[WithoutMe]

Contradiction rules the world
-- Hegel, The Science of Logik.


Where all contradictions meet,
there is a complete man.
-- Osho.

In 1931, the mathematician Kurt Gödel demonstrated that every mathematical system (or set of axioms) contains undecidable propositions. And if the mathematical system is completely decidable, the the mathematical system is simply incomplete. In essence, Completeness & Consistency doesnot go hand in hand. A simplified outline of Godel's brilliant Incompleteness Theorem follows: -


Suppose we have an Universal Truth Machine (UTM) supposedly capable of correctly answering any question. Now consider the following proposition: "The UTM will never say that this sentence is true." Call this sentence G, that is:
G = "The UTM will never say that this sentence, G is true."


Now, what happens when we ask the UTM whether G is true?


If UTM says G is true, then "UTM will never say G is true" is false. If "UTM will never say G is true" is false, then G is false (since G = "UTM will never say G is true"). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements. We have established that UTM will never say G is true. So "UTM will never say G is true" is in fact true. So G is true (since G = "UTM will never say G is true"). We have thus produced a true statement, which UTM cannot make. UTM is not truly universal.

Thus, every self-referential negation is a paradox, or rather, such things neither can be asserted nor can be denied. No wonder, Ludwig Wittgenstein had once said in his Tractacus Logico Philosophicus, - "Whatever can be said at all, can be said clearly, but what cannot be said, must be passed over in silence, for that lies in the range of non-sense". & Godel Incompletes infact pertains to this "range of non-sense", that is, a recursive function of negation is undecidable, resulting in the non-halting of a UTM..

So, for a system is to be complete (i.e., all-in-all), it has to contain all possibilities, even that of the operation of negation, be it on itself, thus, resulting in such states which are in effect undecidable; now, those very undecidable states will make the system inconsistent. So, a complete system is always inconsistent. And to maintain consistency, the system has to be made, by leaving out precisely those self-negation states, thus, making it incomplete. This, a consistent system is always incomplete.

So, there is a tradeoff betwen Consistency & Completeness, & this ratio of one with the other has a connection with Chaitin's Constant, Omega. This puts a limit to how much we can know & infact when applied to such things as Consciouness (with the very concept of "I" or Self, as recursive) can result in remarkable & astonishing results.

Infact this negative-recursion (or a self-referential negation) is found in many other aspects, namely -- Nagarjuna's fourfold theory of logic in Buddhist Metaphysics, the Buddhist theory of Shunyata, Zeno's Paradoxes in Pre-Socraitic Greek Philosophy, the mathematics of Fractals, the Physics of Chaos Theory, the Nietzschean concept of Eternal Recurrence, the Existential concept of Nothingness (esp that of Sartre's), Escher's paintings, Zen Koans in perspective to Linguistics, the Neti-Neti approach of Vendanta (esp pertaining to Advaita), Feedback Systems in Learning Networks ( A.I.), the Big Bang Theory, the Triad Dialectic Theory of Hegel (even in the version used by Marx in his Historic Materialism), the Quantum nature of human consciousness, Computational Theory (including the study of Algorithims), the poetry of Blake, Whitman & many others, Lao Tzu's verses in "Tao Te Ching", the loopiness in Merkaba Geometry (theories relating to the Golden Mean Spiral, with connection of Fractals & Chaos Theory), the Zero-Book concept (like that of "Waiting For Godot"), etc etc.

If something is alive, it has to have a Self-contradiction in it.

Does the Self then really exist, or is it just the fabric of this negative-recursion talking?

Plz comment on this.

Kind Regards,
wM.

PS. Think of the "0-0" state of the NAND flip-flop in Electronic Digital Systems.

[Canute]

I disagree about the incompleteness theorem placing a limit on knowledge. It places a limit on what can be proved by demonstration, but true knowledge, as Aristotle put it, is identical with its object, not a matter of manipulating formal axiomatic systems.

But I agree it's an important issue for any TOE. A TOE cannot be complete and consistent. Hawking's argues physics cannot be completed for this reason (online essay 'Godel and the End of Physics). Therefore, if a complete (or fundamental) TOE is possible the universe must have an inconsistent description. The mystics claim that this is the case, and so it seems to me that theirs is the only worldview in which a complete TOE is possible.

Could you say some more about Nagarjuna's four fold logic? I know his theory of emptiness via the 'Fundamental Verses on the Middle Way' but not this.

[Mr. Nobody]

A true knowledge, identical with an object is impossible. An object is defined by observation (self observation included). For how would the object exist if no one (including the object itself) is there to identify it. What would the world look like if you have no senses at all, no touch, no smell, no sight, no hearing, no taste. How would the world communicate with you and how would you perceive this world. Where would this true knowledge come from and if it existed what would the meaning be, if as mentioned afore, no one is there to absorb it, including the object itself.
Therein lays the inconsistency for completeness

[lodestar]

The most compelling and amazing thing is the irony of truth held in Kurt Godel's theorem. He says you COMPLETELY cannot have a COMPLETE theory, unless it contradicts itself. Well speak of the devil; the theory itself contradicts itself, by pretending to be a complete theory of incompleteness. So the theory simultaneously proves AND disproves it's own conjecture. Therefore we are left with an axiom, of choice. Is the theory complete or incomplete? That is the dillemma which science faces, but really it is a saving grace to an optimist, a true believer. The axiom of choice allows us to have a complete theory, by choice. Imagine that! Godel must have just been a pessimist I think, and that's why he didn't realize his theory was simultaneously proving completeness and incompleteness, and he was supposed to choose. He never got to choose because his pessimistic assumption (that contradiction is not consistent) prevented him from seeing the choice at hand. It is this same pessimism regarding the irony of truth which has caused our teachers to label 1/0 as undefined. What I'm trying to get people to see is that it is really 0 that is undefined, whereas 1/0 is the epitome of all definition.

[Canute]

Originally Posted by Mr. Nobody A true knowledge, identical with an object is impossible. An object is defined by observation (self observation included). For how would the object exist if no one (including the object itself) is there to identify it. What would the world look like if you have no senses at all, no touch, no smell, no sight, no hearing, no taste. How would the world communicate with you and how would you perceive this world. Where would this true knowledge come from and if it existed what would the meaning be, if as mentioned afore, no one is there to absorb it, including the object itself. Therein lays the inconsistency for completeness

Wow. If you'd left out the first sentence I'm sure I would have thought you were quoting from the Upanishads. There's a passage that is astonishingly similar to this in both content and style. I can't find it at the moment but will try later. It concerns the nature of both knowledge and understanding. Who is understanding what once the understander is one with the understanding? What is there to understand except the understander?

Everything you say seems spot on to me, and extremely well put, except for your opening statement. This conclusion does not follow from your argument.

I think you are saying a lot here, all of it worth discussing, but as an objection to Aristitle you seem to be saying that true knowledge cannot be an object, and also that where knower and known are one knowledge cannot be said to exist. This would be perfectly consistent with Aristotle's assertion.

A person who holds the view that true knowledge is identical with its object must also hold the view that the distinction between objects and objects, subjects and subjects, and even between subjects and objects are ultimately, in the final anlaysis, illusory. You are right imo to say that true knowledge cannot be identical with its object if this means that knowledge must somehow become an object to be true. But nobody is suggesting this. True knowledge would be possible precisely because ultimately, at the deepest level of analysis, everything is identical. This 'Unicity' is neither an object nor a subject.

In other words, it does make sense to say that true knowledge is identical with its object, just as long as we don't mean to imply that objects and subjects are fundamentally distinct, nor that true knowledge is not, in some paradoxical way, also the absence of knowledge.


"Very few seek knowledge in this world. Mortal or immortal, few really ask. On the contrary, they try to wring from the unknown the answers they have already shaped in their own minds – justifications, explanations, forms of consolation without which they can’t go on. To really ask is to open the door to the whirlwind. The answer may annihilate the question and the questionner." (Anne Rice, The Vampire Marius, The Vampire Lestat)


The identity of subject and object spoken of by Aristotle does not entail that a subject becomes an object or an object a subject. Where subject and object become identical there is no distinction between them and thus no subject or object of which we can speak. He implies a third state which, as you point out, it would be paradoxical to call either an object or a subject. One might say that true knowledge consists in the annihilation of subject and object.

Does that make sense? Would you be ok with the idea that true knowledge consists in the identity of knower and known? This phrase avoids the subject/object problem to some extent.

For an example of knowledge by identity, or at least it seem to be one to me, consider how you know that solipsism is unfalsifiable. By solipsism I mean the proposition that nothing truly exists apart from my/your own Being or phenomenal consciousness. For unfalsifiable you could also read undecidable. How do you know it is unfalsifiable? Do you know it?

It turns out that our knowledge of the existence of our identity is more secure than our knowledge of the existence of anything else. But on analysis this is a strange kind of knowledge. When we try to work out how we know that solipsism is unfalsifiable we find that it is not entirely by reason that we know it. If we know for certain that we exist as an individual knower this can only be because it is knowledge by identity, knowledge that we cannot help but have because of what we are. In this way we can be more certain that solipsism is unfalsifiable than we can that the phenomenal universe exists. This is the strange but true fact on which the plausibility of the plot of the Matrix rests. It's a funny old world.

Canute

[Mr. Nobody]

Thanks I suppose, but I don't know Upanishad.
My philosophy is shaped by the meaning of quantum mechanics and strict adherence to measurement and a correct interpretation of the afore.
Anton Zeilinger has put it bluntly:
"A world, a reality existing independently of what can be said about it, is void of any meaning. It is obvious that any property or feature of reality "out there" can only be based on information we receive. There can not be any statement whatsoever about the world or about reality that is not based on such information. It therefore follows that the concept of a reality without at least the ability in principle to make statements about it, to obtain information about its features is devoid of any possibility of confirmation or proof. This implies that the distinction between information, that is knowledge, and reality is devoid of any meaning. In other words, it is impossible to distinguish operationally in any way reality from information. Therefore, following Occam's razor, the notion of the two being distinct should be abandoned, as the assumption of the existence of such a difference does not add anything that could not also be obtained without it"

Extend this with an understanding for the true nature of measurement and how to interpret indeterminism and follow that to wherever it leads you

[Canute]

Yes. This is why I like David Chalmer's double-aspect theory of information as the basis for a cosmology. But by the usual definition information requires an information space. As well as atoms there must also be a void.

To confuse the issue further the esoteric view is that true Reality is prior to both information/atoms and space/void. Strictly speaking nothing can be said about reality at this level, since it does not consist of information. This does not mean that reality is 'devoid of any possibility of confirmation', since true knowledge is identical with its object. What it means that any attempt to describe this underlying reality will sound mystical.

Canute