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**Eckard Blumschein**: *on* 11/27/09 at 20:56pm UTC, wrote Dear Enrico, You wrote: ... in the case of CM the C∗-algebra with...

**Lev Goldfarb**: *on* 11/6/09 at 13:25pm UTC, wrote Dear Enrico, I like the general orientation of your essay! However,...

**Enrico**: *on* 11/5/09 at 9:07am UTC, wrote Dear Nath and Terry, thank you for your comments and questions. I would...

**Eckard Blumschein**: *on* 11/4/09 at 19:26pm UTC, wrote Dear Enrico, I am explaining the T-symmetry in QM as due to improper...

**Edwin Klingman**: *on* 11/2/09 at 6:42am UTC, wrote Dear Enrico Prati, To some degree your approach seems to resemble the...

**N Nath**: *on* 10/26/09 at 5:20am UTC, wrote Dear Enrico, The author has yet to choose to respond to my comments of Oct...

**Terry Padden**: *on* 10/14/09 at 23:26pm UTC, wrote Enrico Thanks for responding to my point on how one knows the GNS...

**Enrico**: *on* 10/14/09 at 8:01am UTC, wrote I would like to thank all the readers of my Essay which left a comment with...

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CATEGORY:
What's Ultimately Possible in Physics? Essay Contest (2009)
[back]

TOPIC: The experimental method and the constitutive limits of the mathematical description of physics by Enrico Prati [refresh]

TOPIC: The experimental method and the constitutive limits of the mathematical description of physics by Enrico Prati [refresh]

Nature is believed to be organized by a mathematical fundamental structure. Therefore, the experiments are interpreted through mathematical models. Unfortunately, experiments can only provide macroscopic outputs, even when referred to quantum elementary object. Starting from such observation, I first consider the concept of anomaly as groundbreaking information to falsify a theory. The separability of a system between an experimental equipment and a microscopic object is discussed. Non commutative microscopic observables of elementary entities are postulated from a set of measurements of macroscopic observables interpreted as their eigenvalues. I explain the major role of the Gelfand-Naimark-Segal construction of the representation of classical and quantum abstract $C^*$-algebras to recognize the impossibility of building a theory with a unified domain for the microscopic and the unavoidable macroscopic observables. I discuss implications of the Gelfand theorems on both macrorealism emergence from coarse grained measurements and decoherence programs. Finally I apply the results to determine the fundamental impossibility to identify a Theory of Everything with the mathematical structure attributed to Nature.

Enrico Prati is permanent research scientist of Italian CNR at the Laboratorio Nazionale MDM. His main research interests are quantum transport, spin dynamics and decoherence in nanoscaled quantum devices, and the transition from quantum to classical physics. He is also involved in the broad field of emerging properties of metamaterials. He is particularly interested in the foundations of the physics of time. He has received the 2004 URSI-B Commission Young Scientist Award for his research in metamaterials and the 4th Jury Prize for its Essay on the Nature of Time from FQXI and J. Templeton Foundation.

Dear Enrico,

I have enjoyed your essay very much. I already knew about 2 arguments why a composed classical-QM system is inconsistent, and you made an interesting third case from the GNS construction point of view.

If you are interesed to discus it, I would very much like to see what your opinion on the measurement problem is.

One thing I disagree is that the TOE if it indeed exists, it will be in a closed MSN form. I have 2 strong arguments for this. First, axiomatizing physics is indeed possible, but not in the standard mathematical understanding of the term (I can expand on this if you are interested: see my entry: “Heuristic rule …”). Second, a closed MSN form if it indeed possible, it should include a natural unification of QM and relativity. This mathematical structure was already identified into a non-commutative, non-division algebra, but we already know that this algebra is not enough to construct the full TOE at this time.

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I have enjoyed your essay very much. I already knew about 2 arguments why a composed classical-QM system is inconsistent, and you made an interesting third case from the GNS construction point of view.

If you are interesed to discus it, I would very much like to see what your opinion on the measurement problem is.

One thing I disagree is that the TOE if it indeed exists, it will be in a closed MSN form. I have 2 strong arguments for this. First, axiomatizing physics is indeed possible, but not in the standard mathematical understanding of the term (I can expand on this if you are interested: see my entry: “Heuristic rule …”). Second, a closed MSN form if it indeed possible, it should include a natural unification of QM and relativity. This mathematical structure was already identified into a non-commutative, non-division algebra, but we already know that this algebra is not enough to construct the full TOE at this time.

report post as inappropriate

Dear Florin,

I've appreciated you comment. I'm not sure what do you mean with "what I disagree is that the TOE if it indeed exists, it will be in a closed MSN form". In my paper I've tried to demonstrate that, 1] according to the interpretation of the very general Gelfand theorems in the extremely general framework of the abstract C*-algebras, an extremely weak constraint necessary to have some entity to speak about (observables), and 2] according to the observation that microscopic observables are made of selected sets of macroscopic outputs, THEN there is no way to provide a meaning to the ToE program, neither in the positivist, nor in the strong nor in the weak versions. I conclude that ToEs are not theories with some meaningful definition, and the program to describe a MSN with a mathematical theory crashes with our intrinsic mathematical and experimental limits.

Therefore a ToE will never represent a problem to represent the hypotheical MSN, since it cannot be written.

report post as inappropriate

I've appreciated you comment. I'm not sure what do you mean with "what I disagree is that the TOE if it indeed exists, it will be in a closed MSN form". In my paper I've tried to demonstrate that, 1] according to the interpretation of the very general Gelfand theorems in the extremely general framework of the abstract C*-algebras, an extremely weak constraint necessary to have some entity to speak about (observables), and 2] according to the observation that microscopic observables are made of selected sets of macroscopic outputs, THEN there is no way to provide a meaning to the ToE program, neither in the positivist, nor in the strong nor in the weak versions. I conclude that ToEs are not theories with some meaningful definition, and the program to describe a MSN with a mathematical theory crashes with our intrinsic mathematical and experimental limits.

Therefore a ToE will never represent a problem to represent the hypotheical MSN, since it cannot be written.

report post as inappropriate

Dear Enrico,

Oops, my mistake. I only read carefully the GNS part and completely misunderstood your last section (too many new essays to read I guess). I fully agree with your conclusions. Very good essay.

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Oops, my mistake. I only read carefully the GNS part and completely misunderstood your last section (too many new essays to read I guess). I fully agree with your conclusions. Very good essay.

report post as inappropriate

Hi Enrico,

An interesting essay. Good Luck. I wonder whether you are familiar with N.P.Landsman's "When champions meet: Rethinking the Bohr–Einstein debate", Studies in History and Philosophy of Modern Physics 37 (2006) 212–242, which seems to me relevant to your paper? The paper is available from the author's web site, http://www.math.ru.nl/~landsman/eprints.html (at #60). I also consider that my recent "Equivalence of the Klein-Gordon random field and the complex Klein-Gordon quantum field", EPL 87 (2009) 31002 (available from my web-page, http://pantheon.yale.edu/~pwm22/), has something to say to the idea that classical and quantum are irreconcilable.

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An interesting essay. Good Luck. I wonder whether you are familiar with N.P.Landsman's "When champions meet: Rethinking the Bohr–Einstein debate", Studies in History and Philosophy of Modern Physics 37 (2006) 212–242, which seems to me relevant to your paper? The paper is available from the author's web site, http://www.math.ru.nl/~landsman/eprints.html (at #60). I also consider that my recent "Equivalence of the Klein-Gordon random field and the complex Klein-Gordon quantum field", EPL 87 (2009) 31002 (available from my web-page, http://pantheon.yale.edu/~pwm22/), has something to say to the idea that classical and quantum are irreconcilable.

report post as inappropriate

Enrico

A first rate essay exhibiting breadth and depth of knowledge and comprehension.

Several essays focus on maths as the limit determining the Ultimates for Physics. My essay widens the focus to include Logic, i.e. the limitation is our formalisms, our RATIONAL mental structures. I identify 2 "Ultimates". My first is the same as yours but determined by elementary dualist (Reason v Rationality, Mental / Physical, Maths / Physics) considerations - not the in depth approach you so effectively use. So our reasons for arriving at the same conclusion are different.

You identify one "Ultimate" - that because the mathematical GNS structure is fundamental (to nature ?) it is "Ultimately" impossible to provide a mathematical formulation of a Grand Unified Theory of Physics based on Elementary Particles. ( I hope those words accurately express your claim.)

I would appreciate your comments on what I see as weaknesses in your argument:

1. Mathematical structures are in no way "Natural". they are Platoist abstract mental creations, formalisms. They only become natural when they can be shown by measurement to correspond to some aspect of nature.

2. How do you know the GNS structure is fundamental ? Perhaps it is not.

3. You are accepting "Current" maths as a fixed limit. Essays by e.g. Grgin and Goldfarb show 2 ways that "Current" maths can be (in my terms "progressively") extended. Thus maybe current maths is not a limit. Like G & G I take that position, but argue that the progressive approach is misguided. My second "Ultimate" is that current maths is - and always has been - Incomplete. So we can escape GNS and other limits by completing it. So we can have a maths for unified physics after removing the limits.

4. You confine your GUT* to the physics of elementary particles; i.e. to Reductionist physics. Laughlin & Anderson etc. have shown the reality of physical emergence - and so does Nature. GUT's are immediate challenges for physics; not "Ultimate" The "Ultimate" ones are emergence and consciousness.

PS * Please, please abandon the term TOE. Reductionist physics can never provide a TOE. Even the ones who invented the term have abandoned it and regret they ever used it.

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A first rate essay exhibiting breadth and depth of knowledge and comprehension.

Several essays focus on maths as the limit determining the Ultimates for Physics. My essay widens the focus to include Logic, i.e. the limitation is our formalisms, our RATIONAL mental structures. I identify 2 "Ultimates". My first is the same as yours but determined by elementary dualist (Reason v Rationality, Mental / Physical, Maths / Physics) considerations - not the in depth approach you so effectively use. So our reasons for arriving at the same conclusion are different.

You identify one "Ultimate" - that because the mathematical GNS structure is fundamental (to nature ?) it is "Ultimately" impossible to provide a mathematical formulation of a Grand Unified Theory of Physics based on Elementary Particles. ( I hope those words accurately express your claim.)

I would appreciate your comments on what I see as weaknesses in your argument:

1. Mathematical structures are in no way "Natural". they are Platoist abstract mental creations, formalisms. They only become natural when they can be shown by measurement to correspond to some aspect of nature.

2. How do you know the GNS structure is fundamental ? Perhaps it is not.

3. You are accepting "Current" maths as a fixed limit. Essays by e.g. Grgin and Goldfarb show 2 ways that "Current" maths can be (in my terms "progressively") extended. Thus maybe current maths is not a limit. Like G & G I take that position, but argue that the progressive approach is misguided. My second "Ultimate" is that current maths is - and always has been - Incomplete. So we can escape GNS and other limits by completing it. So we can have a maths for unified physics after removing the limits.

4. You confine your GUT* to the physics of elementary particles; i.e. to Reductionist physics. Laughlin & Anderson etc. have shown the reality of physical emergence - and so does Nature. GUT's are immediate challenges for physics; not "Ultimate" The "Ultimate" ones are emergence and consciousness.

PS * Please, please abandon the term TOE. Reductionist physics can never provide a TOE. Even the ones who invented the term have abandoned it and regret they ever used it.

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Greetings Enrico. Your essay involves interesting questions regarding macroscopic vs. microscopic. Balance and completeness go hand-in-hand in theory. This can include a sort of "neutrality" involving the unification/merger of opposites. Isn't a key component of unifying gravity and electromagnetism/light the demonstration/understanding of scale as balanced by representing space as BOTH invisible and visible?

Do you agree that the fundamental union of gravity and electromagnetism/light necessarily/ideally involves balancing scale by making gravity repulsive and attractive as electromagentic energy/light? Given the pervasive effects of electromagnetism/light, closely consider: 1)No time at light speed. 2)One cannot catch up to a photon. 3)No feeling of gravity in outer space. 4)Our relative/natural immobilization in outer space (and in reference to photons). 5) Now look at the extremes of size/visibility/energy/brightness involving electromagnetism/light (the Sun and photons).

You consider the effectivness of mathematical description. The known mathematical unification of Einstein's theory of gravity (general relativity) with Maxwell's theory of light (electromagnetism) that is achieved by the addition of a fourth dimension of space to Einstein's theory must be plainly and significantly obvious in our direct experience. Do you agree?

We are both addressing very fundamental areas/topics of concern in physics.

My essay is fourth from the top. Thank you for your concern and consideration.

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Do you agree that the fundamental union of gravity and electromagnetism/light necessarily/ideally involves balancing scale by making gravity repulsive and attractive as electromagentic energy/light? Given the pervasive effects of electromagnetism/light, closely consider: 1)No time at light speed. 2)One cannot catch up to a photon. 3)No feeling of gravity in outer space. 4)Our relative/natural immobilization in outer space (and in reference to photons). 5) Now look at the extremes of size/visibility/energy/brightness involving electromagnetism/light (the Sun and photons).

You consider the effectivness of mathematical description. The known mathematical unification of Einstein's theory of gravity (general relativity) with Maxwell's theory of light (electromagnetism) that is achieved by the addition of a fourth dimension of space to Einstein's theory must be plainly and significantly obvious in our direct experience. Do you agree?

We are both addressing very fundamental areas/topics of concern in physics.

My essay is fourth from the top. Thank you for your concern and consideration.

report post as inappropriate

i am incompetent person to comment on this essay, being an experimentalist with low energy nuclear physics background. i however see some innovative points made in this essay concerning the experimental measurements and their comparison with one another. The measurements made in the microscopic world only gives macroscopic values. Also, mathematically different theories based on classical and quantum approach have no meeting ground.

i however note two essays that deal with similar aspects. One is by tejinder Singh and other is by Edwin Klingman. The former talks about a region in between called mesoscopic where neither classical nor quantum mechanics may be valid strictly. The latter harmonize all different interactions into gravity, the first field to emerge in the universe and then introduces another field called 'consciousness' that togather with gravity , are the two commonly sensed/experienced field. The latter further consider that mass matter results in both these. The former represents it as an ensemble while the C-field is attributed to a rotation about the mass ensemble. The two together are claimed to describe the Physics. Both thesae approaches can be tested experimentally!

i personally feel that the absolute truth lies beyond science but it can certainly approach it in a better and better relative manner. That is what Physics is already doing very well. Howver, one needs breadth in all considerations and an open mind to all the ideas that become available either directly or indirctly to an individual scientist.

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i however note two essays that deal with similar aspects. One is by tejinder Singh and other is by Edwin Klingman. The former talks about a region in between called mesoscopic where neither classical nor quantum mechanics may be valid strictly. The latter harmonize all different interactions into gravity, the first field to emerge in the universe and then introduces another field called 'consciousness' that togather with gravity , are the two commonly sensed/experienced field. The latter further consider that mass matter results in both these. The former represents it as an ensemble while the C-field is attributed to a rotation about the mass ensemble. The two together are claimed to describe the Physics. Both thesae approaches can be tested experimentally!

i personally feel that the absolute truth lies beyond science but it can certainly approach it in a better and better relative manner. That is what Physics is already doing very well. Howver, one needs breadth in all considerations and an open mind to all the ideas that become available either directly or indirctly to an individual scientist.

report post as inappropriate

Dear Enrico,

I applaud your paper since I think it has the intention to describe that mathematics cannot be an adequate representation of physics. I was a.o. triggered by your remark "leaving the Standard Model unsatisfactory in terms of consistency".

The trouble I have with your paper however, is that it tries to use that same language of math. That is counterintuitive. So I have a few questions:

1. Why do you use mathematical formalisms to prove something outside that same math language? Natural langage would have suited you fine for this essay.

2. You claim: "a unified description of quantum and classical objects is mathematically forbidden". How do you reconcile that with the macro/microscopic fact that nature seem to have little trouble with that?

3. You conclude with: "Whatever the Grand Unification Theory will be, the application of an aesthetic principle will lead use to choose, if possible, its simplest version". I assume you mean the beaty of the math here. That also seems counterintuitive. Should'nt a GUT just give the correct answer, micro/macro/cosmologically? And would it not be falsified by one incorrect answer (making it a contradiction in terms)?

If you are looking for a simple start with a UFT, check here: Unified Fields in Disguise.

Good luck with the contest!

Steven Oostdijk

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I applaud your paper since I think it has the intention to describe that mathematics cannot be an adequate representation of physics. I was a.o. triggered by your remark "leaving the Standard Model unsatisfactory in terms of consistency".

The trouble I have with your paper however, is that it tries to use that same language of math. That is counterintuitive. So I have a few questions:

1. Why do you use mathematical formalisms to prove something outside that same math language? Natural langage would have suited you fine for this essay.

2. You claim: "a unified description of quantum and classical objects is mathematically forbidden". How do you reconcile that with the macro/microscopic fact that nature seem to have little trouble with that?

3. You conclude with: "Whatever the Grand Unification Theory will be, the application of an aesthetic principle will lead use to choose, if possible, its simplest version". I assume you mean the beaty of the math here. That also seems counterintuitive. Should'nt a GUT just give the correct answer, micro/macro/cosmologically? And would it not be falsified by one incorrect answer (making it a contradiction in terms)?

If you are looking for a simple start with a UFT, check here: Unified Fields in Disguise.

Good luck with the contest!

Steven Oostdijk

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I would like to thank all the readers of my Essay which left a comment with references and questions, as they drive me to clarify a very important aspect connected to the investigation method adopted in my paper.

I believe that the questions "Why do you use mathematical formalisms to prove something outside that same math language? Natural langage would have suited you fine for this essay." of Steven Oostdijk and "How do you know the GNS structure is fundamental ? Perhaps it is not. " of Terry Padden raise a sutble point.

The GNS construction is to my eyes a very fundamental interface between something which is independent from a mathematical representation, like experiments in Nature, with their mathematical description. A set of objects behaving like an algebra manifest an intrinsic property; the fact that it has a mathematical explicit representation it is a consequence of the most general speakable construction one can define, in order to explore the consequences of having such algebric structure. Such algebric structure is *independent* from its representation in mathematical terms.

This justifies in my opinion the use of the GNS as a logic framework capable to bridge the properties of natural objects and experiments (like non commutativity of some quantities), with a mathematical representation, so it consists of the natural environment to ask about the limits of the mathematical description of phyisics.

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I believe that the questions "Why do you use mathematical formalisms to prove something outside that same math language? Natural langage would have suited you fine for this essay." of Steven Oostdijk and "How do you know the GNS structure is fundamental ? Perhaps it is not. " of Terry Padden raise a sutble point.

The GNS construction is to my eyes a very fundamental interface between something which is independent from a mathematical representation, like experiments in Nature, with their mathematical description. A set of objects behaving like an algebra manifest an intrinsic property; the fact that it has a mathematical explicit representation it is a consequence of the most general speakable construction one can define, in order to explore the consequences of having such algebric structure. Such algebric structure is *independent* from its representation in mathematical terms.

This justifies in my opinion the use of the GNS as a logic framework capable to bridge the properties of natural objects and experiments (like non commutativity of some quantities), with a mathematical representation, so it consists of the natural environment to ask about the limits of the mathematical description of phyisics.

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Enrico

Thanks for responding to my point on how one knows the GNS structure is fundamental.

In response you write (3rd para)

1. " ... A set of objects behaving like an algebra manifest an intrinsic property; ....".

How can a set of PHYSICAL objects manifest an ABSTRACT algebraic property unless you have already decided (been forced by the lack of alternatives) in advance that such algebras are applicable. We choose to impose structures on physics. They are in no way intrinsic.

The problem I raise is that our current formalisms limit us to some algebraic / geometric / analytic method BEFORE we even set up the physical system of enquiry. It is a closed loop problem often exemplified by the story of when you lose your keys in a dark place the only way you can hope to find them is to look under the nearest street light. I am suggesting the street lights need replacing with floodlights.

2. then you continue "the fact that it has a mathematical explicit representation it is a consequence of the most general speakable construction one can define,"

I think there is a need for MORE general constructions and we can develop them. I don't accept that the current ones are the MOST general.

3. and " in order to explore the consequences of having such algebric structure. Such algebric structure is *independent* from its representation in mathematical terms. " I agree, but it is not enough.

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Thanks for responding to my point on how one knows the GNS structure is fundamental.

In response you write (3rd para)

1. " ... A set of objects behaving like an algebra manifest an intrinsic property; ....".

How can a set of PHYSICAL objects manifest an ABSTRACT algebraic property unless you have already decided (been forced by the lack of alternatives) in advance that such algebras are applicable. We choose to impose structures on physics. They are in no way intrinsic.

The problem I raise is that our current formalisms limit us to some algebraic / geometric / analytic method BEFORE we even set up the physical system of enquiry. It is a closed loop problem often exemplified by the story of when you lose your keys in a dark place the only way you can hope to find them is to look under the nearest street light. I am suggesting the street lights need replacing with floodlights.

2. then you continue "the fact that it has a mathematical explicit representation it is a consequence of the most general speakable construction one can define,"

I think there is a need for MORE general constructions and we can develop them. I don't accept that the current ones are the MOST general.

3. and " in order to explore the consequences of having such algebric structure. Such algebric structure is *independent* from its representation in mathematical terms. " I agree, but it is not enough.

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Dear Enrico,

The author has yet to choose to respond to my comments of Oct 12. Ofcourse it is his goodwill to find the same worthy of his consideration. i however eagerly await his valuable response.

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The author has yet to choose to respond to my comments of Oct 12. Ofcourse it is his goodwill to find the same worthy of his consideration. i however eagerly await his valuable response.

report post as inappropriate

Dear Enrico Prati,

To some degree your approach seems to resemble the proof that the bumblebee cannot fly.

You state that "The presence of such macroscopic layer which separates the observer from the microscopic structure, made of average quantities and arbitrary combinations of commutative observables, leads to conclude that neither the Version 2 nor the Version 3 of the ToE...

view entire post

To some degree your approach seems to resemble the proof that the bumblebee cannot fly.

You state that "The presence of such macroscopic layer which separates the observer from the microscopic structure, made of average quantities and arbitrary combinations of commutative observables, leads to conclude that neither the Version 2 nor the Version 3 of the ToE...

view entire post

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Dear Enrico,

I am explaining the T-symmetry in QM as due to improper interpreted mathematics. Did you try refuting this?

Regards,

Eckard

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I am explaining the T-symmetry in QM as due to improper interpreted mathematics. Did you try refuting this?

Regards,

Eckard

report post as inappropriate

Dear Nath and Terry,

thank you for your comments and questions. I would like to briefly reply with this post to the points you raise.

Nath: "i however note two essays that deal with similar aspects. One is by tejinder Singh and other is by Edwin Klingman. The former talks about a region in between called mesoscopic where neither classical nor quantum mechanics may be valid strictly....

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thank you for your comments and questions. I would like to briefly reply with this post to the points you raise.

Nath: "i however note two essays that deal with similar aspects. One is by tejinder Singh and other is by Edwin Klingman. The former talks about a region in between called mesoscopic where neither classical nor quantum mechanics may be valid strictly....

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Dear Enrico,

I like the general orientation of your essay!

However, although I have not had time to read it carefully, I noticed that some of the statements, especially the conclusion “that it is fundamentally impossible to both identify a theory with the Mathematical Structure of Nature and demonstrate their compatibility” are not phrased sufficiently accurately. In the latter case, you are probably referring to the existing mathematical structures (depicted in your Fig. 1). But mathematics, including its basic structure, is a *completely open enterprise*, and so we cannot make such general statements about the applicability of mathematics, especially as they refer to the future “mathematical description of physics”.

Best wishes,

Lev

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I like the general orientation of your essay!

However, although I have not had time to read it carefully, I noticed that some of the statements, especially the conclusion “that it is fundamentally impossible to both identify a theory with the Mathematical Structure of Nature and demonstrate their compatibility” are not phrased sufficiently accurately. In the latter case, you are probably referring to the existing mathematical structures (depicted in your Fig. 1). But mathematics, including its basic structure, is a *completely open enterprise*, and so we cannot make such general statements about the applicability of mathematics, especially as they refer to the future “mathematical description of physics”.

Best wishes,

Lev

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Dear Enrico,

You wrote: ... in the case of CM the C∗-algebra with the identity is abelian, while in the case of quantum mechanics it is not.

I agree that one has to carefully distinguish different levels of consideration. However, I would like to humbly ask whether we actually need at the lowest level of physics a C*-algebra if we do not adhere the idea of an a priori existing block universe. I would appreciate if mathematics was really an open enterprise while I rather see it a huge misguided ocean steamer, unable for even taking into consideration a return to Galilei's notion of infinity without any loss of something really valuable.

Regards,

Eckard

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You wrote: ... in the case of CM the C∗-algebra with the identity is abelian, while in the case of quantum mechanics it is not.

I agree that one has to carefully distinguish different levels of consideration. However, I would like to humbly ask whether we actually need at the lowest level of physics a C*-algebra if we do not adhere the idea of an a priori existing block universe. I would appreciate if mathematics was really an open enterprise while I rather see it a huge misguided ocean steamer, unable for even taking into consideration a return to Galilei's notion of infinity without any loss of something really valuable.

Regards,

Eckard

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