All aboard the Atheist Bus
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@CDarklock said:
In your world, the word "axiom" must have the definition you prefer, because you cannot imagine Godel's theorem applying outside the domain of formal systems.
I'm going to restate this again, in the hope that repetition will lead to understanding: <font size="4">EVEN IF YOU APPLY GODEL'S INCOMPLETENESS THEOREM OUTSIDE THE REALM OF FORMAL SYSTEMS, YOU MUST USE A WORD WHICH TRANSLATES PROPERLY OVER TO DESCRIBE THE SAME CONCEPT. THAT WOULD COULD BE "AXIOM", IN THE SAME SENSE IN WHICH WE USE IT. ANOTHER WORD COULD BE "PRESUPPOSITION". USING "AXIOM" TO MEAN "ANYTHING WHICH IS TRUE BUT NOT PROVEN" IS COMPLETELY ERRONEOUS IN THIS CONTEXT. ANY RESULTS YOU GET FROM SUCH USE ARE GARBAGE.</font>
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@CDarklock said:
@morbiuswilters said:
Goedel's incompleteness thereom only applies to formal systems
I do not believe this is true. I believe Godel's theorem is a limited statement of a fundamental truth of the universe: that within any empirically observable problem domain, a complete definition of the problem space cannot be made without reference to entities outside that domain.
It's about logic and formal systems which just happen to be the model that humans use to describe the scientific world. Citing an obscure theorem of formal logic and using it to justify your philosophical beliefs is unreasonable.
@CDarklock said:
@morbiuswilters said:
And within formal systems, words like "axiom" have specific meanings
And that's the failure. In your world, the word "axiom" must have the definition you prefer, because you cannot imagine Godel's theorem applying outside the domain of formal systems.
It applies only to formal systems. Live with it.
@CDarklock said:
@morbiuswilters said:
You cannot use a fairly abstract and theoretical branch of formal logic and arithmetic as a metaphor
I've actually been speaking allegorically. Perhaps you're not qualified to discuss discussion.
Wow, you have no idea what that word means, either. If you were speaking allegorically you would be using a fictional narrative to describe an abstract concept. However, you instead drew parallels between an abstract concept you clearly do not grasp and the fictional narrative of your religion. You were using it as a metaphor and you failed horribly. Please just give this up before you make yourself look more ignorant.
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@morbiuswilters said:
If you were speaking allegorically you would be using a fictional narrative to describe an abstract concept.
I was! I used the fictional idea that someone might deny that 1 is 1 to describe the basic concept of denial being irrelevant to actual truth. It's just that your unreasonable dedication to keeping Godel's theorem in your own little box is in the way, and productive discussion has become impossible.
So I'm just having fun with it. If you'd like to go on frothing at the mouth and demanding that the entire world subscribe to your tiny little limited definitions of words and concepts, feel free. I find it amusing. I'll keep quietly reiterating the facts, while you and morbius can keep calling me names and using big fonts and insisting that your interpretation is the ONLY correct interpretation.
There are other people that do that, too. We call them "fundamentalists". And the irony is over there, in a box. We just can't tell whether it's alive or not.
Or, for that matter, whether it's just having afternoon tea with Godel's theorem.
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@CDarklock said:
I'll keep quietly reiterating the facts, while you and morbius can keep calling me names and using big fonts and insisting that your interpretation is the ONLY correct interpretation.
@CDarklock said:@morbiuswilters said:
If you were speaking allegorically you would be using a fictional narrative to describe an abstract concept.
Do you even know who you quoted? Is this all running together for you because you don't bother to read?
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@CDarklock said:
@morbiuswilters said:
If you were speaking allegorically you would be using a fictional narrative to describe an abstract concept.
I was! I used the fictional idea that someone might deny that 1 is 1 to describe the basic concept of denial being irrelevant to actual truth. It's just that your unreasonable dedication to keeping Godel's theorem in your own little box is in the way, and productive discussion has become impossible.
So I'm just having fun with it. If you'd like to go on frothing at the mouth and demanding that the entire world subscribe to your tiny little limited definitions of words and concepts, feel free. I find it amusing. I'll keep quietly reiterating the facts, while you and morbius can keep calling me names and using big fonts and insisting that your interpretation is the ONLY correct interpretation.
There are other people that do that, too. We call them "fundamentalists". And the irony is over there, in a box. We just can't tell whether it's alive or not.
Or, for that matter, whether it's just having afternoon tea with Godel's theorem.
Alright, you are a serious crackhead. You are not reiterating facts, you are completely misusing a part of formal logic to support your wacky conclusions. Obviously denial has no impact on truth because denial is simply an action taken by a person. However, none of this has a goddamn thing to do with Goedel. You are not some kind of clever person thinking outside of the box and defying stagnant convention; you are just spouting nonsense and hoping you can tie your own biased religious beliefs to a successful therorem from logic and that most people will be ignorant or confused enough to go along with it. You were called out for your flawed understanding of Goedel's theorem as well as for the inappropriateness of trying to use Goedel's theory as an analogy to support your philoso-babble.
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@bstorer said:
Is this all running together for you because you don't bother to read?
Yes. Why would I waste my time reading the same thing over and over?
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@CDarklock said:
Yes. Why would I waste my time reading the same thing over and over?
It might be a good idea to read it at least once, so you understand why your mention of Gödel's theorem is utterly nonsensical here.
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@CDarklock said:
Because you didn't manage to understand it the first dozen times. Since this topic is clearly beyond you, you should just opt for rote learning and call it a day.@bstorer said:
Is this all running together for you because you don't bother to read?
Yes. Why would I waste my time reading the same thing over and over?
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@morbiuswilters said:
you are completely misusing a part of formal logic to support your wacky conclusions.
What conclusions would those be?
@morbiuswilters said:
you are just spouting nonsense and hoping you can tie your own biased religious beliefs to a successful therorem from logic
What religious beliefs are those? How are they biased?
I haven't drawn any conclusions or stated any beliefs. You criticise them without knowing them. You have simply defined any conclusion I draw as "wacky", and any beliefs I hold as "biased". This is entirely your prerogative, but it is certainly not rational. Refusal to examine another's beliefs is, again, simple fundamentalism.
@morbiuswilters said:
You were called out for your flawed understanding of Goedel's theorem
...for sufficiently limited definitions of "understanding".
I understand the theorem to be applicable outside formal logic, and you insist that it is not. Clearly at least one of us has a flawed understanding, but how can we tell which - if either - is accurate?
It's certainly not going to be accomplished by repeating the same thing over and over again. The issue here isn't that I have a flawed understanding of the theorem, it's that your understanding of that theorem defines it as being relevant only in a specific domain, and in that domain a particular symbol - the word "axiom" - has a very specific meaning.
But I don't care about that domain. I'm over here, discussing the notion that things are true whether people believe they're true or not, which plays directly into the question of proof - because proof is a prerequisite of belief for some people. Godel's theorem is relevant to that, even though it's not in the original intended problem domain, because the principle remains true... whether you believe it or not.
In other words, your belief that Godel's theorem does not work where I am applying it has no bearing on whether it actually works there. Insisting that you have defined that theorem as being meaningless outside formal systems is rather like defining a ball as unaffected by gravity. You can define that all you want, but it doesn't change reality.
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@bstorer said:
Because you didn't manage to understand it the first dozen times.
And yet, you keep saying it. What's it called when you expect to get different results from doing the same thing?
@bstorer said:
Since this topic is clearly beyond you, you should just opt for rote learning and call it a day.
No, you can do that yourself. You read in a book somewhere that Godel's theorem only applies to formal systems, and now you refuse to entertain any other proposal. This means if I'm onto something real and valuable, you'll never know. You'll just point to your book and say "it says Godel's theorem doesn't apply" over and over again, like a mantra, and hope it all goes away.
Hey, maybe it will. Or, at the very least, maybe you won't have to look at it because you're staring at some old book that says something you've already read. Who needs to learn anything new?
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@CDarklock said:
Hope. They say it springs eternal.@CDarklock said:@bstorer said:
Because you didn't manage to understand it the first dozen times.
And yet, you keep saying it. What's it called when you expect to get different results from doing the same thing?
Which is proof that you don't read my replies. As I've said, you can try to translate Godel's theorem to other domains, but you have to translate the terms over correctly. You aren't doing that. Yes, you use the word axiom, and so do we, but your definition is too broad. Try restating your opinion while replacing the word "axiom" with "presupposition." You'll find that you suddenly make no fucking sense, which the rest of us have known for a while now.@bstorer said:
Since this topic is clearly beyond you, you should just opt for rote learning and call it a day.
No, you can do that yourself. You read in a book somewhere that Godel's theorem only applies to formal systems, and now you refuse to entertain any other proposal. This means if I'm onto something real and valuable, you'll never know. You'll just point to your book and say "it says Godel's theorem doesn't apply" over and over again, like a mantra, and hope it all goes away.
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@CDarklock said:
The issue here isn't that I have a flawed understanding of the theorem, it's that your understanding of that theorem defines it as being relevant only in a specific domain, and in that domain a particular symbol - the word "axiom" - has a very specific meaning.
I agree with this (that we can use the theorem as an analogy in other areas), and said so above. However, you're still wrong in the way you've used axiom in relation to this concept, and tying it to Godel's theorem. And your arguments (which, I'll grant, may reflect poor communication skills on your part, rather than poor thinking) have shown that you have a poor understanding of what the theorem is really about. Here is my explanation as to why the threorem can be interesting in other domains, such as the real world:But I don't care about that domain. I'm over here, discussing the notion that things are true whether people believe they're true or not, which plays directly into the question of proof - because proof is a prerequisite of belief for some people. Godel's theorem is relevant to that, even though it's not in the original intended problem domain, because the principle remains true... whether you believe it or not.
Most domains do not lend themselves to formal proofs. Mathematics and formal logic, on the other hand, are all about formal proofs. If we can show that even in mathematics, it's not only possible, but the natural order of things to have unprovable truths, then it probably follows that in more 'sloppy' domains, such as the physical world, there are truths that cannot be proved, at least by science as we know it. For such unprovable (whether truly unprovable, or just not feasibly provable by available means) issues, we can only rely on faith, which may be influenced by our knowledge or perceptions, of course, but ultimately are not proof.
What I find interesting is that many (most? all?) religious leaders explicitly acknowledge this with respect to God. That's more than some militant atheists seem to be able to do. Probably including those who have donated money to the bus.
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@bstorer said:
As I've said, you can try to translate Godel's theorem to other domains, but you have to translate the terms over correctly
I didn't translate any terms.
OUTSIDE of formal systems, the word "axiom" is a valid symbol representing something true that cannot be proven. It does not need to be translated. It is already here, and it already means what I am using it to mean.
What confuses you is that there is an entirely coincidental case of the same term existing in the problem domain of formal systems, where it has a different definition. There is nothing wrong with that definition. There is nothing wrong with the same word having two different definitions.
When you are in that problem domain, you are completely correct in that the word "axiom" must hold that definition; it is well-defined as a fundamental element of that problem domain.
But when you are in another problem domain, the word "axiom" is not required to hold the same definition. It would need to hold the same definition if I imported it from that problem domain, but I didn't. I am using it independently of its presence in other problem domains.
@bstorer said:
Try restating your opinion while replacing the word "axiom" with "presupposition."
But "presupposition" is not a synonym of "axiom" in the sense I am using it. It sounds nonsensical because it is. You are replacing symbols that are not equivalent.
Let me try to express this in a way you can understand.
x + 8 = 15; solve for x. The answer is clearly 7.
x + y = 12; solve for y. The correct answer is "12 - x", because you cannot unambiguously solve for y.
But if you were missing the point, you might assume these were related questions, and claim that y is 5. When I point out that the answer is "12 - x", your response would quite likely be "but x is 7, and 12 - 7 is 5".
This is an immensely difficult concept to explain when someone honestly does not understand it. I am of the opinion that failure to understand this is not a factor of intellect; most people fail to understand it. It stems from a failure to establish proper generative mental models, a process which tends to elude almost everyone.
And most notably, the generative model here is improper because there is no generative model. The existence of x in both equations is coincidental. They are not a series; they are not part of the same system. I did not use the word "axiom" because I employed Godel's theorem, or vice-versa. They are independent events. They have no direct correlation to one another. You keep insisting that because Godel's theorem has now been used, the definition of "axiom" must now be the definition from the problem domain of Godel's theorem, but the existence of the term "axiom" in both problem domains is purely coincidental. It was not intended, but neither was it avoidable.
I believe you are capable of understanding this, under the presupposition that you aspire to greater understanding. If you don't, well, I'll just play games with you and giggle about it. It's too much work to explain difficult concepts to people if they simply refuse to listen.
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@CDarklock said:
I didn't translate any terms.
Exactly!@CDarklock said:
OUTSIDE of formal systems, the word "axiom" is a valid symbol representing something true that cannot be proven. It does not need to be translated. It is already here, and it already means what I am using it to mean.
And if you'd left it at that, without dragging dead Austrians into the mix, this thread would have died long ago. But you said something obviously wrong:@CDarklock said:
My point is that you can't prove the axiom no matter who chooses it. This is the very centre of Godel's incompleteness theorem: there exist things you can't prove. They still exist. The fact that they exist is still a fact. The inability to prove it - ironically enough - does not prove anything.
You misstated the theorem, and now say that you didn't. Welcome to the Swamp Shack.@CDarklock said:
But "presupposition" is not a synonym of "axiom" in the sense I am using it. It sounds nonsensical because it is. You are replacing symbols that are not equivalent.
Yes, and the point of everyone who has responded to your posts is that the sense in which you are using 'axiom' is wrong. Just admit that you kept arguing because you didn't want to admit that you were wrong, and let's move on. C'mon, we've all done it. Instead, you just keep digging yourself deeper and deeper.@CDarklock said:
It's too much work to explain difficult concepts to people if they simply refuse to listen.
And you've done an excellent job of proving it in this thread.
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You can indeed use the theorem as an [i]analogy[/i], but you should be aware that any analogy between a mathematical theorem and the physical world is tenuous at best because the realm of mathematics is autonomous and non-physical (or you choose to understand mathematics so that it describes the physical world, but then you don't have absolute certainty in mathematics anymore and they become an ordinary science, both views are valid). @boomzilla said:
If we can show that even in mathematics, it's not only possible, but the natural order of things to have unprovable truths, then it probably follows
change the last phrase to "it isn't surprising" or something else which doesn't suggest a logical implication and I'm with you, although instead of "we can only rely on faith" I would say that we can only reach a provisional judgement based on our best knowledge (which itself may have to be adapted in the light of new observations). @boomzilla said:What I find interesting is that many (most? all?) religious leaders explicitly acknowledge this with respect to God.
Definitely not all, and this acknowledgement has only spread (as far as christianity is concerned) after the enlightenment. @boomzilla said:That's more than some militant atheists seem to be able to do.
Militant being the important word here. Religious nutters, be they theistic or atheistic, tend to consider themselves in the sole possession of truth, they are dangerous and have to be watched.
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@Ilya Ehrenburg said:
You can indeed use the theorem as an analogy, but you should be aware that any analogy between a mathematical theorem and the physical world is tenuous at best...change the last phrase to "it isn't surprising" or something else which doesn't suggest a logical implication and I'm with you, although instead of "we can only rely on faith" I would say that we can only reach a provisional judgement based on our best knowledge (which itself may have to be adapted in the light of new observations).
Yes, I agree with that. If I'd taken more time to write it, it probably would have been phrased closer to your version. I think the reason I used "it probably follows" is that if the real world (or whatever) could ever be expressed as a formal system (theory of everything, etc), then it's trivial. And if it's not a formal system, then you start asking things like "what is a proof?" And if we can't even agree on what it means to prove something, we probably can't turn around and say that nevertheless, we should be able to "prove" all things that are true.@Ilya Ehrenburg said:
@boomzilla said:
Yeah, I can only really go by my experience--raised Catholic, and since lapsed (i.e., I now consider myself to be an atheist). And I recall lots of talk (from my CCD days, especially)about the "mystery of faith." It was quite clear to me that you were simply supposed to accept all of this without proof. I'm not familiar with any prominent religions that argue that they have some sort of proof, rather than faith. And, frankly, I'm not terribly interested in what religious types did hundreds of years ago. Hitler probably loved his dog, too.What I find interesting is that many (most? all?) religious leaders explicitly acknowledge this with respect to God.
Definitely not all, and this acknowledgement has only spread (as far as christianity is concerned) after the enlightenment.@Ilya Ehrenburg said:
@boomzilla said:
Sure, but that wasn't what I was referring to. While the Pope believes that the Catholic Church is the one true church, etc, etc, I'm not aware of them claiming to have definitive proof, as opposed to, for example, Richard Dawkins. I see science and religion, at least at the level of, "Does God exist," to be orthogonal. Certainly, there have been religions that incorporate scientifically falsifiable concepts, but that's not the same thing. For that matter, science has accepted as true things that were eventually falsified.That's more than some militant atheists seem to be able to do.
Militant being the important word here. Religious nutters, be they theistic or atheistic, tend to consider themselves in the sole possession of truth, they are dangerous and have to be watched.
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@boomzilla said:
And your arguments (which, I'll grant, may reflect poor communication skills on your part, rather than poor thinking) have shown that you have a poor understanding of what the theorem is really about.
Poor communication is never the exclusive fault of a single person. Everyone participating in the discussion shares a part in it; no one is blameless. And this is the internet; when you don't understand someone, you can generally just assume they're stupid, and chances are you're right. That can make life difficult for those of us who aren't stupid, but are misunderstood all the same. ;)
The key word in your statement here is "really". Who is to say what the theorem is "really" about? You have the way it was intended, the way it is used, and the way it is proposed to be applicable. Which is "really" the meaning of that theorem? When you nail down a theorem and insist that this is what it means, and all it means, and all it ever can mean - aren't you implicitly closing the door on progress?
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@CDarklock said:
Who is to say what the theorem is "really" about?
Apparently, everyone here except you.@CDarklock said:
You have the way it was intended, the way it is used, and the way it is proposed to be applicable.
And I suppose your claim is that you are proposing a new applicability. And there have been multiple posts showing that what you have been saying doesn't make any sense with respect to the theorem.@CDarklock said:
When you nail down a theorem and insist that this is what it means, and all it means, and all it ever can mean - aren't you implicitly closing the door on progress?
No, it means that maybe you actually are dealing with a theorem. If you want it to apply to something else, you still have to make sense about it, and even then, it wouldn't be the same theorem. It would be something different.For example, suppose that I proved something about the integers. Then you came along and generalized it to apply to all commutative rings. That would be progress. And it would be two different theorems (though the more specific one isn't nearly as interesting as the more generic, and, in fact wouldn't even be necessary any more).
But you're not even trying to preserve the general spirit of the theorem. You're just making stuff up.
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@boomzilla said:
You misstated the theorem, and now say that you didn't.
I did not misstate the theorem. Here, I'll quote it.
"In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory."
And here's what I said was the "centre" of the theorem, by which I mean the central point of it: "there exist things you can't prove."
From the original quote: "there is an arithmetical statement ... not provable".
This is a contextually accurate substatement of the theorem.
Does "is" mean it exists? Yes.
Is an arithmetical statement a thing? Yes.
Does "not provable" mean you can't prove it? Yes.
I do not see how I have misstated anything.
@boomzilla said:
the sense in which you are using 'axiom' is wrong.
I do not believe you can call any accepted definition of a word "wrong". The definition "a principle accepted as true without proof" is an accepted definition of the word "axiom". Your refusal to accept it is not my problem.
@boomzilla said:
Just admit that you kept arguing because you didn't want to admit that you were wrong
I am perfectly willing to admit I am wrong, provided this can be demonstrated without resorting to fallacy.
That is the very core of scientific reasoning. Once you are proven wrong, you must either admit it or sacrifice your integrity.
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@boomzilla said:
You're just making stuff up.
That's called "innovation". Sometimes it works.
"...the innovator has for enemies all those who have done well under the old conditions, and lukewarm defenders in those who may do well under the new." - Niccolo Machiavelli
And I'm okay with that.
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@CDarklock said:
I would argue that attempting to use the non-mathematical definition of "axiom" while simultaneously talking about Gödel's theorems is a misstatement.I do not see how I have misstated anything.
"My dog needs to go outside at least once a day to do her 'business'" might reasonably be considered an axiom in the non-mathematical sense, and few would complain strenuously.I do complain strenuously when you say that "all non-provable true statements are axioms" in the midst of talking about Gödel's theorems.
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@AssimilatedByBorg said:
I would argue that attempting to use the non-mathematical definition of "axiom" while simultaneously talking about Gödel's theorems is a misstatement.
I do appreciate that argument, but it can't be helped. There is no suitable synonym for the definition of "axiom" I intended. I looked, once we started going back and forth over the "right" definition... but there simply isn't one. All the alternatives are worse.
I thought I mentioned that earlier, but I may not have.
@AssimilatedByBorg said:
I do complain strenuously when you say that "all non-provable true statements are axioms" in the midst of talking about Gödel's theorems.
Again, can't be helped. In order to demonstrate that Godel's theorem applies to axioms where axiom is defined as I intended, juxtaposition was unavoidable.
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@CDarklock said:
And here's what I said was the "centre" of the theorem, by which I mean the central point of it: "there exist things you can't prove."
From the original quote: "there is an arithmetical statement ... not provable".
This is a contextually accurate substatement of the theorem.
Does "is" mean it exists? Yes.
Is an arithmetical statement a thing? Yes.
Does "not provable" mean you can't prove it? Yes.
I do not see how I have misstated anything.
It lies mostly in the word "things". Yes, within formal systems--which are highly-artificial and idealized isometric analogs to the physical realm--there exist "things" that are true yet unprovable. Does that make this rule apply to the physical realm or to the existence of a God (theoretically the creator of this physical realm and of formal logic)? No. Might it? Sure, but you're making a big stretch there. The real problem is that you rely on ambiguity and deceit to make your point. You try to use Goedel as an argument but it simply has not been shown to have anything to do with what you are talking about. You are generalizing something that has no known parallel outside of formal systems. You then rely on ambiguities with the term "axiom" to falsely act as if all the other ("domain-specific" in your terms) terminology and supporting evidence relied on by Goedel's theory somehow apply to your "domain".
After all of this, you boil Goedel's very ("domain-") specific theorem down to "there exist 'things' you cannot prove" which is disingenuous at best. I don't know if you are being deceitful or if you are just so hippy-dippy that you believe there is some creamy center of truth in the middle of all that boring math that you can just pluck out and carry back to use as a garnish for your philosophy. And to answer another question: I don't know what philosophy you are pushing and it doesn't matter. You can be Christian, Islamic or atheist, but if you try to find refuge from the storm for you and your pet philosophy in the well-constructed castle of mathematics I will call bullshit on it.
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@CDarklock said:
When you nail down a theorem and insist that this is what it means, and all it means, and all it ever can mean - aren't you implicitly closing the door on progress?
I am not opposed to "thinking outside the box", a concept that must remain vague for it to work its magic. Quirky interpretations can open up doors where once we thought our current systems to be enough.
HOWEVER,
If you reject rigid meaning (which you appear to be doing), you cannot form stable models. Without such models, you cannot set up an experiment or derive any utility form the outcome of one. If any word or theorem can mean anything, you are closing the door on progress much more effectively, and we become Eloi.
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@morbiuswilters said:
The real problem is that you rely on ambiguity and deceit to make your point.
I have used one and only one definition of the word "axiom", which is an accepted definition; this is not ambiguous.
I have demonstrated that Godel's theorem does, in fact, contain language roughly synonymous with my statement; this is not deceitful.
My position has never changed. If we define an "axiom" as a "principle accepted without truth", as I have always defined it in this discussion, any such principle is an axiom - because the definition is not further qualified. No more specific requirement is necessary.
Godel's theorem outright states that there are true principles which cannot be proven. Logically, those principles - being true - must be accepted. Since they cannot be proven, we must accept them without proof. Being principles accepted without proof, they are axioms by the definition I have previously stated. Whether they are axioms by some other definition is irrelevant.
This is all true. It's not the least bit ambiguous. It's not the least bit deceitful. Read every post I've made in this thread; I do not contradict or alter any of the above.
The problem with correct principles and valid logic is that you have to accept the truth of the conclusion whether you like it or not. Godel's theorem does, in fact, say that there exist things you can't prove - principles which I can quite correctly call "axioms" in the English language. You have exerted a tremendous amount of effort trying to falsify something which is logically consistent and correct.
If you could not tell it was consistent and correct, I suggest that perhaps you are not so great at logic.
@morbiuswilters said:
you are just so hippy-dippy
Prejudicial language and ad hominem argument. I expect better from you. If you can't deliver, I claim my right to point at you and laugh.
@morbiuswilters said:
if you try to find refuge from the storm
I invite the storm, because my position is strong and will withstand any onslaught. Remember Jet Li atop the ziggurat at the end of "The One"? That's me. Can't touch this.
Let's look at this from another direction. Pretend Godel never wrote a theorem.
I say there exist things which are true, but that you cannot prove.
Is that an accurate statement? Forget Godel. Forget the word "axiom". Go directly to the substance of the argument.
Can you demonstrate that all things can be proven? That there does not exist even one thing which cannot be proven?
We both know you can't. All the argument about the word "axiom" and the domain of formal systems and whether Godel's theorem means what I said? Pointless. You have wasted all that time and energy on something which cannot possibly have a productive effect. Even if I were to accept your demand that I don't get to this conclusion with mathematics, I can get to it in myriad other ways, and it doesn't change.
But you knew that all along, didn't you? You weren't making a substantive argument in the first place. You just want me to give up. You think if you use big enough words to write long enough posts, I'll run away.
Not happening. See, I think this is fun. I think you're funny. Because I am right, and I have always been right, and the very principles you espouse will eventually force you to recant. The more you bitch and moan, the funnier that ultimate end becomes. The only way you can "win" this argument is to walk away.
But I don't think you can. I see you complaining about ambiguity and deceit and fallacy, all of which you employ in your own arguments, but which cannot be found in mine. And then I look back and see you proposing that I just don't want to admit I'm wrong.
I believe that's your own motivation. I believe you know you're wrong and are just trying like hell to complicate the issue so you won't have to admit it.
Which is simply hilarious.
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@CDarklock said:
Godel's theorem outright states that there are true principles which cannot be proven. Logically, those principles - being true - must be accepted. Since they cannot be proven, we must accept them without proof.
This is where you fail. There is no reason why we have to accept these things. They are true whether or not we accept them as true or false or are undecided. Even if we take your loose version of an axiom, there is still the requirement that we accept the axiom as true.Godels proof does not require anyone to accept the truth of a statement. There would still exist things that we could not prove, but were true. And no one has accepted them, so even if we accept your definition of axiom, you're still wrong about Godel's theorem. So while your general argument, independent of Godel's theorem may be genious or even genius, possibly brillant, it's totally wrong to lump it in with Godel's theorem, for all of the reasons that we've all written.
I'll repeat the key reason why you're wrong, just in case you missed it again: Godels proof does not require anyone to accept the truth of a statement.
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@CDarklock said:
I have demonstrated that Godel's theorem does, in fact, contain language roughly synonymous with my statement;
Homonymous, not synonymous.
@CDarklock said:I have used one and only one definition of the word "axiom", which is an accepted definition; this is not ambiguous.
... If we define an "axiom" as a "principle accepted without truth[recte: proof]" ... Godel's theorem outright states that there are true principles which cannot be proven.
The typo is irrelevant. But to my knowledge and according to my dictionary, the definition of "axiom" as "principle accepted without proof" is only accepted if principle is used in the sense "principle n. a source, root, origin: that which is fundamental: essential nature: a theoretical basis[1]" while the unprovable true statements of Gödel's theorem can only be called principles if that word is used in the much weaker sense "a component"[ibid] or similar.
@CDarklock said:Logically, those principles - being true - must be accepted. Since they cannot be proven, we must accept them without proof.
For the sake of argument accepting your use of the word "principle" here: No, they [b]cannot[/b] be accepted. The simple reason is that they cannot be identified. We know they exist, but don't know which statements belong to this set (a proof that some statement belongs to the set would constitute a proof of that statement).
@CDarklock said:Being principles accepted without proof, they are axioms by the definition I have previously stated.
Which I believe is not an accepted definition of "axiom", but that point is irrelevant by the above.
@CDarklock said:Can you demonstrate that all things can be proven? That there does not exist even one thing which cannot be proven?
Huh? What's going on here? It was a triviality already in Euclid's days that any proof must rest on axioms. Are you suggesting that anybody maintained anything remotely like "all things can be proven"?
@CDarklock said:You just want me to give up.
We want you to give up inappropriately to cite Gödel's theorem as relevant to a position to which it is completely irrelevant. The stubbornness with which you cling to it is tragic, because otherwise you seem to be quite lucid.
@CDarklock said:See, I think this is fun. I think you're funny. Because I am right, and I have always been right
Good for you. I have the impression most everybody else considers you kind of funny in this matter. Because you're wrong, and you've always been wrong [b]in this matter[/b].
A theorem solely concerned with formal theories simply doesn't say anything about things not part of a formal theory.
[1] The Wordsworth Concise English Dictionary, 1994
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@boomzilla said:
Godels proof does not require anyone to accept the truth of a statement.
Wow, we're back on topic, sort of. This is why I brought up the theorem in the first place.
I believe it does require you to accept the statement's truth. Look closely:
"In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory."
If you do not accept the statement as true, then you inherently cannot accept the applicability of the theorem. It may apply in the future, when the statement is accepted as true, but until then the theorem simply isn't applicable.
Which leaves you in the sticky position of what happens when the statement is of indeterminate truth. At this point, the applicability of Godel's theorem is also indeterminate; it will apply if and only if the unprovable statement is true, which - because no proof is possible - will be when and only when that statement is accepted as true. For a provable statement, the statement is accepted as true when a proof is generated, but for an unprovable statement it is an ultimately arbitrary decision.
During this indeterminate stage, there are three things that must be done to legitimately advance the system (formal or otherwise) containing the statement.
First, you must actively attempt to discern the truth of these statements.
Second, where a statement is found not to be true, it must be eliminated from the system.
Third, where a statement is still indeterminate, the system must identify that statement's indeterminacy.
These are the failures of religion. Religion traditionally does not do any of them. Instead, it bluntly asserts the truth of things it cannot determine without identifying the uncertainty, denies that it should make any effort to determine that truth... and when science proves things false, it insists that science is wrong, because "we have this book".
I further believe that any field of study which fails to advance the system in this fashion could be productively viewed as a faith-based or religious field, and any individual within any field of study who fails to challenge the questions of that field in this fashion could be productively viewed as following a religion. I believe these actions are the very things that make a scientific field scientific.
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@CDarklock said:
Fail."In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory."
If you do not accept the statement as true, then you inherently cannot accept the applicability of the theorem. It may apply in the future, when the statement is accepted as true, but until then the theorem simply isn't applicable.
The proof shows that there exists at least one true statement that cannot be proved. It does not identify the statement. It does not care about the details of the statement. It merely shows that it must exist. There are many such proofs in mathematics. As with so many proofs, it doesn't rely on any particular statement, number or equation. It simply shows that given a system that obeys the assumptions, somewhere in the possible space of statements that can be made within the system, there must exist a statement that is true, but cannot be proven to be true.
@CDarklock said:
Third, where a statement is still indeterminate, the system must identify that statement's indeterminacy.
Not sure what you're trying to get at here. The only way to identify such a statement is usually the non-existence of a proof either confirming or refuting it. This is all irrelevant to Godel's theorem, however.Here is an example of something that can probably never be proven to be true:
The digits 8008135 never appear in the decimal representation of pi. Of course, they may very well be in there, I don't know. (For the purposes of argument, assume that I picked some digits that do not appear in the known digits of pi.) This statement is either true or false. If it is false, it is, of course, falsifiable, since we only have to calculate the digits until we find them. If it is true, however, I'm not aware of any way to prove this (of course, someone will now reply with some proof about the digits of transcendental numbers, but hey, it's better than any non-bstorer car analogy) since the digits go on forever, and there's always the possibility that it's still out there. No one has to accept this. But if it is true, there is no way to prove it.
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@Ilya Ehrenburg said:
Homonymous, not synonymous
No, synonymous: "Having the same or a similar meaning."
Homonymous: "Having the same name." Not even close.
@Ilya Ehrenburg said:
the definition of "axiom" as "principle accepted without proof" is only accepted if principle is used in the sense "principle n. a source, root, origin: that which is fundamental: essential nature: a theoretical basis[1]"
I'm uncertain that I would say this is the ONLY accepted definition of "principle" that applies. Among other things, the American Heritage Dictionary lists:
"A rule or law concerning the functioning of natural phenomena or mechanical processes."
This is rather closer to my specifically intended meaning, although I think some argument could be advanced that these are also the "essential nature" of the phenomena or processes in question.
@Ilya Ehrenburg said:
while the unprovable true statements of Gödel's theorem can only be called principles if that word is used in the much weaker sense "a component"[ibid] or similar
First, I am unable to find such a definition of "principle" except in chemistry:
"One of the elements composing a chemical compound, especially one that gives some special quality or effect."
I think that's pretty clearly not what I mean, or what Godel means either. Even in the chemical sense, it is still strongly fundamental - without that component, the compund would not be the same - and qualifies under the original definition. The very word "principle" derives from the Latin principium and almost certainly would not lose its implications of criticality.
Second, let us give the benefit of doubt and accept for the sake of argument that such a definition is in fact what Godel intended. Under that condition, I would say that fundamental principles are still a proper subset of component principles. If a statement is made about component principles, that statement necessarily applies to fundamental principles. The converse is not true. If a thing is said of fundamental principles, that thing is not necessarily applicable to component principles.
@Ilya Ehrenburg said:
Are you suggesting that anybody maintained anything remotely like "all things can be proven"?
I am saying that I looked at Godel's theorem and concluded that there exist things we cannot prove.
Complaining that I can't do that with Godel's theorem is rather irrelevant if you aren't going to dispute the conclusion, isn't it?
Why, you would think I was challenging a tenet of faith or something...
@Ilya Ehrenburg said:
We want you to give up inappropriately to cite Gödel's theorem as relevant to a position to which it is completely irrelevant.
Since my position is directly echoed within the theorem, I fail to see how it could be completely irrelevant.
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@boomzilla said:
It does not identify the statement.
But if the statement is not true, it can't possibly be that statement.
Honestly, man, draw a diagram if you have to - but this isn't particularly complicated.
@boomzilla said:
The only way to identify such a statement is usually the non-existence of a proof either confirming or refuting it.
I suggest that one might identify a thing by placing a label on it, so other people know what it is.
Is that the problem? This idea that other people might want to know things without duplicating your work? Is that idea just foreign to you?
@boomzilla said:
This is all irrelevant to Godel's theorem, however.
Well, yes, because I've moved on. It seemed like maybe you were starting to grasp the idea that I'm not talking about Godel's theorem, just mentioning it in passing on the way to something else. You seem to have missed that part.
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@CDarklock said:
Huh?@boomzilla said:
[Godel's theorem] does not identify the statement.
But if the statement is not true, it can't possibly be that statement.
@CDarklock said:
I guess it was the part where you brought up the theorem again, and told us your "interpretation" of the theorem.@boomzilla said:
This is all irrelevant to Godel's theorem, however.
Well, yes, because I've moved on. It seemed like maybe you were starting to grasp the idea that I'm not talking about Godel's theorem, just mentioning it in passing on the way to something else. You seem to have missed that part.
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@boomzilla said:
Huh?
Godel's theorem says that the statement is true.
If you don't accept that, then you're already denying the correctness of the system. You don't need Godel's theorem to show you that the lack of a statement you can't prove in the system makes it necessarily incorrect and/or incomplete. The presence of a statement you do not accept as true already shows that.
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Homonymous as in "having different significations but the same sound" as I think your usage of "axiom" is so different from the mathematical usage that the word synonymous is inappropriate. I grant you, however, that the usage is not so far away that the word homonym as in "sound the same but have nothing to do with each other" is justified, so I retract that.
Regarding the word principle: axioms are what theories are built upon, the APXAI (faint attempt to simulate greek letters in ASCII), definition in my dictionary is
axiom n. a self-evident truth: a universally received principle: a postulate, assumption.
So in the definition of axiom, in my understanding, the word principle can only be used in the given strong senses. After that, I screwed up. Actually, I think that the unprovable true statements whose existence Gödel's theorem states cannot be called principles at all, because principles are first things while the statements in the conclusion of Gödel's theorem are not first things (the first things being exactly the axioms upon which the theory is built). But I tried to find a meaning of the word principle which would justify its usage there and jumped at the only weak one listed in my dictionary. Let us however be clear that these unprovable true statements are not called principles in Gödel's theorem, neither by Gödel himself nor in the translation you quoted.
@CDarklock said:
I am saying that I looked at Godel's theorem and concluded that there exist things we cannot prove.
I am convinced that you knew there exist things we cannot prove before you even heard of Gödel's theorem. However, it is perfectly understandable to be moved by that theorem which asserts that there are things we cannot prove [i]in a domain where it was believed for centuries that all truths were provable[/i].
@CDarklock said:
Complaining that I can't do that with Godel's theorem is rather irrelevant if you aren't going to dispute the conclusion, isn't it?
If you had only used Gödel's theorem to argue that there are things which we cannot prove, I would only have had your misrepresentation of the theorem to criticise, because Gödel's theorem states indeed that there are things [i]of a specific nature[/i] which we cannot prove. But somewhere in the discussion, you shifted the theorem outside its domain of validity (and relevance):
@CDarklock said:
This is because I am not talking about formal mathematics. I am talking about a very large and comprehensive subject of which formal mathematics is a proper (and miniscule) subset.
@bstorer said:
we're discussing formal systemsNo, we're not. What we are discussing is whether a FACT requires a PROOF.
I've used Godel's theorem to illustrate an overall principle. That principle is that lack of proof does not alter the truth of a statement, and such a lack must be accepted in some cases. This is a valid application of the theorem, albeit outside the original problem domain Godel's theorem covered.
and I strongly object to that. Well, it is legitimate and has been fruitful to move a result outside its domain of applicability, as long as it is used only as an analogy (future experience must show if the analogy is worth anything) or a guideline to explore the other domain. But I got the impression you did more. If that impression was only due to the heated discussion and my imperfect understanding of the English language, all the better.
I maintain that
- facts are facts, regardless of whether they are proven or accepted as facts
- there are unprovable facts
- outside the realm of axiomatic formal theories, conclusive proof is impossible
and Gödel's theorem is completely irrelevant to the above, because they have been established beyond reasonable doubt independently of Gödel's theorem (I am aware of certain philosophical positions which deny any or all of them, though).
@CDarklock said:
Why, you would think I was challenging a tenet of faith or something
Something. In "Fashionable Nonsense" by Alan Sokal and Jean Bricmont you can find many examples of why I'm allergic to people shifting mathematical or scientific results or theories outside their domain, simply had too much of it.
Discussion paused for the weekend, I need some sleep.
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@CDarklock said:
Second, let us give the benefit of doubt and accept for the sake of argument that such a definition is in fact what Godel intended.
Stop. Stop this madness. Now.
Clearly Goedel was referring to an "axiom" as "a fundamental statement assumed to be true for the purposes of constructing a formal system". This is the only possible definition he would have used. Period. You are still making the mistake of taking a highly-technical consequence of formal systems and trying to generalize it to universal truth. You simply cannot do this without showing it is possible. You are attempting to make an analogy that is such a stretch that it will not be accepted without more evidence. You can't just willy-nilly pull conclusions from highly-specific and -technical systems and claim they apply universally to all systems.
Your claimed definition of axiom is wrong, even by the common definition which is "a truth that is taken without proof". Clearly this means a truth that is accepted without proof because it seems self-evident. Assuming Goedel's theory applied, axioms would still have to be truths we knew. Goedel shows us that it is a necessity that any formal system must have unprovable-yet-true statements. Obviously that does not mean axioms. We know axioms exist. We know they must exist. What Goedel showed is that other truths we do not know exist. Therefore your definition would be senseless in this analogy because by the common definition an axiom is something that is accepted or taken without proof. If it was accepted or taken, then it would be known before any statements were even derived from that axiom. Goedel showed there were statements beyond this that were true but not provable. Clearly these statements would have to be unknown and thus unaccepted or untaken. If they were known, then they would probably just become axioms of the system. However, Goedel's theory generalizes to all formal systems which means once we take the newly-discovered true-yet-unprovable statement as an axiom we have altered the system and that this own system will have its own true-yet-unprovable statements, and so on. For awhile, people just assumed there was a limit to the number of axioms and that once you had discovered them all, you could prove all true statements within the system. Goedel showed that there is always at least one true statement that cannot be proven and, if taken as an axiom, will alter the system and create further true-yet-unprovable statements, ad infinitum.
Because of this, you can't even pretend there is some common meaning of axiom that can be taken to support you. We don't know of Goedel's rules even apply outside of formal systems, but we don't need to know to shoot down your argument. That's because the definition of axiom you try to use clearly does not include true-yet-unprovable statements because we don't know if they are there. Due to this, the common definition of axiom only refers to known truths that are taken without proof. Since we do not know if there are non-axiomatic true-yet-unprovable statements outside of formal systems, trying to stuff such phantom statements into a long-standing definition is fairly silly. If such statements do not exist, then Goedel's theory does not apply and your point is null and void. If such statements do exist, then Goedel's theory most likely applies but the common definition of axiom would probably come to mean only a truth that is accepted without proof and would not include true-yet-unprovable statements. However, neither of these outcomes have occurred so for now trying to lump the phantom true-yet-unprovable statements into the common definition of axiom is simply unacceptable. We have not yet proven that these statements do exist outside of formal systems, therefore they cannot exist within our current definition of axiom.
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@Ilya Ehrenburg said:
I think your usage of "axiom" is so different from the mathematical usage that the word synonymous is inappropriate.
However, the language of Godel's theorem is synonymous with my non-mathematical definition of "axiom".
The word "homonym" does accurately apply to the question of mathematical "axiom" and non-mathematical "axiom".
@Ilya Ehrenburg said:
Well, it is legitimate and has been fruitful to move a result outside its domain of applicability, as long as it is used only as an analogy (future experience must show if the analogy is worth anything) or a guideline to explore the other domain.
If you look back in the thread, you'll find me saying this back on the first page:
"My point is that you can't prove the axiom no matter who chooses it. This is the very centre of Godel's incompleteness theorem: there exist things you can't prove. They still exist. The fact that they exist is still a fact. The inability to prove it - ironically enough - does not prove anything."
At this point, I was done with Godel's incompleteness theorem. It was no longer necessary. However, people complained about the word "axiom", which I had to define - "a principle that is accepted as true without proof" - and that Godel's theorem didn't say anything about axioms. So I had to identify that yes, given the definition of "axiom" I intended, Godel's theorem contains the synonymous phrase "an arithmetical statement that is true, but not provable".
That is the extent of my interest in it.
Godel's theorem is largely irrelevant to the larger question that things are true whether you believe them or not. I mentioned it in passing, and people took issue with that. Rather than let the discussion continue on any materially relevant subject, the argument around Godel's theorem and the word "axiom" has consumed pretty much the entire thread. My position has not changed, and no further information has been brought to light on it.
It is demonstrably the case that Godel's theorem centrally identifies the existence of things you cannot prove, and that the word "axiom" may be productively used to identify something true which cannot be proven. If we could mutually accept those two things, or reach an equitable agreement on something we could mutually accept, discussion could continue. But the consensus seems to be that once I've used terms in a manner of which the cognoscenti disapprove, further discussion is forbidden.
Rather a shame, really, but pearls before swine and all that.
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@CDarklock said:
Yes, but this was exactly what we were talking about. You were completely wrong. Godel's theorem doesn't say anything about proving axioms. By saying this, you've ignored all that is interesting about the theorem, not to mention completely misunderstood it. Your claim is trivial and boring. And the more you keep saying it, the more stupid you look.If you look back in the thread, you'll find me saying this back on the first page:
"My point is that you can't prove the axiom no matter who chooses it. This is the very centre of Godel's incompleteness theorem: there exist things you can't prove. They still exist. The fact that they exist is still a fact. The inability to prove it - ironically enough - does not prove anything."
At this point, I was done with Godel's incompleteness theorem.
@CDarklock said:
But the consensus seems to be that once I've used terms in a manner of which the cognoscenti disapprove, further discussion is forbidden.
Look, we clarified that you don't understand the math. If it were something like 2+2, maybe you would have conceded your error, and this thread would have died out long ago. But you've decided to keep arguing your obviously wrong point by either deliberately misunderstanding the theorem, or just simply due to an inability to follow the logic.You've publicly stated something that's obviously false and were called on it. Obviously, math is an important topic to many around here, and we aren't interested in allowing bad information to propagate ignorance.
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@boomzilla said:
Well-stated summations of the argument
I'm going to find out where you live and hug you, good sir.
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£117,331.00, by the way
