Tegmark's Mathematical Universe

Hi to both of you ,Florin and Lawrence and Happy Christmass .

You know I have nothing against your ideas but I think simply that if you superimpose several imaginaries and pure mathematical extrapolations ,you are going to be in a big confusion about our physicality .A lot of theories in this line of reasoning thus are not possible for a verification .I agree it is so far in the extrapolations what it is difficult to verify them. The pure maths are foundamentals for the physicality if and only if the good numbers and the goog limits are utilized ,it is not a question of tools but of referential .With your capacity to play with maths inside a good referential ,your results shall be so important and thus very important for the sciences community .There the complemenatrity seems still an essential .

Best Regards and happy new year too

Steve

I have this idea that loop variables might be a constraint system on spacetime manifold as defined in a stringy sence. The AdS ~ CFT correspondence does indicate that spacetime does have a background structure. The LQG theory is a system of constraints (based on spinorial ADM relativity and Hamilton constraints) that might work to map AdS --> dS where the de Sitter sector is the physical universe, and the AdS is more of a spacetime fiction corresponding to conformal fields. So the LQG equations seem to be one way in which this might be accomplished. My attempts at this ran into considerable difficulties.

With respect to the Tegmark many worlds idea and set theory I found the following Skolem's Paradox, from another discussion. This seems to offer up a way of having first order logic systems which are also ZFC with arbitrary cardinalities. Again, I am not exactly on this sort of track. I think getting quantum gravity and holography worked out is a sufficient challenge. Maybe this does impact how other cosmologies can exist in a multiple outcome situation with quantum gravity. We might if we get good enough we will figure out how to detect some quantum interference between our spacetime and other spacetimes with other cosmologies. Tegmark's ideas are much to "far out" for me to consider as potentially empirical science.

Cheers LC

Dear Lawrence ,

Could you tell me more about the theorem of incompleteness please ?

About the works of Tegmark ,which I respect too like all works ,even if I don't agree,I have a question for the multiverses .These multiverses are in one system thus what is this uniqueness .

The Universe is purely physic ,the maths are just a tool synchronized if the referential ,physical is correct .

The multiverses idea have not limits and a real toplogy .It is just infinite in the extrapolations even for the laws and their invariances and coherences .Thus all looses its sense in the uniqueness and the specific thermodynamical link .On the other side in a human imaginary point of vue ,it is beautiful ,but is it foundamental .

If we imagine a Universe with the hubble law ,I see a big paradox about the expansion if we go behind the limits ,like the speed of the light ,the increasing of the wavelenght and the Doppler effect ,our perception is false about the expansion I think .I think it exists a lot of confusions about the real movements ,if the rotation aroud the cenetr is considered and furthermore the real movement in a specific closed system and its laws of evolution.,thus the real dynamic can be understood since the begining in the physicality.

The spherical objects don't move due to this expansion ??? The perception is different than our reality .

Regards

Steve

Lawrence,

Skolem's paradox in interesting in the way it discredits the realism ideal. In physics the debate about hidden variables and the proper interpretation of QM is old news, but in mathematics the naïve interpretation of set theory would lead one to believe in realism/absolute truths. Set theory is a rather hard area with many unintuitive results.

Steve,

Incompleteness theorem is rather easy and well understood. From 10,000 feet, it goes like this: Consider the liar's paradox: "This sentence is false". If it is true, then we take it at face value, believing what is says and it says it is false. Hence true implies false. In reverse, suppose the statement is false, meaning that the sentence is indeed false, but this is what is says itself, so it is true. Hence false implies true. In conclusion true->false and false->true. Same for any other antinomies, like the barber's paradox: "a barber is the person who shaves precisely the people who cannot shave themselves". Does the barber shave himself?

Now Gödel replaced truth with provability and he formed the following sentence: "This sentence is unprovable". Now if false, it means that it is provable, meaning that there is a sequence of logical steps proving it. But wait a minute, in this case we just proved a false statement and therefore we have an inconsistent system. If the statement is false we have inconsistency. Now if the statement is true, we have incompleteness. Why? Because there is this statement which is both true and unprovable. So what Gödel showed was that one has either incompleteness or inconsistency. Now this is all a big handwaving, to make this airtight and rigorous, there is an entire mathematical construction behind it, but it can be done successfully. The key was to find a rigorous mathematical definition of provability.

Hi dear Florin ,

Thank you very much for this explaination about the incompleteness theorem ,I understand better its meaning.I liked your line of explaination .

I am going to learn more about the works of Godel ,Tegmark ,Wheeler ...I will encircle better this necessity to have these referentials .The incompleteness seems imply a non limits in the system .

I agree it is relevant about the creativity and potential of our brain .But in the physicality ,the ultim axiomatisation seems in one universal system .The only incompleteness is the evolution and thus our step of evolution .The logic and the rationality shows us the pragmatic road even for an idea or a theory or an intuition .If an equation is coherent with its physicality ,thus all recursives axioms shall give an universal correlation .

If the superimposings are inserted in a specific definition of a system with a serie which is personal thus the incompleteness becomes a confusion and an ocean of paradoxs ,just due to the utilization of the imaginaries in the logic complexity of the referential and its laws .The incompleteness apears like an false evolutive point of vue .

On the other side in a specific system invented by humans ,like a computer ,thus the incompleteness takes all its sense in the selectivity of the codes and thus the series .This kind of superimposings imply a specificity where the physicality and its laws are different .There I can agree about some paradoxs because it is correlated with the encoded architectures .The theory of wholes and the paradise of Cantor seems a tool of pure maths which implies the pure confusion about the reality of the physicality .For all x in R or E ....all is a question of referential and synchronization with the real numbers in the physicality in fact in my opinion .Thus how can we interpret the axiomatization ,perhaps only with our evolutive limits in accepting the real whole.The human logic seems far of the universal rational logic .I say that for the physicality of course not for the computing .

The inconsistency of a theory is not a reason to accept the theory like an axiom or in the other side .

I admit it is a little confusing for me .I am going to learn more ,it is very interesting in fact .Thanks dear Florin .

Best Regards

Steve

Hello Steve, Hello Florin,

by thinking about Zurek's Quantum Darwinism i came to an analogy i built some months ago to understand more about the dynamics with which superpositions could create reality. This analogy refers to the barber's antinomy. In its original form it was formulated by Bertrand Russell like this:

„You can define the barber as 'one who shaves all those, and those only, who do not shave themselves.'

The question is, does the barber shave himself?"

One could "solve" this antinomy by simply assuming that both alternatives - shaving himself/not shaving himself - are somewhat superposed in the way that the candidate for the shave shaves one side of his face (this could be the barber himself) and another person is shaving the other side of his face. But that would be against the original spirit of the antinomy.

My "solution" goes a step further and says that

"the barber shaves all those and only those men, who shave the barber and only the barber."

This statement is equivalent with the situation that the barber shaves all those and only those who do not shave themselves - but without having to shave himself. I wrote a paper about why this could indeed be the case and would be happy if anyone interested in it would read it and maybe comment on it. I will attach it as pdf-file here.

I wish the entire fqxi-team, all members and participants of the current fqxi essay-contest and all commentators a happy new near and want to thank all these people for the possibility to exchange interesting ideas and viewpoints.

Best regards

Stefan Weckbach

I am not sufficiently knowledgeable of set theory to comment on much depth. In discussions with somebody else Skolem's paradox came up. The thought occurred to me that this "naïve" approach to set theory (or an countable cardinal interpretation that occurs "paradoxically") might be a way in which Tegmark's approach to his "ultra-verse," or what ever we call it, might work in some ways that fits into deeper mathematical foundations.

My interests are more parochial I suppose. I wonder how it is that LQG and other spinorial approaches to general relativity fit into the string framework. I don't think that dynamic triangulations or LQG are robust enough to describe foundations well on their own. However, they start from completely reasonable assumptions or bases, so I don't think they are wrong in the sense that phlogiston theory was or pre-20th century aether theories were.

Cheers LC

Dear Stefan,

You have quite a paper on the barber's paradox. However, I do not really understand your solution and I feel it is mostly semantics. In my opinion, there are only 3 solutions for self-referencing paradoxes.

1. a standard antinomy is just nonsensical (like liar's or barber's paradox). It is similar with asking: "what is the color of the eyes of the king of USA?" Each word is well defined, but together the sentence is meaningless.

2. time evolution solution: A(t) implies ~A(t+1) implies A(t+2) ...

3. QM solution via superposition. |Psi> = |A> + |~A>

Solution 1 corresponds to no-go theorems in physics like no-time travel paradoxes via either time travel is forbidden, or time travel implies lack of free will and demands "destiny" to avoid all paradoxes.

Solution 2 is realized for example in the basic electromagnetic oscillator.

Solution 3 is possible only under very strict conditions. In a theory with interactions it leads to unitarity violations.

In nature we avoid self-referencing paradoxes because of time. A universe without time (like say with the metric tensor diag (+,+,-,-)) would not be immune from those kind of paradoxes and ontology as we know it would not be possible.

Lawrence,

Dynamic triangulation is OK, but I feel it assumes too much and achieves too little. Its roots are in very early Robb and Zeeman's results. LQG is far richer, but I feel very uneasy with Wheeler-deWitt equation as the starting point.

Tegmark's approach has the big merit of giving respectability to this line of research, but the major problem is extracting mathematical consequences from it. Gordon McCabe had continued his approach into the hard area of model theory, but time till tell if this would get anywhere in a reasonable amount of time.

Dear Florin,

thank you for your important comments on my paper. Yes, i agree, many standard antinomies are in fact truly nonsensical, i agree fully with you. Also the barber antinomy is unsensical, cause a barber cannot shave those and only those who do not shave themselves. This would mean he had to shave all the people of the world! (the universe?:-), who don't shave themselves. So Russell was wrong to define his barber in that way (in my opinion)!

But my interest in such antinomies (self-referencing statements) isn't driven by the search of a specific content, but by the search of their universal form.

I am sure by mentioning the liar's paradox, you didn't refer to the original formulation but more to such sentences as "this sentence is false" etc.

The original formulation of the liar's paradox is indeed decidable ("The Cretans are always liars"). It must be false (a "lie" or whatever) cause there exists another possibility to decipher the paradox, namely the possibility that the Cretans indeed aren't always liars. So Epimenides' sentence is simply false, because if it is considered to be true, this wouldn't make sense. In this case (his famous statement), Epimenides could have been really lied. In contrast, "This sentence is false" seems to me to really being nonsensical.

But back to Russells findings. He was motivated by set-theoretic considerations. Asking, if the class of the classes that are not members of themselves, is a member of itself or not, one comes - according to Russell - indeed into difficulties. Classes that satisfy the criterion of not being a member of themselves are always formulated in explicit form. For example the class of all red cars. In this class there is no room for containing itself.

Classes that do not satisfy the criterion are always formulated in an implicit form, for example the class of all things that aren't red cars. This class is surely one of the things that aren't red cars - and therefore has to contain itself as a member. Those kinds of classes have as a property that they all contain themselves infinitely many times - due to iterations. For Russells classes that are not members of themselves, this property doesn't apply. So in my opinion the main class in question - Russells class of all these classes - cannot be a member of itself (analogically the barber in my solution cannot be a member of the class he is shaving).

My interest in such somewhat stupid things is driven by a main question that is somewhat interwoven with the manner, science comes to knowledge and to conclusions: By observing patterns and building rules out of them. My paper about the barbers antinomy could be headlined with one question that is in my opinion in deep accordance to our current essay contest:

"Has every rule an exception?" or elsewise formulated, is the following true(?):

"No rule without its exception"

Best regards

Stefan Weckbach

Hello Stefan ,Lawrence ,Florin,

Dear Stefan,

Hope you are well ,and happy new year too .

I think you makes a very interesting point about the conscious .In a discussion with Jayakar about the intelligence and extelligence ,we see the link between the rationality which appears like relevant too.The paradox thus is not necessary for the balance between the physicality and the uknew if I can say .

The universal referential and its pure dynamic thus always will be specific with its codes .An imaginary referential ,if it is not synchronized ,will imply paradoxs and incompleteness due to the lack of limits and the lack of real topology of the chosen system .That depends always of the referential and the superimposings in my opinion .There I think it is possible to extrapolate the universal system with a synchronization of the imaginaries if the essentials are respected.

Without this kind of line of direction ,and the time and its specific periodicity,the non coherences appear and thus the confusion too.

Dear Lawrence or Florin ,could you explain me the theory of models ,please?Happy new year

Regards

Steve

Florin,

Here is the issue as I see it. We have two types of theories. We have string/M-theory which is vast in its "theory space." There is too much structure to this to presume it is completely false. It might of course emerge in some new form, but doubtless it will remain a big "space." Then there are the small theories, such as LQG and DT. The one thing which DT has going for it is it avoids some of the chaotic foam issues of LQG. The advantage these theories have is they are closer to the structure of general relativity in its pure form. I suspect these theories suffer from not having the huge domain of field theory that string theory has, which is what is leading to some of its problems, in particular the Barbero-Immirzi ambiguity. LQG does indeed stem from the Wheeler-DeWitt equation, which defines constraints NH = 0 and N^iH_i = 0, the Hamilton and momentum constraints. The quantum analogues are HΨ[g] = 0 and H^iΨ[g] = 0. These are of course not dynamical equations, but rather Lagrange multipliers which define fields on a contact manifold of solution. So far the progress with LQG has been frustrating.

So my modest (or not so modest) proposal is the following. One of the theoretical coups of string/M-theory has been the AdS-CFT correspondence. I will at this point avoid technical discussions on this, but say that this implies that on the boundary of the AdS spacetime there is defined a conformal field on the S^5 sphere. The boundary of AdS_5, with the group SO(3,2) is equivalent to a conformal field theory. The theory is then AdS_5xS^5, which can emerge from an 11 dimensional theory with the extended Anti-de Sitter group SO(4,2) ~ SU(2,2). This is a remarkable and beautiful result which Maldacena worked up 11 years ago. Susskind, Vafa and others have demonstrated that if a BTZ black hole is placed in the AdS that the conformal field theoretic information on the horizon of the BH and on the boundary of the AdS preserve that information, which has been a further recent deep development. Yet the AdS spacetime is not the physical spacetime of our universe. The universe is asymptoting to a de Sitter spacetime, with the group SO(4,1). This spacetime has a Wick rotation on the additional time-like dimension and it recovers a solution to the Einstein field equations which approximates the observable universe. Further, the BTZ black hole is replaced (by hand at this point) with physical black holes which we know exist. My proposal is to use LQG as the system which performs this map between the conformal theoretic AdS spacetime to the physical dS spacetime.

Here is what might be going on. Hyperbolic groups such as SO(m,n) have sequences of moduli points (gauge connections etc) which do not converge in a Cauchy sequence. The Wick rotation SO(3,2) - -> SO(4,1) will then impose this condition on a set of these moduli points. But a straight forwards map is a "violent procedure," which does this in an ad hoc manner. Instead we want to preserve the conformal theoretic information of SO(3,2) when we perform the map. It is my conjecture that the constraints of loop variables might just serve this purpose. This might also provide the additional information required to work the gauge ambiguity with the Barbero-Immirzi parameter. I have tried a number starts, but so far I can't get anywhere. I had thought, maybe still do think, that the quantions you work with might be a tool. I am not clear on just how to proceed with this. Any thoughts or suggestions would be welcome.

Cheers LC

Dear Friends,

The key here is that simplices are the geometrical representation of Spin groups.

A simple example is Spin(4)~SO(4)xSO(4) has 12 operators - the six edges of a tetrahedron (4-simplex & FCC basis) times two directions per edge. Add in the three dimensions in which the tetrahedron exists, and you have the 15 operators of SU(4). Everything has rank-3 and dimension-3.

Similarly, Spin(3), SU(3) and G2 are related to the 2-dimensional 3-simplex (triangular lattice), and Spin(5) and SU(5) are related to the 4-dimensional 5-simplex.

As these dimensions fracture into brane structures, the Lie algebras also fracture, which is why Spin(4,2) becomes so relevant (4 dimensions of Spacetime plus 2 dimensions of AdS).

Of course, these simplices work very well within the framework of CDT.

Have Fun and Have a Happy Blue Moon of a New Decade!

Ray

Lawrence,

I would not deny the impressive results of string theory, but I do not hold this as reason to certify its overall correctness. If for example the same amount of brain power were put in non-commutative geometry, equally impressive results would be there as well.

My interest in quantions, SO(2,4), SU(4), and SU(2,2) is from the point of view of von Neumann algebra factors and Hopf algebras. I feel that quantions may unlock "distinguished" properties which will point the correct way for SU(3) and GUT. If SUSY emerges naturally in this approach, then I will be a believer again in string theory, otherwise I fell that non-commutative geometry is the right path.

I am not sure I can really offer any meaningful guidance except to point out the problems I consider interesting myself. I would like to understand better the metaplectic group, geometrical quantization, and the link with maximal entangled states and Segre embedding. Also I would like to understand in depth the meaning of associativity and Jordan exceptional algebras. The link between quantions, Hopf algebras, and category theory is very interesting as well. The connection between quantions, von Neumann algebras, and non-commutative geometry is worth investigating combined with an in-depth understanding of the spectral triple.

Regards,

Florin

Florin,

I have been working quite a lot on the Jordan exceptional algebra. I think this in connection to quantum error correction codes actually underlies string/M-theory. The 27 dimensional J^3(O) with a light cone constraint reduces to the 26 dimensional bosonic string as 3 octonions plus two independent scalar degrees of freedom. The three octonions are the vector terms plus their supersymmetric pairs as spinors. So the other two additional octonions are those spinor fields and their conjugates. This reduces things to 11 dimensions, or on the light cone condition 10 dimensions. My essay paper goes into more depth on this.

Steve,

String theory is a general spacetime (with curvature) version of the S-matrix theory. The string is a parameterization of fields along a chord or loop. Even early on it was thought that one could parameterize fields in two or more dimensions. This would be the quantum sheet or quantum 3-volume, 4-volume and so forth. This did not turn out to work terribly well. What Witten did was to show how open string with their endpoints attached to a membrane would have its modes coupled to soliton waves on this brane. This permitted one string type to transform into another. I hold to the view that string and d-branes are themselves really emergent from quantum units of information --- related to what Susskind calls the D0-brane. This is a brane of zero dimensions! So in a way we are back to particles. What the d-brane does is from an S-matrix perspective to define the domain of support for the S-matrix so it can work in generalized curved spacetime of up to 11 dimensions. If you read my essay you will find some of this, and the approach with quantum information (quantum error correction codes) leads to a type of Skrymion action. This action operates for particles or quantum fields in topological knots or loops, from which I think strings emerge from --- as well as general d-branes.

Cheers LC

Lawrence,

I am casually learning octonions, but if you already understand them, what is the physical meaning of their lack of associativity (if any)? (From the mathematical point of view the lack of associativity is related to terminating the infinite series of Lie algebras.)

Hi dear Florin ,Lawrence ,

I thank you dear Lawrence .You know I try to be in synchro but this parameterization seems in a infinite serie more the bad tools .

The série in this logic ,never will give good results because the universal gauge between quantum and cosmological spheres are not correlated with our main laws .

You know I liked your essay because your are an exeptional mathematician .But I try still to encircle your method .

Like say Florin ,the associativity seems essential in a physical point of vue .I think when an axiomatization is correct ,it doesn't exist confusions.Just unknews .

In fact the causality is intrinsic and the evolution seems the part of the puzzle whichj builds gravitational stability and its specificities .Thus the fields ,mass ,particules and their rotations have a specific intrinsic code and even at this ultim scale ,we have the same laws with some differences ,foundamentals furthermore .

The transfert of informations is simple thus all parameterizations must be correlated with our laws .The fields too are in this proportionality .

The time and the light thus in this line of develoment can't be rational I think .

Best Regards

Steve

Florin,

Right now I worry less about the nonassociative aspects of E_8 and am more focused on the automorphism and centralizer groups G_2 and F_4.

Nonassociativity is a bit strange. Yet all it says is that (e_ie_j)e_k - e_i(e_je_k) = C_{ijk}^le_l, where the last term is by multiplication table rules e_i(e_je_k). So you think of this as a sort of π/2 phase shift. The physical meaning I think involves the S-matrix. The S-matrix acts on a set of vertices or particles p_i

|φ) = |p_1, p_2, ..., p_i, ..., p_j, ...p_n)

and converts this channel into an S-channel which has some overlap with

(φ| = (p_1, p_2, ..., p_j, ..., p_i, ...p_n|,

so the expectation of the S matrix for these two ordered sets of states is

( S ) = |φ> = (p_1, p_2, ..., p_j, ..., p_j, ...p_n|S|p_1, p_2, ..., p_i, ..., p_j, ...p_n).

By S = 1 2πT this is determined by a transition matrix, which by the exchange of vertices determines the S-T-U relationships or Mandelstam variables. In this case we simply have an exchange or a commutator in a quantum group. This might be represented by (ab)---(ba) as a braid link, and for multiple exchanges the S-matrix determines a braid group. For the exchange of three elements this gives a Yang-Baxter equation which is equivalent to a Jacobi identity on the double commutator [a, [b, c]] and it is equal to zero. The channel is produced by a product of Hilbert spaces for each vertex, so

|p_1, p_2, p_3> = |p_1>|p_2>|p_3>.

Nonassociativity is an ambiguity which says that

|(p_1, p_2), p_3> = (|p_1>|p_2>)|p_3> = |p_1>(|p_2>|p_3>) |C_{123}^4p_4>.

This is a type of Hopf algebraic system, but with a "twist." The noncommutative system is defined by the K-linear map on the vector space V, or between V and V', a multiplication and co-multiplication rule you get the Hopf hexagon. However, for three elements and an associative rule there is a corresponding pentagon (Stasheff polygon), the hexagon and pentagon are fused together to form a general polytope. I can delve into these detail later if you are so interested.

What would this correspond to physically? The ordering ambiguity means there are two S-matrix channels which are not commensurate with each other. For the case of a black hole a string, which is really an S-matrix element, is observed to exhibit completely different physics according to an observer who witnesses it fall towards the black hole from a distance, what Susskind calls a fiducial observer or FIFO, and an observer who falls in with the string, a freely falling observer FREFO. What Susskind has argued is how the physics observed by the two is completely different, but both are physically valid. The general polytope I mention indicates how noncommutative geometry and nonassociative algebra are related (a bit complicated), and physically this means what we think of as a proper ordering of events or trajectories must be "liberalized."

Cheers LC

Dear Lawrence,

You said "This is a type of Hopf algebraic system, but with a "twist." The noncommutative system is defined by the K-linear map on the vector space V, or between V and V', a multiplication and co-multiplication rule you get the Hopf hexagon. However, for three elements and an associative rule there is a corresponding pentagon (Stasheff polygon), the hexagon and pentagon are fused together to form a general polytope. I can delve into these detail later if you are so interested."

Actually, I am interested.

Figure 4 of my "A Case Study..." paper involves a Petrie pentagon that I purposely drew as a distorted Petrie hexagon. This makes a lot of sense if you look closely at the quantum numbers (particularly T'_G). I think this leads to a Spin(6)~Spin(4,2)->Spin(4,1) (related to the 5-simplex, Spin(5) and SU(5)) isomorphism and decomposition, which helps explain the difference between my 12-dimensional model (with 8-dimensional hyperspace - Spin(6) plus G2) and 11-dimensional M-Theory (with 7-dimensional hyperspace - Spin(4,1) plus G2). Higgs theory , the CKM matrix, and the PMNS matix may all be intertwined in G2 and this collapsed dimension.

Please review my recent tiling pattern and enlighten me.

Thanks!

Ray