Re: To John mostly
Dec 16, 1993 06:27 PM
by John Mead
replies to Don (long over due.... sorry)
> <A recent review of Measure theory has caught my attention, in
> the sense that the mathematics of the theory separates out the
> exact requirements (logically speaking) for a system to obtain
> the status of Measure vs truly empty (insignificant is a better
> word) of physical properties such as a definable Length.>
> Could you elaborate on this?
The basic problem arises when you try to define concepts that have
a physical interpretation such as measure (length, volume, mass,
observable quantities that combine and separate and combine and separate
with some notion as to a conservation of whatever the measured quantied
the idea is that mathematically a pea, basketball, and planet
all have the same number of "points" within them. Hence, one can
cut up a Pea into a *finite* number of subsets and then reassemble them
into an object the size of the World. This can be done because
the abstract concept of point is dimensionless. What one wants to do in
measure theory is to define or specify the rules which give things
an objective quality. The best example is Volume of a pea should
be preserved if you disassemble it and reassemble it mathematically.
(if you want it to have a physical presence as we know it on the
hence you want to specify what limits (mathematical requirements)
must be present to dissamble something and then recombine it without
messing up its physical properties like size.
so for lengths (of a line say) you want subsets that behave as:
(I = a fixed INterval, and Sn some subsets (index on n) of its points)
L(I) = L(USn) = Sum L(Sn) (L is a function which assigns a subset of
I a Length or some physical preserved property. you want the length
of the union to always equal the sum of the individual lengths.
In Normal Reiman Integration one always works with subintervals
and there is usually no problem. However, if you allow one to
break the segment into things like Rationals and Irrationals
then it gets to be a problem to assign the length of the
Rationals to a Number and the length of the irrationals to another
number. When do Gaps become important esp regarding cardinality
and how do you gaurantee the size conservation???
Measure theory gives the rules for what types of sets and set algebras
can be successfully constructed to gaurantee a meaningful concept of
One realyy nifty thing is that (Proven not too long ago I think) is that
to build the structures which defy a physical reality requires using
the axiom of Choice. If you do not allow Consciousness to build
an arbitrary structure, and limit yourself to "Normal" operations
which do not require a free will, you will never have a problem with
things becoming "Unphysical". This strikes me as VERY interesting.
It is also interesting that you cannot do QM without Hilbert Spaces
(which are *neccessarily* intertwined with restrictions regarding
Lebesgue measurable Spaces ..... i.e. mathematical reality without
consciousness). Once you introduce free will (Axiom of choice)
you can jump out of the Physical realm easily... indeed it is
*neccessary* if you want to leave the physical plane (mathematically)!
> <... describing the difference between the physical plane and the
> lesser planes>
> Could you elaborate on this some more too? Its my impression that
> the physical plane is a *subset* of the mental plane ...
yes... lesser was a bad (i.e. unconventional choice/use of the
term/word). I meant lesser as less *physical* not a smaller space (it
IS larger in the mental frame.... more freedom and structures can be
> <chaos theory is that is inherently classical and can not explain
> many of the truly QM events that exist.
> Regarding the nondeterminism that *seems* to be present in QM, I
> still am hesitant to buy into the idea that nondeterminism is a
> cornerstone concept. May
A quick example is that Planetary Motion exhibits Chaotic effects,
mostly observed when predicting orbits over very long time frames
(the initial conditions of the equations need accuracy to an
infinite precision). Hence the Chaos of planetary motion makes the
movements chaotic. HOWEVER... the integrals of the motions
will always behave classically (i.e. total energy is conserved etc.)
HENCE... things that are TRULY QM will NEVER be explained through Chaos
theory applied to classical (non QM) mechanics. The QM effects
occur (momentary violation of conservation of energy, allowing
energy tunneling and other effects) only can occur by using
QM. Virtual particles are required to explain reality, and
no chaos theory will create them out of a framework that is
inherently classical. (i.e. deterministic in the various integrals of
motion, conservation laws etc.).
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