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

        Dear Lawrence,

        Thank you for the links, and for the explanations.

        Best wishes,

        Cristi

        Dear Laurence,

        As always, you submitted a challenging and thought provoking essay, and I am glad it is doing very well so far in the ratings. I particularly enjoyed your discussion of numbers too big for a Turing machine in our universe to count to.

        In your conclusion, you write:

        "Chaitan has advanced ideas that mathematics is not something that exists in any sort of coherent wholeness. It is more a sort of archipelago of logically consistent systems that sit in an ocean of chaos. [...] Possibly the quantum vacuum is similar. It may be a tangle of self-referential quantum bits, where some sets of these exist in logical coherent forms. These zones of logical coherence might form a type of universe. These logical coherent forms are then accidents similar to Chaitan's philosophy of mathematics. It is very difficult to understand how this could be scienti fically demonstratedｬ yet maybe regularities in physics described by mathematics exist for no reason at allｮ｢ｼｯpｾｼpｾﾁs you found out when your read ｼﾕﾒﾌ urlｽ｢httpsｺｯｯfqxiｮorgｯcommunityｯforumｯtopicｯｲｴｹｷ｢ｾｼsｾﾛｼｯsｾ my essayｼeｾﾝｨhttpsｺｯｯfqxiｮorgｯcommunityｯforumｯtopicｯｲｴｹｷｩｼｯeｾｼｯﾕﾒﾌｾｬ this is pretty much how ﾉ see our universe in relation to the ﾍaxiverse that results from the ﾍathematical ﾕniverse ﾈypothesisｮ ﾏur universe exists for no specific reasonｬ because all possible universes doｬ and the regularities that we observe between our physics and known mathematics is simply a necessary condition for the existence of selfｭaware substructuresｮ ｼｯpｾｼpｾﾁll the bestｬｼｯpｾｼpｾﾍarc

          Marc,

          Thanks for the encouraging word here, and the upward boost.

          This touches in many ways on the issue of unification of physics. In particular this concerns the plethora of unification schemes. I tend to think that they may all, or most of them, are correct. They may have some probability assignment, but they all may well manifest themselves. There may be cosmologies with very different particles and interactions than what exists in this observable cosmology. I base this on part with what I am working on, which seems to be deriving a type of landscape of string/M-theory. That is of course a good thing that I can recover something known.

          I think the many worlds account of QM has some connection to the string landscape. If the spatial surface of this cosmology is infinite there are an infinite number of us sufficiently far out there. In a cosmology that is infinite, vast distance and lack of causal connection may imply quantum entanglement. This in particular would apply across the particle horizon.

          Cheers LC

          Dear Lawrence,

          I am giving your essay a second read. In addition to my earlier comments above, I wish to take you up on a part of your essay. You said, "...in point set topology there are an infinite number of points between any two points on the real number line with a finite distance between them. This means if they exist in some meaning according to computation there must be a machine that performs any calculation of points separated by any tiny finite set of intervals segmenting the distance between these points"

          If we may interrogate this, I wish to ask:

          1 - Can distance be what separates two points, when distance itself is constituted of points? Or are there some distances constituted of points and other types of distances not so constituted and not having points as their extremities of their extension or segments thereof.

          2 - What is an interval made of? Is it spatial or temporal?

          3 - How many intervals, if such exist can be on a real number line? I ask because of the 'finite' set of intervals in the quote above.

          4 - How can a real number line with an infinite number of points be divided, if points cannot be divided into parts and there is always a point at the incidence of cutting?

          5 - Finally, talking about "existing in some meaning", are points eternally existing objects or can they perish? If the Universe can perish, will points outlive it? If there was a Big bang Universe creation from Nothing, did points precede it?

          Regards,

          Akinbo

          *If you don't mind you may drop me a note on my forum so I get email notice. That is if you are inclined to discuss the above.

            Thanks Lawrence for dropping your comments at my forum. Appreciated.

            If you have the time, you may wish to volunteer direct opinion to the 5 questions I attempted to raise here.

            Regards,

            Akinbo

            I can answer some of these. An interval in relativity is the measure of a clock on a frame bundle and on a certain path. It is the integration of the path length.

            The matter of infinitesimals, a length or displacement along a certain direction that is arbitrarily small, has been a subject of debate and research for a long time. This matter has only been somewhat resolved with the so called Robinson numbers, which have underlying it set theoretic concerns of forcing and the continuum. I am not a great expert on this subject, so I can really only make mention of this in a short post like this. In the end it only works, as I understand, within a certain continuum model. The underpinnings of calculus and questions surrounding the Dedekind cut do not seem to be derived according to a complete axiomatic system. However, with a few basic ideas you can develop a lot of calculus.

            Mathematics in the objective or in some ways the Platonic perspective does not perish with the heat death or end of the universe. If mathematics is nothing more than a pattern system derived from the natural world then in that model it might perish. I am not terribly committed to either perspective. There are troubles with either viewpoint.

            Garrison Keillor has his "Guy Noir," who "On the tenth floor of the Atlas building still seeks answers to life's persistent questions." If you have ever listened to his "Prairie Home Companion" you know this well. There are persistent questions, such as "Does God exist," that will probably never be conclusively answered.

            LC

            Hi Lawrence,

            If it hasn't been brought up yet -- I want to make sure that we get the spelling of Gregory Chaitin's name right. He's among my favorite mathematicians/computer scientists, so the typo jumps out at me.

            There's no getting around the issue of how we differ in our views of foundations. I do not think classical physics is either finished, or emergent from conventional quantum theory -- in fact, I think it is the other way around. Although it is commonly believed , as you say, that "The classical picture of the universe is a continuum of flows [3] ..." this is not true. Continuous functions as described by differential equations or topological methods do not support a physical continuum of space independent of Minkowski spacetime, because space has no physical reality independent of time.

            I think this is easier to see by critical study of Perelman's solution to the Thurston geometrization conjecture -- all singularities on S^3 are extinguished in finite time by continuation (via surgery) of the Ricci flow, on the half open interval [0, oo). This is the mathematical advantage that any simply connected 4-dimensional world -- including Minkowski space-time -- has over a multiply connected space of random functions in 3 dimensions (or in fact, Hilbert space of any dimensionality).

            Nevertheless -- my highest score goes to your essay, for setting up the issues in thoughtful and highly readable terms, even though I couldn't be more opposed to the notion that "Spacetime is built up from entanglements [13]" Classical orientation entanglement explains the phenomenon just as well, when a time parameter (such as that of Hess-Philipp) is included in the dynamics.

            I hope you get a chance to check out my essay as well.

            All best wishes,

            Tom

              Thanks for the positive vote or score.

              Before 1900 it was commonly thought the universe was a continuum, and the idea of atoms was under attack, as this was thought to not conform to the continuum picture of reality. Of course Planck assumed that energy occurred in discrete steps, and Planck and Bohr assumed discrete values of angular momentum as well to model the atom. Quantum physics does have continuum structure, such as the dynamics of the wave function or the system of paths in a Feynman path integral. However, these no longer have the sort of ontology that continuum structures have in classical physics. The existential aspects of the quantum wave function is not longer ontological, and recently it is being found that the epistemological foundation of the quantum wave is not satisfactory either.

              How classical physics emerges is tough to understand. How an einselected basis occurs so that a particular eigenvalues corresponds to a measurement or is associated with a classical value is not solved. The paper by Sax proposes that Goedel's incompleteness theorem plays a role. I had some discussions with him on this on his essay blog page. This is curiously important with D-branes, for these are classical or macroscopic structures. While they are ultimately made of strings, or are similar to Fermi surfaces of electrons or condensates of quantum states, they are nonetheless classical and important for foundations.

              Sorry about the Chaitan for Chaitin. That is a regrettable typo. I don't remember if I read your paper or not. I will try to take a look at it soon.

              Cheers LC

              Dear Lawrence,

              An assumption at the end of your article

              "Mathematics and physics have this curious relationship to each other for purely stochastic or accidental reasons; there ultimately is no reason for this"

              provokes me to note that this possibility is refuted in our essay on the scientific ground.

              Best regards,

              Alexey Burov.

                Dear Alexy and Lev.

                Your paper is well argued. I will admit to being very agnostic about these sorts of ideas. In particular I am very agnostic about Tegmark's hypothesis, which seems not mathematically provable, nor scientifically testable. Even string theory is only at best indirectly testable, but Tegmark's Mathematical Universe Hypothesis seems impossible to test.

                A couple of points I mention first. The WAP as I understand it is the statement that the universe observed must be of sufficient complexity and structure to permit such observers. It does not mean that any cosmology that exists must admit observers. I think that is the strong AP (SAP). The other point is that chaos, at least within the meaning of Hamiltonian chaos or strange attractor physics, means that a system can execute a vast number of complex dynamics, all of them separated by very small initial conditions. This means that phase space is specified to a very small fine grained detail. Given this is cut into N boxes or pieces, and in each is one of the possible states (0, 1), the degree of complexity is 2^N = e^{S/k}. This is the dimension of the Hilbert space corresponding to this classical setting and the entropy S = k ln(2)N = k ln(dim H), H = Hilbert space. Chaos then in fact implies a high level of complexity.

                I did not make much mention of this in my essay. It could be said that mathematics has a body and soul. The body concerns things that are numerically computed and can in fact be computed on a computer. The soul involves things that involve infinitesimals and continua. These tend to be at the foundations of calculus with limits and related arguments. Even though my essay discusses homotopy, this is argued on the basis of continuous diffeomorphisms of loops or paths. However, in the end this is not what we directly compute in mathematics. We are interested in numbers, such as indices or topological numbers, and in physics that is much the same.

                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. This is because the soul of mathematics touches on infinity, or infinitesimals.

                The soul also involves things that quantum mechanically are not strictly ontological. These are wave functions or paths in a Feynman path integral. The existential status of these is not known, for the standard idea of epistemic interpretation is now found to be not complete. This differs from classical physics, where the physics is continuous, with perfectly sharply defined paths and energy values and so forth.

                I am somewhat agnostic about the existential status of the soul of mathematics. In some sense it seems compelling to say it exists, but on the other hand this leads one into something mystical that takes one away from science. So it is not possible as I see it now to make any hard statement about this. We seem to be a bit like Garrison Keillor's Guy Noir, "At the tenth floor of the Atlas building on a dark night in a city that knows how to keep its secrets, one man searches for answers to life's persistent questions, Guy Noir private eye."

                I will give your essay a vote in the 7 to 10 range. I have to ponder this for a while.

                Cheers LC