Pat & All: (01)
Just getting back from vacation and playing catch up ... (02)
Pat Hayes wrote:
>>
>> To avoid the impass resulting from the dichotomy, Feferman proposes
>> the redefinition of a formal system to an ³open ended schematic
>> axiomatic system.² He claims this redefinition is a constructive step
>> towards an ³informative, systematic account at a theoretical level of
>> how the mathematical mind works that squares with experience.²
>
> Yes. His idea was subsequently formalized and then quite rapidly shown
> to suffer from the same limitations as the simpler Goedel notion. So
> considered as an end-run past Goedel, it did not succeed. It is now
> generally accepted that all such attempts to sneak past the Goedel
> result are doomed to failure, and this topic is considered to be closed. (03)
This lecture, in late September of this year, doesn't advocate end runs
or refutations of godel. In fact, Feferman claims there are more
important issues at hand such as defining the role heuristics, analogy,
metaphor. intuition and visualization to develop a systematic account of
how the mathematical mind works. So, by criticizing Feferman's call for
a new definition of formal systems on the grounds that it doesn't refute
Godel, seems at best an incomplete criticism. Could you provide a
reference to the formalization you mention above and its criticism ?
That couldn't have happened since September, so it seems there's some
history for me to catch up on. (04)
> However, none of this has any bearing on AI, contrary to what Penrose
> and others (including Goedel) believe. For a fairly thorough refutation
> of the Goedel-based attack on AI, see: LaForte, Hayes & Ford (1998)
> "Why Godel's theorem cannot refute computationalism" Artificial
> Intelligence 104 (1998) 265-286
> (05)
Many thanks, I'll definitely check this out. (06)
>
>> Seems to me that efforts like <http://cl.tamu.edu/>Common Logic and
>> <http://www.w3.org/TR/lbase/>LBase would either have to a) be defined
>> within this type of an open ended system, letıs say as the natural
>> language description of the constraints to which the axioms that
>> make up the theory of such a system would adhere; or b) evolve
>> into an open ended system that exhibits characteristics
>> of transformation across languages, logics, models and theories.
>
> No, neither. Common logic is simply a modernized Net-savvy restatement
> of first-order logic, in an attempt to get past the interoperability
> problems arising from the huge variety of surface notations in use. (07)
But Feferman's talking about openness of language and you're saying
surface notation. What's the difference? I've lumped in
non-monotonicity, model theories and axiomatic semantics. These might be
a stretch at the empirical level, not to discount the good work already
done by the folks a Kestrel on Specware, and given the solid work of
Barwise, Goguen, Kent & others not so much news at the theoretical level. (08)
> LBase is (was, better, as it seems to have been widely ignored) an
> earlier attempt to do this for the W3C family of semantic web languages.
> Goedel's incompleteness result, which gave such a shock to foundations
> of mathematics, has no relevance to the completeness of first-order
> logic (which was also first proved by Goedel, by the way.) (09)
Interesting, thanks for the info, anything you could refer me to so I
can read up on this ? (010)
>
> Pat
>
>>
>> Rick
>>
>>
>>
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> (011)
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