Marxism and Logic and Science and Comic Books
From: "Justin Schwartz" <jkschw at hotmail.com>
Subject: Re:>Sorry, the question is whether Hegel's Logic is consistent. See
>Bertrand Russell and the Brit rejection of Hegelianism on that one.
>Completeness is out for any system of logic that incorporates simple
>arithmetic.
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>CB: Isn't Hegel famous for Logic with contradiction at the center ? Did
>Russell really solve his paradox ?
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Goedel's incompleteness theorum has nothing to do with Hegel or contradiction.
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CB: I was talking about the reference to Hegel's Logic being _consistent_, "Inconsistency" is often a synonym for contradiction. Wouldn't be surprising if Hegel's Logic was "inconsistent" in some sense.
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The theorem states that in any formal that is powerful enough to formulate simple arithemetic, there is at least one true proposition that cannot be proved within that system. The theorem shows that arithmetic cannot be reduced to logic.
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CB: Formal logic. And the first principle of formal logic is non-contradiction. Ergo, the fact that the most basic form of math cannot be reduced to formal logic and non-contradiction, implies that the most basic form of math is not non-contradictory. This is disturbing and interesting because if we think of anything as internally consistent and logical , it is arithmetic.
Put another way, it is contradictory to say that a proposition within the system is true , but we can't prove it true _in principle_ ( not that it is just too hard to prove it ). I mean how do you know it is true within the system if you can't prove it in the system ? To not see contradiction in Goedel's proof is uh.... I mean why would it be included in a book on Escher, and Bach's leaps in the musical scales if there isn't something paradoxical about it ?
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It does not show that logically incompatible propositions are true.
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CB: The notion the something is true but it can't be proven is certainly epistemologically paradoxical, odd. The notion of truth in math is linked to provability. It is an epistemological paradox to say we know something is true , but we can't prove it. Goedel's proof is famous because it is logically uncomfortable , disturbing.
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As for Hegel, he did not posit that they could. His notion of contradiction is not logical, but ontological. It is a misleading effect of the development of the use of words that we use the term "logic" to describe the properties of formal systems, while Hegel used it to describe metaphysics. He was concerned with the nature of reality; logic, as normally used, is not. In the context of Hegel's logic, a contradiction is a dynamic instability in a phenomenon that makes it tend to change into a phenomenon with a different nature and character. It is not an asserion that p & not-p, as an ordinary logical contradiction is. jks
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CB: Agree. Dialectical contradiction is a way of thinking about motion,movement, change, a la Engels demystification of Hegel. Contradiction is the root of qualititative change. The opposites in a contradiction struggle and one overcomes the other. Their opposition then dissolves.
The opposites are held in strict opposition in formal logic. This is important for definiteness and precision, rigor. However, these strict oppositions always breakdown, & thinking in terms of "yea yea or nay, nay" as in formal logic ain't the whole story.
>From _Anti-Duhring_
quote
To the metaphysician, things and their mental reflexes, ideas, are isolated, are to be considered one after the other and apart from each other, are objects of investigation fixed, rigid, given once for all. He thinks in absolutely irreconcilable antitheses. "His communication is 'yea, yea; nay, nay'; for whatsoever is more than these cometh of evil." [Matthew 5:37. -- Ed.] For him a thing either exists or does not exist; a thing cannot at the same time be itself and something else. Positive and negative absolutely exclude one another, cause and effect stand in a rigid antithesis one to the other.
At first sight this mode of thinking seems to us very luminous, because it is that of so-called sound common sense. Only sound common sense, respectable fellow that he is, in the homely realm of his own four walls, has very wonderful adventures directly he ventures out into the wide world of research. And the metaphysical mode of thought, justifiable and even necessary as it is in a number of domains whose extent varies according to the nature of the particular object of investigation, sooner or later reaches a limit, beyond which it becomes one-sided, restricted, abstract, lost in insoluble contradictions. In the contemplation of individual things it forgets the connection between them; in the contemplation of their existence, it forgets the beginning and end of that existence; of their repose, it forgets their motion. It cannot see the wood for the trees.
For everyday purposes we know and can say, e.g., whether an animal is alive or not. But, upon closer inquiry, we find that this is, in many cases, a very complex question, as the jurists know very well. They have cudgelled their brains in vain to discover a rational limit beyond which the killing of the child in its mother's womb is murder. It is just as impossible to determine absolutely the moment of death, for physiology proves that death is not an instantaneous momentary phenomenon, but a very protracted process.
end quote
etc, and the like
CB