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FQXi FORUM

December 8, 2022

CATEGORY:
Trick or Truth Essay Contest (2015)
[back]

TOPIC: No-Go Theorems in Physics and Maths: from Bell to Gödel with Hidden Propagator Dynamics by Michael James Goodband [refresh]

TOPIC: No-Go Theorems in Physics and Maths: from Bell to Gödel with Hidden Propagator Dynamics by Michael James Goodband [refresh]

The nature of quantum theory seems to suggest that underlying the physical world of experiment is a deeper mathematical reality. This essay considers a physics reason for experiment not to reveal the true reality. The spirit of the EPR and Bell approaches is followed to investigate the physics of an underlying-reality being dynamically hidden from experiment by the physics itself. There being a genuine distinction between experimental-reality and underlying-reality avoids Bell’s no-go result, but is subject to Gödel’s no-go theorem instead. The “unreasonableness” of maths could leave physics with no-where to go, and impose limits on the rest of science.

Michael Goodband has a physics degree from Cambridge University and a PhD in theoretical physics.

Hi Michael Goodband,

"*It all started off so well*." But it has reached the point where, as you say, "… *every student of physics must find their way to make peace with* [the apparent fact that] *quantum theory says that in some mysterious way, experimental-reality is not the same as underlying-reality*."

You describe the key to your hidden propagator dynamics...

view entire post

"

You describe the key to your hidden propagator dynamics...

view entire post

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Hi Edwin

Thanks for your thoughtful comments. There seem to be a number of ways that appear to mathematically give the correct correlation result for the EPR spin singlet state but don't correctly match the physics. The supposition of Hidden Propagator Dynamics is of the same standard as Bell's supposition of Hidden Variable Theory, where having a physical reason for the true physics situation being dynamically hidden from direct experiment, makes it possible to both have the correct physics and match the quantum result.

I noticed that in your essay you have an element of there being a distinction between the underlying physics - spin eigenstate - and the physics measured in experiment- by interaction with a magnetic field. HPD is a generalisation of this sort of distinction, but addresses the issue of discrete eigenstates in experiment. The issue with rotation groups SO(3) and SU(2) is not how they are related to each other but how they are different. The ability of a HPD theory to reproduce QT results depends on the rotation group for the underlying spin objects being SU(2) and explicitly not SO(3). This was the crucial point made by our mutual acquaintance, as we discussed in a previous essay contest.

My point about the sequence N->NDA->N is with regard to the process of modelling experimentally measured discrete particle states that are countable by the natural numbers by normed-division algebra valued fields in quantum theory. QT then makes probabilistic predictions for the discrete particle states, giving the sequence N->NDA->N for experiment to QT model and back to experiment again. This isn't changed to θ->NDA->N; the discreteness is fixed by experiment and the underlying physics. It is the ability of a local deterministic HPD theory - over underlying reality - to reproduce QT results for discrete eigenstates - with the same issues as for QT - that is the significant result.

Regards

Michael

Thanks for your thoughtful comments. There seem to be a number of ways that appear to mathematically give the correct correlation result for the EPR spin singlet state but don't correctly match the physics. The supposition of Hidden Propagator Dynamics is of the same standard as Bell's supposition of Hidden Variable Theory, where having a physical reason for the true physics situation being dynamically hidden from direct experiment, makes it possible to both have the correct physics and match the quantum result.

I noticed that in your essay you have an element of there being a distinction between the underlying physics - spin eigenstate - and the physics measured in experiment- by interaction with a magnetic field. HPD is a generalisation of this sort of distinction, but addresses the issue of discrete eigenstates in experiment. The issue with rotation groups SO(3) and SU(2) is not how they are related to each other but how they are different. The ability of a HPD theory to reproduce QT results depends on the rotation group for the underlying spin objects being SU(2) and explicitly not SO(3). This was the crucial point made by our mutual acquaintance, as we discussed in a previous essay contest.

My point about the sequence N->NDA->N is with regard to the process of modelling experimentally measured discrete particle states that are countable by the natural numbers by normed-division algebra valued fields in quantum theory. QT then makes probabilistic predictions for the discrete particle states, giving the sequence N->NDA->N for experiment to QT model and back to experiment again. This isn't changed to θ->NDA->N; the discreteness is fixed by experiment and the underlying physics. It is the ability of a local deterministic HPD theory - over underlying reality - to reproduce QT results for discrete eigenstates - with the same issues as for QT - that is the significant result.

Regards

Michael

Dear Michael Goodband,

Great essay with interesting content. I like the style and directness "is the universe written in maths, or do we just describe the universe using maths?" and "The truth is that particles are only ever measured as … well … particles."

I have thought a lot about Hidden Variables, but your essay introducing the idea of Hidden Propagators is food for thought. The concept of Hidden Propagators is somewhat new to me. That said, I enjoyed your detailed discussion of spin together with the comment "This standard form of the calculation gives the misleading impression that spin states are static, and correspond to mathematical objects in a Platonic realm of Hilbert space giving underlying reality. But this is not what the physics says. Spin is a dynamic state of relativistic rotation in 3+1 dimensions.". The dynamic nature of spin is a very important concept.

I have one question on your use of the term "hidden". Do you mean hidden in the sense that we haven't figured it out yet? or hidden in the sense that it is necessary to the theory that it not be explained?

In my essay, I directly model the particles of the standard model from which hidden variables could be extracted. I followed with interest your discussion of the SO(3) and SU(2) rotational groups and would be very interested in your read on how they I have presented them in my essay.

A very enjoyable and educational read and you deserve a high ranking.

Best of luck, Ed Unverricht

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Great essay with interesting content. I like the style and directness "is the universe written in maths, or do we just describe the universe using maths?" and "The truth is that particles are only ever measured as … well … particles."

I have thought a lot about Hidden Variables, but your essay introducing the idea of Hidden Propagators is food for thought. The concept of Hidden Propagators is somewhat new to me. That said, I enjoyed your detailed discussion of spin together with the comment "This standard form of the calculation gives the misleading impression that spin states are static, and correspond to mathematical objects in a Platonic realm of Hilbert space giving underlying reality. But this is not what the physics says. Spin is a dynamic state of relativistic rotation in 3+1 dimensions.". The dynamic nature of spin is a very important concept.

I have one question on your use of the term "hidden". Do you mean hidden in the sense that we haven't figured it out yet? or hidden in the sense that it is necessary to the theory that it not be explained?

In my essay, I directly model the particles of the standard model from which hidden variables could be extracted. I followed with interest your discussion of the SO(3) and SU(2) rotational groups and would be very interested in your read on how they I have presented them in my essay.

A very enjoyable and educational read and you deserve a high ranking.

Best of luck, Ed Unverricht

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Dear Ed Unverricht

Thanks for your comments.

The idea of a Hidden Propagator compliments Bell’s consideration of Hidden Variables and doesn’t conflict with it at all. The intended sense of “hidden” is that the true underlying physics happens on a timescale much less than any interaction that could be used in experiments. This renders the details of the dynamics unmeasurable by direct experimental measurement – hence “hidden”. It is only this distinction that is required for the HPD analysis as it proceeds in a way directly analogous to that of Bell. I don’t assume that the details of the dynamics could be found/inferred by other means. If they were, then it wouldn’t make any difference to the HPD analysis.

I will be reading your essay with interest shortly.

Michael

Thanks for your comments.

The idea of a Hidden Propagator compliments Bell’s consideration of Hidden Variables and doesn’t conflict with it at all. The intended sense of “hidden” is that the true underlying physics happens on a timescale much less than any interaction that could be used in experiments. This renders the details of the dynamics unmeasurable by direct experimental measurement – hence “hidden”. It is only this distinction that is required for the HPD analysis as it proceeds in a way directly analogous to that of Bell. I don’t assume that the details of the dynamics could be found/inferred by other means. If they were, then it wouldn’t make any difference to the HPD analysis.

I will be reading your essay with interest shortly.

Michael

Dear Michael Goodband,

I can't see how your distinction between experimental and underlying reality could be of service for a theory that is supposed to confront Bell's theorem and reproduce the predictions of quantum theory. If I follow, you say that the underlying theory is deterministic and (piecewise) causal, but that the experimental (observed) outcomes are not because, in essence, the...

view entire post

I can't see how your distinction between experimental and underlying reality could be of service for a theory that is supposed to confront Bell's theorem and reproduce the predictions of quantum theory. If I follow, you say that the underlying theory is deterministic and (piecewise) causal, but that the experimental (observed) outcomes are not because, in essence, the...

view entire post

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Dear Tim Mauldin

I was hoping that you would read my essay and comment. Firstly, my intention is NOT to confront Bell at all. I’m not seeking to validate the view of EPR, and in fact HPD disproves EPR just as much as Bell. In the exact details my HPD proposition compliments and extends Bell’s hidden variable approach. The distinction between underlying-reality and experimental-reality...

view entire post

I was hoping that you would read my essay and comment. Firstly, my intention is NOT to confront Bell at all. I’m not seeking to validate the view of EPR, and in fact HPD disproves EPR just as much as Bell. In the exact details my HPD proposition compliments and extends Bell’s hidden variable approach. The distinction between underlying-reality and experimental-reality...

view entire post

Dear Michael,

I'm still having trouble following the thought...sorry. Let me try again.

Take a pair of electrons in the singlet state. When you say that the particles are "exploring possible S2 orientations" it sounds as if the state is changing in time "exploring" and therefore the result of a spin measurement will depend on exactly when it is made. But if so, the natural question...

view entire post

I'm still having trouble following the thought...sorry. Let me try again.

Take a pair of electrons in the singlet state. When you say that the particles are "exploring possible S2 orientations" it sounds as if the state is changing in time "exploring" and therefore the result of a spin measurement will depend on exactly when it is made. But if so, the natural question...

view entire post

report post as inappropriate

Dear Tim

The conflict Einstein started with his views of quantum theory is somewhat polarised with entrenched concepts. I’m not interested in one winning over the other, I’m looking for a third way. Determinism cannot be restored – my HPD shows the same – so I’m not looking for it. Quantum theory is just maths without the physics. Bohm’s theory with a non-local QT-like field...

view entire post

The conflict Einstein started with his views of quantum theory is somewhat polarised with entrenched concepts. I’m not interested in one winning over the other, I’m looking for a third way. Determinism cannot be restored – my HPD shows the same – so I’m not looking for it. Quantum theory is just maths without the physics. Bohm’s theory with a non-local QT-like field...

view entire post

Michael - Thanks for the remarkable essay. While much of it is beyond my training, I did appreciate the irony of math playing tricks on physics and vice versa. My essay explores this very issue in the apparent paradoxes in both math and physics and reaches a conclusion similar to, but broader than, yours --- that there is a Hole at the Center of Creation. I would be interested in your view on my thesis and whether it has parallels to your far more technical presentation.

Sincerely - George Gantz

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Sincerely - George Gantz

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Thank you George.

You're right that your essay explores a similar issue. In 2012 I wrote a paper that used Gödel's incompleteness theorem to succeed where Ludwig Wittgenstein failed. His second line in Tractatus Logico-Philosophicus was a statement of maths realism:

1.1 The world is the totality of facts, not of things

that incorrectly led to the conclusion:

7 What we cannot speak about we must pass over in silence

If we instead start with physics realism then we correct his conclusion to read:

What we cannot speak about in direct terms we must describe another way.

The physical systems that suffer this description problem are the self-referential systems that Christine Cordula Dantas discusses in her essay, and another way of describing such irreducible systems is given by Andrei Kirilyuk in his essay. In my 2012 FQXi essay I described how quantum theory itself was also “another way” of describing the physical world because a particle is itself a self-referential dynamic system.

Michael Goodband

You're right that your essay explores a similar issue. In 2012 I wrote a paper that used Gödel's incompleteness theorem to succeed where Ludwig Wittgenstein failed. His second line in Tractatus Logico-Philosophicus was a statement of maths realism:

1.1 The world is the totality of facts, not of things

that incorrectly led to the conclusion:

7 What we cannot speak about we must pass over in silence

If we instead start with physics realism then we correct his conclusion to read:

What we cannot speak about in direct terms we must describe another way.

The physical systems that suffer this description problem are the self-referential systems that Christine Cordula Dantas discusses in her essay, and another way of describing such irreducible systems is given by Andrei Kirilyuk in his essay. In my 2012 FQXi essay I described how quantum theory itself was also “another way” of describing the physical world because a particle is itself a self-referential dynamic system.

Michael Goodband

Dear Michael, After having a look at your ideas in this essay, I find their correlation with my own solution to the problem of mathematical and physical incompleteness. One can probably say that I propose a particular version of "hidden" quantum dynamics you derive by the general incompleteness analysis. Those hidden dynamical dimensions result simply from the unreduced interaction problem solution (never obtained in usual theory) and exist also at all higher complexity levels, giving rise to the omnipresent (but sometimes indeed externally "hidden") dynamic randomness. As to Gödel's incompleteness, it doesn't exist any more within that causally complete mathematical framework, but can also be considered, within any particular system/level study, as the necessary dynamical links to neighbouring interaction/complexity levels. I hope that such kind of extension of traditional science framework can attract the attention of other researchers, leading to various stagnating problem solution and further knowledge progress...

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It's a strange login system here,:) the previous comment was mine, of course.

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Wonderfully excellent Michael..

I like that you spell out both applicable interpretations of Korzybski's map / territory paradigm, and how it relates to the essay question. I also like that you make the connection with the normed division algebras explicitly explained, as part of the backbone of Physics.

And finally; I find your explanation of why the interaction space and observability space have differing domains to be satisfying, and the explanation of why this accounts for things via the hidden dynamics of the propagators mostly lucid. It is good to know there is an actual reality, even if its dynamics are somewhat hidden.

All the Best,

Jonathan

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I like that you spell out both applicable interpretations of Korzybski's map / territory paradigm, and how it relates to the essay question. I also like that you make the connection with the normed division algebras explicitly explained, as part of the backbone of Physics.

And finally; I find your explanation of why the interaction space and observability space have differing domains to be satisfying, and the explanation of why this accounts for things via the hidden dynamics of the propagators mostly lucid. It is good to know there is an actual reality, even if its dynamics are somewhat hidden.

All the Best,

Jonathan

report post as inappropriate

Dear Michael,

You mentioned "Physics realism: physics is the territory and maths is the map

Maths realism: maths provides the territory and physics is a map"

Thats true and its is because there are existing laws of invariance which governs both mathematical structures and physical reality. Its not mathematics describing physics rather their respective laws of invariance match...

view entire post

You mentioned "Physics realism: physics is the territory and maths is the map

Maths realism: maths provides the territory and physics is a map"

Thats true and its is because there are existing laws of invariance which governs both mathematical structures and physical reality. Its not mathematics describing physics rather their respective laws of invariance match...

view entire post

report post as inappropriate

It is a beautifully compelling argument, Michael!

As we already knew, you and I agree on two critical points that cut against the grain of most interpretations of conventional quantum theory:

1. Bell's theorem experimental results only prove their own prior assumptions, such that there can be no correspondence between mathematical theory and physical result.

2. A unified theory is necessarily extra-dimensional.

My own essay deals with (1) and not the other -- so I am happy to see you pick up the topological argument. It was an enjoyable read.

You should get the high score you deserve -- we need more publications that take the EPR argument seriously. I like your strategy of deconstructing Bell.

All best,

Tom

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As we already knew, you and I agree on two critical points that cut against the grain of most interpretations of conventional quantum theory:

1. Bell's theorem experimental results only prove their own prior assumptions, such that there can be no correspondence between mathematical theory and physical result.

2. A unified theory is necessarily extra-dimensional.

My own essay deals with (1) and not the other -- so I am happy to see you pick up the topological argument. It was an enjoyable read.

You should get the high score you deserve -- we need more publications that take the EPR argument seriously. I like your strategy of deconstructing Bell.

All best,

Tom

report post as inappropriate

Thanks Tom.

I would characterise my strategy as extending and complimenting Bell's hidden variable theory approach because his indirect assumptions preclude certain physical possibilities. His main target seemed to be the EPR desire to banish probabilities from physics and restore determinism. My hidden propagator dynamics approach reaches exactly the same conclusion on this point - determinism in experimental predictions is gone forever. I don't have any problem with what Bell did; the issue is that his conclusion doesn't have the status of universal generality claimed for it - it is on this that we agree.

My HPD approach extends Bell's analysis in a new and interesting way that leads to inevitable conclusions when the logic is followed in physics. My intention was to reach for those conclusions, of which one is that a "unified theory" is inevitably extra-dimensional. The other big conclusion that I didn't have room to discuss in my essay is to do with the nature of locality in underlying reality and experimental reality.

All the best

Michael

I would characterise my strategy as extending and complimenting Bell's hidden variable theory approach because his indirect assumptions preclude certain physical possibilities. His main target seemed to be the EPR desire to banish probabilities from physics and restore determinism. My hidden propagator dynamics approach reaches exactly the same conclusion on this point - determinism in experimental predictions is gone forever. I don't have any problem with what Bell did; the issue is that his conclusion doesn't have the status of universal generality claimed for it - it is on this that we agree.

My HPD approach extends Bell's analysis in a new and interesting way that leads to inevitable conclusions when the logic is followed in physics. My intention was to reach for those conclusions, of which one is that a "unified theory" is inevitably extra-dimensional. The other big conclusion that I didn't have room to discuss in my essay is to do with the nature of locality in underlying reality and experimental reality.

All the best

Michael

Dear Michael,

Your effort to re-establish causality in the interpretation of QM measurements should certainly be applauded.

You write "Any serious definition of physics is based on causality" so that for "continuous fields" "if a theory has non-NDA-valued fields, then it’s not physics." But in my understanding of QM (the theory) there is no causality although they may exist some sort of determinism. You are introducing the idea of propagation of the causation. I don't understand how it can be reconciliated with the mere existence of EPR pairs and entanglement unless there exist something like instantaneous causation. In my understanding of QM the concept of preparation of the instruments is fundamental (orientation of the polarizers...) and this is modeled with operators/observables whose ompatibility/commutativity predetermines the possible issues of measurements. The source of the paradoxes lies in the possible incompatibility between the algebra of operators and that of eigenvalues (as well explained by authors like Peres, Mermin and others). In QM the arguments are counterfactual which is just the opposite of causality.

Is there a way to distinguish your 'causal approach' and QM approach? I have to admit that I did not read your other essays on this subject.

Best regards,

Michel

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Your effort to re-establish causality in the interpretation of QM measurements should certainly be applauded.

You write "Any serious definition of physics is based on causality" so that for "continuous fields" "if a theory has non-NDA-valued fields, then it’s not physics." But in my understanding of QM (the theory) there is no causality although they may exist some sort of determinism. You are introducing the idea of propagation of the causation. I don't understand how it can be reconciliated with the mere existence of EPR pairs and entanglement unless there exist something like instantaneous causation. In my understanding of QM the concept of preparation of the instruments is fundamental (orientation of the polarizers...) and this is modeled with operators/observables whose ompatibility/commutativity predetermines the possible issues of measurements. The source of the paradoxes lies in the possible incompatibility between the algebra of operators and that of eigenvalues (as well explained by authors like Peres, Mermin and others). In QM the arguments are counterfactual which is just the opposite of causality.

Is there a way to distinguish your 'causal approach' and QM approach? I have to admit that I did not read your other essays on this subject.

Best regards,

Michel

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Michel

Unfortunately the essay length restriction prevented me from getting to the issue of locality, which is that causal propagation (i.e. less than/equal c) in underlying reality doesn’t necessarily always result in time-like separation of experimentally measured events. In empty space, it does necessarily follow. But that is one of my points – the space of a spin singlet is not...

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Unfortunately the essay length restriction prevented me from getting to the issue of locality, which is that causal propagation (i.e. less than/equal c) in underlying reality doesn’t necessarily always result in time-like separation of experimentally measured events. In empty space, it does necessarily follow. But that is one of my points – the space of a spin singlet is not...

view entire post

Dear Michael,

Thank you, I cannot escape understanding your 2012 essay and other writings from you. I like eccentric ideas as soon as I consider them scientifically sound. Then "it is the rule of the game" I will try to be honest in rating your essay.

Best regards,

Michel

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Thank you, I cannot escape understanding your 2012 essay and other writings from you. I like eccentric ideas as soon as I consider them scientifically sound. Then "it is the rule of the game" I will try to be honest in rating your essay.

Best regards,

Michel

report post as inappropriate

Dear Michael,

You say of the distinction between underlying-reality and experimental reality: “In simple terms, the distinction is because experiments are just too slow to measure what is happening.”

But I ask, isn’t this distinction rather because experimental reality is necessarily incomplete in that it must be at any instance carried out with one and only one base unit of/or measurement out of the infinitely many there can be?

My essay adopts the position that there can be one and only one**de facto** observer. So the human term “observer” is like the mathematician’s “constant” (number bases) or the physicist’s “quantum” [of observables]: there can be one and only one effective.

This does not in principle invalidate every other observer, it only means that the “observer” state is what separates the real from the virtual (it is the Heisenberg CUT so to speak). It is the singularity. Indeed the state “observer” should be what we mean by a conservation law or "stationary state" (think: harmonic oscillator) or simply “invariance” .

This will mean quite frankly that your underlying reality must be ontology wise the "nothing-in-particular" (the virtual or entropy or uncertainty), same in fact as any observer state. It is the "nothing" which defines things. For we cannot actually**define** matter with material attributes. Just as we do not define the coulomb with a coulomb or the joule with a joule etc.

We must define "things" (ontology) with "nothing" (null ontology), and vice versa.

Will appreciate your critical comment at my essay.

Chidi

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You say of the distinction between underlying-reality and experimental reality: “In simple terms, the distinction is because experiments are just too slow to measure what is happening.”

But I ask, isn’t this distinction rather because experimental reality is necessarily incomplete in that it must be at any instance carried out with one and only one base unit of/or measurement out of the infinitely many there can be?

My essay adopts the position that there can be one and only one

This does not in principle invalidate every other observer, it only means that the “observer” state is what separates the real from the virtual (it is the Heisenberg CUT so to speak). It is the singularity. Indeed the state “observer” should be what we mean by a conservation law or "stationary state" (think: harmonic oscillator) or simply “invariance” .

This will mean quite frankly that your underlying reality must be ontology wise the "nothing-in-particular" (the virtual or entropy or uncertainty), same in fact as any observer state. It is the "nothing" which defines things. For we cannot actually

We must define "things" (ontology) with "nothing" (null ontology), and vice versa.

Will appreciate your critical comment at my essay.

Chidi

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Dear Chidi

We have a fundamentally different metaphysical view of reality: maybe the difference between physics and philosophy of physics. In my essay I take the physics of the running coupling constants and the relations between the physical constants h-bar, c and G at face value as saying there is new physics on the Planck scale. Since the Planck time is far less than the interaction time of any experimental measurement, we have the condition that there is effectively an underlying reality that happens on a scale too small and fast to be measured – thus our experimental reality is not directly measuring underlying reality. So my distinction is a conclusion of the physics realism view that there is such a thing as physical reality - independent of our definitions and observations of it - and we measure it: underlying reality is reality, we just don’t see it directly as it is.

Michael

We have a fundamentally different metaphysical view of reality: maybe the difference between physics and philosophy of physics. In my essay I take the physics of the running coupling constants and the relations between the physical constants h-bar, c and G at face value as saying there is new physics on the Planck scale. Since the Planck time is far less than the interaction time of any experimental measurement, we have the condition that there is effectively an underlying reality that happens on a scale too small and fast to be measured – thus our experimental reality is not directly measuring underlying reality. So my distinction is a conclusion of the physics realism view that there is such a thing as physical reality - independent of our definitions and observations of it - and we measure it: underlying reality is reality, we just don’t see it directly as it is.

Michael

Dear Michael,

I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

Joe Fisher

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I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

Joe Fisher

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At it its core, math is about numbers. Natural number arises from counting orders and naming convention for the uniqueness of a places in a sequence. In that sense, the physical world is a book written in natural numbers.

Using an analogy, the English alphabet has 26 letters, and with the alphabet infinite books can be written. We examine the books and find that each book consists at least one of the five vowels, and each word is less than 100 letters long, and so forth.

We are puzzled by how a random book can be that way. But need not be so, if we realize that the rule of writing a book is quite simple although the end product is somewhat complex. We start with a letter, then a word, and then a passage, a chapter and so on. Each step has some simple but irreducible rules. This process masks the simple relationship between a book and the alphabet, if we simple look at them without the steps in between.

rujing_tang at yahoo com

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Using an analogy, the English alphabet has 26 letters, and with the alphabet infinite books can be written. We examine the books and find that each book consists at least one of the five vowels, and each word is less than 100 letters long, and so forth.

We are puzzled by how a random book can be that way. But need not be so, if we realize that the rule of writing a book is quite simple although the end product is somewhat complex. We start with a letter, then a word, and then a passage, a chapter and so on. Each step has some simple but irreducible rules. This process masks the simple relationship between a book and the alphabet, if we simple look at them without the steps in between.

rujing_tang at yahoo com

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