Why Are Metaphysical Questions Undecidable?

[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.

 

[Iacchus32]

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.

Yes, context is crucial to understanding anything. You take anything out of context then you have a misnomer, which are really all these paradoxes are as far I'm concerned.

 

[loseyourname]

I think we're running into the problem here of whether or not there exists any such thing as an absolute, foundational context.

 

[Canute]

I agree about the need for an absolute context. To me this is the real issue here. If the universe, (by 'universe' I mean everything, the Cosmos if you like) is, as scientists and mathematicians assert, representable symbolically as a formally consistent and complete system of terms and theorems, or let's say as a reasonable and complete 'explanation of everything', then this would contradict the incompleteness theorem. Also, if it is the case that the universe is representable as a formal axiomatic system then the universe has a meta-system, something that must always be beyond reasonable explanation, something that is not in the system at all, but which contains it, or which constitutes its environment. Unless that metasystem exists then the universe cannot be represented as a formal axiomatic system.

There's an interesting essay by Stephen Hawking online somewhere called 'The End of Physics' in which he ponders this topic. In the end he just ducks the issue.

By 'foundational context' I take you to mean the level at which we actually decide questions. According to Godel there is no such level. We cannot formally prove that the axioms of any formal system (sufficiently complex etc.) are self-consistent. Therefore all questions are ultimately undecidable. Any search for a foundational level leads to an infinite regress of metasystems, each one examing the one before it.

Yet somehow we decide. It seems to me that the fact that we can decide shows that there is more to deciding than a formally logical process. I see this as being true epistemilogically, in that formal logic cannot decide a question completely so that to decide a question is to transcend logic, and true ontologically, in that it tells us something about the mechanistic processes in our brains. If the physical processes in our brain correlate precisely to our conscious processes when we are deciding questions, then either our brain is not operating according to a formally consistent set of deterministic rules, or we are not deciding, we are just guessing.

 

[Canute]

Pavel

I agree with most of you first few paragraphs, and I think I see what you're saying better now. I think that you're wrong to say that a G-sentence has a truth value in the system, even though we cannot know what it is, but I might be wrong. Either way, it doesn't seem to affect the main issue here.

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.

Hmm. I'm not saying that the language of our discourse or the reasoning behind it is consistent and formal. Rather, I'm saying that if we try to construct a formal and consistent theory, explanation, metaphor, description, account, picture or whatever of the universe then we cannot complete it.

Now this is usually taken to be an epistemilogical issue, some odd quirk of our formal systems of symbols and rules that prevents us from completing them consistently, and which has no implications for the nature of reality. If this is so then we will never be able to fully understand the nature of reality by reason alone. But it could also be an ontological matter. That is, it could be the case that the universe cannot be fully represented by a formally consistent theory, explanation, mataphor, description etc. If this is the case then we still cannot fully understand the nature of the universe by formal reasoning alone.

I am suggesting that we cannot represent the universe symbolically in a formally consistent and complete way for both of these reasons. In other words, I'm saying that to explain the universe completely requires that our explanation has an undefined term in it, a term standing for 'something' about which no question is decidable, a theorem that is not really inside the system. Equivalently a term that refers to something outside of the system. This is the thing that has to be left out of any symbolic representation of the universe. Inevitably formal systems require undefined terms.

Because we cannot conceive of a thing that cannot be defined and about which no question can be decided we cannot even conceive of this 'something'. There is no way that we can conceive of it except by misconceiving it, since a concept is a definiton, and for strictly ontological reasons this ultimate 'something' is indefinable.

This is very roughly the 'non-dual' view of cosmology, in which 'Unicity', the 'Tao', 'emptiness', and so on cannot be defined, represented, conceived, imagined etc. Christian mystics likewise assert that the Godhead must be approached non-conceptually, for it is formless.

Of course I cannot prove that this is the case. If I could demonstrate a formal proof of it then I would have proved that it is not case, for to prove it is the case would require that this ultimate 'something' be symbolised in a manner consistent with two-value logic, which would contradict the proof that it isn't.

However the empirical evidence, for instance the fact that in the opinion of most philsophers there are in principle explanatory gaps in our formally consistent explanations of the origins of the universe, the origins of consciousness and the ontology of matter, and also, crucially, the fact that all questions about what is ultimate (i.e. all metaphysical questions) are undecidable, suggests that it is the most plausible explanation of the existence and nature of the universe.

If it is not the case then it seems to become impossible for us explain why we are unable to construct a reasonable explanation of our existence. We would have to say that this inability was down to some anomaly of our methods of reasoning. But what other method of reasoning is there?

I haven't put that very well. I'm still trying to figure out a straightforward way of saying some of these things. One question worth considering is why, while masters of Advaita, Buddhism, Taoism, etc have long claimed it is possible to know everything, these same masters were completely unsurprised and unruffled by Godel's proof that nothing can be completely known by reason, and that questions (of any kind) can only be answered with certainty from the metasystem. It's what they've been saying for millenia.

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.

No I'm not assuming that. As far as our reasoning systems go I'd say that insofar as they are formal (by our usual definition) they obey the rules of Boolean logic (which was designed to model formal reasoning). But whether they are consistent is an empirical question. All we can say is that if our formally reasoned explanation of everything is consistent then it is not complete, and if it is complete then it is not consistent. (Again, note that Buddhists have said for millenia that there is no such thing as a formally consistent and complete account of reality).

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!

This is a misunderstanding. I'm saying that metaphysical questions are undecidable because it is impossible to represent what is ultimate symbolically, or symbolise it as a true or false theorem within some formal axiomatic system (because such systems are predicated on the idea that all well-formed theorems are either true or false, i.e that all terms can be defined as being this or that).

If I assume that our theories of reality are formal and consistent it is only because I'm assuming that this is the sort of theory that we are trying to construct. If a theory is not formally consistent then yes, all bets are off. But we needn't consider such systems, they are by definition unreasonable and incable of explaining anything.

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.

No, I'm saying the same, that consciousness is the metasystem.

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?

Yes, I'm ok at elaborating, it's clarifying that I have trouble with.

I was saying that the only place a reasoning system can exist is in the mind of a sentient being. (I take reasoning to mean something slightly different to computation). So all formal axiomatic systems exist within an encompassing consciousness. Godel proved this by showing that to decide an undecidable question we must appeal to an infinite regress of extended systems, and it follows from this that in the last analysis we have to decide undecidable questions informally, for if there is an infinite regress of systems then there is no point at which a question can be decided formally. Ultimately we have to decide them informally from the metasystem, a.k.a. our consciousness.

See, that’s exactly what I’m talking about. You’re reasoning about things you claim you can’t reason about.

I'm not saying that one cannot reason about it. I'm saying that we can know things which we cannot know by reasoning alone. We can know what a clarinet sounds like, for instance, which is unknowable by reason alone. Direct experience transcends reason. But we can nevertheless reason about the sound of a clarinet.

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

No amount of reasoning will enable a person to know what a clarinet sounds like, but this does not mean that we cannot reason about the sound of a clarinet. Equivalently while Lao-Tsu says "The Tao that can be talked is not the eternal Tao", he also says "The Tao must be talked". It is simply necassary to be very careful when doing so not to define the term incorrectly, and to remember that when we discuss what it, say the 'Tao', really is we cannot do it, for it must remain an undefined term. Again, we cannot represent the sound of a clarinet within a formal system, it is an experience, and experiences are incommunicable, indefinable, incommensurable etc. beyond a certain point.

If I remember right the only qualities I'm assigning to what is ultimate, the ultimate metasystem if you like, is one of indefinability and non-duality (no dual properties). I'm also suggesting that this is directly connected to our inability to give a scientific definition of consciousness.

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.

This isn't quite right IMO. Godel showed that it is possible to prove that all knowledge gained through reasoning is relative and uncertain. However he proved this with certainty. This is possible because what he proved is true of all formal systems in all possible universes. The reason his proof holds is that the proof is not dependent on any particular set of axioms, but holds for all formal axiomatic systems whatever the axioms. (This is roughly what I meant by saying he went outside of his logical system to construct his proof). So although reason cannot bring certainty, we can be certain by reason that it doesn't. We can be certain because this proof is about reasoning itself, not what we are reasoning about.

To put this another way, the truth or falsity of any statement is dependent on the axioms of the system within which the statement is formed. A true statement in one set of systems will be false in a system differently axiomatised. But a statement that is true independent of any axioms, that is, true in all formal systems, escapes this relativity.

But this isn't the whole answer. Direct experience is the other issue.

INTERLUDE

"In the standard positivist approach to the philosophy of science, physical theories live rent free in a Platonic heaven of ideal mathematical models. That is, a model can be arbitrarily detailed, and can contain an arbitrary amount of information, without affecting the universes they describe. But we are not angels, who view the universe from the outside. Instead we and our models are both part of the universe we are describing. Thus a physical theory is self-referencing, like in Gödel’s theorem. One might therefore expect it to be either inconsistent, or incomplete"

Stephen Hawking
‘Gödel and The End of Physics’

"…since every word in a dictionary is defined in terms of another word… The only way to avoid circular reasoning in a finite language would be to include some undefined terms in the dictionary. Today we must realize that mathematical systems too, must include undefined terms, and seek to include the minimum number necessary for the system to make sense."

Leonard Mlodinow
‘Euclid’s Window’

"When we encounter the Void, we feel that it is primordial emptiness of cosmic proportions and relevance. We become pure consciousness aware of this absolute nothingness; however, at the same time, we have a strange paradoxical sense of its essential fullness. This cosmic vacuum is also a plenum, since nothing seems to be missing in it. While it does not contain in a concrete manifest form, it seems to comprise all of existence in a potential form. In this paradoxical way, we can transcend the usual dichotomy between emptiness and form, or existence and non-existence. However, the possibility of such a resolution cannot be adequately conveyed in words; it has to be experienced to be understood."

Stanislav Grof
The Cosmic Game
State University of New York (1998)

I'm going to miss a chunk of your post out here, because I think it's dealt with in amongst the other issues.

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 !

Again I'm afraid I disagree. If a statement is undecidable then it does not have a truth-value within the sytem. Of course it has one in some other system, but saying it is true or false in some other system doesn't alter the fact that it is neither true or false in the original system. Similarly, a statement that has a truth-value in one system may be undecidable in some other system. Metaphysical questions have been found to be undecidable in all formal systems, and as such are 'meta-undecidable'. They have not yet been shown to have a truth-value in any formal system whatever the axioms. I would argue strongly that they do not, since they are improper questions, equivalent to one that asks 'Is the moon made out of Chedder or Stilton?'

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.

Yes, I agree. But this misses out those question which are not of either of these types. This is why I see the CH and Goldbach's conjecture as not directly relevant. When we ask 'Did the universe arise from something or nothing?' it is a question that we can demonstrate formally to have no non-contradictory answer. In other words such questions do contradict our two-value logic system, and we know that they do not have a true or false answer within any formal system of reasoning.

G-sentence’s truth (or falsity) does NOT contradict the axioms.

But surely they do. Isn't it precisely the fact that both answers give rise to contradictions that makes them undecidable? If a G-sentence was found to be true or false within the system this would contradict the axioms of the system (or its rules, which is the same thing).

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.

I'm afraid I still can't you see how you arrive at that conclusion. If the truth or falsity of a G-sentence is consistent with the axioms of the system then it is not a G-sentence.

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?

I would say that 'Are there ghosts in my house' is not a metaphysical question. In fact, if one believes that ghosts do not exist then it is probably not even a question, for the term 'ghosts' refers to something non-existent. Also, it is a pragmatic matter, for if we can detect the presence of ghosts by means of our senses then ghosts are not metaphysical entities.

Crucially, the answer to the ghost question might be yes or no without contradicting the laws of formal reasoning. But it is not possible to assign a truth value to a metaphysical questions without contradicting those laws. After all, that's why we've never been able to decide any of them.

I hope some of that makes sense. We'll have written a book by the time we've finished (albeit probably an incomprehensible one)

Regards
Canute