Re: Universal grammar

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

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.

So you want to factor some "human" factor X out of mathematical proofs?

I find the idea that there is an indefinable "human" element in
mathematical proofs quite radical. This is something quite new, and
unique to your own work, isn't it?

Still, it is not inconsistent with Goedel's own interpretation, which
was that absolute "truth" existed, but that it must be outside of maths
itself.

I still find it ironic that people should interpret the failure to
observe something (absolute "truth") as evidence that it does still
exist, but that it exists only in some sense which is innate to humans
(or beyond.) Why are we moved to push everything we don't understand
into the "human" category? And then we bemoan the fact that we are
unable to understand humans!

In this case the mystery of the missing "human" element would cease to
exist, if once we accepted absolute "truth" does not exist.

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.

I think the question applies quite generally. We could start by
applying it to patterns defined as "sets of token sequences". If the
sequences are "valid" it is easy. The logical specification is just the
set of rules which specify "validity".

But what if the sequence is not "valid"? Another way of phrasing the
question might be--does there exist a sequence of tokens which is
non-"valid" for all possible sets of rules?

I don't know about you, but I'm starting to think "incompressible
string".

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.

It is ironic that, at least in my understanding, Goedel's proof itself
arguably guarantees that there will be more than one way of seeing
Goedel's proof!

You seem to see it as a paradox of encapsulated infinity, a necessary
incompleteness involved in the paradox of seeing something as
simultaneously finite and infinite. That is fine, and probably very
true. But my understanding emphasizes another aspect. I emphasize not
the encapsulation of infinity, but the change of perspective involved
in encapsulating infinity.

My logical gloss of the proof is that "within a pattern saying one
thing, it is simultaneously possible to have another pattern saying
another." Hence the different levels of understanding possible in the
phrase "I am a liar."

At least we agree the proof establishes "there is no finitistic
bootstrap theory." Even if you would then join Chomsky, and perhaps
Goedel, in moving the weight of missing theory to some external
supervisory element (human or beyond.)

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.

So you are saying it is just a matter of finding a good description,
what I might gloss as "the right rules."

Your position then would be that it _is_ always possible to describe a
distribution, all patterns, in terms of rules?

-Rob

.

Relevant Pages

  • Re: Cantor Confusion
    ... > If actual infinity is assumed to exist, then the second case is valid. ... of limits in mathematics are formulated in such a way that the limit ... Set theory is more basic than analysis. ... sequences of sets are in general not defined. ...
    (sci.math)
  • Re: Uncountable many reals without Cantor
    ... Some people get strange ideas about what "constructivism" should mean. ... |debate of the "actual infinity" is so central to these matters. ... mathematics we know now to past situations. ... free-choice reals, for instance, which are free choice sequences ...
    (sci.math)
  • PANORAMARE
    ... concept of infinity is indefensible - 24 messages ... should be a less interesting>>part of mathematics. ... approximation in your immediate neighbourhood. ... centric solar system is overly complicated, ...
    (sci.physics.relativity)
  • Re: Wheres respect? was Re: Corrective interpretation of real numbers
    ... then denial of the actual infinity ... Those wo are using mathematics in physics often complain about allegedly ... He imagined cardinality of standard continuum much ...
    (sci.math)
  • Re: Earth 8??
    ... The problems you're bringing up come from the fact that you're using common English words that have common English meanings that don't apply to transfinite numbers. ... By definition of _infinity_. ... lay speech, philosophy, and pretty much anywhere but pure mathematics. ... This defined mapping puts the positive integers and the even positive integers into a 1:1 correspondence. ...
    (rec.arts.comics.dc.universe)