I finally got to reading your paper. I have been working to get a piece of instrumentation developed meant to go to another planet. In reading this I think what you say is maybe not that different from what I develop.

Your paper drives home the point on using the Blum, Shub, and Smale (BSS) concept of computability. This is an odd concept for it involves complete computation of the reals to infinite precision and where our usual idea of close approximations are not real computations. This is a certain definition of incomputability. Since these I_U fractal subsets for an underlying fractal system are forms of Cantor sets the p-adic number or metric system is used to describe them. As fractal sets are recursively enumerable their complements are what are incomputable in a standard Church-Turing sense. Since this fractal is really defined in a set of such, there is a set of p-adic numbers or metrics and by Matiyasevich this is not globally computable. By this there is no principal ideal for the entire set or equivalently a single algorithm for all possible Diophantine equations. This is the approach I take with my FQXi paper. As a result, at this time I am relatively disposed to your concept here.

This meaning to incomputability in the BBS system is different, but not that out of line with the standard Church-Godel-Turing understanding. We can see that determinism is not always computable. The Busy Beaver algorithm of Rado has the first five numbers 0, 1, 4, 6, 13, but beyond that things become tough. The 6th is thought to be 4098, though not proven as yet. The 7th is a number greater than 1.29Г--10^{865}. It is not possible to compute higher Busy Beaver numbers. The failure to do so is a form of the Berry paradox or undecidability. The Busy Beaver is then a sort of model idea of a strange attractor with the exponential separation of differing initial conditions for two systems.

We have for coherent states, a general form of laser states of light, the occurrence of states of the form |p, qвџ© that have both symplectic and Riemannian geometry. My mind is pondering what connection this concept of incomputability has to coherent states. The occurrence of Riemannian geometry for spacetime, particularly if spacetime is a large N entanglement or condensate of states, and an underlying quantum geometry may be ordered as such. Einstein in his Annus Mirabilus proposed that states of light have blackbody or Boltzmann thermal distributions with a coherent set of states in his coefficients. This may really describe quantum gravitation as well.

Please take a look at the referenced paper by Simant Dube. He finds essentially the same computability result as Blum et al, studying the fractal attractors of iterated function systems.

5 days later

Dear Tim Palmer,

If you are a physicist rather than an inflexible mathematician, you may hopefully be in position and ready to answer my question:

While I know, "a closed interval is an interval that includes all of its limit points", I guess there might be a fundamental point-based alternative to the "dot-based" mathematics from Dirichlet up to Heine and Borel. Given real numbers constitute Euclidean points, isn't then a discrimination between closed and bounded only justified for rational numbers? Isn't it logically impossible to include a single real number? Is the notion limit point really reasonable?

Well, this request relates to my own essay rather than directly to your essay. I am just curious.

Sincerely,

Eckard Blumschein

    Hello,

    I liked a lot your general essay. Several ideas are very relevant about the links between this quantum mechanics and this GR. I consider personally in my model of spherisation a gravitational coded aether sent from the central cosmological sphere made of finite series of spheres, I tell me that we have a deeper logic than only our relativity and these photons like main essence. This space, vacuum seems more than we can imagine. I have shared your essay on Facebook because it is one of my favorites, regards

      All I can say is that these are deep questions!!

      Thank you.

      I will try and explain at 3385 why I as a layman in mathematics feel forced to deal with fundamentals of mathematics.

      Let me here quote from your abstract something easily understandable to everybody:

      "...undecidability is only manifest in propositions about the physical consistency

      of putative hypothetical states". In my words: Continue calculating as if.

      Just an aside on your Hilbert quote after illustrating Cantor's dust: "The infinite is nowhere to be found." I argue that the property of being infinite is to be seen in every closed loop.

      You demonstrate the point that to synthesize General Relativity (GR) and Quantum theory (QT) seems to require much more complex mathematics. Along the way, for problems of quantum entanglement, Bell inequality, and (I submit) quantum eraser ,a theory would have to be non-local and causal. Also, there are many problem observations in astronomy and cosmology which should be addressed. I prefer to generalize the main goal as finding one theory that describes both big and small of our universe.

      I think, like you, that dealing with non-computability could "...break the road-block in finding a satisfactory theory...". That is, non-computability of the mathematics is a problem. I suggest the novel path to this new theory is to remove the mathematics that make the models complex and then to restructure the principles to explain the problem observation and to correspond to GR and QT with approximations.

      For example, interpret the Bell inequality so to say NO interaction takes place at less than the speed of light or that ALL interactions take place at a speed much greater than light (such as van Flanern and other observations finding the gravity speed is much faster than light). The Newtonian speculation that interference of light includes the aether wave traveling much faster than the photons and then directing the photons. This suggest the matter causes the aether waves and the aether divergence directs the photons - like in GR. Thus, a further unity can be achieved if the aether is the left side of the GRs field equation (space-time) and the medium supporting real waves in QT. Just these two changes/insights can yield a simpler and more complete model.

      I'm unsure how to treat chaos ideas. I think you're correct, we should assume determinism and self-similar (fractal) model even if the reality of the universe may be different.

      Hodge

      Great article Tim. You present your ideas clearly and logically so that even a non-physicist "newbie" such as myself can appreciate the main points.

      I was drawn to your title as my (much less rigorous) theory approaches possible unification through fractal geometry as well. I too suggest that the space-time of general relativity may emerge from the higher dimensional geometry of a quaternionic structure, but come at it from a rather unique way - through the cyclical nature of the prime numbers in base-12 (my article is titled "Primarily True").

      I posit that my "base-12 prime vibration" as a fractal invariant pattern should emerge in space-time at 10-to-the-power-11 logarithmic fractals (in terms of particle density) as that would represent a complete logarithmic cycle or "octave" of a quaternion power cycle in base-12. This might help explain why there are that many galaxies in the observable universe, suns per mature spiral galaxy, atoms per DNA molecule and even neurons in the human brain. In the gap between each such fractal would therefore be where nature would emerge in an unstructured way, much like your conjecture.

      One question: If the quaternion model were purely geometric such as I'm picturing - as simply prime positions on the base-12 circle (thus Euclidean), would it not then become computable after all? If I understand it correctly, Tarski's geometric decidability theorem seems to indicate that would be the case. Any insight would be much appreciated.

      Cheers,

      Michael

      Dear Tim,

      Excellent essay. I agree Chaos Theory offers good insight into nature as well as a predictive tool. I think it brave to major on it. Bill McHarris did so well in 2016 with a poor response. I hope you do better. (I drew more on new foundations & fuzzy sets.) I agree that resistance to non-finite maths is problematic, and thanks for reminding us of Hilbert's quote. Was he blind to 'infinite' Pi, space and time?

      I was interested in your view that a deterministic foundation to QM should exist, one shared by myself and John Bell. I quote Bell this and last yr and describe a mechanism appearing to show we're correct! But of course too shocking for most to countenance! I hope you'll take a close look.

      Beyond your (p6) pairs; if BOTH have N and S poles & parallel axes, and A & B polariser interactions give Poincare sphere surface vector additions, can you think of any reason A & B, by reversing their settings, couldn't reverse their own 'amplitude' outcomes. I found that's NOT a hypothesis Bohr tested!

      Your p8. assertion that inequality violation can emerge from an uncomputable deterministic model seems to preclude a physical ontological understanding of process, as all others assume. Does it? If so I disagree so hope you'll explain why you believe so (if Bells proof can be 'sidestepped' as he suggested).

      I agree gravity is non-computable, but do you agree that may be in the same way weather parameters are? i.e; All low pressure areas have a density gradient due to rotational velocity, after Bernouili, but all differ slightly and constantly evolve.

      Great essay Tim, and I look forward to discussing various matters further, I think best after you've read mine.

      Very Best

      Peter

      Hello to both of you,

      the problem about these strings is that it is a kind of fashion now about what we have at this planck scale and about the 1D main Cosmic field creating our reality by the Waves, fields and oscillations. But in fact a sure thing is that nobody can prove and be certain about this philosophical generality. The same for ,my gravitational coded aether made of spheres sent from the central cosmological sphere. We cannot affirm and all rational deterministic searchers accept the difference between a proved law, equation, axiom or an assumption. Nobody can affirm what we have at this planck scale nor about the philosophy of the generality of this universe. Have we coded particles or Waves creating our geometries, topologies, matters and properties and this emergent space time. We know that inside the theoretical sciences Community, all we are persuaded and that the Vanity is important, but without proofs we cannot affirm, it is a fact.

      What is really this space, this vacuum ? is it still this gravitational superfluid coded aether or fields different , we don t know simply and we must accept this and our limits in knowledges.

      Regards

      Let me support : "slowly and carefully, and not jump to conclusions that may at first sight seem reasonable, but will ultimately turn out to be wrong."

      Having read

      "⊆ means subset, ⊂ will not be used"

      I reconsidered my question "Isn't it logically impossible to include a single real number?"

      6 days later

      Congratulations on a nice essay! I think that chaos theory is an underestimated element of the foundations of physics. I think you may find resonance with your ideas in Hoessenfelder's essay (https://fqxi.org/community/forum/topic/3433) and partly in mine (https://fqxi.org/community/forum/topic/3436).

      Best of luck for the contest!

      Flavio

      Tim. I enjoyed your article and the inclusion of chaos theory into the discussion of the unification of quantum mechanics and relativity. In my essay "Clarification Of Physics--" I introduce a new perspective into the unification efforts. In the essay I propose a creation process that emerges from chaos, unifies quantum mechanics and relativity, and creates "our" finite multiverse and the visible universe. I would appreciate your comments on the essay. John D Crowell

      5 days later

      Hi to both of You, dear Eckard, it is too much complex to find the real meaning of the infinity, we can of course rank the different infinities inside this physicality and still we know just a small number of these infinities, cantor, Godel or Euler or all the maths works are not the problem, the problem is our limitations inside the physicality and philosophically, we cannot understand inside the physicality all the finite series and all the different infinities simply and it is still more complicated to encircle a kind of infinity beyond this physicality, is it conscious or not and how this thing creates this physicality, is it with strings and wavesm fields or points and a geonetrodynamics or in my model with 3D spheres coded and a gravitational coded aether , we cannot affirm, so that implies a pure uncertainty for our foundamental objects and we cannot predict and rank all simply, like we cannot compute all. We are limited in knowledges simply, even in closed loops you cannot find the answers for these infinities inside this physicality and still less this infinity beyond this physicality, we must recognise this simple fact.What do you Think? Regards

      the numbers and maths are not the problems you know Eckard , nor the finite ranked numbers or the different class of numbers, rationals, reals, complex, irrationals or others, or the different infinities inside this physicality, the problem is our limitations in knowledges, you can tell all what you want about the single ral number, all what I said is a fact. The set, the sets, the subsets are not the problem, the partition universal is the problem

      12 days later

      A novel and fascinating idea. It brings to mind Pitowsky's 'Resolution of the EPR and Bell paradoxes' by extending the concept of probability to non-measurable sets.

      Thanks Jeffrey for these kind remarks. I know what you mean about Pitowski's work. However, on (what I call) the invariant set, the relevant measures can be described by elementary finite frequentist probability theory. The mathematics underpinning undecidability arises only when considering counterfactual states which do not lie on the invariant set. My approach, is simply to deny ontic reality to such states by postulating the primacy of the invariant set. Without this, I think one would indeed be drawn to consider non-measurable sets as did Pitowski. However, the concept of non-measurability does not seem to make physical sense to me - as the famous Banach-Tarski paradox clearly indicates.

      PS Reference [18} - my paper arXiv:1804.01734 on invariant set theory - has now been accepted to appear in Proceedings of the Royal Society A.

      Dear Tim,

      You write: "Our inability to synthesise general relativity theory and quantum theory into a satisfactory quantum theory of gravity is legendary and is widely considered as the single biggest challenge in contemporary theoretical physics."...

      Quantum theory and the general relativity theory are phenomenological theories (parametric, operationalistic) without an ontological basification (justification+substantiation). It makes no sense to "unite" them, let each one work in its own "field". Problem 邃-1 for fundamental science and cognition in general is the ontological basification (substantiation) of mathematics, and therefore knowledge in general.

      You conclude: "From where do new ideas come? Do they pop out of the aether as some random flashes of inspiration with no obvious precedent? Or do these ideas mostly already exist, but in a completely separate setting."

      Ideas come to our minds from the primordial (absolute) generating structure that lies both in the "beginning" of the Universum ("top") and in our heads ("bottom"). The task of physicists, mathematicians and philosophers is to understand the dialectics of Nature (catch on the "net" "Proteus of Nature" using the "goddess of form" Eidothea and "crazy" ontological ideas) and build this Superstructure - the ontological basis of Mathematics ("language of Nature") and Knowledge as a whole: ontological framework, ontological carcass, ontological foundation. Today we need a global brainstorming session to "assemble" all the ideas for discussing and creating the Ontological Knowledge Base.

      With kind regards, Vladimir

      10 days later

      Dear Tim,

      It is a very nice and original idea you have presented in your essay. As many others have said I would probably need a few more reads to grasp all the details though.

      Few questions if I may:

      - If an underlying fractal geometry can give rise to quantum-like behaviour, how does classicality emerge from this picture, if it does at all?

      - Would you have any toy-example with the Lorentz attractor of non-computable counterfactual?

      Many thanks.

      Best,

      Fabien