RE: [-empyre-] replying to several posts

[Jim Andrews <jim@vispo.com>]

> > The 'undecidable proposition' was theorized and used by Godel.
> -----
> Sorry to be pedantic. Incompleteness Theorem, the formal refutation of
> Whitehead and Russell's theories. This secondarily led to the development of
> the symbolic logics that made Turing's work possible.

Yes, that is the larger picture, isn't it, as described at http://www.miskatonic.org/godel.html
or, slightly more rigorously, http://www.myrkul.org/recent/godel.htm (I just did a search for
'Godel' and 'incompleteness' and there are scads of pages).

The Incompleteness Theorem establishes that in a system/language of formal logic at least as
powerful as one capable of supporting arithmetic, there will always be 'undecidable
propositions'. That is, one cannot form a system of formal logic in which all well-formed
propositions are either false or provably true. There are *always* truths that cannot be proved
within the system (whatever the system).

By the way, I think the argument on http://www.miskatonic.org/godel.html "that a computer can
never be as smart as a human being because the extent of its knowledge is limited by a fixed set
of axioms, whereas people can discover unexpected truths" is mistaken: Godel's proof says that
even any formal system used by humans is incomplete. So the only question is whether the 'human
system' is necessarily more comprehensive or flexible than a computer's system and it doesn't
seem likely to me.

I wonder whether, given any particular undecidable proposition, there exists an axiom that one
could introduce into the system that would render the undecidable proposition a theorem? Of
course, even if there does exist such an axiom for each undecidable proposition, even then, with
infinitely many axioms, one could still apply the argument to show that that system also was
incomplete.

We are profoundly certain that we will never know all there is to know. We are profoundly
certain that the range of thought and accessible truth can always be expanded via new language,
and that no language will ever have all the answers. We see that the margins are attached to the
foundations.

How does this relate to digital poetry and programming? In any number of ways...

1. we see the intensity of its engagement with language--here we have Godel inquiring into the
possibility of the 'completeness' of knowledge, expressable in language. And he finds that any
conceivable language can never attain completeness. And this (drop-dead gorgeous) work that sees
into the necessary incompleteness of (human and non-human) knowledge thereby provides some
important groundwork for the invention of machines capable of expanding human knowledge in the
way that the computer has and will--but will never be complete.

2. The soaring aspiration of this work--and yet its 'humbling' conclusions--seem to me to be way
poetical.

3. Here we have work concerning the formal properties of language that is sooooooo far beyond
pedantry and grammar rules that it has rocked the world with the full force of knowledge we
associate with the fierce chemistry of the sun and knowledge of it. There is knowledge of atoms
and the material world, physics. Then there is knowledge of the formal properties of language
and thereby some knowledge of the incompleteness of any and all epistemologies. Yet with that
knowledge goes the ability to extend knowledge into the digital age. And the promise of much
more to come.

4. Poetry and programming...

ja

x . .  . . .
. x .  . . .
. . x  . . .

. . .
. . .
. . .