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Essay Abstract

In [Found. Phys. 48.12 (2018): 1669], the notion of 'epistemic horizon' was introduced as an explanation for many of the puzzling features of quantum mechanics. There, it was shown that Lawvere's theorem, which forms the categorical backdrop to phenomena such as Gödelian incompleteness, Turing undecidability, Russell's paradox and others, applied to a measurement context, yields bounds on the maximum knowledge that can be obtained about a system. We give a brief presentation of the framework, and then proceed to study it in the particular setting of Bell's theorem. Closing the circle in the antihistorical direction, we then proceed to use the obtained insights to shed some light on the supposed incompleteness of quantum mechanics itself, as famously argued by Einstein, Podolsky, and Rosen.

Author Bio

Jochen Szangolies acquired a PhD in quantum information theory at the Heinrich-Heine-University in Düsseldorf. He has worked on quantum contextuality, quantum correlations and their detection, as well as the foundations of quantum mechanics. He is the author of "Testing Quantum Contextuality: The Problem of Compatibility".

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Dear Jochen,

What an excellent article, I believe easily the best I have read so far.

I liked this conclusion a lot. "This motivates a proposal of relative realism: assign ‘elements of reality’ only where f(n, k) yields a definite value. In this way, we get as close to the classical ideal of local realism as is possible in a quantum world."

In my article, I wrote of "measurablism", being all that is real is that which is measurable which incorporates Quine "to be is to be the value of a variable" or Ladyman/Dennett "to be is to be a pattern". Your conclusion seems to place a more specific version of that ontology: where what is real is f(n, k) yields a definite value. The support for this seems to be found in the mathematics of Godel, but also in the nature of QM itself. That you point out Godel's rejection of Wheeler here is brilliant.

I think this is a pretty neat correlative argument, as the lines between epistemology and ontology seem to blur and this goes to the heart of the essay contest question.

Re Godel, does this mean that complete systems are real, but incomplete systems (or at least those unaccessible truths in them) are not? You might enjoy some of my 'amalgamated sleuths' in this vein in my article, though they are not meant to be serious, rather just as examples of the kinds of blurring we can do between mathematics, physics and philosophy. I think your article could definitely be published in the vein of Quine in a metaphysics journal. Congratulations.

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Author Jochen Szangolies replied on Jan. 28, 2020 @ 07:19 GMT

Dear Jack,

thank you for your kind words! I'm glad you found something of value in my essay.

The reference to Quine is interesting. I have thought about how to incorporate his take on undecidability and the notion of 'Quining' within my framework, but nothing obvious has come up, so far. I wonder if one could develop things along the direction you suggest, instead of bound variables thinking about definite/decidable propositions (although of course Quine came at this from a different angle). I will have to give this some thought.

As to the question of 'what is real', I purposefully leave a little room for interpretation there. You could take my suggestion literally, amounting to an 'ontic' reading of the arguments I've presented, in which case f(n,k) really tells you 'what there is'. But I want to leave the possibility of an 'epistemic' interpretation open: our theories relate to reality as the map does to the territory, and there may be some inherent limitations to map-making, that is, some parts of the world that can't be modeled.

I had marked your essay already as one that would need further attention; I'm looking forward to engage with it.

"John Wheeler himself proposed the undecidable propositions of mathematical logic as a candidate for a ‘quantum principle’, from which to derive the phenomenology of quantum mechanics ..." Is any mathematical logic that assumes a potential infinity merely theology which might, or might not, be logically consistent? Suppose that I told you that I have a potentially infinite purse with the following property: if the purse can contain n ducats of gold then the purse can contain n+1 ducats of gold — would you believe me? Is nature finite and digital? Ask yourself the following question, "Why do theologians so often discuss infinity?"

"Infinity and the Proofs for the Existence of God" by Glenn F. Chesnut, 2019

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Author Jochen Szangolies replied on Feb. 2, 2020 @ 15:14 GMT

Dear David,

thank you for your interest in my article. Regarding the topic of infinity, I must confess that it does make me somewhat uneasy---but in the end, our most successful current theories of physics (not just quantum field theory and general relativity, but also plain old Maxwellian electrodynamics) depend on the continuum of the real numbers, generally including an uncountable number of degrees of freedom. As these do yield spectacularly accurate predictions in many cases, one might well surmise that they do at least get something right.

That said, there are intriguing arguments that quantum gravity, due to the Bekenstein bound, only includes a finite number of degrees of freedom within a finite volume. But I'm afraid these considerations are rather beyond the scope of the current topic.

In your conclusion of your very fine essay you write: It is as if Schrödinger's student does not know the answer to any questions, as such, but knows each answer only relative to that question being asked.

If true, this may provide the conclusion that the answer is in the system at large and that the student as part of that system is tapping the answer from the system.

In life this is called intuition. If someone relies on it, the person can have a high rate of correct answers. That we may name coincidence when it happens once; if it happens again, psychic; and at always: incredible.

But it is already in the system itself. The system knows itself.

If there is an answer to a question then there is an answer too.

Tautology is simple and true.

It is what it is.

The question therefor is one part of the system and the answer another part. Both are what they are, and belong to the system at large.

The student can only lie if by giving a wrong answer, lying is not making contact to the system at large but to something beside it.

Bests,

Jos

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Author Jochen Szangolies replied on Feb. 2, 2020 @ 15:20 GMT

Dear Joseph,

thanks for taking the time and commenting on my essay. In some sense, you can indeed view things like entanglement as information that's present in the system as a whole, but not reducible to any of its parts: while a classical system, such as the two differently colored cards in the envelopes I briefly consider, contains all of its information in the states of each individual subsystem, that's no longer the case in quantum mechanics---there, the state of each subsystem (of a maximally entangled system) will be maximally mixed, but the total system is not a combination of maximally mixed states.

Hence, there is information within the total state that's not reducible to its components---the basis of what's sometimes called 'quantum holism'.

Whether, of course, that has anything to do with intuition, or other macroscopic phenomena, seems rather doubtful to me. However, you might find this recent paper of renowned physicist Don Page interesting: https://arxiv.org/abs/2001.11331

This is exceptionally deep Jochen...

I am glad you give a little intro to Lawvere's theorem at the end. I am a fan of category theory, but I'm by no means well-versed in the subject, so it may require multiple readings just to grasp the central idea. I will not give up easily, though, because I think my efforts will be rewarded or rewarding.

Good luck with this fine essay.

Regards,

Jonathan

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Author Jochen Szangolies replied on Feb. 2, 2020 @ 15:25 GMT

Dear Jonathan,

thanks for your kind comment. I can't claim myself to be an expert in category theory, and really, the introduction I give to Lawvere's result is a kind of bastardized set-theoretical version, which one can get away with because Lawvere's theorem applies to categories that are cartesian closed, of which Set (the category with sets as objects and total functions as morphisms) is an example.

For a much better introduction into the subject, I can only highly recommend Noson Yanofsky's paper: https://arxiv.org/abs/math/0305282

For the core idea, I think the important part is to grasp the connection between the category-theoretic argument and the diagonalization; everything else flows from there. If you've got any questions, I'd be happy to try and address them!

Dear Jochen Szangolies, after reading your essay, I found you as one of the brightest representatives of the modern paradigm in physics, through which I can inform the scientific world of my discovery. Through new Cartesian reasoning, I came to the conclusion that the probability density of states in an atom depends on the Lorentz abbreviations: length, time, mass, etc. Isn't this a unifying principle for physics?

I invite you to discuss my essay, in which I show the successes of the neocartesian generalization of modern physics, based on the identity of space and matter of Descartes: “The transformation of uncertainty into certainty. The relationship of the Lorentz factor with the probability density of states. And more from a new Cartesian generalization of modern physics. by Dizhechko Boris Semyonovich »

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Author Jochen Szangolies replied on Feb. 2, 2020 @ 15:27 GMT

Dear Boris,

thank you for your comment and for taking an interest in my essay, but you are much too kind.

I will have a look at your essay, and see whether I have anything useful to say on the subject.

If an entangled pair consists of an apple and an orange, and instead of measuring their extrinsic properties (like position and momentum in the original EPR), one instead decides (as Bell subsequently did in his theorem) to substitute a measurement of an intrinsic property (like skin-texture or polarization), then there is going to be a big problem, when you have thus unwittingly assumed...

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Author Jochen Szangolies replied on Feb. 4, 2020 @ 06:07 GMT

Dear Robert,

thanks for highlighting your model. I see you can reproduce the singlet-correlation with a detection efficiency of 72%; out of curiosity, what is the maximal detection efficiency you can allow and still observe a violation of the CHSH inequality?

Furthermore, the identical nature of particles is, of course, a central tenet of quantum mechanics---quantum particle statistics are derived assuming that exchanging any two particles does not lead to a new configuration of the system. This has many observable consequences that seem to be hard to explain otherwise. How do you propose to account for this?

Robert H McEachern replied on Feb. 4, 2020 @ 14:53 GMT

Jochen,

The detection efficiency given in the paper, is not the standard conditional efficiency (single detector) usually reported; it is the product of both detectors. So the corresponding, conditional efficiency is the sqrt(0.72)=0.85, which is already above the theoretical limit for a classical process. With optimization, of the matched-filtering, it ought to be possible to perfectly...

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Jochen,

The detection efficiency given in the paper, is not the standard conditional efficiency (single detector) usually reported; it is the product of both detectors. So the corresponding, conditional efficiency is the sqrt(0.72)=0.85, which is already above the theoretical limit for a classical process. With optimization, of the matched-filtering, it ought to be possible to perfectly reproduce the entire correlation curve (not just the few points evaluated in most Bell tests) at even higher efficiencies.

The "fraternal twins" are statistically identical. So they obey the same probability of detection statistics predicted by quantum theory. But there are two detection distributions that are important, in any detection theory; unfortunately quantum theory only computes one of those two. It only computes the probability that something will be detected, but it never even attempts to predict the probability that that "something" is actually the thing that the system was supposed to detect (Probability of False Alarm). This would not be an issue, if all the things being detected were in fact "identical" particles, as has been assumed. But when the particles are only very similar (statistically identical), rather than perfectly identical, it does matter. This is exactly the problem in a Bell test; the number of detections agrees with the theory, but only because the actual state of the detection is frequently incorrect - enough to change the computed correlation statistics, as the result of non-random, systematic errors in the process. In effect, lopsided, "edge-on", "up" polarized coins are frequently being mistaken for perfect "face-on", "down" polarized coins and vice-versa.

The point is, there is a very important distinction between "assuming that exchanging any two particles does not lead to a new configuration of the system", and assuming that exchanging any two particles does not lead to a new detectable configuration of the system, when the detection process (matched filtering) is not perfect enough to distinguish between "fraternal twins" and "identical twins", in every instance.

Rob McEachern

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Regarding horizons...

I have found another useful analogy for Schwarzschild event horizons to be the virtual ground or amplitude null at the summing junction of an inverting op-amp circuit. This too was suggested by my study of the Misiurewicz point M_3,1, the 'edge of chaos' point. The suggestion here is that other types of black hole horizons could be studied with the toolkit of category theory, by finding the correct circuit diagram analogy, and that the Mandelbrot Set provides somewhat of a map.

I have attached a diagram. Let me know if this makes sense. It would make a black hole horizon like a phase-reversed mirror that appears black or as a black body because whatever strikes it is (energetically or temporally?) inverted.

All the Best,

Jonathan

attachments: MandelAmp.jpg

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Jonathan J. Dickau replied on Feb. 13, 2020 @ 23:51 GMT

Apologies if this is too far off-topic...

I am following up on a comment you made in my essay forum. Still interested in your thoughts.

JJD

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I am starting to get a handle on your paper Jochen...

It appears Lawvere's theorem is a template for quite an array of meaningful conjectures, which makes it very powerful. Not so easy to grasp for those unfamiliar with the language of category theory however. I will stay the course and try to grasp what you are saying.

In fact; I think grasping is a great metaphor here, because the story is about what gets caught before it slips behind the epistemic horizon, and only new information is available. The derivation of the word 'think' is about grasping coming from the word 'tong' a device that lets one pick things up and examine them.

The essence of Lawvere would be caught up in the same idea. If the tongs are used to pick up hot items coming from a forge or kiln, one can examine only one item at a time and the rest are either heating or cooling, so one loses information about or control over the items not in your tongs. So the metaphor of grasping and holding vs. what slips away applies to Lawvere's theorem.

Best,

Jonathan

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Author Jochen Szangolies replied on Feb. 16, 2020 @ 10:14 GMT

Yes, Lawvere's theorem is certainly a very deep result, if anything underutilised in my essay---as I noted, the simple set-theoretical framework in which I present it doesn't really do justice to the full category-theoretic treatment.

There have been attempts of bringing black hole type losses of predictability within the framework of formal incompleteness, such as this one: https://link.springer.com/article/10.1007/s11128-008-0089-2.

I am not sure I see how to connect this to the qualitative similarity you point out in regards to the Misiurewicz point; I'm largely ignorant on that topic, I'm afraid. However, you might be interested in some of the works of Louis H. Kauffmann (http://homepages.math.uic.edu/~kauffman/), particularly on what he calls 'reentrant forms'---for instance, in 'Knot Logic' (http://homepages.math.uic.edu/~kauffman/KnotLogic.pdf), he considers the Koch snowflake as an example, and I have a hunch something similar could be applied to the Mandelbrot set and its recursive dynamics.

Jonathan J. Dickau replied on Feb. 16, 2020 @ 19:15 GMT

Thank you Jochen,

The link to the paper by Srikanth and Hebri is appreciated, and it looks very interesting. I don't know enough about how Godelian incompleteness relates to the BH information paradox. I will check that out when I can, and perhaps discuss here further if timely.

I think the work of Louis Kauffman is pretty amazing. I have had some communication with him, but not in a while. My research has progressed much further since then and perhaps its time for an outreach, and I will check out the linked material which I appreciate your citing.

All the Best,

Jonathan

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I finally read through your paper. It is very interesting that you do make some connection with Gödel incompleteness. I do though appear there is a need to “wring out” violations of Bell’s inequalities --- pun intended. Maybe a form of PR box or Tsirelson bound argument with a diagonalization of possible measurements will work. This is important, for to use the language of my paper this is where the topological obstruction is manifested.

In effect the CHSH or bell inequality may be thought of as a sort of metric. As with geometry non-Euclidean spaces have different metrics, and this is particularly the case for spaces with different topologies. As with Nagel and Newman in their book I see this issue as similar to the incompleteness of geometry as an axiomatic system to determine the truth of the 5th axiom.

I have no clear stance on the matter of counterfactual definiteness. That also seems to be something dependent on various quantum interpretations. I question whether this is something that is simply undecidable. As Palmer puts it this means the statistical independence of state preparation and measurement is not something provable or derivable from QM.

Cheers LC

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Author Jochen Szangolies replied on Feb. 20, 2020 @ 18:09 GMT

To me, the most simple way to think about Bell inequalities is as (hyper-)planes delimiting the set of convex combinations of value-assignments of possible measurements. That is, for e. g. the CHSH-setup, you have four measurements, and hence sixteen value assignments from (0,0,0,0) (or -1, which I use in the paper) to (1,1,1,1). Then, the general state of the system is a 16-dimensional vector of unit 1-norm, i. e. a probability distribution yielding the probability of finding each of the sixteen possible value assignments. The states which have only one entry equal to 1, and the rest equal to 0, then form the vertices of a convex polytope; this convex polytope can equally well be described in terms of its facets, which are the Bell inequalities of this setting.

Given this, I think how Bell inequalities are violated in my setting becomes readily apparent: if all Bell inequalities are obeyed, then you can construct a description in terms of the above, as a convex mixture of fixed value assignments. But the diagonal argument shows precisely that you can't make such an assignment. Hence, in some cases at least, it follows that we can't formulate a description of the system in the above terms; but then, in these cases, some Bell inequality must be violated.

Of course, this doesn't get me anywhere near deriving the Tsirelson bound. Non-computability lurks there, too, as was just recently shown (https://arxiv.org/abs/2001.04383).

As for counterfactual definiteness, I think a strength of my approach is that it gives a straightforward explanation where and when it is applicable---namely, only when reasoning about values explicitly provided by my f(n,k). We can talk counterfactually about the value of the spin (in some particular direction) of a distant particle, reasoning that it would have been the same even had we made a different local measurement, only if there is a definite value provided by the maximum information attainable about the system; but if, for example, that information is instead taken up by yielding a definite value for the correlation between two observables, then such talk becomes meaningless.

So if our knowledge about the system is given by (x-spin 1 is up, x-spin 2 is down), we can consider that x-spin 2 would have been down, even if we had made a different measurement on 1; but if it's instead given by (x-spin 1 is up, x-spin 2 is opposite that of 1), then the fact that the x-spin of 1 is some particular way is a necessary prerequisite for being able to reason about x-spin 2---a prerequisite that we loose if we imagine that we had made some other measurement on 1.

Lawrence B. Crowell replied on Feb. 21, 2020 @ 16:52 GMT

Jochen,

Your chart which I copy below, though it looks clumsy, is a sort of polytope. This is a tesseract in 4-dimensions. The z_A and z_B states form a bipartite entanglement. The same occurs for the states x_A and x_B, where the states z and x are in superposition quantum mechanically. We may then think of some additional spin state that for x_A and x_B these are replaced with the states...

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Lawrence B. Crowell replied on Feb. 21, 2020 @ 16:54 GMT

Jochen,

First off, due to a copy paste error the above does not work well. So here is a better formatted post.

Your chart which I copy below, though it looks clumsy, is a sort of polytope. This is a tesseract in 4-dimensions. The z_A and z_B states form a bipartite entanglement. The same occurs for the states x_A and x_B, where the states z and x are in superposition quantum...

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Author Jochen Szangolies replied on Mar. 2, 2020 @ 17:30 GMT

Dear Lawrence,

sorry for taking so long to respond. Unfortunately, my time is limited at the moment, and your posts always take some careful picking apart for me to understand.

That said, I'm either not quite there yet, or we're not talking about quite the same thing (although perhaps you're suggesting an alternative interpretation of my setting). The CHSH-polytope I reference is a classical entity, living in the probability space spanned by the value-assignments to the observables in the CHSH-experiment (i. e. the hidden variable vectors, and their convex combinations, which just give the probability distributions over experimental outcomes).

The Kirwan polytope lives in a space of (eigenvalues of) quantum states, and contains information about how a given system is entangled. Are you saying that there exists an entanglement polytope that's the same as the CHSH-polytope? If so, I'm afraid that's not quite clear to me. I mean, I can see that the four-qubit entanglement polytope must be contained in it, but that's trivially the case, because it's just the unit (hyper-)cube. Do you think there's more of a connection than that?

Lawrence B. Crowell replied on Mar. 9, 2020 @ 14:55 GMT

It has been a while since I checked FQXi. I am a little disappointed in how the essay contest is developing.

The CHSH polytope is based on the relationship

I_{chsh} = A_1×B_1 + A_1×B_2 + A_2×B_1 - A_2×B_2,

for Alice and Bob experiments with two outcomes. This curiously is a type of metric that can be interpreted as pseudo-Euclidean. This is also a measure of entropy, for it may be expressed according to conditional probabilities. An arbitrary two-qubit state after Schmidt decomposition can always be written as

|ψ_n⟩ = c_0|n_+, n_+⟩ + c_1|n_−, n_−⟩.

We choose the measurement settings in the following way

A_1 = m_1·σ, A_2 = m_2·σ,

B_1 = (1/√2)(m_1·σ + m_2·σ), B_2 = (1/√2)(m_1·σ − m_2·σ).

Here n, m_1 and m_2 are the unit vectors perpendicular to each other. Now find the expectation value of the CHSH operator in the state |ψ_n⟩. We get

⟨ψ_n|I_{chsh}|ψ_n⟩ = 2√2C.

The expectation of I then has this bound.

This CHSH polytope is I think related to the Kirwan polytope. The CSHS comes from the relationship with different basis measurements, while the Kirwan polytope is based on eigenstates. For x and z measurements we can think of there being two copies of the Kirwan polytope. The CHSH is then a discrete lattice for some form of covering space.

Cheers LC

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Steve Dufourny replied on Mar. 13, 2020 @ 11:53 GMT

Hi, All this is very interesting about the polytopes, and the plays of maths. We search after all what are the foundamentals of this universe. The polytops can converge, but for this we must be sure about their properties and if they are foundamental, we know that we have many different polytopes , like the Lie Groups also and this E8 for example,or the infinite polytopes, the abstracts ones or...

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Hi, All this is very interesting about the polytopes, and the plays of maths. We search after all what are the foundamentals of this universe. The polytops can converge, but for this we must be sure about their properties and if they are foundamental, we know that we have many different polytopes , like the Lie Groups also and this E8 for example,or the infinite polytopes, the abstracts ones or the complex polytopes also. And dualities appear also. Now the real question is , must we consider these polytops really considering the QFT ? is it just a tool to rank and study better the fields in our standard model ? the real question is there, and we can extrapolate philosophically deeper, are we sure that all is made of Waves and fields ? like in the strings theory , or in the geometrodynamics, because if we have coded particles instead of fields creating our physicality , so we must consider particles and not fields implyinmg these geometries, topologies,properties of matters and so the effects possible in extrapolating the maths. The maths are Always interesting but they must be utilised with the biggest wisdom considering the interpretations and assumptions, we cannot extrapolate and conclude all what we want.The problem foundamental for me is that we consider still these geometrisations due to fields , like if we had a 1D main field from this Cosmic scale and permitting with the oscillations to create the reality with these 1D strings at this planck scale, all is false if the particles are coded and in 3D, don t forget that we can create all SHAPEs, geonetries, topologies with coded 3D particles, 3D spheres for example, now imagine this, imagine that the codes of geometrisations and properties are inside these particles , imagine a Ricci flow, the Hamilton Ricci flow, a kind of assymetric Ricci flow to create the unique things, imagine too this poincare conjecture and the heat equation and imagine the plays of maths with the topological and euclidian spaces, and the lie derivatives and lie groups, we can create all geometries and topologies also, so we arrive at big philosophical questions about these foundamental objects and the main cause of these objects and their properties. You can tell all what you want with polytopes, we cannot affirm that it is foundamental simply. The same for my reasoning considering these 3D spheres coded at this planck scale considering a gravitational coded aether sent from the central cosmological sphere. I beleive that we must prove what we extrapolate simply and at this moment we are limited simply. The aim is not to create mathematical partitions but to find the real universal partition, it is totally different at my humble opinion. The convex polytopes and the linear transformations must be sure after all, and the vectors and scalars also, the problem is that we cannot affirm in fact, so the same for the extrapolations and assumptions. The secret maybe if I can is to superimpose a deeper logic to this universe , the fields, strings, geometrodynamics and the fact to consider only photons like main essence imply a prison for the majority of thinkers, that is why we cannot explain our unknowns mainly for me.Think beyond the box and maybe consider coded particles, the Waves particles duality is respected because they are inm motions and in contact in a superfluid these particles….Regards

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Hello,

One of my favorites, we recognise a general relevant knowledge about the subjets analysed and extrapolated. I have learnt in the same time several things that I didn t know. Congratulations

ps about the infinity, I beleive that we must rank them and consider this bridge separating this physicality , finite in evolution and this infinity beyond this physicality, a thing that we cannot define and the sciences Community is divided about its philosophical interpretations. I consider personally in my model of spherisation, an infinite eternal consciousness. I beleive returning about this infinity and the infinities and finite systems, that we must rank them, we have a finite universe in logic made of finite systems , coded and we see too this infinity appearing with our numbers and others like pi or the golden number … and we have this infinity beyond this physicality and this eternity even if we go deeper in philosophy. How must we rank and consider these infinities inside this physicality, it is the real question in fact….

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Steve Dufourny replied on Mar. 2, 2020 @ 09:41 GMT

I have shared it on Facebook with the essay of Tim Palmer too, I beleive that your essays merit it, regards

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Author Jochen Szangolies replied on Mar. 2, 2020 @ 16:30 GMT

Dear Steve,

Thanks for your kind words. I'm glad you found something useful for you in my essay!

Infinity is, as many have surmised, a thorny concept, replete with paradoxes. Going back to the Greek philosophers, I believe Anaximander was the first to seriously engage with the notion of 'the infinite' (or apeiron, perhaps more accurately 'the unbounded'). You might think it's a bit of a dodge, but it's not easy at all to get to the notion of infinity from necessarily finite observations.

Of course, since Cantor, we know that there isn't just one infinity, but ranks of them---his argument, of course, being the original form of the argument I present in my essay. We bump into the limits of 'Undecidability, Uncomputability and Unpredictability' precisely because we can't transgress infinitey---to a machine capable of executing infinitely many steps in finite time, the halting problem would be decidable. We're stuck at the bottom of the arithmetical hierarchy, and thus, all of our reasoning is necessarily bounded.

Thanks again for your interest!

Steve Dufourny replied on Mar. 11, 2020 @ 11:22 GMT

You are welcome, I agree fully , it is difficult to understand these infinities inside this physicality that we observe and try to understand. We search a kind of universal partition with these numbers and these foundamental mathematical and physical objects. I consider the 3d coded spheres and a gravitational coded aether sent from this central cosmological sphere, it is there that these finite series of spheres are coded by a kind a infinite eternal consciousness that we cannot define, we can just understand this physicality and its laws. The reals, irrationals, rationals, imaginaries, primes, p adics analyses , harmonics of fourier and this and that seem under a specific universal partition but we know so few still, I consider that these 3D quantum spheres of this aether play between the zero absolute and the planck temperature and they have codes permitting the geonmetries, topologies and properties of matters in this space time, I have considered the Ricci flow, the lie derivatives, the poincare conjecture, the topological and euclidian spaces, the lie groups, the heat equations and other mathematical Tools , and I have invented with a person the assymetric Ricci flow, that permits to create the unique things and all Shapes , I try to find the good mathematics for this formalisation and the good partition, but it is not easy.I have quantize with this logic the quantum gravitation, I have just considered different distances like if our actual standard model was just emergent due to codes farer .

Best Regards

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Dear Jochen, Very interesting paper. To understand Assumption 1 I had to get rid of the idea that measurements are usually repeated under identical conditions; in this case the measurement may change with n and this is necessary to derive the paradox. It may be helpful to think in terms of settings (as in the EPR-Bell-Bohm situation), so that the choice of a measurement is a choice of the settings. This introduces a tacit Free Choice assumption into the argument. The contradiction is reminiscent of, perhaps even equivalent to, the so-called paradox of predictability, see e.g. the review by Rummens and Cuypers, Determinism and the Paradox of Predictability, Erkenntnis 72, 233-249 (2010), https://link.springer.com/article/10.1007/s10670-009-9199-1.
I must admit that I find inferences or suggestions of the kind that an undecidable proposition can be modeled by a quantum superposition suspicious - the former are very general, the latter arise is a very specific mathematical context (Hilbert space) and one needs additional arguments to really make the inference. As far as I know, no one has managed to do this convincingly.

Having said this, I will continue follow your work with great interest. Best wishes, Klaas

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Author Jochen Szangolies replied on Mar. 11, 2020 @ 05:58 GMT

Dear Klaas,

thank you for your comment! I'm glad you found something of interest in my essay.

Regarding the importance of free choice, indeed you can make an argument that one might be constrained to make only measurements for which f(n,k) yields a definite value, thus never running into the sorts of phenomena following from its indefiniteness. That we do, apparently, run into them would then be evidence that there is no such constraint (which of course doesn't entail that we have free choice).

However, one does not need to think about changing measurement conditions to derive the paradox, merely about the totality of all possible measurements on the system---whether they are ever performed or not. The index n is then essentially just an attempt to enumerate these measurements, with the argument then showing that no such enumeration can be complete.

As for superposition, you're right to point out that the structure of linear operators on Hilbert space is a quite specific one; but in the end, the project of deriving a theory from underlying principles is one to derive the specific from the general---for comparison, the structure of Lorentz transformations on Minkowski space is also quite specific, while following from the very general principle of relativity, together with the constancy of the speed of light.

Indeed, if one views Hilbert space as a concrete realization of an abstract propositional structure---the orthomodular lattice of its subspaces---then one can show that this essentially follows from the notion that there exists a maximum amount of information that can be extracted from any system (https://link.springer.com/article/10.1007/s10702-005-1129-0
). This is both connected to undecidability (as in Chaitin's principle, you can't derive---under a suitable measure of complexity---a theorem more complex than the set of axioms), and superposition, with the failure of the distributive law in quantum logic.

For simple (that is, not subject to Gödelian phenomena) axiom systems, this correspondence was demonstrated by Brukner (https://link.springer.com/article/10.1007/s11047-009-9118-z
) and Paterek et al (https://iopscience.iop.org/article/10.1088/1367-2630/12/1/0
13019/meta), who show an explicit way to encode axioms in a quantum system and demonstrate that a given measurement will produce random outcomes whenever the corresponding proposition is not derivable from the axioms. In a sense, my work is simply an extension of this to cases where undecidability is not due to the limitations of the axiom system, but to the inherent limitations imposed by the limitative theorems of metamathematics (although in the treatment using Lawvere's theorem, one can pass over first establishing a correspondence with formal axiomatic systems).

Having said that, I of course don't claim to have a complete reconstruction of quantum theory in hand. There are still different options possible---for instance, it's not easy to see why one should use Hilbert spaces over the complex field, and not over the reals or quaternions. In that sense, perhaps one should think of the connections I point out, as of yet, as 'family resemblances', rather than strict formal equivalences. I view them as enticing prospects that seem sufficiently promising for me to carry on exploring this point of view; but I would not be the first wanderer to be deceived by tantalizing lights in the dark forest.

Cheers

Jochen

Dear Jochen,

You wrote a really excellent Essay, have my sincere congrats. You chose a very hot topic by discussing it from an original point of view. Your approach of reconstructing quantum mechanics is in a certain sense similar to my attempt of reconstructing quantum gravity through its fundamental bricks, that are black holes. In addition, from the philosophical point of view my position is near local realism too. You deserves my highest score, I wish you good luck in the contest.

Cheers, Ch.

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Author Jochen Szangolies replied on Mar. 11, 2020 @ 16:41 GMT

Dear Christian,

thanks for your kind words. I agree that out approaches seem to share the same spirit, of trying to find the 'lynchpin' issues from which everything else may (hopefully) unfold, and come to be explicable. Sometimes, a complicated problem may resist forceful attempts to crack it, while crumbling under a gentle tap at the right place. Let's hope we've found the right place to tap!

Cheers

Jochen

Dear Jochen,

I really enjoyed reading your essay. I particularly liked your clean-cut presentation of the principles of finiteness and extensibility. You might like to have a look at my essay wherein I outline finiteness as a program to (re)construct an alternative, indeterministic classical physics (a program that we are developing with Nicolas Gisin). It would be nice to find an analogous (but of course not completely identical) feature of extensibility in indeterministic classical physics. We can maybe discuss this.

Meanwhile, congratulations again, top rate so far!

Flavio

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Author Jochen Szangolies replied on Mar. 11, 2020 @ 16:46 GMT

Dear Flavio,

thanks for your kind words! I'm aware of your work with Nicolas Gisin, although I haven't yet had the time to study it thoroughly. I'll take your essay as an opportunity to rectify this; if there is a connection between our approaches, maybe we can narrow in on what, precisely, it is that separates the classical and the quantum. I'll get back to you after I've had a look at your essay.

Cheers

Jochen

Jochen. I enjoyed your paper. You may find my essay. “Clarification of Physics—“ interesting. I introduce a self creating system, a new basic level to the current epistemically horizon and show how it fits into the creation of a multiverse that includes “our” physical universe. I would appreciate your comments on my essay. John D Crowell

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Author Jochen Szangolies replied on Mar. 22, 2020 @ 07:35 GMT

Dear John,

thank you for your comment. I'm glad you enjoyed my thoughts. The notion of self-creating systems is a very interesting one, and close to my thoughts in some regards, so I'll definitely have a look at your essay.

Good luck in the contest!

Cheers

Jochen

Dear Jochen.

It is only now that I am daring to make comments on your excellent essay because your approach is quite another one as mine. But I think one is learning most with an open mind, so I made the following notes while reading your essay:

“the Newtonian, classical framework, can no longer be upheld”. Why not, it is describing our daily macro reality quite good and can be...

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Author Jochen Szangolies replied on Mar. 23, 2020 @ 06:15 GMT

Dear Wilhelmus,

thank you for your considered comments. When I say that the Newtonian framework can't be upheld anymore, I mean in an absolute sense---you're right to point out that for almost every practical matter, a Newtonian calculation will yield an adequate answer. But Newtonian mechanics can't be universally valid---necessarily, if my arguments work out. It has to break down at some point, and needs to be amended---or perhaps completed: in the same way as special relativity is a consistent completion of Newtonian mechanics in the realm of velocities approaching that of light, quantum mechanics can be viewed as a consistent completion of Newtonian mechanics in the realm where we're close to extracting the maximum information from a system.

I agree that a theory of everything may apply different concepts to different domains, but these concepts must be consistent, so as to not 'crack at the seems', so to speak. All domains of physical reality ultimately interact, even if perhaps in a mediated way, and thus, our descriptions of each must match up at these interaction points. Hence, Newtonian mechanics needs modification, even if these modifications are practically unnoticeable in everyday life.

You make a good point regarding the embeddedness of the observer within the phenomena they observe. That's in fact another way to think about such phenomena, worked out by Thomas Breuer, Maria Luisa Dalla Chiara, and others.

Regarding measurements, we must surely admit the fact that in a very large number of cases, we can exactly predict what outcome a measurement will yield, and that outcome will indeed be observed. So, to that end, there are some future possibilities that are excluded by the present state of affairs.

I'll have a look at your essay.

Thanks again for your comments, and good luck in the contest!

Cheers

Jochen

Wow Jochen, this was great!

You managed to tie together undecidability and epistemic horizons in a way I have never seen before, and which intuitively rang true to me. For what it’s worth, I believe that the way quantum mechanics will ultimately escape from the problem of supporting so many radically different ontologies is precisely through the kind of reconstruction you propose, and the fact that you managed to tie this to the kinds of epistemic horizons discovered by Godel and Turing blew me away.

I have responded to your kind comment on my piece over there.

Best of luck in the contest!

Rick

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Author Jochen Szangolies replied on Mar. 23, 2020 @ 06:19 GMT

Dear Rick,

thanks for this enthusiastic reply! I'm happy you found something that rang true for you in my essay. The sort of approach to quantum mechanics I pursue has been on the margins for a long time, and still does attract some skepticism (well deserved, in many cases), but I hope that we've gotten to the point that nobody gets thrown out of any offices for making the suggestion (as Wheeler was by Gödel). If my contribution helps with that just a little, I'll be satisfied.

Cheers

Jochen

Hi Jochen,

great essay that demonstrates that empirical data restrict the freedom to extend quantum theory by some deterministic hidden variables already for one and the same kind of QM-experiment. I think you made a very good job to decrease chances for getting thrown out of an office for making a certain suggestion.

If you like i would be happy if you could comment on my essay where i also try to link undecidability to quantum events (although not as elegant as you have done).

Hope you are well an healthy.

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Author Jochen Szangolies replied on Apr. 2, 2020 @ 04:42 GMT

Dear Stefan,

sorry for taking so long to respond. Thanks for your positive comment! If the essay indeed plays a part in making this line of research a little more mainstream, I would consider it to have fulfilled its purpose---while research on the foundations of quantum mechanics has in the past few years become much more respectable than it was in the years of 'shut up and calculate', I think there's still a ways to go before that message has truly percolated through the community.

I'll have a look at your essay as soon as I get the time---but with the situation right now, that might unfortunately be a while.

Take care!

Cheers

Jochen

Dear Jochen,

You write: “Instead of trying to infer the underlying ontology ...”

I believe that in order to overcome the crisis of understanding, the crisis of interpretation and representation in the fundamentals of quantum mechanics and cognition in general, the most profound ontological ideas are needed. Quantum mechanics is a phenomenological (parametric, operationalist) theory without an ontological basis. A. Einstein pronounced the ontological verdict on Quantum Mechanics: “God doesn’t play dice with the universe.”

Yes, Planck and Einstein began the Big Ontological revolution in the basics of knowledge, but it remained incomplete. Gödel’s theorems - this was the answer to the protracted crisis of the foundations of mathematics, which has been going on for more than a hundred years. And this problem for some reason "swept under the carpet." In overcoming the crisis in the philosophical basis of science, one cannot rely on the “classical ideal”, since it is precisely the cognitive attitudes of the “second Archimedean revolution” (“hypotheses non fingo”, “physics, fear metaphysics”), the atomistic paradigm (mechanistic, part paradigm) that prevails in science holds back the necessary ontological breakthrough in philosophical basis of knowledge. Now it is appropriate for all physicists to recall the philosophical precepts of A. Einstein: “At the present time, a physicists has to deal with philosophic problems to a much greater extent than physicists of the previous generations. Physicists forced to that the difficulties of their own science” and of J. Wheeler: “Philosophy is too important to be left to the philosophers.” Carlo Rovelli calls for such a step towards Philosophy in the article Physics Needs Philosophy / Philosophy Needs Physics .

With respect, Vladimir

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Author Jochen Szangolies replied on Apr. 2, 2020 @ 04:53 GMT

Dear Vladimir,

thank you for your comments. I agree (enthusiastically, in fact) that physics needs philosophy, and have often lamented the lack of communication between the two fields. So don't take my remark as suggesting anything else---the comment was a methodological one: the idea of 'backwards-inferring' an ontology to fit the formalism of quantum mechanics---its interpretation, in other words---is on the one hand well-trodden ground, by now, and on the other, has so far failed to produce any large-scale consensus.

Hence, I advocate going the way in the other direction: start with some reasonable assumptions and inferences about ontological matters, and see whether the quantum formalism can be reconstructed from there---the project of finding a foundational principle for quantum mechanics. I'm not saying that this should be pursued to the exclusion of the interpretational project, but merely that it's received comparatively little attention so far, so an investigation might have a chance to dig up something worthwhile. And who knows, maybe the two ways eventually meet up in the middle?

I think that physics still suffers from the hangover of what Feyerabend memorably called the generation of 'savages' in physics, who lack the philosophical depths of Bohr, Einstein, and others. In a sense, this is just a historical pattern that plays out after every major paradigm shift in physics, as you seem to be aware---it seems that after each conceptual revolution, physics retreats to an essentially instrumentalist stance, licking its wounds, only to slowly come back around to the investigation of its conceptual foundations. That's the sort of project I see myself engaged in.

Cheers

Jochen

Vladimir Rogozhin replied on Apr. 13, 2020 @ 11:21 GMT

Dear Jochen,

Thank you for your deep constructive response to my comment.

You write:

"... and on the other, has so far failed to produce any large-scale consensus."

I believe that the FQXi contests are a good “field” for the start of the Big Brainstorming. Look, ideas are already the tenth contests, and physicists and mathematicians have not yet found consensus on...

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Author Jochen Szangolies replied on Apr. 18, 2020 @ 07:21 GMT

Dear Vladimir,

I don't want to let this thread of conversation die, but I find myself more pressed for time than I had anticipated. So I'll have to try and be brief.

First of all, I think I agree with your general concern (oh, and thank you for reminding me of the Hegel quote, it's a hopeful thought in these present times). I have struck up a similar conversation with Fabien Pailluson over at his essay page, about how questions, once 'systematized' and transferred into the domain of mathematically expressed science, may lose some of their original meaning---in other words, how our desire for quantifiable answers may lead to the loss of the original question's substance. Perhaps it is also of interest to you.

I'll have to dash, but I promise to try and find the time to engage with your essay. Thanks again for your kind comments!

Cheers

Jochen

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Dear Jochen Szangelios,

I apologize for misspelling your name and hesitating to read your essay the title of which was deterring to me. Meanwhile I guess, the successful application of QM doesn’t require orthogonal quantum states, and the distrusts of McEachern and of Kadin are not unfounded. Should we still invest more effort into quantum computing?

While I don’t overestimate my...

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Dear Jochen Szangelios,

I apologize for misspelling your name and hesitating to read your essay the title of which was deterring to me. Meanwhile I guess, the successful application of QM doesn’t require orthogonal quantum states, and the distrusts of McEachern and of Kadin are not unfounded. Should we still invest more effort into quantum computing?

While I don’t overestimate my argument that Fourier was partially wrong, I don’t trust in Fraenkel’s ZFC since I read how he supported Cantor’s (as I see it) naïve idea of Überabzänlbarkeit by taking elements of an infinite set of numbers as fixed. I rather trust in Peirce who spoke of mere potentialities and Weyl who spoke of the sauce of real numbers.

Pragmatically, Euclidean spaces are thought to be composed like a set of points, which are defined only by the properties that they must have for forming a Euclidean space.

You Jochen admitted: “it's not easy to see why one should use Hilbert spaces over the complex field”.

Klaas Landsmann wrote: “I would not say that Gödel's theorems imply that mathematics cannot be grounded on logic, except when you mean "grounded" in Hilbert's sense, namely a proof of consistency. Without knowing that e.g. ZFC is consistent, it is still a logical language in which we do our mathematics, most of which is decidable in ZFC.”

I realize minor changes in the language of mathematics. At school I learned “point product” not yet “dot product”. Because I am not a mathematician, I had to naively reinvent the distinction between point and dot which I consider decisive from the perspective of logic and physics.

Let me reiterate: I am suggesting let's learn from Leibniz' success story and calculate as if the unbounded plurality of thinkable references was identical with not just Salviati's notion of being infinite, the logical property of simply being endless. But be careful and understand what you are doing. Don't derive nonsense.

Eckard Blumschein

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Author Jochen Szangolies replied on Apr. 2, 2020 @ 04:59 GMT

Dear Eckard,

thank you for your comment. Regarding quantum computing, to me, this is a 'the proof is in the pudding'-kind of case. Either it will work, which will then legitimize at least some of the foundational principles at the heart of quantum mechanics; or it won't, in which case, we'll likely have learned something excitingly new about the world. Since that appears to be a win-win situation, I don't see why one should not continue to put in the effort.

The reasoning in my article has important similarities to that of Cantor, so if you reject the latter, I can see you having some trepidation regarding the former; however, it does not seem that you have the same reservations against the Gödelian argument, which is in the end yet another example of the same technique (diagonalization, or more generally, the application of Lawvere's theorem). Or do you see an essential difference?

Cheers

Jochen

Eckard Blumschein replied on Apr. 8, 2020 @ 01:43 GMT

Dear Jochen,

What I see is not a difference between Cantor who died in a madhouse, perhaps because his trust in his own point set theory was shuttered by König, and Lawvere who is hopefully still very healthy. I am rather concerned with differences between reality and mathematical models. Many years ago I got health problems because mathematicians rejected "for mathematical reason" my argument that future data are not yet directly available by means of measurement in reality.

Meanwhile, mandatory mathematics seems to require sophisticated efforts to fabricate new definitions of good old ideal notions like point and line. Why?

As I tried to exemplify in my current essay, mathematical pragmatism (calculate as if ...) must not be one to one translated into a universal tool for description of reality. Are orthogonal quantum states still required if I am correct concerning Fourier?

My hint to Kadin's extraordinary courage was meant as a challenge: How to apply your theory?

Eckard

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Author Jochen Szangolies replied on Apr. 10, 2020 @ 08:51 GMT

Dear Eckard,

I can certainly understand being weary about whether mathematics and physics is always in the sort of correspondence many physicists take it to be; mathematical theories may be as spectacularly intricate and aesthetically pleasing as they like, that still doesn't mean there's something in nature that they describe. There can sometimes be cases of 'rigor mortis': a focus on mathematical formalism to the detriment of the underlying physics.

However, to me, mathematics is first and foremost a tool; properly applied, it forces us to be clear and consistent in our reasoning, which unaided reason often is not. In that, it is not different from other tools of scientific inquiry---the microscope is superior to the naked eye in resolving miniscule details, but still, the microscope needs to be pointed at the right spot to tell us what we need to know.

I haven't yet gotten to reading Kadin's essay. I'm skeptical, as you'd perhaps expect, of any proposed return to classical physics---there are arguments that are very simple (I give one in my essay, regarding the existence of a joint probability distribution) that would seem to prohibit something like that. But I'll keep an open mind.

Cheers

Jochen

Jochen,

Great analysis. I appreciate your rare & deep understanding of the issues around QM. I'm reminded of the good advice in your response last year to "focus on observed events", which, yes, I'd done, unfortunately you didn't get to my essay. Yours is flawless (I re-read it to check!) and beautifully written, though QM rarely scores well here, (an exception was my 2015 'Red/Green...

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Author Jochen Szangolies replied on Apr. 2, 2020 @ 05:11 GMT

Dear Peter,

thank you for the kind comments! 'Flawless' is high praise indeed, though I myself keep going back and thinking of ways I could've done better, or be more clear about. But if it all hangs together in the end, I'm willing to be content.

Your project seems engaged in questioning the foundations of logic---revoking the law of the excluded middle invokes comparison with dialetheism, and of course, quantum mechanics has itself been argued to lead to similar revisions, see Reichenbach's three-valued logic, and the more familiar von Neumann/Birkhoff logic. Although of course, in a sense, trying to make logic empirical, or at least, renege on it due to experiment, is sort of putting the cart before the horse.

I will try to get to your essay, in the hope of understanding the rest of your comment better. However, due to recent circumstances, my time for this has rather been slashed, so it might be a while, I'm afraid.

Cheers

Jochen

Peter Jackson replied on Apr. 2, 2020 @ 16:28 GMT

Dear Jochen,

I see it as horse first, as the problems in logic ('paradox') and Philosophy were worse than physics, needing resolving by checking starting assumptions. How do we imagine Aristotle dreampt those up anyway! We now have far better information than he did, but the foundational issues took a long time to dig down to. The sound consequences of the proposed revisions alone seem to confirm veracity.

BUT the most important thing for you to study is the apparent physical solution to QM you suggest your'e looking for on page 1. It's verified by computer, but I trust a good well informed brain more! You didn't get to it last year so I suggest I'm owed a priority look!

Ridiculous Simplicity fqXi 2018.

I do hope your problems aren't family

Very best. Stay safe.

Peter

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Peter Jackson replied on Apr. 4, 2020 @ 10:56 GMT

Dear Jochen,

Thank you kindly for your comment on mine. My reply is here;

Jochen, Thanks, My mentor Freeman Dyson agreed, ANY advancement means all OTHERS will "feel sort of lost", also Lorentz, Feynman etc. And yes I also studied logic & philosophy, both in crisis! Yes I pack a lot in, testing conventional thinkers, but all refs are given.

You wrongly infer I suggest loosing "the absolute identity of quantum particles.", I just suggest they can have different polar axis angles, except when 'paired', but I DO challenge that only a "statistical approach to QM", can work, & show how we can "do better" as Bell suggested! Shocking? Tes. But seems also true (I cited the verification plot). That's what I'd like you to test.

I hope you get a mo as it may be rather important to advancement.

Very Best

Peter

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I've learned that when seeking to explain "existence" using "diagonals" it's more practical (than "uncountable sets") to consider one zero-origin number line from opposite side of a professor's closed window; so I disagree with the foundational math in the title as suggests classical arithmetic like "squaring negatives" and "prioritizing zero". Other than that, I believe one square root of three disproves zero-evidence-cubed, as suggested in my essay, and in your essay here:

"The above has the form of a diagonal argument. Diagonalization was first introduced by Cantor in his famous proof of the existence of uncountable sets, and lies at the heart of Godel’s (first) incompleteness theorem, the undecidability of the halting problem, and many others. "

It looks like I'll be graded as a 1/10 hack but there is at least one good Cantor quote in mine. Help yourself to naming rights on "definition" vs "infinition". I liked another essay I read better for the simple reason it wasn't about being right it was about including our own pretending of "general authority"; otherwise this essay would be perfect so I'll give it a fair 9/10.

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Author Jochen Szangolies replied on Apr. 10, 2020 @ 08:36 GMT

Dear Daniel,

thanks for your comment. I'm glad you found something to like about my essay, and thanks for the generous score. I'm not sure I'd say my essay is about 'being right'; I think about it as an 'essay' in the original sense of the word---it's an attempt, something that may well fail. So it's less 'this is how it is', and more 'wouldn't it be neat if this were how it is'.

Anyway, thanks for reading; I'll have a look at your essay soon.

Cheers

Jochen

Dear Jochen,

Thank you very much for this essay it was a real pleasure to read. Very well written and thought provoking too.

Pardon my slow-mindedness but I would like to grasp the essence of the argument for the non existence of f(n,k) i.e. what are some necessary preconditions for it to hold.

- Is the claim valid even if there are finitely many states?

- If k belongs to a finite interval of integers then I could build a finite set of experiments that would create a f(n,k) table for all possible values of 1 and -1 for each state (like when designing a truth table). In that case the mg operation must bring back an existing row since all rows would have already been exhausted by my truth table.

Many thanks for your help on this.

Best,

Fabien.

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Author Jochen Szangolies replied on Apr. 14, 2020 @ 05:21 GMT

Dear Fabien,

thanks for your comment. I'm glad you found something to enjoy about my essay!

Regarding your questions, I think the most common take would be that, like the halting problem, these difficulties don't occur for finite/discrete systems. So, for instance, the halting problem for finite state automata is commonly said to be decidable---by simple brute simulation, if need...

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Dear Fabien,

thanks for your comment. I'm glad you found something to enjoy about my essay!

Regarding your questions, I think the most common take would be that, like the halting problem, these difficulties don't occur for finite/discrete systems. So, for instance, the halting problem for finite state automata is commonly said to be decidable---by simple brute simulation, if need be.

However, that's not actually a difference to the situation with Turing machines: you need to appeal to a computational system with fundamentally greater computational capacities to solve the halting problem of one with lesser power---you need a system with more states than a given FSA to solve that FSA's halting problem, or a system capable of performing an 'infinite' amount of computational steps to decide a TM's halting problem. Such an augmented system will, however, have itself another halting problem it---or systems of its class---can't solve, and needs a system of fundamentally greater power to solve.

So the situation in the finite and continuous cases isn't really that different; the only distinction is that our intuition does not balk at imagining a system that has more states than a given system with finitely many states, the way it does at imagining one with a 'transfinite' number of states. But the problem only iterates: the halting problem of a system with finitely many states can be solved by one with more states, but what of that system? Either, you have an infinite number of system with a higher amount of states---but then, why disallow infinitely many states in the first place, as the total computational capacity of that collection will equal any Turing machine?

Or, you have some finite system that's at the top of the hierarchy. Then, its halting problem will not be solvable by any concrete system, and the same problems will persist.

This argument can be made more carefully, but in essence, I don't think that it suffices to merely restrict allowable systems to those with finitely many states.

Another possibility would be to invert this reasoning: as the argument I present seems to imply the sort of 'deterministic evolution interspersed with random events' we actually observe, we could take that observation as evidence for the applicability of the argument.

Does this address your question?

Cheers

Jochen

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Fabien Paillusson replied on Apr. 16, 2020 @ 08:48 GMT

Dear Jochen,

Thank you for your reply. That there are hierarchies of halting problems was insightful.

With regards to my initial question, I suppose I was mostly confused by the leap from the two states of coin tossing to infinitely many states at the core of any diagonalisation argument.

As I said, I am just trying to restate what is actually being stated in your proof and in particular what are the necessary assumptions for the proof to hold. That the system can be in infinitely many states appears as such an assumption.

That the system can be in infinitely many states does appear actually reasonable a priori, for we never know what "the" state of a system is before we measure it in some way.

I actually think that the coin example could be used to go beyond the two state system. In fact m1 would still be "Head is showing once the coin has landed" but the state could be something much more complicated linked to the initial setting of the experiment...or to the mechanical state of the coin.

Then I believe what your proof is saying is that:

"Given that a system can be in infinitely many states, there necessarily exist states such that the outcome of some measurements on those states is undetermined"

If that is the case, different physical theories can choose to work only with subsets of the states the system can be in (in the same way that one can decide to work with integers instead of the reals...in some sense, as long as this subclass of states forms a closed set under some dynamical rules and chosen operations, then it seems fine). Classical physics restricts the states to those that are determinable while QM embraces both kinds of states.

Is that a fair restatement of what you are saying?

Many thanks.

Best,

Fabien

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Author Jochen Szangolies replied on Apr. 18, 2020 @ 06:42 GMT

Dear Fabien,

you're correct that in the essay as it is, the infinite number of states is an assumption. I made this somewhat more explicit in the Foundations of Physics article preceding this work, by introducing the notion of 'program world': physical states encode initial data, and programs take these as input, to spit out a measurement result. As there is a denumerably infinite quantity of programs, we get infinitely many possible measurements.

Classical physics essentially emerges from this using coarse-graining: that is, once you're unable to distinguish between certain states, and thus, have only extracted an amount of information about the system that's sufficiently below the finiteness bound, you won't notice any quantum effects. So yes, this is, I think, in some sense what you're saying: you lump all the real numbers in some interval together, and don't distinguish between elements of the resulting set.

Cheers

Jochen

Hi Jochen,

Thank you for your well-written and engaging essay. I agree 100% with your closing thought for the need of a relative realism, except that I would express it as a contextual realism, since context encompasses more than 4D (or 3D+1) inertial reference frames.

You correctly assert that the key attributes of physical reality needed to explain quantum phenomena must include 1) finiteness and 2) extensibility. If physical context includes a non-zero ambient temperature, finiteness and extensibility are immediately accommodated. Absolute zero is an idealization that does not exist in reality, and the universe as a whole has an ambient temperature equal to its 2.7 K cosmic microwave background. For a contextual reality, a finite ambient temperature means that space itself is not infinitely continuous and position-momentum information is finite. A decreasing ambient temperature fine-grains space and creates new information to be discovered. In addition, when perfect measurement is defined with respect to a system’s actual and objective context, empirical classicality is restored.

In my essay I analyze the deeply held assumptions that have led to the widely-held conviction that objective physical reality is non-contextual. I develop a conceptual model of physical reality that is objective and contextual, that is fully compatible with empirical models of physics, and that eliminates quantum paradoxes. I would welcome your thoughts on it.

Harrison Crecraft

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Author Jochen Szangolies replied on Apr. 18, 2020 @ 06:53 GMT

Dear Harrison,

thank you for your comment. I'm happy to see that some of my ideas seem to resonate with you.

It's interesting you should mention contextuality---my earliest work in quantum mechanics was on that notion, and in a sense, you're right, you can think of a sort of 'contextual reference frame' in analogy to (perhaps generalization of) spatiotemporal reference frames. It wasn't primarily relativity theory I had in mind with the notion of relative realism, however, but merely the idea that we can attach the label 'real' to certain events only as relative to others---for instance, the electron's spin value being 'up' could be thought of as a claim about 'reality' only relatively to the measurement apparatus showing an 'up'-reading.

Your notions regarding---if I interpret you correctly---an inherent thermal 'noise' making the acquiring of perfect information about a system impossible remind me of Nelsonian stochastic mechanics. Is there a connection?

I will have a look at your essay---I hope I'll soon find the time to give it a good reading.

Cheers

Jochen

Harrison Crecraft replied on May. 6, 2020 @ 17:40 GMT

Hi Jochen,

Thank you for your response. “Your notions regarding---if I interpret you correctly---an inherent thermal 'noise' making the acquiring of perfect information about a system impossible remind me of Nelsonian stochastic mechanics. Is there a connection?”

Your interpretation is not quite right at a subtle but fundamental level. The idea of thermal noise and stochastic mechanics implicitly assumes random fluctuations of precise coordinates, but precise coordinates are definable only with respect to an assumed ambient temperature of absolute zero. Perfect information for a system contextually defined at a positive ambient temperature is complete. There is no randomness in the actual contextual state. Randomness only comes in during irreversible transition from a metastable state to a more stable (higher entropy) state. As long as a state exists, there is no randomness and the state evolves deterministically.

Harrison

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Dear Jochen,

I loved your essay, very interesting approach to reconstruct quantum mechanics from first principles. I like the epistemic horizons idea, and the diagonal argument making connection with undecidability. Very nice interpretation of Wheeler's suggestion that you mention in the conclusion. I also liked that you try to get close to the classical ideal of local realism by using what you define as relative realism. In this respect, you may find interesting, I hope, a result that QM can be formulated on the 3d space or 4d spacetime, rather than on the configuration space, and especially that this gives a local dynamics as long as there is no collapse (possibility that I think is still on the table, as explained in the attached pdf).

Cheers,

Cristi

attachments: post-determined-block-universe.pdf

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Author Jochen Szangolies replied on Apr. 18, 2020 @ 07:07 GMT

Dear Christi,

I'm happy that you found something you liked about my essay! Thanks for pointing me in the direction of your other papers---I'm always amazed, and a little bit humbled, at the depth and breadth of your ideas. I had, in fact, seen that paper before---one thing I'd been wanting to think about is how this relates to ideas by David Albert, who has proposed to explicitly 'pry apart' physical space and what he calls 'the space of physical determinables' to make sense of quantum weirdness. The latter seems clearly related to configuration space in some sense, so perhaps one could use your formalism to 'pull back' Albert's explanations into a familiar 3+1-arena, and see what they amount to.

But that's hardly even an idea for an idea, so far.

Anyway, you've given me quite the reading list, I'll try my best to eventually catch up, thanks!

Cheers

Jochen

Cristinel Stoica replied on Apr. 18, 2020 @ 08:18 GMT

Dear Jochen,

Thank you for the reply and the interest in my mentioned papers. In the one with the representation of the wavefunction on space or spacetime, the representation, albeit complicated, is just like a classical field theory, with infinite-dimensional vector fields satisfying some global gauge symmetry. But once we put aside the differences in the complexity of the theory, as I...

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Dear Jochen,

Thank you for the reply and the interest in my mentioned papers. In the one with the representation of the wavefunction on space or spacetime, the representation, albeit complicated, is just like a classical field theory, with infinite-dimensional vector fields satisfying some global gauge symmetry. But once we put aside the differences in the complexity of the theory, as I said, we see it's just a classical field theory. This as long as no collapse takes place. I'm not sure if David Albert would find this at odds to his proposal to teach QM and explains how it differs from classical mechanics based on the fact that the wavefunction lives on the configuration space. Now we see that this was only a representation, and the things are as classical as it gets, including being local. The difference occurs if there is a collapse. Collapse, for taking place everywhere in space simultaneously, introduces nonlocality, and because of collapse, entanglement leads to Bell correlations. Now, one can say, "assuming that your representation is true". It is true, I mean a correct representation, and maybe there can be simpler or more natural ones. But both mine and the wavefunction on the configuration space are just representations. Mine, as opposed to the wavefunction one, (1) makes explicit locality and when exactly it's violated, and (2) is consistent with the idea of "ontology on space or spacetime". So there is no need to appeal to ontology on configuration space, not that this was a problem, but I submit that this is NOT the characteristic of quantum mechanics, since there is no difference here. The key difference is brought in by collapse.

Now, about collapse, the paper I attached to my previous comment can be taken as independent on the Phys Rev one I linked in the same comment. But to me they are related. The one about the post-determined block universe makes use of the fact that the ontology of the representation of the wavefunction is on space or rather spacetime. But it's not only this, it proposes an interpretation of QM where there is no collapse and the outcomes are still definite (so it's not MWI, it's just unitary single worlds). But earlier I stated that the key difference between QM and classical is isolated in the collapse. And in the other paper, that collapse is not necessary, it can happen unitarily. And in fact, if there would be discontinuous collapse, it would be undesirable, like breaking conservation laws, relativity of simultaneity, the evolution law, and is in tension with the Wigner-Bragmann derivation of the wavefunction for any spin and its dynamics for free particles, from the Poincaré symmetry. And locality. Now, most people think that all these are a small price to pay. For me, since I care about relativity too, it's a too big price.

OK, so back to isolating quantumness. How can I say that quantumness is not in unitary evolution, since it is equivalent to a classical one, but also say that the collapse may not exist? Isn't there a contradiction between these? Indeed, from the post-determined BU paper follows that the difference is not in the collapse either, at least in my interpretation without discontinuous collapse. The key difference, I think, is deeper, and need to be found, but I hint in that paper that it has to do with some topological constraints which can prevent most of the local solutions to extend globally, leaving to a zero-measure set of solutions which look as fine-tuned to go around Bell's theorem while still being local. Now, you may disagree with the post-determined BU, since I didn't prove that this is the case yet, but at least the paper with the representation on 3d space can only be refuted if some mistake is found in my proof. So at least for now the best way to isolate the quantumness is in the collapse, more precisely, the fact that it takes place globally on all of the degrees of freedom of the fields on 3d which I use to represent the wavefunction. And this can have the same effect on classical fields too, as we can see if we take as example my representation.

Thanks again for the comments,

Cheers,

Cristi

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Cristinel Stoica replied on Apr. 18, 2020 @ 12:16 GMT

P.S. Even if I said earlier that from the representation of the wavefunction on 3d space follows that the quantumness is not due to the wavefunction being on the configuration space, but it's related to the apparent collapse, this is not in conflict to it being related to the quantization of the volume of the cells in the phase space. In fact, I think this is the point of convergence with what I said that I don't think it's in the collapse either, because I don't believe there is discontinuous, nonunitary collapse even for single worlds. If my program will succeed, I expect to be something topological that will result in full interactions being quantized, and here is where I think that the phase space quantization will result. But, as I said, I don't have a proof yet.

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Author Jochen Szangolies replied on Apr. 23, 2020 @ 16:45 GMT

Dear Christi,

thanks for the fascinating discussion. One thing I'd like to understand---one can formulate classical mechanics on Hilbert space, as is done in the Koopmann-von Neumann representation. I wonder if one could apply your 3D-representation to the KvN-wave function; then, one could maybe more explicitly study how that differs from a quantum wave function, giving a hint of where the...

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Dear Christi,

thanks for the fascinating discussion. One thing I'd like to understand---one can formulate classical mechanics on Hilbert space, as is done in the Koopmann-von Neumann representation. I wonder if one could apply your 3D-representation to the KvN-wave function; then, one could maybe more explicitly study how that differs from a quantum wave function, giving a hint of where the quantumness comes from.

Personally, I think my biases are opposite to yours---I tend to think of space and time as ultimately derived entities, and I've got some sympathies with the recent developments regarding the duality between entanglement and spacetime-descriptions. This is an older idea than many realize, going back to (admittedly, somewhat heuristic) arguments of Carl Friedrich von Weizsäcker, and David Finkelstein with his 'spinorism'. But then, I come from a quantum background, so maybe that's to be expected.

Lawrence Crowell, in his essay, has taken the notion of epistemic horizons into a context of topological obstructions, which he seeks to use to explain quantum mechanics/entanglement. If I'm honest, I'm afraid his stuff is a bit too advanced for me, but you might have a better chance---perhaps there's something that could be helpful to you in there.

Regarding the question of collapse, I think to me, that's a (unavoidable) consequence of modeling---I make it more explicit in the Found. Phys. article, but the basic idea is that any function whatsoever can be written as a computable function augmented by an infinite reservoir of random bits---every set is Turing-reducible to a random set. So the best computable description of being faced with such a general world would look a lot like a compressible part with intermittent randomness---i. e. a lot like what we're seeing. But this has to be made more clear, I think.

Anyhow, I'm very enthusiastic about your project---it's not completely aligned with my own predilections, but I think it's the sort of innovative and creative thinking we need in fundamental physics right now. Neither of us can realistically hope of getting it all right, but perhaps it's not too audacious to hope for planting a few seeds that, in time, might yield fruit.

Cheers

Jochen

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Cristinel Stoica replied on Apr. 25, 2020 @ 18:35 GMT

Dear Jochen,

Indeed we can formulate classical mechanics on Hilbert space, as is done in the Koopmann-von Neumann representation. I think they can already be compared on the Hilbert space, as it was intended, but they also can in my representation. Maybe this will lead to some insights, nice idea!

About the relation of QM and GR, I agree there may indeed be some duality involved. The way I imagine the duality is that ultimately there is a structure which looks like something like QFT in some representation, and something like a quantum GR (maybe of the sort I described in the paper I attached to my first comment) in another representation, so something like [QT]~[real QT]=[real GR]~[classical GR] :). I am not very much impressed when I see differences in approaches, because there are many ways to say the same thing. Maybe we'll see someday what's the real deal though. Until then, I hope there are enough biases to cover sufficient research directions to find it ;) Or, as you said it well, seeds.

Cheers,

Cristi

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Dear Jochen,

Your essay is very thought provoking. It is also very well written.

Thank you!

All the best,

Noson

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Author Jochen Szangolies replied on Apr. 18, 2020 @ 07:08 GMT

Dear Noson,

thank you for your comment. I'm glad to get some endorsement from you---after all, some of the core ideas derive directly from your work. So I should really thank you!

Cheers

Jochen

Dear Jochen,

you have written a wonderful essay and thought provoking.I got a question and proposition to make.....

The Godel's law can be written as.........Godel proved that any consistent mathematical theory (formalized as an axiomatic deductive system in which proofs could in principle be carried out mechanically by a computer) that contains enough arithmetic is incomplete (in that arithmetic sentences ' exist for which neither ' nor its negation can be proved)...................

I have few questions about it. This law is applicable to Quantum Mechanics, but will this law be applicable to COSMOLOGY.......?????.........

I never encountered any such a problem in Dynamic Universe Model in the Last 40 years, all the the other conditions mentioned in that statement are applicable ok

I hope you will have CRITICAL examination of my essay... "A properly deciding, Computing and Predicting new theory’s Philosophy".....

Regarding proposition........

Why dont you make a new interpretation of quantum mechanics the 21st one, covering all aspects of it including intelligence, observations, experimental results, etc....I feel that with your knowledge you can definitely accomplish it

Best Wishes ....

=snp

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Author Jochen Szangolies replied on Apr. 23, 2020 @ 16:21 GMT

Dear Satyavarapu,

thank you for your kind comment. I have to admit, I haven't thought about Gödel's theorem, or similar ideas, in a cosmological context explicitly---although I suppose, in some sense, the Gödelian phenomenon can be viewed as a consequence of embeddedness, i. e. of being a subject of the same theory used to describe some object system. That is, if there's no separation between observer and observed, the observer becomes subject to their own observation---and cosmology is really the ultimate framework of embeddedness. So, one might not be surprised to find some connection.

That's not to say that these considerations will be of any import regarding theory building. There's sometimes an argument that Gödelian phenomena preclude a theory of everything---but I think that's misguided: think of something like a universe based on the Game of Life (in memoriam John Conway): you've got a perfectly simple theory of everything---just the update rules of the CA grid---but nevertheless, Gödelian considerations apply, as the system is capable of universal computation (and hence, 'contains enough arithmetic'). These manifest in general in the phenomenon of the question whether certain patterns ever arise being undecidable---so there will be unpredictable phenomena, but that doesn't mean that we can't discover the rules themselves.

As for interpretation, well, in a sense every reconstruction leads to an interpretation, as well---but really, I think the market's kinda saturated.

I will try whether I find something sensible to say about your essay. Best of luck in the contest!

Cheers

Jochen

Satyavarapu Naga Parameswara Gupta replied on May. 6, 2020 @ 10:35 GMT

Dear Jochen,

Yes probably you are correct, Godel theorem may not be applicable to Cosmology.

I hope you will have CRITICAL examination of my essay... "A properly deciding, Computing and Predicting new theory’s Philosophy"..... ASAP

Best Regards

=snp

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Dear Jochen,

I very much enjoyed your essay. I had decided not to comment on your essay because my opinion is that physicists project math structure on the world and then come to believe that physical reality has that structure. ‘Qubits’ are a fine approximation for spins on magnetic domains, but the Stern-Gerlach data on the famous post card shows anything but qubits. Only because Bell insisted on qubits (A,B = +1,-1) in his first equation did he arrive at his no-go theorem. If one uses 3D spin one obtains exactly the correct correlation, but this violates the projected structure. The spins of course have unit magnitude, but their 3D orientation determines their SG deflection that is the actual measurement. The measurements correlate perfectly. Of course, although Bell’s reasoning was based on Stern-Gerlach, all of the experiments have been done with photons. I have not worked out a comparable solution, because I do not fully understand the “exotic quirks like orbital angular momentum”. Regardless, I did not intend to come to your page simply to argue.

But I just saw your comment on Xerxes Arsiwalla’s essay expressing interest in his ‘distributional processing’ of subjective experience, and mentioning your 26 Mar 2020 publication in ‘Mind and Machines’. Congratulations! Do you have a copy that is not behind a paywall?

Anyway, in view of your expressed interest in a model or example of distributional processing, I wish to make you aware that, based on new info appearing 10 days ago, I have rewritten my essay to include such an example that you might find very interesting. I hope you will look at it and would welcome any comments (including argumentative!)

Deciding on the nature of time and space

Cheers,

Edwin Eugene Klingman

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Author Jochen Szangolies replied on May. 1, 2020 @ 09:39 GMT

Dear Edwin,

I'm glad you found something to like about my essay! There is, I think, a subtle point about how 'laws of nature' formulated by physicists track 'the real world'---or perhaps, don't. We're certainly never faced, in the real world, with the sort of idealized systems that feature in our models---no frictionless surfaces, no spherical cows, and yes, no qubits (Nancy Cartwright talks about 'how the laws of physics lie'). On the other hand, experimental data has been predicted to stunning levels of accuracy using these idealized systems, so it's hard to escape the conclusion that the models do get something right (as it's sometimes put, the success of science would otherwise amount to the miraculous).

But one can prove 'unsharp' versions of Bell's theorem, where one does not make any assumptions about the nature of the system upfront---where, indeed, nothing needs to be assumed other than that there exists a well-defined joint probability distribution for all experimentally accessible quantities. Likewise, one can take loss of systems, or mismeasurements, into account.

Thanks, also, for your interest in my paper in Minds and Machines. You can access the full text here. If you have any questions or comments, I would be happy to discuss them!

I will now go and see what your essay can teach me about distributional systems.

Cheers

Jochen

Edwin Eugene Klingman replied on May. 1, 2020 @ 17:04 GMT

Dear Jochen,

Thank you for reading my essay and commenting. I’ve responded there.

My point about Bell’s theorem, based on sharp or ‘unsharp’ versions, is that treating spin as classical yields the actual Stern_Gerlach data distribution seen on the postcard, and also reproduces the desired correlation. Imposing a qubit structure, while statistically appropriate for spins in magnetic domains, destroys locality. I find that too heavy a price to pay, and my opinion seem compatible with the many essays that question whether there is any necessary connection between our projected formal structures and the underlying reality (“whatever that might mean”). As Dascal quoted Curiel:

“just because the mathematical apparatus of a theory appears to admit particular mathematical manipulations does not eo ipso mean that those manipulations admit of physically significant interpretation.”

While Bell reasoned based on Stern-Gerlach, all of the experiments are based on photons. I am trying to grasp the OAM aspects of photons that have been reported for years and am unable to challenge the photon based experiments in the same way that I challenge SG experiments.

I almost decided to burden you with a longer rational supporting the above argument, but, unable to use graphics and equations here, I’ll spare you.

Thanks again, and good luck in this contest.

Warmest regards,

Edwin Eugene Klingman

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Dear Jochen,

I approach from a, perhaps, more intuitive level.

What got me interested in your essay was the idea that quantum mechanics and Godel's theorem might be shown to stem from one and the same first principle. Your rigorous conclusion is eventually my gut feeling hypothesis. And I'm longing for the most economical way to do justice.

My question is what phenomenology or ontology in nature might model such first principle you speak of? Is it Planck's black-body cavity or some cosmic system of waves? Or is there some actual Hilbert Space to be found in nature?

I ask this question because it seems to me that any self-referencing system (as must be Godel's set of all sets or Schrodinger's wave function) should basically model the observer as part of the same system it is aiming to observe/measure.

In short, how would your Epistemic Horizons model the observer proper vis-a-vis its observable(s)?

Chidi Idika (forum topic: 3531)

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Author Jochen Szangolies replied on May. 1, 2020 @ 09:25 GMT

Dear Chidi,

thanks for reading my essay, and taking the time to comment! I think your intuition is spot on, if I understand you correctly. The possibility of self-reference in quantum mechanics comes about due to the universality of the theory---due to the fact that it applies to every system. That's why singling out a specific sort of system as an 'observer', which itself isn't subject to...

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Dear Chidi,

thanks for reading my essay, and taking the time to comment! I think your intuition is spot on, if I understand you correctly. The possibility of self-reference in quantum mechanics comes about due to the universality of the theory---due to the fact that it applies to every system. That's why singling out a specific sort of system as an 'observer', which itself isn't subject to quantum rules (something like Wigner's 'consciousness causes collapse'-interpretation) is one way out of trouble---which is, however, difficult to square with our current understanding of the physical world.

In my model, the self-reference enters in a sort of oblique way. For one, you can write the state of a system by means of the outcomes of certain measurements---that's just like giving a list of properties of the system, like describing an electron as 'charge -1, spin +1/2, mass 511 keV', and so on. So in general, a state can be viewed as a list of measurement outcomes.

A measurement, on the other hand, can be defined by the set of all states for which it yields a particular outcome---in my setting, we only have dichotomic measurements (+/-1 valued), so you can simply just give a list of states for which it has the value +1, say.

So now, you can see where the circle closes: a state can be written in terms of measurements; each of these measurements can be written in terms of states; these states can again be written in terms of measurements; and so on. So, you can have a stat depending on a measurement depending on that very state---so that's where the self-reference comes in.

In a larger sense, I view the general Gödelian phenomena as a mismatch between the world 'out there' and a model of the world. The model, after all, is a feature of the world in any non-dualistic theory---it's something that's concretely there, some particular brain-state, or neuron-firing pattern, or what have you. So we have a model of the world, that's itself part of the world---you can view this like a detailed map of an island laid out on the ground; if it's sufficiently detailed, it will contain a copy of itself at that point where it's placed on the ground, which will itself have a copy within it, and so on.

In this sense, any model including a model of the modeling agent will be subject to phenomena of incompleteness in some sense---or else, succumb to circular inconsistency. I make this point somewhat more in detail in last year's entry, and in a slightly different way in my recent article in Minds and Machines.

I will take a look at your essay, perhaps I'll find something interesting to say.

Cheers

Jochen

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nice work very well done.can finiteness be borne out of infinity through anthropic bias ?kindly read/rate my essay https://fqxi.org/community/forum/topic/3525.all the best to you

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Author Jochen Szangolies replied on May. 1, 2020 @ 09:26 GMT

Dear Michael,

thanks for reading my essay, and taking the time to comment! I'll have a look at yours, and see whether I can find something interesting to say about it.

Cheers

Jochen

Dear Jochen,

please allow me a meta-comment on the opening statement of your essay:

"Almost from the inception of quantum mechanics, it has been clear that it does not merely represent a theory of new phenomena, but rather, an entirely novel way of theory-building."

Science is either a cultivated (refined) way of millennia-old human knowledge acquisition schemes or academic self-entertainment otherwise. Maybe the ENTIRELY NOVEL is the cause of the longest and deepest crisis in physics ever?

Heinz

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Dear Jochen,

finally I've found time to read your essay. Interesting food for thought! Very well-argued that we should reconstruct QM, instead of just "guessing the ontology" (i.e. interpreting it). You draw an interesting analogy between Goedel-type undecidability and the kind of "undecided" outcomes of quantum measurement, in the context of several quantum phenomena.

However, I do have some reservations. All that your diagonalization argument shows is: for any countably-infinite set (of "states"), there are uncountably many binary functions on it. Hence no single algorithm can compute ("predict") them all.

But this is completely true in any possible world -- classical or quantum or post-quantum. In other words, that simple observation cannot be enough to motivate quantumness.

Or what would you say?

Also, in a continuous context (like the continuous phase space that you describe), the naive definition of "any assignment of +1 or -1" will have to be supplemented by some topological or continuity arguments to say what it even means to compute a prediction, or what types of measurements are physically meaningful (not measuring along Cantor sets etc.). There is quite some literature in computer science and philosophy that deals with versions this.

In particular, let me ask you about Bohmian mechanics. This is a well-defined hidden-variable model of QM, and it is computable at least in the sense that people run simulations and compare the outcomes to experiments. (For example, see Valentini's work on equilibration.) I'm not endorsing Bohmian mechanics, but I wonder whether it is a counterargument to your claim. In some sense, there we *can* have a prediction algorithm for any possible measurement setting that we may be interested in...

Finally, are you familiar with Philipp Hoehn's work?

https://arxiv.org/abs/1511.01130

He derives QM in all mathematical detail from postulates of the kind that you mention. Including the two that you mention on page 1.

Best,

Markus

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Author Jochen Szangolies replied on May. 2, 2020 @ 09:15 GMT

Hmm, I have problems getting my comments to post. Initially, I got a 'post is waiting for moderation' or something like that, then I had apparently gotten logged out. I will wait for a while whether the comment appears, and if it doesn't, type a new one sometime later.

Author Jochen Szangolies replied on May. 2, 2020 @ 09:47 GMT

Dear Marcus,

I've decided to try again submitting my comment, as long as my reply is still fresh on my mind.

First of all, thank you for your comments, and criticism! I work on this topic largely in isolation, so it's good to have a little reality check now and then, to be kept on track, and not loose myself down blind alleys. Therefore, I hope to keep this discussion going, in some...

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Member Markus P Mueller replied on May. 4, 2020 @ 13:48 GMT

Dear Jochen,

thank you so much for your detailed and very illuminating answer! And thanks so much also for the links, in particular to Baez’ work — this is highly appreciated!

I feel like I would need to dive deeper into category theory to continue the proper discussion. Nonetheless, perhaps one follow-up question. If I understand the notion of “not being Cartesian closed” well enough, then this property also applies to a structure much more mundane than Hilbert space quantum mechanics: probability distributions.

In fact, since you point out the no-cloning theorem, there is a general no-cloning theorem for probability distributions (and more general probabilistic theories), see e.g. here:

https://arxiv.org/abs/quant-ph/0611295v1

In other words, you cannot copy (clone) probability distributions (trying to do so will introduce correlations, i.e. simply broadcast). Therefore, I would be surprised if the structural property that you point out was in some specific way part of the reason “why” we should expect quantum effects.

However, all that I’m writing here comes with the grain of salt that I don’t have much background in category theory.

Best,

Markus

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Author Jochen Szangolies replied on May. 5, 2020 @ 16:44 GMT

Dear Marcus,

good to hear I could add some clarification! I have to admit, I'm myself insufficiently familiar with category theory to really get into the thick of it---it's too vast a subject for me to really get a general overview over, without expending equally vast amounts of time.

Therefore, I'm not sure, offhand, how to answer your question. However, it seems to me that any assignment of classical probabilities must include the case of certainty---that is, of stipulating for each observable, whether a system possess it, or fails to. (As in the vertices of the CHSH-polytope.) In my framework, this assignment isn't possible, so it seems to me that we can't be left with 'just' classical probability distributions.

But I will have to think about this some more. I have wondered about the role of the classical no-cloning theorem, in particular in light of the Clifton-Bub-Halvorson theorem (I know this also includes a 'no bit commitment'-requirement, but in the end, that just tells you that at least some of the entangled states of the resulting algebra must be physically realizable, I think).

And in a sense, no-cloning is just the measurement problem: if you could clone, you could make a perfect measurement; and if you could make a perfect measurement of every state, you could clone. So if the no-cloning theorem in classical probability theory has the same significance, then why isn't there an equivalent measurement problem? (Or is there?)

Anyway, this seems an interesting line of thought to pursue, thanks for suggesting it!

Cheers

Jochen

Author Jochen Szangolies replied on May. 6, 2020 @ 20:12 GMT

Dear Markus,

thanks for reading my essay, and for commenting!

You are quite right in your observation that my argument, basically, is just equal to Cantor's regarding the fact that the powerset of a set necessarily has a greater cardinality than the set itself, and hence, that there can be no bijection between the two. This is a very familiar fact to us, today, but still, depending...

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Dear Jochen,

such a great essay. Fascinating to reconstruct quantum mechanics from "epistemic horizons".

There are a few points that escaped my understanding in your essay and I would like to use the chance of this blog to ask a few questions and make some remarks.

Classical physics worked pretty well for a few hundred years (and still does) for many phenomena. Also...

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Dear Jochen,

such a great essay. Fascinating to reconstruct quantum mechanics from "epistemic horizons".

There are a few points that escaped my understanding in your essay and I would like to use the chance of this blog to ask a few questions and make some remarks.

Classical physics worked pretty well for a few hundred years (and still does) for many phenomena. Also measurements can be described with classical physics. Quantum mechanics came in slowly in the attempts to explain the blackbody radiation and the discrete atomic spectra and other phenomena. None of these connected directly to limits of measurement or knowability. The point I want to make is: If classical physics/science is principally possible, where did the 'quantum' sneak in, in your argument? Such that the quantum would become necessary for epistemic reasons. I have not seen your two principles of section 1 in your prove by contradiction in section 2.

I sympathise with the aim to use an epistemic horizon for some arguments about the structure of laws or even reality (whatever this means). Specially because I belief that the vieew that things, properties and laws that are completely independent of the relations of the things with the rest is overly onesided. However you certainly know the quote from Einstein, when Heisenberg went to him and told him, that Einstein's theory taught them that only observable elements should enter the theory. Einstein replied that it was the other way around. It is the theory that tells us what can be observed. This means for me, that to use an epistemic horizon of what can be know, must at least be justified.

To advertise my essay: I came to a similar conclusion as you regarding the EPR experiment. You wrote: "Only given that one has actually measured xA is reasoning about the value of xB possible." In my essay I wrote on page 6: "But the very same experimental setup (EPR), shows that the setting of the reference frame far away from the object determines the possible, deﬁned propositions."

Luca

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Author Jochen Szangolies replied on May. 5, 2020 @ 16:57 GMT

Dear Luca,

thanks for your generous comment. You ask some good questions, and I'll do my best to do them justice.

First of all, the point where quantum physics enters into the picture isn't one of 'smallness', exactly, as it's often glossed, but it is one where the amount of information you have about a system nearly or completely exhausts the amount of information it's possible to...

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Dear Luca,

thanks for your generous comment. You ask some good questions, and I'll do my best to do them justice.

First of all, the point where quantum physics enters into the picture isn't one of 'smallness', exactly, as it's often glossed, but it is one where the amount of information you have about a system nearly or completely exhausts the amount of information it's possible to acquire about that system. In other words: if you know the state of a system only in its gross properties, you will not notice any quantum mechanical behavior; and in everyday experience, we only ever know a tiny amount about anything we engage with.

Think about the properties we typically know about a chair---approximate size, weight, shape---in comparison with the complete specification of each of its constituent atoms. The former will be perfectly sufficient for a classical description, yielding accurate predictions; only if we really had access to something approaching the full microstate would we come close to exhausting the information available about a chair, and thus, notice its quantum character. As this is generally only possible for systems with very few characteristics, and these tend to be submicroscopic, quantum theory tends to be glossed as a theory of 'the small'.

Hence, that there is an approximate classical description of macroscopic systems, as long as we don't know very much about them---as long as we don't exhaust the information available due to the 'finiteness' principle---is a consequence of the approach.

Your second point is more difficult to answer precisely, so I'll wave my hands around a little. In a sense, what I'm proposing is that the epistemic horizon is a metatheoretic principle: it's a boundary on what's possible to grasp of the world in terms of a model thereof. Hence, it's not quite the theory that tells us what we can know, but rather, the act of theorizing. This is a little Kantian in spirit: how we view the world is not just a flat consequence of the way the world is, but equally, a consequence of our viewing of it. (This is perhaps explained a bit better, from a different angle, in my contribution to last year's contest.)

Does this make sense to you?

The part you quote from your essay seems certainly not too far from my own views. I will have a look, and try and see whether I find something interesting to say; thanks for highlighting it.

Cheers

Jochen

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Luca Valeri replied on May. 6, 2020 @ 10:13 GMT

Hi Jochen,

Thanks for your reply. However I still don't get where your Finiteness and Extensibility principles enter your prove of the impossibility of Assumption 1.

Do you think your principles are connected to the impossibility of copying the whole information in quantum systems? So in your chair example, I was thinking, the information of weight, form, size etc. actually already exhausts all the properties, that make a chair a chair. What it makes classical is, that this information is available abundantly/redundantly, whereas this is not the case for quantum objects.

The reason for the the necessity of such epistemic restriction remains unclear. And might be not further justifiable than by the empirical evidence.

However the justification of why such epistemic consideration should have an effect on the ontology, cannot be in my opinion, by the way we (only can) view the world. This makes the whole picture a bit to anthropocentric. Don't you think? The objective quantum mechanical phenomena like super conduction, stability of atoms, etc. cannot be because of epistemic limits in the knowability the underlying world.

In my essay I probe the possibility, that the underlying objective 'reality' is emergent from relations between emergent objects themselves. Something like this could justify an epistemic impact on the underlying structure. I'll be curious on your comments on my essay.

Luca

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Author Jochen Szangolies replied on May. 7, 2020 @ 14:51 GMT

Dear Luca,

if I understand you correctly, I think you've gotten something a little mixed up---Finiteness and Extensibility are the starting points for the reconstruction of quantum mechanics in many recent attempts to justify the formalism from first principles (the way the invariance of the speed of light and the relativity principle are for special relativity). However, that just invites the question---but why are we limited in the amount of information we can obtain about a physical system?

That's the question I'm trying to answer---in other words, Finiteness and Extensibility are the output of my approach, they're what I'm arguing must hold, due to the application of Lawvere's theorem to the notion of measurement. That these principles hold is then equivalent to Assumption 1 being false---there isn't a function f(n,k) such that it yields a value for every state and measurement. There are some measurements on certain states such that it doesn't yield a value (Finiteness, although not quite---you need the argument from the Foundations paper based on Chaitin's theorem for that), and for these, we will learn new information upon measurement (Extensibility).

This doesn't really impinge on the objectivity of quantum phenomena, by the way. My proposal of a relative realism---which I don't really develop in the essay, admittedly---assigns values only to those measurements where f(n,k) yields a value, but that's a perfectly objective statement: in a given state, only those properties where the measurement outcomes can be predicted with certainty actually have definite values.

You can, of course, also interpret this as a subjectivist stance---i. e. claim that there's some real values out there, but our descriptions can't include them. But that's an additional interpretational commitment, nothing that's forced on us by my argument.

Cheers

Jochen

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Dear Jochen,

I finally had time to read your essay. I really appreciated the clarity of your arguments.

Your first example of the derivation of Heisenberg's principle from Finiteness and Extensibility is enlightening. Your introduction to superposition from diagonalisation is also very interesting. However, as you pointed out in your essay, "quantum mechanics, itself, does not fall...

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Dear Jochen,

I finally had time to read your essay. I really appreciated the clarity of your arguments.

Your first example of the derivation of Heisenberg's principle from Finiteness and Extensibility is enlightening. Your introduction to superposition from diagonalisation is also very interesting. However, as you pointed out in your essay, "quantum mechanics, itself, does not fall prey to the same issues", notably because of the no-cloning theorem and the fact that the diagonal + state yields a fixed point for the X gate. The fact that the superposition allows to avoid a logical contradiction reminds me of escaping the self-referential paradoxes by invoking a many-valued logic (e.g. trivalent), where '+' (or "indeterminate") would be another kind of truth value, in addition to 0 and 1. But what about complex states ? I don't know if you are familiar with it, I heard about a book entitled ‘Laws of Form’ by Spencer-Brown which presents a calculus dealing with self-reference without running into paradoxes, by introducing an imaginary Boolean algebra. Take the equation x=-1/x , which entails in some way a self-reference, a mimic of the Liar. If x=1 then it is equal to -1 and vice-versa, leading to a contradiction. The solution is to introduce an imaginary number, i , defined by i=-1/i.

Your reading of Bell’s theorem as revealing a counterfactual undecidability was enjoyable to read, as it is in line with my presentation of contextuality as a similar undecidability.

Another point : As you may have read as an epilogue in my essay, I am also interested in the Liar like structure that can emerge from “physical (hypothetical) loops” like CTCs. I am especially interested in quantum-based simulations of such CTCs, as Bennett and Loyd’s P-CTCs. In the literature, e.g. https://arxiv.org/abs/1511.05444, people have studied the relation between logical consistency and the existence of unique fixed-point. I was wondering if you had also some kind of epistemic reading of such loops, if you think that this is also related to Lamvere’s theorem.

Cheers,

Hippolyte

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Author Jochen Szangolies replied on May. 5, 2020 @ 17:11 GMT

Dear Hippolyte,

thanks for your comments! I think we've got a bit of a common direction in our thinking---the 'Laws of Form' has long held some intrigue for me, but I was never quite able to come to any definite conclusions about it. (Perhaps you know the work of Louis Kauffmann, who if I remember correctly has also proposed some interesting connection between the paradoxes of 'reentrant' forms, complex numbers, and the passage of time---perhaps in the paper on 'Imaginary Values in Mathematical Logic'.)

As for the introduction of an 'indeterminate' logical value, this alone probably won't solve Gödelian paradoxes---you can appeal to the 'strengthened liar', the sentence 'this sentence is false or meaningless', which is either true, or not; if it is true, then it must be false or meaningless, and if it is not true, then it is either false of meaningless, hence true. (That's why you also can't get out of trouble postulating 'null' results for measurements as a way out.) Superposition then can't be thought of as another truth value to be added, but rather, the absence of any definite truth value.

The connection between self-referential paradoxes and temporal paradoxes is an interesting one, but I haven't yet found much time (irony?) to spend on exploring it. In a sense, the two most discussed paradoxes---the grandfather paradox and the unproven theorem---bear a close connection to the self-negating Gödel sentence, and the self-affirming Henkin sentence: one eliminating the conditions of its own determinateness, the other creating them.

But as I said, beyond such generalities, I don't have much to offer. But I'll try and spent a little time thinking about this, if I come up with anything worthwhile, I'll make sure to let you know.

Cheers

Jochen

Dear Jochen,

Thank you so much for writing this essay! I enjoyed it very much. I especially enjoyed the idea of an epistemic horizon, and of a potential link between mathematical undecidability and physical unknowability, as we know it in quantum physics. As a very rough summary, would it be fair to say that you are applying the limitations obtained by self-reference and negation in a clever way in order to derive some of the epistemic limitations characteristic of quantum mechanics?

I was also wondering how your view point relates to the principle "universality everywhere"; in particular, to the quantum notions of universality in computation, spin models, etc (as mentioned in my essay). My impression was always that these quantum notions would suffer from the same limitations as their classical counterparts. Perhaps we are just applying the paradox of self-reference and negation in different ways?

Finally, I was wondering about your opinion of this work , in particular in relation to your work.

Thanks again, and best regards,

Gemma

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Author Jochen Szangolies replied on May. 5, 2020 @ 17:38 GMT

Dear Gemma,

I'm glad you found something worth your while in my essay! Your summary, I think, is accurate: in the general sense, there exists a bound on the information obtainable about any given system (which, in the full sense, requires the appeal to Chaitin's quantified incompleteness theorem given in the Foundations-paper), and once one has reached that limit, 'old' information must...

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Dear Gemma,

I'm glad you found something worth your while in my essay! Your summary, I think, is accurate: in the general sense, there exists a bound on the information obtainable about any given system (which, in the full sense, requires the appeal to Chaitin's quantified incompleteness theorem given in the Foundations-paper), and once one has reached that limit, 'old' information must become obsolete---as it does when we try to expand our horizon by walking (on the spherical Earth, or, well, any planet will do) west, losing sight of what lies to the east.

I'm not completely sure I get the gist of your question regarding quantum computation (etc.) right. Are you asking whether my proposal implies that quantum computers should be capable of beyond-Turing computation? If so, then the answer is kind of a 'it depends': any functional model of quantum computation will be able to compute only those functions that a classical Turing machine can compute. But still, if quantum mechanics is genuinely (algorithmically or Martin-Löf) random, then obviously, we can use quantum resources to do something no classical machine can do, namely, output a genuinely random number! Hence, as Feynman put it, "it is impossible to represent the results of

quantum mechanics with a classical universal device."

So in a sense, we need to be careful with our definitions, here---any way of implementing a finite, fixed procedure, whether classically or with quantum resources, will yield a device with no more power than a classical Turing machine (regarding the class of functions that can be computed, if not the complexity hierarchy). The reason for this is that simply outputting something random does not avail any kind of 'useful' hypercomputation, because one could always eliminate the randomness by taking a majority vote (provided the probability of being correct is greater than 1/2), and hence, do the same with a deterministic machine.

In a way, the story with quantum mechanics and computability is like that with non-signalling: it somehow seems like the classical constraint ought to be violated---but then, the quantum just stops short a hair's breadth of actually breaking through.

As for the Deutsch et al. paper, I can't really offer an exhaustive analysis. I'm somewhat sympathetic to the notion of 'empiricizing' mathematics (indeed, Chaitin has made the point that the undecidability results mean that at least to a certain degree, there are mathematical facts 'true for no reason'), but I think that notions such as those in Putnam's famous paper 'Is Logic Empirical?' go a step too far. In a way, different logics are like different Turing machines---you can do all the same thing using a computer based on a three-valued logic (the Russian 'Setun' being an example) as you can do using one based on the familiar binary logic, so in that sense, there can't be any empirical fact about which logic is 'better'.

But again, I'm not familiar enough with the paper to really offer a balanced view.

Cheers

Jochen

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Dear Jochen,

Very interesting. Great to hear from the young talented student like you. I would like to ask you the following point. According to the PBR theorem,

In conclusion, we have presented a no-go theorem, which - modulo assumptions - shows that models in which the quantum state is interpreted as mere information about an objective physical state of a system cannot reproduce the predictions of quantum theory. The result is in the same spirit as Bell's theorem, which states that no local theory can reproduce the predictions of quantum theory.

From your viewpoint, 'epistemic horizon', what do you deal with this theorem?

Best wishes,

Yutaka

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Author Jochen Szangolies replied on May. 5, 2020 @ 17:41 GMT

Dear Yutaka,

thanks for your comment! And I'll take both the 'young' and the 'talented' as compliments...

As for the PBR theorem, it essentially states that there is no more fundamental description than quantum theory that bears the same relation to it as classical mechanics does to quantum mechanics---i. e. quantum mechanics can't be the statistical version of some underlying, more definite theory.

In that sense, it's very well in line with my result---which essentially states that there is no more fundamental description than that given by the quantum formalism; this is, so to speak, all the description that's possible. Whether that means that's all there is, or whether, as in QBism, there is an epistemic element to this description, is something that, I think, is an open question as of now.

Does this make sense to you?

Cheers

Jochen

Yutaka Shikano replied on May. 18, 2020 @ 19:23 GMT

Dear Jochen,

Thank you so much for your clear answer.

This was made clear. This open question will be solved in your future research.

Best wishes,

Yutaka

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Hi Jochen,

Thanks for the really interesting essay! Your introduction to the effects of Finitness and Extensibility was very intuitive and how they may be used to understand Heisenberg's uncertainty principle. I think there could be a very close connection between the amount of energy required to extract information perfect information about the system---I'm thinking squeezed states---and the finitness principle. A perfectly localised measurement requires an infinite amount of energy over an infinitesimally short period of time.

I guess my question is, do you suppose that the epistemic horizon is a physical horizon? However, we might be wading into that age old debate of ontological vs epistemic interpretations of quantum physics!

In any case, it was a terrific essay and I rated it highly! I hope you get a chance to take a look at mine. We certainly have overlap in our ideas, although I fall down on the opposite side of you conclusions if you argue it a thermodynamic angle.

All the best!

Michael

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Author Jochen Szangolies replied on May. 14, 2020 @ 16:28 GMT

Dear Michael,

thanks for your comment! I'm glad you found my intuitive approach to deriving the Heisenberg uncertainty relation approachable. It can be made more rigorous---for a start in that direction, I refer to the Found. Phys.-paper---but I think this is a virtue of this particular approach: it makes an otherwise 'mysterious' phenomenon somewhat more easily palatable, by connecting it with statements that have a readily appreciable intuitive import.

Thinking about the energy needed to extract the information is an interesting direction. In some sense, there ought to be a relation there---thinking about this in terms of energy and time, rather than momentum and position. But I don't have a clear intuition there yet.

You're right to note that this sort of picture sort of straddles the epistemic/ontic divide. In a way, I'm not so sure it's good to think of these as rigidly distinct---certainly, in some sense at least, what we know about something is not something removed, off in some Cartesian realm of 'thinking stuff', from the physical world: the stuff in our brains is ultimately physical, itself. Hence, what we can know, and how we know it, is in the end also a question of what there is, i. e. what sort of stuff supports our knowledge. There's too much of the old 'detached observer' still lingering in this picture.

Regardless, if you're so inclined, I think it's perfectly well possible to interpret my proposal in epistemic as well as ontic ways. This is, to my way of thinking, an issue that only further argument will be able to settle. So, perhaps it's a topic for another contest!

Cheers, and thanks again for your kind words

Jochen

Dear Jochen,

I am curious about entanglement. What do you think of this assumption of mine:

When we break one rod, no matter how far we move them, we will later easily find that those two parts are the same rod. A possible analogy with electrons is: No matter how much the electrons are the same, they differ minimally in mass, say only for 10 ^ -60 part of the mass, but so that every two separated electrons form wholes with identical masses.

Regards,

Branko

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Author Jochen Szangolies replied on May. 14, 2020 @ 16:32 GMT

Dear Branko,

thanks for your comment. I think that your proposal for entanglement would run into trouble with established quantum mechanics, however: for one, quantum particles must be exactly identical---otherwise, quantum statistics would come out wrong, which would lead to easily detectable disagreement with experiment.

But the more important part is that your proposal is essentially a local hidden variable theory---particles have additional properties, not obvious to ordinary measurements, that are responsible for their correlations. But this is just the sort of thing Bell's theorem shows not to be possible---the argument in my essay tells you why: there would then be a probability distribution (whether we know it, or not) for the hidden variables to take on specific values; but this, alone, is enough for all Bell inequalities to be perfectly obeyed. Hence, violation of Bell inequalities tells you that such a picture won't work.

Does this make sense to you?

Cheers

Jochen

Dear Jochen,

I really enjoyed reading your essay.

I particularly like one of your conclusions: "The epistemic horizons the pure mathematician and the experimental physicist find delimiting their perspectives are not separated, but instead, derive from a common thread". I reach similar conclusions in my own essay (https://fqxi.org/community/forum/topic/3523).

One thing that I'm still thinking about after reading it, is whether finiteness and extensibility aren't really incompatible. You argue that they are not, based on the collapse of the wavefunction. However, even if the collapse erases old information, current information still takes space. Therefore, if one invokes finiteness, there is a limit to extensibility. Within the information-theoretic framework I presented in my essay, this follows naturally regardless of the 'overwriting' of obsolete information.

Maybe I'm missing something, so I should probably take a look at your other publications on the topic. Anyways, congratulations for your excellent essay.

All the best,

Rafael

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Author Jochen Szangolies replied on May. 14, 2020 @ 16:41 GMT

Dear Rafael,

thanks for having a look at my essay. I will try to find some time to read yours.

As for finiteness and extensibility, perhaps it helps to consider Spekkens' toy model, which demonstrates many of the features of quantum mechanics. The basic idea there is the 'knowledge balance principle': "the number of questions about the physical state of a system that are answered...

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Dear Rafael,

thanks for having a look at my essay. I will try to find some time to read yours.

As for finiteness and extensibility, perhaps it helps to consider Spekkens' toy model, which demonstrates many of the features of quantum mechanics. The basic idea there is the 'knowledge balance principle': "the number of questions about the physical state of a system that are answered must always be equal to the number that are unanswered in a state of maximal knowledge."

In other words, if your have one bit of information about the state of the system, you must also lack one bit of information that could be gained by additional measurement. But if you perform that measurement, then you'd have two bits of knowledge, and no further knowledge could be obtained; hence, to make things come out right with the knowledge balance principle, the state of the system must change so that the previous knowledge no longer applies.

It's similar with my model. You can think of this as trying to localize a system further within its state space: some amount of knowledge will allow you to perform this localization to a certain degree of precision. If only 'finiteness' were true, then well, that might just be it: you've localized the system as well as it's possible to localize it.

But 'extensibility' implies that you can obtain additional information. For instance, if the system (in state space) is localized to some degree along one axis, you can try to increase its localization along that axis by making a more highly fine-grained measurement; but to cope with the finiteness-requirement, its localization must consequently decrease along another axis. You can visualize this like squeezing a bubble, or a squishy ball: the volume (the total localization/total information you have) stays the same (equal to a power of Planck's constant), but the shape will deform, yielding information gain in one property, compensated by loss along another.

Does this make matters more clear?

Cheers,

Jochen

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Dear Jochen,

Very enjoyable. I think it might be my favourite so far. Ingenious in many ways.

I especially liked the linking between superposition and the liar paradox situations on p. 4. In the other direction, that is a very nice way to understand such self-referential statements: “This sentence is false” is indeed in a kind of superposition.

I agree with the conclusion concerning the “common thread behind mathematical undecidability and physical unknowability”, which is not the usual one mentioned (and frowned upon).

I also like the epistemic horizon concept very much. I note also that this allows you to connect in a more natural way uncertainty and Godel’s theorem - also often frowned upon as one of the many Godel-based overreaches. However, of course, in viewing QM in these terms you are seemingly committing yourself to an epistemic view of quantum uncertainty. Is that correct?

Also, Wheeler’s idea to use the undecidable propositions was not a quantum principle itself, but something deeper: it was to be the stuff of his pre-geometry explaining “why the quantum?” as well as “why spacetime” and “why existence?”. He didn’t stay with this idea very long… [I mention this because Wheeler uses the same term "quantum principle" in a different sense, so it might be worth disambiguating your constructive sense from his.]

Good luck!

Best

Dean

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Author Jochen Szangolies replied on May. 14, 2020 @ 17:00 GMT

Dear Dean,

thanks for your very kind words! I'm happy you got some enjoyment out of my essay.

As for the epistemic vs. ontic question, I'm kinda torn about that. I'm not sure the issue is best framed in these terms---after all, what we know, and can know, depends always on what there is---items of knowledge are, in whatever oblique a way, elements of reality, and even at least...

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Dear Dean,

thanks for your very kind words! I'm happy you got some enjoyment out of my essay.

As for the epistemic vs. ontic question, I'm kinda torn about that. I'm not sure the issue is best framed in these terms---after all, what we know, and can know, depends always on what there is---items of knowledge are, in whatever oblique a way, elements of reality, and even at least supervenient on physical matters of fact, provided one tends towards a materialist metaphysics. So what we call 'epistemic' in that sense is really just a particular way of looking at what's there.

In the other direction, we have of course no unvarnished access to what's out there in the world---whether it's behind the veils of Maya or lurking in Kantian noumena, we (re-)construct the world by means of the phenomena, which are all we can directly access. So what we can know depends on what there is; and what we consider there to be depends on how it is present to us. I'm not sure, then, there's really a hard-and-fast dividing line to be drawn---it may, perhaps, be rather a matter of method: we study the world in an epistemic or ontological way.

So maybe it should not come as too much of a surprise that nobody has yet managed to unscramble the quantum omelette---and perhaps, it's a mistake to try, because the world itself is a mixture of the objective and the subjective. So quantum theory, in my view, can tell us something about what is---for whatever that is, it's something that admits a quantum description at least in some sense---and about what we can know---what information we can hope to obtain about the world.

Hence, I'm not sure if whether the wave function is out there in the world, or in our heads, ultimately makes much of a difference; what's in our heads is in the world, too, after all.

I appreciate the disambiguation regarding Wheeler's 'quantum principle'---but I'd like a bit of elaboration (or perhaps, a pointer to the relevant literature). I don't really know about the connection Wheeler drew between undecidability and space-time (vaguely, in the back of my head, it seems I remember something about building space-time out of a logical calculus, or something along those lines), so I'd love to understand this better!

Cheers

Jochen

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Dear Jochen Szangolies!

Thank you for such an original essay! We agree with almost all of your arguments and assessments. We respect F. William Lawvere very much and refer to it in our essay. Thank you for the serious text. We rated the essay at ten points.

Truly yours,

Pavel Poluian and Dmitry Lichargin,

Siberian Federal University.

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Author Jochen Szangolies replied on May. 15, 2020 @ 17:03 GMT

Dear Pavel and Dmitry,

thanks for the kind comment! I have taken a look at your essay, and have left a few comments of my own.

I'm glad you've found something of interest in my arguments!

Cheers

Jochen

Dear Jochen,

I truly enjoyed reading your essay with its very original path backed by rigorous arguments.

I found it beautiful that you related the undecidability of those values to Bell's results. And that ultimately there there may be deep relationship between mathematical undecidability and physical unknowability Falls out elegantly if one takes the "reconstructing" program.

I also saw that you highlighted the existence of a joint probability distribution regarding the CHSH inequalities. This statement also reminds me of the crucial conditions behind the entropic Bell inequalities. I would be very curious to know how you will advance your program. In particular whether this will have implications to nonlocalities across time (Leggett–Garg inequality and its entropic versions).

Thank you for the wonderful essay and I wish you the best for the contest. I have give you a well deserved top vote!

Cheers,

Del

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Author Jochen Szangolies replied on May. 18, 2020 @ 15:15 GMT

Dear Del,

thank you for your kind words! I'm glad you liked my offering.

You're also pretty much on point with how I plan to advance my 'program'---a study of Kochen-Specker and Leggett-Garg inequalities will be on my agenda sometime soon. I think the precise conditions for co-measurability of observable need to be thrown into sharper relief---fellow contestant Hippolyte Dourdent, in his essay, has pointed out an analogy between non-simultaneously measurable observables and 'chains' of mutually co-referential, incoherent sets of propositions. I am curious whether I can understand this sort of thing from the perspective of Lawvere's theorem/diagonal arguments.

I've also been thinking about the Frauchiger-Renner 'extended Wigner's friend' in this connection. Let's see whether anything will come out of this!

Cheers, and thanks again for your kind comment,

Jochen

Dear Jochen,

Thanks for the thoughtful response to my assumption (speculation). Uncertain answers are common in this competition.

Does this make sense to me?

Yes, that makes sense. Although I don't have an opinion on that because I don't have enough input, so I wouldn't speculate further. What I have an opinion on, I expressed mathematically with formulas that give 100 times better results than the CODATA recommended values.

Regarding your essay, let me make a comparison with Gordian knot. No observers, no dimensions, no shapes - no problem. No Gordian knot, no problem. R=10 + exp( i * pi ).

Regards,

Branko

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Dear Jochen,

I have enjoyed reading your deep essay. Thank you for the insights.

From my side, I tend to agree with Einstein that qm is incomplete. But it turns out that to make it complete one has to also modify relativity and spacetime-structure. I explain this in my recent paper

Nature does not play dice at thePlanck scale

I will value your critique of these ideas.

Many thanks,

Tejinder

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Author Jochen Szangolies replied on May. 18, 2020 @ 15:24 GMT

Dear Tejinder,

thanks for reading my essay, and for your kind comment! Your paper looks fascinating, I will have to carve out some time to delve deeper into it. A 'geometrization' of quantum theory (albeit with some algebraic input, it seems) would certainly have been something of great interest to Einstein!

I wonder if you've seen the recent proposal deriving quantum mechanics from special relativity due to Dragan and Ekert: essentially, they take the (usually discarded) superluminal solutions to the defining equations of the Lorentz transformation, and show that keeping them leads to very quantum-like behavior.

I sometimes wonder: with such proposals to get the quantum from relativity, coupled with the proposals to get relativity from the quantum (as in the recent spacetime-from-entanglement program), perhaps we've been talking about the same thing all along! Maybe, in their own sense, both Bohr and Einstein had it right---after all, as the former is supposed to have said, the opposite of a deep truth may also be a deep truth.

Cheers

Jochen

Member Tejinder Pal Singh replied on May. 19, 2020 @ 05:20 GMT

Dear Jochen,

It is great that you finished at the top in community rankings! Wonderful All the best for the next round too :-)

I have seen the paper by Dragan and Ekert and like it a lot [I was one of their referees for the journal!] : It would be great if someone explored the connection of their idea with Adler's trace dynamics, and my approach.

Indeed I think both relativity and quantum theory have to give in, and be replaced by an underlying theory from which they both emerge. Yes, that would make both Bohr and Einstein right, in a way. I agree :-)

Best,

Tejinder

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Author Jochen Szangolies replied on May. 19, 2020 @ 16:03 GMT

Dear Tejinder,

thanks for the good wishes! Last I checked, I was in second place in the community voting, after Klaas Landsmann's essay---which I would've been more than happy with, it's definitely one of my favorites!---but it seems votes are up for review still, anyway.

I will definitely have to carve out some time to get more familiar with your approach, it seems intriguing. I have some vague familiarity with Adler's trace dynamics, because I was very interested in his quaternionic quantum mechanics at one time.

Cheers, and best of luck to you, too

Jochen

Hi Jochen,

My May 6 response to your question — “Your notions regarding---if I interpret you correctly---an inherent thermal 'noise' making the acquiring of perfect information about a system impossible remind me of Nelsonian stochastic mechanics. Is there a connection?” — was at best misleading.

Thermal randomness applies to random fluctuations in energy levels, as defined by Boltzmann’s partition function at a given temperature. In my dissipative dynamics conceptual model, thermal randomness is contextually defined at the system’s positive ambient temperature(*). The randomness of ground-state energy is “irreducible,” meaning its statistical description is complete and reflects perfect information. There are no hidden variables. So—I do not see any connection with Nelson’s stochastic mechanics. Further discussion can be found in my Medium essay, Reinventing Time.

Harrison

(*) Conventional interpretations (conceptual models) define thermal randomness either at absolute zero (deterministic mechanics) or at the system temperature (thermodynamics) in order to avoid contextuality.

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Hi Jochen,

A clear explanation of a complex topic to a broad audience. Very good.

If we view time as a function of entropy with entropy only having meaning for a collection of particles then time becomes the non-local.

All the best,

Jeff Schmitz

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Author Jochen Szangolies replied on May. 19, 2020 @ 16:08 GMT

Dear Jeffrey,

thanks for your kind comment.

Your point about entropy and time resonates with something like Julian Barbour's approach---in a way, that the past is, well, the past is due to the fact that only higher-entropy states can 'remember' lower-entropy states, so if we view time just as a collection of moments, the ordering emerges just from the way records include that which they record.

Time is then 'non-local' in the sense that we start out with a certain foliation (a sequence of 'nows'), which however still can yield the dynamics of general relativity (as Barbour shows with his 'shape dynamics').

But this is basically just free association.

Cheers

Jochen

Jeffrey Michael Schmitz replied on May. 22, 2020 @ 21:50 GMT

Jochen,

Entropy is meaningless for a single particle. Entropy only exists for a collection of particles, the larger the better (to a point). A sound wave is another example of a collective mode. If time is not fundamental, but a function of entropy then time only has a clear meaning for a collection of particles making it non-local.

All the best,

Jeff

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Dear Jochen,

This is a very sophisticated and impressive work. I had not heard of Lawvere, now I see I should have. I reference related issues of quantum mechanics to the correlations you address, in particular the inability of realistic "signaling" models to explain the strong correlations of entanglement. (https://fqxi.org/community/forum/topic/3548)

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Author Jochen Szangolies replied on May. 19, 2020 @ 16:11 GMT

Dear Neil,

Lawvere's work is very well known in the category theory community, but perhaps not so much beyond it---perhaps less than it ought to be, given its breadth and depth.

I'm sorry to have missed your essay during the voting, it sounds very intriguing. Still, I will try and find some time to give it a read.

Cheers

Jochen