• 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

Dear 'En Passant,'

Regarding your comment on the Liar Paradox, when you mentioned regarding some commenters that "their writing appears to indicate..." which commenters in particular do you mean? If you have a question on a commenter, please ask him/her directly in the same thread as their comment, as it may just be your misunderstanding.

It is very necessary indeed that the Liar Paradox - and more generally a self referential scenario - remains undecided. This allows it to be represented by a superposition of states and physically manifested by examples such as given in my essay, most prominently the qubit. That's a clear point in my essay and the comments above help flush out related concepts; I simply don't see anyone suggesting the Liar Paradox does not remain undecided. So the misconception may be yours (and the only idea that needs to be dispelled is your false assumption of those commenters and your imposed interpretation of their comments onto me without first confirming - hopefully I'm dispelling that now).

What is so profound is that this very undecidability goes from being a limitation to instead an expansion of computability, when the mathematical and physical requirements of undecidability are mutually considered.

Thank you for acknowledging that the self referential operator applied twice may help with the Halting problem - THAT is a major goal of my essay and one that has significant repercussions for computation in general, and one that saliently addresses this forum topic. Whether it takes a second programming statement per se is fine. After all, the physical phenomena indeed requires a second pulse to take the qubit out of superposition. It would be a more subtlely engineered piece of code requiring quantum computing systems (i.e. ones that allow superposed states, qubits) on which to run, and I'd be interested in how you might approach it. Perhaps taking it back to a more linguistic formulation can help. (Also, you're right in that the twice applied SF operator may be considered a nested statement. In the context of my essay, applying the statement again linguistically would require expanding out the full explanation of the statement, in order to work out the logic to completion).

As a side point, "resolving" could even mean "dealing with it" depending on the context. It's important when commenting to try to stick to specific mathematical or physical ideas and not get caught up so much in others' elations or expressions of praise. A specific question then would be "does this suggest the Liar Paradox can be decided?" The answer is it remains undecided. In fact, we're all counting on that.

Steve

...I just realized my last point above might have come out as a pun (and an intriguing one at that). I simply meant: the answer is that the Liar Paradox remains undecided. And that's a useful outcome.

Thanks again, Steve

Hi Jon,

Thanks so much for your very thoughtful questions and contributions. You brought up a lot of topics, so I may need a couple responses to address them all:

1) In the Penrose-Hameroff model of Orchestrated Objective Reduction, consciousness involves brain activities that may utilize quantum computation, but nevertheless which still tie to some higher fundamental reality. Rather than try to fully explain consciousness, my point focuses more on the relationship between consciousness, experience, and causality. Tying consciousness to causality is an interesting explanation given that causality is a key requirement for computation. In The Emperor's New Mind, Penrose discusses some very interesting scenarios relating to causality. Although the Penrose interpretation tends to conflict with the multiverse explanation, nevertheless the connection between nonlocality and superposition may offer some mutual understanding, and the fact that some experiments have been proposed to test some of these explanations (for example FELIX and the table-top optical cavity variation) is refreshing. Perhaps the Platonic approach in Penrose-Hameroff may spring from simply a humble recognition that since we utilize our consciousness to examine these concepts, we would never be able to fully explain consciouness itself - this then takes on a form of Godel incompleteness. Pertinently then the self referential concept plays a role, and this relates to self-awareness. Of course to properly delve into this, one would need to have at least a partial, workable definition of consciousness and what it means to be self-aware (and the correlation exactly between the two). To be aware of oneself perhaps could be casted or modeled as a conscious role outside oneself. Thus, to be conscious of oneself perhaps requires consciousness to be perceived as if from some other state. Thus consciousness applied to itself would be not itself. If this could be represented by some form of superposition, any corresponding physical manifestation would require the statistical nature of quantum mechanics to maintain causal order, which as postulated is needed to maintain the consciousness. Again, this would require some working definitions which overlap with other philosophical concepts. (Thanks for the video link and that's interesting about the zen approach to non-self - in some ways that description addresses the undecidability about self). I want to keep it more technical but the point I'll leave on regarding the consciousness/causality connection is that it could provide some description of the environment necessary for consciousness along the lines of an anthropic principle - namely that if consciousness is a product of causality, or at least requires it, then a physical universe in which conscious beings such as ourselves exist would be the one in which physical principles can at least be perceived by us to maintain causality.

2) Yes, I see how we both discuss the restrictions on how a Turing machine can process a tape, and your question of whether this constraint is physical or informational is the heart of the Holographic Principle as I briefly pointed out. My reference from Bousso is a great resource for that and actually asserts a specific ratio, namely that the area of any surface limits the information content of adjacent spacetime regions, at 1.4Г--1069 bits per square meter. A universal relation between geometry and information is educed that links the number of fundamental degrees of freedom involved in a uniп¬Ѓed description of spacetime and matter. It's quite fascinating.

I'll continue my response to the rest of your questions in another comment.

Thanks,

Steve

Hi Jon,

You asked an excellent question following through on the half pulse manifestation of the self-referential operator. Let me first clear up some of your questions on the technical infrastructure in the Rubidium setup:

A laser whose output power is constant over time is called a continuous wave laser and this is what was used in the setup. This is opposed to pulsed operation in which the optical power appears in pulses of some duration at some repetition rate (such as in a Q-switching or mode-locking laser). For continuous wave operation it's required for the population inversion of the gain medium to be continually replenished by a steady pump source. Even a laser whose output is normally continuous can be intentionally turned on and off at some rate in order to create pulses of light (the modulation rate is on time scales much slower than the cavity lifetime and the time period over which energy can be stored in the lasing medium or pumping mechanism, and it's still classified as a modulated or pulsed yet nevertheless continuous wave laser) - it is this type of pulse that's being shined on the Rubidium atom in the setup. Pulse of laser light thus refers to a duration of time in which the laser is being shined on the atom, necessary to excite the electron to the excited state. The half pulse refers to shining the laser light for half this duration of time on the atom.

Now, one half pulse (i.e. shining this continuous wave laser on the Rubidium atom for this half-duration time) causes the electron to go into superposition of the ground and excited states. Another half pulse excites it to the definitive excited state. This corresponds to two self referential operators, but that's the key: whatever the physical mechanism is, it is one that manifests the self referential operator. In representing a qubit, the most general state of a quantum two-level system can be written in the form ∣ψ〉 = α ∣0〉 + β∣1〉 where α and β are complex numbers. The state has to be normalized, so ∣α∣^2 + ∣β∣^2 = 1, and an overall phase makes no difference, so either α or β can be chosen to be real. This leads to the parametrization ∣ψ 〉 = cos (θ/2)∣0〉+ (e^iφ) sin (θ/2)∣1〉 in terms of only two real numbers θ and φ, with ranges 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π. These are the same as the polar angles in 3-dimensional spherical coordinates, and this leads to the representation of the state as a point on a unit sphere called the Bloch sphere. The Bloch sphere can be traveled with pulses of different lengths and the system driven from the superposition state to the ground or excited states. But this can be done only during a certain amount of time which is called the coherence time. It can vary from several microseconds to several seconds depending on the particular system. When this period is over the system has to be initialized again in the ground state. If the phase of the driving field is fixed, the system will be going in one direction, but if phase is changed by π, the rotation axis in the Bloch representation is changed, and in that case the system might rotate in the reverse direction (for example around -x axis instead of the x-axis). One can also change the phase by π/2 which would result in a mirror image of the state. For example if a π/sqrt(2) pulse is shined and then the axis is changed from x to y, the coefficients of the two states in the superposition are reversed. So the physical manifestation for a self referential operator may not be the same half pulse but rather may change - but this makes sense: there are different physical paths as we see to get back to the ground state; hence there are different physical manifestations for the NOT operator. It then follows there would be different manifestations for the SQR (NOT) and thus for a self referential operator at that particular point. There's lots of research being done on these variations (and part of my own personal research) and I hope this can be shifted to the engineering of self-referential operator gates which would be an extremely fascinating field.

Hi Jon,

You asked an excellent question following through on the half pulse manifestation of the self-referential operator. Let me first clear up some of your questions on the technical infrastructure in the Rubidium setup:

A laser whose output power is constant over time is called a continuous wave laser and this is what was used in the setup. This is opposed to pulsed operation in which the optical power appears in pulses of some duration at some repetition rate (such as in a Q-switching or mode-locking laser). For continuous wave operation it's required for the population inversion of the gain medium to be continually replenished by a steady pump source. Even a laser whose output is normally continuous can be intentionally turned on and off at some rate in order to create pulses of light (the modulation rate is on time scales much slower than the cavity lifetime and the time period over which energy can be stored in the lasing medium or pumping mechanism, and it's still classified as a modulated or pulsed yet nevertheless continuous wave laser) - it is this type of pulse that's being shined on the Rubidium atom in the setup. Pulse of laser light thus refers to a duration of time in which the laser is being shined on the atom, necessary to excite the electron to the excited state. The half pulse refers to shining the laser light for half this duration of time on the atom.

Now, one half pulse (i.e. shining this continuous wave laser on the Rubidium atom for this half-duration time) causes the electron to go into superposition of the ground and excited states. Another half pulse excites it to the definitive excited state. This corresponds to two self referential operators, but that's the key: whatever the physical mechanism is, it is one that manifests the self referential operator. In representing a qubit, the most general state of a quantum two-level system can be written in the form ∣ψ〉 = α ∣0〉 + β∣1〉 where α and β are complex numbers. The state has to be normalized, so ∣α∣^2 + ∣β∣^2 = 1, and an overall phase makes no difference, so either α or β can be chosen to be real. This leads to the parametrization ∣ψ 〉 = cos (θ/2)∣0〉+ (e^iφ) sin (θ/2)∣1〉 in terms of only two real numbers θ and φ, with ranges 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π. These are the same as the polar angles in 3-dimensional spherical coordinates, and this leads to the representation of the state as a point on a unit sphere called the Bloch sphere. The Bloch sphere can be traveled with pulses of different lengths and the system driven from the superposition state to the ground or excited states. But this can be done only during a certain amount of time which is called the coherence time. It can vary from several microseconds to several seconds depending on the particular system. When this period is over the system has to be initialized again in the ground state. If the phase of the driving field is fixed, the system will be going in one direction, but if phase is changed by π, the rotation axis in the Bloch representation is changed, and in that case the system might rotate in the reverse direction (for example around the -x axis instead of the x-axis). One can also change the phase by π/2 which would result in a mirror image of the state. For example if a π/sqrt(2) pulse is shined and then the axis is changed from x to y, the coefficients of the two states in the superposition are reversed. So the physical manifestation for a self referential operator may not be the same half pulse but rather may change - but this makes sense: there are different physical paths as we see to get back to the ground state; hence there are different physical manifestations for the NOT operator. It then follows there would be different manifestations for the SQR (NOT) and thus for a self referential operator at that particular point. There's lots of research being done on these variations (and part of my own personal research) and I hope this can be shifted to the engineering of self-referential operator gates which would be an extremely fascinating field.

Thanks again,

Steve Sax

Dear Stephen,

Thanks for your precipitately brilliant essay.Your allusions on the imperative of a dichotomy free maths-physics relations occasioned by "computation" is commendable.It is even much more pleasurable as it obliquely concurs with the reasoning behind my essay,which I hope you will take a look at.

Keep the flag flying.

Lloyd Tamarapreye Okoko.

I apologize if this appears somewhere more than once.

I could swear I posted this answer somewhere already. I hope it wasn't my bank account, because they are already confused enough.

Dear Steve,

Now that it is clearly stated that self-referential statements remain undecided, there are no further issues.

It might be that I misunderstood a commenter's intended meaning, and perhaps there wasn't anything to clear up at all.

Referring to your "After all, the physical phenomena indeed requires a second pulse to take the qubit out of superposition. It would be a more subtly engineered piece of code requiring quantum computing systems (i.e. ones that allow superposed states, qubits) on which to run, and I'd be interested in how you might approach it."

Firstly, I will qualify the following by saying that I am not a computer programmer, and I know even less about QC. But the logical principle would have to work the way I will describe.

The "general" part of the program will have to only deal with the output coming from that part of the program (however many lines it may be) that causes the whole thing to be self-referential. The self-referential part will have to do its processing in "isolation", and whatever it comes up with will be represented in a statement about it, and that statement will be what can be processed by the "general" part of the program. I will give you an example from MS Excel, since that is what I am familiar with. If you wish, you could type the following into MS Excel, and look at the underlying VBA code (which can be viewed within Excel) to see how the actual programming handles this, but I am not sure about the applicability of that to QC.

Say you have a column (list) of numbers, and a sum of the numbers in that column is at the bottom of the column. But one member of the list has to be calculated on the total of that list (say, as a percentage of 5% the total of the list), including itself. For this example, assume that the list of numbers being added occupies the range from A1.A10, and the sum of the "above" is located in cell A11 (and let's say the member of the list that is causing "self-referential trouble" is in cell A6). What you would do is the following. Within cell A6, you would type a summation formula: =(SUM(A1:A5)+SUM(A7:A10))*0.05+(SUM(A1:A5)+SUM(A7:A10))*0.05*0.05+(SUM(A1:A5)+SUM(A7:A10))*0.05*0.05*0.05+(SUM(A1:A5)+SUM(A7:A10))*0.05*0.05*0.05*0.05

At this point its precision is only 4 decimal points, but you could extend it to any length to obtain any amount of precision (beyond what we can practically handle). You might call it "iterations," or, expansions. I have to disagree with you (or at least alter the language), that you are "processing" the self-referential statement twice (or more). You are processing the statement about the self-referential statement several times, each based on the preceding statement.

Don't bother defending yourself. Nobody cares if what you said is perfect. Your idea is right.

However, the "trick" will be in how to apply this to the "mechanical" example you describe in your essay. That will take a gifted programmer, and a gifted experimenter. I am neither one of those.

En

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