Why Are Metaphysical Questions Undecidable?

[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