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 the two main fundamental issues for physics, mathematics and cognition in general, which Carlo Rovelli writes about in Physics Needs Philosophy / Philosophy Needs Physics : "What is space?", "What is time?" And today, the time is very worrying for all of Humanity, and we must learn to find consensus on all issues. Especially on the basics of knowledge. Here, just Philosophy, "mother of all sciences" comes to the rescue .. Recall Hegel: "The owl of Minerva begins its flight only at dusk"...

"... 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 believe that the search for the "fundamental principle" is necessary not only for quantum mechanics, but for knowledge in general.

In an interview with mathematician and mathematical physicist Ludwig Faddeev ( in the journal "EXPERT" (2007), entitled "The equation of the evil spirit" it is written: «Academician Ludwig Faddeev believes that today mathematical rigor is more important than physical intuition and it is thanks to mathematics that a "unified theory of everything" will be built.

The long-standing debate of scientists about what is more important - mathematical rigor or physical meaning, a correctly solved equation or an intuitive understanding of a natural phenomenon, continued throughout the 20th century, but at some time physicists seemed to win in it: Einstein as the creator of a special and general theory of relativity is better known to the average man than Poincare or Hilbert, Schrödinger is more popular than Weil, and Landau is more popular than Bogolyubov. But in recent decades, the situation began to change: it turned out that successful mathematical techniques have not just technical significance, but deep physical meaning. Mathematical intuition in solving increasingly complex physical problems may be more important than physical. And this caused a noticeable irritation of many great physicists. In the second half of the 20th century, a new generation of scientists appeared who could no longer be called pure physicists or mathematicians. Ludwig Faddeev is one of them. After graduating from the Physics Department of Leningrad University, he gained worldwide fame as a man who, together with his student Viktor Popov, solved the most complicated mathematical problems of the Yang - Mills theory, which later formed the basis of the theory of superstrings. The effects that were discovered were called "Faddeev-Popov spirits" and under this name entered all modern textbooks of theoretical physics. Faddeev is convinced that just as physics solved all the theoretical problems of chemistry, thereby "closing" chemistry, so mathematics will create a "unified theory of everything" and "close" physics. Faddeev is convinced that just as physics solved all the theoretical problems of chemistry, thereby "closing" chemistry, so mathematics will create a "unified theory of everything" and "close" physics. "

But can mathematics, the "language of Nature" "close biology? ... Big doubts ... Big questions .. Everywhere is the problem of the ontological basis of knowledge. I believe that there will be Pavel Florensky: "We repeat: worldunderstanding is spaceunderstanding."

I am very concerned that the crisis of understanding in the fundamentals of knowledge has spread to global society ... Let's begin this brainstorming session ... I score the highest rating for your constructiveness and the ideas of the "epistemic horizons". Please look at my ontological ideas and give critical comments .. .

With kind regards, Vladimir

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

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,

CristiAttachment #1: post-determined-block-universe.pdf

    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

    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

    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

    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

    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 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

    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

    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.

    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

      Dear Jochen

      Thank you very much for your answers, openness and focus on the big discussion. (brainstorm). Today there are many articles, books, conferences, but for some reason there are few brainstorming sessions. FQXi contests give this opportunity to all participants .

      Sincerely, Vladimir

      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

      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

      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

      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

        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)

          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