• Trick or Truth Essay Contest (2015)

  • The Physical Limitations on Mathematical Abstraction, the Representational Effect of Mathematics on Physical Explanation, and the Resulting Expansion of Computability by Steven P Sax

Thanks Michel for your very insightful comments and references. It's amazing how the self-referential operator can take on many different forms. I want to just clarify a couple important distinctions that I sense you were aiming to bringing out: The macroscale "classical" coin flip CF that randomizes the input is indeed idempotent, as any number N of CF operations in sequence are equivalent to a single one: CF^N = CF like you exemplified. But it is not a self-referential operation per se. Idempotence is related to referential transparency with regard to computation in that an expression can be replaced by it's value without changing the behavior of the program and this might be what you meant. But the self-referential operation (SF) results in undecidability; in the representation of qubits this would take the system to a superposition of states. SF^2 = NOT, and would not be equivalent to the original SF operator. In this sense the QCF (quantum coin flip) of Hayes would indeed be a form of the self-referential operator - it's a fascinating distinction. That's very interesting how you combined the Hadamard and Pauli Z gates - I want to make sure we're using the same terminology on the shift gate but I very much like your ingenuity to engineer these gate combinations; it's really quite grand how this can be built up. Just to review for posterity: the Hadamard operation is similar to a SF (and thus SQR(NOT) ) operation in that both take the system to superposition, but a second Hadamard operation returns the system to the original state while a second SF as we know takes it to NOT of the original state. For example,

H(0>) = superposition 0> 1>, H.H(0>) = 0>.

H(1>) = superposition 0> - 1>, H.H(1>) = 1>.

SF(0>) = superposition 0> 1>, SF.SF(0>) = 1>.

SF(1>) = superposition 1> - 0>, SF.SF(1>) = -0>.

That's absolutely stimulating Michel that you connected these concepts to the time entanglement from my other essay and thanks for engaging that. Time entanglement is quite profound and your connection here of that to the CNOT gate in view the SF operator, undecidability, and incompleteness as explained in my current essay is an excellent observation - thanks very much.

I very much enjoyed your comments, and look forward to reading and rating your essay very soon.

Steve Sax

Dear Steven,

I am glad that you found my reading of your excellent essay useful. Our discussion shows how much a 'correct' interpretation of what is going on in a physical experiment depends on the 'correct' maths. I am enthusiastic in your view that Goedel's incompleteness is (at least partially) related to the classical language and that the QM language is helpful on that matter, and similarly for the issue of self-reference.

As a clever physicist, I am sure you are also sensitive to the ongoing work about rubidium and the CNOT gate where entanglement between ligth and atoms has been established, e.g.

http://www.cos.gatech.edu/news/Researchers-Report-First-Entanglement-between-Light-and-an-Optical-Atomic-Coherence

I am also happy that you were not frightened by my (may be too ambitious) topic and I thank you for your high mark. I already rated your essay highly at the time I studied it.

My best regards,

Michel

Steven,

Thanks for your kind comments and for flagging up your essay. I found myself in agreement with most all, and more importantly found further new aspects. I have a comprehensive chapter in draft on the matters of predicate logic which you may be interested in previewing. Another chapter goes deeper into the issues with Cartesian systems and motion, suggesting non point particles and Pauli (etc) exclusion confound understanding. More to the point I hope you may read my (joint) recent (related) quasi classical reproduction of QM's prediction, causal but not entirely deterministic. The false assumptions used leading the the logical failure are identified (without socks).

Quasi-classical Entanglement, Superposition and Bell Inequalities.

I do hope you'll read and comment as you may be one of the few who'll even understand it ('knowing QM' is a modest encumbrance).

But this is about your essay; applause and full marks, if only for; "never PROVE.." (I live in a recurring nightmare being told the contrary!) in line with Dyson "there are no 'facts' in physics'. But not only for that.

Thanks and congratulations

Peter

Hey Steve,

Thanks for the thorough response.

1)All that consciousness and causality talk made my head hurt:) The only thing that really sunk in was how you tied it into the anthropic principle at the end. Although your statement that "conscious of oneself perhaps requires consciousness to be perceived as if from some other state. Thus consciousness applied to itself would be not itself," reminded me of a Cantor diagonalization over the denumerable, computable reals to construct a non-computable real...in a computable kind of way. Diagonalization has always bothered me. In my opinion, it presupposes non-computability into existence. I guess this relates to my essay.

2) I am going to check out that Bousso reference! Thanks!

3)Regarding your second reply: Thanks for the explanation on what a "half pulse" is. I think I understood a significant amount of your explanation (and the mathematics that goes along with it), but since I have never actually seen the experiment, I still feel a little lost. One last question... A "digital physics"/deterministic question: How do you know that the Rubidium electron is in a superposition state with the half pulse? How do you know that it isn't either definitively in the excited state, or the ground state? Is there something analogous to an interference pattern that is observable?

Thanks again for your thoughts!

Jon

Dear Steve,

Very enjoyable and interesting essay. I like that you explained and illustrated the dance between math and physics, and the result of this. I wish you do well in the contest!

Best regards,

Cristi

Dear Steve,

I found your essay very interesting, especially your treatment of self-reference. Being partial myself to the Mathematical Universe Hypothesis and the multiverse it implies, I found quite intriguing your idea of using the MUH to devise mathematically a universe that wouldn't allow for the physical explanation of an infinite multiverse! If the MUH is true, the most difficult challenge is to explain why our universe is so lawful and stable, and I like your take that only mathematical structures allowing an infinite multiverse explanation would "live out"...

I agree with you that self-reference is of great importance, because ultimately, if we are to devise an explanation of reality that is self-contained, it will have to invoke self-reference one way or the other.

All the best,

Marc

    Dear Marc,

    Thank you for your kind words about my essay and for appreciating the very special role of self-reference. Indeed the representation of self-reference and the mechanics of self-referential operators in particular, especially as applied to the physical modeling of undecidability, is a major focus of the essay - and hopefully a springboard for much further research. It's amazing how what appeared to effect a limitation is in fact an expansion when the requirements of math and physics are mutually considered with respect to a self-referential operator.

    And thanks for noticing the importance of the third concluding point regarding MUH. As I pointed out earlier in the threads, a self referential operation so applied (i.e. utilizing MUH to devise mathematically a universe that wouldn't allow for the physical explanation of an infinite multiverse) suggests a superposition of "MUH compliant" and "not MUH compliant" , but MUH already should incorporate the concept of superposition. If superposition must be invoked even within an explanation that supposedly already incorporates superposition, then this would have to be explained by a theory in which superposition is yet but an emergent concept.

    I've begun your essay which is already very interesting and enjoyable. I look forward to finishing and rating it later today, at which point I will gladly write more on your page.

    Thanks again,

    Steve Sax

    Dear Steve,

    thanks for the comments and for reading my essay. As you correctly saw, I'm not a fan of Platon and his idea about the world of ideas (independent of us).

    I see math as part of humanity and of our brain. Aliens will also use math but (because of their other abilities) in another fashion.

    I had other the chance to read your essay and rated it high.

    You took agreat circle to explain your point of view: the problem of self-referentials, causality as main part of a computation and you dismissed the infinite universe of Tegmark.

    Points about which we can agree.

    Good luck for the contest

    Torsten

      Dear Steven,

      Congratulations on a brilliant essay. It is well-argued and well-written. I share your opinion about computations, as you can see in my essay .

      Best regards and good luck in the contest.

      Mohammed

        Thanks again Torsten, and as I elaborated on your page, your essay has many great points about which we would agree - including our discussions of mathematical abstraction and the relationship to physical explanation - and a lot of our ideas would support each other.

        One point I just want to clarify: my essay did not dismiss the infinite universe per se, much as it did reign it in a little. Utilizing MUH to devise mathematically a universe that wouldn't allow for the physical explanation of an infinite multiverse really explores moreso the fundamentality of superposition and the role of the self-referential operator, as I explained further in the thread just above this.

        Best regards, and thanks again,

        Steve Sax

        Thank you Mohammed for your kind words about my essay, and I'm glad you enjoyed it. I look forward to soon reading yours as well.

        Best regards,

        Steve Sax

        Dear Steven,

        Thanks for the nice essay, it was a very enjoyable read. I am wondering if you think that self-referential systems can be instantiated in a manner other than as a quantum superposition. Many have suggested that self-reference is a defining feature, for example, of living systems (something I explore in my essay). My impression that the distinction might be whether you require the self-reference to be manifest at an instant in time or whether it is distributed through time - in particular in biology it shows up in the dynamics, but perhaps does not make so much sense when you look at the system in any given instant. Have you thought on this at all?

        Best,

        Sara

          Dear Steve,

          I'm glad I made it to your essay on time. As there are just a few minutes to the closing and I'm not sure I can still comment after, I just wanted to let you know that I found your essay to be a very enjoyable read, full of great ideas and you have my vote. I hope to be able to return later today and elaborate on a few ideas.

          Warmest regards,

          Alma

            Dear Steve,

            I appreciated your writing because it made me aware of a fact that I somehow ignored, or perhaps forgot. Your examples illustrate how the physics itself gained more ground due to changes in the mathematical structure of the theory. This idea seems to clash with my own, that physical insight was the light that guided the way to progress, but the more I think of it, the more it is obvious that our ideas sit in a neat superposition, because the physical insight itself had to originate somewhere! Noticing the inconsistencies, intuiting their meaning and then solidifying the theory was the key for Einstein's photoelectric effect. This is the true meaning of what people have in mind that Einstein approached the problems like a wise man seeking reconciliation between what the experiments were showing and Newton's theory of light and, later, gravity. I enjoyed how you have placed together Holography, Halting and Incompleteness to work your way to the striking statement that "it is only when the state is considered with respect to itself that it is no longer itself". This principle at the heart of quantum mechanics may very well be the key to adding physical meaning and understanding how self-measurement makes space arise from entanglement. I think this is a very forceful point and it deserves further study. I also enjoyed your easy proof on the Halting Problem. Its simplicity greatly benefits understanding because it's removing all but the logic skeleton of the construction.

            The real gem is however hidden in the last two sentences of the paper. "The [...] nature of quantum mechanics is needed to balance the nonlocality brought on by a self referencing operation, to maintain causality. Perhaps this could be the basis of self awareness?". I removed the word "statistical" on purpose because I feared it may distract from the point. I consider it shocking because it intuitively makes sense and self referencing indeed is the key to self awareness. As far as I am aware, not even the smallest hint of any other sensible explanation for self awareness exists to this date; also this theoretical model does not require an advanced neurological theory to pair with(!). I urge you to continue to think of this idea and develop it further if possible.

            My best wishes,

            Alma

            a month later

            Hi Sara,

            Thanks for your very fascinating questions. That's an excellent distinction regarding requiring the self-reference to be manifest at a given instant versus allowing it to dynamically distribute through time. Since a system can be analysed from the perspective of other observables besides time, perhaps a way to generalize this distinction then is to consider whether the self-reference applies to a specific state, or instead manifests as an emergent phenomena throughout a distribution of states and their paths. The former scenario can be more active; that is a self referential 'operator' can be applied to a given state, and this physically can be accomplished by a specific physical event like the half pulse of laser light shining on the rubidium atom. Looking at it that way - from instantiating self reference at a given state - quantum superposition lends itself as a compelling description because it can explain efffects that we observe. The latter scenario (emerging through a distribution of states and paths) seems more passive, but need not necessarily be the case. In a path dependent multi-state system like that described in your paper, perhaps the particular state path (which may be nonlinear, as your paper suggests) in which a self reference emerges can be looked at as one macrostate M. The key would be what defines the self reference for such a macrostate, and what physical phenomena led to it. Did it evolve and if so is there any way to predict or assign probability to it happening? Could it be 'pushed' or 'tweeked' to emerge that way from an added physical event? That macrostate M applied to itself would be a NOT function of itself, but there would be many possibilities in a path dependent system that could satisfy NOT [M]. It may be that if such a self reference were interpreted to yield a quantum superposition, or a system of superpositions of the particular microstates involved, then it would have to be so complex that the observable effect yielded could perhaps be inferred as many discernible effects. This could be difficult if they in some way interfere with each other or are not fully addressed depending on the measuring environment. Even more revealing is if together they could indeed be perceived as one phenomena or concept. I'm thinking out loud at this point and would need to consider this much more, but needless to say your question has been very inspiring! It's an intriguing suggestion that self reference is a defining feature of living systems. I approached it from the idea of consciousness and its connection to causality, and this is a good example where the specific consideration of time (i.e. self reference manifest at an instant in time versus distributed in time) would especially apply. The analysis in this discussion here adds yet another dimension; perhaps there are environments in which the complex macrostate superposition mechanism described would yield phenomena that yet triggers other self referencing. If so, a modifed defining feature of life could be not just self reference, but a framework in which self reference is constantly or iteratively being invoked in a coordinated manner, (for example such as by these complex mechanisms).

            Thanks again Sara for some very thought provoking questions; these are just some of my thoughts on it, and I definitely want to explore this distinction of instant versus emergent self reference much more!

            Kindly, and best regards,

            Steve Sax

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