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From: igl@ecs.soton.ac.uk (Ian Glendinning)
Newsgroups: comp.ai
Subject: Re: Comments on "The Emperor's New Mind
Message-ID: <3887@ecs.soton.ac.uk>
Date: 12 Sep 90 16:11:00 GMT
References: <47000003@uxa.cso.uiuc.edu>
Organization: University of Southampton, UK
Lines: 86

In <47000003@uxa.cso.uiuc.edu> xhg0998@uxa.cso.uiuc.edu writes:

>	Let us pick up some examples to elaborate here. Maybe I should not
>mention that the example in Page 255 (Fig. 6.18) can be well explained
>with classic Maxwell electromagnetic wave theory, without troubling us with
>the uncomfortable belief that a photon can appear in two distant locations
>at once (Qigung Master Yen Xin even thinks that an eminent Qigung master can

It is true that a classical electromagnetic wave would also interfere
with itself so as to emerge wholely at detector A.  (Fig. 6.18 - page
330 in my paperback edition.)  However, the key point here is that we
are dealing with a *single* photon in the apparatus at any time.  That
is, a single *particle*.  If we were to put a detector in the path of
each beam, after the splitting by the first half-silvered mirror, then
we would detect photons arriving at *either* one detector *or* the
other one.  *Never* both at the same time.  This is quite different
from the classical electromagnetic case, in which case the light wave
would take both paths and *always* be detected by both detectors.

This nicely illustrates the basic non-intuitive feature of the
behaviour of particles in quantum mechanics.  That is, left to their
own devices they behave like waves (spread out) but as soon as you
try to look at them they behave like particles (appear to be in one
place).  The idea was introduced by Penrose in the previous section,
in terms of the U and R evolution procedures.

In summary then, quantum mechanical particles behave neither like
classical waves (since they are always detected as being in one place
at one time) nor like classical particles (since they can interfere
with each other like waves).  Instead, they combine properties of
both in a way that is non-intuitive in the light of macroscopic
experience.  But don't be fooled by intuition.  For "quantum
mechanical particles" read "physical particles".  This is not just
theory - it really describes the way the world is experimentally
observed to behave, whether you like it or not!

>	At last, let us spend length to go over the core problem:
>undecidability. Penrose piled up Goedel's theorem, Turing's Theorem, Russell 
>paradox together, but, I would shout that in this particular important point 
>he lacks genuine and original understanding of the meaning of these theorems 
>and paradoxes.

Really!

>	 Actually, Goedel's incompleteness theorem and Turing's theorem on
>unceasingness of Turing Machine, can all reduce to Russell paradox, which 
>again is another formalised version of the knight and knave story of the 
>ancient Greek sophists. Actually as Penrose mentioned, both Goedel and 
>Turing obtained their theorems after studying the Russell paradox. This 
>paradox can be stated as follows:

>1.	S: Statement S is not true.
>	*  ==========#===============

This is not a statement of Russell's paradox (which is phrased in the
language of set theory, remember) but is a simple contradiction.
True, Russell's paradox leads to a contradiction, but that is
precisely why the form of reasoning which led to it can not be
permitted - because contradictions (by definition) are not allowed
within self consistent formal systems.  The Goedel argument actually
runs something like:

G: There is no proof of G in this system.

which involves no contradiction at all.  If G is assumed true, then
there is no proof of it within the system - which is perfectly ok,
since not all propositions must be decidable - but more to the point,
neither is there a contradiction in the above statement.  If, on the
other hand, G were assumed false, then the right hand side says there
is a proof of G (meaning it's 'true') so we would then have a
contradiction.  Thus, we simply reject the second alternative, and we
have a consistent system within with G is true but not provable.

>more alternate negative and positive statements. Therefore, the very 
>definition of S has violated the law that a variable can have exactly one 
>value at once in a two value logical system. The Goedel's proposition 
>involves such a definition, so does Turing's theorem. It is nothing else 
>but to say "yes = no, and yes and no are two different values". Why this later 

Yes, S is a contradiction, but no, Goedel's proposition G does not
involve one.
--
Ian Glendinning                               igl@uk.ac.soton.ecs
Electronics and Computer Science              Tel: +44 703 595000
University of Southampton
Southampton SO9 5NH England