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**Armin Nikkhah Shirazi**: *on* 4/24/15 at 4:24am UTC, wrote Dear Bill, I just wanted to let you know that I posted a relatively...

**Sylvia Wenmackers**: *on* 4/22/15 at 20:35pm UTC, wrote Dear Bill, I am very grateful for your reply, which demonstrates that our...

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October 16, 2019

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

TOPIC: Are Boltzmann Brains running Hilbert's Hotel? by William T. Parsons [refresh]

TOPIC: Are Boltzmann Brains running Hilbert's Hotel? by William T. Parsons [refresh]

In a light-hearted way, we employ Hilbert’s Infinite Hotel and a vast collection of Boltzmanns, including Boltzmann Brains, to explore some of the more disconcerting aspects of infinity. We compare and contrast mathematical infinity and physical infinity. We conclude, not regretfully, that physical infinity should be banished from the field of physics.

Bill Parsons is a physicist-in-residence in the Department of Physics at American University, Washington, DC.

Dear William,

From my essay

"The relative size of a subgroup H of G is called the index which means that there are n inequivalent copies (called cosets) of H that fill up G. The action of generators on these cosets creates the permutation group P by the Todd-Coxeter algorithm " that is there are n rooms in the hotel with infinitely many slots.

I have been asking myself if this is related to Cantor's trick.

If you know something, please let me know.

Your text is unique in the sense that it introduces the discrepancy for the infinity in maths and physics.

I intend to read it in more details.

Best.

Michel

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From my essay

"The relative size of a subgroup H of G is called the index which means that there are n inequivalent copies (called cosets) of H that fill up G. The action of generators on these cosets creates the permutation group P by the Todd-Coxeter algorithm " that is there are n rooms in the hotel with infinitely many slots.

I have been asking myself if this is related to Cantor's trick.

If you know something, please let me know.

Your text is unique in the sense that it introduces the discrepancy for the infinity in maths and physics.

I intend to read it in more details.

Best.

Michel

report post as inappropriate

Hi Michel--

Thank you for your comments and question. Unfortunately, I am not familiar with your field, and thus do not feel qualified to render a judgment. I apologize that I could not be more helpful.

Best regards,

Bill.

Thank you for your comments and question. Unfortunately, I am not familiar with your field, and thus do not feel qualified to render a judgment. I apologize that I could not be more helpful.

Best regards,

Bill.

Gentlemen, If I may, I might have a hint to a possible answer.

This sounds very much like a problem of ordering. The guests can be assigned to the empty rooms by a pairing function, but in this case the guests would need to be numbered uniquely. Alternatively (and more interestingly now) it can be done without numbering the guests, if the axiom of countable choice is used. AoCC allows to arbitrarily extract one element from each set and then pair them, which results in creating a sequence, therefore ordering. It works for countable infinities but I think that using the axiom of choice it can be generalized to uncountable infinities as well.

I think this is the equivalence you are looking for because choice is equivalent to ordering and in this respect the permutation group is similar to the AoCC possibility to distribute guests inside the hotel.

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This sounds very much like a problem of ordering. The guests can be assigned to the empty rooms by a pairing function, but in this case the guests would need to be numbered uniquely. Alternatively (and more interestingly now) it can be done without numbering the guests, if the axiom of countable choice is used. AoCC allows to arbitrarily extract one element from each set and then pair them, which results in creating a sequence, therefore ordering. It works for countable infinities but I think that using the axiom of choice it can be generalized to uncountable infinities as well.

I think this is the equivalence you are looking for because choice is equivalent to ordering and in this respect the permutation group is similar to the AoCC possibility to distribute guests inside the hotel.

report post as inappropriate

Bill,

"Bravo". This was a very entertaining essay. It is very easy to mistakenly equate mathematical infinity with physical infinity. You show this clearly and present a method of recognizing and avoiding it.

I'm curious, is Tegmark's nearest twin within our Hubble Bubble? That distance looks pretty big to me.

Many of the essays deal with variations of what you present. If you have the infinite time needed, you should have some interesting reading.

Best Regards and Good Luck,

Gary Simpson

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"Bravo". This was a very entertaining essay. It is very easy to mistakenly equate mathematical infinity with physical infinity. You show this clearly and present a method of recognizing and avoiding it.

I'm curious, is Tegmark's nearest twin within our Hubble Bubble? That distance looks pretty big to me.

Many of the essays deal with variations of what you present. If you have the infinite time needed, you should have some interesting reading.

Best Regards and Good Luck,

Gary Simpson

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Hi Gary--

Thank you for your kind words. You asked a great question!

So, the radius of the Observable Universe is about 47 billion light years (see, e.g., Egan & Lineweaver, arXiv:0909.3983v3, 25 Jan 2010). That works out to a radius of about 4.45 x 10^26 m. Tegmark estimates that his nearest twin is about 10^10^29 m away. We thus have an absolutely huge, but finite, delta in distance.

Some groups have tried to estimate how much larger the Universe is (in terms of homogeneity, etc.) outside the Observable radius, using data from WMAP and PLANCK. For example, Castro et al. estimate a lower limit of 10^3 times larger than what we can see directly (arXiv:astro-ph/0309320v1). Pereira and Silva calculate 87 to 10^5 times larger (arXiv:1304.1181v1). Even with these expanded estimates, it is very unlikely that a twin Tegmark would be within the volume.

However, in principle, a twin Tegmark could be in a galaxy nearby--it's just highly unlikely.

Best regards,

Bill.

Thank you for your kind words. You asked a great question!

So, the radius of the Observable Universe is about 47 billion light years (see, e.g., Egan & Lineweaver, arXiv:0909.3983v3, 25 Jan 2010). That works out to a radius of about 4.45 x 10^26 m. Tegmark estimates that his nearest twin is about 10^10^29 m away. We thus have an absolutely huge, but finite, delta in distance.

Some groups have tried to estimate how much larger the Universe is (in terms of homogeneity, etc.) outside the Observable radius, using data from WMAP and PLANCK. For example, Castro et al. estimate a lower limit of 10^3 times larger than what we can see directly (arXiv:astro-ph/0309320v1). Pereira and Silva calculate 87 to 10^5 times larger (arXiv:1304.1181v1). Even with these expanded estimates, it is very unlikely that a twin Tegmark would be within the volume.

However, in principle, a twin Tegmark could be in a galaxy nearby--it's just highly unlikely.

Best regards,

Bill.

Dear William Parsons,

As Gary said,*very* entertaining. I don't believe in physical infinity, and therefore don't spend much time worrying about mathematical infinity, but I can certainly enjoy the originality and verve of your essay, and can appreciate your arguments and your conclusions.

It*is* time to dump physical infinity, and although you're not the first to suggest it, you certainly are the first to suggest *Bravo*, which has a whiff of Bell's FAPP.

Thank you for providing much more fun than most discussions of infinity provide, without in any way sacrificing rigor.

My very best regards,

Edwin Eugene Klingman

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As Gary said,

It

Thank you for providing much more fun than most discussions of infinity provide, without in any way sacrificing rigor.

My very best regards,

Edwin Eugene Klingman

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Dear William T. Parsons,

I agree with your view on infinity. By adding finite numbers we can never reach infinity. We can introduce an infinite loop: go on adding infinitely, and it becomes a never ending process. Regarding the relation between physics and mathematics, I invite your attention to my essay: A physicalist interpretation of the relation between Physics and Mathematics

Hilbert's Hotel is based on wrong assumptions or axioms. (i). “Hilbert’s Hotel consists of an infinite set of rooms”. Never. Hilbert can start from zero, constructing rooms one by one but can never complete infinite rooms; here, construction is a never ending process. (ii). “an infinite set of buses arrives”. Never. Buses keep coming; it is a never ending process. Hilbert’s Hotel is actually an infinite loop (of finite processes). Thus in fact, there is no paradox.

I agree with you that infinity has no role in physics. However, I think the best method to avoid infinity is to 'finitise' one by one all the concepts and equations by suitably modifying the existing ones. Newton's straight-line motion, force- acceleration relation and equation for gravity, all lead us to infinity. Thus infinity crept into physics from the time of Newton (his third law is an exception). I have brought out a model devoid of infinity (refer: finitenesstheory.com).

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I agree with your view on infinity. By adding finite numbers we can never reach infinity. We can introduce an infinite loop: go on adding infinitely, and it becomes a never ending process. Regarding the relation between physics and mathematics, I invite your attention to my essay: A physicalist interpretation of the relation between Physics and Mathematics

Hilbert's Hotel is based on wrong assumptions or axioms. (i). “Hilbert’s Hotel consists of an infinite set of rooms”. Never. Hilbert can start from zero, constructing rooms one by one but can never complete infinite rooms; here, construction is a never ending process. (ii). “an infinite set of buses arrives”. Never. Buses keep coming; it is a never ending process. Hilbert’s Hotel is actually an infinite loop (of finite processes). Thus in fact, there is no paradox.

I agree with you that infinity has no role in physics. However, I think the best method to avoid infinity is to 'finitise' one by one all the concepts and equations by suitably modifying the existing ones. Newton's straight-line motion, force- acceleration relation and equation for gravity, all lead us to infinity. Thus infinity crept into physics from the time of Newton (his third law is an exception). I have brought out a model devoid of infinity (refer: finitenesstheory.com).

report post as inappropriate

Hi Bill,

I thoroughly concur with the idea of replacing infinity with Bravo, and suggest that the definitive enumeration of**B** is the number of Planck volumes in the observable universe, with **B** increasing as that which we are able to observe increases. I like to equate Planck volumes with Leibnitz's 'monad', and to have them capable of replicating like von Neumann's automata. We can always make Hilbert's *physical* hotel arbitrarily larger, but it always remains finite (the monads are countable). The infinite capacity resides in the mathematical (computational) composition of the monad itself.

Cheers

Rowan

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I thoroughly concur with the idea of replacing infinity with Bravo, and suggest that the definitive enumeration of

Cheers

Rowan

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Hi Rowan--

Thank you for your comments. I certainly agree that slicing space up into Planck volumes is about the smallest you can go--and yet you are still left with a finite approach.

I'm not familiar with Leibnitz's "monad" concept. Heretofore, I had always thought of him as the co-inventor of calculus (much to Newton's intense displeasure!). Thanks for drawing my attention to this point of view.

Best regards,

Bill.

Thank you for your comments. I certainly agree that slicing space up into Planck volumes is about the smallest you can go--and yet you are still left with a finite approach.

I'm not familiar with Leibnitz's "monad" concept. Heretofore, I had always thought of him as the co-inventor of calculus (much to Newton's intense displeasure!). Thanks for drawing my attention to this point of view.

Best regards,

Bill.

Dear William T. Parsons,

In your essay you wrote, “Currently, the best theoretical extension of the ΛCDM model is “cosmological inflation”. It was originally formulated by Alan Guth and others to iron out problems concerning the “Hot Big Bang”, i.e., the early moments surrounding physical creation of our Observable Universe. Word count constraints prohibit churning through all the details. We cut to the bottom-line. Inflation postulates the existence of an infinite number of universes, such as ours, all of which may be physically infinite in one way or another.” Guth's theory of inflation, WHICH IS MERELY ONE VERSION OF NEWTONIAN-EINSTEINIAN INFLATION, postulates the existence of an infinite number of universes. However, the EMPIRICAL FINDINGS of Milgrom, McGaugh, Kroupa, and Pawlowski suggest that Newtonian-Einsteinian inflation is incorrect and Milgromian inflation is correct (whatever Milgromian inflation might be). According to Kroupa the ΛCDM concordance cosmological model has been ruled out by empirical observations. Google "kroupa dark matter". The space roar and the photon underproduction crisis also indicate that something is seriously wrong with the ΛCDM model (despite its successes). I MAKE THE FOLLOWING CLAIM: NASA's space roar science team affirms that 3 independent empirical data sets confirm the space roar. Do you agree or disagree with my claim?

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In your essay you wrote, “Currently, the best theoretical extension of the ΛCDM model is “cosmological inflation”. It was originally formulated by Alan Guth and others to iron out problems concerning the “Hot Big Bang”, i.e., the early moments surrounding physical creation of our Observable Universe. Word count constraints prohibit churning through all the details. We cut to the bottom-line. Inflation postulates the existence of an infinite number of universes, such as ours, all of which may be physically infinite in one way or another.” Guth's theory of inflation, WHICH IS MERELY ONE VERSION OF NEWTONIAN-EINSTEINIAN INFLATION, postulates the existence of an infinite number of universes. However, the EMPIRICAL FINDINGS of Milgrom, McGaugh, Kroupa, and Pawlowski suggest that Newtonian-Einsteinian inflation is incorrect and Milgromian inflation is correct (whatever Milgromian inflation might be). According to Kroupa the ΛCDM concordance cosmological model has been ruled out by empirical observations. Google "kroupa dark matter". The space roar and the photon underproduction crisis also indicate that something is seriously wrong with the ΛCDM model (despite its successes). I MAKE THE FOLLOWING CLAIM: NASA's space roar science team affirms that 3 independent empirical data sets confirm the space roar. Do you agree or disagree with my claim?

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Hi David Brown--

Thank you for your comments. In truth, I do not have enough expertise regarding the ARCADE mission and its data set(s) to either agree or disagree with your claim. What I can say is that I find the "Space Roar" issue fascinating. As to its cause, I have not a clue. I understand that you have put forward a very creative and original theory as to its causation. Hopefully, there will be follow-on missions (with, in particular, a much greater FOV), which should provide greater insight into this phenomenon.

Best regards,

Bill.

Thank you for your comments. In truth, I do not have enough expertise regarding the ARCADE mission and its data set(s) to either agree or disagree with your claim. What I can say is that I find the "Space Roar" issue fascinating. As to its cause, I have not a clue. I understand that you have put forward a very creative and original theory as to its causation. Hopefully, there will be follow-on missions (with, in particular, a much greater FOV), which should provide greater insight into this phenomenon.

Best regards,

Bill.

Bill -

Thanks for a great essay (one of the most entertaining this year!) and an excellent exposition of the conceptual difficulties with infinity. There are parallels to other difficulties (e.g. Godel and Turing) that raise similar problems. In one way or another all deal with self-referential properties - in the case of infinity, the fact that whatever arithmetic operations you perform, you still end up with the same cardinal infinity.....

It is reassuring to believe that the physical universe is finite, and I hold that belief myself. But I wonder at the usefulness of the concept of infinity - why is it so damn helpful in physics? Would you agree that mathematical infinity has a form of existence? Perhaps we can only see it in the formulas or think it in our minds, but it is real!

Many thanks - George Gantz

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Thanks for a great essay (one of the most entertaining this year!) and an excellent exposition of the conceptual difficulties with infinity. There are parallels to other difficulties (e.g. Godel and Turing) that raise similar problems. In one way or another all deal with self-referential properties - in the case of infinity, the fact that whatever arithmetic operations you perform, you still end up with the same cardinal infinity.....

It is reassuring to believe that the physical universe is finite, and I hold that belief myself. But I wonder at the usefulness of the concept of infinity - why is it so damn helpful in physics? Would you agree that mathematical infinity has a form of existence? Perhaps we can only see it in the formulas or think it in our minds, but it is real!

Many thanks - George Gantz

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Hi George--

Thank you very much for the kind words. I, too, think that there are parallels to other difficulties, such as those posed by Godel and Turing. Very briefly, I thought about discussing those additional issues in my essay, but then sanity kicked and I realized that it would definitely be going "a bridge too far"!

I certainly believe in the existence of mathematical infinity--at least until such time as the mathematical community tells me different. The concept of infinity is extremely useful for certain types of computations. The problem, it seems to me, is when we take the infinity concept and reflexively import it into all manner of physical problems. At that point, infinity stops helping us do computations and, instead, starts hindering us. Ironic, is it not? My conclusion is that infinity is a very powerful but very specialized weapon in our analytical arsenal. Infinity is to physics what nuclear weapons are to politics.

Best regards,

Bill.

Thank you very much for the kind words. I, too, think that there are parallels to other difficulties, such as those posed by Godel and Turing. Very briefly, I thought about discussing those additional issues in my essay, but then sanity kicked and I realized that it would definitely be going "a bridge too far"!

I certainly believe in the existence of mathematical infinity--at least until such time as the mathematical community tells me different. The concept of infinity is extremely useful for certain types of computations. The problem, it seems to me, is when we take the infinity concept and reflexively import it into all manner of physical problems. At that point, infinity stops helping us do computations and, instead, starts hindering us. Ironic, is it not? My conclusion is that infinity is a very powerful but very specialized weapon in our analytical arsenal. Infinity is to physics what nuclear weapons are to politics.

Best regards,

Bill.

Bill,

Congratulations, this essay has everything needed for a great essay: accessible to a general audience, humor, enlightening figures and new ideas.

Infinite and infinitesimal are useful non-numbers that can be placed in functions to show limits and relationships between functions (as used in calculus). The philosophical implications of these convenient non-numbers are normally ignored. It should come as no surprise that relationships first found with amber, rabbit’s fur and pith balls have trouble at very small scales and relationships first found by observing the planets in our solar system, would not find the expansion rate of the universe. Infinite and infinitesimal push equations far beyond the limits of our current understanding of physics. The idea of a “Bravo”, placed right at the observable limit, is a great idea. Maybe you should have a “speculative” a place where there might be dragons, just beyond our current observable limit where we can push things just a little bit. The nature of “real” infinite (if it exists) is unimportant, expanding what we got a little bit farther is important.

One final note, a “speculative” would be larger than a “Bravo” most of the time, but thinking of the three body problem, the trillions of particles in Saturn’s ring would be the “Bravo” and the “speculative” would be three.

Jeff Schmitz

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Congratulations, this essay has everything needed for a great essay: accessible to a general audience, humor, enlightening figures and new ideas.

Infinite and infinitesimal are useful non-numbers that can be placed in functions to show limits and relationships between functions (as used in calculus). The philosophical implications of these convenient non-numbers are normally ignored. It should come as no surprise that relationships first found with amber, rabbit’s fur and pith balls have trouble at very small scales and relationships first found by observing the planets in our solar system, would not find the expansion rate of the universe. Infinite and infinitesimal push equations far beyond the limits of our current understanding of physics. The idea of a “Bravo”, placed right at the observable limit, is a great idea. Maybe you should have a “speculative” a place where there might be dragons, just beyond our current observable limit where we can push things just a little bit. The nature of “real” infinite (if it exists) is unimportant, expanding what we got a little bit farther is important.

One final note, a “speculative” would be larger than a “Bravo” most of the time, but thinking of the three body problem, the trillions of particles in Saturn’s ring would be the “Bravo” and the “speculative” would be three.

Jeff Schmitz

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Hi Jeff--

Thank you very much for your comments. Nicely said. I really like your idea of "speculative". Barrow and Rudy Rucker refer to three types of infinity: mathematical, physical, and the "Absolute". I prefer your term "speculative", which to me equates to "metaphysical infinity". In this latter approach, we would be "pushing the envelope", in a logical way, but would still be going beyond the bounds of (current) science.

Best regards,

Bill.

Thank you very much for your comments. Nicely said. I really like your idea of "speculative". Barrow and Rudy Rucker refer to three types of infinity: mathematical, physical, and the "Absolute". I prefer your term "speculative", which to me equates to "metaphysical infinity". In this latter approach, we would be "pushing the envelope", in a logical way, but would still be going beyond the bounds of (current) science.

Best regards,

Bill.

William -

Clever title, congrats. I can see that infinity causes lots of trouble, but as for banishing from physics: despite possible alternatives as you note, the simplest interpretation of current curvature ranges and expansion is that space is probably flat and not closed, and "goes on forever" in spatial extent (and perhaps temporal as well)? Although "for all practical purposes" we can neglect the existence of that unending range of space-time, it would still be a foundational aspect of our reality, in principle, and thus presumably "meaningful" as part of why things are the way they are. Sure, it doesn't have to be true, but neither can the lack of proof that it is, be called a banishment or grounds for positive doubt either. Your thoughts?

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Clever title, congrats. I can see that infinity causes lots of trouble, but as for banishing from physics: despite possible alternatives as you note, the simplest interpretation of current curvature ranges and expansion is that space is probably flat and not closed, and "goes on forever" in spatial extent (and perhaps temporal as well)? Although "for all practical purposes" we can neglect the existence of that unending range of space-time, it would still be a foundational aspect of our reality, in principle, and thus presumably "meaningful" as part of why things are the way they are. Sure, it doesn't have to be true, but neither can the lack of proof that it is, be called a banishment or grounds for positive doubt either. Your thoughts?

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Hi Neil--

Thank you very much for your comments. You ask a superb question. I actually had a section in my essay that addressed your question, but I deleted it for reasons of space constraints. So, I welcome to the opportunity to revisit it here in the comments section.

As I mentioned in my previous post, in response to Jeff Schmitz, one may envision three types of infinity:...

view entire post

Thank you very much for your comments. You ask a superb question. I actually had a section in my essay that addressed your question, but I deleted it for reasons of space constraints. So, I welcome to the opportunity to revisit it here in the comments section.

As I mentioned in my previous post, in response to Jeff Schmitz, one may envision three types of infinity:...

view entire post

Dear William Parson,

You convincingly explained that Hilbert’s hotel has been improperly based on oo*20=oo. Why do you hide this criticism behind pretended lightheartedness? Those who fabricated or defended the mathematics of Hilbert’s hotel did it very emotionally. Dedekind hesitated for a decade. Cantor even claimed having got CH immediately from God and got insane. Hilbert behaved rude toward Brouwer.

I quoted Galileo Galilei, D. Spalt, and the ultra-finitist W. Mückenheim. Russians made me aware of Zenkin too. Meanwhile, I understood that Fraenkel 1923 is sufficient for a critical reader who perfectly understands German as to grasp what went wrong, and I see the main necessity in correcting the notion of number, in particular correcting the interpretation of aleph_1.

By the way, C. S. Peirce just failed publishing the definition of the “mathematical” infinity you mentioned which has therefore been ascribed to Dedekind. Because I see infinity an ideal property, not a number, I would hesitate speaking of a subset of infinity.

You wrote: “Meaningless use of infinity includes invoking computational set-ups such as oo+oo, oo*0, and oo/oo.“ Only the latter two invite to use Bernoulli/Hospital.

While I will feel honored if you are ready to substantially criticize my essay(s), I already confirm you to support and possibly extend the suggestion that your essay offers for the sake of physics.

With my best regards,

Eckard

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You convincingly explained that Hilbert’s hotel has been improperly based on oo*20=oo. Why do you hide this criticism behind pretended lightheartedness? Those who fabricated or defended the mathematics of Hilbert’s hotel did it very emotionally. Dedekind hesitated for a decade. Cantor even claimed having got CH immediately from God and got insane. Hilbert behaved rude toward Brouwer.

I quoted Galileo Galilei, D. Spalt, and the ultra-finitist W. Mückenheim. Russians made me aware of Zenkin too. Meanwhile, I understood that Fraenkel 1923 is sufficient for a critical reader who perfectly understands German as to grasp what went wrong, and I see the main necessity in correcting the notion of number, in particular correcting the interpretation of aleph_1.

By the way, C. S. Peirce just failed publishing the definition of the “mathematical” infinity you mentioned which has therefore been ascribed to Dedekind. Because I see infinity an ideal property, not a number, I would hesitate speaking of a subset of infinity.

You wrote: “Meaningless use of infinity includes invoking computational set-ups such as oo+oo, oo*0, and oo/oo.“ Only the latter two invite to use Bernoulli/Hospital.

While I will feel honored if you are ready to substantially criticize my essay(s), I already confirm you to support and possibly extend the suggestion that your essay offers for the sake of physics.

With my best regards,

Eckard

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Hi Eckard--

Thank you very much for your comments. Your grasp of the history of mathematical infinity is much better than mine. Your points are well taken: Discussions involving any type of infinity can be very vexing and emotional.

You asked why I took a light-hearted approach to the subject. Good question. I typically use humor whenever I take a position, professionally or personally, that is likely to provoke disagreement. The purpose of the humor is to try to defuse anger so that the disagreement can be analyzed on the merits. In my opinion, disagreement is absolutely essential for the sound operation of physics--but ad hominem attacks should always be avoided. Toward the back of my essay, I call into question the way in which many cosmologists use the concept of physical infinity. I figured that might provoke a heated response. I wrote the entire essay in a humorous way in order to defuse such potential unhappiness. Sometimes that strategy works--and sometimes it doesn't!

Best regards,

Bill.

Thank you very much for your comments. Your grasp of the history of mathematical infinity is much better than mine. Your points are well taken: Discussions involving any type of infinity can be very vexing and emotional.

You asked why I took a light-hearted approach to the subject. Good question. I typically use humor whenever I take a position, professionally or personally, that is likely to provoke disagreement. The purpose of the humor is to try to defuse anger so that the disagreement can be analyzed on the merits. In my opinion, disagreement is absolutely essential for the sound operation of physics--but ad hominem attacks should always be avoided. Toward the back of my essay, I call into question the way in which many cosmologists use the concept of physical infinity. I figured that might provoke a heated response. I wrote the entire essay in a humorous way in order to defuse such potential unhappiness. Sometimes that strategy works--and sometimes it doesn't!

Best regards,

Bill.

Physics abhors infinity as something directly observed. This may not necessarily mean that the universe is finite. If the space of the universe is infinite, we are still prevented from measuring or observing anything out beyond a certain distance. Eventually everything is so red shifted the wave length is longer than the cosmological horizon length. I am not committed to the case that the space of the universe is R^3 or S^3, infinite or finite, but if the space is infinite we as observers are limited by the dynamics of spacetime so we can only observe a finite part of the universe.

If you go to my essay you might find some similar ideas. I discuss the prospect for superTuring machines that are able to compute beyond the limits of computability by first order λ-calculus. Certain spacetimes appear to permit this to occur, but these conditions are found in impossible places such as black holes. In some ways what you argue about Boltzmann brains is similar to the existence of superTuring machines. Both are able to compute infinite problems, and superTuring machines can even do this in a finite time.

Cheers LC

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If you go to my essay you might find some similar ideas. I discuss the prospect for superTuring machines that are able to compute beyond the limits of computability by first order λ-calculus. Certain spacetimes appear to permit this to occur, but these conditions are found in impossible places such as black holes. In some ways what you argue about Boltzmann brains is similar to the existence of superTuring machines. Both are able to compute infinite problems, and superTuring machines can even do this in a finite time.

Cheers LC

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Hi LC--

Thank you for your comments. I agree with your sentiments, exactly, regarding the size of space. Sure, it could be infinite, or finite, for that matter. Perhaps, someday, we will actually be able to observe that space is finite, in the sense of, say, detecting some sort of torus-type topology or measuring K>1. On the other hand, our best theories may continue to point in the direction of infinity. But we would never know, for sure, because, as you say, "we can only observe a finite part of the universe".

Best regards,

Bill.

Thank you for your comments. I agree with your sentiments, exactly, regarding the size of space. Sure, it could be infinite, or finite, for that matter. Perhaps, someday, we will actually be able to observe that space is finite, in the sense of, say, detecting some sort of torus-type topology or measuring K>1. On the other hand, our best theories may continue to point in the direction of infinity. But we would never know, for sure, because, as you say, "we can only observe a finite part of the universe".

Best regards,

Bill.

The I^∞ of exterior spacetime is conincident with r_- in the Kerr-Newman spacetime. This physically means there is a piling up of null geodesics. As a result a set of qubits sent from the outside that compute an infinite problem can be realized by an observer as they cross r_- in this black hole. So a supertask might be solved here. This can be seen with the problem of flipping a switch every 1/2, 1/4, 1/8, 1/16 ... of a second to ascertain whether the switch is on or off after the end of this sequence within a second.

Of course there are some problems with this. This assumes the solution is eternal, which means no Hawking radiation. Also the r_- is a Cauchy horizon that has a type of singularity. This might not be survivable. I have been intrigued by whether these second order λ-calculus systems are signatures of black holes or event horizons that shield exterior observers from computing problems that circumvent Godel and Turing results.

Cheers LC

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Of course there are some problems with this. This assumes the solution is eternal, which means no Hawking radiation. Also the r_- is a Cauchy horizon that has a type of singularity. This might not be survivable. I have been intrigued by whether these second order λ-calculus systems are signatures of black holes or event horizons that shield exterior observers from computing problems that circumvent Godel and Turing results.

Cheers LC

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Hi LC--

Your latest comments focus on super-Turing machines and the physics of super-tasking. I don't address these matters in my essay. However, I read your essay, and see that you do. I'm not quite sure of the etiquette here, but I think the best bet is for me to move to your essay threads and pursue the discussion there. In the meantime, thanks again for your comments here.

Best regards,

Bill.

Your latest comments focus on super-Turing machines and the physics of super-tasking. I don't address these matters in my essay. However, I read your essay, and see that you do. I'm not quite sure of the etiquette here, but I think the best bet is for me to move to your essay threads and pursue the discussion there. In the meantime, thanks again for your comments here.

Best regards,

Bill.

Dear Dr. Parsons,

I do not wish to be disrespectful, but I do not think Boltzman abstract brains and Hilbert’s abstract hotel have anything to do with how the real Universe is occurring for the following real reason:

Do let me know what you think about this: This is my single unified theorem of how the real Universe is occurring: Newton was wrong about abstract gravity; Einstein...

view entire post

I do not wish to be disrespectful, but I do not think Boltzman abstract brains and Hilbert’s abstract hotel have anything to do with how the real Universe is occurring for the following real reason:

Do let me know what you think about this: This is my single unified theorem of how the real Universe is occurring: Newton was wrong about abstract gravity; Einstein...

view entire post

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Hi Joe--

Thank you for your comments. You are not being disrespectful at all. I don't think Boltzmann Brains or Hilbert's Hotel have anything to do with the real universe, either. As I wrote in the final paragraph of my essay: "And, no, Boltzmann Brains aren't running Hilbert's Hotel. There are no Boltzmann Brains, and there is no Hilbert's Hotel, because there is no such thing a physical infinity".

Best regards,

Bill.

Thank you for your comments. You are not being disrespectful at all. I don't think Boltzmann Brains or Hilbert's Hotel have anything to do with the real universe, either. As I wrote in the final paragraph of my essay: "And, no, Boltzmann Brains aren't running Hilbert's Hotel. There are no Boltzmann Brains, and there is no Hilbert's Hotel, because there is no such thing a physical infinity".

Best regards,

Bill.

Dear Bill,

Thank you for not reporting my comment to FQXi.org as being inappropriate in order to have it classified as Obnoxious Spam.

Joe Fisher

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Thank you for not reporting my comment to FQXi.org as being inappropriate in order to have it classified as Obnoxious Spam.

Joe Fisher

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Dear William,

I loved your essay which managed to be both funny and rigorous at the same time. I also think your solution to the infinity problem was amazingly clever. Boltzmann Brains have always given my nightmares and your essay is now like a flashlight under the bed.

Please take some time to check out and vote on my essay:

http://fqxi.org/community/forum/topic/2391

All the best in the competition!

Rick Searle

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I loved your essay which managed to be both funny and rigorous at the same time. I also think your solution to the infinity problem was amazingly clever. Boltzmann Brains have always given my nightmares and your essay is now like a flashlight under the bed.

Please take some time to check out and vote on my essay:

http://fqxi.org/community/forum/topic/2391

All the best in the competition!

Rick Searle

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William,

This is a valiant effort to defeat physical significance of infinities of number and extent (as of space-time and its contents). However, the aspects of infinity considered most problematic in physics are actually those regarding what could be called intensity or density of energy etc. One example is the case of the QED infinities that are handled by the suspiciously kludges of renormalization, another is the compressed singularity of GR. Ironically, the first one is caused by aspects of quantum mechanics, and the other kind might be ameliorated by QM! Your thoughts?

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This is a valiant effort to defeat physical significance of infinities of number and extent (as of space-time and its contents). However, the aspects of infinity considered most problematic in physics are actually those regarding what could be called intensity or density of energy etc. One example is the case of the QED infinities that are handled by the suspiciously kludges of renormalization, another is the compressed singularity of GR. Ironically, the first one is caused by aspects of quantum mechanics, and the other kind might be ameliorated by QM! Your thoughts?

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This surreptitious log-out is really getting annoying. I wrote the above comment, sorry. BTW thanks again for your comments at my essay. Note to any readers: my essay tries to make a true, specific contribution to physical knowledge (about why space is three-dimensional) and not just general points of principle.

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Hi Neil--

Thanks for your question. I agree that physical infinities have terrorized both QED and GR from the get go. Like you, I look forward to QM (or its follow-on) eventually solving the singularity problem in GR. As for QED, I see the research involving string theory, etc., as one extended exercise in defeating physical infinity. What they seem to have done is replaced physical infinity with a type of "Bravo". However, they have paid a high price for the eradication of physical infinity, in that they have a "Bravo landscape" on the order of 10^500.

By the way, for anyone else reading this thread, I encourage you to read Neil's essay. I thought that it was excellent!

Best regards,

Bill.

Thanks for your question. I agree that physical infinities have terrorized both QED and GR from the get go. Like you, I look forward to QM (or its follow-on) eventually solving the singularity problem in GR. As for QED, I see the research involving string theory, etc., as one extended exercise in defeating physical infinity. What they seem to have done is replaced physical infinity with a type of "Bravo". However, they have paid a high price for the eradication of physical infinity, in that they have a "Bravo landscape" on the order of 10^500.

By the way, for anyone else reading this thread, I encourage you to read Neil's essay. I thought that it was excellent!

Best regards,

Bill.

Dear Conrad,

I liked much your essay, I gave it a high rate and I included it in the (second) list of best essays of my review. You seem to gather unanimity here, so I think the interesting question is: who would think otherwise (believe in physical infinity) ? You wrote in your comment that you "started out believing in physical infinity" yourself. Was it just a default position by lack of...

view entire post

I liked much your essay, I gave it a high rate and I included it in the (second) list of best essays of my review. You seem to gather unanimity here, so I think the interesting question is: who would think otherwise (believe in physical infinity) ? You wrote in your comment that you "started out believing in physical infinity" yourself. Was it just a default position by lack of...

view entire post

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Hi Sylvain--

Thank you very much for your kind words (and high rating!). I am especially honored that you reviewed my essay and considered it to be one of the better ones. You ask a number of excellent questions, too. Let me try to answer them in order.

First, my initial belief in physical infinity was basically a "default position", as you put it. Over the years, I have asked...

view entire post

Thank you very much for your kind words (and high rating!). I am especially honored that you reviewed my essay and considered it to be one of the better ones. You ask a number of excellent questions, too. Let me try to answer them in order.

First, my initial belief in physical infinity was basically a "default position", as you put it. Over the years, I have asked...

view entire post

Dear William,

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

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

Joe Fisher

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

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

Joe Fisher

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Dear Bill Parsons,

I agree that Hilbert's hotel is unphysical. But do you think that potential infinity is unphysical?

Best,

Lou Kauffman

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I agree that Hilbert's hotel is unphysical. But do you think that potential infinity is unphysical?

Best,

Lou Kauffman

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Hi Lou--

Thank you for your question.

Technically, I think that the answer to the question, "Is potential infinity unphysical?", depends upon how one defines "potential infinity". As I define it, I would say that potential infinity is unphysical. For example, one type of potential infinity involves math, such as N, the set of positive integers. Some say that this set is potential; others argue, actual. However, we can all agree that it is unphysical. Similarly, some people make statements like, "God is infinite love". I consider such statements to be a type of metaphysical statement; they may be debatable in terms of truth, but I do not see how they can be classified as statements about physical infinity. They are inherently unphysical.

A trickier issue concerns potential infinities in physics. As I point out in my essay, we encounter potential infinities all the time in cosmology. I offer two fundamental points in this regard. First, many of these potential infinities are really mathematical or metaphysical statements (e.g., statements about some types of "multiverse") masquerading as physical statements. As such, they are by definition unphysical. Second, sometimes our theories really do seem to be telling us that, potentially, some aspect of Nature may be infinite in physical extent (e.g., infinite spatial sections). I argue that these types of potential infinity are neither necessary or useful physically. For example, do I really believe in physically infinite spatial sections in cosmology? Not only "No!", but "Heck, no!". Why? Because it is a long way from Here to Infinity--and I would bet all my money in the bank that something changes along the way. But, of course, this is just speculation on my part.

I hope this helps.

Best regards,

Bill.

Thank you for your question.

Technically, I think that the answer to the question, "Is potential infinity unphysical?", depends upon how one defines "potential infinity". As I define it, I would say that potential infinity is unphysical. For example, one type of potential infinity involves math, such as N, the set of positive integers. Some say that this set is potential; others argue, actual. However, we can all agree that it is unphysical. Similarly, some people make statements like, "God is infinite love". I consider such statements to be a type of metaphysical statement; they may be debatable in terms of truth, but I do not see how they can be classified as statements about physical infinity. They are inherently unphysical.

A trickier issue concerns potential infinities in physics. As I point out in my essay, we encounter potential infinities all the time in cosmology. I offer two fundamental points in this regard. First, many of these potential infinities are really mathematical or metaphysical statements (e.g., statements about some types of "multiverse") masquerading as physical statements. As such, they are by definition unphysical. Second, sometimes our theories really do seem to be telling us that, potentially, some aspect of Nature may be infinite in physical extent (e.g., infinite spatial sections). I argue that these types of potential infinity are neither necessary or useful physically. For example, do I really believe in physically infinite spatial sections in cosmology? Not only "No!", but "Heck, no!". Why? Because it is a long way from Here to Infinity--and I would bet all my money in the bank that something changes along the way. But, of course, this is just speculation on my part.

I hope this helps.

Best regards,

Bill.

Dear Bill,

Thank you ever so much for leaving such a positive comment about my essay.

One real Universe can only be occurring in one real infinite dimension. Unfortunately, scientists insist on attempting to measure the three abstract dimensions of height, width and depth, with completely unrealistic results. The real Universe must be infinite in scope and eternal in duration.

Gratefully,

Joe Fisher

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Thank you ever so much for leaving such a positive comment about my essay.

One real Universe can only be occurring in one real infinite dimension. Unfortunately, scientists insist on attempting to measure the three abstract dimensions of height, width and depth, with completely unrealistic results. The real Universe must be infinite in scope and eternal in duration.

Gratefully,

Joe Fisher

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Bill,

Your Hilbert Hotel is an esoteric location steeped in meaning. Do Boltzmann Brains have physical baggage of a type 0 civilization that restrict a Hilbert Hotel in a type 2 civilization?

My connections of mind, math, and physics are quite pedestrian in producing advances in quantum biology, DNA mapping and simulation of the BB: http://fqxi.org/community/forum/topic/2345.

Thanks for sharing your imaginative hotel.

Jim

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Your Hilbert Hotel is an esoteric location steeped in meaning. Do Boltzmann Brains have physical baggage of a type 0 civilization that restrict a Hilbert Hotel in a type 2 civilization?

My connections of mind, math, and physics are quite pedestrian in producing advances in quantum biology, DNA mapping and simulation of the BB: http://fqxi.org/community/forum/topic/2345.

Thanks for sharing your imaginative hotel.

Jim

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Bill,

You are very kind, not only in being engaged in my essay but also engaging in your interest.

Quick question: Is the equation involving Gt on page 3 your work? If so, how did you derive it? Having such a meager math background, I thought it somewhat primitive but applicable, starting with a compound interest formula, the principal of dynamic growth.

On a more personal note, as a pilot, I've always respected Boeing aircraft. Did you ever work on the Triple7? A truly fantastic airplane. I worked on the military side mostly, only occasionally doing cost-benefit on the commercial side, including the 777.

Jim

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You are very kind, not only in being engaged in my essay but also engaging in your interest.

Quick question: Is the equation involving Gt on page 3 your work? If so, how did you derive it? Having such a meager math background, I thought it somewhat primitive but applicable, starting with a compound interest formula, the principal of dynamic growth.

On a more personal note, as a pilot, I've always respected Boeing aircraft. Did you ever work on the Triple7? A truly fantastic airplane. I worked on the military side mostly, only occasionally doing cost-benefit on the commercial side, including the 777.

Jim

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I disagree with it.

Since I believe "It Takes Two Hands Clapping to Make a Noise"

-Best regards

Miss. Sujatha Jagannathan

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Since I believe "It Takes Two Hands Clapping to Make a Noise"

-Best regards

Miss. Sujatha Jagannathan

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Dear William,

Great essay! It is well-argued and well-written. You explained the difference between mathematical and physical infinity. You also gave strong arguments for how to deal with physical infinity, and I strongly agree with them and give you highest rating. I would be glad to take your opinion in my essay.

All the best,

Mohammed

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Great essay! It is well-argued and well-written. You explained the difference between mathematical and physical infinity. You also gave strong arguments for how to deal with physical infinity, and I strongly agree with them and give you highest rating. I would be glad to take your opinion in my essay.

All the best,

Mohammed

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Dear Bill,

What a delightful essay! You should consider moonlighting as a science writer (what is a physicist-in-residence, anyway?)

"..I reject physical infinity, for three reasons. First, mathematically, it

makes computations intractable. Second, operationally, I do not know how—even in principle—how

to observe, measure or manipulate physically infinite objects...

view entire post

What a delightful essay! You should consider moonlighting as a science writer (what is a physicist-in-residence, anyway?)

"..I reject physical infinity, for three reasons. First, mathematically, it

makes computations intractable. Second, operationally, I do not know how—even in principle—how

to observe, measure or manipulate physically infinite objects...

view entire post

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Hi Armin--

Thank you very much for your comments and questions. Actually, I quite agree with you. Often physical infinities are telling us that we are not looking at the problem in the right way. In this regard, physical infinities are thus the proverbial "canary in the 'physics' coal mine" (if I may be permitted to mix metaphors). And this characteristic is under-appreciated. I appreciate you taking the time to highlight it.

I try to take a nuanced approach to infinities. It is important to distinguish between mathematical infinity and physical infinity. Obviously, in physics, sometimes that can be hard to do. I see "orthogonality" as a mathematical construct. As such, not troubling. However, when it gives rise to statements about the physical world which imply infinities, then I think that we are heading for trouble. If I understood your comments correctly, we seem to be in agreement on this point as well.

As to "physicist-in-residence", it is a research position. I am pretty much out of the teaching business at this point in my career.

Thanks again for sharing your thoughts and point of view.

Best regards,

Bill.

Thank you very much for your comments and questions. Actually, I quite agree with you. Often physical infinities are telling us that we are not looking at the problem in the right way. In this regard, physical infinities are thus the proverbial "canary in the 'physics' coal mine" (if I may be permitted to mix metaphors). And this characteristic is under-appreciated. I appreciate you taking the time to highlight it.

I try to take a nuanced approach to infinities. It is important to distinguish between mathematical infinity and physical infinity. Obviously, in physics, sometimes that can be hard to do. I see "orthogonality" as a mathematical construct. As such, not troubling. However, when it gives rise to statements about the physical world which imply infinities, then I think that we are heading for trouble. If I understood your comments correctly, we seem to be in agreement on this point as well.

As to "physicist-in-residence", it is a research position. I am pretty much out of the teaching business at this point in my career.

Thanks again for sharing your thoughts and point of view.

Best regards,

Bill.

Dear Bill,

I just wanted to let you know that I posted a relatively lengthy response to your comments on my blog and that it matters to me to know whether I was able to express my ideas intelligibly...if you did not understand it, that's OK, I won't attempt any additional explanations unless you want me to, but I would like to know if that is the case because it helps me find out which way of explaining my ideas is effective and which is not.

The problem that I seem to be facing is that I have much more comprehensive picture in mind in which different concepts from different areas interlock tightly, resulting in an overall picture that, to the extent I can tell, fits nicely together. But the whole picture adds up to a worldview that is in some ways substantially different from the current one, so that trying to explain individual parts of the puzzle (like in my response to your comments) which can only give a partial view, will, due to unfamiliarity, have a risk of failing to convey even just that limited part of the picture. Finding the "right" parts to explain first as well as the ways to do this may help make it more likely that eventually I can intelligibly convey the entire picture.

Best,

Armin

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I just wanted to let you know that I posted a relatively lengthy response to your comments on my blog and that it matters to me to know whether I was able to express my ideas intelligibly...if you did not understand it, that's OK, I won't attempt any additional explanations unless you want me to, but I would like to know if that is the case because it helps me find out which way of explaining my ideas is effective and which is not.

The problem that I seem to be facing is that I have much more comprehensive picture in mind in which different concepts from different areas interlock tightly, resulting in an overall picture that, to the extent I can tell, fits nicely together. But the whole picture adds up to a worldview that is in some ways substantially different from the current one, so that trying to explain individual parts of the puzzle (like in my response to your comments) which can only give a partial view, will, due to unfamiliarity, have a risk of failing to convey even just that limited part of the picture. Finding the "right" parts to explain first as well as the ways to do this may help make it more likely that eventually I can intelligibly convey the entire picture.

Best,

Armin

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Dear William,

I have read your essay quite a while ago, but only realized that I forgot to comment yesterday, when I wanted to post something in reply to Michel’s question. My apologies for this; I realize the rating itself is the utmost expression of appreciation but I also know it’s very satisfying when people interact with your work.

You’re making an unexpected and original analysis for the infinite hotel and I admire your argumentation when drawing parallels to physics as you are taking into consideration possible objections. I think Fitzgerald was saying that “the test of a first-rate intelligence is the ability to hold two opposed ideas in mind at the same time and still retain the ability to function”. Here I mean of course how you develop the point about the laws of physics and the operation of the hotel in finite time, two competing points brought together in short sequence. You are making a profound analysis about the placement of the rooms and in general about the topographical properties that may or may not impact how and iff the hotel works, thus developing your idea completely. Not a single thread is left out of place because you explain how Boltzmann brains come into picture. Thank you for a good read and wish you best of luck in your work and in the contest!

Warm regards,

Alma

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I have read your essay quite a while ago, but only realized that I forgot to comment yesterday, when I wanted to post something in reply to Michel’s question. My apologies for this; I realize the rating itself is the utmost expression of appreciation but I also know it’s very satisfying when people interact with your work.

You’re making an unexpected and original analysis for the infinite hotel and I admire your argumentation when drawing parallels to physics as you are taking into consideration possible objections. I think Fitzgerald was saying that “the test of a first-rate intelligence is the ability to hold two opposed ideas in mind at the same time and still retain the ability to function”. Here I mean of course how you develop the point about the laws of physics and the operation of the hotel in finite time, two competing points brought together in short sequence. You are making a profound analysis about the placement of the rooms and in general about the topographical properties that may or may not impact how and iff the hotel works, thus developing your idea completely. Not a single thread is left out of place because you explain how Boltzmann brains come into picture. Thank you for a good read and wish you best of luck in your work and in the contest!

Warm regards,

Alma

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Hi Alma--

Thank you very much for your kind words. In particular, thank you for taking the time to comment on my essay. You are correct. Interaction is very important. The primary reason I wrote and posted my essay was so that it could serve as a "test vehicle" for my approach to physical infinity, especially with respect to cosmology. I had no illusions about winning. I was just hoping for critical responses, whether positive or negative. Of course, I'm thrilled by your positive response.

I thought your comments were incredibly well-written. So much so that you have compelled me to go read your essay!

Best regards and good luck in the contest,

Bill.

Thank you very much for your kind words. In particular, thank you for taking the time to comment on my essay. You are correct. Interaction is very important. The primary reason I wrote and posted my essay was so that it could serve as a "test vehicle" for my approach to physical infinity, especially with respect to cosmology. I had no illusions about winning. I was just hoping for critical responses, whether positive or negative. Of course, I'm thrilled by your positive response.

I thought your comments were incredibly well-written. So much so that you have compelled me to go read your essay!

Best regards and good luck in the contest,

Bill.

Dear Bill,

Thank you very much for your visit and for your words! I just dropped by to tell you that I answered your question.

My sincerest appreciation!

Alma

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Thank you very much for your visit and for your words! I just dropped by to tell you that I answered your question.

My sincerest appreciation!

Alma

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Dear Bill,

Infinity, set theory and relativity are some of the most difficult concepts in math and physics. Not difficult per se, but difficult to grasp to the level where you can work with them properly. You are going at the heart of the problem when you are distinguishing infinity as the biggest plague of physics. It is both clever and brave to bring these together in your essay and also a strong proof of very original thinking. You are also making a strong point about the difference between physical and mathematical infinity. I particularly enjoyed your analysis of the FLRW metric and the cosmological implications and possibilities to measure physical infinity. This metric is an old friend of mine. I enjoyed reading your well written and well argued essay because you take a point and follow it to the end of the line, not unlike a mathematical proof.

Cheers,

Cristi Stoica

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Infinity, set theory and relativity are some of the most difficult concepts in math and physics. Not difficult per se, but difficult to grasp to the level where you can work with them properly. You are going at the heart of the problem when you are distinguishing infinity as the biggest plague of physics. It is both clever and brave to bring these together in your essay and also a strong proof of very original thinking. You are also making a strong point about the difference between physical and mathematical infinity. I particularly enjoyed your analysis of the FLRW metric and the cosmological implications and possibilities to measure physical infinity. This metric is an old friend of mine. I enjoyed reading your well written and well argued essay because you take a point and follow it to the end of the line, not unlike a mathematical proof.

Cheers,

Cristi Stoica

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Hi Cristi--

Thank you very much for your kind words. As I mentioned to Alma, it is wonderful to receive critical feedback, especially of a positive sort. I was intrigued to read that the FLRW metric was an "old friend". I feel the same way. I now have no choice but to go read your essay. Once again, thank you for your kind comments.

Best regards and good luck in the contest,

Bill.

Thank you very much for your kind words. As I mentioned to Alma, it is wonderful to receive critical feedback, especially of a positive sort. I was intrigued to read that the FLRW metric was an "old friend". I feel the same way. I now have no choice but to go read your essay. Once again, thank you for your kind comments.

Best regards and good luck in the contest,

Bill.

Dear William T. Parsons,

I like the topic and the style of your essay and I agree with your main stance (mathematical infinity is fine, physical infinity is something else). However, I do think there are a few flaws as well.

One minor thing is that you state that the lemniscate is the common and correct symbol for mathematical infinity. You use it both for infinity in the sense...

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I like the topic and the style of your essay and I agree with your main stance (mathematical infinity is fine, physical infinity is something else). However, I do think there are a few flaws as well.

One minor thing is that you state that the lemniscate is the common and correct symbol for mathematical infinity. You use it both for infinity in the sense...

view entire post

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Hi Sylvia—

Thank you very much for your comments. In particular, I am grateful for your constructive criticism.

I don’t know about you, but I find these essays quite difficult to write. There is a strict page limit; yet we are to assume that our readers are not experts. Accordingly, I endeavor to explain everything as I go and, hopefully, in a readable and entertaining...

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Thank you very much for your comments. In particular, I am grateful for your constructive criticism.

I don’t know about you, but I find these essays quite difficult to write. There is a strict page limit; yet we are to assume that our readers are not experts. Accordingly, I endeavor to explain everything as I go and, hopefully, in a readable and entertaining...

view entire post

Dear Bill,

I am very grateful for your reply, which demonstrates that our opinions on these matters are even closer than I had realized initially!

You are right, getting these essays right is difficult: they are supposed to be for a general audience, but in these forums there are mainly people with highly specialized backgrounds and it remains impossible to please everyone. :) As I indicated before, I do think your piece is exciting for the intended audience.

So, you actually considered mentioning non-standard numbers? How cool. :)

I did notice that you were not really defending the 50-50 argument, but my response was that there are better options available for this. Whether the alternatives are applicable to physics is indeed a different matter. I would say that the natural numbers are a helpful concept for modelling physical situations, but not literally applicable to the physical world. Yet, currently, they are really entrenched in physics: the natural numbers occur in the construction of the real numbers, they occur in the definition of limit and series,... Similarly, also natural density and related probability measures are approximations at best.

So, I agree that "In the physical world, infinity has got nothing to offer." Yet, in physical reasoning, it is considered to be a helpful abstraction for unspecified large quanties. Trouble arises when we forget that we are working with an unphysical idealization. The Bravo card can be an alternative for the idealization step. Like you, I would prefer this option. So, indeed, we are on the same page here.

O, and I will send you some references by e-mail!

Best wishes,

Sylvia

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I am very grateful for your reply, which demonstrates that our opinions on these matters are even closer than I had realized initially!

You are right, getting these essays right is difficult: they are supposed to be for a general audience, but in these forums there are mainly people with highly specialized backgrounds and it remains impossible to please everyone. :) As I indicated before, I do think your piece is exciting for the intended audience.

So, you actually considered mentioning non-standard numbers? How cool. :)

I did notice that you were not really defending the 50-50 argument, but my response was that there are better options available for this. Whether the alternatives are applicable to physics is indeed a different matter. I would say that the natural numbers are a helpful concept for modelling physical situations, but not literally applicable to the physical world. Yet, currently, they are really entrenched in physics: the natural numbers occur in the construction of the real numbers, they occur in the definition of limit and series,... Similarly, also natural density and related probability measures are approximations at best.

So, I agree that "In the physical world, infinity has got nothing to offer." Yet, in physical reasoning, it is considered to be a helpful abstraction for unspecified large quanties. Trouble arises when we forget that we are working with an unphysical idealization. The Bravo card can be an alternative for the idealization step. Like you, I would prefer this option. So, indeed, we are on the same page here.

O, and I will send you some references by e-mail!

Best wishes,

Sylvia

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