>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 a fallacy based on a misunderstanding. Goedel showed that arithmeric (indeed any formal system) is _incomplete_, that it contained at least one true proposition not provable within the system. That does not show that it contains as theorems two inconsistent propositions. Read Newman and Nagel, Goedel's Theorem, for a medium length exposition, or for a short one, read theentry in Edwards' or Audi's Encyclopaedia of Philosophy. In fact, it's another theorem that arithmetic is consistent, that it contains only propositions that are mutually true.
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 ).
No. We can prove the propositions that make arithemetic incomplete, just not within arithmetic. We have use set theory or some such.
>I mean how do you know it is true within the system if you can't prove it
>in the system ?
Prove it using another formal system you know to be true.
>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 ?
>
Paradoxical is not inconsistent, just odd.
>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.
Formal logic isn't anybody's idea of the whole story. Indeed, that's a theorem--Goedel's theorem to be precise.
>
>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.
>
>
I don't see how how any of this illuminates either Hegel or any other sort of claim about contradictions. The examples just show that there are borderlines cases with concepts like "alive." It doesn't how that I can be alive and not alive at the same time and in the same respect.
jks
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