From: ptaylor-AT-trombone.demon.co.uk (Paul Taylor) Subject: Re: Godel's Incompleteness Theorem Date: Sun, 25 Apr 1999 18:53:37 +0000 Ken Mackendrick wrote: >Godel has two main incompleteness theorems. Taken >together they demonstrate that except for trivial (finite) cases, >undecidable propositions exist in mathematics and that the >consistency of formal systems is never provable within these >same systems. What these theorems affect - is the >possibility of that one can have rigorous hypothetical >deductive knowledge in outside of the trivial. So 2 +2 = 4 is >not a problem, since it resides within a closed system. I'm not sure that these are the implications. I hesitate to mention his name here in Lacania (!?), but Sokal has a concise description of the theorems: "Godel's first theorem exhibits a proposition that is neither provable nor refutable in the given formal system, provided that this system is consistent. (One may nevertheless see, using reasoning that cannot be formalized within the system, that this proposition is true.) Godel's second theorem asserts that, if the system is consistent, it is impossible to prove this property by means that can be formalized within the system itself." (Intellectual Impostures, p.45) It does not seem to follow from the theorems that it is impossible to have rigorous deductive knowledge. Hence my remark that Godel did not abolish mathematics. 1. It is not that mathematics is littered with propositions that are altogether undecidable, rather that it is possible to generate propositions that are undecidable within the same formal system. They may be decidable by other means. 2. It is not that a formal system's consistency cannot be determined: the consistency is a premise already. It is just that the proof of the consistency would have to be sought in another system. >Now maybe I'm completely out to lunch, but it seems to me >that this can be translated into Hegel's critique of Kant. >Hegel's dialectic confronts Kant's formalism on the level of >radical contingency. What Hegel demonstrates, if you read >him through a theory of ideology, is that the act of naming >something is also a retroactive choice. In other words, a >justification is always an 'after the fact of' explanation (an >action might appear to be valid only insofar as one constitutes >it as valid after the fact). There is no deduction or induction >here - simply the imposition of meaning on something, x, that >is. And this is a highly tautological process. The gap here is >an expression of a closed system. The critique always >stands in favour of contingency conceived of as radical. > >I really stand to be corrected about this. > >Anticipating embarrassment. Well, I'm embarrassed to admit that I haven't yet absorbed and digested Kant's Critique of Pure Reason and Hegel's Logic(s), so I can't say much. All I can say is that bringing in Godel may be bad for your health. Sokal (p.168) quotes Debray as admitting that "Godelitis is a widespread disease" and that "extrapolating a scientific result, and generalizing it outside of its specific field of relevance, can lead . . . to gross errors." (Debray himself had been previously "guilty".) Anticipating chastisement. Paul Taylor
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