I’ve been pondering realism for the past few months. In particular I’ve been thinking about Bell’s theorem, and FQXi member Joy Christian’s claimed disproof of it. I recently posted a paper on this (http://arxiv.org/abs/1107.1007) and I’d like to discuss the topic here.

You can watch Joy Christian describing his research at FQXi’s 2009 conference in the Azores below:



(After reading my post, you should also read Joy Christian's response: here.)

Some background: John Bell coined the word “beable” to capture the essence of any realistic theory of nature. It is made of two parts: “Be” as in Hamlet’s tortured question, “To Be or Not To Be?”, and “able”, an ending shared with “Observ-able”. Beables were supposed to be the building blocks of the observable. If beables exist, however, then they are at odds with quantum mechanics because quantum mechanics on predicts only probabilistic outcomes and also it claims to be complete. (There is a fundamental theorem by Gleason which states that the only prescription of obtaining predictions in quantum mechanics is what quantum mechanics is using. What this means is that quantum mechanics is unique and cannot be improved upon any further.)

For the strong believer in realism there was only a way out: quantum mechanics is incomplete. This is the conclusion reached by Einstein, Podolski, and Rosen in now their famous EPR paper. If quantum mechanics is incomplete, then there must be some hidden variable, some “beable”, theory which captures the essence of nature. These beables cannot be directly measured in any experiments carried out by us, as clumsy macroscopical beasts in the flimsy world of atomic domain. The experimenter must be a hopeless “bull in a china shop,” trampling delicate beables when making any measurement.

For a long time after the EPR paper, neither the realists, nor the supporters and practitioners of quantum mechanics could convince the other that their viewpoint was right. Then Bohm found a hidden variable theory, which claimed to recover all predictions of quantum mechanics, and yet it was a realistic theory. That may sound too good to be true, but the price Bohm paid for this was introducing instantaneous action at a distance. Later on, more serious problems were discovered in his model, like its inability to have a realistic description for spin. (For me what I find disturbing in the Bohmian model is how the hydrogen atom is described: two opposite electric charges standing still at a fixed distance from each other.)

However, after Bohm introduced his model, John Bell studied it in detail and played with other hidden variables models, attempting to find a conclusive proof in favor of realism. Specifically, John Bell was studying the correlations of two physical systems separated at a large enough distances that relativity forbids communication between them during the measurement. What he found in the end (and not without irony) was a result now called Bell’s theorem, which was a turning point in the demise of all hidden variable theories.

So why is Bell’s theorem special? After all, we see correlations in our daily lives and no quantum mechanics is required. For example, if I have one glove in each pocket, and I extract a left glove from my right pocket, I know with certainty without looking that the right glove is left in my left pocket. But here is the catch: if quantum mechanics is right, there are residual effects of its inherent randomness which would result in additional correlations that cannot be explained by any hidden variable theory. We may be clumsy macroscopic objects, but John Bell found a way to circumvent this limitation. And now the discussion between realists and quantum mechanics can come down from the rarefied air of philosophical discussions into experimental verification. The experiment was performed by Aspect and nature ruled in favor of quantum mechanics. Quantum systems exhibit this funny behavior called entanglement where the best knowledge of the whole system does not contain the best knowledge of the subsystems. Hidden variable theories are dead; case closed and let’s move on. Or so it seemed.

But the story does not end there. Along comes FQXi’s Joy Christian, who recently made a very bold and provocative statement: Bell’s theorem is wrong.

Christian argues that Bell proved his theorem by making a “topological error” and by demanding that beables must commute with each other. To see he argument, take rotations in 3-dimensions, for example. It can qualify for the beable status because rotations are trivial macroscopic classical operations. Yet, if you perform two rotations around different axes on a body, you’ll find that the end result depends on the order you carry out the rotations. (If you don’t believe it, there’s a cute demonstration here using a rotating panda.)

Rotations, then, are non-commutative. Therefore it looks like one of the assumptions in John Bell’s theorem is artificial and not physically justified. So what happens if this assumption of commutativity is dropped? Here is the remarkable thing: Joy Christian managed to prove in several important cases that entanglement is an illusion due to our commutative prejudice, and complete separable descriptions—that is descriptions of the two physically separated subsystems independent of each other—are possible.

It was at this point that I became very interested in his result, and recently I had completed an analysis of it attempting to put the result in the proper context (http://arxiv.org/abs/1107.1007). Let me try to explain Joy Christian’s results and why, in my opinion, the claim that Bell’s theorem is wrong in not fully justified.

Key-Ring Model of Hopf Fibration

The way Joy Christian proceeds is by counting all elements of reality in a given experimental setting, meaning that he counts both factuals and counterfactuals. This is not what usually is understood in the context of Bell’s theorem, which is supposed to explain correlations between actual experimental results, but let’s follow along. The space of all actuals and counterfactuals forms a topological space. For example, in Bohm’s variant of the EPR experiment, the experimenter measures spin on one direction, and the outcome can be plus or minus one. The topology of the actual results is a disconnected set of two points (plus and minus one), while the topology of all actuals and counterfactuals is an ordinary two-dimensional sphere corresponding to measuring spin on all possible directions in space. This “S2” sphere is embedded into an “S3” sphere where the so-called Hopf fibration construction is possible.

(The reason “S3” is used is because “S3” is parallelizable. To see what this means let’s start with a regular 3-dimensional sphere S2 and use the so-called “hairy ball theorem”. Imagine adding hair at any point on this sphere. Then try to comb the hair continuously with no singularities anywhere. This cannot be done: there will always be at least one singularity (in an actual head called a cowlick). S3 is different because this combing can always be performed. The “combed hair” in S3 can represent for example a vector field arising from tracing a particular dynamical evolution of the system. When you can always comb without singularity a manifold, that manifold is called parallelizable.)

So here is Joy Christian’s prescription: start with an experimental setting, count all potential experimental outcomes, and then embed this result in a parallelizable manifold. Call the resulting topological space Omega and this represents elements of reality or beables. The particular parallelizable manifolds Joy Christian used were spheres. There are only 4 spheres which are parallelizable: S0, S1, S3, and S7 corresponding to the four normed division algebras: real numbers, complex numbers, quaternions, and octonions. There are other parallelizable manifolds, but the spheres enjoy an additional property critical for disproving Bell’s theorem: in any normed division algebra, the norm of the product is the product of the norms. This core property is the key of proving entanglement wrong because one can “disentangle” or separate the whole into parts neatly. This method was applied for the original spin one half EPR-Bohm experiment, for GHZ states, and for Hardy-type theorems. So it seems that Bell’s theorem was “disproved” by counterexample.

The original reaction of the physics community was to search for mathematical errors in Joy Christian’s arguments. Joy Christian uses geometric algebra for his results, and many original criticisms were plagued by mistakes in using this unusual mathematical formalism.

So is Joy Christian right or wrong? Both actually. His counterexamples are valid entangle-free realistic models. This is were he is right. But one still cannot exceed Bell’s classical correlations using only local resources. Rotations as beables are not “actionable beables” and any mechanical device realizing them in forms of gears like the ones used in clocks, for example, cannot exceed Bell’s correlations for spatially separated subsystems because relativity forbids rigid bodies. In fact there are two theorems by Clifton (arXiv:quant-ph/9711009v1) demanding commutativity for beables in two very critical cases: relativistic field theories with bounded energies, and for what I would call “non-contextual ontologies”. Joy Christian counterexamples are in fact contextual “environmental” hidden variable theories. What do we mean by that? Broadly speaking hidden variable theories falls into two main classes: contextual and non-contextual, while the non-contextual ones sub-separate into two more subclasses: “algebraic” and “environmental”. It takes too long to explain in detail here, so I will take a shortcut instead.

Let’s take a look at the space Omega. The topology of this space depends critically on which experiments are about to be performed. For the same initial preparation of the system, select different measurements and you may get different separable beables and hidden variables. The ontology of the realistic model is contextual (dependent) on the experimental setting. Contextual realism is almost a contradiction of terms. All interpretations of quantum mechanics aimed at explaining it in usual classical terms suffer from a problem or another: Bohm’s theory demands a faster than the speed of light “quantum potential”, transactional interpretation uses signals from the future, etc. Joy Christian’s local realistic model demands a contextual ontology which is at least just as strange as quantum mechanics itself. Common sense everyday activities have no place for contextual ontology—althoughpoliticians love contextual ontology as they usually claim: “I was not flip-flopping, I was merely cited out of context”, to an average person this is a sign of dishonesty and not a valid explanation. (Who knows, maybe if Nixon had known about quantum mechanics, he would have said: but this is how quantum mechanics and nature works, I am not a crook.)

Joy Christian’s examples cannot be non-contextual hidden variable theories as they are ruled out by the Kochen-Specker theorem. They cannot be algebraic hidden variables theories because all predictions of quantum mechanics are recovered in the particular experimental setting. The only possibility left are environmental contextual hidden variable theories. And indeed, one can find a particular example where this becomes explicit: an EPR-Bohm experiment with spin one particles (the Heywood-Redhead experiment). The beauty of this experiment is two-fold. First, for spin one, we can use the Kochen-Specker theorem and prove that all hidden variable theories in this case are contextual. Second, Howell and collaborators (arXiv:quant-ph/0105132v2) showed the equivalence of this experiment with two spin one-half EPR-Bohm settings where we can apply directly Joy Christian’s analysis. All that is left to do is to marry the two lines of argument and show explicitly that indeed Joy Christian’s model for spin one-half is a contextual environmental hidden variable model.

So now is time to tally up the pluses and minuses of Joy Christian’s claims.

On the plus side:

Joy Christian’s results are important because they show that entangled particles can be disentangled sometimes in a particular mathematical formalism. Entanglement is no longer universal as it depends on the mathematical formalism used. Bell’s theorem alone cannot rule out all contextual hidden variable theories obeying local realism. Joy Christian’s result does chip away some of the importance and glamour from Bell’s theorem, as it cannot be regarded anymore as the silver bullet which killed all hidden variable theories.

On the minus side:

Bell’s limit from Bell’s theorem still stands and local realistic models of quantum mechanics are still not possible for ALL experimental contexts. Bell’s theorem is not “disproved” mathematically because Joy Christian starts with different mathematical assumptions. Bell’s theorem importance is not “disproved” because Bell’s limits continue to stand for all local operations and classical communication processes. Bell’s and Clifton's theorems together now do the job of what people originally attributed to Bell’s theorem alone. Hidden variable theories remain a dead end in physics.

I'm grateful to Joy Christian for posting his response to my thoughts: "Quantum Music from a Classical Sphere".

this post has been edited by the forum administrator

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The issue with the parallelizable spheres and the topology which Joy Christian cites as the error in Bell’s theorem may suffer from a little problem. The Hopf fibration

S^3 -- > S^7 --- > S^4,

defines the Hopf invariant, which tells us how to link 3 dimensional spheres in 7 dimensions. The S^7 satisfies the Hopf fibration at the next level

S^7 -- > S^15 --- > S^8

Where the three spinors which span S^3 exist in 7 combinations according to the Fano plane “rules,” and these are dual to the S^4 elements which include the unit plus three e_i elements, again in 7 possible combintations. This has some deep implications, which includes this matter of commutative beables. We may then say that the quaternion field data on the S^4 is equivalent to the spinor field data on S^3. This is why the Weyl equations do reproduce the same information as the more compact Dirac equation. By way of illustration I propose how this might fit into field theory, which then brings us to the issue here.

A particular basis of S^3 will obey an associator condition, as will the dual quaterions on S^4. The associator means these two rules then operate in a single system with two groups of 4 qubits which form an exceptional algebra of 8 qubits. These are octonion valued with a basis 1, e_i i = 1, …,7. The nontrivial elements define associative triplets which satisfy multiplication rules on the Fano plane. The triplets define an associative 3-form

Ω^(3)(x, y, z) =

The remaining components are then nonassociative elements of S^8. This nonassociativity means that the structure constant for any algebraic structure is not constant so [e_i, e_j] = 2e_ie_j = t_{ijk}e_k. The associator gives

(e_ie_j)e_k - e_i(e_je_k) = t_{ijl}e_le_k - e_i t_{jkl}e_l

and there is a commutator [e_i, t_{ijk}] =! 0. There is then a teleparallel connection or torsion associated with nonassociativity. To define the nonassociative half of S^8 according to an associative algebra define the four-form

Ω^(4)(w, x, y, z,) = 1/2(< w, x(y-bar z)> - < w, (x, z-bar y>),

over the coassociative algebra dual to the associative subspace . The full eight dimensional space has the two dual operators Ω ± iΦ as the twistor description of the spacetime.

The spacetime is Ω^{(4)} = Re C^4, which has the group action U(4). This is decomposed into U(4) = spin(6)x U(1). The 15 dimensional Spin(6) is a conformal group that contains the Spin(5) ~ SO(3,2) de Sitter isometry group. The remaining 5 parameters from the Spin(6) are a diliton factor and four conformal factors. This is a form of the AdS/CFT correspondence.

The data on the S^3 is given by a Hopf invariant ∫Ω^(3) = 4πn, as the link number. This information is dual to the data on S^4, which is not a parallizable space. Hence the data which is presumed to be missing in the Bell theorem could be argued to be inconsequential.

Cheers LC

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happy to see people focus on my theory, don't stop.

Insert physical spheres and also the volumes, more my equations ,you shall see the real proportions if the pure serie of the quantum number is inserted also.

The COMMUTATIVITY AND THE ASSOCIATIVITY of course like a torch of foundamentals.

Interesting discussion that said dear Florin and Joy, interesting work and extrapolations. Congratulations.

Steve

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Florin,

Just a tiny detail, not affecting the overall exposition, but I have to tell you, because you may, unwillingly, initiate a misconception about what "parallelizable" means.

S³ is indeed parallelizable, but not just because it can be combed. "Uncombable" implies "non parallelizable", but the reverse is not true, hence it is not true that "When you can always comb without singularity a manifold, that manifold is called parallelizable". Take as an example the Moebius strip: one can comb its hair, but one can't find n global vector fields which form a basis to the tangent space at any point. So, to be parallelizable, a manifold has to wear simultaneously n hair-styles which are linear independent at any point.

Cristi

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Since commenting on Joy's reply, I thought I should comment here. Florin makes some very good points. I want to agree with Florin that Joy's result does not mar the importance or significance of experiments on non-locality and entanglement. I disagree with Joy's claim that his idea is a disproof of Bell's theorem, per se.

Instead; he provides an alternate description which renders some of the paradoxes resolved and some of our contradictory dual descriptions unnecessary. At the same time; it is a far cry from a complete theory - robust enough to replace the current view. As TH Ray says, he does not have an alternate structure to 'subsume all of known Physics.'

But I think he is on to something important. My guess is that it has to do with extended vs compact dimensions, which relates to our concept of locality. A 4-sphere has maximal hypervolume. When we add spatial dimensions beyond 5, they must therefore be compact! I agree with Joy's comment that octonions are simply the most general case (numbers and algebras), and feel that if we treat them as fundamental some of the paradoxes are seen as alternate projections of one framework.

So I believe his approach could work, and explains much, but takes nothing away from the validity of other work or the utility of other constructions.

Regards,

Jonathan

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Dear Peter Van Gaalen,

You placed a post in here a day or so ago to which I responded. Unfortunately, several posts were deleted in an effort by the administrator to keep these blogs on topic.

As I recall, your models equate 'fundamental units' with 'fundamental dimensions'. Is your model now up to 16 dimensions? That could simply be two bundled octonions - exactly double the space with which Joy is working.

It almost seems like all of our 'dimensions' need to be 'modulo 8' because of the periodicity of the normed divisor algebras. Thus, to get 4-D of Spacetime, we need 12, 20, 28, 36,... total dimensions, where we must include all dimensional effects from topology, supersymmetry, and scales.

Have Fun!

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Dear Joy, Florin, and Lawrence,

I made some comments earlier with minor errors and never received a proper response, so I am correcting those errors and reposting those ideas. Joy said that the devil is in the details, and yet some of those details bother me.

Joy is working with a parallelizable and potentially-stable 7-sphere, but he also has these weird statistics that continuously range from negative one (an anti-symmetric fermion wave-function) to positive one (a symmetric boson wave-function). This looks like the real projection of Anyonic statistics with phases of exp(i*theta), where theta ranges from zero to 2*pi. Anyons are quantum particles on 2-dimensional surfaces that don’t behave exactly like fermions or bosons (and may thus be a good place to start in an effort to unify quantum statistics into a TOE). Some examples of possibly important 2-D surfaces are the M2 Blackbrane and the effectively 2-D Graphene lattice of Holographic Theory.

Now suppose that the potentially-stable 7-sphere were to spontaneously decay into an M2 Blackbrane, its dual NS5-brane, and an eighth dimension of time. The M2-brane could explain the occurrence of Anyonic statistics, while the NS5-brane explains why the five-fold Pentagon color graph (see Florin’s Color Me Surprised post with the Kochen-Specker Theorem) is a minimal model for quantum behavior.

The devil may be in the details, but these details seem consistent with M-Theory…

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Florin,

Having read your article and Joy's response, I look forward to your next response.

You suggest that Bell's theorem "retains its full importance as the key result which suggested experimental tests for settling the hidden variable theory questions."

Is it correct to say that "violation of Bell's inequality is *the* basis for the rejection of local realism in physics"?

Edwin Eugene Klingman

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The body/eye enjoins invisible/visible space where vision/space begin -- inside the body/eye. See the black space(INERTIAL) beginning of space/vision in the eye/body? Then FULL GRAVITY is felt at the feet/ground -- the end of space/vision. The top of the body/head is also enjoining invisible space while waking/standing.

When we have EQUALLY (and both) visible and invisible space -- as we do in dreams -- then it is all the same space -- with half inertia and half gravity; as space/vision begins [equivalently] as/where it all ends.

This is why space is semi-detached from touch in dreams. Half gravity and half inertia.

This is why the manifestations of force/energy in dreams are at [a maximum of] one half of that of the force/feeling that is felt at the feet/ground while waking/standing.

Sorry folks, the math games are a joke for the most part, as I have fundamentally unified physics.

DREAMS DEMONSTRATE SPACE MANIFESTING AS INERTIAL/GRAVITATIONAL/ELECTROMAGNETIC ENERGY.

I DEMAND ACTION FQXI.ORG. STOP THE LIES IN PHYSICS.

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The self ultimately represents, forms, and experiences a comprehensive approximation of the totality of experience. This means that being/experience is ultimately integrated and interactive, of course. This requires a fundamental and profound revision of physics/physical analysis.

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Again FQXi.org ACTION!

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The primairy discussion between Florin and Joy is at the basis of our scientific reasoning, as we are (like at every moment in history) at the entry of an important new era, these discussions contribute to the addition of more coulours in the rainbow of physiscs and mathematics, be aware of staying inside your "sphere" and staying centered in the middle, perhaps it is wiser to think that the twodimensinal surface of the sphere is the origin of the holographic experience (pub) we are living in.

keep on thinking free

Wilhelmus

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Steve is right about the rampant misdirection and diminishment of physics/truth/reality here at FQXi.org. It is shameful.

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Why ignore this: ULTIMATE TRUTH IN PHYSICS DEPENDS UPON BODILY EXPERIENCE.

The self ultimately represents, forms, and experiences a comprehensive approximation of the totality of experience. This means that being/experience is ultimately integrated and interactive, of course. This requires a fundamental and profound revision of physics/physical analysis.

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I may be stepping deeply into it, to do this, but I already opened my mouth, so I would like to offer this partial draft of a paper I have been working on for quite a while that I think is on topic for employing algebraic, topological and analytical techniques -- toward analytical continuation over topological manifolds. I decided that I might as well reveal it while the subject is hot.

Joy identifies S^7 as the "special" topology. I think, however, that S^3 is the one, and that we need all the rational functions of the closed algebra.

Tom

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Dr. Ray Monroe,

Have appreciated reading your posts and hope that you have a good day?

James

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Ray You have misundertood what I said simply, and of course all my posts are deleted still no but frankly it's what that Brendan. I made a comparaison Ray , never I said that about you no but frankly .

Reread my post in french but they have disappeared. Incredible that.Where are my french posts ? still a stategy because you fear of my language.It is what That???? In fact you are a real band of what??? I can ask me several questions with all these comportments.

Steve

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WHERE ARE MY POSTS IN FRENCH???? YOU FEAR OR WHAt of my french ???It is incredible that,some people are really bad people there, just for this monney, but perhaps they want my recognizings and my monney also, never!!!!You fear of my french and my sciences. In fact you thoughts never that about me and now yoyu are in a bizare position, I can understand indeed. Never you thought that my theory was so complete. I understand your desire to profit of my future capitals. Never it's my works.

Steve

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Very interesting article in NEW SCIENTIST , from Lee Smolin and others, his phse space needs 8 dimensions, I do not understand why, but I see some paralels with the topological space OMEGA.

keep on thinking free

Wilhelmus

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I like your points of vue dear Wilhelmus,

How I react, I think it's perhaps the only one who understands really my theory, Joy seems a good scientists, I don't know his generality,nor his universality at this momment. I think he wants simply make a kind of work towards my theory, if it is that and if his work is good, why no? after all my theory is there for a pure sharing indeed.But with a deeper analyze,they want my prizes,that's all wilhelmus, have you seen their strategy, frankly it is just for monney.But after all they are just humans who makes errors. The most important now Ray, you want my prizes, never, you can copy after all, all people knows now it is my theory, and come on APS linkedin, the discussions are relevant also. Wilhelmus, they try that's all for my future capitals and the future billions, never. That's I must be strong. These kinds of people have already stolen me and that continues, it's sad.In fact you know ,they thoughts that my language, my sceinces, and my maths were noty reality.Now they understand and they are in a bizare position for their future credibility. But I can understand them and i can understand that they have a business team behind, you know there people who implies the chaos. And what I am going to say thanks also and accept that people profits of my economical situation. Never, that's why I have written on several platforms for a kind of protection, I was obliged.

Ray, if you say that the works of joy are more important, there yiou proof in live that it is just a business. Don't fear of my french I know I write well in all humility,lol but after all you oblige me to be like that, I am nice, but don't play with my kindness and my spirituality nor my universality. Frankly you are very bad fallen dear Team.You can use all strategies what you want , never that will change. I have nothing against you but You lisi, To and Lawrence, and Florin more now Joy , are just a team which supports the works of each other.Any sense. If now you want make a kind of super discussion between Joy and Florin for a kind of pseudo mathematical complexity, you are very bad fallen also, if you want imply confusions for publics, you are very bad fallen also. If you have no others strategies, I CAN SUGGEST THAT YOU LEARN MY EQUATIONS. E=(c²o²s²)m and mcosV=constant.where of course the number must be inserted with the volumes.Like the entanglement of cosmological spheres, you see the center ray,me yes. Don't copy but be rational.It is not a parano, no I am just conscient of this human nature and his taste for monney and power.I dislike these things simply.

Steve

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My dear Ray,

You mention : infinite Big Bang singulairity.

Isn't that a "contradictio in terminis", surely when you apply this in a causal 4D universe, because when you accept a singulairity as "beginning" in fact it is "nothing", an infinite nothing is a very difficult idea for me.

If you place this origin after the lower limit (as I see it) of our universe the Planck length or the INTEGRAL length (10^-48m) then there is no longer a singulairity problem if you accept that after one of these lengths there is no more causality, but a new dimension wher every possibillity is present (read possible, see my essay wher I call that Total Simultaneity), so in fact your infinite singulairity is a way of expressing something that we cannot understand, perhaps it is comparable with Joy Christian's OMEGA.

keep on thinking free

Wilhelmus

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Florin,

I searched through the messages. I didn't see one of yours about the 'von Neumann theorem'. I went through it message by message twice. I did use 'von Neuman' as my key words to spot. Perhaps I should be using other key words to look for?

James

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