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FQXi FORUM
October 15, 2019

CATEGORY: Trick or Truth Essay Contest (2015) [back]
TOPIC: The Physical Limitations on Mathematical Abstraction, the Representational Effect of Mathematics on Physical Explanation, and the Resulting Expansion of Computability by Steven P Sax [refresh]
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Author Steven P Sax wrote on Mar. 19, 2015 @ 15:59 GMT
Essay Abstract

The relationship between mathematical representation and physical explanation is discussed. The limitations physics places on mathematical abstraction is exemplified, as well as the effect that changing mathematical representation has on physical understanding. Computation is utilized conceptually to further illustrate these connections. The limits of computation are potentially expanded regarding the role of a self referential state, and physical insights are gleaned from this.

Author Bio

I'm a physicist at the U.S. Department of Commerce. I received my Graduate and Bachelors Degrees in Applied Physics from Cornell University. I have been a visiting lecturer at University of Maryland, and was an Adjunct Professor at the New Jersey Institute of Technology.

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Thomas Howard Ray wrote on Mar. 19, 2015 @ 20:05 GMT
Wow! All the way down the line -- from special relativity and metric spaces, to the foundations of Planck's constant, to the liar's paradox, 3-valued logic and computability -- I think we agree in full. This is an essay I am certain to read several times, and the first vote I'll cast.

If you don't mind revealing your email address, I have something else I'd like to share privately. thomasray1209@comcast.net

All best,

Tom

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Author Steven P Sax replied on Mar. 20, 2015 @ 22:13 GMT
Thank you Tom for your kind words, and I'm enjoying your essay very much as well (which I also plan on reading again in depth when I get back to town). Fascinating take on Bell, and your explanation of Popper is very enlightening. Looking forward to corresponding with you further. Steve

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Author Steven P Sax replied on Mar. 20, 2015 @ 22:25 GMT
Also congratulations on your being published by Springer in Conflict and Complexity!

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Thomas Howard Ray replied on Mar. 21, 2015 @ 13:49 GMT
(reposted from my essay forum)

Thanks, Steve -- as I posted in your forum, we are in accord on many things, and the foundations of computability is, I think, the most important issue in frontier science.

Beyond the scope of the essay question, the growing fields of brain science and artificial intelligence depend strongly on resolving the issues of network robustness and integrity -- i.e., the amount of information that can be effectively used at each decision point such that positive feedback doesn't overpower the computing function.

It's a key point -- the number (1) in your concluding remarks, that twice applying the self-referential operation generates a true statement. It's the identical point I was making with the Popper example of pairwise correlations followed by a single result that may or may not be correlated with the pairwise value. Length restrictions kept me from exploring the basis of Popper's program -- which is Richard von Mises's theory of the independence of collectives -- Popper notes (p. 196) in *Realism and the Aim of Science*:

"von Mises's 'axiom' (which postulates the existence of a limit of the relative frequency of the occurrence of a property P in any probabilistic sequence of events or 'collective') may be written as a universal-existential-universal-existential-universal statement, of the following form: '*For every* probabilistic sequence, *there exists* a real number x between 0 and 1, called the limit of the relative frequency, such that *for every* given fraction y, however small, for which y > 0 holds, *there exists* a natural number n, such that *for every* natural number n (for which n > m holds) the relative frequency of m/n, of m occurrences of the property P up to the nth event of the sequence does not deviate from x by more than y, that is to say, - y =/< x - (m/n) =/< + y."

In network terms, adding a time parameter, this implies that information lost to one decision node is not lost to the network hub at which it originated, such that continuously shifting hubs of decision activity are self-organized in the same context that you take to be self-referential.

As I think it is pertinent to the content of both of our essays, if you don't mind, I am going to repost this in toto in your forum.

Thanks again and all best,

Tom

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George Gantz wrote on Mar. 19, 2015 @ 23:09 GMT
Steven - A brilliant essay, thank you. Remarkably, it seems designed to answer some questions Marc Seguin and I were just asking about in the essay that appears just prior to yours in the competition. That is so curious...

I was speculating, in response to Mark's essay, on the fact that you can respond to a Godellian truth - the ones that cannot be proved - by adding it as an axiom and getting to a higher level mathematical theory. Or you could add it's negation as an axiom, which would then split the Mathematical universe in two, just as the physical universe gets split in two when a superposition collapses. So at the fifth postulate in geometry, mathematics splits into Euclidean or Reimmanian space.

But Godel's problem is tied to the self-referential nature of mathematics, and as you point out, that seems to resolve if you do the self-referencing twice... Beautiful!

In your conclusion #2, you note that physical reality can not be entirely reduced as "there will never be one measurement paradigm that can be used to explain everything else."

Would you agree as well that mathematics can never be reduced as there is never one formulation that will ever explain everything - no universal set of axioms (finite or infinite) that can produce all the answers. And,of course, if you open the door to the MUH with the ultimate "no axioms" (infinite degrees of freedom) you also get no answers.

Your conclusion #4 introduces consciousness, causality, non locality and self-awareness. That's a lot to cover in one paragraph and I'm not sure I follow your leaps. But would you agree that these concepts will share in the indeterminacy you describe for math and physics?

That would suggest - no TOE?

I would be delighted if you have a chance to read my essay "The Hole at the Center of Creation."

Many thanks! - George Gantz

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Author Steven P Sax replied on Mar. 20, 2015 @ 23:20 GMT
Thanks George, and your discussion on Godel is very fascinating! A key factor regarding Euclid vs. Riemann as a mathematical representation of space, is the physical explanation one attempts for physical observation (including the assumed explanation of how those observations are measured, for example), when using such a representation. If the fifth postulate forces the sum of angles of a triangle to be 180, and then one measures them to be 200, (and your measuring device has previously given results consistent with all your other physical experience to the point you're confident in it) - then something has to change. When the physical explanation is compelling enough, it forces us to abandon the postulated assumptions of the mathematical representation. For example, the sum of the angles of a triangle on a sphere is 180°(1 + 4f), where f is the fraction of the sphere's surface that is enclosed by the triangle. That understanding based on new intuitive observation around a sphere shows is not compatible with a Euclidean representation, based on other assumptions we already maintain. All that being said, in MUH it may be possible to have such split physical universes based on the particular restrictions of the additional axioms. It's an interesting exercise to consider how that would play out.

I'm traveling now, but hope to address your other two points this weekend.

Thanks again, Steve

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Author Steven P Sax replied on Mar. 23, 2015 @ 08:14 GMT
George, thanks again for your stimulating questions. Regarding your second one, please see the discussion further down the thread initiated by Lawrence Crowell. Our discussion brought out the point that just like there is not one measurement paradigm that could be used to explain everything (every observable, interaction, etc.), so there is not one mathematical basis in which everything is decidable.

These two restrictions go hand in hand, and from the interfaced perspective of computation they in fact would be equivalent. Both can be related to Gödel incompleteness: a) the basis change and corresponding superposition representing a new undecidability resulting from a new self-referential operation, and b) a new environmental interaction setup physically corresponding to a self-referential action changing the measuring paradigm.

This, especially in view of your question, ties back to Hilbert's Entscheidungsproblem which asks for an algorithm that takes as input a statement of a mathematical formalization of axioms and determines whether the statement is universally valid in every structure satisfying the axioms. Thus it asks whether it's possible to have a fixed procedure to determine whether any specific mathematical statement can be proven within that system. This as we know was answered by Church and Turing in the negative - it can't be done and for example a Turing machine can't solve the self-referential nature of the Halting problem. Closely related is Hilbert's second problem, which asks for a proof that arithmetic is consistent (free of any internal contradictions). Gödel's second incompleteness theorem shows this can't be done. Interestingly, Gentzen showed a consistency proof for Peano arithmetic (a first order axiomatic formalism of arithmetic for natural numbers) that isn't a stronger form of first order arithmetic theory per se, but it is not finitistically within the same formalism either. Maybe this ties in to your speculation above.

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Jose P. Koshy wrote on Mar. 20, 2015 @ 07:36 GMT
Dear Steven P Sax,

Yours is the last essay posted in this contest. I was eager to know what it contained. You say, "But even those postulates rely on some type of intuition, which ultimately must come to terms with our knowledge of physical reality if they are to represent it”. Everyone will agree with this, though each will have his own version of 'physical reality'. What is your version of the 'physical reality'?

“When he presented his theory, most scientists (including Planck!) didn’t consider this quantum concept to be realistic but believed it to be just a mathematical trick.” This should have been the stand even now, a 'physicalist' stand, as I would like to call it. Please go through my essay: A physicalist interpretation of the relation between Physics and Mathematics.

“But if you shine the laser for only half the time – a half pulse – the electron goes into a superposition of both the ground and excited states”. A superposition? The actual physical situation has been explained by you clearly: “What this means physically is if you attempted to measure the energy of the electron you would have a 50% chance of measuring the energy at the ground state, and a 50% chance of measuring it at the excited state”. Explaining the '50% probability' as superposition is illogical, especially when you argue that this 'superposition' is a 'physical situation' that quantum computer offers. In my opinion, superposition goes against physical reality.

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Author Steven P Sax replied on Mar. 22, 2015 @ 03:43 GMT
Jose, the point I meant is that even the most reduced version of mathematical assumptions we attempt to make must yield predictions for measurements and interactions that are consistent with the explanations we give for our observations, if those explanations rely on those assumptions or are represented by a mathematical formalism based on those assumptions. Looking at it the other way around (and perhaps even more astounding), the explanation(s) we give to our observations and the predictions we make when physically interacting with the world, change based on the way we mathematically represent those observations and interactions. Physical reality is understood through the dynamic volleying between these two reckonings. Thanks for helping to flush that out with your question.

To appreciate the rigorous patience of this dynamism, consider how many centuries it took to reconsider Euclid's assumption about the flatness of space, or 'absolute direction.'

I liked your essay and am curious based on your 'physicalist' stand what your issue is with the quantum concept. In particular, what explanation would you alternatively give to explain the discrete energy representation? Or what else would you have changed from the assumptions and representations to account for the observed data of blackbody radiation?

Superposition, especially in view of the multiverse concept, offers an explanation based on the probabilistic representations given for measurement, while considering our other physical assumptions about measurement. What makes superposition compelling is the physical interpretation it offers to explain interference effects, which are very much the physical situation we appear to observe. (Consistent at least with the other assumptions and representations we afford to those observations).

Steve Sax

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Jose P. Koshy replied on Mar. 23, 2015 @ 05:11 GMT
Dear Steven P Sax,

From what you have stated, I interpret your 'physical reality' as follows: From observations to explanations to mathematical relations to correct predictions, such a sequence gives us some idea, and from the mathematical relations (that make correct predictions) back to observations and explanations, the reverse sequence helps to fine-tune that idea, and thus we can...

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Lawrence B Crowell wrote on Mar. 20, 2015 @ 12:08 GMT
I am not sure about how you treat undecidability. The two state system with eigenstates |0> and |1> in a certain dressed state configuration can be said to be in the undecided logical state if the total state is singlet or triplet or |ψ> = c(|0> + e^{iφ}|1>), c = 1/sqrt{2}. This undecidability is relative to a certain basis, which is unitarily equivalent to any other basis. By this it means that a basis with

|ψ> = c(|0> + e^{iφ}|1>), |ψ’> = c(|0> + e^{i(φ+π)}|1>),

are related by a π pulse or ½ pulse of the laser (or what ever system adjusts these states), and if the system is in one of these states it is decided within that basis. What is then decidable is relative to the choice of basis uses.

Using the superposed nature of quantum states as a measure of decidability might be relevant for einselection problems. The measurement outcome or classical einselected states might in some formal sense have a level of undecidability that is described in this manner.

Cheers LC

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Author Steven P Sax replied on Mar. 22, 2015 @ 05:13 GMT
Thanks Lawrence, that's very insightful and ties in well with the end of my essay. When one measures a complementary operator, the basis changes for that measurement - and what was once undecided is now a decided measurement. For example you measure Spin x to be X up for an electron. So Spin x is decided but it's in the basis of Spin z, namely X up =(Z up + Z down)/√2, and Spin z is undecided. Now you measure Spin z. You changed the basis to Spin x, and in doing so you're able to obtain a decided value for Spin z. Let's say it's Z up. But now with the changed basis, namely Z up =(X up + X down)/√2, Spin x is undecided. This entails changing the paradigm of what's being focused on. Physically this amounts to changing the environment namely the measuring interactions (such as a magnetic field in a different direction, etc.) so yes, I see how this can be looked at from an environmental decoherence (and thus einselection) perspective. Thanks for filling in that gap, I like it.

The key point then is there is not one basis in which everything can be decided. This will change depending on the particular measurement paradigm and I related it to Gödel incompleteness. Modeling physically how a self referential operation may change the particular measurement paradigm opens many possibilities for quantum computing (pun intended ;) ). As I pointed out above, superposition in light of the multiverse concept offers an explanation for how the undecidability might be represented, and the physical interpretation it offers to explain physical observations like interference effects makes it very compelling.

Thanks again for an excellent comment, and I look forward to reading you essay.

Steve Sax

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Lawrence B Crowell replied on Mar. 22, 2015 @ 23:56 GMT
This leads to the prospect there is no complete system for computing einselected bases. This could be an interesting thing to work on. Curiously this has some bearing on the nature of energy gaps. We usually think of quantum bases that are “natural” as ones that are diagonalized in an energy eigenbasis. This is largely because we measure quantum systems this way and the outcome to use Copenhagen language is collapsed in the way. However, quantum systems in a pure form have no intrinsic recognition of this basis over any other. A quantum system of an atom and photon that is Rabi oscillating between two energy states has no preferred basis.

This seems to have some bearing upon the problem of energy or mass gaps. How a nonabelian gauge field generates a mass-gap, which some associated Hilbert space of bounded states, is maybe not decidable.

I will try to elaborate later when I have a bit more time.

LC

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Lawrence B Crowell replied on Mar. 24, 2015 @ 10:48 GMT
I am glad that you enjoyed my essay. The Einscheidungsproblem of Hilbert turned out to have this strange impact on mathematics that Hilbert never imagined at the time. On the other hand I have read that Goedel discussed with Einstien on how he was fairly unhappy that his result seemed not to have practical impact on mathematics. However, in some ways that may now be the case. The formulation of mathematical physics might involve recognition of these matters.

Your recognition that a quantum system in a superposition of two states in a qubit has undecidable nature is interesting. I think a quantum system in a superposition of states could reflect a Goedelian undecidable situation in some problem involving einselection, or maybe even deeper with problems with quantum error correction codes (QECC) in black holes. It discuss hypercomputing in my essay, and this could involve some aspect of how QECC in black holes and the erasure of quantum bits that accumulate. This may be an undecidable problem, and hypercomputing might indicate something that is concealed from observability.

I will try to look up Gentzen's proof of consistency for Peano axioms. I thought I had scored your essay earlier, but I had not, so I just now scored it.

Cheers LC

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Jonathan Khanlian wrote on Mar. 20, 2015 @ 16:49 GMT
Hi Steve,

I’m glad to see your essay here. Are you familiar with Penrose and Hameroff’s work? Your quantum consciousness discussion, although different in the points it brings up, reminds me of their work. I’d be interested to hear a comparison of their work and some of your ideas.

In your essay, you talk about two half pulses equaling a NOT, and one half pulse equaling SQR(NOT). I do like this interpretation better than SQR(-1), but I am still a little confused… but this may be due to my lack of familiarity with many technical aspects of physics. You say that two half pulses (Is this the same thing as a half or quarter cycle?) will definitely put the rubidium electron into the excited state. Would a 3rd half pulse put it back into superposition and a 4th put it back into the ground state? If so (and my understanding may be wrong), why would a “continuous” wave polarized laser keep the electron only in the excited state? Wouldn’t it be changing states according to your analysis?

Other parts of your essay reminded me of some of the questions I posed at the end of my essay. Questions such as…

B) How quickly could a tape be processed through a Turing machine and is this constraint physical or informational in nature?

…among others related to Gödel, formal systems, the multiverse, etc. may interest you.

I am still pondering your statement:

“Now, consider again the self referential situation. It is only when the state is considered with respect to itself that it is no longer itself. That’s the paradox.”

It reminds me of this Zen analysis of “non-self”, as well as a discussion I was having with another FQXi member on the objective nature of color only existing in some relational sense..

Overall, this is a very enjoyable and thought-provoking essay. I like how it does not get bogged down in the details, yet it is technical enough to be taken seriously. I think it is the kind of foundational questioning that can lead to real leaps in understanding. It also reminds me of your “It from Bit” essay in that it takes some of these philosophical ideas and ties them into the physical world. I hope it scores well in the essay contest, and I hope you discuss it some more in the discussion thread here.

Jon

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Author Steven P Sax replied on Apr. 13, 2015 @ 05:46 GMT
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 1.4×1069 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 unified 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

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Author Steven P Sax replied on Apr. 13, 2015 @ 09:24 GMT
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...

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Author Steven P Sax replied on Apr. 13, 2015 @ 09:45 GMT
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

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Constantinos Ragazas wrote on Mar. 20, 2015 @ 20:35 GMT
Hello Steven,

I enjoyed reading your essay and I agree with much of what you write. But one thing that peaked my interest is your discussion of Planck's Law for blackbody radiation. The very beginnings of Quantum Theory. You write,

”Five years later Einstein rederived Planck’s [blackbody spectrum] results by changing the physical assumptions of the cavity oscillations of the electromagnetic field. He proposed they were quantized themselves, and thus light and all electromagnetic radiation were quantized.”

Einstien's Quantum Hypothesis used to derive Planck's Law is not, however, needed to mathematically derive this result! But it was the “first” explanation. And in Physics the “first” successful explanation is often taken as “physically true”!

In my FQXi 2010 Contest essay, “A World Without Quanta?”, I show how Planck's Law can be derived using continuous methods and not needing the physical assumption of “quanta”. In fact, I show Planck's Law is actually an exact Mathematical Truism (like the Pythagorean Theorem) and is not really a Physical Law. (see also, “The Thermodynamics in Planck's Law”)

Further, Einstein's Constant Speed of Light hypothesis contradicts his Quantum Hypothesis! Since it can be mathematically proven that “if CSL, then light is a wave”! (Proposition 11 in “The Thermodynamics in Planck's Law”)

Often, the remarkable fit of the experimental curve for Cosmic blackbody radiation with the theoretical curve using Planck's Law is used to argue for the existence of “quanta”! But such confirmation is not proof. What can only explain why we have such a remarkable fit between the experimental spectra and Planck's Law is my proof that Planck's Law is an exact Mathematical Identity. Such remarkable fit we would also expect for a finely drawn circle with the Pythagorean Theorem.

In my current Contest essay, ”The 'man-made' Universe”, I argue all Basic Laws of Physics are Mathematical Truisms. But I also argue we cannot know “what is” the physical Universe. But can only know our measurements and understanding of “what is”. Thus, the only Universe we can know is the Universe we conceive! To that end, we should be guided by an Anthropocentric Principle: Our Knowledge of the Universe is such as to make Life possible.

I welcome your thoughts on this!

Constantinos

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John R. Cox wrote on Mar. 21, 2015 @ 16:31 GMT
Steven,

I don't know who in the FQXi community sandbagged the high rating your essay enjoyed, but there are several individuals whom make themselves conspicuous by a jealous regard of their own personal equations. That might serve to guard their position in the informal ranking, but also serves to warn others that if such an equation or applicable argument is indeed of independent merit and is then included and cited in another's work, they become burdened with defense of the source as well. Being employed in the federal bureaucracy (a thankless job) you are no stranger to the syndrome.

Being a member of the public myself, my rating is rather meaningless here, but I do thank you for professionally contributing to the public good. And I found your essay, informative, insightful and comprehensible. It is apparent that some original work is incorporated in the section on limits of computability, but presented in keeping with the Contest Topic and inclusive of real research that impacts the cornerstone of quantum mechanics and the role of mathematics. Overall, your essay addresses the Essay Contest Topic in the most appropriate manner of any that I have personally read. All politics aside; Excellent! jrc

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Constantinos Ragazas replied on Mar. 22, 2015 @ 20:40 GMT
John R. Cox,

"I don't know who in the FQXi community" you are targeting but since your post here followed mine, let me quickly dispel any sly implication you may wish to make I had anything to do with your "sandbagging" accusations. The truth of the matter is I have yet to cast any vote for any essay!

Further, the ratings are meaningless to me. That is not the reason I participate in FQXi Contests. The reason is simple! I realize, however, it may be beyond your comprehension.

I seek open and honest discussions with others so interested on some ideas I have been thinking for years. My references in my posts are entirely for this reason.

Constantinos

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John R. Cox replied on Mar. 23, 2015 @ 01:51 GMT
Constantinos,

I really don't know who has trashed some high ratings, but when clicking to the page to select contest essays, which shows top ratings, it was noticeable that sudden drops occurred while comments clearly opposed to the metaphysical paradigm had been made. And like yourself, I don't that assume that I know anything for certain. And if I disagree I don't vote. Who am I to rate anyone down? Simply a matter of timing that my post in a separate thread followed yours in another, and anyone reading your essay would or should recognize you only come to learn.

Incidentally, 2010 was pre-computer for me and I hadn't seen your essay that year. For what its worth, I agree that the Planck Quantum is an action quantity, not an indivisible fundamental entity. Just when physics was getting interesting, the photo-electric effect was interpreted as a photon being an indivisible 'chunck' of energy, and the instantaneous quantum leap defied realism. The ad hoc inventions of quantum mechanics are in my view a failure of classical mechanics to rationalize the quantum, define electrical charge in terms which do not result in a singularity, and determine what a particle is. We can't fault success, so the results of QM can be used as clues to a realistic paradigm.

In his list of references, Prof. Sax gives #(10) a research paper at the University of Maryland to which he briefly refers concerning laser experiments with Rubidium. Other research there involves counting photons down to a mere (4) four. The Rubidium experiments he cites found that a half-pulse would not pump the single electron in the outer shell to a level of an excited state of emission, but a second half-pulse (presumably within the window of relaxation time) would! That suggests that the energy of the 1st half-pulse is stored in the region between the shells and the 2nd half-pulse supplies the rest needed for an electron mass quantity to condense into a volumetric particle, at least long enough to dissipate the full pulse energy as a shower of light. It is coming close to experimental capability to determine the quantum leap as time dependent. And the photon to be described as a bundle of volumetric wavetrains.

The quantum when viewed as a preferred quantity observed as the action through the duration of any single wavelength, would allow a component of that action energy to be determinant of spatial length of wave event. In reality it would be more like a Quantum Stroll. I see no reason not to take it leisurely. :-) jrc

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Constantinos Ragazas replied on Mar. 23, 2015 @ 02:54 GMT
John R.

Thank you for clarifying! Yes, I do come to learn! And to discuss and exchange ideas. The Rubidium experiments you mention is especially interesting. This fits well my explanation of the double-slit experiment. Central to this (and most all I write about in Physics) is the "accumulation of energy" (what in my writings is the quantity 'eta') and my claim that "before discrete manifestation there is continuous accumulation of energy".

Eric Reiter has done some experiments which also seem to confirm such "accumulation of energy". You may like to read his 2012 FQXi essay,

"A Challenge to Quantized Absorption by Experiment and Theory".

Constantinos

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David Brown wrote on Mar. 22, 2015 @ 08:52 GMT
Dear Steven P Sax,

In your essay you wrote, "Photons, quantum mechanics, and the basis of modern technology arose by shifting the terrains of infinity itself. We see now that the two pillars of modern physics ‐ general relativity and quantum physics – owe their discovery in part to purely mathematical changes in representation." I claim that there are at least 3 empirical disproofs of the Lambda-CDM concordance cosmological model: the space roar, the photon underproduction crisis, and the empirical evidence for Milgrom's MOND. Google "Witten Milgrom" for more information. My guess is that string theory needs to explain the cosmological problems BUT with "mathematical changes in representation" — using the monster group and the 6 pariah groups. One puzzle is the GZK paradox. Could the explanation for the GZK paradox be inverse Compton scattering from relativistic jets? Do you have any guesses for the explanation of the GZK paradox?

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Chidi Idika wrote on Mar. 23, 2015 @ 22:45 GMT
Dear Steve,

Your essay is to me a precise and comprehensive treatment of the questions raised by the contest topic.

Just one question: couldn't what you called the self referential state be BETTER interpreted to be simply in any system of observables the observer proper? Even more so when you also allow ultimately that the self referential trait may actually explain self awareness (consciousness).

You say also: “The limit of computability thus marks the ultimate interface of mathematics and physics.”

This assumption in one wrap is the whole thesis of my essay: Observer as the Mathematician’s “constant” and the physicist’s “quantum".

But am yet to see just one professional who has actually read through it. I can understand that I being neither a mathematician nor a physicist the attitude is that not much worth can come from my end, especially when people have got their time to optimize.

But let me say, Mr Sax, you seem to see the kind of spark I myself see (forgive my grandstanding). Yet could you read and comment frankly on my line of argument. I think your far-going insight will be rewarded.

Bests,

Chidi Idika

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Michel Planat wrote on Mar. 24, 2015 @ 18:45 GMT
Dear Steven,

You decided to illustrate the maths/phys correspondance with a few well choosen examples.

I like the case of a qubit where you talk about self-reference.

Classically you have the CF (coin flip) gate (that randomizes the inputs) and is self-referential (that is idempotent) in the sense that CF^2=CF.

For qubits QCF= H.Z where H is the Hadamard gate and Z the Pauli (phase) gate (here H or QCF create the superposition of the input qubits). And it is easy to calculate QCF^(1/2)= X (the Pauli shift gate that you may well call NOT because it switches the input qubits). A good account is in "The square root of not by Bryan Hayes"

http://bit-player.org/wp-content/extras/bph-publicatio
ns/AmSci-1995-07-Hayes-quantum.pdf

To conclude you write "Self referential operations, first seen to matematically limit a physical computer, indeed can be the underpinning of qubit manipulation, and the physical foundation of quantum computing".

I just polished your argument but I agree that passing from the classical bit to the qubit allows to reconsider the Halting Problem in a fresh (quantum) way and this has consequences on Goedel's theorem.

You had an essay on time entanglement (here the CNOT gate is relevant) so that you know that the (claimed) incompleteness of quantum theory relates to the EPR paradox. My view is that it can be further clarified by the use of advanced mathematics.

Best,

Michel

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Author Steven P Sax replied on Apr. 13, 2015 @ 22:18 GMT
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

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Edwin Eugene Klingman wrote on Mar. 25, 2015 @ 18:38 GMT
Dear Steven Sax,

I very much enjoyed your essay. While ordinarily I tend to think of coordinates as simply a labeling convention, I liked your discussion of inertia and the fact that "Any mathematical representation still depends on physical assumptions, and changing the mathematical representation changes the physical explanation we use." And, per gravity, changing the physical assumptions changes the mathematical representation of space-time.

This perspective certainly applies to Bell's theorem. When one changes the physical assumption from "precession in a constant field" to "scattering in a non-constant field" the representation changes from Pauli's provisional binary map to a continuum-based scattering spectrum, with consequent changes in correlation.

Your treatment of computation is excellent, beginning with "every finitely realizable physical system can be perfectly simulated by a … computing machine…" My Automatic Theory of Physics explores this point and [page 10 in my essay] I show how the automaton's 'next-state-address' corresponds to the physics 'potential' by linking the canonical automaton to a typical Feynman QFT kernel.

Your explanation that undecidability of self-referential statements can be traced to endless loops that destroy causality is excellent. And, although I tend to resist certain interpretations, I found your discussion of half pulses, NOT gates, and your insights for future research especially fascinating. Thank you for your excellent essay.

My best regards,

Edwin Eugene Klingman

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Akinbo Ojo wrote on Mar. 29, 2015 @ 12:40 GMT
Hello Steve,

You are obviously very knowledgeable about what you have written about.

While one will not like to disagree with an expert, I will like you to consider that the domain of the theory of Special relativity which is said to be flat space cannot apply to Earth-based experimental findings, since the Earth is not without gravity. Indeed, space on the Moon would be much flatter than space on Earth given its lower gravity. Do you agree?

Then talking about the relationship between mathematical representation and physical explanation that you chose as your topic, based on your expertise can you explain how the mathematical representation of a line can be divided, if it consists an infinite number of points and I presume a point cannot be mathematically or physically divided or can it?

My essay has more of a philosophical flavor and so may not be your cup of tea. If it is however, you may drop me some comments as I have agonized there how a line can be cut when what it consists of cannot. Eckard Blumschein and one other drew my attention to proposals from Dedekind and C.S. Peirce on how to attempt this.

Regards,

Akinbo

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Sujatha Jagannathan wrote on Mar. 31, 2015 @ 10:33 GMT
You have brought out a good point in your work that there is not one mathematical basis in which everything is decidable.

This extends many branches of thoughts for fruition.

Liked it.

- Sincerely,

Miss. Sujatha Jagannathan

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Christian Corda wrote on Apr. 1, 2015 @ 09:20 GMT
Hi Steve,

As I told you in my FQXi page, I have read your intriguing Essay. Here are some comments:

1) I agree with your statement that "physical theories are never “proven” per se". I add that theories are not perfect, in the sense that they do not exactly represent the reality, but only an ideal approximation of it.

2) The statement "gravity became geometry, forming the bedrock of general relativity" is the core of my Essay!

3) Concerning Planck substituting discrete values for changing the Rayleigh‐Jean law, he claimed that he introduced discrete values in despair! I recently did something similar in black hole physics finding discrete values for Hawking radiation. In fact, non strict thermality implies countability of subsequent emitted quanta. I have been strongly attacked for this by various "orthodox" physicists. In any case, your discussion in pages 4-7 is interesting also in the framework of my recent research in black hole physics.

4) Your concluding insight 3) seems consistent with anthropic principle.

Finally, I found very interesting and enjoyable the reading of your nice Essay. It surely deserves the highest score that I immediately give you.

I wish you best luck in the Contest.

Cheers, Ch.

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En Passant wrote on Apr. 4, 2015 @ 16:46 GMT
Steven,

I was impressed with the fact that your writing style was rigorous and yet entertaining at the same time. The historically influenced developments of certain important milestones in physics and mathematics that you discussed were interesting and instructive.

I was not as impressed with the latter parts that got speculative, but everyone is entitled to speculate. After all, speculations indicate what needs to be worked on next (in the mind of the one speculating).

But a separate point I wished to make here is to address what I think is a misconception on the part of some commenters above. Their writing appears to indicate that you have resolved the Liar’s Paradox (and you did not work hard to dispel that idea in your replies). All you have done in your essay is show that a statement about the Liar’s Paradox will have a decided truth value. The Liar’s Paradox still stands undecided. In other words, if you make a “statement” that the Liar’s Paradox (stated in its normal form) is false, then you will be able to say that that “statement” is false (since the LP is undecided). The LP can be nested (you call it expanded) within the “statement” about it, making it appear as a single statement, essentially making the “statement” non-self-referential. But then you have not dealt with self-referential statements! A minor criticism, perhaps.

It may help with the Halting problem, but it does not get around the Liar’s Paradox in a logical sense (you just will have to have a 2nd statement about it (i.e. about a self-referential statement in a program), integrated in such a way as to bypass the self-referential line).

All in all, good work.

En

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Author Steven P Sax replied on Apr. 9, 2015 @ 23:09 GMT
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.

--------

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.

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Author Steven P Sax replied on Apr. 12, 2015 @ 22:17 GMT
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

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Author Steven P Sax replied on Apr. 13, 2015 @ 01:06 GMT
...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

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Ted Erikson wrote on Apr. 8, 2015 @ 17:16 GMT
FYI:My Essay 2408 error corrections @

Chicago Section AAPT

Spring Meeting 2015 - Glenbrook South High School

April 11, 2015

8:15-8:45

Registration and Continental Breakfast

8:50-9:00

Welcome and Introductions - John Lewis - Host

9:00 -9:15

Dimensionless Dualities

Ted Erikson - R/E UnLtd. - sdog1@sbcglobal.net

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Author Steven P Sax wrote on Apr. 9, 2015 @ 23:40 GMT
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!

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RJ Tang wrote on Apr. 12, 2015 @ 16:19 GMT
Simplicity is a result of equilibrium, and seeking simplicity is a human specific preference

Simplicity is that a relative few theories and mathematical models can explain a number of phenomena. While complexity is the opposite where there seems to be an unending need to invent new theories. By this definition, physics and astronomy are in the former camp and social science and biology...

view entire post


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LLOYD TAMARAPREYE OKOKO wrote on Apr. 13, 2015 @ 13:57 GMT
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.

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Michel Planat wrote on Apr. 15, 2015 @ 08:13 GMT
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

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Peter Jackson wrote on Apr. 16, 2015 @ 23:06 GMT
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

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Cristinel Stoica wrote on Apr. 20, 2015 @ 06:13 GMT
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

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Member Marc Séguin wrote on Apr. 21, 2015 @ 03:45 GMT
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

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Author Steven P Sax replied on Apr. 21, 2015 @ 06:44 GMT
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

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Torsten Asselmeyer-Maluga wrote on Apr. 21, 2015 @ 07:26 GMT
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

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Author Steven P Sax replied on Apr. 22, 2015 @ 07:20 GMT
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

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Mohammed M. Khalil wrote on Apr. 21, 2015 @ 16:23 GMT
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

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Author Steven P Sax replied on Apr. 22, 2015 @ 07:29 GMT
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

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Member Sara Imari Walker wrote on Apr. 22, 2015 @ 23:13 GMT
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

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Author Steven P Sax replied on May. 26, 2015 @ 23:39 GMT
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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Alma Ionescu wrote on Apr. 23, 2015 @ 03:54 GMT
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

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Alma Ionescu replied on Apr. 23, 2015 @ 17:07 GMT
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

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