Why Are Metaphysical Questions Undecidable?

[Iacchus32]

It's a reasonable suggestion, but it can be shown to be untrue. Idealism, which implies immaterial origins for the universe, is not just scientifically untestable. As a cosmological doctrine, in the form it takes within western philosophy, it is logically inconsistent. This is why scientists, who are not all narrow-minded or dogmatic, cannot really be criticised for not adopting it. (Although one might criticize them for not applying the same standards of reasoning to materialism).

Well, it suggests that either can only be falsifiable when set in contrast to the other. We can't deny that the physical world exists. And yet, the only way we can truly acknowledge it is internally. Which, is why I propose dualism. How do we acknowledge the truth without a brain ... or, the mind which resides therein?

 

[Canute]

The question then is; "Why are all metaphysical questions undecidable?" period, as a stand alone question having nothing to do with materialism or physicalism. They are undecidable even within the framework of metaphysics using formal logic and reasoning. This statement is supported by the fact that there has never been a complete and satisfactory answer in over 3 thousand years of thinking and reasoning by thousands of thinkers. Is this correct so far?

Damn it, why didn't I just say that at the start? Sorry for muddling the issues by being so longwinded. Yes, this is correct as far as I'm concerned, and I should have said it as concisely and clearly as you have.

I mentioned materialism and idealism alongside asking the question because it seemed to me that many people overlook the significance of the undecidability of metaphysical questions. This undecidability means that according to reason the philosophical doctrine of materialism contradicts reason, just as does the doctrine of idealism, for it is precisely this fact that makes the question of which of them is true or false undecidable in the first place. I thought it would stir up a more interesting debate if l said this along with asking the question. In hindsight I can see that all it was derail the discussion. My apologies.

As you say, the question is simply 'Why are metaphysical questions undecidable?'

Alternatively, to make it sound less like just another pointless 'philosophical' question, it could asked as 'What is it about the universe that makes it impossible to construct a formally consistent explanation of its existence?

 

[Canute]

Iacchus32

If a proposition is unfalsifiable it's unfalsifiable. It's nothing to do with setting it in contrast with anything. Also, although we cannot deny that the physical world exists in some sense, as you say, we can still wonder in what sense it does, whether it is an epiphenomenon of mind, whether it is made out of something material, and so on.

Despite considerable research there is no evidence yet showing that our consciousness resides in our brains. Rather, it seems to have no extension at all. I must say my consciousness feels for most of the time as if it is somewhere right behind my eyes, but this is just how it seems to me, not scientific evidence of its location.

 

[Iacchus32]

Iacchus32

If a proposition is unfalsifiable it's unfalsifiable.

So it sounds like both materialism and idealism are unfalsifialble, which is to say self-referential? ... unless, as I suggest, you understand them in context with each other.

It's nothing to do with setting it in contrast with anything. Also, although we cannot deny that the physical world exists in some sense, as you say, we can still wonder in what sense it does, whether it is an epiphenomenon of mind, whether it is made out of something material, and so on.

The thing is you can't have an inside without an outside, and that's the only difference between materialism and idealism as far as I'm concerned. Materialism represents the outward manifestation of reality, as determined through the five senses, whereas idealism correpsonds to the inner-experience which is alive and well and perceives that outer-reality. So, we can hoot and holler all we want about the existence of materialism, and yet we don't actually know outside of we perceive of it. In which case the two are wholly contingent upon each other.

 

[Royce]

Damn it, why didn't I just say that at the start? Sorry for muddling the issues by being so longwinded.

LOL, No apology necessary. It served your purpose and got the discussion started. I felt like a fool when I finally realized what you where asking after 3-4 days of talking about the wrong thing.

I am glad to read that physical-ism or Materialism suffers from the same plight.
I have long contended that Materialism is flawed and incomplete.

As you say, the question is simply 'Why are metaphysical questions undecidable?'

Alternatively, to make it sound less like just another pointless 'philosophical' question, it could asked as 'What is it about the universe that makes it impossible to construct a formally consistent explanation of its existence?

Ans: Obviously we are missing something or leaving out something basic
and crucial to the question. I want to answer God or the Universal
Conscious; but I think that it is more fundamental than that. I think that
it has to do with reality itself in that we are mired in it, within it, and
Trying to get a bird's eye(God's eye) view of it while we are stuck
here on the bottom like a fish swimming in the bottom of a murky pool
trying to learn about not just the water itself but what lies above it too.

We are looking at reality upside down. As the material universe is the medium in which we live and all that we can know with any certainty we look at it first as all there is and secondly as the ultimate cause and ultimate effect.
I think that this is wrong. The material universe is the effect, the result of the non-material reality. IMHO this means subject/mental realm as well as the so called spiritual realm. It is all one reality and all realms or aspects of reality are real but there is an hierarchy of at least cause and effect. This is why so many say that the material universe and our lives within it are in reality an illusion. It is not an illusion but the impression that it is the Ultimate Reality and all the rest is not really real but just subjective, spiritual hogwash is an illusion.
This leads me to believe that the questions being undecidable is because we are asking them from an illusional, invalid, viewpoint.
I have no solution but to try asking them from another viewpoint or mind set.

Example: Ask what it is about consciousness that allows or make the material to come into existence?

I know this is no answer for you; but as I am in way over my head now it is the best that I can do. After all it took this long just to figure out what the hell the question was!

 

[Canute]

Royce

I'd go along with most of your post. I suppose that the question becomes 'what other perspective?'

So it sounds like both materialism and idealism are unfalsifialble, which is to say self-referential? ... unless, as I suggest, you understand them in context with each other.

I don't see that. In what way does understanding them 'in context with each other' change anything? I roughly agree about the inside outside thing. It raises the question of what perspective we should adopt on the question if both the view from the inside and the view from the outside are wrong.

 

[Iacchus32]

I don't see that. In what way does understanding them 'in context with each other' change anything? I roughly agree about the inside outside thing. It raises the question of what perspective we should adopt on the question if both the view from the inside and the view from the outside are wrong.

How do you know what black is unless you contrast it with white? Of course in that respect, here's what looks to be an interesting http://web.mit.edu/persci/people/adelson/checkershadow_illusion.html. Which is amazing when you click on "the proof."

 

[Pavel]

As you say, the question is simply 'Why are metaphysical questions undecidable?'

In what sense? From what I vaguely remember, the undecidability really has to do with a recursive algorithm that cannot spit out “yes” or “no” as its output, given an infinity of instances at its input. The famous Halting Problem on a Universal Turing machine (UTM) illustrates the issue (there’s no general UTM that accepts an algorithm (i.e. another UTM) and can tell with a “yes” or “no” answer whether the given UTM will halt (i.e. recursively solve a problem)). If you’re dealing with only problem instance, e.g. find a solution to Fermat’s conjecture, the problem is actually considered to be decidable. Godel’s truths belong to this “decidable” category, the single instance problems. The implication of the incompleteness theorem is not that there are statements that don’t have a “yes or no” answer, the implication is that there are statements that are true, but you don’t know which ones, since there’s no algorithm to prove them to be true. But they can be proved from a meta system. So, what I’m asking is are you aware of some metaphysical statements that are true, but we can’t prove them with our logic, or that the metaphysical statements in general don’t have a yes-no answer? If the former, then it’s either a tautology, since you’re reiterating the incompleteness theorem and merely mentioning that metaphysical statements are part of the system; or, you’re assigning the ontological status to metaphysical statements as being G statements but I don’t see on what grounds.

If you mean metaphysical statements don’t have a yes-no answer, then I don’t see how you differentiate metaphysical and physical from a formal point of view. After all, just like logic, the inability to resolve a problem recursively is demonstrated with variables, not parameters. The Turing machine doesn’t care whether the input variable is metaphysical or not, it cares about halting. So, why do you care whether it’s metaphysical or not? For example, I don’t see what’s undecidable in the latter sense about, let’s say, “my consciousness is undetectable froth that floats on top of my brain”. It’s perfectly decidable, yet metaphysical, isn’t it?

Alternatively, to make it sound less like just another pointless 'philosophical' question, it could asked as 'What is it about the universe that makes it impossible to construct a formally consistent explanation of its existence?

Either that, or our explanatory system is weak and ill-formed. Now, I’m nowhere even close to the position of criticizing Godel’s incompleteness theorem, and I do realize that to do so, you’d have to put the whole number theory upside down, but still. I’m a big fan of Godel, but I always wondered: did he discover something inherently profound about the Universe, or simply showed that our formal system is weak to explain itself. The history is filled with examples of the latter. Newton didn’t have strong mathematics to describe his theory, so he invented his own! Newton’s description of reality was considered to be so firm for a couple of centuries that to question it was unthinkable. Well, we all know what Einstein did to it. My point is that through the history, both physics and math have not been discovered a priori. They have to make sense, and if they don’t, they are modified and expanded. What makes you think there won’t be Einstein in logic (even though Godel has already been considered to be one) who will change our formal system and the next you know we can fulfill Hilbert’s wish – to reduce everything to mathematical axioms and provable theorems.

Pavel.

 

[Pavel]

. I think that it has to do with reality itself in that we are mired in it, within it, and Trying to get a bird's eye(God's eye) view of it while we are stuck here on the bottom like a fish swimming in the bottom of a murky pool trying to learn about not just the water itself but what lies above it too.

We are looking at reality upside down. As the material universe is the medium in which we live and all that we can know with any certainty we look at it first as all there is and secondly as the ultimate cause and ultimate effect.

Your claim is a bird's eye view, isn't it? Isn't it subject to the same criticism as Kant's noumenon?

 

[honestrosewater]

Can someone give an example of a metaphysical question (MQ) so we can analyze its form?
Do all MQs have the same form?
Do all MQs share a combination of (non)reflexive, (non)symmetric, or (non)transitive relations/operators?

 

[Iacchus32]

Can someone give an example of a metaphysical question (MQ) so we can analyze its form?
Do all MQs have the same form?
Do all MQs share a combination of (non)reflexive, (non)symmetric, or (non)transitive relations/operators?

If we understood that the truth is evidenced within, then every question becomes a metaphysical question. Why? Because it doesn't pertain to the physical world, but only to what we perceive of it. This is my whole point by the way. We can't separate the metaphysical from the physical, because our consciousness (that which observes the physical) resides within the metaphysical reality of the mind.

 

[Canute]

In what sense? From what I vaguely remember, the undecidability really has to do with a recursive algorithm ... SNIP... If you’re dealing with only problem instance, e.g. find a solution to Fermat’s conjecture, the problem is actually considered to be decidable. Godel’s truths belong to this “decidable” category, the single instance problems.

The implication of the incompleteness theorem is not that there are statements that don’t have a “yes or no” answer, the implication is that there are statements that are true, but you don’t know which ones, since there’s no algorithm to prove them to be true. But they can be proved from a meta system.

Great post. Just to be clear, I agree so far.

So, what I’m asking is are you aware of some metaphysical statements that are true, but we can’t prove them with our logic, or that the metaphysical statements in general don’t have a yes-no answer? If the former, then it’s either a tautology, since you’re reiterating the incompleteness theorem and merely mentioning that metaphysical statements are part of the system; or, you’re assigning the ontological status to metaphysical statements as being G statements but I don’t see on what grounds.

Yes, I'm surprised nobody picked me up on that earlier. It is my opinion that they are G-sentences, undecidable within any formally consistent system of reasoning, but it would take a few thousand words to attempt a proof, and I don't know whether I could make it stick.

However given the ongoing failure of analytical philosophers to decide these questions there is some evidence on my side. This is not a proof of it of course. However all the answers to these questions give rise to contradicitions, and that does seem to clinch it. Mathematician George Spencer-Brown sees our attempts to answer them as the same kind of iterative process as that which is used in the trembler circuits of an electric bell. If you say materialism is true (input) then you conclude that it must be false (output) so then you assume that idealism is true (input) only to find that it cannot be (output) and thus we do round and around for all eternity.

If you mean metaphysical statements don’t have a yes-no answer, then I don’t see how you differentiate metaphysical and physical from a formal point of view.

Well, in a way I'm saying that metaphysical question have only yes-no answers and that this is the whole problem with them. I'm not saying that metaphysical questions are undecidable just because they're metaphysical, (while I do think that it does follow I haven't tried to argue it). They are undecidable because they have no non-contradictory answers, and metaphysical because they are about what is ultimately real or ultimately true or false, and ultimates and absolutes lie beyond our senses, beyond science and beyond formal reasoning. This is Kant's transcendent reality I suppose, which requires us to transcend metaphysics to understand it.

After all, just like logic, the inability to resolve a problem recursively is demonstrated with variables, not parameters. The Turing machine doesn’t care whether the input variable is metaphysical or not, it cares about halting. So, why do you care whether it’s metaphysical or not? For example, I don’t see what’s undecidable in the latter sense about, let’s say, “my consciousness is undetectable froth that floats on top of my brain”. It’s perfectly decidable, yet metaphysical, isn’t it?

Clearly there are an infinite number of undecidable questions which are not metaphysical, but I believe that all metaphysical questions (if they are well-formed within the system) are undecidable. This partly because of the way we define 'metaphysical', and also because in my opinion what is ultimate cannot be represented by true and false theorems within any formal system. (The Tao cannot be named etc).

More controversially, or perhaps I should say even more controversially, I would argue that fundamental questions about consciousness are both metaphysical and undecidable. Again, a proof would take me long time, but the evidence is on my side.

Now, I’m nowhere even close to the position of criticizing Godel’s incompleteness theorem, and I do realize that to do so, you’d have to put the whole number theory upside down, but still. I’m a big fan of Godel, but I always wondered: did he discover something inherently profound about the Universe, or simply showed that our formal system is weak to explain itself.

A good question. Personally I believe that the incompleteness theorem holds for all sentient beings in all possible universes, and that it is inevitable that absolute truths cannot be derived from assumptions. I couldn't prove it, but I feel it's already been proved.

Also I'd say it is true for ontological reasons. In other words all the reasonable answers to metaphysical questions are wrong because such questions are based on a false assumption about reality. These are questions about the meta-system which gives rise to the formal system we call the universe, and they can only be resolved or understood (but not decided) from the meta-system, by transcending the system.

What makes you think there won’t be Einstein in logic (even though Godel has already been considered to be one) who will change our formal system and the next you know we can fulfill Hilbert’s wish – to reduce everything to mathematical axioms and provable theorems.

Funnily enough I believe this can be done. But only by creating a formal system which has a (formally) undecidable axiom its heart (one that is formally acknowledged to be undecidable), and which embodies (apparent) contradictions. This is the epistemilogical structure of Buddhism, Taoism, and so on. That would be a daunting topic to get into. However, roughly speaking, it's very similar to the epistemilogical system used in QM, in which the question of whether a wavicle is a particle or a wave is undecidable, and two complementary/contradictory formal systems arise as a result, one in which they are waves, one in which they are particles.

On this it's worth noting that undecidable metaphysical questions do not arise in these doctrines. No Buddhist or Taoist batted an eyelid when Goedel produced his proof. What he proved is what they've been asserting for millenia.

 

[Canute]

If we understood that the truth is evidenced within, then every question becomes a metaphysical question. Why? Because it doesn't pertain to the physical world, but only to what we perceive of it. This is my whole point by the way. We can't separate the metaphysical from the physical, because our consciousness (that which observes the physical) resides within the metaphysical reality of the mind.

That seems true. I've got a good quote somewhere from a respectable physicist arguing that we cannot do physics without doing metaphysics, but I've mislaid it. I'll post it if I find it. However, I'm not sure it's right to say that scientific questions are metaphysical.

Although come to think of it I suppose this is ultimately the reason why science can never provide absolute proof of any of its theories. But I haven't really thought this through.

Honestrosewater

The trouble with these questions is that they can be asked in different ways. Thus the question 'why does anything exist' is pretty much the same as 'did the universe arise from something or nothing'. But they can all be asked in such a way as to force us to choose between alternative and opposite answers, and it is questions in this form that are most relevant here. Perhaps we could use that one as a test case - did the universe arise from something or nothing? Alternatively - is matter made out of something or nothing?

 

[Sho'Nuff]

I'm still digging trough the discussion in this post but I want to see if I got it right so far since my english is a bit lagging behind...help me out here...

A metaphysical question is a question that can have multiple answers acquired with reason but the best answer cannot be decided upon though reason.

This is because none of the answers are either falsifiable of verifiable.

Did I get it ?

 

[Royce]

Your claim is a bird's eye view, isn't it? Isn't it subject to the same criticism as Kant's noumenon?

As it is attempting to look from the sky down into the pool, then yes its a bird's eye view. from the ideal to the material rather than just the material.

If I knew what a Kant's noumenon was and the criticism that it is subject to, I might me able to answer that. I am at best a lay philosopher. Its been 30 years since I read any Kant and then it was in a intro philosophy class.

 

[Pavel]

As it is attempting to look from the sky down into the pool, then yes its a bird's eye view. from the ideal to the material rather than just the material.

If I knew what a Kant's noumenon was and the criticism that it is subject to, I might me able to answer that. I am at best a lay philosopher. Its been 30 years since I read any Kant and then it was in a intro philosophy class.

No big deal, I'm not a pro either, I just happen to know about it because it is the kind of stuff I love learning.

Seems like you didn't understand my pick on you though. Briefly, Kant proposed that reality has a different form or structure than what we perceive it to be. Our knowledge of the world or what we perceive is phenomena. The real stuff is noumena. We can never know the true nature of noumenon because our perception of it is "distorted" by our senses and mind. This is an oversimplified version but you get the idea. The obvious criticism of this is that Kant makes an observation about the system from the outside the system and he has no epistemological right to do so. In other words, if you claim that your perception distorts truth about reality, then your own claim is a distorted view of reality, it’s phenomenon. You trapped yourself. It’s the same fallacy the determinists commit when then say “my thoughts are determined”…..

Similarly, you claimed we’re fish at the bottom of the pool trying to have a bird’s eye view. That very claim seems to be quite a bird’s eye view. But if you’re fish, you can’t make it. You’re assuming an outsider position, a bird, if you will, look down and say “oh, we’re fish” and then jump right back in. I don’t think you can do that unless you show you can fly. Anyway, I was not trying to start a debate about it, that was just an observation….

 

[Pavel]

It is my opinion that they are G-sentences, undecidable within any formally consistent system of reasoning, but it would take a few thousand words to attempt a proof, and I don't know whether I could make it stick.

Heh, I'm in the same boat, in a sense that what I stated is also my opinion and that it'd take a few thousand words to attempt a proof, even though I believe I know how to proceed with it. I'm not saying you're wrong, and I believe your sources have a lot of merit. I'm just stating there's some inconsistency with my understanding of Godel and I wanted to contribute to this discussion to keep it more informative and detailed.

I hope we're going to avoid coding natural language into binary strings here but, as I said in my early post, I don't believe the metaphysical statements presented here are undecidable. Let me take another shot, but this time in a little more detail.

How did Godel prove incompleteness? He mapped natural language to an axiomatic arithmetic system. He then translated, I believe, a version of Liar’s paradox – “this statement is not provable” or “the statement on the other side of the page is not true….” into this arithmetic system. He thus produced a mathematical version of the Liar paradox. Note, it’s the same kind of contradictory paradox you’re referring to when discussing materialism. The translation is a statement G that says “G is not provable”. So, you have If G is provable, it is not provable, a contradiction. However, if G says it is not provable and it really is not provable, then G is true, but not provable. The first choice makes the system inconsistent, that’s not what we want, so logicians settled for the second choice. The proof is quite interesting and not that complicated but requires focus, in case you haven't indulged yourself yet. Anyhow, the bottom line is the Liar paradox is translatable into a finite formal language, and thus into a finite binary string that can be accepted or rejected by a Turing machine! Let me give you an analogy. Nobody proved Golbach’s conjecture (every even number above 2 is a sum of two primes). However, I can say, let’s define number P as 67 if Golbach’s conjecture is true, and P is 97 if the conjecture is false. We know the number exists, we just don’t know which one it is! Similarly, I can feed the conjecture into 2 Turing machines, one that accepts all input, the other one that rejects. One if them might have the solution, but we don’t know which one. That’s Godel’s G statement. Note that the conjecture, just like the paradox can be coded into a finite binary string. This class of statements is considered to be decidable!

Now, consider the famous Halting problem – is there a general algorithm that can determine if a Turing machine will halt on a given input. Think about it, the general algorithm is a Universal Turing Machine (UTM) that accepts as its input a pair of strings – one is the Turing machine to be tested, the other - that machine’s input binary string. But there is an infinity of the input binary strings and Turing machines. So, the input to the UTM is an infinite set of strings, meaning there’s no effective way to determine if the UTM will halt or not. If you changed the problem to a specific pair of strings, not general, the problem would be decidable. Well, it depends on the nature of the input string to the machine under test. If it’s infinite, like in your infinite regress with causality example, the situation gets convoluted, at least for me, because you start dealing with countable infinity, Cantor set and the whole continuity problem. And speaking of which, here’s another good example of undecidable statement – the continuity hypothesis (uncountable infinity of real numbers “between” a pair of rational numbers, which in their turn form a countable infinity line). Godel proved that the assertion of the hypothesis is consistent with an axiomatic set theory. In fact, I think he made it an axiom, but then Cohen came along and proved that the negation of the hypothesis is also consistent with the set theory. So, now you have truth and negation of the same statement consistent within the same axiomatic system! This is not the same assertion and negation of idealism because it’s unfalsifiable. If translated into formal language, I’m certain the negation will be inconsistent with its assertion, but please don’t make me do it, it’ll be quite a home work.

Funnily enough I believe this can be done. But only by creating a formal system which has a (formally) undecidable axiom its heart (one that is formally acknowledged to be undecidable), and which embodies (apparent) contradictions. This is the epistemilogical structure of Buddhism, Taoism, and so on.

But are you sure such system will be consistent? I'm not familiar with Buddhist and Tao epistemological structure, they have a consistent system?

However, roughly speaking, it's very similar to the epistemilogical system used in QM, in which the question of whether a wavicle is a particle or a wave is undecidable, and two complementary/contradictory formal systems arise as a result, one in which they are waves, one in which they are particles.

Somehow I considered QM simply to show that waves and particles are not mutually exclusive but there's really no undecidability about it. The only other thing I thought was that QM showed to intuitionists that a number can exist even if there is no way to construct it, until you prove it, just like the position of an electron is unknown, until you look. If anything, there's an existence of possible physical worlds, but without corresponding formal systems accompanying each one of them. That's how I view it, but to be quite honest, I'm not that sure about it, as I haven't given it too much thought What's the general consensus among the logicians?

Pavel.

 

[Iacchus32]

Our knowledge of the world or what we perceive is phenomena. The real stuff is noumena. We can never know the true nature of noumenon because our perception of it is "distorted" by our senses and mind. This is an oversimplified version but you get the idea. The obvious criticism of this is that Kant makes an observation about the system from the outside the system and he has no epistemological right to do so. In other words, if you claim that your perception distorts truth about reality, then your own claim is a distorted view of reality, it’s phenomenon. You trapped yourself.

And yet in not knowing you know. So at least you know that much about reality overall. Which, in fact is what Kant is saying isn't he? ... Reality, in all its absoluteness -- yes -- must be greater than what I perceive of it. Why is that so difficult to understand?

 

[honestrosewater]

And yet in not knowing you know. So at least you know that much about reality overall. Which, in fact is what Kant is saying isn't he? ... Reality, in all its absoluteness -- yes -- must be greater than what I perceive of it. Why is that so difficult to understand?

"Noumena exists independently of phenomena." Isn't that statement undecidable in a phenomenal system?

Pavel- Thanks for those explanations. Well, thanks to everyone, but those were especially nice

Did the universe arise from something or nothing?

I suspect we can answer this question by making one statement and analyzing the relation:

Nothing caused something.

Substituting "$" for "caused", the cause relation:

N $ S

1) Is $ reflexive? (Can A cause A?)
2) Is $ symmetric? (Is it true that if A caused B, then B caused A?)
3) Is $ transitive? (Is it true that if A caused B, and B caused C, then A caused C?)

Does anyone think this approach could be productive?
___
Okay, I should add a few thoughts. There seem to be similarities between physical causation and mathematical order (>). Mathematical order also seems to be at work in other questions, esp. about god. As in, can an omnipotent being limit its own power? Do you see a connection between this and something like "for all x in A, there exists some y in A such that y > x." Of course, because > is not reflexive, y cannot exist (it is never true that y > y). But if you change > to $\geq$ then y can exist. Well, I could have phrased that example better, those are just some rambling thoughts, and I have reached no conclusions yet.

 

[Iacchus32]

"Noumena exists independently of phenomena." Isn't that statement undecidable in a phenomenal system?

Yes, but does phenomena give rise to itself? No. Even an illusion needs something tangible to prop it up ... unless of course everything was an illusion to begin with, but then again that's not possible, because even an illusion is something (tangibly based) compared to nothing.

 

[honestrosewater]

A metaphysical question is a question that can have multiple answers acquired with reason but the best answer cannot be decided upon though reason.

This is because none of the answers are either falsifiable of verifiable.

Did I get it ?

Yes, that is being assumed by some, but they have yet to prove it

Someone could argue that it depends on how you ask the question and in what system you ask the question. Canute and Pavel said they think that metaphysical questions are G(ödel) statements (or sentences). I couldn't find much info, but it seems G statements are defined as being undecidable in their system (the system in which they appear). C&P seem to use them to apply to all systems. I'm not sure what is common practice. I imagine if you can prove that all sufficiently complex, consistent formal systems contain G statements, you have a shot at proving that all G statements have certain properties in common, regardless of their system. Perhaps that is precisely what you are proving, I don't know, I can't think that quickly. Perhaps this is precisely how Gödel's proof proceeded. Perhaps someone can enlighten us.
____
http://homepages.which.net/~gk.sherman/baaaaab.htm explains briefly the steps of the proof.
http://home.ddc.net/ygg/etext/godel/godel3.htm the horse's mouth ;)

 

[Canute]

A metaphysical question is a question that can have multiple answers acquired with reason but the best answer cannot be decided upon though reason.

This is because none of the answers are either falsifiable of verifiable.

Did I get it ?

Sort of, but not quite. Officially (my dictionary) metaphysical questions are questions about what lies 'beyond nature' and beyond ordinary knowledge or experience.

Such questions cannot be answered by reason because the reasonable answers to them are inconsistent with the formal rules of the system of logic in which they are asked. In other words, the question 'Is matter made out of something or nothing' cannot be answered because the only available 'reasonable' answers are something or nothing, and both give rise to logical contradictions (neither answer makes sense to someone asking the question who reasons that these are the only two possible answers).

So this question is metaphysical in that it is about what is ultimately true or false (which puts it beyond science) and undecidable in that neither answer makes sense (which puts it beyond the system of formal reasoning that was used to construct the question).

As it appears that all formal systems of reasoning about reality give rise to undecidable metaphysical questions we can say that undecidable metaphysical questions are beyond reason, rather than just beyond some particular system of reasoning. (I would say this is an error, but I won't risk muddling the issues by going into why here).

 

[Canute]

Briefly, Kant proposed that reality has a different form or structure than what we perceive it to be. Our knowledge of the world or what we perceive is phenomena. The real stuff is noumena. We can never know the true nature of noumenon because our perception of it is "distorted" by our senses and mind. This is an oversimplified version but you get the idea. The obvious criticism of this is that Kant makes an observation about the system from the outside the system and he has no epistemological right to do so. In other words, if you claim that your perception distorts truth about reality, then your own claim is a distorted view of reality, it’s phenomenon. You trapped yourself. It’s the same fallacy the determinists commit when then say “my thoughts are determined”…..

I read Kant the other way around, as making a comment about what is ouitside the system from within the system. This is logically allowable since it is in the nature of formal systems that they have a meta-system outside themselves. In other words we can formally infer a metasystem without having to actually get out of the system, epistemilogically speaking.

Iow, Kant argued that there was a meta-system not knowable through our senses or by reason ('transcendent' reality), and this seems allowable to me since it seems logically inevitable. I suspect Kant would not have been suprised by Godel's theorem.

Also, I only half agree about the noumenal. It's true that we cannot trancend the system to confirm Kant's assertion by observation or reason. But it is perfectly possible, in principle at least, to transcend it non-conceptually, by direct experience of that transcendent reality. It seems relevant to mention that Buddhism is sometimes characterised as 'the view from nowhere'.

 

[Royce]

No big deal, I'm not a pro either, I just happen to know about it because it is the kind of stuff I love learning.

Seems like you didn't understand my pick on you though. Briefly, Kant proposed that reality has a different form or structure than what we perceive it to be. Our knowledge of the world or what we perceive is phenomena. The real stuff is noumena. We can never know the true nature of noumenon because our perception of it is "distorted" by our senses and mind. This is an oversimplified version but you get the idea. The obvious criticism of this is that Kant makes an observation about the system from the outside the system and he has no epistemological right to do so. In other words, if you claim that your perception distorts truth about reality, then your own claim is a distorted view of reality, it’s phenomenon. You trapped yourself. It’s the same fallacy the determinists commit when then say “my thoughts are determined”…..

Well I don't and never have agreed with everything Kants said, especially the above. It may be just the words that he(you) used that we misinterpret. What I am saying and have been saying for some time here is that; if our view
of reality is distorted it is because we limit our view to the physical objective universe and ignore or dismiss the rest of reality, the subjective, mental and spiritual (for want of better words) and that we look at the physical as the cause of our reality whereas it is IMO the effect, hence up-side-down.

Similarly, you claimed we’re fish at the bottom of the pool trying to have a bird’s eye view. That very claim seems to be quite a bird’s eye view. But if you’re fish, you can’t make it. You’re assuming an outsider position, a bird, if you will, look down and say “oh, we’re fish” and then jump right back in. I don’t think you can do that unless you show you can fly. Anyway, I was not trying to start a debate about it, that was just an observation….

Again you are making an invalid assumption, as most physicalist do, that I am talking about an outside viewpoint. This is why I used the analogy that I did. The sky and the bird are all part of our nature the same, one ecology.
There is only one reality. our error is by looking at only one aspect of it and thinking that that is all that there is. How could we, or God for that matter, be outside of reality looking in. If God and we are real then we obviously are in reality.

 

[Canute]

How did Godel prove incompleteness? He mapped natural language to an axiomatic arithmetic system. He then translated, I believe, a version of Liar’s paradox – “this statement is not provable” or “the statement on the other side of the page is not true….” into this arithmetic system. He thus produced a mathematical version of the Liar paradox. Note, it’s the same kind of contradictory paradox you’re referring to when discussing materialism. The translation is a statement G that says “G is not provable”. So, you have If G is provable, it is not provable, a contradiction. However, if G says it is not provable and it really is not provable, then G is true, but not provable. The first choice makes the system inconsistent, that’s not what we want, so logicians settled for the second choice.

I'm uncomfortable with the term 'provable' here, and would rather use 'decidable'. But I don't disagree with what I think you mean.

The proof is quite interesting and not that complicated but requires focus, in case you haven't indulged yourself yet.

Ha. Not that complicated to you perhaps. I can't follow the mathematics. However it strikes me that G constructed his proof by departing from his formal system of proof and then re-entering it, which is suggestive.

The two para's you wrote on the continuum hypothesis and the Goldbach conjecture seem about right to me. But I'm not sure why they're relevant here. I cite Goedel simply to show that formal axiomatic systems always have metasystems, so that any formally consistent description of the universe must leave something out (must be incomplete - cf. Kant's transcendent reality), and to show that all formal axiomatic systems must contain at least one undecidable question.

But are you sure such system will be consistent? I'm not familiar with Buddhist and Tao epistemological structure, they have a consistent system?

I'm not sure that this is a decidable question. It depends how you look at it. The peculiar epistemilogical structure of Buddhism stems from the undecidability of all questions about what it takes to be ultimately real, what it takes as axiomatic if you like.

Thus, for instance, what is ultimate cannot be characterised as existing or not-existing. Never mind the counterintuitiveness of this for the moment, what it means is that in Buddhism there are two complementary descriptions of reality, one in which what is absolute exists, one in which it doesn't. Each of these explanations is perfectly consistent in themselves, but both are equally true and false, since what is ultimate transcends such distinctions. This applies to all 'dual' properties that might be assigned to what is ultimate. (The problem here is over the definition of 'exist' - there are at least two ways of defining it, so at least two ways of answering the question. Buddhists tend to say what is ultimate is, rather than that it exists, to avoid making a false assertion).

What this entails is that a Buddhist explanation of reality (and those of Taoism, Advaita, mystical Christianity, Sufism etc.) are really two complementary explanations that contradict/complement each other. Each is entirely consistent in itself, but each contradicts the other and neither is complete. This is not some trick to confuse non-believers, and definitely not 'mysticism' in the disparaging sense. It is just the way things are. What is ultimate transcends distinctions and must remain undefined in any formally systematic description of reality. This is why it can only be approached non-conceptually. (Hence the 'Tao that cannot be named', the avowed danger of naming God or 'idolising' him, and the impossibilty of explaining what Buddhists mean by 'emptiness').

I feel too much has been made of the link between Buddhism and QM, but in this epistemilogical sense they are very similar systems of explanation. A wavicle can be consistently explained as a wave, and can be consistently explained as a particle. But neither explanation is complete, for neither explains what a wavicle is, and contradictions are only avoided by keeping the two contradictory/complementary explanations separated. What a wavicle is is neither a wave nor a particle. What ultimate reality is is neither something nor nothing.

On this view any question that asks whether what is ultimate is something or nothing would be undecidable, for it's an improper question, an artefact of dual thinking. It embodies a false assumption, like the question 'have you stopped beating your wife?' It's unanswerable (unless, that is, you have been beating your wife!).

If that doesn't make sense I won't be surprised. It's a very confusing topic to discuss.

 

[honestrosewater]

Canute,
That compliment/contradict idea reminds me of 1 and 0. You know, how you have to include that 1 does not = 0 in the field axioms.
I forgot to mention hierarchies are also ordered. When you say "ultimate", do you mean the root(s) of a hierarchy?

 

[Canute]

Canute,
That compliment/contradict idea reminds me of 1 and 0. You know, how you have to include that 1 does not = 0 in the field axioms.
I forgot to mention hierarchies are also ordered. When you say "ultimate", do you mean the root(s) of a hierarchy?

I'm afraid I don't know what 'field axioms' are. But it sounds like you're saying that 1 has to be defined as ~0 in the same way as we have to define something as ~nothing, thus giving ourselves a cosmological dilemma. Is that it? Yes, by 'ultimate' I do mean the root of a hierarchy. Or alternatively, that without which nothing else would exist.

 

[Pavel]

And yet in not knowing you know. So at least you know that much about reality overall.

Well, yeah, but how coherent is that knowledge? That's the problem. As thinkers, we want to be consistent in how we understand the reality. If my knowledge allows me to believe that I'm a programmed robot and that I can choose at the same time, the knowledge is flawed.

Reality, in all its absoluteness -- yes -- must be greater than what I perceive of it. Why is that so difficult to understand?

I have no difficulty understanding it, I'm having a problem seeing on what grounds you believe there's something else. Why not add green men and the perfect island to the mix as well?

 

[honestrosewater]

Canute,
Right. The field axioms define normal addition and multiplication. I think I'm ducking out of this discussion for a while. Have fun

 

[Pavel]

I read Kant the other way around, as making a comment about what is ouitside the system from within the system...
Iow, Kant argued that there was a meta-system not knowable through our senses or by reason ('transcendent' reality)

See, I told you I'm not a pro. That's right, it's the other way around.

This is logically allowable since it is in the nature of formal systems that they have a meta-system outside themselves. In other words we can formally infer a metasystem without having to actually get out of the system, epistemilogically speaking.

Hmmm, interesting, I'm not sure about it, but I'll keep my mind open. Do you have any reference I can read on how we can infer a meta system that transcends our level of consciousness? See, to me, Godel's meta systems or any other formal systems in arithmetic are within our level of abstraction or complexity, if you will, just like a 2D space is a subset of a 11D space. That's what allows us to transcend them. All Godel did was to prove there are truths that can be proved only from a meta level. There are two assumptions: the system is consistent, and that it's formal. Such system, he proved, is incomplete. You can't assume that about the system on which our consciousness operates. All these post Godel's speculations about the necessity of meta systems that transcend our being are just that - speculations, in my understanding, but I'd be happy to read a good article about the inference of meta systems, ad infinitum? To put it yet in other words, how do you know, that WE are not at the root of the hierarchy? I don't think Godel proves we are not.

On the same note, I have a problem with the classical agnosticism claiming "we can't know if God exists". I don't have a problem if you say "I don't know if He exists". But when you say "can't", that's a heck of a claim! It's a claim about the reality you're claiming you can't know. Kant said the knoweldge of God, or noumena is unknowable, hidden from the scientific inquiry. If it's unknowable, there's no sense in talking about it, you're talking about something you don't know. (isn't that what "talking out of your butt" means?) How much sense is that making?

It's true that we cannot trancend the system to confirm Kant's assertion by observation or reason. But it is perfectly possible, in principle at least, to transcend it non-conceptually, by direct experience of that transcendent reality. It seems relevant to mention that Buddhism is sometimes characterised as 'the view from nowhere'.

I can't comment on that. WIth all due respect, I think that's highly subjective to personal experience, as my transcendental experience can take me in totally different place than yours, if you know what I mean

Pavel.

 

[Pavel]

Ha. Not that complicated to you perhaps. I can't follow the mathematics. However it strikes me that G constructed his proof by departing from his formal system of proof and then re-entering it, which is suggestive.

Heh, maybe I got that impression because I read a some kind of "Godel for dummies" version, where they pretty much spoon fed you the steps. Seriously though, if you have understanding of predicate calculus and basic number theory, you're good to go. Knowledge of the diagonalization method would be helpful in understanding his mapping of the natural language to an axiomatic system.

I'm not sure where he departs from his system and then reenters it, specifically?

The two para's you wrote on the continuum hypothesis and the Goldbach conjecture seem about right to me. But I'm not sure why they're relevant here.

Ok, I'll try to be brief this time and take yet another, more general approach. I had a debate with my buddy a couple of years ago (which led me to study Godel in more detail) about our logic. I was trying to convince him that we're stuck with a binary logic, we don't have a choice. Speaking about other logics ultimately brings the same question to the table "but is that logic true"? Denying or questioning our fundamental axioms of logic is meaningless, as in doing so, you have no other tool to do it but the very axioms in question... You get the idea. He was arguing there are propositions that don't have a boolean yes or no answer to them, they're undecidable. I said wait a second, you can't be talking about Godel, because while we can't prove the statement to be true, it's ontological status, if you will, is still true or false, we just don't know which one it is, as there's no mechanical way to prove it. That's why I mentioned Golbach's and Fermat's conjectures. There might not be a computational way to prove or disprove the conjectures, they're still either true or false. Unfortunately for me, my friend taught an Automata and Intro to computation class in college and he was more up to speed. He showed that problems like the two conjectures are actually trivially decidable; when traslated into a formal language, they are recursive and there's a computational way to decide them. Because of a wide acceptance of Church-Turing thesis, any computable function must be computable by a TUring machine, decidability is tested on Turing machines (not physically of course; TM is just an algorithm). That's how they come into play. Both conjectures can be coded into a finite string and recursively solved, but we don't know if the machine will ever halt or not. There's another set of propositions, however, like the continuim hypothesis, which can't be even coded into a recursive language. I mentioned that the Continuim hypothesis was showed to be consistent with the axiomatic set theory by Godel, and not consistent by Cohen. Unlike Godel's G statement, this one is neither true or false. The assertion and the negation of the proposition is consistent within the same formal system. There's no computational way to reduce the problem to a yes or no answer. And that was my point. Paradoxes like The Liar, or examples of contradictions with materialism are examples of Godel's G statements - they're not provable, but they're true (or false). All they're indicating is that we need a meta system to prove them. BUt they're not making this profound claim about the mystery of the Universe. The undecidable statements of the Cont. Hypoth. might, but I was saying that not all metaphisical statements are of that kind, which seemed to be your claim. Besides, I'm not even sure I could agree even if we stayed within the context of G statements. I still don't understand what is unprovable (in a formal sense of course) about the statement "there are ghosts in my house"?

Thanks,

Pavel.

 

[Canute]

Hmmm, interesting, I'm not sure about it, but I'll keep my mind open. Do you have any reference I can read on how we can infer a meta system that transcends our level of consciousness?

No I afraid I don't. But it seems to me that if every formal axiomatic system has within it theorems which do not have a truth value relative to the axioms of that system, but can be decided from outside the system by extending the axiom-set, then all for every formal axiomatic system there is a metasystem. Does that seem incorrect to you?

See, to me, Godel's meta systems or any other formal systems in arithmetic are within our level of abstraction or complexity, if you will, just like a 2D space is a subset of a 11D space. That's what allows us to transcend them. All Godel did was to prove there are truths that can be proved only from a meta level. There are two assumptions: the system is consistent, and that it's formal. Such system, he proved, is incomplete. You can't assume that about the system on which our consciousness operates.

I'm not asssuming that. I'm assuming that it applies to all possible systems of formal reasoning based on our usual laws of formal logic.

To put it yet in other words, how do you know, that WE are not at the root of the hierarchy? I don't think Godel proves we are not.

But I do think we are at the root of the hierarchy, and feel that Godel proved it. What I'm suggesting is that for any systems of reasoning there is a consciousness within which the system of reasoning exists, and which is capable of deciding questions that cannot be decided within any system of reasoning.

On the same note, I have a problem with the classical agnosticism claiming "we can't know if God exists". I don't have a problem if you say "I don't know if He exists". But when you say "can't", that's a heck of a claim! It's a claim about the reality you're claiming you can't know. Kant said the knoweldge of God, or noumena is unknowable, hidden from the scientific inquiry. If it's unknowable, there's no sense in talking about it, you're talking about something you don't know. (isn't that what "talking out of your butt" means?) How much sense is that making?

Makes sense to me. What is transcendent is certainly hidden from scientific enquiry, since science studies appearances and phenomena. It is also hidden from reason and our senses, as Kant concluded. However he was quite wrong to conclude that the transendent cannot be not be known becuase of this. He showed only that the noumenal and the transcendent has to be known non-conceptually if at all.

WIth all due respect, I think that's highly subjective to personal experience, as my transcendental experience can take me in totally different place than yours, if you know what I mean

I wasn't really making a claim about my personal experience. I was just saying, if I remember right, that in principle at least it is possible to know things that cannot be known by reason, and thus transcend Kant's notion of the limits to knowing (or Goedel's notion of the limits to deciding) .

 

[Canute]

Ok, I'll try to be brief this time and take yet another, more general approach. I had a debate with my buddy a couple of years ago (which led me to study Godel in more detail) about our logic. I was trying to convince him that we're stuck with a binary logic, we don't have a choice. Speaking about other logics ultimately brings the same question to the table "but is that logic true"? Denying or questioning our fundamental axioms of logic is meaningless, as in doing so, you have no other tool to do it but the very axioms in question... You get the idea.

I do get the idea, and I agree. This is the point really, logic has to be transcended in order to attain certain knowledge. All knowledge gained though logic and reason is relative and uncertain. This is no more than Aristotle said when he wrote that 'true knowledge is identical with its object', or words to that effect. Thus knowledge is gained by 'becoming', not by formally logical reasoning. In this Aristotle anticipated Godel.

He was arguing there are propositions that don't have a boolean yes or no answer to them, they're undecidable. I said wait a second, you can't be talking about Godel, because while we can't prove the statement to be true, it's ontological status, if you will, is still true or false, we just don't know which one it is, as there's no mechanical way to prove it.

This is tricky because like you I'm no pro. However in my layman's opinion you've slightly misunderstood Godel. If a statement is undecidable it does not have a truth value within the system. Of course you can say that it does have a truth-value within some other system, but you cannot say that the ontological status of such statements is true or false. If you change your axiom-set in order to demonstrate a proof of the statement's truth or falsity then it is different statement, since you have derived it from a different axiom-set.

That is, the statement in the original system would say 'this statement does not have a truth-value within this system', whereas the new statement would be 'this statement does not have a truth-value within that system', and as such it becomes decidable. A precisely equivalent statement would be undecidable in the new system.

That's why I mentioned Golbach's and Fermat's conjectures. There might not be a computational way to prove or disprove the conjectures, they're still either true or false.

Yes, but this is a pragmatic issue relating to these particular conjectures. We do not yet know whether they are undecidable or not. However we are talking here about statements which we can formally prove are undecidable.

Paradoxes like The Liar, or examples of contradictions with materialism are examples of Godel's G statements - they're not provable, but they're true (or false).

Again I disagree. This is partly for the reasons given above. The statement 'this sentence is not a theorem of T' is not decidable within any formal system T. Both answers give rise to contradictions. It is decidable only by creating a system that encompasses T, but is not T. Let's call this expanded system U. In U we can decide the statement 'this sentence is not a theorem of T', but we still cannot decide one that says 'this sentence is not a theorem of U'. So a statement that says of itself that is not a theorem within any formal axiomatic sytem' is undecidable full stop.

Statements can be decided only be reference to ones axioms, and thus can be only relatively proved. So no statements have the 'ontological status' of being true or false. Statements are derived from axioms, and, as Godel showed, we can never prove that our axioms are self-consistent. In an absolute sense there is no such thing as a logically-demonstrable true or false statement.

On the continuum hypothesis it seems to me that there is a fundamental difference between its undecidability and that of a G-sentence. The C.H. is undecidable because neither its truth or falsity contradicts the axioms of set theory. But for a G-sentence both its truth and falsity do contradict the axioms. The two situations do not seem to be equivalent.

All they're indicating is that we need a meta system to prove them. BUt they're not making this profound claim about the mystery of the Universe.

No, but consider, metaphysical questions have the characteristic that their answers contradict reason. This is why they are undecidable. They have been found to be undecidable in all the systems of reasoning tried out by all western philsophers, however they have chosen to axiomatise their formal reasoning systems. Here it is not a question of extending ones axioms, that has been tried many times with no success.

Because these questions arise in all systems of reasoning they must have the 'ontological status' of being undecidable. If so then it would suggest that their (reasonable) answers are neither true nor false, or rather, they do not have reasonable answers, and the reason for this may be that 'transcendent reality' is non-dual, and thus impossible to represent truthfully within any formal system of reasoning, as Taoists et al assert. Perhaps this cannot be proved, but if it were true it would at least be consistent with the facts, and it would explain why metaphysical questions are undecidable.

The undecidable statements of the Cont. Hypoth. might, but I was saying that not all metaphisical statements are of that kind, which seemed to be your claim.

Not quite. I see the CH as a different case.

Besides, I'm not even sure I could agree even if we stayed within the context of G statements. I still don't understand what is unprovable (in a formal sense of course) about the statement "there are ghosts in my house"?

That doesn't seem undecidable to me either.

Btw I'm finding this discussion very useful, but if my posts are too long just tell me and I'll cut them down.

 

[Pavel]

No I afraid I don't. But it seems to me that if every formal axiomatic system has within it theorems which do not have a truth value relative to the axioms of that system, but can be decided from outside the system by extending the axiom-set, then all for every formal axiomatic system there is a metasystem. Does that seem incorrect to you?

The statement 'this sentence is not a theorem of T' is not decidable within any formal system T. Both answers give rise to contradictions. It is decidable only by creating a system that encompasses T, but is not T. Let's call this expanded system U. In U we can decide the statement 'this sentence is not a theorem of T', but we still cannot decide one that says 'this sentence is not a theorem of U'. So a statement that says of itself that is not a theorem within any formal axiomatic sytem' is undecidable full stop.

100% agree. I’m afraid I didn’t communicate my point well enough then. The above holds true for a formal, as you said, system. But there’s another premise – consistent system. I believe that, because of the undecidable statements like the Continuum Hypothesis, our own system [that we employ in evaluating simple formal arithmetic systems] is not consistent, and far from being formal. Again, as I was trying make this clear, Godel’s statements are formalizable, translatable, or computable, however you want to say it. That is, they’re legitimate true/false statements in our systems, we just can’t prove them to be one way or the other! They are theorems, but not provable by the axioms of the system. However, there’s another class, what I call undecidable, is the statements that can’t be even formalized. Godel would not be able to map such a statement into his arithmetic. The CH is such an example. Another example is determination of halting of any Turing machine on any input (Halting Problem dealing with infinity of instance problems on the input to the UTM)) You can’t even formalize them to determine whether you can prove them or not, whether they’re examples of G statement or not. I believe that is precisely the reason Godel simply made the CH an axiom and showed that it plays well with other axioms, thus preserving the consistency of the system. But that is, of course, the way I see it. So, to continue with your line of thought, I’d like you to demonstrate to me that our system that we use to have this very discourse is consistent and formal, just like an arithmetic system that Godel proved to be incomplete. If you successfully demonstrate it to me, then by Godel's theorem, I’ll completely agree with you – we can infer a meta-system that transcends our own consciousness, or the level of its complexity. I understand it’s not an easy task, so if you can't, we’ll just have to agree on reaching an impasse and leaving it simply as a matter of personal opinion.

See, to me, Godel's meta systems or any other formal systems in arithmetic are within our level of abstraction or complexity, if you will, just like a 2D space is a subset of a 11D space. That's what allows us to transcend them. All Godel did was to prove there are truths that can be proved only from a meta level. There are two assumptions: the system is consistent, and that it's formal. Such system, he proved, is incomplete. You can't assume that about the system on which our consciousness operates.

I'm not asssuming that. I'm assuming that it applies to all possible systems of formal reasoning based on our usual laws of formal logic.

Well, I get an impression you are assuming that all possible systems of formal reasoning based on our usual laws of formal logic are formal and consistent. That’s how you try to infer a meta system with the help of Godel’s theorem. If they are not, then all bets are off, why do you even bring Godel? The incompleteness theorem deals with consistent and formal systems only. That is really important!

But I do think we are at the root of the hierarchy, and feel that Godel proved it. What I'm suggesting is that for any systems of reasoning there is a consciousness within which the system of reasoning exists, and which is capable of deciding questions that cannot be decided within any system of reasoning.

OK, now I’m getting confused. What I meant by the root of the hierarchy is that our consciousness is final, there’s no meta system that transcends it, the one you’re trying to infer. I’m not sure I see what you mean by “there’s a consciousness within a reasoning system..” which contains a reasoning system in itself?? You have a “total” reasoning system containing subreasoning systems? Where do we fall? Am I on the level of total system? Can you please elaborate a little?

I was just saying, if I remember right, that in principle at least it is possible to know things that cannot be known by reason, and thus transcend Kant's notion of the limits to knowing (or Goedel's notion of the limits to deciding)

See, that’s exactly what I’m talking about. You’re reasoning about things you claim you can’t reason about. How can you assign, even in principle, these properties to an object which is hidden from your reason? It’s meaningless, don’t you think? It's one thing to try to infer a meta system via Godel's theorem (what you're trying to do), but it's totally something else to be assigning properties to it. Or perhaps I'm putting too much of a functional value into your notion of "knowing". Perhaps an example on your part might help

I do get the idea, and I agree. This is the point really, logic has to be transcended in order to attain certain knowledge. All knowledge gained though logic and reason is relative and uncertain.

There you go again, jumping out of the system. If your knowledge gained through logic and reason is relative and uncertain then what about your very claim itself? How did you come to transcend and “see” that our logic and reason is relative and uncertain, what else did you use to come to this conclusion? Please be specific. Because if you used logic and reason, then to believe you, I need to conclude that what you told me is also relative and uncertain. This seems to me like an obvious fallacy, what is it that I don’t understand here?!

This is tricky because like you I'm no pro. However in my layman's opinion you've slightly misunderstood Godel. If a statement is undecidable it does not have a truth value within the system. Of course you can say that it does have a truth-value within some other system, but you cannot say that the ontological status of such statements is true or false. If you change your axiom-set in order to demonstrate a proof of the statement's truth or falsity then it is different statement, since you have derived it from a different axiom-set.

That is, the statement in the original system would say 'this statement does not have a truth-value within this system', whereas the new statement would be 'this statement does not have a truth-value within that system', and as such it becomes decidable. A precisely equivalent statement would be undecidable in the new system.

I know exactly what you’re saying, but just like you said, I think you misunderstood Godel. I don’t mean to pull some kind of “argument from authority”, but I think you’d change your mind if you went through the proof itself, or at least read a close interpretation of it. There are numbers on the real number line that do exist, yet incomputable! In fact, there’s an uncountable infinity of them. Square root of 2 is an example. There is no recursive way to solve the number. All we can do is brute force it and find more digits in the decimal. But just because we can't mechanically compute it, it doesn't mean it doesn't exist. In fact, it gave a big headache to the Pythagorians because they couldn't express it as a rational number. They knew the number existed (by the Pythagorian theorem) but they couldn't figure out how to compute it. And as far as computational devices are concerned, such as computers and calculators, they use a computable number that is an aproximation to the square root of 2. Anyhow, this was all known way before Godel and if you want to dig in it, read about Canter sets, diagonilization method (mapping rational numbers to real numbers), and of course, the Cont. Hypothesis. Godel mapped the number theory into logical propositions, that’s the genius of his work, and showed that just like there are numbers that can’t be computed, there are statements that can’t be proven. These statements are theorems. In other words, they are true! But you can’t prove them to be true with the axioms given, just like the set theory can’t map to certain numbers with axioms within the set theory. It’s worth repeating that Godel’s statements are theorems, meaning they are true propositions about the system, just like axioms.
Axioms are true by definition, we stipulate them to be true, they’re self evident and atomic truths. We then deduce theorems from them, which are also necessarily true. How do we deduce? By rules of transformation, which are also axioms, they’re stipulated, but they operate on other axioms. (I know you agree so far, I’m reviewing this to make sure we’re still on common grounds here). These explicit definitions make the system formal. Moreover, the axioms have to play well with each other, i.e, not contradict each other, which makes the system consistent. Godel showed, that given consistency and formality of any powerful enough system, there will be theorems in that system that can’t be deduced from the axioms of that system, but those are nevertheless true statements about the system. In order to understand why it’s a theorem, and not some incoherent “square circle”, you really need to look at his proof. He constructs a wff in predicate logic that shows there is a necessary relationship between two numbers but that relationship is not an axiom in the system. Which is also why I disagree with your following comment:

Statements can be decided only be reference to ones axioms, and thus can be only relatively proved. So no statements have the 'ontological status' of being true or false. Statements are derived from axioms, and, as Godel showed, we can never prove that our axioms are self-consistent. In an absolute sense there is no such thing as a logically-demonstrable true or false statement.

So, yes, there is such a thing as a true statement that you can’t prove to be true by the axioms and rules of transformation of a any [powerful enough] formal and consistent system. That is exactly what Godel’s theorem is all about !

Yes, but this is a pragmatic issue relating to these particular conjectures. We do not yet know whether they are undecidable or not. However we are talking here about statements which we can formally prove are undecidable.

Again, this is because we mean different things by “undecidable”. I say we don’t know if Goldbach’s conjecture can be proven or not, but we do know it’s either true or false. The class of what I call “undecidable” is the one in which neither true or false status can be assigned to a statement, yet the statement is consistent within our two value logic system. The CH is an example.

On the continuum hypothesis it seems to me that there is a fundamental difference between its undecidability and that of a G-sentence. The C.H. is undecidable because neither its truth or falsity contradicts the axioms of set theory. But for a G-sentence both its truth and falsity do contradict the axioms. The two situations do not seem to be equivalent.

Exactly, and that’s my point. They’re different classes of statements, but you keep calling both of them “undecidable”.
G-sentence’s truth (or falsity) does NOT contradict the axioms. If the statements contradicted them, the system would be inconsistent. Again, that’s how the theorem reads - in a consistent system, there are unprovable truths, which make the system incomplete (you don’t have enough to prove its own truths) If it was your way, it would read “there are statements that render the system inconsistent”! That’s not the case. The G-statements truths or falsities are consistent within a system, and can’t be both true and false at the same time either, like the CH.

Besides, I'm not even sure I could agree even if we stayed within the context of G statements. I still don't understand what is unprovable (in a formal sense of course) about the statement "there are ghosts in my house"?

That doesn't seem undecidable to me either.

Well, but that’s a metaphysical statement, isn’t it? Ghosts in a sense of physically impossible to detect beings that explain what I perceive to be weird behavior of some objects in my house. I thought you suggested that all metaphysical statements are undecidable. But the “there are ghosts in my house” statement is not. What did I miss?

Btw I'm finding this discussion very useful, but if my posts are too long just tell me and I'll cut them down

Ha! I think I just beat your record for the longest post. But seriously, as long we don’t branch off and start talking about 10 different things at the same time, I finid it a productive discussion as well. It seems like we know where we disagree, and that’s a progress!

Pavel.

 

[Pavel]

I reread my comments after posting them and realized I misspoke in the first section:

I believe that, because of the undecidable statements like the Continuum Hypothesis, our own system [that we employ in evaluating simple formal arithmetic systems] is not consistent, and far from being formal.

duh. I didn’t mean to make the CH an example of a system being inconsistent. In fact, the opposite is true – the CH is an example of the system being consistent, yet the CH itself being neither true or false, hence undecidable. I need to think more carefully about examples of our own natural language, but I’m quite certain that, at least as far as formality is concerned, nobody yet translated our own natural language system into a formal language. So, I still stand by the claim that in order to show that our own natural system has a meta system (by the incompleteness theorem), you have to demonstrate that it is formal and consistent! Otherwise, the argument for the meta system that transcends our cosncousness doesn't hold water.

Pavel.