Graduate Does the Bell theorem assume reality?

[Demystifier]

Maybe like a (hypothetical) physical version of the Goedel incompleteness theorem? Which would explain why Bohmian Mechanics is doomed to fail (maybe Lorentz invariance is exact :)

I think that's an abuse of Godel.
https://www.amazon.com/dp/1568812388/?tag=pfamazon01-20

 

[Demystifier]

Xiao-Gang Wen's textbook quotes the Tao Te Ching:
"The Dao that can be staled cannot be eternal Dao. The Name that can be named cannot be eternal Name. The Nameless is the origin of universe. The Named is the mother of all matter."

And has fun translating it as:
"The physical theory that can be formulated cannot be the final ultimate theory. The classification
that can be implemented cannot classify everything. The unformulatable ultimate theory does exist
and governs the creation or the universe. The formulated theories describe the matter we see every
day."

Well, my Bohmian theory of everything in "Bohmian mechanics for instrumentalists" is fully compatible with this. Note that I haven't written down the fundamental Hamiltonian from which the relativistic Standard Model is supposed to emerge.

 

[atyy]

Well, my Bohmian theory of everything in "Bohmian mechanics for instrumentalists" is fully compatible with this. Note that I haven't written down the fundamental Hamiltonian from which the relativistic Standard Model is supposed to emerge.

Or the lattice chiral fermion problem cannot be solved. Though Wen has also been working on this - maybe he's a secret Bohmian.

 

[Demystifier]

Or the lattice chiral fermion problem cannot be solved.

Maybe it can be solved, but not in the mathematical form.

 

[DrChinese]

He is not assuming that. He infers that from locality and perfect correlations (here he has to assume R4 as well, this indeed is unstated assumption).

And he states this quite clearly before he even starts his derivation:
"Since we can predict in advance the result of measuring any chosen component of $\vec{\sigma}_2$, by previously measuring the same component of $\vec{\sigma}_1$, it follows that the result of any such measurement must actually be predetermined."

This is not Bell's argument, it's strictly EPR's. Bell essentially refutes it.

 

[stevendaryl]

This is not Bell's argument, it's strictly EPR's. Bell essentially refutes it.

Yes, but the point is that Bell's argument (or a similar argument) refutes not just deterministic local hidden variables theories, but also nondeterministic local hidden variables theories. A deterministic local hidden variables theory would explain the quantum probabilities by proposing functions $F_A(\lambda, a)$ (giving the result that Alice would get if she measured her particle's spin along axis $a$) and $F_B(\lambda, b)$ (giving the result that Bob would get if he measured his particle's spin along axis $b$) such that:

$P(A, B | a, b) = \sum'_\lambda P(\lambda)$

where $P(\lambda)$ is the probability distribution on the hidden variable values, and where the sum is over all values of $\lambda$ such that $F_A(\lambda, a) = A$ and $F_B(\lambda, b) = B$.

A nondeterministic local theory would more generally assume the form:

$P(A, B | a, b) = \sum_\lambda P(\lambda) P_A(A|a, \lambda) P_B(B |a, \lambda)$

where $P_A(A|a, \lambda)$ is the probability of Alice getting result $A$ given that she chose orientation $a$ and the hidden variable has value $\lambda$ and similarly for $P_B$.

Although the second form seems more general than the first, it actually leads to the same inequality. EPR's argument about perfect correlation/anti-correlation implies that any local hidden variables theory that agrees with QM must actually have the less general (deterministic) form.

 

[zonde]

This is not Bell's argument, it's strictly EPR's. Bell essentially refutes it.

Indeed it is EPR argument and it is starting point for Bell argument. Bell refuted nothing, he just derived prediction for certain type of models. It is experimental tests of this prediction that falsified the type of models which Bell considered.
Theories are not falsified by theoretical arguments, but by experiments. Using theoretical arguments you can only argue that a theory is flawed (ambiguous, inconsistent and so on) but not false.

 

[DarMM]

Fine, but in such interpretations it is not only that the world is not nonlocal. In such interpretations the world is not even real. And yet, adherents of such interpretations deny that they are solipsists. But I think they are solipsists, even if they hide under euphemisms such as "relationalists".

This is the hardest part to understand about QBism, but if they were solipsists that would mean they think only their experience is real. They don't act like that – they act like everybody's own experience is just as real as their own, but only to one's self. So my best guess is they do assume there is a shared arena for experiences to interact but it is not conducive to a mathematical description and certainly isn't QM.

To be clear what these interpretations say is not that there is no world or reality external to humanity, but that:

1. That reality is not described directly by QM, i.e. saying $\psi$ isn't real doesn't mean you think there is no world, it just means you think $\psi$ doesn't give a representational account of it. Making $\psi$ being unreal equal to solipsism assumes a $\psi$-ontic view is correct as far as I can tell.
   
2. In some versions they also think the reality underneath QM is not mathematically comprehensible. QM is sort of "as best as you can do" with a scientific model. The world isn't completely mathematizable.

See Adán Cabello's recent paper: https://arxiv.org/abs/1801.06347, where he obtains the Born rule and all of $\psi$'s properties and the entire Hilbert space and operator structure purely from consistent probability assignments to outcome graphs of experiments. You get a large chuck of QM (essentially everything but the Hamiltonian of a given theory) purely from operational concerns of agents gambling on experimental results. QM results from dropping the assumption that the graphs have any "laws" controlling their outcomes, where as you get classical probability if you assume they do. This might give a better idea of what these interpretations mean, i.e. that most of the objects in QM are just Bayesian tools not real physical objects. Again this isn't the same as saying the world isn't real, it's AntiRealism about $\psi$ not the actual world.

Saying $\psi$ only predicts an agent's experiences just ties back to a subjective Bayesian view of probability and the idea that the underlying reality being relational/participatory.

 

[*now*]

(snip) A and B can only exist together, not separately, which is why the quantites P ( A | . . . ) P(A|...) and P ( B | . . . ) P(B|...) do not make sense. (snip)

Could examples like the first few pages of this paper provide reasonable descriptions that make sense?
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.88.084027

A relational view could be that physical interactions are real and any physicality in interaction has an observational role.

 

[akvadrako]

To be clear what these interpretations say is not that there is no world or reality external to humanity, but that:

1. That reality is not described directly by QM, i.e. saying $\psi$ isn't real doesn't mean you think there is no world, it just means you think $\psi$ doesn't give a representational account of it.
2. In some versions they also think the reality underneath QM is not mathematically comprehensible. QM is sort of "as best as you can do" with a scientific model. The world isn't completely mathematizable.

The QBists are definitely saying there is an external reality. I wanted to get a better picture of their view so I looked through Fuch's self-published email archive, which is quite a nice thing for him to have done BTW. Here are a few relevant bits:

• It is rather that my point of view admits too many things into the world—too many things of an independent and self-sustaining reality, things for which there are no equations; realities of which I am only willing to point to and say effectively, “Yeah the world includes that too.”
• Far from thinking the world is an empty place, a place only with me in it. I think it is full of things, overflowing with things. All distinct things, from head to toe. And literally so. It is not a world made of six flavors of quarks glued together in various combinations. It is not a world that maps to a single algorithm running on Rob Spekkens’s favorite version of Daniel Dennett’s mechanistic cellular automaton. It is a world of heads and toes and doorknobs and dreams and ambitions and every kind of particular. (And that is not a typo: It is a world in which even dreams and ambition have substance.) It is a world in which Vivienne Hardy is a distinct entity, not “constructed” of anything else, but a true-blue crucial piece of the universe as it is today—no less crucial than spacetime itself.
• Instead of a starkly empty world, as a quantum reductionist might have it (“the world is nothing but the universal wave- function undergoing unitary evolution”), the world of QBism is overflowingly full—it is a world whose details are beyond anything grammatical or rule-bound expression can articulate.

The problem is that if you assume multiple agents can use QM as a subjective tool it places pretty stringent constraints on what that reality can be. If it's describable by math it probably needs to contain at least a universal wavefunction; that's what the Bell, PBR, FR and CR theorems suggest to me. Is it even a reasonable thing to suggest it's outside the realm of math? Yet that seems to be the only other suggestion on the table.

See Adán Cabello's recent paper: https://arxiv.org/abs/1801.06347, where he obtains the Born rule and all of $\psi$'s properties and the entire Hilbert space and operator structure purely from consistent probability assignments to outcome graphs of experiments. You get a large chuck of QM (essentially everything but the Hamiltonian of a given theory) purely from operational concerns of agents gambling on experimental results. QM results from dropping the assumption that the graphs have any "laws" controlling their outcomes, where as you get classical probability if you assume they do. This might give a better idea of what these interpretations mean, i.e. that most of the objects in QM are just Bayesian tools not real physical objects. Again this isn't the same as saying the world isn't real, it's AntiRealism about $\psi$ not the actual world.

It's suggestive that QM is so closely related to rational agents and logical constraints, but that doesn't mean reality is something different. It could be that they share the same structure because it's a universal structure.

Saying $\psi$ only predicts an agent's experiences just ties back to a subjective Bayesian view of probability and the idea that the underlying reality being relational/participatory.

How does the idea reality is relational/participatory give any guidance in helping to show what it could be, if not QM?

 

[DarMM]

If it's describable by math it probably needs to contain at least a universal wavefunction; that's what the Bell, PBR, FR and CR theorems suggest to me

To me they have a universal wavefunction as among a few of the possibilities not eliminated by them, but I don't think they suggest it more strongly than any others. The wavefunction being real has its own problems, so depending on taste you'll choose one of the others.

The problem is that if you assume multiple agents can use QM as a subjective tool it places pretty stringent constraints on what that reality can be

An example?

It's suggestive that QM is so closely related to rational agents and logical constraints, but that doesn't mean reality is something different. It could be that they share the same structure because it's a universal structure.

No it doesn't mean it, my intent wasn't to prove these interpretations are correct. It was to say here is a derivation of QM purely from agential concerns. It shows how it is possible to hold their view. Also to be frank, as somebody who isn't decided on interpretations, it's a pretty damn strong argument. How many other interpretations derive QM without postulating large chunks of it? Many seem more like reactions to the formalism.

Usually in probability theories the epistemic space doesn't have the same structure as the ontic space over which it is built and epistemic states tend to obey very different theorems.

Could you explain a bit more what you mean?

What kind of world do you have in mind where states of reasoning and belief updating function identically to the actual "stuff" down to obeying the same theorems.

That naively sounds further than QBism, that the stuff is agent belief.

Is it even a reasonable thing to suggest it's outside the realm of math?

Again this is all based on taste, but if you take Godel's theorem or other lines of reasoning to suggest mathematics is purely in our heads, then sure there might be layers of reality that don't map to any structure like mathematics that comes from human thought.

At a weaker level, there are mathematical structures for which there is no algorithm to compute them. So even in a mathematically describable world the world mightn't be algorithmic.

 

[DarMM]

How does the idea reality is relational/participatory give any guidance in helping to show what it could be, if not QM?

Is that really a problem with these interpretations? They seek only to explain QM, their explanation is that it is a generalisation of the Bayesian reasoning framework. Some of them then say that the underlying reality is relational or participatory or both. The guidance is then that you now know reality is like that and can use that fact to guide future developments.

What I mean is if they hypothetically succesfully showed reality is relational/participatory and QM is just a form of Bayesian reasoning, then that isn't invalidated by them not also suggesting the complete underlying theory.

 

[akvadrako]

The problem is that if you assume multiple agents can use QM as a subjective tool it places pretty stringent constraints on what that reality can be.

An example?

I just mean the usual restraints those theories produce, like making it hard to combine locality and single outcomes while reproducing QM. QBism claims reality can do all three.

Usually in probability theories the epistemic space doesn't have the same structure as the ontic space over which it is built and epistemic states tend to obey very different theorems.

Could you explain a bit more what you mean?

What kind of world do you have in mind where states of reasoning and belief updating fubction identically to the actual "stuff" down to obeying the same theorems.

That naively sounds further than QBism, that the stuff is agent belief.

I mean a quantum world of course . One where anything can happen except inconsistencies. I'm not saying it's made of agent belief, just that they happen to have the same structure. It seems QM can be mostly if not totally derived from constraints on consistent reasoning. That doesn't imply it can't also be derived from logical constraints on ontic models. Though it's hard to rule out that it's made of the union of all "agent" beliefs.

What I mean is if they hypothetically succesfully showed reality is relational/participatory and QM is just a form of Bayesian reasoning, then that isn't invalidated by them not also suggesting the complete underlying theory.

The problem isn't that their model is incomplete. I am willing to accept that QM is a form of Bayesian reasoning and reality is relational/participatory – those are at least reasonable claims. But they don't address the issue, which is that a reality which satisfies the constraints of QBism doesn't seem possible due to Bell's theorem and the others. And they don't suggest any way it might work.

 

[DarMM]

I just mean the usual restraints those theories produce, like making it hard to combine locality and single outcomes while reproducing QM. QBism claims reality can do all three.

Retrocausal and acausal theories also escape these theorems and have one world, locality and replicate QM. It's not impossible, although QBism doesn't take the retrocausal route.

But they don't address the issue, which is that a reality which satisfies the constraints of QBism doesn't seem possible due to Bell's theorem and the others. And they don't suggest any way it might work.

Following from the above. They reject the existence of $\lambda$, mathematical variables controlling the world.

This is what Cabello discusses in his paper, no laws.

If you have no $\lambda$ Bell's theorem has no effect on you.

Now I don't like this either, but they do have reasons to think it. Just even reflect on the fact that all actual derivations of QM (Cabello's is my favourite, but there are others) make no ontic assumptions beyond measurements and agents existing.

They take this to mean there is no point in thinking of $\psi$ as real, since you can derive it from epistemic considerations. This is one reason why they'd dismiss Many Worlds and Bohmian Mechanics. There are other reasons as well if you want to know them.

Okay so $\psi$ is epistemic, about what?

The only options from the various ontological framework theorems are a retrocausal world, nonlocal world, superdeterministic world or a non-mathematical world.

First two are out from fine tuning arguments, links if you want. Third is out because it means everything is a massive conspiracy, maybe minerals just happened to concentrate in the shape of dinosaur bones.

That leaves only the fourth option.

QM is the Bayesian reasoning you must use for parts of the world admitting no fundamental mathematical description.

 

[zonde]

Following from the above. They reject the existence of $\lambda$, mathematical variables controlling the world.

This is what Cabello discusses in his paper, no laws.

If you have no $\lambda$ Bell's theorem has no effect on you.

Well, maybe we can imagine that Bell's $\lambda$ is not the only way how EPR determinism can be modeled or we can imagine that perfect correlations is false prediction of QM. But there is elegant Eberhard's derivation of inequalities without $\lambda$ and perfect correlations. You can look at Eberhard's derivation here: https://www.physicsforums.com/threa...y-on-probability-concept.944672/#post-5977632

 

[Demystifier]

The QBists are definitely saying there is an external reality.

But then they must accept the Bell theorem that reality obeys nonlocal laws. Yet they don't accept it.

 

[DarMM]

But then they must accept the Bell theorem that reality obeys nonlocal laws. Yet they don't accept it.

Reality doesn't have to be nonlocal from Bell's theorem, as I mentioned above retro or acausal theories also remain local (although this isn't what QBism supposes).

 

[Demystifier]

Reality doesn't have to be nonlocal from Bell's theorem, as I mentioned above retro or acausal theories also remain local (although this isn't what QBism supposes).

So how exactly is QBism local? Is it just signal locality or something more? If it is just signal locality, then Bohmian mechanics is local too.

 

[zonde]

Reality doesn't have to be nonlocal from Bell's theorem, as I mentioned above retro or acausal theories also remain local (although this isn't what QBism supposes).

It seems that in retro and acausal models even superluminal communication device can satisfy locality. So I'm not sure you can use theories like that as an argument.

 

[akvadrako]

This is what Cabello discusses in his paper, no laws.

No laws doesn't necessarily imply something indescribable happens. Another option is everything computable happens, within logical consistency requirements like those for agents. For example along the lines of Tegmark's mathematical universe.

They take this to mean there is no point in thinking of $\psi$ as real, since you can derive it from epistemic considerations.

I think you got my point, but what I'm saying is just because it's derivable this way doesn't mean it has to be only epistemic. It's suggestive, but it doesn't logically follow that the ontic space needs to be something different. Along the lines of the ontology theorems, one can even derive the ontic space from the epistemic considerations and a few extra assumptions.

QM is the Bayesian reasoning you must use for parts of the world admitting no fundamental mathematical description.

Thanks - that was more clear than any explanation of QBism I can remember. I can see how they've found reason to believe reality is non-mathematical. But it's hard to distinguish that conclusion from saying they've reached a contradiction. Since their assumptions are questionable it seems much more likely one of them should be dropped.

 

[akvadrako]

But then they must accept the Bell theorem that reality obeys nonlocal laws. Yet they don't accept it.

If you say something can't be described mathematically or as Fuch's said above, beyond anything grammatical or rule-bound expression can articulate, then I suppose theorems become powerless.

 

[Demystifier]

If you say something can't be described mathematically or as Fuch's said above, beyond anything grammatical or rule-bound expression can articulate, then I suppose theorems become powerless.

Yes, for me that's nothing but mysticism. Bohr, indeed, has often been classified as a mystic. Given that Fuchs said the above, can we conclude that QBism is just an euphemism for mysticism?

 

[Lord Jestocost]

Bohr, indeed, has often been classified as a mystic.

In “Quantum Reality: Beyond the New Physics”, Nick Herbert remarks on Bohr:

“Einstein and other prominent physicists felt that Bohr went too far in his call for ruthless renunciation of deep reality. Surely all Bohr meant to say was that we must all be good pragmatists and not extend our speculations beyond the range of our experiments. From the results of experiments carried out in the twenties, how could Bohr conclude that no future technology would ever reveal a deeper truth? Certainly Bohr never intended actually to deny deep reality but merely counseled a cautious skepticism toward speculative hidden realities.”

 

[DarMM]

No laws doesn't necessarily imply something indescribable happens. Another option is everything computable happens, within logical consistency requirements like those for agents. For example along the lines of Tegmark's mathematical universe.

First, why the exclusion of non-computable mathematical objects?

Secondly, in Cabello's paper he has no restrictions on the behaviours, any consistent probability assignment occurs, that's what QM is in his derivation. How would this restrict the ontic space to "everything computable"? In fact I don't see the link with computability.

Also assuming a restriction down to something like "computable actions" is a step Cabello doesn't take, you'd have to show that that restriction doesn't affect his proof.

However that would be hard to imagine as he gets out the QM structure exactly, a restriction would have to reduce the assignments in some way and thus close off some parts of QM.

I think you got my point, but what I'm saying is just because it's derivable this way doesn't mean it has to be only epistemic. It's suggestive, but it doesn't logically follow that the ontic space needs to be something different. Along the lines of the ontology theorems, one can even derive the ontic space from the epistemic considerations and a few extra assumptions.

Regarding the first part, certainly, I'm not saying QBism or similar views are logically forced on us otherwise I'd be a committed QBist, I'm simply explaining them. However regarding the second part, I cannot think of an example in physics where the space of ontic objects is equivalent to the space epistemic estimates of them. Typically if the former is $Q$ let's say, the later is something like $\mathcal{L}^{1}(Q)$. Do you have an example or perhaps a sketch of what you mean?

But it's hard to distinguish that conclusion from saying they've reached a contradiction.

What contradiction do you mean?

 

[DarMM]

So how exactly is QBism local? Is it just signal locality or something more? If it is just signal locality, then Bohmian mechanics is local too.

Well you're not going to like it, but they say the world is local because there are no mathematical variables describing it, i.e. no $\lambda$, so no implications from Bell's theorem. Same as Copenhagen as viewed by Bohr and Heisenberg.

Yes, for me that's nothing but mysticism. Bohr, indeed, has often been classified as a mystic. Given that Fuchs said the above, can we conclude that QBism is just an euphemism for mysticism?

I suppose it depends on what you take mysticism to be, but QBism is what I said above, it sees QM as a Bayesian toolkit for managing expectations of a fundamentally lawless world. (This is not to say that the whole of the world is lawless, e.g. GR would still be correct, just that there are lawless components)

I don't think it's completely mad, as Cabello for example has a good derivation of QM assuming some observations obey no laws.

 

[DarMM]

It seems that in retro and acausal models even superluminal communication device can satisfy locality. So I'm not sure you can use theories like that as an argument.

No, there are retro and acausal models that have locality, but forbid superluminal signals. Propagating signals into the past light cone is fundamentally different to propagating them in spacelike directions.

Acausal theories would be even more different, no propagation at all just 4D consistency conditions.

 

[N88]

Does the Bell theorem assume reality? According to me: No!

Well, for me at least: not any meaningful version of physical reality when you are writing in the context of EPRB.

Here's my reason. From high-school algebra, without any refence to EPRB, Bell, etc, we irrefutably obtain:

$|E(a, b) - E(a,c)| \leq 1 - E(a,b)E(a,c). \qquad (1)$

Compare this with Bell's famous 1964 inequality:

$|E(a, b) - E(a,c)| \leq 1 + E(b,c). [sic] \qquad (2)$

Given [as I read him. p.195] that Bell's aim was to provide "a more complete specification of EPRB by means of parameter $\lambda$": I suggest that his supporters should pay more attention to his 1990 suggestion that maybe there was some silliness somewhere.

For example, let's rewrite (2). We find:

$|E(a, b) - E(a,c)| - E(b,c) \leq1. [sic] \qquad(3)$

But, under EPRB, that upper bound is $\tfrac{3}{2}.$

Thus, in that Bell uses inequality (2 ) as proof of his theorem: I believe that Bell's writings need to be challenged --- without any reference to nonlocality, QBism, BWIT, AAD, MW, etc [which, in my view, are also silly].

PS: While I am against Bell [who, in a dilemma in 1990, was against locality], I am for Einstein [and for Einstein-locality].

Thus, in that "Einstein argued that the EPR correlations could be made intelligible only by completing the quantum mechanical account in a classical way," Bell (2004:86): that's what I work on.

It being my hope that QFT would not reject my ideas.

 

[akvadrako]

First, why the exclusion of non-computable mathematical objects?

Secondly, in Cabello's paper he has no restrictions on the behaviours, any consistent probability assignment occurs, that's what QM is in his derivation. How would this restrict the ontic space to "everything computable"? In fact I don't see the link with computability.

It's just an example of how "no rules" can be satisfied with mathematically describable objects. If you haven't seen Markus P. Mueller's Law without Law, it's probably the paper I think back to most over the past year. It doesn't fully reproduce QM but some aspects of it, based on computational complexity. It's also subjective but I think with an extra assumption of multiple observers it could still work.

However regarding the second part, I cannot think of an example in physics where the space of ontic objects is equivalent to the space epistemic estimates of them. Typically if the former is $Q$ let's say, the later is something like $\mathcal{L}^{1}(Q)$. Do you have an example or perhaps a sketch of what you mean?

Well QM is special, so the existence of other examples doesn't matter much; it could be the only theory that works this way. Anyway, my current guess is subjective QM is basically Copenhagen but the combination of all subjects is MWI. So not exactly the same, but of the same structure.

What contradiction do you mean?

I mean saying something can't be described by math is equivalent to saying it's contradictory. If no formal rules apply, even roughly, then it seems like anything goes, even inconsistencies. At least I don't understand the difference.

 

[DarMM]

Well QM is special, so the existence of other examples doesn't matter much; it could be the only theory that works this way. Anyway, my current guess is subjective QM is basically Copenhagen but the combination of all subjects is MWI. So not exactly the same, but of the same structure.

Okay I see what you mean from the last bit. The epistemic objects and the ontic objects are the same type of thing, i.e. you can use a wavefunction epistemically as in standard QM for "local" events, but the actual ontic global wavefunction is the same type of object.

I have doubts about this though, for typical $\psi$-epistemic reasons. The most basic being that classical uncertainty about $\psi$ doesn't manifest as something like $\mathcal{L}^{1}(\mathcal{H})$, which you'd expect if $\psi$ was a real object you were ignorant of (because this is purely classical ignorance). Rather $\mathcal{H}$ is a subset of the observable algebra's dual (its boundary) and some of that dual has terms that mix classical and quantum probability in odd ways. So you can have a mixture $\rho$ which could be considered a mix of two states $\psi_1$ and $\psi_2$ or a mix of $\psi_3$ and $\psi_4$ and it's the exact same mixture. Hard to understand if $\psi$ is ontic (though not a killing argument of course), it makes pure states like $\psi$ just seem like a limiting type of probability assignment, not ontic.

I mean saying something can't be described by math is equivalent to saying it's contradictory. If no formal rules apply, even roughly, then it seems like anything goes, even inconsistencies. At least I don't understand the difference.

I don't think so. Inconsistency means that two contradictory mathematical properties would be assigned. Not relevant if the thing isn't mathematically describable.

It's just an example of how "no rules" can be satisfied with mathematically describable objects. If you haven't seen Markus P. Mueller's Law without Law, it's probably the paper I think back to most over the past year. It doesn't fully reproduce QM but some aspects of it, based on computational complexity. It's also subjective but I think with an extra assumption of multiple observers it could still work.

That's a cool looking paper, thanks!

Just to help orient a reading of it, what does he think the world is like underneath the reasoning of agents? I see our typical "laws" are seen to come about as a limiting behaviour in subjective probability assignments, but does he make any conjecture about the underlying world?

You are right, "no rules" doesn't logically mean "mathematically indescribable", again otherwise I'd be a QBist. However the QBist view is just as valid, that QM exposes a deep indeterminism, a not fully mathematically comprehensible nature. When I originally said it, it was in the context of Cabello's paper which does show that this way of thinking does work out, he derives QM from a world without laws, in the sense of no underlying mathematical variables describing things. The phrase "no laws" in abstract doesn't imply no mathematical variables controlling it, but when Cabello is saying it that's what he means. Just to be clear if I was vague:

Following from the above. They reject the existence of $\lambda$, mathematical variables controlling the world.

This is what Cabello discusses in his paper, no laws.

If you have no $\lambda$ Bell's theorem has no effect on you.

What I meant was Cabello deals with this view (non-mathematical world), not so much a comment on what the phrase "no laws" must mean in general.

 

[N88]

Does the Bell theorem assume reality? According to me: No!

Well, for me at least: not any meaningful version of physical reality when you are writing in the context of EPRB.

However, to be clearer: IF we accept Bell's 1964 move from the eqn after eqn (14) -- call it (14a) -- to the next equation, call it (14b): THEN it seems to me that Bell could be theorizing about these realities:

1. A set of objects (subject to non-perturbative testing) that are available for re-testing. Maybe a set of paired-billiard balls with their color tested under various lights?
2. A duplicate set of objects tested pertubatively. Say: pairs of linearly polarized photons that can be reproduced on demand.
3. I'd be pleased to learn of other possibilities; including quantum ones.

In these cases, (1, 2), Bell's inequality would be valid. But I find it hard to accept that Bell expected that a return to such classicality would provide a more complete specification of EPRB. In my view, the classicality that Einstein sought -- and that Bell acknowledged -- is more subtle, and available. But this gets us into interpreting EPR's "elements of physical reality" and their use of "corresponding" in this context. By my interpretation, EPR's "elements of physical reality" are such as we meet in 1 and 2 above. But they are also more than those in quantum settings: as in EPRB.