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Topic: TOE - Thoughts
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chimp
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posted 14. January 2004 04:00
Quantum entities are described as probability distributions, which are attributes of an underlying phase space, where the properties-attributes such as "spin" and "charge" are not the attributes of individual particles, but they are universally distributive entities, being the attributes of a "coherent wave function". It is this wave-distribution property that then "decoheres" into the ostensible "wave function collapse", as waves become localized particles that are "in phase" creating standing-spherical-wave resonances, which are condensations of space itself. The continual collapse-condensation of space into matter-energy is the continual "change", i.e. the property called "time". The spherical waves, or probability distributions are represented by the Schrodinger wave function, "psi".
The continual intersection and collapse of probability distributions, also known as quantum phase entanglement, is a continual increasing of the "total" combined information of the universal wavefunction itself. Information density. With more information, more complex structures can be created.
Quantum mechanics leads us to the realization that all matter-energy can be explained in terms of "waves" or probability distributions containing information. In a confined region(i.e. a closed universe or a black hole) the waves exist as STANDING WAVES In a closed system, the entropy never decreases.
The analogy with black holes is an interesting one but if there is nothing outside the universe, then it cannot be radiating energy outside itself as black holes are explained to be. So the amount of information i.e. "quantum states" in the universe is increasing. We see it as entropy, but to an information processor with huge computational capabilities, it is compressible information.
The information density of the universal system must be increasing. The increase of information density is analogous to a pressure gradient.
[density 1]--->[density 2]--->[density 3]---> ... --->[density n]
[<-[->[<-[->[U]<-]->]<-]->]
Intersecting wavefronts = increasing density of spacelike slices
As the wavefronts intersect, it becomes a mathematical computation:
2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...2^n
According to conventional theories, the surface area of the horizon surrounding a black hole, measures its entropy, where entropy is defined as a measure of the number of internal states that the black hole can be in without looking different to an outside observer, who must measure only mass, rotation, and charge. Another theory states that the maximum entropy of any closed region of space can never exceed one quarter of the area of the circumscribing surface, with the entropy being the measure of the total information contained by the system.
S' = S_m + A/4
So the "black hole" theorists came to realize that the information associated with all phenomena in the three dimensional world, can be stored on a two dimensional boundary, analogous to the storing of a holographic image.
Since entropy can also be defined as the number of states, that particles can be in within within a region of space, and the entropy of the universe must always increase, the next logical step is to realize that the spacetime density, i.e. the information encoded within a circumscribed region of space, must be increasing in the thermodynamic direction of time.
Of course, thermodynamic entropy is popularly described as the disorder or "randomness" in a physical system. In 1877, the physicist Ludwig Boltzmann defined entropy more precisely. He defined it in terms of the number of distinct microscopic states that the particles in a system can be configured, while still looking like the same macroscopic system. For example, a system such as a gas cloud, one would count the ways that the individual gas molecules could be distributed, and moving.
In1948, mathematician Claude E. Shannon, introduced today's most widely used measure of information content: entropy. The Shannon entropy of a message is the number of binary digits, i.e. "bits" needed to encode it. While the structure, quality, or value, of the information in Shannon entropy may be an unknown, the quantity of information can be known. Shannon entropy and thermodynamic entropy are equivalent.
The universal laws of nature are explained in terms of symmetry. The completed infinities, mathematician Georg Cantor's infinite sets, could be explained as cardinal identities, akin to "qualia" [Universally distributed attributes] from which finite subsets, and elements of subsets [quantum decoherence-wave function collapse] can be derived.
Completed infinities, called "alephs" are distributive in nature, similar to the way that a set of "red" objects has the distributive property of redness[qualia]. Properties, or "attributes" like red are numbers in the sense that they interact algebraically according to the laws of Boolean algebra. Take one object away from the set of red objects and the distributive number "red" still describes the set. The distributive identity[attribute] "natural number" or "real number" describes an entire collection of individual objects.
The alephs can be set into a one to one correspondence with a proper subset of of themselves. The "infinite" Cantorian alephs are really distributive[qualia].
Yet, if we have a finite set of 7 objects, the cardinal number 7 does not really distribute over its individual subsets. Take anything away from the set and the number 7 ceases to describe it[wave function collapse-condensation into specific localization].
Symmetry is analogous to a generalized form of self evident truth, and it is a distributive attribute via the laws of nature, being distributed over the entire system called universe. A stratification of Cantorian alephs with varying degrees of complexity. Less complexity = greater symmetry = higher infinity-alephs. So the highest aleph, the "absolute-infinity" distributes over the entire set called Universe and gives it "identity".
The highest symmetry is a distributive mathematical identity[also a total unknown but possibly analogous to a state of "nothingness"]. This fact is reflected in part, by the conservation laws.
So an unbound-infinite-potentia and a constrained-finite-bound-actuality, are somehow different yet the same. The difference and sameness relation is a duality. Freedom(higher symmetry) and constraint-complexity-organizational structure(lesser symmetry) form a relation that can be described by an invariance principle.
On a flat Euclidean surface, the three angles of a triangle sum to 180 degrees. On the curved surface of a sphere, the three angles add up to more than 180 degrees. On the hyperbolic surface of a saddle they sum to less than 180 degrees. The three types of surface are not equivalent.
There is a rotational invariance for a triangle, that seems to hold for the three types of surface though.
ABC = BCA = CAB
CBA = BAC = ACB
According to Einstein, and the CTMU of Langan, www.ctmu.org , "space and time are modes by which we think, and not conditions in which we live". Space becomes abstract, a relation that is perceptual and "mental", where distance interval between two points becomes a mental perception.
[ abstract representation]--->[semantic mapping]--->[represented system]
An abstract representation is exactly that, "abstract". It is not a space, or time, but is instead a product of consciousness, or a mental construct. Topologically it is equivalent to a "point". The abstract description contains the concrete topology. Likewise, the concrete contains the abstract.
A duality.
A point contains an infinite expanse of space and time?
Could it be, that the "absolute" infinity, is actually a dimensionless point? Or more correctly, an "infinitesimal"?
Universe? = Zero?
On one level of stratification, two photons are separate. On another level, of stratification, the photons have zero separation.
Instantaneous communication between two objects, separated by a distance interval, is equivalent to zero separation[zero boundary] between the two objects.
According to the book "Gravitation", chapter 15, geometry of spacetime gives instructions to matter telling matter to follow the straightest path, which is a geodesic. Matter in turn, tells spacetime geometry how to curve in such a way, as to guarantee the conservation of momentum and energy. The Einstein tensor[geometric feature-description] is also conserved in this relationship between matter and the spacetime geometry. Eli Cartan's "boundary of a boundary equals zero."
A point can be defined as an "infinitesimal". The Topological spaces are defined as being diffeomorphism invariant. Intersecting cotangent bundles[manifolds] are the set of all possible configurations of a system, i.e. they describe the phase space of the system.
Waves are then abstract distributions and particles are convergent "concrete" localizations.
http://en.wikipedia.org/wiki/Singleton_set
quote:
In mathematics , a singleton is a set with exactly one element. For example, the set {0} is a singleton. Note that a set such as {{1,2,3}} is also a singleton: the only element is a set (which itself is however not a singleton).
A set is a singleton if and only if its cardinality is 1. In the set-theoretic construction of the natural numbers the number 1 is defined as the singleton {0}.
The problem with set theory appears to be the primitive concept of : inside / outside
A or not-A
Set theory is based on 2-valued logic of course. Since the absolute infinite is the power set of everything that exists, ergo, an element cannot be removed from it by definition, hence the talk of the removal of elements becomes wishful thinking. A fruitless exercise in futility, for the imagination. Also, the problem arises where, if, an element such as a singleton, is removed from the absolutely infinite, and it[absolute infinity] somehow becomes less than its previous absolute condition, that seriously implies that the absolute infinity is totally non-replenishable, but, actually, if the absolute is generalized as the Platonic state of affairs known as total nothingness, then it no longer has this unprecidented weakness inherited from ZF set theory which is dooomed to die a slow and agonizing death, as die hard academians continue to flog the dead ZF "horse" for all it's worth.
quote:
Structures built on singletons often serve as terminal objects or zero objects of various categories
The statement above shows that every singleton S is a terminal object in the category of sets and functions. No other sets are terminal in that category.
Any singleton can be turned into a topological space in just one way (all subsets are open). These singleton topological spaces are terminal objects in the category of topological spaces and continuous functions. No other spaces are terminal in that category.
Any singleton can be turned into a group in just one way (the unique element serving as identity element These singleton groups are zero objects in the category of groups and group homomorphisms . No other groups are terminal in that category.
An "excellent idea":
http://www.mmsysgrp.com/stefanik.htm
quote:
Thesis: Scientific objectivity is best characterized by the concept of invariance as explicated in category theory than the concept of truth as explicated in mathematical logic.
If reality is self referential, it observes itself, through particle interaction AND conscious observers?
[ 14. January 2004, 09:11: Message edited by: Moderator ]
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chimp
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posted 18. January 2004 01:12
A - B
= [A^[1/2] - B^[1/2]]*[A^[1/2] + B^[1/2] ]
= [A^[1/4] - B^[1/4]]*[A^[1/4] + B^[1/4]]*[A^[1/2] + B^[1/2]]
= [A^[1/8] - B^[1/8]]*[A^[1/8] + B^[1/8]]*[A^[1/4] + B^[1/4]]*[A^[1/2] + B^[1/2]] ... [A^[1/n] - B^[1/n]] ... which becomes an infinite product...
A + B is also infinite?:
A + B = [A^[1/2] - i*B^[1/2]]*[A^[1/2] + i*B^[1/2]]...etc.
A^[1/n] - B^[1/n]
A---> 1, B---> 1, as n---> oo
So as n --->oo , A^[1/n] - B^[1/n] approaches zero...
Interesting
Roll over Zeno
A^[1/n] - B^[1/n]]*[A^[1/n] + B^[1/n]] ...
As n--->oo
The expression approches [1 - 1]*[1 + 1] ---> 0*2
At infinity, does mathematical existence become a self similar pattern?
02020202020202020202020202020202020202020202020202020202 0202020202020202020202020202020202
0202020202020202020202020202020202 02020202020202020202020202020202020202020202020202020202
020202020202 0202020202020202
Phoenix at
http://207.70.190.98/scgi-bin/ikonboard.cgi?s=98893505469010cf3cb332ca23d129ef;act=ST;f=13;t=228;st=40;&#entry866
discovered this mathematical relation for the Absolute:
we also have UžUžUžUžUžUžUžUžUžUžUžUžU&isin
;UžUžUžUžUžUžUž UžUžUžUžUžU&isi
n;UžUžUžUžUžUžUžUžUžUžUžUžU& is
in;UžUžUžUžUžUžU with no beginning and no end.
end quote.
Those symbols could represent something analogous to the "yin - yang" or Tao?
2 - valued logic?
TFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTF TFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTF TFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFT...
An infinite number of mathematical relations intersect at infinity. Infinite symmetry.
Since entropy can also be defined as the number of states, that particles can be in within within a region of space, and the entropy of the universe must always increase, the next logical step is to realize that the spacetime density, i.e. the information encoded within a circumscribed region of space, must be increasing in the thermodynamic direction of time.
Of course, thermodynamic entropy is popularly described as the disorder or "randomness" in a physical system. In 1877, the physicist Ludwig Boltzmann defined entropy more precisely. He defined it in terms of the number of distinct microscopic states that the particles in a system can be configured, while still looking like the same macroscopic system. For example, a system such as a gas cloud, one would count the ways that the individual gas molecules could be distributed, and moving.
In1948, mathematician Claude E. Shannon, introduced today's most widely used measure of information content: entropy. The Shannon entropy of a message is the number of binary digits, i.e. "bits" needed to encode it. While the structure, quality, or value, of the information in Shannon entropy may be an unknown, the quantity of information can be known. Shannon entropy and thermodynamic entropy are equivalent.
If the locality principle is not going to be thrown into the trash heap, then a viable option is that space is something analogous to homogeneously distributed probability density functions(a perfect fluid?) i.e. increasing density gradients, giving the observed thermodynamic arrow of time. The observed cosmic expansion is a "relative" one! A "perspective effect" from our local vantage point. A shrinking object gives the illusion of receding motion. Increasing *refractive* density gradients give the appearence of a doppler-red-shift. Space increases density as matter is re-sized.
Spacetime then "remembers" the input! A quantum measurement is made, the action principle demands the shortest distance between two points be taken, whatever that may be. There is no instantaneous action at a distance!
A universal function of symmetry?
It seems that in order to continue with the idea of a physically infinite "multiverse" instantaneous communication at a distance must be accepted as an absurd axiom. But we must remember, Newtons classical reality was superceded by Albert Einstein's relativity! There is no action at a distance!
Instantaneous action at a distance was shown to be unecessary, by Einstein's theories of special and general relativity. How can an infinite multiverse be in accordance with probability theory if all of the separate universes are not in a type of corrspondence with each other? Who observes the entire multiverse?
So it can be shown that the multiverse DOES carry to much metaphysical, and heavy conceptual baggage. Goodbye spatially infinite multiverse? we cannot really know...
It appears that the best solution is that our universe is self contained. Or better yet, the concepts of inside/outside are an aspect of a symmetric logic functor forming the basis of mathematical existence, therefore, the locality principle is not violated.
Space is at right angles to time.
Electricity is at right angles to magnetism.
Gravity is at right angles to inertia.
All are aspects of one unified field.
Photons are wave-fronts of processed space-time, alot like the frames on a movie reel. Time is a process. We don't move through time, we are processed BY time.
|-->|-->|-->|-->[photon]
Wavefronts ==> cotangent vectors = one forms
As a thought experiment, imagine a perfectly smooth homogenous liquid or "fluid" that stretches out to infinity. An open universe of infinite space-time radius, is equivalent to a closed finite "big shrink", conspanding, or compression universe. There are as many fractions from zero to one as there are natural numbers from zero to infinity.
The fluid has a small distortion or vortice moving in in it. How can the velocity of the "vortice" in the fluid be measured? Its motion can only be relative to another vortice.
Total space-time energy is given by the Einstein-Pythagorean equation:
E^2 = {mc^2}^2 + {pc}^2
Space is at right angles to time:
S
|||||---->T
The thermodynamic arrow of time, points in the direction of continually increasing space-time density. Increasing density gradients. It is a ratio adaption:
{S/T}_n = {S/T}_n+1
S and T are reducing in tandem, such, that their ratio remains a constant c, for the velocity of a photon of light.
S<--{energy}---------->T
Energy compresses{resists} space and dilates{stretches} time.
Since potential energy equals kinetic energy, gravitational mass must equal inertial mass. An increase in kinetic energy always causes resistance in space-time.
Absolute truth can exist within certain limited contexts, as we define it according to very limited human knowledge. But if absolute truth is defined as "the whole truth", then forget it. The total sum of human knowledge is just one small droplet in an infinite sea of TRUTH.
[url= http://www.wikipedia.org/wiki/G%F6del's_incompleteness_theorem]http://www.wikipedia.org/wiki/G%F6del's_incompleteness_theorem[/url]
quote:
In mathematical logic , Gödel's incompleteness theorems are two celebrated theorems proven by Kurt Gödel . Somewhat simplified, the first theorem states that:
in any consistent formal system of mathematics sufficiently strong to allow one to do basic arithmetic , one can construct a statement about natural numbers that can be neither proven nor disproven within that system.
[...]
Gödel's second incompleteness theorem, which is proved by formalizing part of the proof of the first within the system itself, states that any sufficiently strong consistent system cannot prove its own consistency.
Actually, spacetime does not really need to be "sliced up" in that it can proceed in discrete steps, yet, still be continuous.
[density 1]--->[density 2]--->[density 3]---> ... --->[density n]
The information density[i.e. Shannon entropy] of space increases. This is a relation and its inverse.
For example, unity is a constant, representable by:
[1 = c ] = [1/2 + 1/2] = [1/4 + 3/4] = [1/5 + 4/5] = [1/6 + 5/6]
The left fraction represents [energy/momentum] and the right fraction represents compressed [space/time] density, where space means "distance interval" , a relative measurement.
[E/p]<--->[S/T]
[1/R]<--->[R]
The physics for a circle of radius R, is the same for a circle of radius 1/R
E/p = S/T = c
[Space/time] and [energy/momentum] are two different forms of the same invariant quantity [c].
[E/p]_n = [E/p]_n+1 = [S/T]_n = [S/T]_n+1
Yes, c+c = c
[c + c]/[1 + c^2/c^2] = c
So E/p + S/t = 2c/2 = c
S/T = E/p = S/T + E/p = c
A simple diagram-equation representing the unified force:
|{real n-dimensions = ct }
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|<--------{quantum vacuum ~ ict}
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|{real numbers}
Reality is a damped harmonic oscillation, a system at resonance:
{ict}^2 + {ct}^2 ~ 0
One force
F ~ m*m'/d^2
F ~ A/d^4 ~ Casimir effect
F ~ q*q'/d^2
m = mass
A = 2 dimensional area
q = electric charge
d = distance
Solve the Schwarzschild solution for the entire universe, since the universe can be postulated to be a closed system with nothing outside itself. The condition of "nothingness" leads one to ask "What are the properties of nothingness?" Of course there are no measurable properties, but nothingness in itself must be a type of massless solid. A condition that has no distance - metric scales. In other words, there is no outside to the universe, no measurable border between something and nothing.
Nothing then becomes analogous to a perfectly symmetrical pressure force on the surface of existence.
-[F]^2 ---->|U|<---- +[F]2
Simple harmonic oscillation given by the equation [F]^2 = -[K*X]^2
What is K ? What is X ?
U stands for universe. So it becomes reasonable to assume that the entire universe is analogous to that which is inside the event horizon of a black hole. The cosmos becomes a quantum superposition of states, collapsing under the crushing force of "nothingness".
Analytically continue the Schwarzschild solution to the imaginary values of the time variable. The Schwarzschild solution becomes periodic in the imaginary time direction.
All waves would then be standing waves in the closed universe. A Schrodinger wave equation in one dimension is of the form:
d^2 psi/dx^2 + [2m/hbar^2] [E - U{x}] psi[x]
U{x} is the potential energy and E is the total energy.
psi[x]is the wave function for a state in which the energy E is constant in time. Such states are called stationary states. Certain definite vibration frequencies are allowed multiples of fundamental wavelengths
lambda = h/p
|psi[x]|^2 dx, is the probability of finding a particle{universe} in a certain state between the region x and x+dx
psi^2 = psi psi* . When psi is complex, psi* is the complex conjugate of psi. psi^2 [x] is the probability density.
An equation for the damped oscillator in one dimension:
X = A[exp[-{b/2m}t]]*cos[wt + theta]
Why not describe Einstein's equation as a rule that tells the geometry of space how to evolve as function of time? Lorentzian manifolds M, diffeomorphic to R x S, where the manifold S represents space, and t, an element of R, represents time. So spacetime is sliced into instants of time as an arbitrary choice, or possibly boundary limits, imposed by Planck's constant.
F: M---> R x S
Spacetime becomes quantized or "sliced up" but that could be what nature really does. According to relativity, an objects position and momentum can only be defined with respect to a frame of reference, i.e. another object. Yet the universe as a whole has no frame of reference outside of itself, so how can its momentum be defined? It can only be defined with reference to itself. Worldlines fill up spacetime and the criss crossing of world lines mark events beyond the need for coordinate systems or coordinates. Points in spacetime are given the name "events" so there is a coordinate independence.
The geometric view of physics means that the laws of physics are the same in every Lorentz reference system. Local Lorentz invariance. But since the universe has no exterior reference frame, and it must refer to itself, its world line intersects with itself. This quantized-evolution of spacetime dictated by GR and QM, means that the world line of the past intersects with the world lines of the present, for the universe. A geometric stacking of space like slices, parameterized by t, The universe is a function of itself. Spacetime becomes compressed. As the time evolution proceeds in the thermodynamic direction of t, the space like sheets continually increase in density. The information storage of space time.
[->[->[->[U]<-]<-]<-]
This increasing refractive spacetime density must be background independent. The increasing density functions are, in a sense, equivalent to the non-Euclidean geometry of Riemann and Einstein.
Is gravity a statistical probability distribution?
Yes, the force called gravity is actually geometry, non-Euclidean geometry, where spacetime becomes anisotropic and inhomogeneous in the presence of mass-energy. Also representable as a Gaussian distribution.
Then the question becomes "what is space?" "What is time?"
Space is relational. Time is a process.
Heisenberg Uncertainty:
DxDp >= hbar/2
The relation becomes totally "chaotic" below the Planck length. So, space could be described as a self similar relation which is generated by the quantum foam, and forms Penrose's "spin networks". Something possible analogous to 010101010101010101...TFTFTFTFTFTFTFTFTFTFTF...
The curvature of spacetime could be represented as a Gaussian distribution. If mathematics only is an approximation of reality, then the mathematics of probability corresponds "exactly" with reality.
The Riemann tensor explains how a tangent vector, parallel translated around a tiny parallellogram is changed. So, to say that spacetime is "curved" means how much a tangent vector changes during parallel transport around a loop. Parallel transport is the translation of an infinitesimal tangent vector along a geodesic.
Yes, the probability distribution should agree exactly with the Riemann tensor of Einstein's relativity, or better.
Theoretical physicist Richard Feynman derived the "sum over histories" interpretation of quantum mechanics, where a system does not have a single history, but it has every possible history, and each history has its own probability amplitude. For example, an electron travels from point A to point B by every possible route at once. Each possible route or "path" corresponds to a history.
http://www.maths.usyd.edu.au:8000/u/hughl/PI.html
The amplitude for each history defines the probability of that particular path being followed. The number involves the "action" associated with the history-path, which seems to determine that the path taken, will be the history closest to the "classical" trajectory, in accordance with the law of conservation of energy.
[ 18. January 2004, 01:16: Message edited by: Russell E. Rierson ]
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chimp
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posted 20. January 2004 01:34
From the book "Mystery of the Aleph", the axioms of set theory:
3rd Axiom, The Axiom of Specification:
quote:
To every set A and every condition S(x) there corresponds a set B whose elements are exactly those elements x of A for which S(x) holds. This is the axiom that leads to Russell's paradox. For if we let the condition S(x) be: not(x element of x), then B = {x in A such that x is not in x}. Is B a member of B? If it is, then it isn't; and if it isn't, then it is. Therefore B cannot be in A, meaning that nothing contains everything.
While all of mathematics sees it as a paradox, I see it as an absolutely true statement:
Nothing contains everything
So it seems to me, that the Absolute infinity is analogous to a condition of total Nothingness... also total Beingness...
.
To explain existence as a "Being" seems to be to say nothing specific about existence at all. The concept of Being is an empty concept, whose content is nothing. Yet also, there is the concept, of Nothing, which has equivalently, the same content as the concept of Being, but which seems to stand in diametric opposition to it. Nothingness is the negation of existence and Being is the essence of existence, ergo, Being and Nothing must form a synthesis, a "Hegelian Dialectic"? where the new synthesis is what we call the "Set of all Sets" or the "Absolute Infinity". Therefore Nothingness is of the same determination, or rather, an abscence of determination, and thus altogether the same as the pure essence of Beingness. On the other hand, paradoxically, Being and Nothing are not the same. So we have the difference of Being and Nothing passing into identity, and the identity passing into difference. Ergo, their truth is a dynamic, the dynamic is an immediate vanishing of one in the other: becoming, a movement in which both are distinguished, yet both are a aspects of a higher level of symmetry.
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Evan
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posted 20. January 2004 07:44
Yin/yang - try the Tao Te Ching for a change of reading material, Russell. All opposites are two aspects of a complementary duality that form a more fundamental unity. However, this unity cannot be known through rational knowledge, because rational knowledge (and the infamous two-value logic often mentioned in these discussions of logic) can only see the duality, not the unity. The Tao that is known is not the true Tao, they say.
[ 20. January 2004, 07:48: Message edited by: Evan ]
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Kastl
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posted 14. February 2004 19:55
Russell,
You say that "The information density of the universal system must be increasing". What if information density were constant? Is this a possibility? If so, how would that alter your thoughts?
This 'constant' would effectively equate space with information. I see all kinds of fallout - such as a theory of what gave birth to the big bang and an explanation for the equivalence principle and a mechanism for GR.
Your thoughts please.
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