Re: Universal grammar

In article <1161318846.624985.11180@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, "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?

For a theorem prover, not at all, this is just something one does because
it is quite impossible for a computer to interpret a human written proof.
:-) But it strikes me that it might be an interesting linguistic question
trying to let a computer interpret human written math, for the reasons
that the language used is very limited, plus that the semantics ought in
principle be quite clear.

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 this context I use to bring up a "theory of contradictions" that the
Dadaists are claimed to have made. It is typical that a theory uses its
own logic to determine its consistency. So a theory of contradictions will
have contradictions in it, but there is no way to communicate that to a
believer, as as the contradictions will be viewed as being consistent.

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.

As for math. I just see the infinities as an attempt to model reality. By
experience, we have found these theories. Also, mathematicians usually do
not work with the whole broad spectrum of math at once, but usually just a
local theory, where consistency in the large and the particular method
of constructibility is irrelevant. For example, working math rarely
indicates which axiomatic set theory it relies on.

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?

If one is just out to describe the formally correct strings of words in a
language as a logical model, that is formally easy: just list the valid
sentences. If one goes the other direction, and want all that
is compatible with human cognition, that is essentially impossible,
at least for now. Logical models using grammars etc., fall somewhere in
between.

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

I do not know what you mean by this term. - I start thinking about
computer data compression. :-)

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!

I am not sure what you have in your mind here. - You can elaborate if you
so like. :-)

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.

Formally, it just means that one needs to have the axioms for the natural
numbers. There, mathematical induction allows one to prove formulas valid
for all integers, thus giving one the ability to treat the infinite set of
numbers as a singles object. There are generalizations of this to higher
infinities, called "axiom of choice", with equivalents called
"transfinite induction", and some other variants.

That is fine, and probably very
true.

The Goedel theorem is sort of a metamathematical variation of Cantor's
proof that the set of real numbers is uncountable. Let's prove that the
set of real numbers [0, 1) = {x| 0 <= x < 1} is uncountable. Each such
number has a binary expansion 0.a_1a_2... where a_1, a_2, ... have the
values 0 or 1. Suppose now we can list all the numbers in [0, 1) in a
countable way:
  x_1 = 0.x_11x_12...
  x_2 = 0.x_21x_22...
  ...
If m = 0 (resp. 1), let m' denote 1 (resp. 0) - just change 0's into 1's
and vice versa. Then the number 0.x_11'x_22'... is not in the list, as it
disagrees with number x_k in binary expansion position k. Contradiction.
Thus, it is not possible not list the numbers of [0, 1) in a countable
way. As we agreed to view [0, 1) as an object: a set, it must be an
uncountable set.

Goedel made his proof by introducing numbering of formulas and proofs, and
somehow got an explicit formula for a statement, where it, nor its logical
negation, cannot have a proof.

But my understanding emphasizes another aspect. I emphasize not
the encapsulation of infinity, but the change of perspective involved
in encapsulating infinity.

I just view this as a mathematical modeling, be experience found useful.

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."

There is a related mathematical paradox in axiomatic set theory called
the Russell paradox. If {x| P(x)} is the set of all x such that P(x) is
true, then define
  U = {x| x not in x}
Then U in U <=> U not in U, a logical contradiction. Axiomatic set theory
resolves this by saying that U cannot be a legal set: NBG calls this a
class, and in ZF, U can not be interpreted as a set.

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.)

All one can say is that the logical foundations of mathematics have
not yet been cleared up. There is a movement, where one tries to reverse
engineer working math, in order to get closer to this problem.

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?

Well, pinning down a human language in terms of grammar and semantical
rules might be like figuring out how many trees there should be for it to
be considered a forest. :-) But there should not be a problem of logical
foundations.

--
Hans Aberg
.

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