Essay Abstract

What if the paradoxical nature of quantum theory could find its source in some undecidability analog to that of Gödel's incompleteness theorem ? This essay aims at arguing for such Gödelian hunch already suggested by Szangolies via two case studies. Firstly, using a narrative based on the Newcomb problem, the theological motivational origin of quantum contextuality is introduced in order to show how this result might be related to a Liar-like undecidability. A topological generalization of contextuality by Abramsky et al. in which the logical structure of quantum contextuality is compared with ``Liar cycles'' is also presented. Secondly, the measurement problem is analyzed as emerging from a logical error. A personal analysis of the related Wigner's friend thought experiment and and a recent paradox by Frauchiger and Renner is presented, by introducing the notion of ``meta-contextuality'' as a Liar-like feature underlying the neo-Copenhagen interpretations of quantum theory. Finally, this quantum Gödelian hunch opens a discussion of the paradoxical nature of quantum physics and the emergence of time itself from self-contradiction.

Author Bio

I am a 2nd year Phd student, studying quantum foundations. During my master studies, I worked on quantum contextuality, supervised by Alexei Grinbaum in CEA Paris-Saclay ; and I did my master thesis on the superpositions of quantum causal orders, supervised by Cyril Branciard in Institut Neel (Grenoble). My thesis, supervised by Cyril Branciard, aims at clarifying theoretically a conceptual connexion between quantum contextuality and quantum causality.

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Dear Hippolyte Dourdent,

Very nice essay on discussing Godels Hunch, you wonderfully discussed good history to disprove Godel. After seeing all these i got a small question, whether this hunch is applicable to Cosmology also?

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

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

Best Regards

=snp

    Dear Hippolyte,

    you've produced a fantastic and highly readable essay. The linkage of contextuality graphs to inconsistent cycles of propositions seems, in hindsight, almost obvious---but requires a great creative leap to realize. (Perhaps, to make this more vivid, you could've chosen a 'Penrose pentagon', the structure of which should essentially yield the KCBS-incompatibility graph. But I think the point is clear either way.)

    Likewise, your introduction of levels of meta-description, paralleling the introduction of meta-languages in mathematical logic, is highly illuminating---ultimately, as 'paradox' in mathematical logic enters through the ability to formulate meta-language predicates within the object-language (for sufficiently expressive systems), it enters into physics with the application of the same theory to object and observer ('meta-object'). And ultimately, neither is, of course, paradoxical in the true sense: the Gödel sentence, unlike the liar, does not talk of truth, but of provability within some formal system---it is only through the misguided attempt of identifying this notion with truth that something genuinely paradoxical emerges. Likewise, quantum mechanics only appears to be paradoxical upon the---similarly misguided---attempt to assign truth values to all propositions, regardless of context.

    My original idea for this contest was to try and apply my own ideas to the Frauchiger-Renner paradox, but I felt this would get to unwieldy. Hence, I'm more than happy to see it given such a capable discussion from your perspective. I hope this essay will do well in the contest!

    Cheers

    Jochen

      Dear Hippolyte Dourdent,

      I realized that Landsman seems to be interested in his Landsmann Brouwer.

      Just another hint concerning contextuality:

      My essay tries to show that Fourier was wrong when he claimed that complex FT is exactly as comprehensive as nature. I rather see it introducing an unnecessary redundancy that seems to play a key role in QM too.

      Eckard Blumschein

        Dear Satyavarapu Naga Parameswara Gupta,

        Thank you for your interest !

        At first sight, I don't really know indeed how the Gödelian hunch might be found in Cosmology. If the Universe is treated as the largest system that an observer can studied (what the cosmologists do if I am not mistaken) , and not as "everything" included the observer itself, then no self-referential issue occurs. Maybe, on a very speculative note, the hunch would rather be found in the study of astrophysical objects at the frontier of relativity and quantum theory, e.g. black-holes or CTCs.

        I will definitely have a look at your essay, thank you.

        Best,

        Hippolyte

        Dear Eckard Blumschein,

        Thank you for your interest. I will take a look at your essay too !

        Best,

        Hippolyte Dourdent

        Dear Jochen,

        Thank you very much for your comment.

        I came accross this idea that self-referential problems might be at the core of quantum weirdness while I was studying contextuality with Alexei Grinbaum in 2017. I was delighted to read your paper on ``Epistemic Horizons'' the next year, which was formulating so clearly my intuition and added a lot of fuel to it as well. I learned a lot from it, and as you may have guessed it was my principal source of inspiration for this essay.

        Indeed a "Penrose pentagon" might have been enlightening to do the parrallel with the KCBS inequality !

        I deliberately avoided to read your essay until now because I was afraid that I might get too influenced by it while I was writing my own essay. But now that it has been submitted, I am really going to enjoy the reading !

        Best,

        Hippolyte

        The question is whether different cuts can be studied systematically (mathematically). Which one are possible, which aren't?

          I agree that a simple "postulated" cut might left hungry for more. This analysis is a justification for a Gödelian hunch, and is more of a first step of a program rather than a definite claim.

          Such mathematical study of cuts would nevertheless need to be done at a meta-theoretical level, otherwise it would lead to the return of logical inconsistencies.

          Dear Hippolyte,

          You exhibit mastery of the topic and argue it well. You conclude:

          "Quantum theory does not only defy common sense, but it also defies classical logic, i.e. our common language and semantic. (...) But is nature itself paradoxical?"

          I believe that physicists project math structure onto the world, and then come to believe that the physical world actually has that structure. Your Penrose triangle is a perfect example. The 2D structure projected on paper is not a 3D reality.

          Similarly, qubits are reasonable structure for spins in magnetic domains, but do not match the Stern-Gerlach measurement data shown on the famous postcard, of single spins in an inhomogeneous field. Under the influence of "quantum universality" (that is, absolute commitment to the projected structure) Bell forced qubit structure (A, B = +1,-1) onto SG, and derived his theorem. In actuality, 3D spins produce exactly the data shown on the postcard and the deflection measurements yield exactly the correlation that Bell claims is impossible.

          This is ontological error based on absolute belief in projection, which works well statistically in some cases and leads to entanglement in others.

          I put this heresy out for you to consider, some day, years from now, when things still don't make sense.

          I invite you to read my essay which focuses on ontology.

          My best regards,

          Edwin Eugene Klingman

            Dear Hippolyte Dourdent,

            To me, arguments are more important than votes. Don't hunch. Well, you have a hunch that what you learned is always true. Maybe, Alexei Grinbaum himself may challenge my unwelcome reasoning?

            Best hope,

            Eckard Blumschein

            Dear Eckard Blumschein,

            I think you misunderstood my brief reply.

            "To me, arguments are more important than votes." I completely agree !

            I was (and still am) going to have look at your essay in order to give you a more constructive answer, whether it is on my forum or yours. I just did not find the time yet.

            "Well, you have a hunch that what you learned is always true. Maybe, Alexei Grinbaum himself may challenge my unwelcome reasoning?"

            I am sorry that you have this hunch, and I am afraid that this is pretty far from the truth. This is my essay and not Alexei's. Again, I will glad to give you an answer whenener I found the time to give a constructive comment.

            Best,

            Hippolyte

            Dear Edwin,

            Thank you for your comment.

            "I believe that physicists project math structure onto the world, and then come to believe that the physical world actually has that structure. Your Penrose triangle is a perfect example. The 2D structure projected on paper is not a 3D reality."

            Indeed, I believe this might be another way to go. I prefer another metaphysical approach, that would rather see this projection as epistemological (projection of the meta-theoretical on the theoretical).

            I guess your conclusions on Bell are not so far from my Gödelian hunch, but instead of analyzing this in the relationship between meta-theoretical (observer, measurement) and theoretical (quantum systems), you prefer to have a more realistic / ontological approach, and thus be sceptical towards the projection (which is equivalent to dropping the universality assumption ?).

            I will have you a look at your essay as soon as I can,

            Best,

            Hippolyte

            Dear Hippolyte,

            sorry for taking so long to respond. I'm afraid I've somewhat overstretched my time budget in starting so many threads of correspondence in this essay contest.

            One thing I've been thinking about, which I think needs some more thinking about (?), is how one could formulate these self-referential 'chains' of observables within my framework. Basically, I construct an 'inconsistent' observable by means of Lawvere's theorem---something not too dissimilar from Russell's set that contains itself iff it does not contain itself. It would be interesting to see whether one could extend this to yield something like the 'liar-cycles' which have no consistent assignment of truth values. Perhaps one could 'daisy-chain' the Lawvere argument.

            Another question, it seems, might be whether there's an analogue to something like Yablo's infinite set of paradoxical sentences, too. This concerns sentences of the form:

            (S_n): For each i > n, S_i is not true

            Assuming S_n to be true, we get some later S_k, k > n, such that it is both true and not true; but then, assuming that each S_i is not true, yields the conclusion that S_n must be true, because that S_i for i > n is not true is exactly what it asserts. Hence, we obtain a contradiction.

            This is an 'indirect' sort of self-reference, in that each sentence does not refer to itself, either directly or via a circle of intervening propositions, but rather, to the whole set of sentences, with a contradiction arising from that. I'm not sure, however, how one would go about finding an analogue of this in terms of observables.

            Anyway, that's still sorta open-ended speculation on my part. I'd be very interested to hear your thoughts on my essay!

            Cheers

            Jochen

            Dear Hippolyte,

            Hopefully you understood my reply. I would just like to add that I don't argue against idealization but against taking ideals for reality.

            And I am sure any reply by a Greenbaum was not more convincing to me than your one.

            Best,

            Eckard

            Dear Jochen,

            No problem, I myself struggle to find the time to read and discuss other essays.

            Concerning building a "Liar cycle" within your framework, I guess an idea would be to try to retrieve a proof of the Kochen-Specker theorem (e.g. Mermin-Peres square or the KCBS inequality) using Lawvere's theorem as you did with Bell inequality ? The major difference being that in general, these proofs (but not KCBS) are state-independent.

            Concerning the Yablo paradox. I also happened to ask myself the same question. Indeed, the Yablo paradox is a kind of indirect Liar, and I also wondered if one could find some proof of quantum contextuality that would share a similar structure. Unfortunately, so far, I also have no idea how such referential structure could be formulated with observables.

            I have finally found the time to read your essay, I'll leave a comment on your section !

            Best,

            Hippolyte

            Dear Hippolyte,

            thank you for a well-written essay, that I enjoyed very much reading.

            It was a pleasant surprize to find that your main arguments come from the (semantic approach to) the quantum measurement problem analyzed through "Wigner's Friend Paradox". In fact, this is my main research topic, despite my essay focuses on something else. I think that your analysis based on objectes, meta-objects and meta-meta-objects is promising. If anything, although I intuitively understand how these problem realte, I would have liked a more explicit comparison between the quantum paradoxes you reassess and the logical paradox of the liar.

            Anyways, congratulation on a great essay and I wish you good luck for the contest!

            Flavio

            Hi Hippolyte,

            Thank you for the very beautifully written essay. The introduction section discussing the topology of contextuality, the tension relating between local and global observations combined with Penrose Triangle was really very intuitive! Also the similarities between Hardy's paradox and the Liar's paradox were very clearly articulated.

            The Wigner's friend paradox and freindification has always sat a little uneasily with me, although that is the experimental physicist in me coming out. Your conclusion ``Quantum paradoxes are not physical, but emerge from a lack of metaphysical distancing'' is certainly on the right track to explain these theoretical paradoxes.

            Again, thanks for the great essay. It was really well written and I really enjoyed it! I hope you have enough time to take a look at my essay too.

            Thanks,

            Michael

              Hi Michael,

              Thank you very much for your nice comment.

              I will try to read your essay as soon as I can !

              All the best,

              Hippolyte

              Dear Hippolyte,

              I enjoyed a number of things in your essay. Indeed, collaborators and I are presently (finally) constructing a response to the Frauchiger-Renner issue, and there is a small section of our paper that concerns a question of self-reference, which we dismiss summarily. Perhaps though there is a more solid argument for what we want to assert, and perhaps your paper, even if not directly addressing our concern, will inspire an idea in us. I encourage you to continue these lines of thought. "Although it can describe anything, a quantum description cannot include everything." - Asher Peres. Hear hear!

              You correctly characterize QBism as having a "movable cut [which] is functional and not ontological." Jacques Pienaar has recently made a very nice contribution toward mathematizing the movability of the cut from a QBist perspective. In case it is of interest to you, it can be found here:

              https://arxiv.org/pdf/2004.14847.pdf

              I don't know if any of that will be expressible in your framework, but it might be worth thinking on.

              All the best,

              Chris Fuchs