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RECENT POSTS IN THIS TOPIC

**manik ji**: *on* 6/13/19 at 7:27am UTC, wrote Thank you for your explanations. Good luck, Felix. visit...

**basudeba**: *on* 3/20/11 at 6:21am UTC, wrote Sub: Possibility of manipulation in judging criteria – suggestions for...

**Felix Lev**: *on* 3/12/11 at 7:24am UTC, wrote Tom, Thank you for your explanations. Good luck, Felix.

**T H Ray**: *on* 3/11/11 at 14:38pm UTC, wrote Fixing link: ICCS 2006

**T H Ray**: *on* 3/11/11 at 14:30pm UTC, wrote Felix, Briefly, my personal preference is for quantum field theory in a...

**Felix Lev**: *on* 3/8/11 at 21:54pm UTC, wrote Tom, Thank you for your compliments. The ideas that the ultimate theory...

**T H Ray**: *on* 3/7/11 at 13:38pm UTC, wrote Felix, Fascinating essay. You give me new hope that a true algebraic...

**Steve Dufourny**: *on* 2/28/11 at 15:18pm UTC, wrote 5 and more and of course it will be relevant...forget your competition...

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FQXi FORUM

October 23, 2019

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

TOPIC: What Mathematics Is Most Pertinent For Describing Nature? by Felix M Lev [refresh]

TOPIC: What Mathematics Is Most Pertinent For Describing Nature? by Felix M Lev [refresh]

We argue that principles of quantum theory inevitably imply that any fundamental physical theory can be based only on a finite mathematics. A version of a quantum theory based on a Galois field (GFQT) with a characteristic p is described. Since any Galois field is finite, there are no infinities in this theory and all operators are well defined. In a formal limit p->\infty GFQT reproduces the results of standard theory based on continuity. In GFQT the notion of particle-antiparticle and the conservation of such additive quantum numbers as the electric, baryon and lepton charges can be only approximate if de Sitter energies are much less than p.

Graduated from the Moscow Institute for Physics and Technology, got a PhD from the Institute of Theoretical and Experimental Physics (Moscow) and a Dr. Sci. degree from the Institute for High Energy Physics (also known as the Serpukhov Accelerator). In Russia Felix Lev worked at the Joint Institute for Nuclear Research (Dubna). His major area of research was relativistic quantum theory and a quantum theory over a Galois field. Since 1999 Felix Lev lives in Los Angeles, California and works at a software company on mathematical algorithms for the IC industry.

Dear Felix,

Interesting essay. For some time I wanted to understand QM based on p-adic numbers. In the light of Jordan algebra classification, I am extremely skeptical of this approach. Yes, it may solve some infinite dimensional field theory problems, but how does it stack up with simple textbook QM problems? You mention ref 4. Is there any good archive paper on this?

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Interesting essay. For some time I wanted to understand QM based on p-adic numbers. In the light of Jordan algebra classification, I am extremely skeptical of this approach. Yes, it may solve some infinite dimensional field theory problems, but how does it stack up with simple textbook QM problems? You mention ref 4. Is there any good archive paper on this?

report post as inappropriate

Dear Florin,

Thank you for your note. You say that you are skeptical about the p-adic approach. Does this imply that you are skeptical about my approach as well? One of the main reasons why I prefer a finite field approach is that it does not contain infinities at all. In my approach there is no problem with standard QM since, as explained in the essay (see also my cited papers and/or my papers in the arXiv), in the formal limit p->\infty my approach recovers the results of standard quantum theory. I am not the author of [4], so you would better ask them about problems bothering you. Let me only note that the authors of the p-adic approach also state that they have a correspondence principle when p->\infty.

Thank you for your note. You say that you are skeptical about the p-adic approach. Does this imply that you are skeptical about my approach as well? One of the main reasons why I prefer a finite field approach is that it does not contain infinities at all. In my approach there is no problem with standard QM since, as explained in the essay (see also my cited papers and/or my papers in the arXiv), in the formal limit p->\infty my approach recovers the results of standard quantum theory. I am not the author of [4], so you would better ask them about problems bothering you. Let me only note that the authors of the p-adic approach also state that they have a correspondence principle when p->\infty.

Dear Felix,

Thank you for your reply. I suspect that in the p-adic approach, simple problems like harmonic oscillator, or hydrogen atom have solutions which differ from standard QM. I can see how when p goes to infinity the usual results are obtained, but in that limit you are effectivey doing complex QM, In this sense, this approach is not different than say a dimensional regularization approach, but the problem still remains of proving that the infinities go away nicely. What I am saying is that this seems to be a case of having the cake and eating too: on one hand the lack of infinities collides with simple results, on the other hand, taking the limit recoveres the standard case but you don't want to reach the limit. I see thus as no different than arbitrarily truncating the Taylor series in the quantum gravity case; in other words it is an attempt of having a regularization technique, but with no proof of renormalizability.

It is clear that for quantum gravity something has to give. String theory is one way. Noncommutative geometry is another. p-adic and Galois approaches to QM seem to be an unnecessarily radical departure from the standard approach compared with those two approaches. Also an unproven approach as I don't see any serious renormalization results in the literature (for only a few days of reading the references on the archive), but I could be mistaken about that.

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Thank you for your reply. I suspect that in the p-adic approach, simple problems like harmonic oscillator, or hydrogen atom have solutions which differ from standard QM. I can see how when p goes to infinity the usual results are obtained, but in that limit you are effectivey doing complex QM, In this sense, this approach is not different than say a dimensional regularization approach, but the problem still remains of proving that the infinities go away nicely. What I am saying is that this seems to be a case of having the cake and eating too: on one hand the lack of infinities collides with simple results, on the other hand, taking the limit recoveres the standard case but you don't want to reach the limit. I see thus as no different than arbitrarily truncating the Taylor series in the quantum gravity case; in other words it is an attempt of having a regularization technique, but with no proof of renormalizability.

It is clear that for quantum gravity something has to give. String theory is one way. Noncommutative geometry is another. p-adic and Galois approaches to QM seem to be an unnecessarily radical departure from the standard approach compared with those two approaches. Also an unproven approach as I don't see any serious renormalization results in the literature (for only a few days of reading the references on the archive), but I could be mistaken about that.

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Dear Florin,

Thank you for your opinion. If you look at the beginning of the discussion section in [11] (in the arXiv this is the paper http://xxx.lanl.gov/abs/1011.1076 ) you will see that I am aware of this opinion. This is a typical opinion and of course, everyone has a right to have his or her own preferences (“De gustibus non disputandum est!”). I will try to comment this opinion....

view entire post

Thank you for your opinion. If you look at the beginning of the discussion section in [11] (in the arXiv this is the paper http://xxx.lanl.gov/abs/1011.1076 ) you will see that I am aware of this opinion. This is a typical opinion and of course, everyone has a right to have his or her own preferences (“De gustibus non disputandum est!”). I will try to comment this opinion....

view entire post

If an assumption leads to a contradiction one generally concludes that there's a flaw somewhere in the assumption. Zero-probability events (per mathematics) nonetheless do in fact occur (per reality). This contradiction appears to result from the assumption that the concept of infinity is objectively true. Ergo the concept of infinity (and by extension the continuum and an analog reality) looks like it might have a problem. Or is there another possible explanation?

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In my essay and papers I argue that only a finite mathematics can describe reality. In particular, the notion of probability can be only approximate. Indeed, this notion implies that we should carry out an infinite number of experiments within an infinite time interval. In reality this can be never done and when one says about probabilities, he or she believes that a finite number of experiments gives a value close to a hypothetical limit when the number of experiments is infinite. For example, we can never guarantee that the probability is exactly zero since if some event has not been found even in a very large number of experiments, there is no guarantee that it will not be found in a greater number of experiments.

Dear Rick P,

can you give an example for

"Zero-probability events (per mathematics) nonetheless do in fact occur (per reality"?

Thank you.

report post as inappropriate

can you give an example for

"Zero-probability events (per mathematics) nonetheless do in fact occur (per reality"?

Thank you.

report post as inappropriate

Anonymous,

Say you spin a fair pointer from a randomly chosen starting position by applying a randomly selected force. In an analog world there'd be an infinity of possible directions for it to end up pointing and so probabilistically it couldn't end up pointing anywhere. (1/Infinity)=0. Yet it does end up pointing somewhere.

Or, per [ http://mathforum.org/dr.math/faq/faq.prob.intro.html ]:

"Note that when you're dealing with an infinite number of possible events, an event that could conceivably happen might have probability zero. Consider the example of picking a random number between 1 and 10 - what is the probability that you'll pick 5.0724? It's zero, but it could happen."

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Say you spin a fair pointer from a randomly chosen starting position by applying a randomly selected force. In an analog world there'd be an infinity of possible directions for it to end up pointing and so probabilistically it couldn't end up pointing anywhere. (1/Infinity)=0. Yet it does end up pointing somewhere.

Or, per [ http://mathforum.org/dr.math/faq/faq.prob.intro.html ]:

"Note that when you're dealing with an infinite number of possible events, an event that could conceivably happen might have probability zero. Consider the example of picking a random number between 1 and 10 - what is the probability that you'll pick 5.0724? It's zero, but it could happen."

report post as inappropriate

In probability theory, probability is defined as a measure of sets belonging to a sigma-algebra. Those sets can have only a positive nonzero measure. In physics, a definition of a physical quantity is a description how this quantity should be measured. So a question arises whether in physics it is possible to define probability in accordance with mathematics. Mathematics prompts us that we cannot define such a quantity as “the probability to find 5.0724” since the set containing only the point 5.0724 has measure zero and does not belong to the sigma-algebra. But we can try to define the probability to find a number in some interval. As I noted in the previous note, the only known way of defining probability in physics is that we should carry out an infinite number of experiments within and infinite time interval and this is problematic. Also, in quantum physics probability can be zero if there are superselection rules.

Hi all,

Congratulations for your beautiful essay dear Felix,The finite groups of Galois are relevants in my humble opinion when we want calculate rationally the quantic number and all its proportionalities.This system has a finite serie at my opinion.

To all, very relevant discussions.Don't stop dear Friends, hhihihi Laplace, Poisson and Gauss shall be happy to see these discussions and they shall say,; don't forget the theory of errors and the dispersions.....a kind of precison and sorting appears in the same rational logic.Like an Occam Raozr applied to maths for rational physics.

That permits to see better the serie towards the Planck scale and its finite number.

The infinity , the 0 and the - must be rationalized in the pure physicality and its pure laws in 3 Dimensions and a time constant of evolution.I d say even ,they doesn't really exist, if we add them yes, but not in our pure uniqueness, and their finite system and their pure number.

We can for example add or multiplicate our cosmological spheres, that doesn't mean that their number changes...their pure number inside an evolutive Unievrse rests like it is.It's the same for our quantum number, we can add or multiplicate them ,their pure number rests.It's a little like a proportional approximation in fact with rational limits.

Regards

Steve

report post as inappropriate

Congratulations for your beautiful essay dear Felix,The finite groups of Galois are relevants in my humble opinion when we want calculate rationally the quantic number and all its proportionalities.This system has a finite serie at my opinion.

To all, very relevant discussions.Don't stop dear Friends, hhihihi Laplace, Poisson and Gauss shall be happy to see these discussions and they shall say,; don't forget the theory of errors and the dispersions.....a kind of precison and sorting appears in the same rational logic.Like an Occam Raozr applied to maths for rational physics.

That permits to see better the serie towards the Planck scale and its finite number.

The infinity , the 0 and the - must be rationalized in the pure physicality and its pure laws in 3 Dimensions and a time constant of evolution.I d say even ,they doesn't really exist, if we add them yes, but not in our pure uniqueness, and their finite system and their pure number.

We can for example add or multiplicate our cosmological spheres, that doesn't mean that their number changes...their pure number inside an evolutive Unievrse rests like it is.It's the same for our quantum number, we can add or multiplicate them ,their pure number rests.It's a little like a proportional approximation in fact with rational limits.

Regards

Steve

report post as inappropriate

Dear Steve,

Thank you for encouraging words about my essay. Some your remarks are not clear to me and, probably, we have different opinions on some issues (e.g. on the role of geometry, whether the theory should be based on finite groups or Lie algebras over finite fields etc.). We could discuss them via email if you are interested. Happy New Year!

Felix.

Thank you for encouraging words about my essay. Some your remarks are not clear to me and, probably, we have different opinions on some issues (e.g. on the role of geometry, whether the theory should be based on finite groups or Lie algebras over finite fields etc.). We could discuss them via email if you are interested. Happy New Year!

Felix.

Hi all,

dear Felix,

You are welcome.

Indeed we have different points of vue(as many here on FQXi,the sharing of ideas seems essential), but the most important is this universality behind.

I like finite groups, and I think that maths must be analyzed with the biggest rationality when we analyze physics in its details.

I utilize algebras with an ocaam razzor,it exists several methods ,interestings and relevants.I add or superimpose them.

But I don't rest in one method.

In fact lie algebras, Clifford's alg.,.....are interestings when they respect the foundamental theorem of algebras.Now of course the physicality is the physicality.And the number is the number.

I see the quantum entanglement a little as our universe.Now if the entanglement of spheres is specific....the volumes are important and the number is the same and finite as the serie of volumes.The begining is a fractal of the main central sphere.Now I ask me how is the serie between 1 and our number of cosmological spheres.My problem is about the spheres between the center and our planets.And between 1 and 2 and 3........the volumes decrease on a specific harmonous serie.

Yes of course here is mine , a simple google mail.We can speak here you know I am transparent.

Ps sorry for my poor litteral english.

dufournybionature@gmail.com

Regards

Steve

report post as inappropriate

dear Felix,

You are welcome.

Indeed we have different points of vue(as many here on FQXi,the sharing of ideas seems essential), but the most important is this universality behind.

I like finite groups, and I think that maths must be analyzed with the biggest rationality when we analyze physics in its details.

I utilize algebras with an ocaam razzor,it exists several methods ,interestings and relevants.I add or superimpose them.

But I don't rest in one method.

In fact lie algebras, Clifford's alg.,.....are interestings when they respect the foundamental theorem of algebras.Now of course the physicality is the physicality.And the number is the number.

I see the quantum entanglement a little as our universe.Now if the entanglement of spheres is specific....the volumes are important and the number is the same and finite as the serie of volumes.The begining is a fractal of the main central sphere.Now I ask me how is the serie between 1 and our number of cosmological spheres.My problem is about the spheres between the center and our planets.And between 1 and 2 and 3........the volumes decrease on a specific harmonous serie.

Yes of course here is mine , a simple google mail.We can speak here you know I am transparent.

Ps sorry for my poor litteral english.

dufournybionature@gmail.com

Regards

Steve

report post as inappropriate

Dr. Lev ... thank you very much for your thoughtful reply. One problem with being a nice person is that people tend to take advantage of you, as I shall do now. This is a quote from a paper by Brukner and Zeilinger (see cite below). I'm wondering if you agree or not, or agree/disagree in part ...

“Clearly, a number of important questions remain open. Of these, we mention here two. The first refers to continuous variables. The problem there is that with continuous variables, one has in principle an infinite number of complementary observables. One might tackle this question by generalizing the definition of (3.4) to infinite sets. This, while mathematically possible, leads to conceptually difficult situations. The conceptual problem is in our view related to the fact that we wish to define all notions on operationally verifiable bases or foundations, that is, on foundations which can be verified directly in experiment. In our opinion, it is therefore suggestive that the concept of an infinite number of complementary observables and therefore, indirectly, the assumption of continuous variables, are just mathematical constructions which might not have a place in a final formulation of quantum mechanics.

“This leads to the second question, namely, how to derive the Schrödinger equation. ….”

from: "Quantum Physics as a Science of Information" (2005)

http://tinyurl.com/26dwfel

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“Clearly, a number of important questions remain open. Of these, we mention here two. The first refers to continuous variables. The problem there is that with continuous variables, one has in principle an infinite number of complementary observables. One might tackle this question by generalizing the definition of (3.4) to infinite sets. This, while mathematically possible, leads to conceptually difficult situations. The conceptual problem is in our view related to the fact that we wish to define all notions on operationally verifiable bases or foundations, that is, on foundations which can be verified directly in experiment. In our opinion, it is therefore suggestive that the concept of an infinite number of complementary observables and therefore, indirectly, the assumption of continuous variables, are just mathematical constructions which might not have a place in a final formulation of quantum mechanics.

“This leads to the second question, namely, how to derive the Schrödinger equation. ….”

from: "Quantum Physics as a Science of Information" (2005)

http://tinyurl.com/26dwfel

report post as inappropriate

Dear Dr. Lev,

Thanks again. I certainly thought so, which is why I made the connection. There's plenty of material out there to choose from but very little of it anywhere near as apposite. But of course the devil (or God, if you're Mies van der Rohe) is in the nevertheless-not-entirely-spiritual details.

I know that both of these guys are Community members and possible contest voters so not to press you further.

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Thanks again. I certainly thought so, which is why I made the connection. There's plenty of material out there to choose from but very little of it anywhere near as apposite. But of course the devil (or God, if you're Mies van der Rohe) is in the nevertheless-not-entirely-spiritual details.

I know that both of these guys are Community members and possible contest voters so not to press you further.

report post as inappropriate

Dear Felix,

You did a nice exploration and exposition of a possible application of finite fields in quantum theory. Your article on arxiv shows that you developed extensively this idea. Since I do not know any field in fundamental physics which is closed, or at least which accounts for all observations, I think that we should not demand new-born theories to be perfect and answer all questions. Let's let them grow up so that we can really compare them with others which were developed during one century by so many scientists. I think it is good to question them and to compare them with experiment even from the beginning, but I don't think that their value should be judged before their maturity.

I will ask some questions about your essay, if you don't mind. Please, if you feel that we disagree at some points, consider my questions as a proof of interest and curiosity.

Best regards,

Cristi Stoica

report post as inappropriate

You did a nice exploration and exposition of a possible application of finite fields in quantum theory. Your article on arxiv shows that you developed extensively this idea. Since I do not know any field in fundamental physics which is closed, or at least which accounts for all observations, I think that we should not demand new-born theories to be perfect and answer all questions. Let's let them grow up so that we can really compare them with others which were developed during one century by so many scientists. I think it is good to question them and to compare them with experiment even from the beginning, but I don't think that their value should be judged before their maturity.

I will ask some questions about your essay, if you don't mind. Please, if you feel that we disagree at some points, consider my questions as a proof of interest and curiosity.

Best regards,

Cristi Stoica

report post as inappropriate

Dear Felix,

I confess that I do not perceive the standard mathematics as being wrong, and the discrete or finite one as the only justified. I don't think I have enough information to decide whether our world is discrete or continuous. This is why I salute both directions of research, and I am interested in the arguments or evidence of each of them.

You said: "Standard mathematics is based on axioms about infinite sets (e.g., Zorn's lemma or Zermelo's axiom of choice), which are accepted without proof. Our belief that these axioms are correct is based on the fact that sciences using standard mathematics (physics, chemistry etc.) describe nature with a very high accuracy."

This triggered in my head the following questions (I would be pleased to receive answers from other readers too):

[?] As far as I know, Zorn's lemma and the axiom of choice are independent of the other axioms in set theory (although they are equivalent for example in Zermello-Fraenkel's system). Would it be possible to interpret one of the known experiments, or to devise a new one, so that we can check if they are valid from our world?

[?] All mathematical physics uses mathematics based on some axioms. Were some of these axioms tested directly, or only through their consequences (predictions)? Would it be possible, at least in principle, a physics based on axioms which are tested directly?

Best regards,

Cristi Stoica

report post as inappropriate

I confess that I do not perceive the standard mathematics as being wrong, and the discrete or finite one as the only justified. I don't think I have enough information to decide whether our world is discrete or continuous. This is why I salute both directions of research, and I am interested in the arguments or evidence of each of them.

You said: "Standard mathematics is based on axioms about infinite sets (e.g., Zorn's lemma or Zermelo's axiom of choice), which are accepted without proof. Our belief that these axioms are correct is based on the fact that sciences using standard mathematics (physics, chemistry etc.) describe nature with a very high accuracy."

This triggered in my head the following questions (I would be pleased to receive answers from other readers too):

[?] As far as I know, Zorn's lemma and the axiom of choice are independent of the other axioms in set theory (although they are equivalent for example in Zermello-Fraenkel's system). Would it be possible to interpret one of the known experiments, or to devise a new one, so that we can check if they are valid from our world?

[?] All mathematical physics uses mathematics based on some axioms. Were some of these axioms tested directly, or only through their consequences (predictions)? Would it be possible, at least in principle, a physics based on axioms which are tested directly?

Best regards,

Cristi Stoica

report post as inappropriate

Dear Cristi,

Thank you for encouraging words about my works.

I am not saying that standard mathematics is wrong. The question is whether we

i) accept a principle that only those statements have a physical significance, which can be experimentally verified (at least in principle) or ii) we agree that some statements (axioms) can be accepted without proof (for some reasons). Since you pose two questions [?] in your note, you probably think that we should accept i), right? But then we should acknowledge that standard axioms cannot be verified. For example, how can we verify that a+b=b+a for any natural numbers a and b?

Thank you for encouraging words about my works.

I am not saying that standard mathematics is wrong. The question is whether we

i) accept a principle that only those statements have a physical significance, which can be experimentally verified (at least in principle) or ii) we agree that some statements (axioms) can be accepted without proof (for some reasons). Since you pose two questions [?] in your note, you probably think that we should accept i), right? But then we should acknowledge that standard axioms cannot be verified. For example, how can we verify that a+b=b+a for any natural numbers a and b?

Dear Felix,

could my question "Would it be possible, at least in principle, a physics based on axioms which are tested directly?" be answered positively by a physics based on finite fields?

Could the habitants of a finite universe know everything about their world, just because there is a finite number of things to be known? It seems to me that they are "more finite" than the knowledge about their world. So, I would incline towards the second possibility you mentioned.

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could my question "Would it be possible, at least in principle, a physics based on axioms which are tested directly?" be answered positively by a physics based on finite fields?

Could the habitants of a finite universe know everything about their world, just because there is a finite number of things to be known? It seems to me that they are "more finite" than the knowledge about their world. So, I would incline towards the second possibility you mentioned.

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Dear Felix M. Lev,

Mathematics is a blind tool designed mainly to follow physics and describe the products made by physisists, but not a tool to discover something in physics; There are a few examples only when mathematical methods discovered something in physics, but a many thousands of erroneous mathematical papers and false mathematical "proofs". It is dangerous for physics because a lot of people mask their false and erroneous papers under mathematical formulas and mathematical theories.

I have examined some your papers in order to find what this mathematical instrument can really discover in PHYSICS; However, the most of your papers deals mainly with Galois fields. Even in your paper "A POSSIBLE MECHANISM OF GRAVITY" I don't found any physical mechanism of gravitation - mathematics only; Gravity is a manifestation of Galois fields? Can you explain how this manifestation of Calois fields can curve spacetime and slow down time? Also some your papers repeats the same information, for example the figure "Relation between Fp and the ring of integers" I saw in 3 your different papers.

Sincerely,

Constantin

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Mathematics is a blind tool designed mainly to follow physics and describe the products made by physisists, but not a tool to discover something in physics; There are a few examples only when mathematical methods discovered something in physics, but a many thousands of erroneous mathematical papers and false mathematical "proofs". It is dangerous for physics because a lot of people mask their false and erroneous papers under mathematical formulas and mathematical theories.

I have examined some your papers in order to find what this mathematical instrument can really discover in PHYSICS; However, the most of your papers deals mainly with Galois fields. Even in your paper "A POSSIBLE MECHANISM OF GRAVITY" I don't found any physical mechanism of gravitation - mathematics only; Gravity is a manifestation of Galois fields? Can you explain how this manifestation of Calois fields can curve spacetime and slow down time? Also some your papers repeats the same information, for example the figure "Relation between Fp and the ring of integers" I saw in 3 your different papers.

Sincerely,

Constantin

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Dear Constantin,

You write: "Mathematics is a blind tool designed mainly to follow physics and describe the products made by physisists, but not a tool to discover something in physics" and so on. This is really nonsense.

I noticed that sometimes the authority argument works better than a more rational one.

Isaac Newton was Lucasian Professor of Mathematics at the University of Cambridge. His contemporary and archenemy Gottfried Wilhelm Leibniz was a mathematician as well. His Monadology, considered to be utterly unphysical, started the research program which ended with the invention of the computer and of the www (namely the http).

Coming back to Lev paper, it is certainly nothing wrong, IN PRINCIPLE, to suggest that a physical phenomenon "is a manifestation of Galois fields", because, look, Newton himself had to invent differential calculus in order to establish his theory.

(Leibniz invented the same mathematical "blind instrument" simultaneously and better than Newton, for reasons which were independent of physics.)

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You write: "Mathematics is a blind tool designed mainly to follow physics and describe the products made by physisists, but not a tool to discover something in physics" and so on. This is really nonsense.

I noticed that sometimes the authority argument works better than a more rational one.

Isaac Newton was Lucasian Professor of Mathematics at the University of Cambridge. His contemporary and archenemy Gottfried Wilhelm Leibniz was a mathematician as well. His Monadology, considered to be utterly unphysical, started the research program which ended with the invention of the computer and of the www (namely the http).

Coming back to Lev paper, it is certainly nothing wrong, IN PRINCIPLE, to suggest that a physical phenomenon "is a manifestation of Galois fields", because, look, Newton himself had to invent differential calculus in order to establish his theory.

(Leibniz invented the same mathematical "blind instrument" simultaneously and better than Newton, for reasons which were independent of physics.)

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Dear Constantin Leshan,

In my understanding, you think that my works contain only mathematics but not physics, right? For example, in my essay I describe the following results of my works. In contrast with standard theory, in my approach based on finite fields there are no independent irreducible representations (IRs) for a particle and its antiparticle but one IR describes an object such that a particle and its antiparticle are different states of this object. As a consequence, there are no neutral elementary particles, the electric, baryon and lepton charges can be only approximately conserved and even the notion of particle-antiparticle is only approximate. In your opinion, these results are only mathematical or they have something to do with physics?

The problem of describing gravity is out of the scope of this contest; we could discuss this problem via email if you are interested.

In my understanding, you think that my works contain only mathematics but not physics, right? For example, in my essay I describe the following results of my works. In contrast with standard theory, in my approach based on finite fields there are no independent irreducible representations (IRs) for a particle and its antiparticle but one IR describes an object such that a particle and its antiparticle are different states of this object. As a consequence, there are no neutral elementary particles, the electric, baryon and lepton charges can be only approximately conserved and even the notion of particle-antiparticle is only approximate. In your opinion, these results are only mathematical or they have something to do with physics?

The problem of describing gravity is out of the scope of this contest; we could discuss this problem via email if you are interested.

In addition to what Marius said, I would mention:

- Riemannian geometry preceded its applications to general relativity

- Hilbert spaces preceded their applications to quantum theory

- Clifford algebras and spinors preceded their applications to relativistic quantum theory

- connections on fiber bundles preceded their applications to Yang-Mills theory

- holonomy groups preceded their applications to gauge theory, to Wilson loops and to loop quantum gravity

- representations of Lie group preceded their applications to particle physics

- topos theory preceded its applications to quantum theory obtained by Chris Isham

- the particular Kaehler manifolds named Calabi-Yau manifolds preceded their applications to string theory

I agree that mathematics originated from practical necessities, which come from the physical world. But mathematicians are playful species, and they like to explore platonic worlds as well. For some reason, their explorations anticipated many of the necessities of physics. Or maybe physicists find easier to borrow from mathematics, rather than making their own tools ;-). Or when they do, the tools are often full of divergences and singularities, and are inconsistent. As John Baez said once, it is the job of mathematicians to eliminate these inconsistencies. So I would say that both physicists and mathematicians have their equally important role.

___

Lebniz's monadology finds also applications in Haskell (programming language). It also influenced Whitehead, and through him some applications to quantum theory.

Best regards,

Cristi

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- Riemannian geometry preceded its applications to general relativity

- Hilbert spaces preceded their applications to quantum theory

- Clifford algebras and spinors preceded their applications to relativistic quantum theory

- connections on fiber bundles preceded their applications to Yang-Mills theory

- holonomy groups preceded their applications to gauge theory, to Wilson loops and to loop quantum gravity

- representations of Lie group preceded their applications to particle physics

- topos theory preceded its applications to quantum theory obtained by Chris Isham

- the particular Kaehler manifolds named Calabi-Yau manifolds preceded their applications to string theory

I agree that mathematics originated from practical necessities, which come from the physical world. But mathematicians are playful species, and they like to explore platonic worlds as well. For some reason, their explorations anticipated many of the necessities of physics. Or maybe physicists find easier to borrow from mathematics, rather than making their own tools ;-). Or when they do, the tools are often full of divergences and singularities, and are inconsistent. As John Baez said once, it is the job of mathematicians to eliminate these inconsistencies. So I would say that both physicists and mathematicians have their equally important role.

___

Lebniz's monadology finds also applications in Haskell (programming language). It also influenced Whitehead, and through him some applications to quantum theory.

Best regards,

Cristi

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Dear Marius Buliga and Cristi Stoica,

I agree that there are a few examples of successful predictions in physics made by mathematics, against the thousands of wrong predictions on the other hand; I can show that about 70 percents of all theoretical papers made by mathematicians in physics are wrong. For example, about hundreds of different theories of gravitation have been published in the academic journals, but it is self-evident that one or two similar gravitational theories only can be true at the same time, but not hundreds of theories. It is self-evident that 99 percents of all published gravitational theories are erroneous. Meanwhile all these erroneous theories have the "PERFECT MATHEMATICS" and "mathematical proofs" and are accepted by peer reviewed journals and physics community. Also the same situation arise in other areas of physics: about 70 percents of all theoretical papers in physics made by mathematicians are wrong. For example, let us analyze this paper this paper published in Physical Review Letters: I found tens of errors here whereas this paper has been supported by Physical Review Letters and NASA. You see, the authors try to prove their erroneous papers by help of mathematics; Thus, the mathematical proofs in physical theories must be in doubt. All the Standard Model is a mathematical model only that can compute only but explain nothing. The invasion of mathematicians will stop the development of Physics. That situation arise because peer reviewed Journals accepts papers with mathematical content only. Journals should accept that a physical logic (reasoning) must have equal rights with mathematical proofs.

Dear Felix M Lev

Yes, your mathematical approach tries to describe the EXISTING already physical phenomena only. Your results confirm my point of view that mathematics must be a tool to describe quantitatively products (theories and phenomena) made/discovered by physicists only, but it is not a indicator for physicists what they must do. Because physics is directed by mathematics, it is a cause of modern crisis in physics.

Sincerely,

Constantin

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I agree that there are a few examples of successful predictions in physics made by mathematics, against the thousands of wrong predictions on the other hand; I can show that about 70 percents of all theoretical papers made by mathematicians in physics are wrong. For example, about hundreds of different theories of gravitation have been published in the academic journals, but it is self-evident that one or two similar gravitational theories only can be true at the same time, but not hundreds of theories. It is self-evident that 99 percents of all published gravitational theories are erroneous. Meanwhile all these erroneous theories have the "PERFECT MATHEMATICS" and "mathematical proofs" and are accepted by peer reviewed journals and physics community. Also the same situation arise in other areas of physics: about 70 percents of all theoretical papers in physics made by mathematicians are wrong. For example, let us analyze this paper this paper published in Physical Review Letters: I found tens of errors here whereas this paper has been supported by Physical Review Letters and NASA. You see, the authors try to prove their erroneous papers by help of mathematics; Thus, the mathematical proofs in physical theories must be in doubt. All the Standard Model is a mathematical model only that can compute only but explain nothing. The invasion of mathematicians will stop the development of Physics. That situation arise because peer reviewed Journals accepts papers with mathematical content only. Journals should accept that a physical logic (reasoning) must have equal rights with mathematical proofs.

Dear Felix M Lev

Yes, your mathematical approach tries to describe the EXISTING already physical phenomena only. Your results confirm my point of view that mathematics must be a tool to describe quantitatively products (theories and phenomena) made/discovered by physicists only, but it is not a indicator for physicists what they must do. Because physics is directed by mathematics, it is a cause of modern crisis in physics.

Sincerely,

Constantin

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[I apologize to Felix, I do not want to monopolize this thread. I would kindly ask a FQXi admin who validates the comments to move the discussion to a different thread, if it is off topic. Or perhaps to move this comment and the father comment as children to the discussion opened by Constantin Leshan, so that the discussion gets collapsed and does not occupy too much of this page.]

Dear...

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Dear...

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Dear Cristi Stoica,

The border between mathematicians and physicists is very thin; therefore by papers of mathematicians I mean papers were the percentage of mathematics is more than 40 of volume, or all proofs are mathematical; From this point of view, almost all gravitational theories are mathematical theories, made by mathematicians. Also, I do not accuse all mathematical world; I say that all false physical theories have mathematical proofs, consequently mathematical proofs in physical theories must be in doubt. Therefore, it is mathematics' fault that the most of published physical papers are wrong.

It is because journals accept papers with mathematical content only. I'm sure that the percentage of false theories may fall, if the journals allow physical reasoning instead of mathematical proofs. Thus, Journals must allow to physicists to publish their papers with physical proofs instead of mathematical. Since the Standard Model is more mathematical model than physical, therefore I accuse mathematics. We'll never find any Higgs boson because SM is a mathematical model only that may fall in nearest future. The future Physics will be based on physical reasoning rather than on mathematical proofs.

About moving this comment to the discussion opened by Constantin Leshan: soon I'll send my essay to FQXI. I invite you to find logical errors in my theory.

Sincerely,

Constantin Leshan

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The border between mathematicians and physicists is very thin; therefore by papers of mathematicians I mean papers were the percentage of mathematics is more than 40 of volume, or all proofs are mathematical; From this point of view, almost all gravitational theories are mathematical theories, made by mathematicians. Also, I do not accuse all mathematical world; I say that all false physical theories have mathematical proofs, consequently mathematical proofs in physical theories must be in doubt. Therefore, it is mathematics' fault that the most of published physical papers are wrong.

It is because journals accept papers with mathematical content only. I'm sure that the percentage of false theories may fall, if the journals allow physical reasoning instead of mathematical proofs. Thus, Journals must allow to physicists to publish their papers with physical proofs instead of mathematical. Since the Standard Model is more mathematical model than physical, therefore I accuse mathematics. We'll never find any Higgs boson because SM is a mathematical model only that may fall in nearest future. The future Physics will be based on physical reasoning rather than on mathematical proofs.

About moving this comment to the discussion opened by Constantin Leshan: soon I'll send my essay to FQXI. I invite you to find logical errors in my theory.

Sincerely,

Constantin Leshan

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Dear Constantin,

I am sympathetic to your viewpoint that these day physics is too much math without physical content. I stop now because I took too much space from Felix with my off topic comments. We can continue by email. Good luck with your forthcoming essay.

Cristi

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I am sympathetic to your viewpoint that these day physics is too much math without physical content. I stop now because I took too much space from Felix with my off topic comments. We can continue by email. Good luck with your forthcoming essay.

Cristi

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I like the idea of using the Galois fields in physics. Are you aware of the result that it is possible to construct a complete set of mutually unbiased basis for finite dimensional quantum systems if and only if the base is indexed by a Galois field?

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A direct generalization of mutually unbiased bases to the case of Galois fields is meaningless for several reasons. For example, ½ in Galois fields is (p+1)/2, i.e. a huge number if p is huge. In standard theory, probabilities are normalized to one but this is only a matter of convention since not the probability itself has a physical meaning but only ratios of probabilities of different experimental outcomes have (that’s why Hilbert spaces in quantum theory are projective). In addition, as noted in my essay and papers, in theories over Galois fields the notion of probability can be only approximate. However, I believe that in situations when probability is meaningful, it is not difficult to modify the definition of mutually unbiased bases such that the main idea of the definition will be implemented. But the question that the base should be indexed by a Galois field is not clear to me. For example, when we have a finite dimensional linear space over a Galois field, we don’t say that the basis elements are indexed by a Galois field, right?

I read your paper with interest. I do have a couple of questions about the idea of Galois QFT. The cyclotomic numbers of F_4 z = e^{i2πn/3} describe the root space of D_4. F_4 is the Dynkin diagram for D_4 ~ SO(8). The D_4 root lattice is the dual of the F_4 and a subring of Hurwitz quaternions. In this way Galois groups can characterize the symmetries of a QFT, or a YM gauge theory.

Cheers LC

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Cheers LC

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PS

I realized I used F_4 in two different contexts. At first I use F_4 as the Galois field, but then in reference with the D_4 root lattice I am referring to the exceptional F_4.

Cheers LC

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I realized I used F_4 in two different contexts. At first I use F_4 as the Galois field, but then in reference with the D_4 root lattice I am referring to the exceptional F_4.

Cheers LC

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Thank you for your interest to my essay. Here I argue that any fundamental physics can be based only on a finite math and consider an approach based on a Galois field. So I cannot have e^{i2πn/3} or SO(8). Also, I cannot use Dynkin diagrams for describing Lie algebras over Galois fields since the latter are not algebraically closed. Your first F_4 is not a Galois field; probably you mean a Galois group. Galois groups are used for describing field extensions and in general the fields are not assumed to be necessarily Galois ones. Since you are talking about cyclotomic numbers, you probably mean extensions of Q. So, in my understanding, your questions refer to standard theory but not to my approach.

Best regards, Felix Lev.

Best regards, Felix Lev.

Hi to both of you,

Happy dear lev to see this rationality, indeed the finite systems must respected their own limits.

We can superimpose but with rationality of course.

Finite maths....galois field.very relevant indeed , very relevant.

Steve

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Happy dear lev to see this rationality, indeed the finite systems must respected their own limits.

We can superimpose but with rationality of course.

Finite maths....galois field.very relevant indeed , very relevant.

Steve

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Dear Lev,

in your essay you make the clear point that finite mathematics (such as GFQT) is the most pertinent choice for describing physical reality.

However, you do not seem to take an equally clear position about the ultimate nature of reality: is the universe (discrete and) finite or infinite? In fact, are both possibilites still open, under GFQT?

Do you perhaps envisage a third possibility, namely that GFQT works very nicely for just making accurate experimental predictions in a QM setting, without still resolving this finite vs. infinite universe puzzle, which is perhaps only of philosophical relevance?

If the question sounds indeed too philosophical (but that's essentially the title of the contest...), I could reformulate it as follows: would your theory be compatible, incompatible, or neutral, with a statement such as 'there are 10^234 atoms of spacetime in the universe'?

A second question. You talk about parameters p and n, defining the size of the GF, as universal constants. Would it make any sense to rather imagine them as changing, I mean on a cosmological scale?

Thanks!

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in your essay you make the clear point that finite mathematics (such as GFQT) is the most pertinent choice for describing physical reality.

However, you do not seem to take an equally clear position about the ultimate nature of reality: is the universe (discrete and) finite or infinite? In fact, are both possibilites still open, under GFQT?

Do you perhaps envisage a third possibility, namely that GFQT works very nicely for just making accurate experimental predictions in a QM setting, without still resolving this finite vs. infinite universe puzzle, which is perhaps only of philosophical relevance?

If the question sounds indeed too philosophical (but that's essentially the title of the contest...), I could reformulate it as follows: would your theory be compatible, incompatible, or neutral, with a statement such as 'there are 10^234 atoms of spacetime in the universe'?

A second question. You talk about parameters p and n, defining the size of the GF, as universal constants. Would it make any sense to rather imagine them as changing, I mean on a cosmological scale?

Thanks!

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Dear Tommaso,

Thank you very much for your very important questions. Probably I need a few days to describe what I think.

Best regards, Felix Lev.

Thank you very much for your very important questions. Probably I need a few days to describe what I think.

Best regards, Felix Lev.

Dear Tomasso,

I will try to answer your questions.

First of all, let me note that in my understanding, the question “Is Reality Digital Or Analog?” is meaningful only if it is understood as a question about mathematics describing reality. Some contest participants argue that e.g. mathematics might be continuous but physics - discrete but I don’t understand such arguments. In my...

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I will try to answer your questions.

First of all, let me note that in my understanding, the question “Is Reality Digital Or Analog?” is meaningful only if it is understood as a question about mathematics describing reality. Some contest participants argue that e.g. mathematics might be continuous but physics - discrete but I don’t understand such arguments. In my...

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If you work together, Lawrence, Tommasi and Hector and Florin...you shall ponder very interestings things...

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De brogle maths togther with Einxstins dice programmed to obey the rules 1 ODD THROW+ 1 EVEN THROW= 2 ODD THROWS.

And 2 ODD THROWS+ 2 EVEN THROWS= 4 EVEN THROWS.

Can be used to generate a simulated quantum universe on computer where everything is determined.

And we cna develop an algorythm to predict random numbers in our real universe.

This maths apples before the big bang where foru states are one and this determines the fact that the four forces are one.

Everthing is determined when the four forces are one.

And Einsteins thoery determines everything that happens before the big bang................

Do you appreciate this contribution let me know.

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And 2 ODD THROWS+ 2 EVEN THROWS= 4 EVEN THROWS.

Can be used to generate a simulated quantum universe on computer where everything is determined.

And we cna develop an algorythm to predict random numbers in our real universe.

This maths apples before the big bang where foru states are one and this determines the fact that the four forces are one.

Everthing is determined when the four forces are one.

And Einsteins thoery determines everything that happens before the big bang................

Do you appreciate this contribution let me know.

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Dear Felix ,

I will not download your paper to read as I can see from your discussion with Florin that it is concerned with the mathematical description of reality rather than the physics of what is real.I have read your argument that it is a valid approach to the essay question. As I have no mathematical background and am not even a physics specialist I could not begin to comprehend what you have written in your essay. Likewise I would be unable to decipher a physics paper written by Lawrence Crowell.

With respect Sir can I just ask, Is reality digital or analogue? Did you come to a conclusion? or did you argue something entirely different?

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I will not download your paper to read as I can see from your discussion with Florin that it is concerned with the mathematical description of reality rather than the physics of what is real.I have read your argument that it is a valid approach to the essay question. As I have no mathematical background and am not even a physics specialist I could not begin to comprehend what you have written in your essay. Likewise I would be unable to decipher a physics paper written by Lawrence Crowell.

With respect Sir can I just ask, Is reality digital or analogue? Did you come to a conclusion? or did you argue something entirely different?

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Dear Felix,

I had not read the entire content of your thread, which is now rather lengthy. On looking to see if you had posted a reply to my question I noticed your reply to Tomasso above, which actually does answer my question very well.

You have said "First of all, let me note that in my understanding, the question "Is Reality Digital Or Analog?" is meaningful only if it is understood as a question about mathematics describing reality." Which is probably because you are mathematician and think like a mathematician. Which is not a criticism but a possible reason for your particular thinking style and approach to the problem.

I actually agree that all we can do is create models which we hope describe reality as we can not fully construct the reality itself or know what lies beneath our mental interpretation and the descriptions created by human minds, verbal or mathematical.

I can see that you have given a full explanation to Tomasso. I can not grasp all of it but the last paragraph caught my attention. You say you agree with Heisenburg that a fundamental physical theory should not involve space-time at all. I agree that at the most foundation level space-time does not exist but that it is an emergent reality produced subsequent to interception of data by an observer, whether that is a conscious entity or an inanimate reality interface such as a camera or other recording device.

I wish I was able to discuss your essay in more detail. I have found some of the conversation in your thread most interesting.

Kind regards and good luck, Georgina.

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I had not read the entire content of your thread, which is now rather lengthy. On looking to see if you had posted a reply to my question I noticed your reply to Tomasso above, which actually does answer my question very well.

You have said "First of all, let me note that in my understanding, the question "Is Reality Digital Or Analog?" is meaningful only if it is understood as a question about mathematics describing reality." Which is probably because you are mathematician and think like a mathematician. Which is not a criticism but a possible reason for your particular thinking style and approach to the problem.

I actually agree that all we can do is create models which we hope describe reality as we can not fully construct the reality itself or know what lies beneath our mental interpretation and the descriptions created by human minds, verbal or mathematical.

I can see that you have given a full explanation to Tomasso. I can not grasp all of it but the last paragraph caught my attention. You say you agree with Heisenburg that a fundamental physical theory should not involve space-time at all. I agree that at the most foundation level space-time does not exist but that it is an emergent reality produced subsequent to interception of data by an observer, whether that is a conscious entity or an inanimate reality interface such as a camera or other recording device.

I wish I was able to discuss your essay in more detail. I have found some of the conversation in your thread most interesting.

Kind regards and good luck, Georgina.

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Dear Georgina,

Thank you for your interesting remarks. The notion of spacetime is now one of the most debated questions of modern physics. I need probably a few days to describe my understanding of this notion.

Best regards, Felix.

Thank you for your interesting remarks. The notion of spacetime is now one of the most debated questions of modern physics. I need probably a few days to describe my understanding of this notion.

Best regards, Felix.

Dear Georgina,

Thank you for your interest to my essay. First of all, let me note that I am not a mathematician since am not working on mathematical theories. Mathematicians work with theories based on sets of axioms; typically they don’t discuss how their theories apply to reality. But physics cannot be without math. In my essay I argue that any fundamental physical theory can be based...

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Thank you for your interest to my essay. First of all, let me note that I am not a mathematician since am not working on mathematical theories. Mathematicians work with theories based on sets of axioms; typically they don’t discuss how their theories apply to reality. But physics cannot be without math. In my essay I argue that any fundamental physical theory can be based...

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We conclude that the very notion of particle-antiparticle is approximate

and the electric, baryon and lepton charges are only approximately conserved quantities.

The non-conservation of the baryon and lepton quantum numbers has been

already considered in models of Grand Unification but the electric charge has been

always believed to be a strictly conserved quantum number. The non-conservation

of these quantum numbers also completely changes the status of the problem known

as ”baryon asymmetry of the Universe” since at early stages of the Universe energies

were much greater than now and therefore transitions between particles and

antiparticles had a much greater probability.

Felix,

This quote from your essay seems to support the analogue nature of particles, the inclination to change energies and nature like the electron neutrinos changing to tau neutrinos after emerging from the sun's fusion.

Your argument is esoteric, yet convincing, especially to the mathematically challenged like myself.

Jim Hoover

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and the electric, baryon and lepton charges are only approximately conserved quantities.

The non-conservation of the baryon and lepton quantum numbers has been

already considered in models of Grand Unification but the electric charge has been

always believed to be a strictly conserved quantum number. The non-conservation

of these quantum numbers also completely changes the status of the problem known

as ”baryon asymmetry of the Universe” since at early stages of the Universe energies

were much greater than now and therefore transitions between particles and

antiparticles had a much greater probability.

Felix,

This quote from your essay seems to support the analogue nature of particles, the inclination to change energies and nature like the electron neutrinos changing to tau neutrinos after emerging from the sun's fusion.

Your argument is esoteric, yet convincing, especially to the mathematically challenged like myself.

Jim Hoover

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Jim,

It is not clear to me why, in your opinion, this quote from my essay "seems to support the analogue nature of particles". As shown in the essay, there is a correspondence principle between my approach and standard theory. In particular, there are no obstacles for recovering the results on neutrino oscillations in my approach.

Felix.

It is not clear to me why, in your opinion, this quote from my essay "seems to support the analogue nature of particles". As shown in the essay, there is a correspondence principle between my approach and standard theory. In particular, there are no obstacles for recovering the results on neutrino oscillations in my approach.

Felix.

Felix,

Fascinating essay. You give me new hope that a true algebraic theory can subsume field representations.

I hope you get a chance to read my entry as well.

Good luck!

Tom

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Fascinating essay. You give me new hope that a true algebraic theory can subsume field representations.

I hope you get a chance to read my entry as well.

Good luck!

Tom

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Tom,

Thank you for your compliments. The ideas that the ultimate theory will not be based on local fields were very popular in 60th (e.g. the Heisenberg S-matrix program) but now those ideas are almost forgotten.

I read your essay and tried to understand your position. You quote many well known scientists who had different opinions. However, so far I could not understand what your preferences are. Probably a more careful reading is required.

Felix.

Thank you for your compliments. The ideas that the ultimate theory will not be based on local fields were very popular in 60th (e.g. the Heisenberg S-matrix program) but now those ideas are almost forgotten.

I read your essay and tried to understand your position. You quote many well known scientists who had different opinions. However, so far I could not understand what your preferences are. Probably a more careful reading is required.

Felix.

Felix,

Briefly, my personal preference is for quantum field theory in a continuous function model. I find this possible only in an extradimensional theory. I discussed my own preference only in the technical endnote, because my intent in the essay was to survey how subtle the question of continuous vs. discrete really is.

I agree with you on the problematic nature of Zorn's lemma (axiom of choice). On that issue, you might be interested in my ICCS 2006 paper in which I show (see particularly 5.6 - 5.9) how a well ordered sequence is derived without appeal to AC.

Best,

Tom

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Briefly, my personal preference is for quantum field theory in a continuous function model. I find this possible only in an extradimensional theory. I discussed my own preference only in the technical endnote, because my intent in the essay was to survey how subtle the question of continuous vs. discrete really is.

I agree with you on the problematic nature of Zorn's lemma (axiom of choice). On that issue, you might be interested in my ICCS 2006 paper in which I show (see particularly 5.6 - 5.9) how a well ordered sequence is derived without appeal to AC.

Best,

Tom

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Sub: Possibility of manipulation in judging criteria – suggestions for improvement.

Sir,

We had filed a complaint to FQXi and Scienticfic American regarding Possibility of manipulation in judging criteria and giving some suggestions for improvement. Acopy of our letter is enclosed for your kind information.

“We are a non-professional and non-academic entrant to the Essay...

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Sir,

We had filed a complaint to FQXi and Scienticfic American regarding Possibility of manipulation in judging criteria and giving some suggestions for improvement. Acopy of our letter is enclosed for your kind information.

“We are a non-professional and non-academic entrant to the Essay...

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Thank you for your explanations.

Good luck, Felix.

visit here:https://www.jaipurgirlsescorts.com/independent-girls-in

-jaipur/

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Good luck, Felix.

visit here:https://www.jaipurgirlsescorts.com/independent-girls-in

-jaipur/

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