Monday, February 10, 2020

Guest Post: “Undecidability, Uncomputability and the Unity of Physics. Part 2.” by Tim Palmer

[This is the second part of Tim’s guest contribution. The first part is here.]

In this second part of my guest post, I want to discuss how the concepts of undecidability and uncomputability can lead to a novel interpretation of Bell’s famous theorem. This theorem states that under seemingly reasonable conditions, a deterministic theory of quantum physics – something Einstein believed in passionately – must satisfy a certain inequality which experiment shows is violated.

These reasonable conditions, broadly speaking, describe the concepts of causality and freedom to choose experimental parameters. The issue I want to discuss is whether the way these conditions are formulated mathematically in Bell’s Theorem actually captures the physics that supposedly underpins them.

The discussion here and in the previous post summarises the essay I recently submitted to the FQXi essay competition on undecidability and uncomputability.

For many, the notion that we have some freedom in our actions and decisions seems irrefutable. But how would we explain this to an alien, or indeed a computer, for whom free will is a meaningless concept? Perhaps we might say that we are free because we could have done otherwise. This invokes the notion of a counterfactual world: even though we in fact did X, we could have done Y.

Counterfactuals also play an important role in describing the notion of causality. Imagine throwing a stone at a glass window. Was the smashed glass caused by my throwing the stone? Yes, I might say, because if I hadn’t thrown the stone, the window wouldn’t have broken.

However, there is an alternative way to describe these notions of free will and causality without invoking counterfactual worlds. I can just as well say that free will denotes an absence of constraints that would otherwise prevent me from doing what I want to do. Or I can use Newton’s laws of motion to determine that a stone with a certain mass, projected at a certain velocity, will hit the window with a momentum guaranteed to shatter the glass. These latter descriptions make no reference to counterfactuals at all; instead the descriptions are based on processes occurring in space-time (e.g. associated with the neurons of my brain or projectiles in physical space).

What has all this got to do with Bell’s Theorem? I mentioned above the need for a given theory to satisfy “certain conditions” in order for it to be constrained by Bell’s inequality (and hence be inconsistent with experiment). One of these conditions, the one linked to free will, is called Statistical Independence. Theories which violate this condition are called Superdeterministic.

Superdeterministic theories are typically excoriated by quantum foundations experts, not least because the Statistical Independence condition appears to underpin scientific methodology in general.

For example, consider a source of particles emitting 1000 spin-1/2 particles. Suppose you measure the spin of 500 of them along one direction and 500 of them along a different direction. Statistical Independence guarantees that the measurement statistics (e.g. the frequency of spin-up measurements) will not depend on the particular way in which the experimenter chooses to partition the full ensemble of 1000 particles into the two sub-ensembles of 500 particles.

If you violate Statistical Independence, the experts say, you are effectively invoking some conspiratorial prescient daemon who could, unknown to the experimenter, preselect particles for the particular measurements the experimenter choses to make - or even worse perhaps, could subvert the mind of the experimenter when deciding which type of measurement to perform on a given particle. Effectively, violating Statistical Independence turns experimenters into mindless zombies! No wonder experimentalists hate Superdeterministic theories of quantum physics!!

However, the experts miss a subtle but crucial point here: whilst imposing Statistical Independence guarantees that real-world sub-ensembles are statistically equivalent, violating Statistical Independence does not guarantee that real-world sub-ensembles are not statistically equivalent. In particular it is possible to violate Statistical Independence in such a way that it is only sub-ensembles of particles subject to certain counterfactual measurements that may be statistically inequivalent to the corresponding sub-ensembles with real-world measurements.

In the example above, a sub-ensemble of particles subject to a counterfactual measurement would be associated with the first sub-ensemble of 500 particles subject to the measurement direction applied to the second sub-ensemble of 500 particles. It is possible to violate Statistical Independence when comparing this counterfactual sub-ensemble with the real-world equivalent, without violating the statistical equivalence of the two corresponding sub-ensembles measured along their real-world directions.

However, for this idea to make any theoretical sense at all, there has to be some mathematical basis for asserting that sub-ensembles with real-world measurements can be different to sub-ensembles with counterfactual-world measurements. This is where uncomputable fractal attractors play a key role.

It is worth keeping an example of a fractal attractor in mind here. The Lorenz fractal attractor, discussed in my first post, is a geometric representation in state space of fluid motion in Newtonian space-time.

The Lorenz attractor.
[Image Credits: Markus Fritzsch.]

As I explained in my first post, the attractor is uncomputable in the sense that there is no algorithm which can decide whether a given point in state space lies on the attractor (in exactly the same sense that, as Turing discovered, there is no algorithm for deciding whether a given computer program will halt for given input data). However, as I lay out in my essay, the differential equations for the fluid motion in space-time associated with the Lorenz attractor are themselves solvable by algorithm to arbitrary accuracy and hence are computable. This dichotomy (between state space and space-time) is extremely important to bear in mind below.

With this in mind, suppose the universe itself evolves on some uncomputable fractal subset of state space, such that the corresponding evolution equations for physics in space-time are computable. In such a model, Statistical Independence will be violated for sub-ensembles if the corresponding counterfactual measurements take states of the universe off the fractal subset (since such counterfactual states have probability of occurrence equal to zero by definition).

In the model I have developed this always occurs when considering counterfactual measurements such as those in Bell’s Theorem. (This is a nontrivial result and is the consequence of number-theoretic properties of trigonometric functions.) Importantly, in this theory, Statistical Independence is never violated when comparing two sub-ensembles subject to real-world measurements such as occurs in analysing Bell’s Theorem.

This is all a bit mind numbing, I do admit. However, the bottom line is that I believe that the mathematical definitions of free choice and causality used to understand quantum entanglement are much too general – in particular they admit counterfactual worlds as physical in a completely unconstrained way. I have proposed alternative definitions of free choice and causality which strongly constrain counterfactual states (essentially they must lie on the fractal subset in state space), whilst leaving untouched descriptions of free choice and causality based only on space-time processes. (For the experts, in the classical limit of this theory, Statistical Independence is not violated for any counterfactual states.)

With these alternative definitions, it is possible to violate Bell’s inequality in a deterministic theory which respects free choice and local causality, in exactly the way it is violated in quantum mechanics. Einstein may have been right after all!

If we can explain entanglement deterministically and causally, then synthesising quantum and gravitational physics may have become easier. Indeed, it is through such synthesis that experimental tests of my model may eventually come.

In conclusion, I believe that the uncomputable fractal attractors of chaotic systems may provide a key geometric ingredient needed to unify our theories of physics.

My thanks to Sabine for allowing me the space on her blog to express these points of view.


  1. By the end of 1990s we knew of the existence of an infinite subset of periodic orbits inside any chaotic attractor and the possibility of enforcing a periodic behavior via a judicious sequence of perturbations, so called, "Chaos Control". If the universal phase space truly hosts such a genuine fractal attractor wouldn't it be natural to ask for similar control methods for Bell like experiments?

  2. Very good posts about your excellent essay, Tim. I wish I could add anything to the essay itself, but I have read it many times and will certainly read it many more times to begin to grasp the subject. The only fractal I actually use is the Prime Graph, even then... I don't know any math, so... don't hold your breath.

    Good luck! Hope you win!

  3. Maybe we should reject the counterfactual entirely-- the idea of a "possible alternative". If an event takes place, there was never an alternative. If I throw a rock at a window, or arrange a test of Bell's theorem, I am not making any free choices, selecting "freely" among various options. There were no alternatives and there are no alternatives. And in fact, what I "think" about alternatives or free choices is also determined. For some reason I don't find that objectionable at all. And that's really true-- for SOME reason. There is a cause, even if I am not aware of it. For SOME reason, I like that idea!

  4. "the differential equations for the fluid motion in space-time associated with the Lorenz attractor are themselves solvable by algorithm to arbitrary accuracy and hence are computable", which, from the FQXi essay, apparently means Liouville flow. Introducing measure theory conflicts nicely with the axiom of choice, so OK. The next step, to address the relationship between Liouvillian CM and QM, is now, with some failings, in

    1. It should be analogous to the halting problem: deciding "if given program will halt in time t" can be checked in time t.
      However, asking "if given program will halt" could need testing t -> infinity sequence this way.
      Even worse, it is uncomputable/undecidable problem - assumption of existence of such program can be used to build a sentence which cannot be true/false, like "this sentence is false" sentence.

      For a fractal, question "is given point in distance epsilon from the fractal" can usually be computed in a finite time, going to infinity for epsilon -> 0.
      Question "is given point in fractal" would require to use epsilon -> 0 infinite sequence of tests this way.
      However, to be uncomputable we would additionally need to build such self-contradictory sentence with it, e.g. by reduction to the halting problem ...
      We have studied this "Complexity and Real Computation" book a decade ago in computer-assisted proof group, but I don't remember such proof (?)

      However, even reading the linked paper, I still don't see connection with Bell theorem here.
      We have inequalities derived from assumptions of realism and locality, which are violated by physics, so at least one of these assumptions is nonphysical - which one?
      Does e.g. general relativity satisfy these assumptions?

  5. Hi all:

    I am still absorbing the contents, but here is a question, concerning Lorenz' attractor, which had struck me years ago, but didn't find discussed anywhere---not at least in the beginning text-book treatments or pop-sci literature on nonlinear dynamics, chaos etc. The question is this:

    Consider the two black ovals ``inside'' all the trajectories shown in the above diagram. Are their sizes finite? I mean to say, is there some region of this space which will never be cut through by the trajectories (given a fixed set of governing parameters) on the inside? Can it be proved?

    Another, obviously closely related, question: How about on the ``outer'' side? Are all trajectories bound to a finite region so that there is an infinite region of this space through which the trajectories would never pass? Can it be proved?

    Thanks in advance for any pointers...


    1. Ajit: after trying some different terms in a well-known search engine, I found an article that may serve as a starting point for researching the topic, assuming no specific answer is forthcoming here:

      Unfortunately the article is paywalled. I hope this helps.

    2. Hi Jeff,

      Thanks for taking all the trouble. Yes, the article does seem to be quite relevant.

      And yes, the question does turn out to be well-posed in the first place! I wasn't even sure of that! :)


  6. Sounds as a very interesting idea to be followed up seriously.
    I'll try to understand the cited articles, but am afraid it will not be easy stuff.

  7. Quote:
    "In such a model, Statistical Independence will be violated for sub-ensembles if the corresponding counterfactual measurements take states of the universe off the fractal subset (since such counterfactual states have probability of occurrence equal to zero by definition)."

    I am merely an amateur in physics but I have some years back made some computer simulations to try to violate Bell's Theorem. With up to 83% of simulated particles undetected it is well known that it is possible to break the inequalities reasonably well. If the undetected 17% of particles actually had zero probability of existence, then a simulation could match experiment without obvious slight of hand in data massaging/selection.

    I know of a case where some simulated particles are supposed to have zero probability of existence becase they exist in R^3 [the simulation space] but do not exist in S^3 [the supposed experimental space]. The theoretical model behind this has met opposition. Are you in a position to set up a simulation where you can identify which simulated particles are in and which are not in the fractal subset?

    I am not clear about whether particles are only tending (with time) to be on the fractal subset, or are they always on the subset. If particles are not always in the fractal subset then that might reduce the performance in a real Bell experiment?

    Austin Fearnley

    1. It's not only the particle that makes up the prepared state which defines a point in state space but, at least technically, a point in state-space is defined by a complete set of degrees of freedom of all particles (or fields, if you wish) in the universe.

      In Tim's model, this complete state is always on the attractor. On that attractor, some combinations of prepared state and detector settings just do not exist, resulting in a violation of Statistical Independence.

      I suspect you have Joy Christian's model in mind. This doesn't have any relation to Tim's model, or at least I cannot see any.

    2. This comment clarifies things nicely.

    3. "I suspect you have Joy Christian's model in mind. This doesn't have any relation to Tim's model, or at least I cannot see any."

      Yes, I had Joy Christian's model in mind, but only because it is the model I am most familiar with with respect to Bell's Theorem simulations. I do not support Christian's model, whereas I am merely uncertain about the fractal attractor/p-adic model.

      There are a few points in common though, at least to my naive understanding. Joy used geometric algebra to express his model. He appears to believe it could not be expressed in normal algebra. I learned enough geometric algebra to check his model to my own dissatisfaction. Tim is using p-adic numbers to express his model.

      A simple computer simulation of 'Bell' would launch particles into R^3 space which would not correspond to the real world space, in Tim's case, of fractal attractors. As I read the model, going from R^3 to fractal attractor space would be uncomputable (so bang goes any hope of a simulation) but movement entirely within fractal attractor space is computable only using p-adic numbers?

      Austin Fearnley

  8. I'm so glad there are others who are exploring the superdeterminism alternative to the Bell inequality despite the likely ridicule from the rest of the physics community, as 't Hooft has experienced. Re-deriving quantum mechanics in a deterministic (and local!) framework might lead to a deeper understanding the same way that kinetic theory of gas improved on what came before. At the very least it should give an adequate answer to the measurement problem. Who knows what other insights will emerge once we stop thinking of the universe as inherently probabilistic--the possibilities are endless.

    1. "Who knows what other insights will emerge once we stop thinking of the universe as inherently probabilistic--the possibilities are endless."

      Very funny!

  9. I guess the thing I am having trouble with is understanding exactly what is meant by a fractal that is not computable. Fractals are generated by recursive enumeration, or equivalently are recursively enumerable sets, which are perfectly computable. The complement of a recursive set is recursive, but the complement of a recursively enumerable set is not recursive nor is it recursively enumerable. This complement is what is incomputable.

    With the fractal sets of orbits that nest into each other there is a Cantor structure and it is not possible to define a field structure or metric geometry. To define such one appeals to p-adic numbers. The result of Matiyasivich is there is no general method for computing these sets, which are equivalent to Diophantine equations. Then for different entanglement geometries there are different obstructions between them. For a massive N-entangled system these obstructions are given by the fact this p-adic measure is only local, not global, and so there is no universal map between all possible entanglements.

    1. Lawrence,

      The fractal in question here is the attractor of some (unspecified) differential equation. If you wanted to calculate it by integrating the equation, you'd have to integrate it to all eternity. It may be that this fractal can *also* be generated (or at least be defined) in a way that does not require you to actually integrate the equations, but whether or not that's the case is somewhat hard to say if you don't have the differential equation in the first place.

    2. That one has to compute endlessly to find a more exact description is not incomputability. If it were then because trigonometric functions are computed by Taylor series would mean they are incomputable. Fractals are recursively enumerable sets. A Julius set can be computed to finer detail based on the floating point precision of a computer.

      An incomputable set is one that fails to be provable to exist within some axiomatic system. This is a different matter. Now what something such as a Cantor set can have as an incompleteness is a metric description the works for all other such sets. This in the p-adic setting means there are no universal or global solution methods. Matiyasivich, following work by Post, Robinson etc, demonstrated this is a form of Gödel'a theorem.

    3. Part of the problem here seems to be the question of the model of computation in use in the Tim Palmer theory. This model of computation, the BSS model, is a form of "real computation" which assumes that the computer can calculate with infinite precision real numbers. Of course in Turing based computations, computable reals would physically require an infinite amount of computation time to be determined. Also calculations involving them would take an infinite amount of computation time (is this the same as "physical time" here?) to "complete".

      With BSS computation the results (can) become available immediately. In this theory the terms like "computable" and "decidable" therefore are different terms. In BSS theory the Mandelbrot set is provably non-computable; the result for Turing computation is still unknown, I believe. According to Wikipedia if a certain conjecture is true then the Mandelbrot set is actually Turing computable.

      Nevertheless Tim Palmer is promoting an intriguing idea that of making a distinction between : "computable real trajectories" and "non-computable phase space trajectories".

      However thinking about it further there may be a need (as mentioned in a comment in Part 1) that there should be a Part 0. For example I assume that the spaces here are all based on regular real numbers (and not rationals, say). If so then most "points" will be non-turing-computable, since there are only a countable number of Turing computable reals. So trajectories may or may not pass through non-computable points. Does this matter?

      Also the terms "fractal" and "differential equation" do not sit well in the same sentence, since a fractal is non-differentiable everywhere!

      Finally I am not convinced that physics has established that all differential equations will preserve (Turing) computability in any case.

    4. Lawrence,

      To build an effective field structure and a metric geometry on non-smooth manifolds like the Cantor set, one appeals to either 1) fractional calculus (FC) or 2) the theory of multifractal sets.
      The study of chaos and strange (fractal) attractors is not so much about solving differential equations in closed form, as it is about analyzing their corresponding phase space behavior. FC applied to generic dynamical systems leads to fractional dynamics, whereas FC applied to QFT leads to fractional field theory.
      Also, one does not typically compute individual trajectories in chaos theory, but the global ensemble of trajectories bounded inside the strange attractor. The study of global properties of chaotic ensembles parallels the study of equilibrium statistical mechanics, when one appeals to traditional concepts such as invariant probability distributions, statistical independence and the ergodic theorem.

    5. The BBS computation renders the Mandelbrot set incomputable, but this is then different from undecidable. With Gödel’s theorem we are talking about the latter. Hence, my issue I think still remains.

      Fractals are RE sets and their complements are undecidable. In a p-adic setting a fractal set, at least when mapped to or in a functor sense the same as a Cantor set, the undecidable setting with with the set of such sets. In this you have a set of p-adic numbers and these have no decidable system, or equivalently there is no universal method for solving Diophantine equations.

    6. Lawrence,

      Note that part of my comment above was strictly related to the question of building an effective field-theoretic structure on fractal sets via fractional differential and integral operators.
      There is a vast literature on this topic demonstrating the successful application of such operators in science and engineering alike.

    7. @Ervin: I have been pondering whether there is a breakdown of ergodicity in QED. The ergodic principle

      Lim_{T→∞}1/T ∫dt f(ωt) = ∫dωF(ω)dω

      has the time average on the left and expectation on the right. For extremely short laser pulses, or conditions where the photon envelop is shorter than the wavelength, say a resonance, it seems to me this condition fails.

      There is a lot underneath this. The Cantor set is a subset of the interval [0. 1], and it has some paradoxical properties. In particular the Cantor set is an immesurable subset, or one that defines a translation set that is not completely measureable. This is the paradox of the immesurable measure. The Banach-Tarski paradox is a case of this for the 3-ball in R^3.

    8. Lawrence,

      The violation of the ergodic and fluctuation-dissipation theorems is bound to happen in dynamic regimes that are far-from-equilibrium. Such is the case of nonperturbative behavior of QED in the far UV limit and in the presence of strong or highly fluctuating fields (see e.g. ). Same conditions lead to Landau pole scenarios, dynamic instabilities in terms of Poincaré resonances and the breakdown of the decoupling theorem. And this is where fractional calculus and fractional dynamics come in.

      Indeed, there are peculiarities in dealing with Cantor sets and nowhere differentiable functions, which are differentiable only in the fractional sense. Fractional dynamics is not compatible with standard QFT as it manifestly violates its consistency conditions driven by unitarity and locality. Fractional dynamics targets specifically nonequilibrium processes that display scale-invariant properties, dissipation and long-range correlations. It is only asymptotically compatible with QFT when the order of fractional differentiation smoothly approaches integer values.

      We can continue the discussion elsewhere as it involves a fairly complex subject.

    9. @Lawrence Crowell

      The Cantor set is a Borel set hence Lebesgue measurable. What are you saying? It has indeed interesting and not paradoxical topological properties but it is a measurable set.

      Banach-Tarski has nothing to do with the Ergodic theorem.

    10. @Bourbaki, I said immesuarable.

      The discussion on Banach-Tarski is a separate topic. Sorry for the confusion.

    11. @Ervin, Bourbaki complained I wrote measurable. I mistyped here. Elsewhere I wrote immeasurable.

      The fractional calculus can reflect a utility function or momentum renormalization function that is d(p,α) = (p^(1-α} – 1)/(1 – α), that for α = 0 gives a linear expression and as α → 1 this is logarithmic. The linear expression would correspond to the standard ergodic case. In between the bounds of the linear and logarithmic conditions the operators would be fractional differentials, and integrals of momentum would be gamma functions.

    12. Lawrence,

      I am afraid your comment fails to do justice to fractional calculus, in general, and fractional dynamics, in particular.

      This is a rich field of research with lots of subtleties and ramifications into chaos and complexity theories. We can barely skim the surface here and that's why the discussion should continue elsewhere.

    13. I am not that experienced with fractional operators, though I have read on them a bit. My email is at

      I have just had this idea for a while on how for ultrashort pulses of light that the ergodic assumption should breakdown. I have thought there might be some new types of physics or new forms of quantum states in this regime.

  10. Apologies for this not being directly about the math & physics - but for someone like myself who suspects that consciousness is the empty chatter of paralysis and inhibition, your counterfactual account of free will is gorgeous. Free will is listing all the things you could have done :-)

  11. There are uncountably many strictly monotonic functions from [0,1] to [0,1]. The image of the Cantor set C_2 under each of these functions is a different fractal set, so almost-all fractal sets are not as simply describable as C_2. On the other hand, if you're interested in a constructive proof of the existence of fractal sets, it's of course simpler to just exhibit C_2.

    Quite differently, if you're interested in a useful metric on [0,1] suitable for useful arithmetic, then your only choices are p-adics metrics and the Archimedean metrics.

    My question for Dr. Palmer is the following: are the p-adic based attractors he has described more like the first case (i.e. they're useful as an existence proof but there's no reason to suspect that reality actually has those surfaces as opposed to other more-difficult-to-describe fractal surfaces) or more like the second case (i.e. there really aren't mathematically-possible surfaces with the required property)?

    1. As I indicate above what is undecidable is whether any method for solving one p-adic field or metric works with any others. Matiyasivich proved a form of Gödel's theorem that there does not exist a global p-adic solution. This was an answer to Hilbert's 10th problem on whether there existed a universal method for solving Diophantine equation, and Diophantine equations are p-adic sets as proven by Julia Robinson and others.

  12. If uncomputability is central to Palmer's idea, then his idea may be fatally flawed, as Palmer seems unaware that he is invoking two distinct and incompatible concepts of computability on the reals.

    Palmer writes, "the differential equations for the fluid motion in space-time associated with the Lorenz attractor are themselves solvable by algorithm to arbitrary accuracy and hence are computable."

    Here Palmer is invoking Type-2 computability, which focuses on being able to compute arbitrarily close rational approximations to the exact answer.

    He also writes, "the attractor is uncomputable in the sense that there is no algorithm which can decide whether a given point in state space lies on the attractor"

    As explained in his essay, here Palmer is invoking the Blum, Shub, and Smale (BSS) notion of computability on the reals, which allows for infinite precision real constants, infinite precision arithmetic operations, and infinite precision comparisons of two real numbers, but has no notion of computing arbitrarily good approximations.

    The problem with BSS-computability is that it is both absurdly strong and absurdly weak. As Braverman shows, the Halting Problem itself is BSS-decidable! And yet the trigonometric functions are not BSS-computable, so even the motion of the classical harmonic oscillator is BSS-uncomputable.

    1. Thanks to all who have brought up BSS. I do remember something about this in the 90s. The issue with what is meant by decidability and computability here is clearer. This is not really Gödel incompleteness. I have though posted a paper on FQXi on how something related to this is connected to Gödel incompleteness,

    2. Correction: I misread Braverman, and he does not actually show that the Halting Problem is BSS-decidable. However, here is a proof. Note that the infinite-precision arithmetic operations and comparisons allowed by the BSS model allow one to extract arbitrary bits from the binary expansion of a real number. Note also that you can use arbitrary real constants in your program, including real constants that encode arbitrary oracles. So the following algorithm in the BSS model solves the Halting Problem:

      1. Input the index i of a program p.

      2. Let x be the real constant in the interval (0,1) representing the halting oracle: bit j of x is 1 if the program whose index is j eventually halts, and 0 otherwise.

      4. Extract and return bit i of x.

  13. Ah... Music to my ears! Essentially, Tim contradicts the assumption (ubiquitous in philosophical circles) that the set of physically conceivable worlds is identical with the set of physically possible worlds. This always seemed dubious to me.

  14. If there is always smaller and always bigger components, with the only bigger parts and always smaller parts only communicating by gravity and not electromagnetism, how could there be superdeterminism?

  15. My degree in biochemistry and 40 year banking career did nothing to help me understand this article... but I sure enjoyed reading it. I should stick to Sabine’s YouTube videos!

  16. It seems that one can think in terms of at least two kinds of universe, or categories of universe. One category would be the universe as simulation, in which everything that occurs in finite time must be computable, since what we experience is running on some sort of device, according to some "program". But that sort of model can be implicit in theories that make no reference to simulation at all. They are simply invoking computable processes at every point. These are models of a universe that COULD be simulations. On the other hand we could have a category of models in which the universe is not computing anything-- the universe as a thing-in-itself manages to be what it is, and contain whatever events it contains, with no process akin to computation at all, and then we see that our theories and models are attempts to simulate such a universe. But in this case we might find that our theories are never complete and never are able to accurately simulate the workings of the universe because we are required to model processes that are fundamentally different than anything we can express using the mathematical tools available. I hope this makes some sense. I just see that some models implicitly assume the sort of universe that COULD be simulated, that could, in effect, "run on a computer".

  17. I get the gist of the mathematical argument here, but will leave it to others to determine its soundness. However, if we can go so far as to reference some physical reality, doesn’t the universe make its own calculation as to what happens next? Do we find there that causality is continuous or discrete? From the viewpoint of biology and its myriad dissipative systems we find a multitude of instances where, roughly, one part of the universe is turned back upon another in a contest of possible futures. It has been argued that such counter currents are a necessary feature of stable dynamical systems. (Positive and negative feedback: striking a balance between necessary antagonists) (

    Consider the contest between male and female Birds Of Paradise. Why, in a deterministic and parsimonious universe, would the male Bird of Paradise dance. Why would a universe governed by least-energy paths engage in the material expenditure of flamboyant plumage and caloric dissipation of energetic dance unless the female had agency to make an actual in the moment choice that determines which genes move forward. Would the universe have reached its present level of complexity without the ratcheting upward of these determinative events?

    1. Don Foster writes "Why, in a deterministic and parsimonious universe, would the male Bird of Paradise dance[?]"

      This seems understandable in terms of blind trial-and-error evolution. It doesn't proceed systematically, but randomly within the limits of path-dependency, so it tries many strange things. It turns out some strange things help or at least don't hurt, such as an expenditure of energy to demonstrate some sort of fitness (perhaps stamina, perhaps neurological soundness, perhaps coordination--not much of a dancer myself) to potential mates.

      Leaving aside the evolutionary argument, which is probably not unfamiliar to Dr. Foster, perhaps his issue is one of determinism vs. agency?

      As argued recently at this site, there is no contradiction between determinism and agency. AlphaGo Zero operates within a completely deterministic (computer) environment, and is responsible for winning or losing games depending on how well it makes choices. We praise it for its achievements, while lesser programs are erased from hard drives. The same is arguably true for female Birds of Paradise (Bird of Paradises?). The key point for me is that without determinism, there is no way to determine good choices. Whereas with determinism (or at least some) we can compute the consequences of choices (as best we can), and learn from our mistakes, rather than repeating the same choice and hoping it will turn out differently, as it could without determinism.

      (There goes another fine for unsolicited and possible irrelevant thoughts spamming the Internet.)

  18. Your argument fails to a very basic flaw. Chaotic systems are deterministic systems. The mathematical theory of dynamical Systems do no study random behaviors. That is not what it is all about. Ergodic Theory studies the statistical behavior of complex deterministic systems: that is where the random part comes from. Statistics. Averages. Correlations.

    Those systems are not computable for the simple reason that they are the result of a limit to infinity. A limit we are not yet able to take in the general case.

    My background is Dynamical Systems (phd in Ergodic Theory) but I also have a Master in theoretical physics. I have tried during many years to find the bridge between Dynamical Systems and Quantum theories. It makes totally sense. Dynamical Systems in its most general sense is a Spectral Theory (see for example Lyapounov exponents which are essencially eigen values of the limit average of the transformation). This is conceptually so close to operators acting on a Hilbert space... Actually it is the same in certain situations (see Von Neumann ergodic theorem). There should be a way to find a bridge between those two.

    Well, there is no bridge whatsoever between those two. That has been allready solved by Alain Connes...

    1. The only possible bridge is with undecidabilty --- it is not computable or decidable. I agree there is no dynamical connection or one that involves analytic solutions.

    2. By mistake, I clicked the wrong reply-button, and so it appeared below at:

      Applies to both "Bourbaki" and Lawrence Crowell.


    3. @Lawrence Crowell

      Well I think that most people (mathematicians and physicists included; including me) have no clue about what randomness means.
      Dynamical Systems (chaos theory as some people want to call it here) deals with complexity and how a complex behavior can be modelled as a random behavior.

      So basically the point of this article come to this: can quantum randomness (Born axiom) be modelled as a complex, dynamical randomness. The answer is yes and no.

      Yes because of decoherence: unfortunately, this setting is already studied by Statistical Physics (thermodynamics). This is the bridge between classic physics and statistical physics. The apparent randomness only appears because we are dealing with a lot of particles (complexity).

      No because if we want to explain quantum randomness: why a particle has a state modelled by a superposition of a spin up and spin down? Where is the complexity here? Where is the "immeasurable" world behind this superposition? One that would explain Born's axiom? There is none.
      Well, there is one actually, QFT is the correct setting to put the problem. Unfortunately, Alain Connes has allready classified all non-commutative probability measures there can be in the Hilbert space setting. So there is nothing below.

      Nothing below. Nothing above. We are pumping air in a subject that is empty. Quantum Mechanics is a black box we do not have yet neither the tools nor the theory to open.

      About undecidability. As far I know, in theoretical physics, there are NO problems of undecidability in the mathematical and logical sense.

    4. Cubitt and others posted proofs of the undecidability of the spectral band gap and of phase diagrams in arXiv:1810.01858 and arXiv:1910.01631. Does this count as a problem of decidability in the mathematical and logical sense, in theoretical physics? If not, why not?

    5. Randomness is undefinable. If there were a way of codifying random sequences they could be compressed by some algorithm. Chaitin's halting probability makes some light on this.

      While there is decoherence and the rest, this just reduces a density matrix to a diagonal with probabilities. It does not tell us which outcome will be found. The observed eigenvalue is then something undecidable and simply obtains for no computable reason. It is a form of spontaneous emergence.

  19. TINA : it reminds me of something...

  20. > find the bridge between Dynamical Systems and Quantum theories
    The simplest case to see this difficulty is: what stationary probability distribution should we expect for dynamics in [0,1] range?

    QM says we should expect localized distribution rho~sin^2.
    Standard diffusion says we should expect uniform rho=1.
    Maximal Entropy Random Walk (uniform path ensemble) says rho~sin^2 as QM.

    Can we explain this rho~sin^2 in [0,1] with deterministic dynamical systems, chaos?


    1. My friend, you are very connected with the realism and locality of Einstein; I don't know why Einstein saw the behavior of the pair of quantum particles spooky, when you can do classical experiments where the same rarity happens and nobody is astonished at the result

  21. Dear, err, Bourbaki,

    Do you mean to say that something which Alain Connes has proved can be shown to reduce to the proposition/conclusion that there can't be any valid conceptual relation between the nonlinear dynamics theories and the non-relativistic Schrodinger equation?

    Please correct me if I am wrong in understanding what you say in this way; thanks in advance.

    [BTW, my knowledge of QM is limited only to the non-relativistic theory.]

    1. Dear Ajit R.Jadhav,

      The correct setting for asking those question IS QFT and therefore is the special relativity setting (since we have no quantum gravity). Undergraduate quantum mechanics is fun. There are dozens of thousands of articles about the interpretation of QM, complexity in QM (quantum information theory, quantum computers bla bla bla). Most of it (not to say all of it) are based on circular arguments on how to obtain quantum randomness assuming quantum randomness.
      You can even find this kind of logical vicious circle in works of figures like T'Hooft (one 200 page article where he tries to explain QM with automatas).

      What Alain Connes did was classify non-commutative probabilities in Operator's Algebra (a special case of those being Von Neumann's algebra). So there you go. He has all of them. The question asked in this article and on the thousands of articles mentioned above, Alain Connes allready asked, and answered. It's a shame this is not the answer we all wanted to hear.
      This summarizes in the following way: QM is a black box. It works. It explains a lot (at least QFT does). But. And its a big BUT, we have no tools to open that box. There is no hidden physical interpretation inside QM. It is just a theory that produces the right answers (for now).

      We need other tools. And one of the possible tools is Gravity. A Quantum theory of gravity might give us the correct setting to ask sensical questions about Quantum theories. You see? We do not even have yet the right language to ask questions. Trying to find answers is a waste of time. Well, not completely since it is used to publish articles and keep our jobs.

    2. Dear "Bourbaki",

      1. No. You didn't even begin addressing my specific question above.

      2. From your replies (above, and to Cromwell further above), I gather this much:

      Connes classified non-commutative probabilities in the context of some operator algebra which is a generalization of von Neumann's (I guess, as used in non-relativistic QM).

      Now, in general, I know that: SR (special relativity) is a generalization of NM (Newtonian mechanics), and that QFT is a generalization of QM (non-relativistic quantum mechanics).

      You introduce SR and QFT in your comments, but while replying to me, you don't connect them with the algebra that Connes took up, and Connes' conclusions.

      3. So, my next questions, just to get your position right:

      3.1. Does Connes' work hold in reference to QFT? (Just a simple yes/no would be much appreciated before any further explanations.)

      3.2. If yes, is there any consideration why Connes' work might hold for QFT but not for QM? Just because of that speed of light limitation?

      I mean to say: Just because $c$ is a finite number, therefore, does there arise some possibility that nonlinear dynamical theories can in principle be conceptually connected with the QM probabilities when the QM theory is non-relativistic, but not when it is non-relativistic? How precisely does such a circumstance come about?

      See if you want to answer *my* questions.

      4. Your comments on others' works (including Prof. Palmer's) did not answer my questions, and I think they wouldn't, either.

      If you think I am being too demanding with too little a knowledge-base to understand what you are saying, then do feel free to leave my questions alone. But in any case, no beating around the bush, please! Thanks in advance.

      Best wishes, and goodbye (in case you wish to leave this thread alone right here),

    3. Bourbaki wrote:
      >This summarizes in the following way: QM is a black box. It works. It explains a lot (at least QFT does). But. And its a big BUT, we have no tools to open that box. There is no hidden physical interpretation inside QM.

      Bohmian mechanics works: Bohm showed how to apply it to QFT way back in his second paper in 1952.

      It's not pretty but it works, and it is realist and deterministic.

      Bohmian mechanics is not merely a pie-in-the-sky speculative idea: It is well-defined, you can (and people have) done definite numerical calculations to get results out of it, which of course necessarily agree with the predictions of quantum theory.

      Any claims of impossibility concerning QM (and QFT) need to deal with this fact.

      A similar point can be made about Edd Nelson's "stochastic mechanics" and various other realist models of QM.

      Personally, I find those models aesthetically unappealing: I agree with Einstein that they seem "too cheap."

      However, the fact that Einstein and I do not like them is no excuse for pretending that they do not exist.

      Any claims about possible models of QM and QFT that are proven wrong by Bohmian mechanics are simply false.

  22. Everything changes everything, but it still looks the same as what is expected.

  23. Here's me touching yet another part of the elephant (not sure if it's the elephant in the room though, to mix metaphors).

    Navier-Stokes is one of the Millennium Problems. Which is fine for the ocean under Europa's icy crust, and the tiny droplet which carries the virus which gives you Covid-19.

    But the universe is not "turtles all the way down" (or up) ... a fluid with enough mass becomes a black hole, where Navier-Stokes does not apply. Likewise, a fluid is composed of molecules (not turtles ;-)) which are composed of atoms which are ... and quarks and gluons could care less about Navier-Stokes.

    So does the universe contain a Lorenz fractal attractor? Can the universe contain a Lorenz fractal attractor? Aren't such attractors "turtles all the way down"?

    1. You are right. A fractal attractor seems to be the last straw to save locality and determinism and avoid the consequences of Bell's theorem. Of all attempts at making sense of Quantum Theory I find this one the least intelligible.

  24. I hope this video passes the moderator as it seems to be relevant:

    It shows a video of a T-shaped handle in zero gravity in a bi-stable state due to intermediate moments of inertia. The handle spins and flips backwards and forwards regularly along an axis. The handle's motion reminds me of Tim's diagram of the Lorenz attractor, except that Tim's figure is for state space while the video is in R^3 space. The lorenz diagram was easier for me to understand in the wiki site at
    which shows a point moving between the two 'butterfly wings'.

    Austin Fearnley

    1. I think the origin of spinning and flipping is in the fact that matter structure cannot be absolutely rigid. Fluctuations of tension swing the nutation axis over 180 degree to the next relaxation of tension waves by coherent interference with each others.

      Build the T-handle from the mix of different molecule structure and see how its behavior changes...

    2. Building the T-handle differently could affect the behaviour of the T-handle but the reason that the T-handle is moving in an unstable manner (or bistably) is because it is spinning about its intermediate-length axis. See for a discussion of intermediate-axis spin; though the spinning T-handle in zero g shows the bistable/unstable motion much clearer.

      The butterfly pattern in Tim's article is only an exemplar but it does show a bistable effect where a particle can move (in a calculable way) from one butterfly wing to the other, say Left wing to right wing.

      Some comments, here and on other blogs, say that it is difficult to see the relevance of this article to Bell's Theorem, but in my naive view it is relevant. A pair of 'Bell' particles would need to move in two separate butterfly patterns [of state variables] where one butterfly is antisymmetric to the other. Alice and Bob could detect particles in two positions, Left or Right, and any individual outcome would be apparently random as every particle would oscillate between left and right wings. But in a calculable way so that the wing positions of the two particles are coordinated.

      My main point was that if the bistable pattern were to be essential to the article, then it describes rotation about an intermediate axis in classical physics. What if electrons/photons are not spinning about an intermediate axis? In the Standard Model isn't it nonsense even to ask if elementary particles have intermediate-length axes? (It is, however, technically possible that this could happen in my preon model.)
      This is just a naive view, but it does seem relevant to 'Bell', though not going on to calculate correlational outcomes in a simulated Bell Test.

      Austin Fearnley

  25. Attractors may be the answer to the existence.

  26. If you would, imagine a Disney cartoon that features as protagonist a C12 atom and traces its path from its nucleosynthesis within a long dead star, its ejection into space at high velocity and its travels through space across billions of miles and like number years. It animates the atom's trajectory as it devolves into smaller orbits and eventually into the zigzag of Brownian motion in the earth’s upper atmosphere. It illustrates that for perhaps three billion years it has been eddying in the earth’s biosphere, here in the chlorophyll molecule of a Jurassic fern, there in the breath of a polar bear cub and finally coming to temporary ‘rest’ as one of the twenty carbon atoms in a retinol molecule within the slice of organic carrot in your salad.

    It is not an affront to reason to question whether this is the only possible outcome for this bit of carbon. Within the same timeframe, could it reasonably have landed some place entirely different? Yet, as I understand it, a super deterministic universe would insist that is the only possible location the time evolution of the physical universe will allow. And, so it is with every other atom, every thought, every action.

    For your average pedestrian with their wealth of experience that things don’t always go as planned, this is hard to accept. Perhaps the physical universe is not so precise in making extended bank shots. Perhaps paths in the universe more like that of a beer cooler drug across beach sand, dodging sunbathers and running children and coming to rest at a point decided by a semi-sober committee of five. Well, maybe not that bad, but certainly less precise than our mathematics suggests.

    That possibility has been argued in a recent paper: “Physics without Determinism: Alternative Interpretations of Classical Physics” –

    Is that argument sound?

    1. Don Foster asked:
      >That possibility has been argued in a recent paper: “Physics without Determinism: Alternative Interpretations of Classical Physics”...
      >Is that argument sound?

      Well, I've read the paper, and to call it an "argument" is overly charitable. I'll be polite and say that the paper is "speculation," unwarranted speculation.

      First, we already know that classical physics is not true. Classical physics is a set of models, based on the real numbers, which is often adequate and, in any case, is generally mathematically consistent. To try, as the authors do, to strip the real numbers out of classical physics is just... pointless.

      You still will not end up with a theory that accords with all the quantum features of the world (and the authors make no pretense that you will). However, what you will do is take some theories that are, within their domain of application, quite useful and reasonably accurate, and make them enormously more complicated and, indeed, undefined (the authors never really spell out their novel models).

      Why do this? It is as if we were to "improve" Italian cooking by requiring that henceforward every Italian dish must include one femtogram of cyanide and one attogram of arsenic! Too little poison to actually harm anyone, but why bother?

      Further, their only serious justification is that quantum considerations (the Bekenstein bound) require a limit on information. Yeah, but we already know that classical physics is not quantum physics, and their weird attempts to change classical physics will not turn it into quantum physics. Classical physics is an approximation for when you do not need to worry about quantum mechanics.

      In short, some people have way too much time on their hands!

      If physicists cannot figure out something useful to work on that is actually physics, they need to find a new line of work: perhaps they could learn coding.


    2. >In short, some people have way too >much time on their hands!

      Says the guy who reads people's PhD theses.

      Your rants might be more coherent if you learned the meaning of words like "true", "model", "theory", etc.

      I'm just here to Poke the Parrot!

    3. Reply PhysicistDave

      Thanks very much for taking time to reply. Clearly, I am not a physicist. My engagement here is more that of a humanitarian: I am concerned that people who believe in super determinism will experience a greater incident of accident in crossing the street!

      Perhaps your dismissal of the paper is justified. I know that an article about the paper on was shared 929 times. To what purpose, I don’t know. As a first effort at addressing the possibility that the current mathematics of physics skews our sense of how the world works, it seems cogent and collegial.

      What is your assessment, charitable or no, of the narrative of the carbon atom, its complex path from long ago past to evolving present? Does it have only one possible path, one beginning with certain end? What is the principle that would compel such absolute constraint on possible outcome? Does constraint have its limits, have a counterpoise such that neither rules absolutely?

      Do you admit to being, in the overall nature of things, an exceedingly rare and exotic construct? Do you see yourself as a sentient, adaptive biological system with a time binding and elegantly associative neural network? Are you somehow more than the sum or your parts, carbon etc.? Would you feel comfortable in a universe where you were the pilot wave with your hand upon a working tiller and a course yet to be determined?

    4. Greg Feild wrote to me:
      >Says the guy who reads people's PhD theses.

      Says the guy who seems to think no one should ever read anyone's Ph.D. thesis.

      Which is sad.

      But predictable.

    5. Don Foster wrote to me:
      > I don’t know. As a first effort at addressing the possibility that the current mathematics of physics skews our sense of how the world works, it seems cogent and collegial.

      Well... my point was simply that since we already know the limited domain of applicability of classical physics, trying to fool around with classical physics is rather silly. Better to fool around with quantum mechanics.

      Don also asked:
      >What is your assessment, charitable or no, of the narrative of the carbon atom, its complex path from long ago past to evolving present? Does it have only one possible path, one beginning with certain end?

      Well, in a narrow technical sense, the answer is a question: what is really going on in quantum mechanics?

      No one really knows.

      If you want a broader metaphysical/theological answer, I fear that this is a bit like asking a dog to solve a differential equation -- beyond (current) human capabilities.

      Don also asked:
      >Do you admit to being, in the overall nature of things, an exceedingly rare and exotic construct? Do you see yourself as a sentient, adaptive biological system with a time binding and elegantly associative neural network? Are you somehow more than the sum or your parts, carbon etc.?

      I am tempted to give Popeye's answer: “I yam what I yam, and that's all what I yam.”

      More seriously, no one (yet) understands consciousness, and it seems foolish to me to pretend otherwise.


  27. This is parallel to the issue of computing recursively enumerable sets or languages. Quantum computing, entanglement, and theorem provers

    "We show that the class MIP* of languages that can be decided by a classical verifier interacting with multiple all-powerful quantum provers sharing entanglement is equal to the class RE of recursively enumerable languages."

    by Scott Aaronson
    January 14th, 2020


    Verifying proofs to very hard math problems is possible with infinite quantum entanglement
    by Tom Siegfried
    February 17, 2020

    A quantum strategy could verify the solutions to unsolvable problems — in theory
    by Emily Conover
    January 24, 2020

  28. I agree that noncomputability may play a role underneath Quantum theory. I have read the paper for the essay contest and I have a question for Tim:
    In page 8 you said that based on real computational processes in spacetime, a (uncomputable) deterministic local model can violate Bell's theorem without violating the freedom of choice assumption.
    On the other hand, you also mentioned that from a computational perspective, equation (7) is true. Together with locality assumption, doesn't this implies the Bell's inequalities?

    1. I read Palmer’s paper on the FQXi that makes a clear point on using the Blum, Shub, and Smale (BSS) concept of computability [–Shub–Smale_machine ]. 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. I have on FQXi a paper on this. As a result, at this time I am relatively disposed to Palmer’s concept here.

      We have for coherent states, a general form of laser states of light, the occurrence of states of the form |p, q⟩ 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.



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