Applet for The Unknowable & Limits of Math

Quote:

> > 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.