From: hougen@uirvld.csl.uiuc.edu (Darrell Roy Hougen)
Newsgroups: sci.physics,sci.math
Subject: Re: Computability of the universe
Date: 10 Sep 1992 03:46:30 GMT
Organization: Center for Reliable and High-Performance Computing, University of Illinois at Urbana-Champaign
NNTP-Posting-Host: uirvld.csl.uiuc.edu

lfoard@Turing.ORG (Lawrence C. Foard) writes:

I'm afraid you are laboring under several mistaken or naive
assumptions.  This is not meant as a flame, but one should be wary of
thoughts that everything is simple and solved and knowable.

) I've been arguing with several people over the "reality" of irrational
) numbers. By reality I mean a process in the real world that "operates"
) on irrational numbers. This is not to say that irrational numbers are
) meaningless since they represent the limiting case of many "real" things as
) the number of trials go to infinity. 

First of all, processes in the real world don't operate on numbers at
all.  Numbers are just a mental abstraction of quantities that exist
in the world.  Before dismissing irrational numbers, try to think of a
process that operates on rational numbers. 

If I give you a piece of metal and ask you how long it is, what will
you answer?  If you measure it, you will undoubtedly arrive at a
rational number, but that is only because rational numbers have finite
representations in the usual representational scheme.  Therefore,
measurements are always rounded to the nearest number of a given
precision.  In reality, however, the length may not even be well
defined in a mathematical sense.  As one magnifies the piece of metal,
substructure will emerge showing not one length but many lengths and
as the structure is magnified still further, fluctuating electron
clouds will appear and the boundary will become fuzzy.  Thus, length
is an abstraction and measurements of length are only meaningful at
some scales.  The length of an object is neither a rational nor an
irrational number, but it is convenient to represent it by the
rational number that best approximates it.

In some cases, however, it may be more convenient to think in terms of
an irrational number, such as when comparing the diameter of a
circlular object to its circumference.  In reality, the diameter and
circumference of an object cannot be strictly assigned either rational
or irrational values, but an idealized abstract circle possesses the
property that the ratio of the two quantities is an irrational number.
In this case, the particular irrational number is used so often that
it has a name and therefore a finite representation.

) My argument is that any given single expermental trial must be
) able to be >>exactly<< simulated by a finite turing machine in finite
) time. 

This obviously fails to hold due to the fact that measurements cannot
be made exactly.  Every physical quantity is an abstraction of the
complex and varied nature of the real world.  These abstractions may
be very accurate but the accuracy of every abstraction is always
limited.  I would go so far as to say that there are always an
infinite number of potential quantities that are left out of every
experimental (or everyday) measurement.  This is impossible to prove
but not hard to support with a little thought.  Because there are
always unmeasured quantities, the outcome of an experiment, which is
just another measured quantity, is never precisely predictable and it
wouldn't be precisely measureable even if it were.  All of this is
true without even having to consider quantum mechanics which puts
limits on the accuracy with which measurements can be performed.

) If this is not the case we have to ask what we have left out of logic
) such that its not sufficient to explain the universe. In other words
) a world that cannot run on a finite turing machine is no better than
) assuming that some God above logic and reason pushs the particles
) around :-)

The question might be better stated, what have we left out of formal
logic?  However, the answer may not be that we have left anything out
of formal logic.  Remember, every measurement is an approximation and
there is nothing in what I have said that prevents a person or a
machine from arriving at better and better approximate models of the
world.  However, most of the present approaches to AI are doomed to
failure because they do not properly account for the ambiguity of
concepts, ie., the fact no particular measurement or finite set of
measurements tells the whole story about an object or event, and the
consequent fact that no set of real objects is composed of objects
having exactly the same properties.

Darrell
Newsgroups: sci.physics