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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 is...) 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 physical plane.) 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 physical measure. 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 created). > <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.). more later... Peace --- John Mead

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