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

    Thanks Fabien. Good questions.

    My fractal model has a free parameter N. In the singular limit N=infinity all the fractal gaps close up and the state-space geometry is classical. However, for any finite value of N, no matter how big, the Bell counterfactuals lie in the fractal gaps and the state-space geometry is non-classical. Michael Berry has written about how old theories of physics are often the singular limits of new theories as some parameter of the new theory is set to zero or infinity.

    The Cantor Set underpins the Lorenz attractor. Imagine a point X on the Cantor Set and perturb it with a perturbation delta X drawn randomly using the measure of the Euclidean line in which the Cantor Set is embedded. Then the perturbation almost certainly perturbs the point off the Cantor Set.

    Such a perturbation can be thought of as corresponding to one of my counterfactuals: although I live in a world where I did X, what would have happened if I had instead done X+delta X? Suppose the delta X is dynamically unconstrained - i.e. something you just make up in your head without consideration of whether it satisfies the laws of physics - then if the world associated with X lies on the invariant set, the world associated with X+delta X almost certainly does not and so the answer to the question "what would have happened?" is undefined.

    Dear Tim Palmer,

    Any essay combining general relativity and Bell's theorem is a 'must read'.

    In it you show that it's possible to violate Bell's inequalities with a locally causal but uncomputable deterministic theory for locally causal spacetime computations. Chaos is powerful, but I'm unsure what the ontological implications are.

    A number of authors are concerned whether 'classical physics' is truly deterministic, and if not, how is this explained.

    If one assumes that the deBroglie-like gravitomagnetic wave circulation is induced by the mass flow density of the particle [momentum-density], then the equivalent mass of the field energy induces more circulation. This means that the wave field is self-interacting. For 'one free particle' a stable soliton-like particle plus wave is essentially deterministic. But for many interacting particles, all of which are also self-interacting, then 'determinism' absolutely vanishes, in the sense of calculations or predictions, and the statistical approach becomes necessary.

    This theory clearly supports 'local' entanglement, as the waves interact and self-interact, while rejecting Bell's 'qubit'-based projection: A, B = +1, -1 consistent with the Stern-Gerlach data (see Bohr postcard). For Bell experiments based on 'real' spin (3-vector) vs 'qubit' spin (good for spins in magnetic domains) the physics easily obtains the correlation which Bell claims is impossible, hence 'long distance' entanglement is not invoked and locality is preserved.

    This is not a matter of math; it is a matter of ontology. I believe ontology is the issue for the number of authors who also seem to support more 'intuition' in physics. My current essay, Deciding on the nature of time and space treats intuition and ontology in a new analysis of special relativity, and I invite you to read it and comment.

    Edwin Eugene Klingman

    4 days later

    Tim Palmer re-uploaded the file Palmer_FXQi_Palmer_1.pdf for the essay entitled "Undecidability, Fractal Geometry and the Unity of Physics" on 2020-04-17 07:52:00 UTC.

    Hello Tim --

    Wow, this rather blew my mind, and I'm still digesting it.

    Let me ask you a really basic question. A chaotic system implies that even approximate knowledge about a path into the far future depends upon the initial conditions--you need to keep going to more and more decimal places in the expansion.

    This, in turn, means that we should expect meaningful facts about the future evolution of a chaotic system to be uncomputable. To be really specific, there should be many facts along the lines of "will these three objects collide with each other eventually" whose answer will be uncomputable.

    Would it be fair to say that your results here (Eq 6) draw their power from this feature of chaotic dynamics? This would help me in understanding your results better.

    So many lovely things here. I had never thought about Lewis' notion of "neighbourhood" in modal logic could be so usefully transposed to counterfactual thinking in physical systems. The idea that the p-adic metric is the "right" notion of "nearby", i.e., modally accessible, is extremely cool.

    Yours,

    Simon

    PS, minor remark Re: finite time singularities in Navier-Stokes--we know (for sure) that they exist in General Relativity, and if you're a hardcore physical Church-Turing thesis person, this is one way we know that (classical) GR is incomplete.