From: ·······@watson.ibm.com
Subject: Applet for The Unknowable & Limits of Math
Date:
Message-ID: <81v2bo$auh$1@nnrp1.deja.com>
Hi Everybody, this it to let you know about new software
for my two books The Unknowable & The Limits of Mathematics
(published by Springer-Verlag and also available for
browsing on my web site). For these books I invented a
version of LISP, and wrote an interpreter in C and another
one in Mathematica. Now it is available as a Java applet.
Just go to
http://www.umcs.maine.edu/~chaitin/unknowable/lisp.html
http://www.cs.auckland.ac.nz/CDMTCS/chaitin/unknowable/lisp.html
Enjoy!
Rgds,
GJC
Sent via Deja.com http://www.deja.com/
Before you buy.
·······@watson.ibm.com wrote:
> For these books I invented a
>version of LISP, and wrote an interpreter in C and another
>one in Mathematica.
And on the seventh day, I rested. Your Nature article was splendid. Could
we have an equally approachable executive summary? Please?
_______________________________
Oliver Sparrow
Oliver Sparrow wrote:
> ·······@watson.ibm.com wrote:
>
> > For these books I invented a
> >version of LISP, and wrote an interpreter in C and another
> >one in Mathematica.
>
> And on the seventh day, I rested. Your Nature article was splendid. Could
> we have an equally approachable executive summary? Please?
> _______________________________
>
> Oliver Sparrow
This is a quote from an earlier paper by Prof. Chaitin. It seems to say
that omega is noncomputable
(and cannot ever be computed exactly) and I am wondering if this is the same
kind of 'does not compute'
that arises in discussions of: can the human mind perform (noncomputable)
functions that a computer
cannot(only computable functions) since biological import is suggested(the
brain has a noncomputable
aspect which will generate thinking that cannot be duplicated other than by
the original, that is unique
algorithmically). So how far does the "terms analogous" extend? That
evaluation is how I perceive
the decision of when is a logical structure sufficiently rich to follow
under the sway of Goedelian mast.
I think Prof. Chaitin has concluded that AI can be engineered to produce
something like CyC or the
grandmaster chess playing program, perhaps in other areas. But I dont know
if this approval covers
the older idea of strong AI(sentient) self-referencing and creative enough
to meet the challenge
that Penrose contended humans could demonstrate: recognizing mathematical
truths not provable
(I think that is accurate enough) and if these truths are inexpressible are
they also noncomputable?
Are there other problems in other fields of science that can benefit from
these insights into the foundations of mathematics? I believe algorithmic
information theory may have relevance to biology. The regulatory genes of a
developing embryo are in effect a computer program for constructing an
organism. The ``complexity'' of this biochemical computer program could
conceivably be measured in terms analogous to those I have developed in in
quantifying the information content of Omega.
Although Omega is completely random (or infinitely complex) and cannot ever
be computed exactly, it can be approximated with arbitrary precision
given an infinite amount of time. The complexity of living organisms, it
seems to me, could be approximated in a similar way. A sequence of Omegan's,
which approach Omega, can be regarded as a metaphor for evolution and
perhaps could contain the germ of a mathematical model for the evolution of
biological complexity.
At the end of his life John von Neumann challenged mathematicians to find an
abstract mathematical theory for the origin and evolution of life. This
fundamental problem, like most fundamental problems, is magnificently
difficult. Perhaps algorithmic information theory can help to suggest a way
to
proceed.