| Subject: Addendum: Logic, the Uncertainty Principle |
Author:
mi chamocha
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Date Posted: 08:43:42 07/23/01 Mon
In reply to:
Lover of wisdom
's message, "Since wer'e talking quantum" on 23:30:05 07/22/01 Sun
An addendum to my previous response to LoW:
I don't think that any physical theory will ever have implications on the "laws of logic", as you say. The truths established by first order logic, set theory, mathematical logic, etc., are determined completely independently from the physical world. THOSE truths have implications on physics, but not vice versa.
Physics has obvious implications for how we understand the physical world, and this has implications for certain aspects of philosophy, but not logic proper.
The Heisenberg Uncertainty Principle (HUP) essentially says that we are unable to measure both the position of a particle AND its momentum to an arbitrary precision. In other words, there is some base, unavoidable limit on our ability to know the exact initial conditions of any system.
More generally, a particle in quantum mechanics is defined by a kind of function called a "wave function", which contains all possible information about a particle or a system of particles. Various pieces of info are extracted from the wave function by applying different "operators" to the wave function. For example, the momentum is found by taking the spatial derivative of the function. The energy i found by taking the time derivative. The average position is found by squaring the function, multiplying by the position coordinate, and integrating. Given any two of these operators, if they don't commute with each other (ie, if switching the order of operation yields a different result), then there will be some lower limit on the ability to simultaneously measure both of these observable quantities.
For example, the momentum in the 'x' direction is (essentially) found by taking the derivative with respect to 'x'. The 'x' position is (essentially) found by multiplying by 'x'. For those versed in calculus, it's immediately apparent that the ordering of these two processes changes the result - taking the derivative then multiplying gives a different result than multiplying then taking the derivative. Hence the position-momentum uncertainty principle.
Qualitatively, this can be understood as the following: in order to gain information about a system, you need to disturb the system in some fashion. You can't simply watch from the outside and get a value for some observable quantity. Trying to measure the position *demands* that you disturb the momentum of the system, and vice versa. Hence these two measurements can't be made simultaneously and exactly.
The obvious implication is that this means that we, as observers within a quantum mechanical system (ie, the universe), are *totally* unable to construct, even in theory, a sort of simulation described above by omnifinite. This is frequently used by believers in free will to invalidate the deterministic Newtonian mechanics, and restore some degree of indeterminism into the model of the physical universe, thereby allowing the possibility for free will. The problem, however, is that the uncertainty principle, in my opinion, says nothing about what is *in principle* knowable to a non-physical "observer" from outside the system. The question of what is in princple knowable is left to various interpretations of QM, which have yet to be sorted out.
Fair enough? ;-)
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