Mainly, but not completely.
the problem with English, or any other language OTHER than the actual math
involved is:
it's not the math involved.
I would recommend perusing Lesniewski's work in the area.
More below:
On 7/9/07, Jose Mario Quintana <[EMAIL PROTECTED]> wrote:
> -----Original Message-----
> [EMAIL PROTECTED] On Behalf Of John Randall
>
> Raul Miller wrote:
> On 7/5/07, Terrence Brannon <[EMAIL PROTECTED]> wrote:
> >> But then again, math is limited by Goedel's Incompleteness
Not limited. Perhaps bounded, certainly structured.
>> theorem whereas computers can do more?
No. Turing's test, the halting problem, all are homologically proper
_subsets_ of Godel's Theorem:
which has been unfortunately MISnamed the Incompleteness Theorem.
Kvetch: Did Godel call it that? Or soemthing slightly different in
German?
>
> > That does not match my understanding, of Goedel's
> > Incompleteness theorem, nor of the character of computers.
Agreed.
>
> > As I understand the incompleteness theorem: a mathematical
> > system of sufficient complexity will allow an infinite number
> > of statements which are not resolvable simply by recourse to
"of sufficient complexity", "an infinite number": not really relevant.
> previous axioms (definitions).
>
>
> I largely agree with Raul, and differ from Terrence.
>
> Completeness and consistency are not mutually exclusive.
In axiomatically based context free languages of sufficient power to express
even arithmetic:
yes, they are.
In much more limited domains, occasionally, if the two are CAREFULLY defined
with much much less than full generality:
No, they are not.
At least, as I understand most mathematicians
who actually understand or WORK in this field
have come to believe
and accept certain proofs as adequate to maintain that.
> The only mathematical systems that are considered are consistent ones:
No, actually. In fact, not at all:
working with INconsistent ones is extremely difficult.
Which may be why most mathematicians are scared to attempt to.
hence the extreme sluggishness in working with partial logics, such as
"probably true" ones.
To the extreme detriment of the field of mathematics. The mos talented
mathematicians are
NOT YET
always encouraged to do so. Sadly.
ANY one who deals with NNP or semiotic cognitive studies ALREADY knows this,
and
there is a quiet evolution/revolution underway that may end up redefining
mathematics non-axiomatically.
(But also, alas, IMOHO, also incorrectly)
they may or may not be complete.
Depends on the exact definition of completeness.
Usually anything less than dealing with arithmetic is regarded as cheating
on the topic;
and at that point ALL axiomatic context free languages are
at least as interpreted as a result of Russell & Whitehead's inability
to find flaw with Godel's Theorem
_regarded_ as EITHER incomplete OR inconsistent.
But the EXACT nature of those definitions is not always the same among all
mathematicians
or scientists.
Hence, arguments.
> Boolean algebra is complete: ordinary arithmetic on the natural
> numbers is not. However this does not mean that every statement on
> the natural numbers is undecidable (for example, x<:x^2 is decidable
> and true), only that there are some which are not.
Agreed.
Yet, as far as I understand, a consequence of Godel's Incompleteness Theorem
(found by himself) states that the consistency of the ordinary arithmetic
cannot be proven (at least, an effective proof cannot exist). In other
More correctly: as stated.
words, believing in the consistency of the arithmetic is, ultimately, an act
of faith. (Then again, what is not?)
More completely: what is, or is not, so?
Your version and mine are NOT co-equivalent.
To complicate matters, at least for me, there is a result by Tarski implying
that (an axiomatic version of) the field of real numbers is complete. I
am
I am a little puzzled as to how arithmetic could be incomplete but the field
of real numbers is.
Unless one jumps into the middle and does not provide an
enumerator/generator;
bu then avoiding the "Axiom" of Choice has always been a hot button, as far
as I can see.
not familiar with its proof and I do not know how to reconcile it with the
former statement. Can you, or anybody else, provide some enlightenment?
(My
Nor I on both counts; and as such (even were I to claim or even CORRECTLY
_prrove_ so)
tyhe statement you just made may be entirely correct AND true.
Can't be? Only in a _consistent_ AND complete mathematics:
look carefully before you try to abnegate the above statement, and
remember:
the mathematics YOU are working with IS either incomplete OR
inconsistent.
Side note: the capitals are intended as emphasis to the general audience,
not the specific author
who is largely on the nose with the terribly difficult problems being
dealt with.
only guess is that the above mentioned axiomatic description of the field of
real numbers is insufficient to imply the axioms of the ordinary
arithmetic
of the natural numbers presumed to be embedded.)
Which feels really contradictory, ne? Hurts to ponder it for long.
But then real number generation is always slippery.
there is, for example, a really bad weakness in Cantor's Diagonalization
argument,
which is (the first and only ONE of) the argument(s) for the existence of
unnameable real numbers.
> Completeness does not have a huge impact on most of mathematics except
Wrong. It _currently_ is REGARDED as such.
Back when first discovered, it threw 3 generations of mankind's best
Thinkers
Scientists, mathematicians, epistemologists, semanticians, and
philologists
for a loop.
And for a while, the Ostrich reflex took ovr.
Luckily, some took up the challenge again by the 1940s:
among them Stanislaw Lesniewski: who had _a_ resolution AND solution to
it.
in areas which directly depend on it, such as the word problem and the
> halting problem. Computable functions are much smaller than functions
Agreed.
from the natural numbers to the natural numbers: there are only
> countable many of the former, and uncountably many of the latter.
Countably only against the natural numbers; but mebbe not the reals.
it gets Dutch real fast.
A related problem to halting is finding (using a proper definition of) the
shortest version of an arbitrary self-contained algorithm (which it is not
an unusual pastime of some members of this forum, myself included). This
problem prompted Chaitin's work (according to himself).
minimal or optimal orthonormal form. mappings get to this really quickly.
it's just spectacularly hard to do.
hence my humble appreciation of the brilliance of the work of j's
progenitors and practitioners.
> As to learning mathematics, I believe that proofs are a post hoc
> formalization, and that almost all mathematics proceeds from
Sometimes yes, sometimes no. Though I agree mostly al my own eurekas came
coalescently,
followed by a horrid struggle to DEMONSTRATE the truthfulness so realizaed.
examples. Indeed, theorems without concrete examples are dismissed as
> "general nonsense".
Nope. Not by those who have come over the divide of learnign how to THINK
that way.
But that is a helluva long uphill battail; and one I am not claiming I have
done.
But efforts such as APL and J have made it a LOT easier to approach than
ever before IMOHO.
> Best wishes,
>
> John
Comments continue in this thread on other notations.
I will note the following:
a) Russell & Whitehead's efforts were to be praised, and are to be praised.
b) Godel's Theorem is _correct_ AS STATED in the original mathematics used
at least inasmuch as I understand it.
c) The later English (and other natural language) interpretations
ARE INCORRECT.
d) Lesniewski's demonstration of this has been shunted to the side in favor
of
context free languages
when his solution uses
context sensitive ones
to overcome the problem.
There is a simple resolution to the difficulty that might be best expressed
in English
as the following precept:
A language to be of use MUST have a field of discourse to discuss upon.
This field of discourse can NOT bre part of the language.
Thus a context free language discussed WITHOUT A FIELD OF DISCOURSE
ATTACHED
is USEless: you can NOT TALK about anything in it.
Thus, even WITH a context FREE language you MUST include the field of
discourse to be of value.
A context SENSITIVE language ALWAYS has a field of discourse attached to
ruminations about it;
otherwise you can NOT achieve any concrete result ABOUT that language:
thatis what makes it context SENSITIVE.
So perhaps all that Godel's Theorem REALLY says is:
With any consistent system
there is always more (incompleteness) outside of it to talk about.
Which if curse, leads to the unending infinite expansion of knowledge.
Which doe snot seem all that contradictory to me at all.
Just one that perhaps has not yet percolated down to the mathematics
and science communities who gave up on R&W's dream and
have spent the last 90 years ignoring the field of language development
and all those thorny problems still growing there.
BUT: that wil have to change
if math and science AND FAITH
are gonna grow up.
I do not see any contradiction between them
...except for what people have forcibly thrown up along the way.
Cheers.
>
>
>
>
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