Re: Universal grammar

In article <1161269148.965379.301930@xxxxxxxxxxxxxxxxxxxxxxxxxxx>, "Rob
Freeman" <groups@xxxxxxxxxxxxxxxxxxx> wrote:

Well, thanks for trying.

Personally I am not convinced it has nothing to do with Goedel's
theorem:

Is it possible to reduce all of mathematics to logic?

In principle, yes, of the part of mathematics that deals with theorems and
proofs, though there are practical problems when trying to implement it
into a computer. Working math also consists much of human cognitive
information, which is then lost, in as much it is not described in formal
logical terms. The best hope for the immediate future is though a program
that aids the human in writing proofs, proving things that might be
technical but not depending so much on cognition and is not overly
structured.

/Is it possible to
reduce all patterns to logic?

What to you mean by pattern? A language formally is just a set of valid
word (or token) sequences.

Apparently Goedel himself interpreted his proof as a demonstration of
philosophical idealism. If mathematical "truth" could not be reduced to
logic, then it must have some independent existence... He used this to
contrast himself with the logical positivists (who felt a theory only
needed to be consistent with the evidence?)

If one wants to prove consistency of a theory that admits infinities as
object, one must rely on a more powerful metamathematics. So there is no
finitistic bootstrap theory.

If nothing else, as an interpretation of evidence, the parallel with
Chomsky is amusing.

Independently of any connection with Goedel's proof, I think the
question is interesting. Perhaps you are right and it is beyond
mathematics to prove either way whether all distributions can be
described in terms of rules.

I would like to know though. My hunch is there are many which cannot.

I think this is merely a practical problem, to find a good description. To
me, the practical approach seems to be find a description where syntax and
semantics flow together. With human languages, it is hard to figure out
what might be a suitable logic description of semantics.

--
Hans Aberg
.

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