How a wrong logic could affect mathematics?

[Speakpigeon]

Yes, mathematical logic is a good example of non-Aristotelian logic.

But it's better to think of it as non-logic.
EB

 

[Speakpigeon]

Thanks for your examples but, again, that's not what I meant. This thread is about the possible consequences of mathematical logic being wrong. I'm asking you for examples of important theorems based on mathematical logic, not trivial examples where I can see that the conclusion follows and therefore where nobody is wrong.
EB

Yes, of course, I understand what you are saying. I'm talking about important theorems not derivable in Aristotelian logic. For that matter, I already provided links so that interested readers can find that important theorems are not derivable when one restricts logic in different senses. The point is that if I show examples, you can simply say that I'm wrong.

I seem to have missed the posts where you give important examples and where you explain how the examples are only derivable from mathematical logic. All I remember is a trivial example about cows.
EB

 

[Speakpigeon]

So EB 'How bad is it?' Bad as in the looming collapse of human civilization because of bad logic?

I don't know yet. I have to pull teeth out of these people, you know.

I asked for a justification that the definition of validity used in mathematics is correct, and couldn't get an answer.

I asked for examples of important theorems proved in mathematical logic that couldn't be proved in Aristotelian logic and only got a trivial example about cows.

So, I'm still expecting without much hope.
EB

 

[Speakpigeon]

And your link to intuitionistic logic on Wikipedia gives examples of the kind of contradictions I was talking about and you denied that they existed between different methods of mathematical logic, here between intuitionistic and non-inutitionistic methods:

https://plato.stanford.edu/entries/intuitionism/

Thus in the context of the natural numbers, intuitionism and classical mathematics have a lot in common. It is only when other infinite sets such as the real numbers are considered that intuitionism starts to differ more dramatically from classical mathematics, and from most other forms of constructivism as well.

3.4 The continuum
In intuitionism, the continuum is both an extension and a restriction of its classical counterpart. In its full form, both notions are incomparable since the intuitionistic real numbers possess properties that the classical real numbers do not have. A famous example, to be discussed below, is the theorem that in intuitionism every total function on the continuum is continuous. That the intuitionistic continuum does not satisfy certain classical properties can be easily seen via weak counterexamples. That it also contains properties that the classical reals do not posses stems from the existence, in intuitionism, of choice sequences.

Weak counterexamples
The weak counterexamples, introduced by Brouwer in 1908, are the first examples that Brouwer used to show that the shift from a classical to an intuitionistic conception of mathematics is not without consequence for the mathematical truths that can be established according to these philosophies. They show that certain classical statements are presently unacceptable from an intuitionistic point of view.

I really like the euphemism...They show that certain classical statements are presently unacceptable from an intuitionistic point of view.

I could learn something from these guys, how to be less rude.

But you have a lot to learn from them apparently...

EB

 

[fromderinside]

So do mathematicians subscribe to empirical validity or rational validity. Given that gains in math are in sciences I suspect math validity is onto empirical validity does it do what it prescribes to do.

I guess one could argue both or otherwise, but that would be saying nothing.

One might see the structure of the clock as the target of rational validity while the function of the clock as the target of empirical validity.

 

[steve_bank]

Ultimately math is validated by usage and test cases.

Logic is a set of rules and definitions. So is algebra. There are no rules as to how to apply both algebra and logic. That comes from experience.

Both logic and algebra are subject to GIGO, garbage in garbage out. Write a valid algebraic equation, it may or may not adequately model and compute the problem at hand. Same with logic. You can create a valid logical equation or statement in formal logic and it may or may not adequately model the problem at hand.

Testing in both cases are required. Logical expressions alone do not lead to truth.

BTW EB, please let me know if there are problems in the math underlying aerodynamics, just in case jets may just stop flying. Point being the validation of math is not in proofs alone, it is centuries of applications without any hint of a problem.

You can not argue what demonstrably works.

 

[Angra Mainyu]

Thanks for your examples but, again, that's not what I meant. This thread is about the possible consequences of mathematical logic being wrong. I'm asking you for examples of important theorems based on mathematical logic, not trivial examples where I can see that the conclusion follows and therefore where nobody is wrong.
EB

Yes, of course, I understand what you are saying. I'm talking about important theorems not derivable in Aristotelian logic. For that matter, I already provided links so that interested readers can find that important theorems are not derivable when one restricts logic in different senses. The point is that if I show examples, you can simply say that I'm wrong.

I seem to have missed the posts where you give important examples and where you explain how the examples are only derivable from mathematical logic. All I remember is a trivial example about cows.
EB

Yes, you seem to have missed much of the exchange. I posted links to SEP articles on the matter, considering both intuitionistic logic (which accepts explosion, but not A or ¬A), and also relevance logics that also exclude explosion (and, of course, anything that implies it). The former result in significant changes even in the continuum (i.e., the real numbers), whereas the latter results in very different natural numbers too. The articles also give you citations so that interested readers can find the papers.

 

[Angra Mainyu]

And your link to intuitionistic logic on Wikipedia gives examples of the kind of contradictions I was talking about and you denied that they existed between different methods of mathematical logic, here between intuitionistic and non-inutitionistic methods:

https://plato.stanford.edu/entries/intuitionism/

Thus in the context of the natural numbers, intuitionism and classical mathematics have a lot in common. It is only when other infinite sets such as the real numbers are considered that intuitionism starts to differ more dramatically from classical mathematics, and from most other forms of constructivism as well.

3.4 The continuum
In intuitionism, the continuum is both an extension and a restriction of its classical counterpart. In its full form, both notions are incomparable since the intuitionistic real numbers possess properties that the classical real numbers do not have. A famous example, to be discussed below, is the theorem that in intuitionism every total function on the continuum is continuous. That the intuitionistic continuum does not satisfy certain classical properties can be easily seen via weak counterexamples. That it also contains properties that the classical reals do not posses stems from the existence, in intuitionism, of choice sequences.

Weak counterexamples
The weak counterexamples, introduced by Brouwer in 1908, are the first examples that Brouwer used to show that the shift from a classical to an intuitionistic conception of mathematics is not without consequence for the mathematical truths that can be established according to these philosophies. They show that certain classical statements are presently unacceptable from an intuitionistic point of view.

I really like the euphemism...They show that certain classical statements are presently unacceptable from an intuitionistic point of view.

I could learn something from these guys, how to be less rude.

But you have a lot to learn from them apparently...

EB

You just grossly misrepresent what I said. Let us explain what happened to readers, to reduce the risk of their making the mistake of thinking that your claim about what I said is correct. Let us take a look. The source is the other thread:

https://talkfreethought.org/showthr...-conclusion-follows-necessarily-from-premises

Here's your claim:
https://talkfreethought.org/showthr...-from-premises&p=676777&viewfull=1#post676777

This view isn't a foregone conclusion. The practice of Mathematical logic today suggests on the contrary that logic is arbitrary. Mathematical logic itself is a branch of mathematics, not a method or a theory of logic. As a branch of mathematics, it brings together a very large number of theories and methods (calculus) which are all different from each other and in effect mutually contradictory.

My question:

https://talkfreethought.org/showthr...-from-premises&p=679602&viewfull=1#post679602

Could you give an example of those contradictions?

Your answer:

Sure. For example, "when some mathematicians argue in support of intuitionistic logic and argue against ¬¬P->P, whereas others disagree and argue for classical mathematical logic (which accepts ¬¬P->P), then you have a disagreement about the correct logic" (Angra Mainyu, 05-31-2019, 04:07 AM).

Here's my reply:
https://talkfreethought.org/showthr...-from-premises&p=682824&viewfull=1#post682824

That is not a contradiction between different areas of math. That is a philosophical disagreement between mathematicians. It is not the same. Moreover, those who argue against ¬¬P->P do not argue that it is false. They argue that it is not proper to use it, for philosophical reasons. Depending on the person, they might say it's neither true nor false.

Still, now that I see what you have in mind when you say that " The practice of Mathematical logic today suggests on the contrary that logic is arbitrary. Mathematical logic itself is a branch of mathematics, not a method or a theory of logic. As a branch of mathematics, it brings together a very large number of theories and methods (calculus) which are all different from each other and in effect mutually contradictory.", I would say that philosophical disagreement and debate about what the correct logic is is more common among philosophers than it is among mathematicians. Would you agree also that the practice of philosophy today suggests that logic is arbitrary?

Your answer:
https://talkfreethought.org/showthr...-from-premises&p=683598&viewfull=1#post683598

By contradiction, I mean just that. One theory will say something is true, the other will say something is false.

My answer:

https://talkfreethought.org/showthr...-from-premises&p=683653&viewfull=1#post683653

However, that is not what happens between mathematical theories. It is what happens between philosophical theories about mathematics. Some philosophers and some mathematicians adhere to some of those theories; others, adhere to other such theories.
In any case, if you think that this suggests logic is arbitrary, it seems it's not so much the practice of mathematical logic today (no contradiction there), but the philosophical practices of some mathematicians and some philosophers. However, by the same disagreement criterion, we could say the practice of, say, physics today suggest that physics is arbitrary; the practice of ethical philosophy or ethical debates today suggests ethics is arbitrary, and so on. I think it's an extremely weak case.

Your answer?
No answer. As I pointed out, your point was extremely weak. And now you falsely claim that I denied the existence of some contradictions that allegedly exist. I simply showed that that particular claim of yours (alongside many others) was ill-founded.

 

[steve_bank]

Now I see where EB is coming from and why he does not post specific examples.

Something like that used to come up on science forum. Science has got it all wrong...science does not get it.

Logic is a metaphysical human creation. It is not arbitrary in a general sense. You might say historically logic evolved from observation of reality and predicting outcomes. Just like math.

I'd say EB lacks a reference point which comes from experience.

When I stared as an engineer I knew theory, but I had a feeling there was some deep revelation to be had. There was none.

Theory comes from observation and modeling. There is nothing beneath that. Logic evolved and has been tested through usage for thousands of years. A list of logical fallacies have been found, misapplication of logic and impossible logic or paradoxes.

 

[Speakpigeon]

Now I see

No you don't. You're talking from ignorance. You have zero idea what my experience is. Don't make up stuff you can't justify. It makes you look really stupid.

And it is precisely my point that mathematical logic isn't science. A proper method of logic can only be founded on the empirical evidence we have.

Don't you agree with that?

Well, you don't seem to even understand the English of what I say...
EB

 

[Speakpigeon]

You just grossly misrepresent what I said.
(...)
Your answer?
No answer. As I pointed out, your point was extremely weak. And now you falsely claim that I denied the existence of some contradictions that allegedly exist. I simply showed that that particular claim of yours (alongside many others) was ill-founded.

???

Who, Sir, I just provided one example. You now reply that I didn't answer... Beats me.

I think there's a language barrier here. I just don't understand what you say and you clearly don't understand much of what I say. I thought my English was good, though.

So, I can only repeat below the one answer I already provided, which is a publicly available SEP web-page. It's not me making up stuff... This example shows that "certain classical statements are presently unacceptable from an intuitionistic point of view". Me I read that as saying there's a contradiction, literal contradiction, between what "classical" mathematical logic says and what intuitionist logic says.

Both claim to be mathematical logic methods of logic. Both say they are correct. One must be wrong, surely.

And correct relative to what?! Surely, if it's correct at all, it has to be correct in representing the way the human brain does logic. But this is an empirical and therefore scientific question, not a mathematical one. Mathematicians are not scientists, according to me and most people anywhere and on this forum (I did a poll a while ago on that). So, why should anyone accepts that any of these dudes is correct as they claim? Where's the empirical evidence? All we have are mathematical theories and zero evidence.

In fact, we have evidence to the contrary. For example the principle of explosion and the paradoxes of material implication, whose paradoxical nature shows material implication doesn't match the kind of implication the human brain does. And now what the quote below says, that these people are contradicting each other.

These people don't even agree on what is logic and they don't even say what is logic. Read any textbook on logic and quote me where it explains what is logic. Not one ever does that nowadays. All you get in this respect now are at best pathetically fluffy considerations on the logical concepts discussed by historical logicians. And that's it. No mathematicians has been able to justify mathematical logic as a correct model of logic as a human performance and an inherent capacity of (at least) the human brain.
EB

And your link to intuitionistic logic on Wikipedia gives examples of the kind of contradictions I was talking about and you denied that they existed between different methods of mathematical logic, here between intuitionistic and non-inutitionistic methods:

https://plato.stanford.edu/entries/intuitionism/

Thus in the context of the natural numbers, intuitionism and classical mathematics have a lot in common. It is only when other infinite sets such as the real numbers are considered that intuitionism starts to differ more dramatically from classical mathematics, and from most other forms of constructivism as well.

3.4 The continuum
In intuitionism, the continuum is both an extension and a restriction of its classical counterpart. In its full form, both notions are incomparable since the intuitionistic real numbers possess properties that the classical real numbers do not have. A famous example, to be discussed below, is the theorem that in intuitionism every total function on the continuum is continuous. That the intuitionistic continuum does not satisfy certain classical properties can be easily seen via weak counterexamples. That it also contains properties that the classical reals do not posses stems from the existence, in intuitionism, of choice sequences.

Weak counterexamples
The weak counterexamples, introduced by Brouwer in 1908, are the first examples that Brouwer used to show that the shift from a classical to an intuitionistic conception of mathematics is not without consequence for the mathematical truths that can be established according to these philosophies. They show that certain classical statements are presently unacceptable from an intuitionistic point of view.

I really like the euphemism...They show that certain classical statements are presently unacceptable from an intuitionistic point of view.

I could learn something from these guys, how to be less rude.

But you have a lot to learn from them apparently...

EB

 

[steve_bank]

Now I see

No you don't. You're talking from ignorance. You have zero idea what my experience is. Don't make up stuff you can't justify. It makes you look really stupid.

And it is precisely my point that mathematical logic isn't science. A proper method of logic can only be founded on the empirical evidence we have.

Don't you agree with that?

Well, you don't seem to even understand the English of what I say...
EB

That is an old philosophical debate. Is knowledge solely based in empirical experience. Can the brain ever form an hypothesis not linked to previous empirical knowledge. I do not know, one of these perennial philosophical unanswerables.

Pragmatically both math and science in the end are validated empirically in one way or another. That fact that there are an infinite, uncountable, number of real numbers between 1.0 and 2.0 is not empirically proven. Which brings in the Incompleteness Theorem. in any consistent axiomatic system there are improvable truths in the system.

Can you give practical examples of real world problems you gave worked using either inductive, deductive, or any form of logic? Other than short syllogisms.

There is a saying attributed to Confucius 'I hear and I forget. I see and I remember. I do and I understand'. Try a bit of doing. [VBMISC][/VBMISC]

It took me years of applied work to develop understanding. That logic and reasoning works is evidenced by what is accomplished by it. The efficacy of math is evidenced by what it accomplishes used by us humans.

So far there have been no contradictions in math that surfaced. Arithmetic, algebra, trig, calculus, linear algebra and so on.

You are like someone who never pitched in baseball shouting at a pitcher in a game he has got it all wrong. You are an observer, I have been a lifelong participant and applied mathematical as most engineers are.

Again, get that small book How To Read And Do Proofs then do the problems then come back. If you have never tied to strike out a batter you really do not know pitching.

And again you said two different geometries conflict, please be specific as to what that is. Otherwise you are just going by random net links to form a conclusion.

Still do not see what you are all jazzed about.

 

[Speakpigeon]

Can you give practical examples of real world problems you gave worked using either inductive, deductive, or any form of logic? Other than short syllogisms.

There are publicly available examples in Wikipedia and probably SEP. They're not difficult to find. You're the expert at Wiki here.

And a while ago, I did a thread on one well-known, publicly available example. Short term memory only?

Try a bit of doing.

Try a bit of not making stuff up. You have zero idea of what I do.

So far there have been no contradictions in math that surfaced. Arithmetic, algebra, trig, calculus, linear algebra and so on.

I'm not sure what you're talking about here. There are contradictions between the different methods of logic that exist in mathematical logic. See above. As for areas outside mathematical logic, I didn't say there were contradictions. I'm asking how a wrong mathematical logic could affect mathematical theorems. If there was such an impact, you wouldn't notice any contradiction. Even in terms of applied mathematics, it is possible we would just miss an opportunity to understand nature rather than find some contradiction with empirical facts. And our missing would go unnoticed. Or look exactly like Quantum Physics today, with no major theoretical advance for a long while.

You are like someone who never pitched in baseball shouting at a pitcher in a game he has got it all wrong. You are an observer, I have been a lifelong participant and applied mathematical as most engineers are.

You are someone who speaks from his arsehole. You have zero idea what I do now or did all my life. Try a little bit of humility. When you don't know something, don't speak as if you did.

And again you said two different geometries conflict, please be specific as to what that is.

That's common knowledge but you're an ignoramus so you don't know.

Still do not see what you are all jazzed about.

Well, no jazz here. I just asked a question and you haven't a clue what the answer is and yet you keep pretending you can provide meaningful comments. All you're doing here is spurt very cheap philosophy.
EB

 

[Angra Mainyu]

???

Who, Sir, I just provided one example. You now reply that I didn't answer... Beats me.

That is a gross misrepresentation of what I just said in the post you were replying to. And you have the gall of saying "". Have you actually failed to realize that the claim you are making is false? You ought to know better. It should be obvious to you that the claim that you are making - namely, that I now reply that you did not answer when you did - is a false claim, made with reckless disregard for the truth. Let me quote the post I just made, so as to reduce the chance that readers might believe your report of what I said, which is false, and it should be obvious to you that it is false.

Here is what I just said:

And your link to intuitionistic logic on Wikipedia gives examples of the kind of contradictions I was talking about and you denied that they existed between different methods of mathematical logic, here between intuitionistic and non-inutitionistic methods:

https://plato.stanford.edu/entries/intuitionism/

Thus in the context of the natural numbers, intuitionism and classical mathematics have a lot in common. It is only when other infinite sets such as the real numbers are considered that intuitionism starts to differ more dramatically from classical mathematics, and from most other forms of constructivism as well.

3.4 The continuum
In intuitionism, the continuum is both an extension and a restriction of its classical counterpart. In its full form, both notions are incomparable since the intuitionistic real numbers possess properties that the classical real numbers do not have. A famous example, to be discussed below, is the theorem that in intuitionism every total function on the continuum is continuous. That the intuitionistic continuum does not satisfy certain classical properties can be easily seen via weak counterexamples. That it also contains properties that the classical reals do not posses stems from the existence, in intuitionism, of choice sequences.

Weak counterexamples
The weak counterexamples, introduced by Brouwer in 1908, are the first examples that Brouwer used to show that the shift from a classical to an intuitionistic conception of mathematics is not without consequence for the mathematical truths that can be established according to these philosophies. They show that certain classical statements are presently unacceptable from an intuitionistic point of view.

I really like the euphemism...They show that certain classical statements are presently unacceptable from an intuitionistic point of view.

I could learn something from these guys, how to be less rude.

But you have a lot to learn from them apparently...

EB

You just grossly misrepresent what I said. Let us explain what happened to readers, to reduce the risk of their making the mistake of thinking that your claim about what I said is correct. Let us take a look. The source is the other thread:

https://talkfreethought.org/showthr...-conclusion-follows-necessarily-from-premises

Here's your claim:
https://talkfreethought.org/showthr...-from-premises&p=676777&viewfull=1#post676777

This view isn't a foregone conclusion. The practice of Mathematical logic today suggests on the contrary that logic is arbitrary. Mathematical logic itself is a branch of mathematics, not a method or a theory of logic. As a branch of mathematics, it brings together a very large number of theories and methods (calculus) which are all different from each other and in effect mutually contradictory.

My question:

https://talkfreethought.org/showthr...-from-premises&p=679602&viewfull=1#post679602

Could you give an example of those contradictions?

Your answer:

Sure. For example, "when some mathematicians argue in support of intuitionistic logic and argue against ¬¬P->P, whereas others disagree and argue for classical mathematical logic (which accepts ¬¬P->P), then you have a disagreement about the correct logic" (Angra Mainyu, 05-31-2019, 04:07 AM).

Here's my reply:
https://talkfreethought.org/showthr...-from-premises&p=682824&viewfull=1#post682824

That is not a contradiction between different areas of math. That is a philosophical disagreement between mathematicians. It is not the same. Moreover, those who argue against ¬¬P->P do not argue that it is false. They argue that it is not proper to use it, for philosophical reasons. Depending on the person, they might say it's neither true nor false.

Still, now that I see what you have in mind when you say that " The practice of Mathematical logic today suggests on the contrary that logic is arbitrary. Mathematical logic itself is a branch of mathematics, not a method or a theory of logic. As a branch of mathematics, it brings together a very large number of theories and methods (calculus) which are all different from each other and in effect mutually contradictory.", I would say that philosophical disagreement and debate about what the correct logic is is more common among philosophers than it is among mathematicians. Would you agree also that the practice of philosophy today suggests that logic is arbitrary?

Your answer:
https://talkfreethought.org/showthr...-from-premises&p=683598&viewfull=1#post683598

By contradiction, I mean just that. One theory will say something is true, the other will say something is false.

My answer:

https://talkfreethought.org/showthr...-from-premises&p=683653&viewfull=1#post683653

However, that is not what happens between mathematical theories. It is what happens between philosophical theories about mathematics. Some philosophers and some mathematicians adhere to some of those theories; others, adhere to other such theories.
In any case, if you think that this suggests logic is arbitrary, it seems it's not so much the practice of mathematical logic today (no contradiction there), but the philosophical practices of some mathematicians and some philosophers. However, by the same disagreement criterion, we could say the practice of, say, physics today suggest that physics is arbitrary; the practice of ethical philosophy or ethical debates today suggests ethics is arbitrary, and so on. I think it's an extremely weak case.

Your answer?
No answer. As I pointed out, your point was extremely weak. And now you falsely claim that I denied the existence of some contradictions that allegedly exist. I simply showed that that particular claim of yours (alongside many others) was ill-founded.

As anyone can see (and that includes you, if you choose to see), I acknowledged every one of your replies, including the "example" you provided. The part of our exchange that I quoted ends with my answer:


https://talkfreethought.org/showthr...-from-premises&p=683653&viewfull=1#post683653

However, that is not what happens between mathematical theories. It is what happens between philosophical theories about mathematics. Some philosophers and some mathematicians adhere to some of those theories; others, adhere to other such theories.
In any case, if you think that this suggests logic is arbitrary, it seems it's not so much the practice of mathematical logic today (no contradiction there), but the philosophical practices of some mathematicians and some philosophers. However, by the same disagreement criterion, we could say the practice of, say, physics today suggest that physics is arbitrary; the practice of ethical philosophy or ethical debates today suggests ethics is arbitrary, and so on. I think it's an extremely weak case.

To that, you give no reply at all. Your previous answer was already factored in and quoted.

 

[Angra Mainyu]

I think there's a language barrier here. I just don't understand what you say and you clearly don't understand much of what I say. I thought my English was good, though.

That is not the problem. You grossly and repeatedly misrepresent what I say, you refuse to acknowledge when your claims are proven false, you insist on argument-free claims, etc. Those are some of the problems. If you just stopped it and discussed in a civil manner, there would be no problem.

So, I can only repeat below the one answer I already provided, which is a publicly available SEP web-page. It's not me making up stuff... This example shows that "certain classical statements are presently unacceptable from an intuitionistic point of view". Me I read that as saying there's a contradiction, literal contradiction, between what "classical" mathematical logic says and what intuitionist logic says.

You are making stuff up when you attribute to me a claim or implication that everything that follows classically follows also intuitionistically. Obviously, I never suggested - let alone said - so. What I did was to ask for an example of a contradiction, and when you provided an example of an alleged contradiction, I showed that if that is what you mean, then your claim

This view isn't a foregone conclusion. The practice of Mathematical logic today suggests on the contrary that logic is arbitrary. Mathematical logic itself is a branch of mathematics, not a method or a theory of logic. As a branch of mathematics, it brings together a very large number of theories and methods (calculus) which are all different from each other and in effect mutually contradictory.

was extremely weak.


Now, of course "certain classical statements are presently unacceptable" from an intuitionistic point of view. I actually used that as an example repeatedly. Obviously. Obviously. Obviously!!!!, I'm not suggesting otherwise. But if that is for you good evidence that " practice of Mathematical logic today suggests on the contrary that logic is arbitrary", well you are clearly mistaken. With the same argument, there are different theories about the correct metaethics, and some things are true according to some theories and false in others. Is that good evidence that the practice of philosophy today suggests metaethics is arbitrary? But forget metaethics, and let's go to first order ethics. Again, there are different theories. Thomism (in each of its variants) says one thing. Consequentialism (in each of its variants) another, and so on. Obviously, some moral statements follow from some of the theories, but not from others - in fact, their negation follows. Is that good evidence that the practice of philosophy today suggests morality is arbitrary?
Obviously, we can go to metaphysics, or epistemology, or pretty much anything in philosophy, and you will find different theories that often contradict each other. It's all over the place. Does then the practice of philosophy suggest that all of those things are arbitrary?

But forget philosophy. Let's go to science. Of course, different theories in physics are mutually incompatible. Does the practice of physics suggest physics is arbitrary?

Both claim to be mathematical logic methods of logic. Both say they are correct. One must be wrong, surely.

I already addressed that. When it comes to the philosophical claim, certainly. But it does not follow that either intuitionistic theorems are invalid, or classical theorems are invalid. In fact, I hold that all of them are valid. Why? Because I reckon classical logic is correct. As I already explained, every deduction that is valid with respect to a truth-preserving system, is also classically valid. In particular, every deduction that is intuitionistically valid, is also classically valid. The converse is false. But that is not the point.

And correct relative to what?! Surely, if it's correct at all, it has to be correct in representing the way the human brain does logic.

Actually, the claim seems to be about when it is correct to derive statements in mathematical contexts. The reasons given (if you read the debates) are not always (or usually) applicable to other contexts, such as the world around us. For example, many intuitionists would not say that A v ¬A does not hold in the world around us. They see the difference as one of construction (of mathematical realms) (plus, in a limited manner, discovery) vs. only discovery of concrete things (the world around us).

So, it would be a matter of what the proper method for deduction in the context of mathematics is ("proper" here is not defined, but used intuitively). It is silent about whether that corresponds to what the human brain (or the untrained human brain) does (though, of course, human mathematicians are doing the deductions, so surely the human brain can deduce by more than one method; it's not what the argument is about, though).

But this is an empirical and therefore scientific question, not a mathematical one. Mathematicians are not scientists, according to me and most people anywhere and on this forum (I did a poll a while ago on that). So, why should anyone accepts that any of these dudes is correct as they claim? Where's the empirical evidence? All we have are mathematical theories and zero evidence.

That is a really bad argument. It is like saying:

Different ethical theories claim to be methods of finding moral truth. But surely, at most one is correct, given that they are pairwise mutually incompatible. And correct relative to what?! Surely, if it's correct at all, it has to be correct in representing the way the human brain does ethics. But this is an empirical and therefore scientific question, not an ethical one. Ethicists and people who debate ethics in general are not scientists. So, why should anyone accepts that any of these dudes is correct as they claim? Where's the empirical evidence? All we have are ethical theories and zero evidence.​

Can you see why that would be a bad argument?
First, different people disagree about whether moral truth is about what the human brain does. Now, they might be mistaken. But the point is that methods about how to find moral truth are not claims about the human brain (in most cases). They are silent (in most cases) about whether morality is what the human brain does. And methods of mathematical logic are also generally silent.

Second, there is a way of testing ethical theories (and that goes for both first-order ethics and metaethics, to the extent they make predictions were we can use our sense of right and wrong): we can use our own faculties, including our ability to reason and our own sense of right and wrong to test them. And we can do the same in mathematics. Now, doing that will not eliminate all disagreement. There will be plenty left, more in ethics than in math. But so what?

There are more problems with your claim, though that should suffice.

Now you asked:

So, why should anyone accepts that any of these dudes is correct as they claim? Where's the empirical evidence? All we have are mathematical theories and zero evidence.

Well, that's not the matter of the thread. What I did show is that under the assumption that you accept that every statement is true or false, and the further assumption that classical mathematical logic is the wrong logic in the sense it fails to match human logic, then it's a superior method for finding mathematical truth. The fact that you fail to realize that my arguments show that is a problem with you, not with me.

In fact, we have evidence to the contrary. For example the principle of explosion and the paradoxes of material implication, whose paradoxical nature shows material implication doesn't match the kind of implication the human brain does. And now what the quote below says, that these people are contradicting each other.

Actually, clearly the human brain does that. Mathematicians are humans. And most mathematicians find CML very intuitive, even when they first take a course in it. But regardless, that was not the matter I was discussing in the thread. My point was that granting for the sake of the argument that CML is wrong in the sense that it fails to match human logic, it is superior as a method of finding mathematical truth. And I showed why this is so. You only denied it, but have no counterargument - just counterclaims, whereas I explained my arguments.

These people don't even agree on what is logic and they don't even say what is logic. Read any textbook on logic and quote me where it explains what is logic. Not one ever does that nowadays.

You could say the same about science, or morality, or metaphysics, or whatever. But you're just going on a tangent. I don't have time to address everything you claim, and on top of that defend myself against your misrepresentations (note: self -defense will always take priority over any discussion about logic, math, or whatever; consider that before attributing claims to me that I have not made or even suggested).

 

[fromderinside]

Eric? Eric Hofer? that you?

 Validity

One probably should conclude putting validity into context vis a vis empirical and non empirical is only the beginning since validity really only means 'does the thing described as valid do what it is intended to do in context of what the thing happens to be'.

So all this flamework over the validity logical argument being similar to that of validity of mathematical statement comes down to how logic and mathematics are used. Those can only be determined in the context of what they were intended to do or be.

Most think they are different since one arose and is mostly determined from rational argument while the other mostly determined from empirical argument. What rationalism and empiricism are intended to do is therefore distinct and clear depending on how it is used.

Clearly, a scientific hypothesis is no different than is a rational argument until one assigns valuation to it. Yet, when evaluated entirely different Criteria are applied.

To view them differently than thus is clearly irrational and not empirical.

 

[Speakpigeon]

So, it would be a matter of what the proper method for deduction in the context of mathematics is ("proper" here is not defined, but used intuitively). It is silent about whether that corresponds to what the human brain (or the untrained human brain) does (though, of course, human mathematicians are doing the deductions, so surely the human brain can deduce by more than one method; it's not what the argument is about, though).

These people don't even agree on what is logic and they don't even say what is logic. Read any textbook on logic and quote me where it explains what is logic. Not one ever does that nowadays.

You could say the same about science

No. We know what is logic and we know what is science.

Logic 1. The study of principles of reasoning, especially of the structure of propositions as distinguished from their content, and of method and validity in deductive reasoning.

Science a. The observation, identification, description, experimental investigation, and theoretical explanation of phenomena.

OK?

What we won't know, even by reading logic textbooks, is what is mathematical logic. And yet, everybody behaves as if mathematical logic was something like the study of human reasoning. You yourself is so confused in your explanations I still don't know what your view is. Mine can be expressed in one sentence: To produce a proper formal model of logic as a performance of human beings and a capacity of the human brain, there should be a proper empirical science of it and there is none at the moment.

We're going nowhere, anyway.

You reply but you never answer my questions. Not about the justification as correct of the definition of validity used in mathematical logic, not about what important theorems can be regarded as specifically derived from mathematical logic, and your answer to this thread as to the possible consequences of a using a wrong logic in mathematics is plain nonsense. To everybody else, to use an incorrect logic by definition will lead to invalid conclusions. Not to you apparently. You argued that these invalid conclusions will nonetheless be valid according to the notion of validity used in this incorrect logic. Alright. I've enough. Please ignore me from now on.
EB

 

[A Toy Windmill]

After following these threads, I understand the main complaint concerns "the principle of explosion" or "ex falso quod libet". This is the rule that lets you derive anything from a contradiction.

The rule is not a violation of Aristotle's logic, because Aristotle's logic doesn't say anything about such inferences. It only covers a simple argument structure called a syllogism. These are arguments made of three sentences, two premises and a conclusion, sharing three term variables, and such that no two sentences have all the same variables. Each sentence has one of the forms:

1) All A are B
2) No A are B
3) Some A are B
4) Some A are not B

Since the order of the premises doesn't matter, that gives you 256 possible syllogisms. With the obvious interpretation, we find that 15 of them are valid and the remaining 241 are invalid.

This classification of syllogisms into valid and invalid doesn't tell us about the principle of explosion. That would require a syllogism whose premises are contradictory, but the rules given above do not admit such things as syllogisms. They don't say they are valid. They don't say they are invalid. They say nothing about them.

This is just a fact of Aristotle's logic, which doesn't cover many argument forms and deals with only a few logical particles. The logic is too simple to have formalized even the the simplest geometry proofs from Aristotle's time, so if you want to know how mathematician's have been reasoning for the last 2000 years, you won't find it in Aristotle. Aristotle only considers a relatively small number of trivial inferences.

Over the centuries, scholastic philosophers sought to advance Aristotle's logic, extending it to account for particulars and things like disjunctions. And consequently, by the middle ages, and possibly as early as the 12th century, we get the first proofs of "the principle of explosion." The first such proof can be condensed down to this:

Premise 1) P
Premise 2) Not-P
3) P or Q (from 1)
4) Q (disjunctive syllogism from 2 and 3)

Thus, any Q follows from P and Not-P.

This proof is not formalizable in Aristotle's original syllogisms, and so was not classified by his logic. But once certain extensions are made, it became possible for some scholastic philosophers to write it and declare it valid.

This is many centuries before mathematical logic.

 

[Speakpigeon]

After following these threads, I understand the main complaint concerns "the principle of explosion" or "ex falso quod libet".

First, there's no "complaint". Just facts.

Second, it's not mainly about EFQ or ECQ. It's about mathematical logic being wrong in the sense that it proves valid implications that human beings see as invalid. It's a fact that there are in effect an infinity of such implications. EFQ and ECQ are wrong in this sense but they are they are the less problematic of them.

The rule is not a violation of Aristotle's logic, because Aristotle's logic doesn't say anything about such inferences.

Aristotle's logic is the same as that of any human being and it is intuitively clear that these implications are invalid. What Aristotle himself said also confirm this.

It only covers a simple argument structure called a syllogism.

No, it doesn't. Aristotle's logic is universal. But I would grant you that most people are so myopic they fail to understand that. Aristotle showed us the Moon and you guys look at the finger.

This classification of syllogisms into valid and invalid doesn't tell us about the principle of explosion. That would require a syllogism whose premises are contradictory, but the rules given above do not admit such things as syllogisms. They don't say they are valid. They don't say they are invalid. They say nothing about them.

Yes, true. I'm talking about Aristotelian logic, not about specific syllogisms.

This is just a fact of Aristotle's logic, which doesn't cover many argument forms and deals with only a few logical particles. The logic is too simple to have formalized even the the simplest geometry proofs from Aristotle's time, so if you want to know how mathematician's have been reasoning for the last 2000 years, you won't find it in Aristotle. Aristotle only considers a relatively small number of trivial inferences.

As does any system of axioms.

The fact that Aristotle didn't do the logic of geometry doesn't validate what other people may have done in this respect. Given what mathematical logic does, I'd be worried there.

I'm trying to identify important theorems derived from mathematical logic rather than mathematicians' natural logical intuitions. If you know of such, I'd be interested.

Over the centuries, scholastic philosophers sought to advance Aristotle's logic, extending it to account for particulars and things like disjunctions. And consequently, by the middle ages, and possibly as early as the 12th century, we get the first proofs of "the principle of explosion." The first such proof can be condensed down to this:

Premise 1) P
Premise 2) Not-P
3) P or Q (from 1)
4) Q (disjunctive syllogism from 2 and 3)

Thus, any Q follows from P and Not-P.

No, it doesn't. In fact, there's just one logician at the time, William of Soissons who proposed this proof. Most Scholastic logicians disagreed.

This proof is not formalizable in Aristotle's original syllogisms

Arguments with contradictory premises, or generally false premises, are easily formalised in Aristotelian logic. Such arguments are not valid. And intuitively, they are not.
EB

 

[steve_bank]

There are publicly available examples in Wikipedia and probably SEP. They're not difficult to find. You're the expert at Wiki here.

And a while ago, I did a thread on one well-known, publicly available example. Short term memory only?



Try a bit of not making stuff up. You have zero idea of what I do.

So far there have been no contradictions in math that surfaced. Arithmetic, algebra, trig, calculus, linear algebra and so on.

I'm not sure what you're talking about here. There are contradictions between the different methods of logic that exist in mathematical logic. See above. As for areas outside mathematical logic, I didn't say there were contradictions. I'm asking how a wrong mathematical logic could affect mathematical theorems. If there was such an impact, you wouldn't notice any contradiction. Even in terms of applied mathematics, it is possible we would just miss an opportunity to understand nature rather than find some contradiction with empirical facts. And our missing would go unnoticed. Or look exactly like Quantum Physics today, with no major theoretical advance for a long while.

You are like someone who never pitched in baseball shouting at a pitcher in a game he has got it all wrong. You are an observer, I have been a lifelong participant and applied mathematical as most engineers are.

You are someone who speaks from his arsehole. You have zero idea what I do now or did all my life. Try a little bit of humility. When you don't know something, don't speak as if you did.

And again you said two different geometries conflict, please be specific as to what that is.

That's common knowledge but you're an ignoramus so you don't know.

Still do not see what you are all jazzed about.

Well, no jazz here. I just asked a question and you haven't a clue what the answer is and yet you keep pretending you can provide meaningful comments. All you're doing here is spurt very cheap philosophy.
EB

It is incumbent on you to provide detailed examples, which you are obviously incapable of doing. Any nath that were contradictory would get a lot of attention. It would be reconciled or the math rejected.

There may be math that has been proposed or never widely accepted that may have problems, but that is how both scince and math advance.

Your problem is that Classical Logic in philosophy and general principles of reasoning are in wide use, and syllogism's are more of a etching tool than a practical tool of reasoning. Logic and reasoning are well understood.

Math and logic are evidenced by what they accomplish.

Again specific examples and the context.