb#20 (so, if you like, you have an implicit premise). But if you prefer, just say "farm animals" instead of "livestock" in the second sentence, and my point remains.

On the other hand, if you have another objection, then I disagree because I can see that the argument works, but given that you do not explain your counterargument, I have no way of further engaging.

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 are welcome, but of course I disagree. Under the assumptions that CML fails to match human logic and also that all mathematical statements are either true or false, I have made a compelling argument to the conclusion that CML is a superior tool for finding mathematical truths than human logic is.

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.

It is strange, but I do not believe it is true. Rather, I am assuming for the sake of the argument that CML is the wrong logic in the specific sense of failing to match human logic. Under that assumption - which, again, I'm not buying -, plus the assumption that all mathematical statements are true or false (which you agree with, so no problem there) my arguments establish that there are mathematical truths proven with the wrong logic - not just with any wrong logic, but with the particular wrong logic CML.

There's just one unique deductive logic and it's probably a universal (Aliens would have the same, inevitably).

That is extremely improbable under the assumption that CML is the wrong logic. But regardless, I could assume for the sake of the argument that even aliens would have the same universal logic. And then, my arguments in this thread show that under that assumption plus the assumption that all mathematical statements are true or false (which you agree with, so no problem there), we have discovered a tool for finding mathematical truths that is superior to the universal logic - namely, CML. Weird? Perhaps, but then, I'm not suggesting there actually is a tool for finding mathematical truths that is superior to even human logic. I'm just discussing the consequences of this particular wrong logic under the assumption that it is wrong, which I grant only for the sake of the argument but do not buy at all.

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.

Indeed, if it is not true, it is not a mathematical truth. However, if a statement P CML-follows from mathematical truths, then it is true, and thus a mathematical truth, regardless of whether it human-follows, Aristotelian-follows and/or universal-logic-follows from mathematical truths.

In other words, if we start with mathematical truths M1, .., Mk, and by the CML-valid argument A1 we obtain a conclusion M_{k+1}, then M_{k+1} is true, and thus a mathematical truth, regardless of whether A1 is Aristotelian-valid, human-valid, alien-valid, or universal-valid, and regardless of whether all of those notions of validity are the same or not. CML is the most powerful truth-finding method in mathematics, even if it is - by assumption - the wrong logic.

So, if you persuaded me that it's against human logic, or even against some universal logic, I would just stop saying the arguments are valid - but I would keep making them, finding mathematical truths with them, and call them CML-valid.

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

You're welcome.
Btw, our disagreements aside, I'm enjoying the exchange. I like civil discussions.

 

[Speakpigeon]

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.

If you mean that the problem is the use of "livestock" instead of "farm animals", that was used as a synonym by Bomb#20 (so, if you like, you have an implicit premise). But if you prefer, just say "farm animals" instead of "livestock" in the second sentence, and my point remains.

No, I wouldn't make such a cheap point. Assuming the argument was made all formally proper, his reasoning that the argument cannot be shown valid in Aristotelian logic is wrong and obviously so.

On the other hand, if you have another objection, then I disagree because I can see that the argument works, but given that you do not explain your counterargument, I have no way of further engaging.

Yep, sorry about that.

Btw, our disagreements aside, I'm enjoying the exchange. I like civil discussions.

You're welcome. I'm sorry I can't say the same. You're not the cause. You are just representative of apparently most people trained academically and probably most people in general. I've shopped around to see if I could find people like you willing at least to explain themselves. Few in fact are, but those who are show they follow the same sort of reasoning pattern as you do. I guess I would call that "formal reasoning". Formal reasoning in this sense is that you reason from a set of formal rules. A lady put it succinctly on radio the other day: there's a difference between reasoning and thinking. People like you don't seem interested in thinking. They're obviously good at reasoning and that's something. I guess 99.9% of humans belong to this category. Talk to a bureaucrat and you'll see what it does. I would guess it's the consequence of natural selection whereby reasoning by rules saves time. Thinking sure takes an awful amount of time but without at least a few people doing it, humanity would still apply the rules of the caveman. I guess that's why we can remember the names of the few who changed the world by doing it: Aristotle, Copernicus, Kepler, Galileo, Newton, Leibniz, Einstein, and indeed Russell among others. Thinking doesn't necessarily work and I'm sure many more have tried it without any result to show for it. I'm sure, too, we need both kinds of people and I certainly need you all to make my life easier and more comfortable. So, thanks. But people like you do look myopic and this despite their higher education. There is something even awful to think of these hundreds of millions of people around the world who have received a higher education and yet apparently go through life without doing much more than reason from rules they don't understand. And they don't even think about that. I think I now have most of the few reasons mathematicians have for doing things the way they do. Not one of them is conclusive and it is rather easy to understand why. Yet, you all keep repeating these reasons as if they had a value. I understand what you say. I also understand what you say has no conclusive value. Why don't you? Given that it would be rather easy, I guess you're just not interested and if millions of people are not interested, there has to be a reason. Natural selection seems a good explanation. Some thinker came up with this idea a while ago now. It's a rule now. Saves time. So, given I understand your reasons, there's nothing else to say about the subject. Thanks again.
EB

 

[repoman]

I was thinking about this for the extrapolation of our 3d geometry to higher dimensional (4d+) geometry and polytopes,

Could some logic or mechanism of extrapolation be wrong? Who would be able to tell? Are 5d life forms going to pop out and say "Hey, you did a pretty good job, but there a couple errors in the standard textbooks. Actually a reclusive guy got it right, but none of you believed him." Probably not.

 

[steve_bank]

I was thinking about this for the extrapolation of our 3d geometry to higher dimensional (4d+) geometry and polytopes,

Could some logic or mechanism of extrapolation be wrong? Who would be able to tell? Are 5d life forms going to pop out and say "Hey, you did a pretty good job, but there a couple errors in the standard textbooks. Actually a reclusive guy got it right, but none of you believed him." Probably not.

Any mathematical system has to be consistent. This means that no matter how you apply the rules to a problem you will always get the same answer. I'd use the word extension rather than extrapolation.

Euclidean geometry is consistent but it is not perfect. In Euclidean geometry the shortest distance between two points is a straight line. In Relativity it is not. Einstein had help from a mathematician devaluing new math.

There is plane geometry, spherical geometry and so on.

 

[Angra Mainyu]

No, I wouldn't make such a cheap point. Assuming the argument was made all formally proper, his reasoning that the argument cannot be shown valid in Aristotelian logic is wrong and obviously so.

I disagree, but you can show that I am mistaken by deriving the conclusion using Aristotle's syllogisms.

Yep, sorry about that.

Okay, so nothing else I can do. He gave the example. I provided a link. I'm willing to engage in a discussion if you give an argument, but you're just denying it. If it's so easy, you could simply take up his challenge and derive the conclusion using Aristotle's syllogisms.

Btw, our disagreements aside, I'm enjoying the exchange. I like civil discussions.

You're welcome. I'm sorry I can't say the same. You're not the cause. You are just representative of apparently most people trained academically and probably most people in general. I've shopped around to see if I could find people like you willing at least to explain themselves. Few in fact are, but those who are show they follow the same sort of reasoning pattern as you do. I guess I would call that "formal reasoning". Formal reasoning in this sense is that you reason from a set of formal rules. A lady put it succinctly on radio the other day: there's a difference between reasoning and thinking. People like you don't seem interested in thinking. They're obviously good at reasoning and that's something. I guess 99.9% of humans belong to this category. Talk to a bureaucrat and you'll see what it does. I would guess it's the consequence of natural selection whereby reasoning by rules saves time. Thinking sure takes an awful amount of time but without at least a few people doing it, humanity would still apply the rules of the caveman. I guess that's why we can remember the names of the few who changed the world by doing it: Aristotle, Copernicus, Kepler, Galileo, Newton, Leibniz, Einstein, and indeed Russell among others. Thinking doesn't necessarily work and I'm sure many more have tried it without any result to show for it. I'm sure, too, we need both kinds of people and I certainly need you all to make my life easier and more comfortable. So, thanks. But people like you do look myopic and this despite their higher education. There is something even awful to think of these hundreds of millions of people around the world who have received a higher education and yet apparently go through life without doing much more than reason from rules they don't understand. And they don't even think about that. I think I now have most of the few reasons mathematicians have for doing things the way they do. Not one of them is conclusive and it is rather easy to understand why. Yet, you all keep repeating these reasons as if they had a value. I understand what you say. I also understand what you say has no conclusive value. Why don't you? Given that it would be rather easy, I guess you're just not interested and if millions of people are not interested, there has to be a reason. Natural selection seems a good explanation. Some thinker came up with this idea a while ago now. It's a rule now. Saves time. So, given I understand your reasons, there's nothing else to say about the subject. Thanks again.

You are very, very mistaken. In fact, my arguments in this thread are rationally compelling. Why don't you get it?

But - purely for example -, you repeatedly made the claim that there were infinitely many proofs that were valid in the Aristotelian system but not in CML (and to be crystal clear, when talking about 'classical mathematical logic' or CML , we are using the definition of validity you provided in this thread). I pointed out that that is false: in fact, there is not even one proof that is valid in the Aristotelian system but not in CML. But moreover, I proved that if there were any such proof, then the Aristotelian system would fail to be truth-preserving. Now, you say you understand what I'm saying. If you do, then you understand my conclusive argument on the subject. Yet, instead of conceding that you were seriously mistaken about a crucial point, you make comments about me that are both unwarranted and false. Why do you do that?
Or consider my - again, conclusive - argument showing (assuming that every mathematical statement is either true or false, as you believe) that anything that CML-follows from mathematical truths is also a mathematical truth, which you earlier denied. You have not yet acknowledged that you were mistaken about that crucial point. Instead, you said nothing about such a serious error, and replied with false, unwarranted and disparaging comments about me - no longer talking about the subject matter, but about me.
Why do you behave like that? In my assessment, the most likely scenario by far is that you have in fact failed to understand my arguments, even though you believe you did. The alternative interpretation is that you're being dishonest. However, that's very improbable, given your previous track record and frankly previous human track record. In fact, your behavior matches what is common of nearly all humans I have encountered on the internet. Your reaction is unsurprising, though I have to say, the conversation is no longer enjoyable, because it's no longer about math, logic, philosophy, etc., but instead it's about me, and now - in response - about you, etc.

In light of this latest reply, I will avoid engaging you in any new threads, except if you decide to say something about me in them even though I'm not taking part. Please, do not do that. On the other hand, I will defend any of my points in any of the threads in which I'm already taking part, provided that you choose to engage them rather than just ignore them or declare them wrong (while they're right and backed by conclusive arguments) without argumentation.

 

[Speakpigeon]

I was thinking about this for the extrapolation of our 3d geometry to higher dimensional (4d+) geometry and polytopes,

Could some logic or mechanism of extrapolation be wrong? Who would be able to tell? Are 5d life forms going to pop out and say "Hey, you did a pretty good job, but there a couple errors in the standard textbooks. Actually a reclusive guy got it right, but none of you believed him." Probably not.

Any mathematical system has to be consistent. This means that no matter how you apply the rules to a problem you will always get the same answer. I'd use the word extension rather than extrapolation.

Euclidean geometry is consistent but it is not perfect. In Euclidean geometry the shortest distance between two points is a straight line. In Relativity it is not. Einstein had help from a mathematician devaluing new math.

There is plane geometry, spherical geometry and so on.

Yeah, but this has nothing to do with logic and all to do with the evidence we have available to us as ordinary human beings that the shortest path between two points is a straight line. If our assumption is wrong, our conclusion may well be wrong two.
EB

 

[Speakpigeon]

I was thinking about this for the extrapolation of our 3d geometry to higher dimensional (4d+) geometry and polytopes,

Could some logic or mechanism of extrapolation be wrong? Who would be able to tell? Are 5d life forms going to pop out and say "Hey, you did a pretty good job, but there a couple errors in the standard textbooks. Actually a reclusive guy got it right, but none of you believed him." Probably not.

Suppose you have a method of logic which is wrong. For example, let's assume it says that some well-know syllogism is in fact not valid. For example, let's assume that your logic says that it is not valid that if x is F and that all Fs are P, then x is P. Apply that to any concrete case, say that copper is a conductor. So, assume you know, from experience, that copper has high electrical conductivity. Assume you have copper wires aplenty in your stock, and let's assume you need a good conductor for the system you are developing. Of course we all understand we could use any copper wire because it will be a very good conductor. But your logic is faulty so in fact you don't understand that. The result is you will just fail to understand the world around you. OK, all methods of logic give the correct assessment of the validity of ordinary syllogisms and possibly all inferences we routinely come across in science and engineering. Yet, if your logic is correct for all these formulas but wrong for some other, you will just never realise it. You will do as you usually do without taking advantage of the world as it is. You would be like a cow in front of a computer, not knowing what to do with it. Maybe it's unlikely, maybe it's a marginal problem if the formulas concerned maybe are very complicated or somewhat special, but the result might be vital in the future, when we get to deal with the survival of humanity as problems start to pile up and we have no solution for them. Your choice.
EB

 

[Speakpigeon]

You are very, very mistaken. In fact, my arguments in this thread are rationally compelling. Why don't you get it?

I already told you I understand your argument. Your logic is not faulty here. It's your assumptions. As I already told you. And all of the argument mathematicians use to support the validity of their method of logic, those I've been able to find anyway, are similarly based on faulty assumptions. Good logic, though. Ain't it ironic? A good logic to infer a wrong logic? Yeah, can happen. Mathematicians did it.
EB

 

[fromderinside]

The problem with logic is that it is drawn from a priori presumption, divorced from material evidence. No reasonable person would suggest assuming is empirical and no empiricist would accept other than empirical evidence as basis for decisions. Consequently your arguments are moot re attempting to arrive at good logic from rational argument.