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RECENT POSTS IN THIS TOPIC

**Eckard Blumschein**: *on* 4/2/11 at 16:14pm UTC, wrote Dear Lev, I see present mathematics based on a mutilated self-deceptive...

**Author Yuri Danoyan+**: *on* 3/31/11 at 17:38pm UTC, wrote Dear Eckard "For practical use Planck length, time and energy are...

**Eckard Blumschein**: *on* 3/31/11 at 16:20pm UTC, wrote Dear Yuri, Even ZFC includes the Archimedian axiom of infinity. Do we...

**Author Yuri Danoyan+**: *on* 3/30/11 at 19:47pm UTC, wrote Dear Eckard "My logic is the logic used by Euclid.." . Are you sure your...

**Eckard Blumschein**: *on* 3/29/11 at 18:08pm UTC, wrote Dear John Benavides, 1JB: “I think, you are misunderstanding something....

**Lev Goldfarb**: *on* 3/19/11 at 19:03pm UTC, wrote John, As promised, but with some delay ;-) , here is my (very brief)...

**Alan Lowey**: *on* 3/19/11 at 11:16am UTC, wrote Dear John, Congratulations on your dedication to the competition and your...

**Constantinos Ragazas**: *on* 3/17/11 at 2:11am UTC, wrote Congratulations John! I am pleased that you made it to the final round! ...

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FQXi FORUM

December 1, 2022

CATEGORY:
Is Reality Digital or Analog? Essay Contest (2010-2011)
[back]

TOPIC: Leaving Cantor’s Paradise through Paul Cohen’s Golden Door by John Benavides [refresh]

TOPIC: Leaving Cantor’s Paradise through Paul Cohen’s Golden Door by John Benavides [refresh]

All different formulations of Quantum Mechanics show that Quantum Reality do not obey some laws of classical logic. So far, this has been seen more as a curiosity or as a deficiency in our models. However, Quantum Computation results show that these non-classical logic features are not just a deficiency in our models, but the true essence of Quantum Reality. Once we accept this, we should start using non classical logics, not just to measure, but also to understand Quantum Reality. The first consequence of this, it is that widespread notions like those of Planck Scale limit, the Measurement problem and the Discrete vs Continuum duality, become non-sense concepts in a non-classical logic conception of reality. Until now it seems we have unconsciously ignored that properties of the structures we study depend on the principles of logic we employ in studying them. Fortunately, when we use this non classical logics to UNDERSTAND reality, their semantics open new shocking perspectives to relate General Relativity and Quantum Mechanics.

I am a mathematical physicist mainly interested in Geometry and the Foun- dations of Physics and Mathematics. Nowadays I am finishing my Ph.D. in Mathematics at the University of Florence. I am working on the simplification of a possible Topos theoretical foundation of physics and I am also trying to understand if topology could be a dynamical feature of reality using results from low dimensional topology .

Dear John Benavides,

I wish all the best to you. You will certainly successfully finish your PhD. After a while you will get aware that the already announced some years ago delivery of the first functioning as promised quantum computer will be further delayed. Do not worry, you will find excusing technicalities in a good essay from a previous contest.

Do you have an alternative? Perhaps not a rewarding one. You could read my essay that deals with the same matter as yours but arrives at a rejection of the usual topoi and the like. I see a considerable part of progress in physics available by revelation of very basic mistakes even in mathematics rather than lazy mutilation at will of logics. So you might train your ability and refute my apparently bold

ideas . Please do not hesitate. Any factual attack is welcome.

Regards,

Eckard

report post as inappropriate

I wish all the best to you. You will certainly successfully finish your PhD. After a while you will get aware that the already announced some years ago delivery of the first functioning as promised quantum computer will be further delayed. Do not worry, you will find excusing technicalities in a good essay from a previous contest.

Do you have an alternative? Perhaps not a rewarding one. You could read my essay that deals with the same matter as yours but arrives at a rejection of the usual topoi and the like. I see a considerable part of progress in physics available by revelation of very basic mistakes even in mathematics rather than lazy mutilation at will of logics. So you might train your ability and refute my apparently bold

ideas . Please do not hesitate. Any factual attack is welcome.

Regards,

Eckard

report post as inappropriate

Dear Eckard Blumschein

Thank you for read my essay.

I think, you are misunderstanding something. You cannot separate mathematics and logic. If you are saying you have found some problems with some basic notions in mathematics it is because the logic that you are using to think about these notions do not agree with the logic that define and govern them. I read your essay and what you see as problem is because you are unconsciously denying the excluded middle principle of classical logic that allows to abstract numbers with the limit point of the extension they represent. I agree with you extensions are more fundamental, but what you are ignoring is that if you say we should not identify extensions with its points, what are you saying is that we shouldn't use classical logic to describe the continuum, that kind of misconceptions is what my essay is about. On the other hand you don't need a quantum computer to justify that quantum reality is ruled by a non classical logic, you just need a half-silvered-mirrow and light to construct a a machine that cannot be described using classical logic.

Regards,

J.B.

Thank you for read my essay.

I think, you are misunderstanding something. You cannot separate mathematics and logic. If you are saying you have found some problems with some basic notions in mathematics it is because the logic that you are using to think about these notions do not agree with the logic that define and govern them. I read your essay and what you see as problem is because you are unconsciously denying the excluded middle principle of classical logic that allows to abstract numbers with the limit point of the extension they represent. I agree with you extensions are more fundamental, but what you are ignoring is that if you say we should not identify extensions with its points, what are you saying is that we shouldn't use classical logic to describe the continuum, that kind of misconceptions is what my essay is about. On the other hand you don't need a quantum computer to justify that quantum reality is ruled by a non classical logic, you just need a half-silvered-mirrow and light to construct a a machine that cannot be described using classical logic.

Regards,

J.B.

Hi to both of you,

I agrre with Eckard, the maths are cool and essential but when the physicality want be explained, of course the rationalism of our continuity and our discreteness become so important.

The topos of Mr Baez is for computing and its sortings, for the physicality it's an other story.All that to say that in fact we can't confound a simulation on computer and on the other side the real physical dynamic in 3 Dimensions. The maths imply so many confusions about our physicality, a time travel, the external cause of mass as higgs, the multiverses, the strings and this and that...all that is a pure irony for the rationalism and its axiomatization of physics and its pure laws.

The computing is a tool, an of course we can invent topologies or the topology...that is the question.

This difference is essential at my humble opinion.

Regards

Steve

report post as inappropriate

I agrre with Eckard, the maths are cool and essential but when the physicality want be explained, of course the rationalism of our continuity and our discreteness become so important.

The topos of Mr Baez is for computing and its sortings, for the physicality it's an other story.All that to say that in fact we can't confound a simulation on computer and on the other side the real physical dynamic in 3 Dimensions. The maths imply so many confusions about our physicality, a time travel, the external cause of mass as higgs, the multiverses, the strings and this and that...all that is a pure irony for the rationalism and its axiomatization of physics and its pure laws.

The computing is a tool, an of course we can invent topologies or the topology...that is the question.

This difference is essential at my humble opinion.

Regards

Steve

report post as inappropriate

Hi John,

Thank you for sharing your essay. I liked it, but found myself catching on one point that I'm keen to hear your opinion on. While the kind of mathematical tools you describe sound like an excellent way to understand and extend thinking on quantum mechanics, are you *certain* that, as you put it to Eckard, 'you only need a half silvered mirror and light to create a machine that can't be described using classical logic'.

If there existed an algorithmic model using classical logic which produced the kinds of result seen in the double slit experiment, or a Bell-inequality violation, or some other clear example, would that not constitute a refutation of the *need* for such forms of logic. (Please note that I make a distinction between need and usefulness. The approach you outline certainly seems useful.)

My thanks in advance for any opinions you may choose to share,

Alex

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Thank you for sharing your essay. I liked it, but found myself catching on one point that I'm keen to hear your opinion on. While the kind of mathematical tools you describe sound like an excellent way to understand and extend thinking on quantum mechanics, are you *certain* that, as you put it to Eckard, 'you only need a half silvered mirror and light to create a machine that can't be described using classical logic'.

If there existed an algorithmic model using classical logic which produced the kinds of result seen in the double slit experiment, or a Bell-inequality violation, or some other clear example, would that not constitute a refutation of the *need* for such forms of logic. (Please note that I make a distinction between need and usefulness. The approach you outline certainly seems useful.)

My thanks in advance for any opinions you may choose to share,

Alex

report post as inappropriate

Dear Alex

Thank you for reading my essay.

About your question, the problem is the following: using a machine based on half silvered mirrows, we can create a machine that simply takes a proposition and as result give its negation. You can create a machine without quantum mechanics which does the same ruled by classical logic. The problem is that you cannot explain with classical logic why the first machine gives the results it gives. To understand how the first machine works you need quantum mechanics i.e a new logic. I recomend you to read the paper of David Deutsch that I cite in my essay, it is a divulgative work very short that explains everything very clear I am sure you will enjoy it a lot.

Thanks again for your comments, let me know if I answer your question or if I misunderstand something.

J.B.

Thank you for reading my essay.

About your question, the problem is the following: using a machine based on half silvered mirrows, we can create a machine that simply takes a proposition and as result give its negation. You can create a machine without quantum mechanics which does the same ruled by classical logic. The problem is that you cannot explain with classical logic why the first machine gives the results it gives. To understand how the first machine works you need quantum mechanics i.e a new logic. I recomend you to read the paper of David Deutsch that I cite in my essay, it is a divulgative work very short that explains everything very clear I am sure you will enjoy it a lot.

Thanks again for your comments, let me know if I answer your question or if I misunderstand something.

J.B.

Hello John, I was only able to skim read your essay as I'm not as technically or mathematically minded as you are. I was impressed with your biography in stating that you are considering a dynamical feature of reality w.r.t a topology. I have a simple idea which might interest you. Have you ever considered an Archimdedes screw as the mechanical model of a GRAVITON? If this particle/wave helical structure travelled around a wraparound universe then it would emerge on the other side as an ANTI-GRAVITON, or in other words, DARK ENERGY. It's too good a model to be ignore for any longer imo. Did Newton miss this simple trick?

Best of luck.

Alan

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Best of luck.

Alan

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Dear Alan

Thank you for your comments

I think at the end we will discover that we should not quantize gravity, probably there is nothing like a graviton. In the terms of my essay what I think is gonna be the solution is that the intrinsic structure of quantum reality will tell us why on the big scale we see a spacetime where matter is ruled by the curvature of the spacetime. I hope you will try to read again my essay to understand what I am trying to say with this.

J.B.

Thank you for your comments

I think at the end we will discover that we should not quantize gravity, probably there is nothing like a graviton. In the terms of my essay what I think is gonna be the solution is that the intrinsic structure of quantum reality will tell us why on the big scale we see a spacetime where matter is ruled by the curvature of the spacetime. I hope you will try to read again my essay to understand what I am trying to say with this.

J.B.

Dear John,

I read your paper with interest. I have a question about an intriguing statement you make: ‘For the first time we will be able to understand that if quantum reality were not chaotic and non deterministic, probably our classical world would not be so smooth and well behaved.’ I wonder if you could elaborate on this. It seems plausible, but I would like to understand better why you think this should be so. Perhaps it is related to the example you give immediately after, the example of DNA as a “lower level simple universe” expanded into the complexity of a living organism? If so, I would point out that DNA acts not unilaterally, but in concert with the environment, to “unfold” the organism. It is already, then, a very complex “universe”!

I suspect that relations of one form or branch of mathematics to another can provide important insights about the physical world, by analogy, as you are pointing out. But unless one believes that mathematics is identical to the physical world, any exact mapping of mathematical truths to physical reality remains an assumption, an idealization.

Thanks and best wishes,

Dan

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I read your paper with interest. I have a question about an intriguing statement you make: ‘For the first time we will be able to understand that if quantum reality were not chaotic and non deterministic, probably our classical world would not be so smooth and well behaved.’ I wonder if you could elaborate on this. It seems plausible, but I would like to understand better why you think this should be so. Perhaps it is related to the example you give immediately after, the example of DNA as a “lower level simple universe” expanded into the complexity of a living organism? If so, I would point out that DNA acts not unilaterally, but in concert with the environment, to “unfold” the organism. It is already, then, a very complex “universe”!

I suspect that relations of one form or branch of mathematics to another can provide important insights about the physical world, by analogy, as you are pointing out. But unless one believes that mathematics is identical to the physical world, any exact mapping of mathematical truths to physical reality remains an assumption, an idealization.

Thanks and best wishes,

Dan

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Dear Dan

Thank you for your comments.

About your question what I am trying to say is that we should stop thinking that at some limit, reality is not anymore classic to become quantum, this is a notion that has sense if we conceive reality within classical Logic. But once we accept other logics, for example a intuitionist one, the classic limit notion looses any sense. The example I give in the essay show then how to relate two realities (logics) in a more fundamental level a new kind of limit if you want. In this conception is the intrinsic structure of the non-classical logic world (fluctuating quantum reality) which determined what the classical-logic world sees (smooth, well behaved classical reality). You are right about the DNA example, what I am trying to say is that this new way of relate two different level realities could be useful to explain some phenomena that we see now as emergent phenomena, the DNA case is probably more complicated as I said but the idea opens new perspectives.

Finally your concern about mathematics is valid, but for example Einstein was able to develop general relativity because Minkowski gave him a mathematical powerful abstraction, this is what quantum mechanics is missing and is what I am trying to develop.

Let me Know if I answered your questions.

J.B.

Thank you for your comments.

About your question what I am trying to say is that we should stop thinking that at some limit, reality is not anymore classic to become quantum, this is a notion that has sense if we conceive reality within classical Logic. But once we accept other logics, for example a intuitionist one, the classic limit notion looses any sense. The example I give in the essay show then how to relate two realities (logics) in a more fundamental level a new kind of limit if you want. In this conception is the intrinsic structure of the non-classical logic world (fluctuating quantum reality) which determined what the classical-logic world sees (smooth, well behaved classical reality). You are right about the DNA example, what I am trying to say is that this new way of relate two different level realities could be useful to explain some phenomena that we see now as emergent phenomena, the DNA case is probably more complicated as I said but the idea opens new perspectives.

Finally your concern about mathematics is valid, but for example Einstein was able to develop general relativity because Minkowski gave him a mathematical powerful abstraction, this is what quantum mechanics is missing and is what I am trying to develop.

Let me Know if I answered your questions.

J.B.

Yes, thanks, John.

I am still trying to see if there is more to understand in the relationship between a chaotic non-deterministic quantum world and the smoothness and orderliness of the macro world.

Dan

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I am still trying to see if there is more to understand in the relationship between a chaotic non-deterministic quantum world and the smoothness and orderliness of the macro world.

Dan

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Dear John,

Quantum mechanics is only mysterious theory as long as we cling to causality. The flaw of causality is that if we understand something only if we can reduce it to a cause, and we can understand this cause only as the effect of a preceding cause etcetera, to end at some primordial cause which cannot be reduced to a previous cause, then causality ultimately cannot explain anything.

If in a universe which creates itself out of nothing, particles have to create themselves, each other, then (the properties of) particles must be as much the product as the source of their interactions, their energy exchange. As the same holds for force between them, a force in principle cannot be either attractive or repulsive. That is, particles can only exist, have properties (attract/repulse) if they have some kind of backbone so they can, within limits, absorb energy in an increase of their kinetic energy rather than in a change of identity. However, if the rest energy of particles ultimately is as much the product as the source of their energy exchange, of their behavior, then interaction energies obviously never can become infinite at infinitesimal distances.

A universe which finds a way to create itself without any outside intervention can hardly stop doing so: gravity, the contraction of masses and the related expansion of spacetime between the mass concentrations they form, is the expression of this continuing creation process. As long as we cling to the causal, classical 19th century ideas about mass and charge, to the belief that particles only are the source of fields and forces, we condemn ourselves to waste our time on string theory and Higgs bosons. I'm afraid that physics has become too opaque a mixture of truths, half-truths and inconsistencies, to be able to solve any fundamental problem at all. For details, see my essay.

Best regards, Anton

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Quantum mechanics is only mysterious theory as long as we cling to causality. The flaw of causality is that if we understand something only if we can reduce it to a cause, and we can understand this cause only as the effect of a preceding cause etcetera, to end at some primordial cause which cannot be reduced to a previous cause, then causality ultimately cannot explain anything.

If in a universe which creates itself out of nothing, particles have to create themselves, each other, then (the properties of) particles must be as much the product as the source of their interactions, their energy exchange. As the same holds for force between them, a force in principle cannot be either attractive or repulsive. That is, particles can only exist, have properties (attract/repulse) if they have some kind of backbone so they can, within limits, absorb energy in an increase of their kinetic energy rather than in a change of identity. However, if the rest energy of particles ultimately is as much the product as the source of their energy exchange, of their behavior, then interaction energies obviously never can become infinite at infinitesimal distances.

A universe which finds a way to create itself without any outside intervention can hardly stop doing so: gravity, the contraction of masses and the related expansion of spacetime between the mass concentrations they form, is the expression of this continuing creation process. As long as we cling to the causal, classical 19th century ideas about mass and charge, to the belief that particles only are the source of fields and forces, we condemn ourselves to waste our time on string theory and Higgs bosons. I'm afraid that physics has become too opaque a mixture of truths, half-truths and inconsistencies, to be able to solve any fundamental problem at all. For details, see my essay.

Best regards, Anton

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Dear Anton

You are right, there are a lot of problems to conceive reality if we insist using notions based in our classical-logic conception of reality. The causality problem arises because we describe evolution problems using Cauchy problems that cannot be conceived without initial conditions. Once we accept that classical logic (and then classical mathematics) cannot be used to abstract more profound problems, we will find new ways to understand causality and many other phenomena that cannot be explained in the classical framework.

J.B.

You are right, there are a lot of problems to conceive reality if we insist using notions based in our classical-logic conception of reality. The causality problem arises because we describe evolution problems using Cauchy problems that cannot be conceived without initial conditions. Once we accept that classical logic (and then classical mathematics) cannot be used to abstract more profound problems, we will find new ways to understand causality and many other phenomena that cannot be explained in the classical framework.

J.B.

Hi John,

Thanks for pointing me at the David Deutsch paper. I greatly enjoyed it, and went on to make sure I understood the idea of the Mach–Zehnder interferometer upon which the work appears to be based. The interferometer seems a tidy application of the idea of particle self-interference.

Particle self-interference is something I believe I can model. By this I mean that I have a simulation in which a particle travels through two slits and interferes with itself to produce a pattern of fringes on a screen of detectors. When I move a detector two one of the slits, I get occasional detections at that slit, and the interference pattern goes away. This program uses no complex numbers or wave equations. All its logic is classical and algorithmic.

You seem like a someone with a strong grasp of the logical implications of QM, and also someone for whom my claim will sound deeply unlikely, or at least that's what I'm hoping.

Would you be interested to see a short video of interference in action or discuss the ways in which my simulation might be flawed. I'd like to refute it's potential if possible.

My thanks in advance in any thoughts or opinions you may have.

Alex

(An outline of the algorithm used is provided in my essay if you're interested. The effect is achieved by employing a different, but still entirely classical, notion of locality.)

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Thanks for pointing me at the David Deutsch paper. I greatly enjoyed it, and went on to make sure I understood the idea of the Mach–Zehnder interferometer upon which the work appears to be based. The interferometer seems a tidy application of the idea of particle self-interference.

Particle self-interference is something I believe I can model. By this I mean that I have a simulation in which a particle travels through two slits and interferes with itself to produce a pattern of fringes on a screen of detectors. When I move a detector two one of the slits, I get occasional detections at that slit, and the interference pattern goes away. This program uses no complex numbers or wave equations. All its logic is classical and algorithmic.

You seem like a someone with a strong grasp of the logical implications of QM, and also someone for whom my claim will sound deeply unlikely, or at least that's what I'm hoping.

Would you be interested to see a short video of interference in action or discuss the ways in which my simulation might be flawed. I'd like to refute it's potential if possible.

My thanks in advance in any thoughts or opinions you may have.

Alex

(An outline of the algorithm used is provided in my essay if you're interested. The effect is achieved by employing a different, but still entirely classical, notion of locality.)

report post as inappropriate

Dear Alex

I read your extraordinary essay. The coincidences with what I am trying to propose are so shocking that my legs are trembling. I try to explained why, I think you have the information I am missing. The discrete models that you are proposing are just a partial description of the order I am trying to find, that it explains why we see the properties of the classical world we see. Particularly, the fact of no locality in my approach is expressed by the fact that when we collapse to a classical world we are taking a generic ultrafilter on the order topology which is a global fact. The way you relate the nodes of the graph is just the structure of the order, for example, the Continuum Hypothesis example shows that if the order is not choosen right, we don't get the result in the classical world, this is what it is happening in your models. When you say how we should iterate your models what are you doing is describing how the order topology should behave locally, i.e. you are choosing the ultrafilter.

Finally your concern about the logic we should use it is missing something. We already use classical logic to describe and model quantum reality, it is what the classical approach does, but we can't understand very well quantum reality. What you do is construct your model by try and error and you try to explain why some iterating model gives the result you are looking for and others not. Why I am trying to say is that, if we introduce non classical logics, in my case a intuitionist one, we can explain these phenomena perfectly.

I would like to hear your opinions again.

J.Benavides

I read your extraordinary essay. The coincidences with what I am trying to propose are so shocking that my legs are trembling. I try to explained why, I think you have the information I am missing. The discrete models that you are proposing are just a partial description of the order I am trying to find, that it explains why we see the properties of the classical world we see. Particularly, the fact of no locality in my approach is expressed by the fact that when we collapse to a classical world we are taking a generic ultrafilter on the order topology which is a global fact. The way you relate the nodes of the graph is just the structure of the order, for example, the Continuum Hypothesis example shows that if the order is not choosen right, we don't get the result in the classical world, this is what it is happening in your models. When you say how we should iterate your models what are you doing is describing how the order topology should behave locally, i.e. you are choosing the ultrafilter.

Finally your concern about the logic we should use it is missing something. We already use classical logic to describe and model quantum reality, it is what the classical approach does, but we can't understand very well quantum reality. What you do is construct your model by try and error and you try to explain why some iterating model gives the result you are looking for and others not. Why I am trying to say is that, if we introduce non classical logics, in my case a intuitionist one, we can explain these phenomena perfectly.

I would like to hear your opinions again.

J.Benavides

John,

To one with a meager understanding of higher math, your argument is esoteric and difficult to comprehend.

Mathematical models certainly help to understand the parts of reality we are modelling but I can't see them as a substitute for reality, and that reality takes on the models characteristics.

I contend that we can't truly know reality but that simulations can aid in the effort, but my deficiencies in math may taint my opinion.

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To one with a meager understanding of higher math, your argument is esoteric and difficult to comprehend.

Mathematical models certainly help to understand the parts of reality we are modelling but I can't see them as a substitute for reality, and that reality takes on the models characteristics.

I contend that we can't truly know reality but that simulations can aid in the effort, but my deficiencies in math may taint my opinion.

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Dear James

Thank you for your comments.

I hope you will give me another chance and you will try to read my essay again. I understand your concern about the role of mathematics. Mathematics can become dangerous if they become the objective, and their beauty, an intrinsic justification. This for example is what it is happening with string theory and topological quantum field theory. What I am trying to propose is different because the motivation is to understand better reality, in this sense mathematics could be the key, as the curvature was the Key that Einstein used to create General Relativity. I know that what I am proposing sounds very weird because it have never been used in physics before, like Lorentzian manifolds where never used before Einstein, but I think I am giving strong arguments to justify my choice I think what I exposed is what we are missing and the main reason because the unification program has failed.

J. Benavides

Thank you for your comments.

I hope you will give me another chance and you will try to read my essay again. I understand your concern about the role of mathematics. Mathematics can become dangerous if they become the objective, and their beauty, an intrinsic justification. This for example is what it is happening with string theory and topological quantum field theory. What I am trying to propose is different because the motivation is to understand better reality, in this sense mathematics could be the key, as the curvature was the Key that Einstein used to create General Relativity. I know that what I am proposing sounds very weird because it have never been used in physics before, like Lorentzian manifolds where never used before Einstein, but I think I am giving strong arguments to justify my choice I think what I exposed is what we are missing and the main reason because the unification program has failed.

J. Benavides

John

I found your essay highly intuitive and logical, even as one who'se looked elsewhere but maths for logical solutions. You said; "If we can get a model of quantum reality in this context, for example taking as the base space some kind of causal set, we will have a complete description of the hardware."

I have tried using what we CAN be reasonably sure of at the small scale, to pattern match with the classical, and derived what looks worryingly like a toe, and which I can't disprove. I hope you may rad my essay and can follow the logic. Yours deserves a higher rating and I'll oblige. I hope you may feel the same of mine, though poles apart in viewpoint we observe the same reality.

Best wishes

Peter

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I found your essay highly intuitive and logical, even as one who'se looked elsewhere but maths for logical solutions. You said; "If we can get a model of quantum reality in this context, for example taking as the base space some kind of causal set, we will have a complete description of the hardware."

I have tried using what we CAN be reasonably sure of at the small scale, to pattern match with the classical, and derived what looks worryingly like a toe, and which I can't disprove. I hope you may rad my essay and can follow the logic. Yours deserves a higher rating and I'll oblige. I hope you may feel the same of mine, though poles apart in viewpoint we observe the same reality.

Best wishes

Peter

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Dear Peter

Thank you for your comments.

Reading your essay and also reading the essays which propose a digital model based on a discrete lattice and computation, I see that the discrete features that you describe together with the discrete features of the digital-computer approach are converging with my ideas but in my context can be explained in more deep level as the construction of a variable-set structure on a partial order. It would be interesting if you try to considerate your approach from this point of view, to see if you find new interesting explanations or motivations.

J. B.

Thank you for your comments.

Reading your essay and also reading the essays which propose a digital model based on a discrete lattice and computation, I see that the discrete features that you describe together with the discrete features of the digital-computer approach are converging with my ideas but in my context can be explained in more deep level as the construction of a variable-set structure on a partial order. It would be interesting if you try to considerate your approach from this point of view, to see if you find new interesting explanations or motivations.

J. B.

Dear John,

Your essay interested me with its desire to explain reality through a pure analysis of quantum behavior by eliminating remaining classical concepts left in the theory. That seems like a logical and perhaps necessary step if it is to produce a satisfactory unifying theory. The introduction mentions gravity but it is not mentioned again in the rest of the essay. Is gravity expected to be explained by the history of the causal sets? Would that be the same as considering gravity as something we cannot evaluate in the non-collapsed quantum state, and gravity only appears in the collapsed humanly visible state?

I see your essay making a valid logical argument which makes it very interesting. My own approach is very similar in requiring the theory to be explained simply in terms of its core assumptions. It is interesting how our core assumptions are different, but I want to say I highly value your ideas.

Kind regards, Russell Jurgensen

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Your essay interested me with its desire to explain reality through a pure analysis of quantum behavior by eliminating remaining classical concepts left in the theory. That seems like a logical and perhaps necessary step if it is to produce a satisfactory unifying theory. The introduction mentions gravity but it is not mentioned again in the rest of the essay. Is gravity expected to be explained by the history of the causal sets? Would that be the same as considering gravity as something we cannot evaluate in the non-collapsed quantum state, and gravity only appears in the collapsed humanly visible state?

I see your essay making a valid logical argument which makes it very interesting. My own approach is very similar in requiring the theory to be explained simply in terms of its core assumptions. It is interesting how our core assumptions are different, but I want to say I highly value your ideas.

Kind regards, Russell Jurgensen

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Dear Russel

I appreciate very much your comments. About your question concerning gravity, You have expressed very well my idea of how gravity should appear in this context. I think gravity is just a consequence of the internal structure of quantum reality. In other words to obtain a satisfactory model unifying QM and GR, we should treat gravity and our perception of classical reality as an emergent phenomena in the sense of my essay, i.e as a classical logic structure arising from a non-classical logic ground model. An inevitable conclusion will be that gravity is not a fundamental force but just the result of how the interaction of matter and the other forces create our classical perception of spacetime ruled by the Einstein's equations.

I appreciate very much your comments. About your question concerning gravity, You have expressed very well my idea of how gravity should appear in this context. I think gravity is just a consequence of the internal structure of quantum reality. In other words to obtain a satisfactory model unifying QM and GR, we should treat gravity and our perception of classical reality as an emergent phenomena in the sense of my essay, i.e as a classical logic structure arising from a non-classical logic ground model. An inevitable conclusion will be that gravity is not a fundamental force but just the result of how the interaction of matter and the other forces create our classical perception of spacetime ruled by the Einstein's equations.

Excellent essay!

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Dear Florin

I had the opportunity to read your essay of the previous contest, I enjoyed very much and I am very flattered by your comment thanks.

J.B.

I had the opportunity to read your essay of the previous contest, I enjoyed very much and I am very flattered by your comment thanks.

J.B.

Dear John,

I too am flattered by your comments. I was pondering for some time questions along the lines of your essay and your essay resonated strongly with what I am thinking. Hope you will win a prize.

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I too am flattered by your comments. I was pondering for some time questions along the lines of your essay and your essay resonated strongly with what I am thinking. Hope you will win a prize.

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Interesting essay John. I appreciate in particular the emphasis you give to emergence, and emergence in computation. Also, the closing quote by Deutsch shines. I did not know it, and I am very pleased by his mentioning, among the deepest explanatory theories, (i) quantum theory, (iii) evolution theory for living organisms, and (iii) theory of computation, side by side. Surprisingly, Relativity is not in the group of four. Why do you think he excluded it?

I am worried/confused by the fact that the set P at the right of your central Figure 1 can be equated to two very different things such as (i) 'a space of boolean algebras whcih represent history propositions...' , or (ii) just a causal set modeling spacetime. I agree that 'richness can become an enemy', as you write -- and I could add 'meta-theories eventually need instantiation', or 'the devil is in the details', etc...

Finally, you write that 'The duality between discrete and continuum is just one more of the misunderstandings caused by a classical logic reasoning'. I am not sure I can retain a clear and strong argument explaining why this would be the case, after reading your essay. Does it mean that I have missed your main point?

Tommaso

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I am worried/confused by the fact that the set P at the right of your central Figure 1 can be equated to two very different things such as (i) 'a space of boolean algebras whcih represent history propositions...' , or (ii) just a causal set modeling spacetime. I agree that 'richness can become an enemy', as you write -- and I could add 'meta-theories eventually need instantiation', or 'the devil is in the details', etc...

Finally, you write that 'The duality between discrete and continuum is just one more of the misunderstandings caused by a classical logic reasoning'. I am not sure I can retain a clear and strong argument explaining why this would be the case, after reading your essay. Does it mean that I have missed your main point?

Tommaso

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Dear Tommaso

Thank you for reading my essay. I think Deutsch does not mention general relativity just because, he is assuming that soon or later we will understand it within the quantum formalism. On the other hand, which kind of order or topological space could be appropriate to describe quantum reality in the sense I propose is something that I don't know yet. I think it depends on the approach you choose to interpret the classical formalism, i.e if we used the classical Copenhagen approach the more natural order is the Boolean algebra that I mention, but I am more interested in the digital approach that you propose because a lattice-order simplifies considerably the models I have proposed.

Finally discreteness is not fundamental, because discreteness on quantum mechanics is mainly related with the measurement, which is the tool that make work the classical logic approach to model quantum reality. You are right this is a more deep issue that I haven't clarified on the essay.

Regards,

J.B.

Thank you for reading my essay. I think Deutsch does not mention general relativity just because, he is assuming that soon or later we will understand it within the quantum formalism. On the other hand, which kind of order or topological space could be appropriate to describe quantum reality in the sense I propose is something that I don't know yet. I think it depends on the approach you choose to interpret the classical formalism, i.e if we used the classical Copenhagen approach the more natural order is the Boolean algebra that I mention, but I am more interested in the digital approach that you propose because a lattice-order simplifies considerably the models I have proposed.

Finally discreteness is not fundamental, because discreteness on quantum mechanics is mainly related with the measurement, which is the tool that make work the classical logic approach to model quantum reality. You are right this is a more deep issue that I haven't clarified on the essay.

Regards,

J.B.

You have written an interesting essay. I give Topos theory some discussion in my essay

http://fqxi.org/community/forum/topic/810

However, that is more mentioned with respect to the Zariski topology.

I am not entirely sure what the continuum hypothesis אּ_1 = 2^{אּ_0} has to do with physics. The Cohen Bernay theorem indicates this is consistent with the ZF set theory by Godel’s theorem. I am somewhat familiar with these developments, but they are not entirely my area of expertise. Your paper gets a good thumbs up from me.

Cheers LC

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http://fqxi.org/community/forum/topic/810

However, that is more mentioned with respect to the Zariski topology.

I am not entirely sure what the continuum hypothesis אּ_1 = 2^{אּ_0} has to do with physics. The Cohen Bernay theorem indicates this is consistent with the ZF set theory by Godel’s theorem. I am somewhat familiar with these developments, but they are not entirely my area of expertise. Your paper gets a good thumbs up from me.

Cheers LC

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Dear Lawrence

Many thanks for your comments. I think that topos theory will play a fundamental role on physics in the next years, unfortunately Isham's work have not received the attention It deserves. This is mainly because the language of Category theory is still too abstract and difficult to handle. On the other hand, the importance of Cohen's forcing resides in how this technique allow to relate two models ruled by different logics, unfortunately the categorical tools of topos theory blur this fact, and because of this, many people working on topos theory have not noticed this important feature. What I am doing is trying to simplify this approach based on the importance of how a structure of variable sets can connect two theories ruled by different logics like quantum mechanics and general relativity.

Regards,

J. Benavides

Many thanks for your comments. I think that topos theory will play a fundamental role on physics in the next years, unfortunately Isham's work have not received the attention It deserves. This is mainly because the language of Category theory is still too abstract and difficult to handle. On the other hand, the importance of Cohen's forcing resides in how this technique allow to relate two models ruled by different logics, unfortunately the categorical tools of topos theory blur this fact, and because of this, many people working on topos theory have not noticed this important feature. What I am doing is trying to simplify this approach based on the importance of how a structure of variable sets can connect two theories ruled by different logics like quantum mechanics and general relativity.

Regards,

J. Benavides

Hello John,

I make references to Topos theory, largely with connection to the algebraic or projective varieties with Zariski topology. This is the basis of what might be called “pre-topos” theory. It is a form of pre-sheaf construction, which can be used to build a sheaf theory. One main interest is in a twistor geometry with E_6 subroup (or E_6xE_8) with a sheaf or pre-sheaf construction for twistor geometry.

There is a relationship between lightcones and Heisenberg groups within this Kleinian quotient system. I tend to see this as a precursor for the far more generalized system you present in your paper. I am also interested in the prospect for monster-moonshine structure, which is are projective varieties in 26 dimensions (eg the bosonic string) with Lorentzian structure. These projective varieties form the pre-sheaf construction for topos or grothendieck-Etale structure.

The space of lightlike geodesics is a set of invariants and then due to a stabilizer on O(n,2), so the space of lightlike curves L_n is identified with the quotient O(n,2)/P, where P is a subgroup defined the quotient between a subgroup with a Zariski topology, or a Borel subgroup, and the main group G = O(n,2). This quotient G/P is a projective algebraic variety, or flag manifold and P is a parabolic subgroup. The natural embedding of a group H - -> G composed with the projective variety G - ->G/P is an isomorphism between the H and G/P. This is then a semi-direct product G = P x| H. For the G any GL(n) the parabolic group is a subgroup of upper triangular matrices. An example of such a matrix with real valued elements is the Heisenberg group of 3x3 matrices (sorry for the inconvenient representation, but I have bad luck with these html-TeX systems)

|1 & a & b|

|0 & 1 & c|

|0 & 0 & 1|

which may be extended to n-dimensional systems to form the 2n+1 dimensional Heisenberg group H_n of n + 2 entries

|1 & a & b|

|0 & I_n & c|

|0 & 0 & 1|

where for O(n,2) the Heisenberg group is H_{2n+3}. The elements a and c are then n+2 dimensional row and column vectors of O(n,2). These are Borel groups, which emerge from the quotient space AdS_n/Γ, where the discrete group Γ is a manifestation of the Calabi-Yau 3-cycle, and which as it turns out gives an integer partition for the set of quantum states in the AdS spacetime. So both spacetime and quantum structure as we know them are emergent.

This of course can exist in more general setting, which is the type of construction you are presenting with the continuum hypothesis. I will not write that with alephs, for the Unicode representation does not work well. This suggests that the universe (the system of multi-cosmologies or multiverse) has this underlying system of topos between different structures. The system above illustrates how lightcones and Heisenberg groups emerge from the same quotient structure, where the topology indicates this is a form of topoi.

Cheers LC

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I make references to Topos theory, largely with connection to the algebraic or projective varieties with Zariski topology. This is the basis of what might be called “pre-topos” theory. It is a form of pre-sheaf construction, which can be used to build a sheaf theory. One main interest is in a twistor geometry with E_6 subroup (or E_6xE_8) with a sheaf or pre-sheaf construction for twistor geometry.

There is a relationship between lightcones and Heisenberg groups within this Kleinian quotient system. I tend to see this as a precursor for the far more generalized system you present in your paper. I am also interested in the prospect for monster-moonshine structure, which is are projective varieties in 26 dimensions (eg the bosonic string) with Lorentzian structure. These projective varieties form the pre-sheaf construction for topos or grothendieck-Etale structure.

The space of lightlike geodesics is a set of invariants and then due to a stabilizer on O(n,2), so the space of lightlike curves L_n is identified with the quotient O(n,2)/P, where P is a subgroup defined the quotient between a subgroup with a Zariski topology, or a Borel subgroup, and the main group G = O(n,2). This quotient G/P is a projective algebraic variety, or flag manifold and P is a parabolic subgroup. The natural embedding of a group H - -> G composed with the projective variety G - ->G/P is an isomorphism between the H and G/P. This is then a semi-direct product G = P x| H. For the G any GL(n) the parabolic group is a subgroup of upper triangular matrices. An example of such a matrix with real valued elements is the Heisenberg group of 3x3 matrices (sorry for the inconvenient representation, but I have bad luck with these html-TeX systems)

|1 & a & b|

|0 & 1 & c|

|0 & 0 & 1|

which may be extended to n-dimensional systems to form the 2n+1 dimensional Heisenberg group H_n of n + 2 entries

|1 & a & b|

|0 & I_n & c|

|0 & 0 & 1|

where for O(n,2) the Heisenberg group is H_{2n+3}. The elements a and c are then n+2 dimensional row and column vectors of O(n,2). These are Borel groups, which emerge from the quotient space AdS_n/Γ, where the discrete group Γ is a manifestation of the Calabi-Yau 3-cycle, and which as it turns out gives an integer partition for the set of quantum states in the AdS spacetime. So both spacetime and quantum structure as we know them are emergent.

This of course can exist in more general setting, which is the type of construction you are presenting with the continuum hypothesis. I will not write that with alephs, for the Unicode representation does not work well. This suggests that the universe (the system of multi-cosmologies or multiverse) has this underlying system of topos between different structures. The system above illustrates how lightcones and Heisenberg groups emerge from the same quotient structure, where the topology indicates this is a form of topoi.

Cheers LC

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Dear Lawrence

What I have in mind is very different and is more close to the use of topos made by Isham and not in the context of twistor theory or string theory, I am sure there could be some connections but the approach, that I am looking to propose, is very different. Particularly I am trying to understand emergence, from a different perspective that cannot be based on models based on classical set theory like Heisenberg groups or lightcones on any kind of classical manifold. For example a first step would be just to redefine the measurement postulate of the classical formulation of QM, in terms of the collapse of a structure of variable sets, on a presheave on a poset, to the classical Hilbert structure. I wonder If the emergence you describe can be put in these terms.

Regards,

J. B.

What I have in mind is very different and is more close to the use of topos made by Isham and not in the context of twistor theory or string theory, I am sure there could be some connections but the approach, that I am looking to propose, is very different. Particularly I am trying to understand emergence, from a different perspective that cannot be based on models based on classical set theory like Heisenberg groups or lightcones on any kind of classical manifold. For example a first step would be just to redefine the measurement postulate of the classical formulation of QM, in terms of the collapse of a structure of variable sets, on a presheave on a poset, to the classical Hilbert structure. I wonder If the emergence you describe can be put in these terms.

Regards,

J. B.

Dear John,

My approach is a bit more “conservative” I suppose. I think that this sort of topos construction may pertain on a deeper layer than what I am working on. I happen to be more focused on hypothetical connections with physics we understand currently. I would say a relationship between what I am doing and the sort of topos you advocate is similar to the relationship between pre-sheaf and sheaf theory.

Cheers LC

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My approach is a bit more “conservative” I suppose. I think that this sort of topos construction may pertain on a deeper layer than what I am working on. I happen to be more focused on hypothetical connections with physics we understand currently. I would say a relationship between what I am doing and the sort of topos you advocate is similar to the relationship between pre-sheaf and sheaf theory.

Cheers LC

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John,

I find your idea that theory limits what we 'see' of Nature very interesting and also very relevant to the raging debates going on. Let me add my voice to that debate and point you to a result along the same lines in my essay.

I show that Planck's constant is a necessary boundary to our ability to 'see' the Universe because h is the 'measurement standard' that defines Kelvin temperature. The entire theoretical regiment through which we measure, observe and understand the Universe breaths existence into h. Planck's constant is the 'focal point' to the physics we have created beyond which we cannot 'see' the world. It is NOT a necessary fact of Nature!

Just the other day I posted a very short paper, “If the speed of light is constant, then light is a wave”, that mathematically proves that light must be a wave.

I look forward to your comments and your support for these significant results.

Best wishes,

Constantinos

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I find your idea that theory limits what we 'see' of Nature very interesting and also very relevant to the raging debates going on. Let me add my voice to that debate and point you to a result along the same lines in my essay.

I show that Planck's constant is a necessary boundary to our ability to 'see' the Universe because h is the 'measurement standard' that defines Kelvin temperature. The entire theoretical regiment through which we measure, observe and understand the Universe breaths existence into h. Planck's constant is the 'focal point' to the physics we have created beyond which we cannot 'see' the world. It is NOT a necessary fact of Nature!

Just the other day I posted a very short paper, “If the speed of light is constant, then light is a wave”, that mathematically proves that light must be a wave.

I look forward to your comments and your support for these significant results.

Best wishes,

Constantinos

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Dear Constantinos

What you are saying about Planck constant is very interesting. I am looking for this kind of arguments to understand better what kind of limitations or features are closely related to the character of our theories related with classical logic. I will read your paper about the wave character of light I hope to give you some comments soon.

Regards,

J. Benavides

What you are saying about Planck constant is very interesting. I am looking for this kind of arguments to understand better what kind of limitations or features are closely related to the character of our theories related with classical logic. I will read your paper about the wave character of light I hope to give you some comments soon.

Regards,

J. Benavides

I promised in my reply to your post that I will read your essay and let you know what I think. Sorry for taking so long. It is a very interesting paper. I like the way you relate mathematics progess to physics progress.

I do have a feedback about your feedback to me earlier. You mentioned in your post to my essay that "... what you call zero-distance connection in my context is the fact that the each point in the order determine a universe which structure depends ...". I think there is a missunderstanding. The connectiviy mentioned in my paper was between two space-time positions that may be planck-scale apart. The link in your Figure 1 is between individual universes. They are quite different both in scale and in concept. I hope I haven't missundertood what you presented in your paper.

Thanks!

Honda

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I do have a feedback about your feedback to me earlier. You mentioned in your post to my essay that "... what you call zero-distance connection in my context is the fact that the each point in the order determine a universe which structure depends ...". I think there is a missunderstanding. The connectiviy mentioned in my paper was between two space-time positions that may be planck-scale apart. The link in your Figure 1 is between individual universes. They are quite different both in scale and in concept. I hope I haven't missundertood what you presented in your paper.

Thanks!

Honda

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Congratulations John! I am pleased that you made it to the final round!

Now that this frenzy is over, I like to pick up our conversation concerning Planck's constant and also the short post I linked you previously regarding light.

Good luck with the panel!

Constantinos

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Now that this frenzy is over, I like to pick up our conversation concerning Planck's constant and also the short post I linked you previously regarding light.

Good luck with the panel!

Constantinos

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Dear John,

Congratulations on your dedication to the competition and your much deserved top 35 placing. I have a bugging question for you, which I've also posed to all the potential prize winners btw:

Q: Coulomb's Law of electrostatics was modelled by Maxwell by mechanical means after his mathematical deductions as an added verification (thanks for that bit of info Edwin), which I highly admire. To me, this gives his equation some substance. I have a problem with the laws of gravity though, especially the mathematical representation that "every object attracts every other object equally in all directions." The 'fabric' of spacetime model of gravity doesn't lend itself to explain the law of electrostatics. Coulomb's law denotes two types of matter, one 'charged' positive and the opposite type 'charged' negative. An Archimedes screw model for the graviton can explain -both- the gravity law and the electrostatic law, whilst the 'fabric' of spacetime can't. Doesn't this by definition make the helical screw model better than than anything else that has been suggested for the mechanism of the gravity force?? Otherwise the unification of all the forces is an impossiblity imo. Do you have an opinion on my analysis at all?

Best wishes,

Alan

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Congratulations on your dedication to the competition and your much deserved top 35 placing. I have a bugging question for you, which I've also posed to all the potential prize winners btw:

Q: Coulomb's Law of electrostatics was modelled by Maxwell by mechanical means after his mathematical deductions as an added verification (thanks for that bit of info Edwin), which I highly admire. To me, this gives his equation some substance. I have a problem with the laws of gravity though, especially the mathematical representation that "every object attracts every other object equally in all directions." The 'fabric' of spacetime model of gravity doesn't lend itself to explain the law of electrostatics. Coulomb's law denotes two types of matter, one 'charged' positive and the opposite type 'charged' negative. An Archimedes screw model for the graviton can explain -both- the gravity law and the electrostatic law, whilst the 'fabric' of spacetime can't. Doesn't this by definition make the helical screw model better than than anything else that has been suggested for the mechanism of the gravity force?? Otherwise the unification of all the forces is an impossiblity imo. Do you have an opinion on my analysis at all?

Best wishes,

Alan

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John,

As promised, but with some delay ;-) , here is my (very brief) comment which you asked for.

As I mentioned in my reply to your message (on my essay's page), I believe in the priority of considerations related to (object or data) representation in science over logical considerations: the latter should follow the "logic" of the chosen representation, i.e. the logic should emerge during our analysis of "data "manipulation". Mathematics so far has been dealing basically with the numeric representations (for over four thousand years), and it is quite possible that the way out of our present difficulties in physics and other sciences require a radical revision of this basic, numeric, form of representation rather than any escapades into new forms of logic that are not motivated by any new forms of object/data representation. Again, to see the nature in a new light we need to change the form of data representation, which we have never done so far.

I'm sorry if I have not been as helpful as you expected.

My best wishes to you!

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As promised, but with some delay ;-) , here is my (very brief) comment which you asked for.

As I mentioned in my reply to your message (on my essay's page), I believe in the priority of considerations related to (object or data) representation in science over logical considerations: the latter should follow the "logic" of the chosen representation, i.e. the logic should emerge during our analysis of "data "manipulation". Mathematics so far has been dealing basically with the numeric representations (for over four thousand years), and it is quite possible that the way out of our present difficulties in physics and other sciences require a radical revision of this basic, numeric, form of representation rather than any escapades into new forms of logic that are not motivated by any new forms of object/data representation. Again, to see the nature in a new light we need to change the form of data representation, which we have never done so far.

I'm sorry if I have not been as helpful as you expected.

My best wishes to you!

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Dear Lev,

I see present mathematics based on a mutilated self-deceptive logic: Either smaller, or equal to, or larger than. What is wrong in my essay?

Dear Yuri,

You wrote: "All Planck finding relevant and useful. Moreover, transcendental pi has a finite number of digits when the side of the inscribed polygons became equal to the Planck length."

My argument was: Planck time, length, and energy are outside any applicable size.

Planck mass is within the measurable range but was never measured so far.

What about pi and the inscribed polgons, I also do not see any relevance.

Regards,

Eckard

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I see present mathematics based on a mutilated self-deceptive logic: Either smaller, or equal to, or larger than. What is wrong in my essay?

Dear Yuri,

You wrote: "All Planck finding relevant and useful. Moreover, transcendental pi has a finite number of digits when the side of the inscribed polygons became equal to the Planck length."

My argument was: Planck time, length, and energy are outside any applicable size.

Planck mass is within the measurable range but was never measured so far.

What about pi and the inscribed polgons, I also do not see any relevance.

Regards,

Eckard

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Dear John Benavides,

1JB: “I think, you are misunderstanding something. You cannot separate mathematics and logic. If you are saying you have found some problems with some basic notions in mathematics it is because the logic that you are using to think about these notions do not agree with the logic that define and govern them. “

1EB: My logic is the logic used by Euclid,...

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1JB: “I think, you are misunderstanding something. You cannot separate mathematics and logic. If you are saying you have found some problems with some basic notions in mathematics it is because the logic that you are using to think about these notions do not agree with the logic that define and govern them. “

1EB: My logic is the logic used by Euclid,...

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Dear Eckard

"My logic is the logic used by Euclid.."

.

Are you sure your logic only rigt logic?

How about non-Archimedian logic noted by Winterberg?

"Archimedes believed he could determine the value of K through a limiting process, by drawing a sequence of polygons inside a circle with an ever increasing number of sides. This "exhaustion" method though must fail if there is a smallest length. It was Planck who in a 1899 paper had shown that the fundamental constants of physics, h, G and c, give us such a small length, the Planck length

These three quantities are sufficient for the architecture of a non-Archimedean geometry for a finitistic formulation of physics. The square root in the expression for m0 gives us only the freedom to have two possible signs for m0, but nothing more.

In such a finitistic formulation one can, in an arbitrary number of space dimensions, replace differentiation operators by finite difference operators"

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"My logic is the logic used by Euclid.."

.

Are you sure your logic only rigt logic?

How about non-Archimedian logic noted by Winterberg?

"Archimedes believed he could determine the value of K through a limiting process, by drawing a sequence of polygons inside a circle with an ever increasing number of sides. This "exhaustion" method though must fail if there is a smallest length. It was Planck who in a 1899 paper had shown that the fundamental constants of physics, h, G and c, give us such a small length, the Planck length

These three quantities are sufficient for the architecture of a non-Archimedean geometry for a finitistic formulation of physics. The square root in the expression for m0 gives us only the freedom to have two possible signs for m0, but nothing more.

In such a finitistic formulation one can, in an arbitrary number of space dimensions, replace differentiation operators by finite difference operators"

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Dear Yuri,

Even ZFC includes the Archimedian axiom of infinity. Do we really need a special logic as to set a more or less speculative finite limit to resolution? For practical use Planck length, time and energy are obviously irrelevant.

I feel not yet in position to comment on Winterberg's interpretation of mass as a small difference between positive and negative Planck masses.

Regards,

Eckard

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Even ZFC includes the Archimedian axiom of infinity. Do we really need a special logic as to set a more or less speculative finite limit to resolution? For practical use Planck length, time and energy are obviously irrelevant.

I feel not yet in position to comment on Winterberg's interpretation of mass as a small difference between positive and negative Planck masses.

Regards,

Eckard

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Dear Eckard

"For practical use Planck length, time and energy are obviously irrelevant."

I am not agree with you. All Planck finding relevant and useful.

Moreover, transcendental pi has a finite number of digits when the side of the inscribed polygons became equal to the Planck length.

All the best

Yuri

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"For practical use Planck length, time and energy are obviously irrelevant."

I am not agree with you. All Planck finding relevant and useful.

Moreover, transcendental pi has a finite number of digits when the side of the inscribed polygons became equal to the Planck length.

All the best

Yuri

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