[om-list] Re: Fuzzy Set Theory

[Mark Butler]

Tom wrote:

>     My main question:  Why do they call it a wave function?

It is called a wave function because it propagates through free space in a
manner very similar to any other wave.  The complex valued wave function
carries both position and momentum information that evolves through time in a
manner specified by Schroedinger's equation.

The spatial distribution of the wave corresponds to position and the frequency
distribution corresponds to momentum.  It is straightforward to demonstrate
that any signal that is very narrow in the time domain is very broad in the
frequency domain and vice versa, a fact which leads to the Heisenberg
uncertaintly principle for quantum mechanics.

The controversial part about quantum mechanics is the matter of wave function
collapse where an external observer supposedly causes a wave function to
randomly collapse to one of a family of eigenfections determined by the
measurement being performed.  This has come under intense criticism for very
good reasons - mostly that there is no physical definition of a measurement.

The Bohm alternative, originally suggested by Einstein, proposes that in
addition to the wave function, there is also a core particle with a definite
position and momentum that follows a trajectory strictly determined by its
local wave function.  This is also known as the pilot wave model.

The big difference is that the wave function never collapses and measurement
processes are defined in strictly physical terms.  It has been shown that this
model produces statistical results identical to traditional quantum
mechanics.  However, no one has yet demonstrated that it is experimentally
superior.  If I were a physicist, that is precisely what I would test.

It is important to note that multiple identical particles share a common 3*N
dimensional wave function, which leads to superluminal influences between
particles in any deterministic version of quantum mechanics.

It is also useful to consider that if a particle did not have an associated
wave, it would have no mechanism for ever affecting or being affected by other
particles any considerable distance away.  A wave function acts as a very
effective distributed information system for a particle - effectively causing
it to adjust its trajectory in a collision based on advance information
returned by its own reflected wave.

It is well known that particles can "tunnel" through very high energy barriers
provided they are thin enough.  This is basically because a small portion of
an incident wave is transmitted through rather than reflected from any such
barrier.  This leads to a rule of thumb that says that a particle can "borrow"
any amount of energy provided it is for a short enough period of time so that
the product is less than Planck's constant.

I speculate that an intelligent particle could navigate very effectively with
a similar line of credit and the ability to use it to selectively modify its
trajectory.  Of course, since any energy borrowed would have to be paid back
in short order, the behavior of an ensemble of such particles would be
difficult to distinguish from an ensemble of "dumb" ones.


> 
>     Isn't the LEM:
> 
>     p(x) and not p(x) = false
> 
>     Maybe they are different forms of the same idea.

I believe they are DeMorgan transformations of each other.

>     Regarding time, you still look at time as being more different from
> space than I do.  You say that if we add time dimension to an object, then
> the statement about it being in the box does not follow LEM.  But if we give
> any spatial dimension to the same objection, the same statement about the
> box again does not follow LEM.  It can be too big to fit in the box, or big
> enough to fit in two boxes, etc.

Of course.  The big difference is that even single valued attributes have this
behavior in time, where they do not in space.  To make a useful model of
reality, storing the time variation of object relations is much more important
than modeling the spatial distribution of a single object.  The reason is that
most real objects (say a building) can just be decomposed into relations
between smaller components, where time variance of a relation cannot be
decomposed unless you declare that each object is changing its identity to a
new object at each moment in time.
 
>     In other words, if we say that LEM really says "p(x) cannot be both 100%
> true and 100% false at the same time" we have no problem.  Where, the
> assumption that anti-fuzzy logicians have conveniently assumed is that LEM
> says "p(x) cannot be both true and false to any degree", which is just a
> silly assumption when you haven't ever though about the possibility that
> p(x) could be anything other than 100% true or 100% false.

I agree, although linguistically speaking, I would say that in English a
statement is only strictly true when it is 100% true, whereas it is strictly
false if it is not 100% true.

For example:  "I have lived in Farmington all my life" is about 70% correct,
but everyone would agree that it is a false statement even though the negation
"I have never lived in Farmington", is even less true.

This reminds me of the sentiment "God created integers; real numbers are the
work of man".
 
>     Back to the box and time: If you define the object as having extension
> in time, then of course it is possible that p(x) (= "In the box") and not
> p(x) can both be partially true and partially false at the same "time".  And
> the "true and living" definition of LEM is not violated.

I agree, my whole point was that it was violated using Boolean logic. The
problems with a fuzzy logic LEM are that no one has demonstrated one that
works.

For example, using the standard min()/max() your form the LEM is:

w and not w = False  =>

min(w,1-w) = 0 

This is obviously not the case for any w except 0 and 1. 

>     I think DeMorgan's Law is actually:
> 
> 1.  (not a) and (not b) = not (a or b)
> 2.  (not a) or (not b) = not (a and b)

Yes, my mistake.

>     Anyway, I have been interested in finding a good function for fuzzy
> logical ands and ors, but I haven't researched it yet.  I'm sure we can find
> something a lot better than max and min -- although probably not as simple
> and efficient, computationally.

Due to the ability to do symbolic formula reduction before evaluation, an
analytic form may be much more efficient in many applications.  When I first
read about fuzzy logic, the first thing I thought was what kind of
mathematical masochist chooses functions like min() and max() for the core
operators of his theory?

- Mark