Dear Michel,

That might be interesting. I have been pondering how it might be that ﾎ"^+_0(5) is related to the tiling and permutation of links on AdS_5. The quotient SO(4,2)/SO(4,1) = AdS_5 is not an entanglement group, at least not as I know, but this might have some relationship to entanglement. This might be through the ﾎ"^+_0(5). Particularly if this is related to Langlands in some way.

Cheers LC

I want to give your paper some time Lawrence..

But I want you to know that your essay is on my radar of important papers to read for detail (and I have skimmed it), while the contest is still underway. I see that you mention Bott-periodicity, which is a topic I would have touched on in my essay - had I allowed myself adequate time. My entry this year is briefer than I intended, because I did not.

I was happy to see that you mentioned the HOTT program, which I also find to be interesting and relevant. I especially like that their pursuit of univalent foundations is geometrically constructive, but it is tied to a rigorous analytic proof checking engine. I find this usage of constructivist Math as program code particularly elegant.

More later,

Jonathan

    In your Bio you wrote: "I think it is likely there is some subtle, and in some ways simple, physical principle that is not understood, or some current principle that is an obstruction."

    Einstein's constant-speed-of-light postulate is an obstruction. In a paper published in Science Miles Padgett showed that the speed of light (in a vacuum) is not a constant:

    "The speed of light is a limit, not a constant - that's what researchers in Glasgow, Scotland, say. A group of them just proved that light can be slowed down, permanently."

    Pentcho Valev

      Jonathan,

      I am working my way through reading these essays. I will try to get to yours before too long.

      The HOTT program does put mathematical foundations closer to algorithmic structures. It might be a way to address what I call the "body" of mathematics, which is that part of mathematics that is reduced to a computation. This can be computed in some way on a computer. The part of mathematics that involves infinitesimals and set theoretic infinities are what might be called the "soul." I don't deny the existence of this per se, but I don't think it has a direct connection to physics.

      I am working right now to find out how Bott periodicity applies with exceptional groups. The intention is to find a way that nilpotent sets can be mapped to max compact subsets as with the Kostant-Sekiguchi theorem.

      Cheers LC

      Pentcho,

      You spend a lot of time thumping this theme. Sadly, mostly this is just a demonstration that you don't know what you are talking about. I have no intention of getting into an argument over this, any more than I intend to argue for evolution to a committed creationist or global warming to a climate denialist.

      The speed of light is different in media, and some exotic media have been developed that can trap light. This does not falsify relativity.

      LC

      "The speed of light is different in media, and some exotic media have been developed that can trap light. This does not falsify relativity."

      They slowed down light IN A VACUUM:

      "Physicists manage to slow down light inside vacuum (...) ...even now the light is no longer in the mask, it's just the propagating in free space - the speed is still slow. (...) "This finding shows unambiguously that the propagation of light can be slowed below the commonly accepted figure of 299,792,458 metres per second, even when travelling in air or vacuum," co-author Romero explains in the University of Glasgow press release."

      Pentcho Valev

      This does not have a bearing on relativity, but is a quantum effect. One might say that the action of this mask that slows down photons can persist with a photon in much the same way as with the Wheeler Delayed Choice Experiment.

      LC

      Hi LC--

      I loved your essay. You covered an immense amount of ground--and did so in a cogent yet concise manner. Congratulations!

      I now turn to discuss some comments that you made in response to my essay. You raised the issues of super-Turing machines and the physics of super-tasking. I am not an expert on either. However, I have looked at several examples of physical super-tasking (e.g., carrying out an infinite number of physical operations within a finite time period). I did so because super-tasking appeared to be one place where physics might really need the concept of "physical infinity". As you know from my essay, I call into question the necessity and desirability of relying upon physical infinity.

      In fact, for me, super-tasking was the "tipping point" against physical infinity. In every example I looked at, I found that either: (a) the super-tasking scenario was unphysical and could not work realistically (e.g., because of friction, chaos, cannot propagate a signal faster than c, etc.); or (b) the underlying physics was so murky that I couldn't tell whether the scenario was physically realistic or not. I place super-tasking via Malament-Hogarth spacetimes in the latter category. Regarding super-tasking via M-H spacetimes, I strongly recommend Earman's book, "Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes". His Chapter 4 includes an excellent review of M-H super-tasking.

      Best regards,

      Bill.

        • [deleted]

        • Post #73

        I am glad you liked my essay. I threw in the subject the MH spacetimes and supertasking because that seems to be something that needs to be considered for a number of reasons. I think that non-eternal black holes can't be supertasking machines. The black hole decays by Hawking radiation and disappears before i^в€ћ, so there is no continuous stream of infinite amount of information that can approach an observer as they approach r^-. However, this probably means that NP-complete problems can be quickly solved for the internal observer and the exponential time is replaced with ~ r - r_- near r_-. This may mean that the NP-complete problem of compactifying all CY manifolds is computed by black holes. I do agree that it may be unlikely that superTuring computing is possible in a way that the output can be read by an exterior observer.

        It is possible still that black holes are MH machines, even if they are finite in duration. This might be the case if black hole singularities are all the same thing. It could well be that black holes are all connected in a single quantum state that defines the singularity, and in a multiverse setting it could be that this is a great MH machine. The universe might then has underlying it a supertasking computer that is the ultimate quantum error correction code. I can go into this in detail if you want, though I will avoid that for now. Supertasking process in this setting is then associated with what were called shadow states. Shadow states are an old idea going back to the 1970s with S-matrix bootstrap physics. These are states which have T-matrix realizations, but they have no Born interpretation as associated with observables. The output of the MH spacetime machine can't be read!

        Cheers LC

        Hi Lawrence,

        Nice historical introduction. Interesting new maths. Most enjoyed philosophical concerns, which is more "down my street". Good that your essay is getting noticed. Good Luck, Georgina

        Hi Lawrence. Thank you for your reply to Georgina. Sorry I forgot to reply to you earlier.

        As I said it is not true that "a universal Turing machine is not able to enumerate all Turing machines". Turing machines can be automatically enumerated, but what is not possible is to find a general algorithm always correctly able to prove for any other algorithm, whether or not it will ever stop.

        You wrote that "Peano's number theory is incomplete, and so something funny does happen with N+1". It may be funny but it is not anything wrong with the existence of N+1. You seemed to mistake incompleteness with inconsistency, which are 2 very different things. All we need for a theory of arithmetic is that it is consistent, and indeed it is (even if we cannot have any formal proof for it). It is not a problem for a theory of arithmetic to be incomplete, anyway it remains a valid theory, and since it cannot be completed we must satisfy ourselves with this fact. There is no problem with the axioms, we only can never have enough axioms for all arithmetical truths to be deduced from them.

        I am not asking for symbolism, I consider the possibility to explain things with words as well. I only ask the words, whatever the details level, to stay in agreement with the logical structure of things as they actually are, a requirement which I found to be lacking in your essay ; and when one does not properly understand the logical structure of some issue, then better would be to avoid telling any story about it than telling a probably incorrect one. For example if you know a result but you are not familiar with the proof, it may be wiser to just tell the result but not try to give any sketch of proof that may not be the correct one, so as to better develop instead something else you would know better to do it correctly. Making things short to give the intuition of something can be good only if the intuition you provide is indeed a correct intuition, i.e. in coherence with the correct understanding.

        There is a concept of Polish set, of course, but what I meant is that this never aimed to constitute a "Polish set theory" as a candidate for the foundations of mathematics.

        Hi LC--

        I think that we are in agreement on the issue of super-tasking via M-H spacetimes. It's amazing the kinds of things that show up in our discussion threads! Thanks for taking the time to set out your position on this issue.

        Best regards,

        Bill.

        Some of my discussions were meant to illustrate something of the divide between computational mathematics and pure mathematics. With Peano arithmetic we know that Goedel's theorem indicates that something is not complete, even though much of it involves N ---> N 1. There are then numbers, such as some between 10^{10^{10^{10}}} and 10^{10^{10^{10^{10}}}} that have no description. There is a Berry paradox or self-referential form of incompleteness, based on the complexity or unnamable property of such numbers between these two, in not being able to describe numbers.

        I had a limited amount of space to describe this, and maybe I did not do the best at it. I tried to explain some of these ideas in physical terms without getting into depth on set theory or logic. It is also best I have found that keeping these essays on a level accessible to general readers to be a good strategy.

        LC

        There is not likely to be any way that supercomputing machines such as from MH spacetimes will produce readable output. This does not mean it is absent, but it may simply not involve quantum information that is directly read.

        LC

        Hi Lawrence,

        Your essay is a real wealth of knowledge, and I thoroughly enjoyed reading it. I liked your analysis of Godel and relating it to the cardinality of the continuum not being decidable, and your approach to the limits of computability and the Berry paradox. Your relation of topology to computation is very fascinating as is your subsequent in depth perspective through holographic principles. "What is fundamental are topological quantum numbers, such as those here associated with the two slit experiment or black hole horizon units of area." I'm going to think about this foundationally, and see how I can relate it to other foundational concepts, including my self-referential operators. Finally your discussion on continuous mathematics and attempting non-computable problems is interesting. I'm very intrigued by this as you know and my essay explores transcending Turing machines as well. We of course are in agreement that the physical aspects are quite important.

        Thanks again very much your comments on my essay and the dialogue we had on undecidability; see also there the thread (above yours) where I tie this back to incompleteness and the undoing of Hilbert's Einscheidungsproblem. Also note Gentzen's proof of consistency for Peano axioms using transfinite induction, which affirms some of the concepts in your paper.

        Thanks again, it's a great contribution to this essay topic and I rated it very highly. Please also take a moment to rate mine, especially now that we've been through them both and share a number of topics. Best, Steve

          I am glad that you enjoyed my essay. The Einscheidungsproblem of Hilbert turned out to have this strange impact on mathematics that Hilbert never imagined at the time. On the other hand I have read that Goedel discussed with Einstien on how he was fairly unhappy that his result seemed not to have practical impact on mathematics. However, in some ways that may now be the case. The formulation of mathematical physics might involve recognition of these matters.

          Your recognition that a quantum system in a superposition of two states in a qubit has undecidable nature is interesting. I think a quantum system in a superposition of states could reflect a Goedelian undecidable situation in some problem involving einselection, or maybe even deeper with problems with quantum error correction codes (QECC) in black holes. It discuss hypercomputing in my essay, and this could involve some aspect of how QECC in black holes and the erasure of quantum bits that accumulate. This may be an undecidable problem, and hypercomputing might indicate something that is concealed from observability.

          I will try to look up Gentzen's proof of consistency for Peano axioms. I thought I had scored your essay earlier, but I had not, so I just now scored it.

          Cheers LC

          8 days later

          Dear Lawrence,

          I finally got to read your essay, and I loved it! As usual, you make excellent and deep connection between various things, connections that allow us to see relevant subtleties. You made interesting connections between computation, quantum theory, homotopy, black holes, and proved that HOTT may be very well the way to the next stage of physics.

          Best wishes,

          Cristi Stoica

            Dear Dr. Crowell,

            I thought that your engrossing essay was exceptionally well written and I do hope that it fares well in the competition.

            I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

            All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

            Joe Fisher

              Christinel,

              Thanks for the positive assessment of my paper. I gave your paper a pretty high score a few weeks ago. I did this while I was on travel and I don't think I had time to write a post on your blog page. I will try to write a comment, which will probably require rereading your paper.

              There is a paper by Schreiber on directly applying HOTT to physics. This is a difficult and in some ways foreign way of doing physics. I am less sure about the role of HOTT directly in physics, but rather that a simplified form of mathematics that connects to HOTT will become more important. It is in much the same way that physicists do not employ set theory a whole lot in theoretical physics. However, behind the analysis used by physicist there is point-set topology. We generally reduce the complexity of this mathematics. If I were to actually engage in this I would study the HOTT, and an introduction to HOTT with physics and related web pages on this site, are worth going through.

              To be honest it has been a while since I have studied this. I have been working on a homotopy approach to quantum gravity. I mention some of that in my essay. This concerns Bott periodicity with respect to holography. The connection though is rather apparent. There are also some similarities to C* algebra. This work of mine connects with what is called magma, which constructs spacetimes as the product on RвЉ•V, for V a vector space,

              (a, x)в--¦( b, y) = (au + bv, [x|y] - ab)

              where the square bracket is an inner product. This is a Jordan product and the right component is a Lorentz metric distance. This is also the basis for magma, which leads to groupoids and ultimately topos. A more convenient "working man's" approach to HOTT is needed.

              There is my sense that mathematics has a body and a soul. The body concerns things that are computed, such as what can run on a computer. The soul concerns matters with infinity, infinitesimals, abstract sets such as all the integers or reals and so forth. If you crack open a book on differential geometry or related mathematics you read in the introduction something like, "The set of all possible manifolds that are C^в€ћ with an atlas of charts with a G(n,C) group action ... ." The thing is that you are faced with ideas here that seem compelling, but from a practical calculation perspective this is infinite and in its entirety unknowable. This along with infinitesimals, or even the Peano theory result for an infinite number of natural numbers, all appears "true," but much of it is completely uncomputable.

              Cheers LC