Posted on 12/07/2005 3:31:28 AM PST by snarks_when_bored
In "Clan of the Cave Bear" (which I **don't** recommend!) the Cro-Magnon heroine is "pleasured" by a Neanderthal and gives birth to a half-breed. Then she puts two and two together...
and goes on to discover agriculture, tame horses, whatever. (It's a real soap opera)
You got me interested in Lascaux, so I did a bit of surfing on the caves. Fascinating art work - and very revealing of the culture.
you: That would be the last question.
On Hawking's black hole entropy, the issue was physical entropy. Strominger and Vafa used string theory to make the calculation (Bekenstein/Hawking). The issue of information (reduction of Shannon entropy) being lost in black holes came up separately. AFAIK, none of the physicists confused the two entropy formulations.
Also, the "ergod" is not the inverse of the void.
As before, if we are going to explore ergodic theory - we need to distinguish between physics and mathematics.
And concerning ergodic theory and singularities, the zero point is the poison pill:
Boltzmann thought of the proper average values to identify with macroscopic features as being averages over time of quantities calculable from microscopic states. He wished to identify the phase averages with such time averages. He realized that this could be done if a system started in any microscopic state eventually went through all the possible microscopic states. That this was so became known as the ergodic hypothesis. But it is provably false on topological and measure theoretic grounds. A weaker claim, that a system started in any state would go arbitrarily close to each other microscopic state is also false, and even if true would not do the job needed.
The mathematical discipline of ergodic theory developed out of these early ideas. When can a phase average be identified with a time average over infinite time? G. Birkhoff (with earlier results by J. von Neumann) showed that this would be so for all but perhaps a set of measure zero of the trajectories (in the standard measure used to define the probability function) if the set of phase points was metrically indecomposable, that is if it could not be divided into more than one piece such that each piece had measure greater than zero and such that a system started in one piece always evolved to a system in that piece.
But did a realistic model of a system ever meet the condition of metric indecomposability? What is needed to derive metric indecomposability is sufficient instability of the trajectories so that the trajectories do not form groups of non-zero measure which fail to wander sufficiently over the entire phase region. The existence of a hidden constant of motion would violate metric indecomposability. After much arduous work, culminating in that of Ya. Sinai, it was shown that some "realistic" models of systems, such as the model of a gas as "hard spheres in a box," conformed to metric indecomposability. On the other hand another result of dynamical theory, the Kolmogorov-Arnold-Moser (KAM) theorem shows that more realistic models (say of molecules interacting by means of "soft" potentials) are likely not to obey ergodicity in a strict sense. In these cases more subtle reasoning (relying on the many degrees of freedom in a system composed of a vast number of constituents) is also needed.
If ergodicity holds what can be shown? It can be shown that for all but a set of measure zero of initial points, the time average of a phase quantity over infinite time will equal its phase average. It can be shown that for any measurable region the average time the system spends in that region will be proportional to the region's size (as measured by the probability measure used in the microcanonical ensemble). A solution to a further problem is also advanced. Boltzmann knew that the standard probability distribution was invariant under time evolution given the dynamics of the systems. But how could we know that it was the only such invariant measure? With ergodicity we can show that the standard probability distribution is the only one that is so invariant, at least if we confine ourselves to probability measures that assign probability zero to every set assigned zero by the standard measure.
We have, then, a kind of "transcendental deduction" of the standard probability assigned over microscopic states in the case of equilibrium. Equilibrium is a time-unchanging state. So we demand that the probability measure by which equilibrium quantities are to be calculated be stationary in time as well. If we assume that probability measures assigning non-zero probability to sets of states assigned zero by the usual measure can be ignored, then we can show that the standard probability is the only such time invariant probability under the dynamics that drives the individual systems from one microscopic state to another.
As a full "rationale" for standard equilibrium statistical mechanics, however, much remains questionable. There is the problem that strict ergodicity is not true of realistic systems. There are many problems encountered if one tries to use the rationale as Boltzmann hoped to identify phase averages with measured quantities relying on the fact that macroscopic measurements take "long times" on a molecular scale. There are the problems introduced by the fact that all of the mathematically legitimate ergodic results are qualified by exceptions for "sets of measure zero." What is it physically that makes it legitimate to ignore a set of trajectories just because it has measure zero in the standard measure? After all, such neglect leads to catastrophically wrong predictions when there really are hidden, global constants of motion. In proving the standard measure uniquely invariant, why are we entitled to ignore probability measures that assign non-zero probabilities to sets of conditions assigned probability zero in the standard measure? After all, it was just the use of that standard measure that we were trying to justify in the first place.
In any case, equilibrium theory as an autonomous discipline is misleading. What we want, after all, is a treatment of equilibrium in the non-equilibrium context. We would like to understand how and why systems evolve from any initially fixed macroscopic state, taking equilibrium to be just the "end point" of such dynamic evolution. So it is to the general account of non-equilibrium we must turn if we want a fuller understanding of how this probabilistic theory is functioning in physics.
In the book she is repeatedly raped as a form of humiliation, not pleasured.
When I said Spiritual revelation is the most certain knowledge, I was speaking of Christian Spiritual revelation. In such revelations, there is no "observer".
The first Spiritual revelation a Christian receives is that "Jesus Christ is Lord". It doesn't come from his own mind, rather the Truth appears in him. But because of the revelation, we believe, He "knows" us and we are born again by and in the Spirit.
We are no longer "observers" - we abide in Him and He in abides in us. (Gospel of John, etc.)
What happens BTW if some supernatural entity puts His finger on the scales and violates the law of equal a priori probabilities of degenerate states?
I was serious, though, and not series. My beeber is not stuned.
How does one know that the artwork with phallic adornment was part of the piece as originally designed and executed?
You said: In the book she is repeatedly raped as a form of humiliation
Like I said, I don't recommend the book.
The notion of setting a book in the Ice Age was kinda cool (ahem), but I didn't like it.
I was just trying to give BB an example of "primitive" people who didn't make the connection between coitus and birth. The anthropological literature (and the Straight Dope) are much better than fictional Ice Age romances.
I know thats right.. Just finished watching PollyAnna and am in a mood.. Pity the fool that would hurt one of my kids, or grand children.. it would not be nice.. a mother bear would seem tame..
O.K.. Bring out the big guns...
Al Gores Law..
Atomic theory doesn't mention God does it? So that makes it atheistic. The idea that all matter follows natural laws and is not guided by an intelligent hand is the basis of atheist philosophy.
Ok I have another one - the Atheistic Germ Theory of Disease. I think that if we just taught it objectively in the classroom then kids could make up their own minds. Im sure a lot more of them would prefer to believe that disease is caused by immorality. Why should atheist teachers force them to believe germs cause disease?
My experience on spiritual revelation is varied.. Sometimes it can be as simple as an intellectual breakthrough or as mind boggling as an epiphany or even a vision.. I see a vision as a vision.. ugh!.. more than mere a clearing of thoughts or the fog that clouds my idea about some thing.. But a "seeing" on a grand scope, a vision of "something" far greater than you were in the market for, since didn't even know that a "sight" like that Could exist..
Amazing (once you get used to it) that a human mind could be transported to another place "not on your own".. "not on purpose". Other than that it could be scary.. I would say revelation is progessive.. Some revelation needs other relevation as foundation.. Because without a proper foundation the revealment of the revelation would not be emergent.. Inspiritation is only revelation if it is spiritual..
Discovery of awareness and what awareness is, might be an epiphany but its not a vision.. as I see it so far.. For awareness is a synonym for gratitude.. Seems like the more aware you are the more gratitude overflows your cup.. The less gratitude you are filled with the more awareness you are in need of.. And where gratitude is King, a loveing faithful sacrificial attitude is Queen..
A revelation of what a proper family is, is indeed a "vision"..
Leakey, of course, is entitled to his evaluation of the accessibility of the "content" of Lascaux. Frankly, I don't think "cultural blindness" makes an understanding of its meaning impossible. What most impresses me about Lascaux is its ability to effectively communicate with people living so many millennia later: The artist speaks to me in a language I can understand.
I certainly wholly agree with Leakey's last sentence: "In seeking to understand our origins, we come away from a place like Lascaux with a deep conviction of connectedness, and a humility at the power of the human mind...." Thanks for writing, Virginia-American!
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