Phillip,

Your essay was interesting. I appreciated the description in the supplement of what is meant by the necklace. Also your discussion on Turing's theorem is a nice compact version of that.

I have some points of difference with it. I would say a departure is with the implication of some sort of infinitude. I think that for any observer the number of quantum states available for observation is finite. We may think of the horizon of a black hole with S = kA/4â„"_p^2 = NK for N the number of Planck unit quantum states. The Bekenstein bound and related results imply the number of quantum states available to an observer is finite. So even if the global Hilbert space is infinite dimensional any observer can witness only a finite subset. That might be where the necklace or enveloping algebra can come it, where this could be some general form of separable set of states in tensor products. This may in principle extend infinitely, but any observer has access to a finite quantity. The existence of horizons is a way that Hilbert space available for measurement is finite but unbounded.

Your approach makes the implicit assumption that action = entropy, which is in a Euclideanized time sense t â†' it = ħβ = ħ/kT.the path integral maps into a partition function, where an energy E of a system is computed according to all possible combinations of microstate energies. This is a form of integer partition function. The only extension I can think of is where the measure μ(U) = e^{-S(U)} needs to be extended to μ(U) â†' μ(U)/diff(G), where G is the group of diffeomorphisms of U. However, this action = entropy or an equivalency between

TdS = dW + dU â†" dS = pdq - Hdt,

links quantum information with entropy

My essay concerns how Gödel incompleteness is associated with different entanglement geometry. A heuristic comparison would be with Nagel and Newman's book on Gödel's theorem and Euclid's fifth axiom. The undecidability of the fifth axiom leads to two possible model systems for geometry, Euclidean flat space vs more general geometries of Gauss, Lobachevski and Riemann. This is a case of consistency and incompleteness vs inconsistency vs completeness. Loosely we can think of the Euclidean case as consistent but incomplete, while the more general geometries are more complete, but not consistent with each other. This without details of ω-completeness/consistency is how different entanglements have different topologies.

Szangolies describes this as an epistemic horizon. His paper is worth reading. This has I think connections with my work through the locality of solutions for Diophantine equations, or equivalently p-adic sets.

At any rate the part about the measure μ(U) â†' μ(U)/diff(G) as mod-diffeomorphisms I think is important. This is in gauge theory these diffeomorphisms are what define a moduli. The moduli space describes the topology such as the ADMH construction. In a duality gauge theory â†" entanglement symmetry, here on the right hand side in a SLOCC meaning, there may be a correspondence between topology of entanglement with gauge fields and topological gauge fields.

Anyway, things to think about. I would like to see what you think of my paper.

Cheers LC

    I will leave others to decide if my work should be interpreted as computational art, but I can try to answer both questions anyway.

    My motivation is the observation that reductionism must ultimately fail. Each time your knowledge is reduced to something more fundamental a new set of how and why questions arises. Let's not pretend that we are not trying to answer that kind of question, of course we are.

    So reductionism must be abandoned at some point, perhaps from the start. Instead we must look at the whole, at what information and experience is. Everything must be equally real or unreal, but reality is relative to the observer.

    This conclusion then provides the problem I am trying to solve. How can our experience of physics arise from that starting point? Many physicists talk about how information is fundamental, and reality must be emergent. If you assume that space and time emerge then perhaps you can argue for gravity as an emergent entropic force, but what about the deeper origins? How do we get from pure information and self-reference to space and time and some real physics? I want to go beyond all this nice philosophy and actually so something with the maths.

    If I tell you that a random three digit number is even I have given you one bit of information. If I tell you only that it is not 137 I have given you much less information.

    Quantum theory leaves open the possibility of a deterministic underlying theory but it requires a big stretch. Everything suggests the opposite, but if your philosophical position is that reality must be deterministic then by all means try to exploit the loop holes. I wish you luck.

    I hope you will submit a good essay to support your case and I look forward to it.

    You cannot transmit (and thus give to me) the first sentence you stated, in a message only one bit in length. The same is true of your second sentence. Your conception of "information", differs substantially, from that of Shannon. Luck is not necessary, where understanding is sufficient. An essay is not necessary to support my case, since a sufficient demonstration has already been published. Q.E.D.

    Rob McEachern

    "An alternative possibility is a theory that predicts what we see and nothing else."

    What we see includes the standard model of particles, neutrino masses, gravity, galaxy rotation curves etc. Theorists don't set out to find models that add more to the list. They just look for models that provide a consistent framework for some or all of these things together. It may turn out that those models have additional particles or other observable or unobservable effects, and this is often the case. These additional things provide some testability, but if they found a consistent model for everything that had nothing new I am sure we would all be equally happy with it.

    You are right that I start with an ensemble of possible universes. The way I differ from the usual MUH is that I think these different universes combine into a single whole that is accessible in our reality. According to the MUH we live in just one mathematical external reality. I think my approach may be closer to yours. My "multiverse" of universes is actually indistinguishable from the "multiverse" of wavefunctions in quantum mechanics which is really just one field.

    Dear Philip E. Gibbs,

    "Information is the basic material of reality, but the processing of information raises paradoxes due to the self-referential nature of computability."

    Perhaps I overlooked this statement and something that may substantiate it at the beginning, the end or even any part of your essay on "Undecidability ..."

    I was lazy and looked for where you referred to [ 4] Shannon at your p. 2. Maybe you merely corrected Shannon by mentioning that the DNA is a material code. Otherwise, one is tempted to read your theory of theories as new-Hegelian idealism. Anyway, I see you obliged to take McEathern's critical arguments more seriously. He is complaining at my own essay.

    Eckard Blumschein

      Thanks for your comments, references and questions.

      You make a fair point that when defining computability, infinite strings of output are required. Any finite string is computable. However for the halting problem it is easier to work with just finite output. The computation of a universe might make more sense if it is allowed to be infinite, but I am not sure I could have cleared this up in the limited space.

      For the Wick rotation I accept that this was too quick. Working with probabilities between 0 and 1 is not very good when trying to define and algebra so a switch to complex numbers works better. More work would be required to determine whether this is justified in that it would give an equivalent formulation.

      The amount of information conveyed by a statement is not the bit-length of the string that communicates it. That only sets an upper limit. What I have said at the start about information content is standard stuff, but a lot of people have a misconception that information always comes in discrete bits so I had to explain that it doesn't. It is covered in wikipedia at https://en.wikipedia.org/wiki/Information_content

      The negative log of a probability gives the information content for the outcome of a random process and it can take any real value from zero to infinity.

      That information is the basic material of reality is an axiom here. I would justify it by observing that everything is described by information in physics, mathematics and any other subject. I don't need any other substrate as a foundation and seek to derive everything else from information alone.

      I am not an expert on the classifications used by philosophers but I find much in common with both Hume and Kant and others. I would say that the underlying structure of our reality can be derived from the way we must experience it in terms of processing uncertain information. Probably this is some form of idealism but I am not sure which variety it is.

      "What I have said at the start about information content is standard stuff..." In Algorithmic Information theory, but it has little to do with Shannon's Information theory.

      As Shannon states in the first sentence of his A Mathematical Theory of Communication, he is concerned with:

      "various methods of modulation... which exchange bandwidth for signal-to-noise ratio... in a general theory of communication."

      In other words, a trade-off (exchange) can be made, that increases bandwidth in order to offset the effect (on information capacity) of a decreased signal-to-noise ratio and vice-versa. That is what he is interested in characterizing.

      On page 43, where he gives his famous Capacity theorem (Theorem 17), he states: "This means that by sufficiently involved encoding systems we can transmit binary digits at the rate (C) bits per second, with arbitrarily small frequency of errors."

      Thus, unlike any other "flavor" of "Information theory", Shannon is primarily concerned with the circumstances, under which one can correctly (without errors) recover the values of each, individual bit, encoded into a continuous, noisy waveform. It is meaningless to talk about the error-free value, of an unrecoverable fraction of a bit, precisely because a fraction-of-a-bit has no "correct" value.

      The significance of this, is that it pertains directly to "uncertainty" in physics. Namely what is the maximum number of discrete (quantized) bits of information, that can ever be recovered, without any errors, from any set of measurements of any continuous function/signal, that has a limited duration, bandwidth and signal-to-noise ratio (SNR)? In other words, what is the maximum number of bits, that you can ever be "certain" of, in measurements of any received "signal"? What happens if that maximum number happens to be exactly equal to one? What happens is - you get the Heisenberg Uncertainty principle. That is the origin of the HUP. It has nothing to do with any "weird physics", it is simply a property of any mathematical function, with a severely limited duration, bandwidth and SNR; the very properties that Shannon said he was concerned with, in his very first sentence. There are circumstances for which, there is no possible "exchange of bandwidth for signal-to-noise ratio", that will ever enable an observer to be certain of the value of any more than one, single-bit-of-information, within an entire set of measurements, of some continuous signals. That is what "entanglement" is all about.

      Rob McEachern

      Hegel (1770-1831) strongly rejected the atomism: "The atomists are thinking too materialistic". Materialism was seen close to atheism. Maybe believer in God feel attracted from your idealism.

      On your p.8 you stated like an axiom: "Physicists have so far failed ... because they still cannot relinquish certain cherished philosophical beliefs ... objective physical reality ... and causality".

      I am ready to agree concerning reality because I see it just a successful conjecture. However by questioning causality you lost me. Aren't you aware what you are saying with words like process and processing?

      Have you any idea how to apply your axioms?

      It is perceptive of you to pick up on the philosophical side of this essay. It is very important. I am not going to allow myself to be identified with any particular philosopher because there is none that matches my whole outlook. As you point out, much of philosophy is historically built around the debate of religion vs atheism rather than what physics needs or tells us.

      I wrote a whole essay on causality under the topic of giving up assumptions. It was my least successful essay in the community voting, but that does not mean it is wrong.

      I also reject materialism, determinism and reductionism, for starters. People don't like that. They want to keep these things. These are concepts that are built into our education, our language and therefore our psych. You say that using words like processing implies that I really accept causality, but I disagree. You can process and compute the universe in any order. It is hard to express this in language because our everyday language is built around the assumptions that I am giving up. It is only in the mathematics that the real picture can be understood.

      "You can process and compute the universe in any order."

      The essence of mathematics is its freedom (Cantor). This makes it just a tool that even allows, to some extent, reasonable backpropagation.

      You raise an interesting point about the finitude of information. In an idealist philosophy my reality is determined only by the finite amount of information in my mind. The external world is an ensemble of possible worlds consistent with that information. I dont allow that there is one external physical reality for the state of the whole universe around me because that would require an infinite amount of information, but each possible state of the universe would require a large or infinite amount of information to describe it. What this provides is an illusion of an external reality described by more information than I really possess in my mind.

      I think that symmetry is important to how this works. Suppose that all objects in the universe were distinguishable so they can be labelled in some way. If the universe is infinite and I know that the object with label X is in the room with me then I have an infinite amount of information about it. This can't be right. The paradox is resolved because elementary particles are indistinguishable. All I can really know is that there is an object in the room made up of elementary particles in some configuration. This requires only a finite amount of information. The indistinguishability of particles is a permutation symmetry.

      You are right that I have identified information entropy with action, even if I did not say it explicitly. It is an interesting conclusion, and not one that I would have expected.

      At least I am not alone in linking action to entropy

      https://arxiv.org/abs/physics/0106081

      https://www.researchgate.net/publication/239614590_The_concept_of_entropy_Relation_between_action_and_entropy

      https://physics.stackexchange.com/questions/483580/is-it-there-any-relation-between-an-action-and-entropy

      etc

      Philip,

      philosophy has always been discussed along contemporary hot topics and of course by using contemporary vocabulary. The only constant I can find in it right from its start is:

      - Time versus Time-lessness (not eternity!)

      - a posteriori versus a priori

      - empiricism versus rationalism

      which all mean the same.

      The other point I want to address is that REAL progress has been made when a scientist cannot decide whether he has discovered or invented something. Then what he discovered/invented is Absolutely free from contradiction. The formal description of this state of 'universal validity' exceeds the capacity of logic, but not the capacity of the scientific metaphor.

      Heinz

      Phil,

      Thanks for clarifying the difference between yours and MUH. And thanks for pointing out that your approach is "indistinguishable from the'multiverse' of wavefunctions in QM which is really just one field." I'm actually glad to hear that as I like your ideas, but felt they were the extreme opposite of mine. I do believe that good ideas should generally converge to a "correct" idea.

      Best,

      Edwin Eugene Klingman

      In the light of some of the great comments above, I want to clarify the philosophical position in this essay.

      When physics students learn quantum mechanics they often feel unsatisfied with the formulation. I know I did. In-determinism is not the problem. The real difficulty is the role of the observer. In the standard interpretation the wave function collapses when an observer makes a measurement. The problem with this is that our traditional philosophy of science is materialism. Everything is made of particles and fields which exists independently of us. we are just configurations of these material structures that happen to have a highly evolved brain capable of self-awareness. How can we even define what we mean by an observer that is consistent with this philosophy? This is the question that leads to the famous statement by Feynman that nobody understands quantum mechanics.

      Different people come to terms with this in different ways. Some people claim that the wave-function does not collapse and it is just decoherence. This is nonsense. Others look for an underlying deterministic hidden variable theory to eliminate the probabilistic part altogether. Some develop a pluralistic philosophy where consciousness is something more than just high level brain power so that the observer can be counted as something special. For me it was the many worlds interpretation that allowed me to accept a more comfortable materialistic view where the wave function collapse is an illusion of our own experience.

      I now think that all these ideas are wrong.

      The only viable solution to the measurement problem is to replace materialism with idealism. The observer is not just something different, it is everything. Before quantum theory philosophers proposed various types of idealism: subjective idealism, objective idealism, transcendental idealism, etc. what we are considering now is best termed quantum idealism. This is not some religious or spiritual philosophy. It is pure science.

      In this view our reality is determined by the information we have in our mind. This is a slightly fuzzy quantity because we are not conscious of everything at once and the boundary of our mind is not absolute but we can allow for that. Our brains can only hold a hundred terabytes of information which is tiny compared to what is required to determine the physical position of all the particles around us, and yet it is everything.

      In a sense the detail in the world around us is an illusion. There are really many possible worlds. All the worlds that are consistent with the information we possess. The real amount of information out there is no more than the amount in our head. It is as if a giant quantum computer is continuously computing all universes consistent with what we know, not just in the present but also in the past and future. New information from these worlds comes back into our mind through our senses when we make observations providing the illusion of a world containing much more information than is really there. This computer is not a real thing. That would be a materialistic interpretation. It is not part of our mind either. Our brain is not powerful enough. It is just the force of consistency which makes our reality operate in this way.

      I know this will not be a popular view, just as my views on causality and many other things are not popular, but if I am right then the very nature and structure of reality must be determined by the mathematics of the information calculus required to describe this process. In my essay I tried to show that we can get to the emergence of quantum mechanics, space, time, particles and fields just starting from an algebraic analysis of information and how it works in a self-referential system. Obviously there is more work to do but lie to think that I have made a promising start.

        Thank you for your kind words and encouragement.

        For the FLRW metric with k = 0 the spatial surface of the Hubble frame is R^3. A similar situation occurs with how one defines coordinates in de Sitter spacetime. There is no serious problem I see with that sort of infinitude. The limit to observing an infinite amount of stuff, first pointed out by Olber in his paradox, is the expansion of the universe and cosmological horizon. The expansion factor e^{Ht} in the FLRW, or cosh(Ht) in de Sitter, defines a limit for the expansion of a Planck length at the earliest quantum gravitation stage to the scale of the CMB or cosmological horizon. This is an enormous factor e{Ht} = 8Г--10^{60}, but Ht = 140. This gives r = 140Г--в€љ{3/О›} ≈ 1800 light years.

        The hunt for B-modes in the CMB is a form of this. Graviton production or gravitational waves produced in inflation should leave a polarization signature on the CMB. In this way the most distant and earliest quantum information we may receive or observe is on the CMB as a sort of cosmic quantum gravitation detector. Anything beyond this scale is sealed off from view. As a result, any observer can access only a finite amount of information.

        There is a curious sort of renormalization of scale with information. As we observe further outwards the number of qubits available decreases, until at some final scale of the Planck wall there may only be one qubit. Of course, observers, or we, find themselves in a local world filled with information. The cosmological horizon at the time of inflation has an area of around 10^{15} Planck units of area. This is the amount of information potentially available from this epoch. Pushing further back to the Planck era this approaches one unit.

        Cheers LC