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

(Response to March 25th post) That's right, just looking at the very nature of measuring a single observable is very fundamental to undecidability. My comparison mentioned above is focused on complementary observables. In that analysis the phenomenological experience of nature involved in measuring one observable would have as its underpinning some self-referential operation of the other observable - this is manifested physically in the specific measuring action (such as direction of the magnetic field, the way the detector interacts, etc.). A bit subtle is the linking between the two observables - the SR operation relates to one observable in the act of measuring the other.

Regarding a single observable, consider that every observable has a complementary conceptual observable out there. So the |+> and |-> oscillating superposition in some way perhaps corresponds to another observable with an already decided value. If a measurement specific to this 'another observable' were made, it would not affect the already oscillating superposition of |+> and |->. Then, physically doing what is necessary to measure the|+> and |-> system (i.e. to determine if in the excited or unexcited state) would effectively be a SR operation on that 'another observable.' The undecidability of |+> and |-> is now swapped for undecidability in this 'another' domain. This is the same reasoning applied to pairs of observables already known to be complementary, like Sx and Sz. This again shows not everything can be decided together.

Now, one might say this was still a physical undecidability; if so then by our new measurement we physically transferred the undecidability so to speak. But what did we really do? We changed the nature of our measuring environment, that is we sought to explain the system differently by conceptually distinguishing a different observable. Remember that changing a mathematical representation requires a change in physical explanation. "Change in physical explanation" is more vague than "change a mathematical representation" so more is needed to state this conversely: By distinguishing out a different observable, we're changing how we physically explain a system. If we seek to explain by distinguishing an observable that can't be described in a particular mathematical basis, then we must represent by a different basis. But what if we just try to change the representation to begin with? This can be done symbolically. If we have a measurement in Sx (let's say it's UP) and now we wish to represent that using a basis of (X up, X down), well we can't - that's mathematically undecidable as you brought out above. For example, (X up + X down)/в€љ2 does not give you a decided measurement of Sx. It was an attempt to represent it with respect to itself, and is thus self-referential. This mathematical self-referential operation is undecidable, but can be used to describe a decided value of a new concept Sz whose value would be UP. When physically manifested this changes the paradigm of explanation - it destroys any measured value of Sx and instead produces a new decided observable Sz.

That not everything can be decided together squashes Hilbert's attempts as we noted above. (Maybe the Halting Program can be reformulated with the basis B and states T(n) as the inputs). That not everything can be explained together reflects complementarity. That each of these leads to each other shows the equivalence of mathematical incompleteness to physical uncertainty.

(Response to March 26th posting) Thanks Lawrence this is a very interesting article. Yes perhaps the exact nature of where the self-referential operation manifests physically could be considered at this contact point. I definitely want to explore that more. I tend to not use the term "classical world" per se but I see how the article uses it, and that contact point is really where the environment provides just enough decoherence to explain it through emergent properties, like the point of einselection as you described earlier. In this case it appears the environment is associated with the attenuator and the number of bi photon pairs. It's very fascinating, and I want to check out the specific physical mechanism and see how self-referencing may be interpreted. Thank you :)

Thanks for enjoying my essay. The last section "Several Concluding Insights for Further Research" is laid out such that each successive point explores and overlaps with a wider range of research. Hopefully I can continue to use the comment threads to expand upon those four points, and I'll use this response as part of that.

Point #1 derives a significant conclusion of how a self referential operation applied twice does indeed lead to a definite state. This provides confidence to the physical explanation of one half pulse bringing the qubit into superposition and then another half pulse bringing it back to a definite state; furthermore this gives additional insight not only into the self referential representation for such physical phenomena, but into the physical requirements necessary to understand a self referential operation. Point #1 is thus a direct extension from the previous section which introduces the self referential operation and lays out its significance in seeking the limits of computability. There are some excellent questions raised above expanding on this point which are also part of my research.

Point #2 is already brought out more in the comments above. See in particular the dialogue I have with Lawrence Crowell. This develops into an equivalence between (Godel's) Incompleteness and (Heisenberg's) Uncertainty - in its own right this interfaces math and physics.

Point #3 might be a fun exercise if anything to consider, especially if more rigorously analyzed. It also may tap into how deep the self referential concept lies in the most fundamental understanding of the universe, because a self referential operation so applied 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.

Point #4 aptly then is perhaps the most exploratory but is relevant to consider. I received feedback from others on it and I see there are other essays throughout FQXI forums that touch on consciousness. The key postulate in this point ties consciousness to causality, which is an interesting explanation given that causality is a key requirement for computation. The idea opens the door for discussion on the connection between nonlocality and a deeper understanding of superposition, as well as a thorough explanation of the balancing between nonlocality and statistics in order to maintain causality. I think it then 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.

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Most importantly, it seems that others as well have been interested in these points and helped develop them throughout the threads, so I'm glad to see these ideas pursued and utilized. Including here, I hope to continue in that dialogue.

You mentioned your other comment on the Liar Paradox as separate, so I will address it in another response.

Traveling extensively, but will try to answer several of the comments here by Monday morning. Lots of good ideas and questions, thank you and good luck!

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

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