Gödel’s incompleteness theorem, consciousness and possible explanations

Is the mind the same as the body? What is consciousness? Can machines have it?

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Dimebag
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Gödel’s incompleteness theorem, consciousness and possible explanations

Post by Dimebag »

Kurt Gödel, a mathematician described a theorem which supposedly showed that:
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.
If we apply this to consciousness, and we ask ourselves, how can we ever understand consciousness, from within consciousness, this seems to me to draw upon a parallel to the second incompleteness theorem by Gödel.

Now I know that consciousness isn’t exactly an algorithm, it doesn’t consist of (to my knowledge or understanding) an effective procedure for calculating numbers. But, it seems to share a similar problem to the above description.

When we are trying to understand consciousness, we are trying to understand it from within consciousness.

But we attempt to step “outside” of consciousness, to understand it, science proceeds to explore the brain as an avenue of explanation.

But the problem is, despite what science would like us to think, science is still contained within consciousness. It may be stripped of much of the content of consciousness, but it is still undeniably contained within the abstract conceptual layer of consciousness. It does not exist in some external objective realm outside of consciousness. It may attempt to describe that hypothetical realm outside of consciousness, but, it can only ever use what is contained within consciousness to attempt to explain what’s out there in that objective realm.

So getting back to the problem of consciousness. Could our problem in trying to explain consciousness be due to the fact that, we are attempting to essentially do as Gödel has shown is impossible, to prove or explain a system with the very system we are trying to understand?
SteveKlinko
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Re: Gödel’s incompleteness theorem, consciousness and possible explanations

Post by SteveKlinko »

Dimebag wrote: Mon Nov 16, 2020 11:46 am Kurt Gödel, a mathematician described a theorem which supposedly showed that:
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.
If we apply this to consciousness, and we ask ourselves, how can we ever understand consciousness, from within consciousness, this seems to me to draw upon a parallel to the second incompleteness theorem by Gödel.

Now I know that consciousness isn’t exactly an algorithm, it doesn’t consist of (to my knowledge or understanding) an effective procedure for calculating numbers. But, it seems to share a similar problem to the above description.

When we are trying to understand consciousness, we are trying to understand it from within consciousness.

But we attempt to step “outside” of consciousness, to understand it, science proceeds to explore the brain as an avenue of explanation.

But the problem is, despite what science would like us to think, science is still contained within consciousness. It may be stripped of much of the content of consciousness, but it is still undeniably contained within the abstract conceptual layer of consciousness. It does not exist in some external objective realm outside of consciousness. It may attempt to describe that hypothetical realm outside of consciousness, but, it can only ever use what is contained within consciousness to attempt to explain what’s out there in that objective realm.

So getting back to the problem of consciousness. Could our problem in trying to explain consciousness be due to the fact that, we are attempting to essentially do as Gödel has shown is impossible, to prove or explain a system with the very system we are trying to understand?
I think the incompleteness theorem says you can't know Everything about a system of numbers. But you certainly still can know a lot about a system of numbers. So I think if the completeness theorem applied to Consciousness, we still should be able to know a lot about it, rather than the nothing that we are stuck with. I would be happy to have a first Clue even if I could not ultimately know everything about it.
Scott Mayers
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Re: Gödel’s incompleteness theorem, consciousness and possible explanations

Post by Scott Mayers »

Dimebag wrote: Mon Nov 16, 2020 11:46 am Kurt Gödel, a mathematician described a theorem which supposedly showed that:
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.
If we apply this to consciousness, and we ask ourselves, how can we ever understand consciousness, from within consciousness, this seems to me to draw upon a parallel to the second incompleteness theorem by Gödel.

Now I know that consciousness isn’t exactly an algorithm, it doesn’t consist of (to my knowledge or understanding) an effective procedure for calculating numbers. But, it seems to share a similar problem to the above description.

When we are trying to understand consciousness, we are trying to understand it from within consciousness.

But we attempt to step “outside” of consciousness, to understand it, science proceeds to explore the brain as an avenue of explanation.

But the problem is, despite what science would like us to think, science is still contained within consciousness. It may be stripped of much of the content of consciousness, but it is still undeniably contained within the abstract conceptual layer of consciousness. It does not exist in some external objective realm outside of consciousness. It may attempt to describe that hypothetical realm outside of consciousness, but, it can only ever use what is contained within consciousness to attempt to explain what’s out there in that objective realm.

So getting back to the problem of consciousness. Could our problem in trying to explain consciousness be due to the fact that, we are attempting to essentially do as Gödel has shown is impossible, to prove or explain a system with the very system we are trying to understand?
The very logic of the 'theorem' requires using a 'complete' system that can exhaust all possible problems on its domain to have a valid proof. The point of the theorem is to suggest that certain problems that are asked for a 'finite' solution to the question of whether you can prove with closure THAT some possible logic system (mathematics being a subset of it) can PROVE definitively any and all possible problems regarding logical inquiry, such as scientific ones as well as math. But if we cannot prove definitively with closure that we can, this means that there exists at least some real incapacity to prove everything, because everything, especially things we use math and logic for proof, such as scientific matters, are DEPENDENT upon this factor. As such, the theorem speaks more broadly beyond just a math.

However, given you are making a comparison to a subset of the universe, of which 'consciousness' represents, it is not comparable because you can still prove specific proofs about specific defined subsets of reality of which consciousness is contained in.

Remember though that the logic Gödel uses for the proof, though itself is complete, it may be confined to a relatively finite domain. As such, IF there is some other system OR a set of distinct but co-operating systems of reasoning that can cover a more exhaustive domain, a proof MAY exist using that system. A lot of the problem deals with our finite capacities. When dealing with infinites, things like Set Theory are needed, for instance. The particular proposed set theories themselves are not yet proven 'closed' and why they remain 'theories' and not 'theorems'. As such, we can potentially discover reasoning that may require more complexity than we expected that utilizes 'para-consistent' logic or other proposed non-consistent or conditionally consistent logics.

I can define "consciousness" as "the effect that some unity of sensation exists of something with multiple points of entanglement." This would be more broadly inclusive of non-animals but helps to understand the more specific instance of "human (defined) consciousness" (as subset of conscious things in totality). An example may be "collective set of all atoms of the Earth that contribute to the rise of the electromagnetic field surrounding the Earth", in general, that acts as a unified deflecting force of the sun's most dangerous flares, for a specific instance of behavior.
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DanDare
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Re: Gödel’s incompleteness theorem, consciousness and possible explanations

Post by DanDare »

Before you can worry about the completeness theorem applying, you need to solve the empirical problem. That is, your direct knowledge of consciousness is limited to a single observer, yourself. The evidence for other people having consciousness at all is indirect via reports, behavior and apparent similarity to yourself, and applying the idea that things are uniform. You do not know, however, if a rock has conscious experience or not, merely assume based on the previous factors that it does not.
I have always preferred Daniel Dennett's approach, simply looking at competence in behavior and at apparent comprehension. These are measurable and observable and can be reasonably allowed to "bound" the realm in which consciousness is plausibly found, so rocks are excluded. There is no way to show that these bounds are correct, so I accept them based on utility only.
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