Are Boltzmann Brains running Hilbert\'s Hotel? by William T. Parsons

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?

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

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

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?

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

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

The I^в€ћ 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 О»-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

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 was wrong about abstract space/time, and Hawking was wrong about the explosive capability of abstract NOTHING. Proof exists that every real astronomer looking through a real telescope has failed to notice that each of the real galaxies he has observed is unique as to its structure and its perceived distance from all other real galaxies. Each real star is unique as to its structure and its perceived distance apart from all other real stars. Every real scientist who has peered at real snowflakes through a real microscope has concluded that each real snowflake is unique as to its structure. Real structure is unique, once. Unique, once does not consist of abstract amounts of abstract quanta. Based on one's normal observation, one must conclude that all of the stars, all of the planets, all of the asteroids, all of the comets, all of the meteors, all of the specks of astral dust and all real objects have only one real thing in common. Each real object has a real material surface that seems to be attached to a material sub-surface. All surfaces, no matter the apparent degree of separation, must travel at the same constant speed. No matter in which direction one looks, one will only ever see a plethora of real surfaces and those surfaces must all be traveling at the same constant speed or else it would be physically impossible for one to observe them instantly and simultaneously. Real surfaces are easy to spot because they are well lighted. Real light does not travel far from its source as can be confirmed by looking at the real stars, or a real lightning bolt. Reflected light needs to adhere to a surface in order for it to be observed, which means that real light cannot have a surface of its own. Real light must be the only stationary substance in the real Universe. The stars remain in place due to astral radiation. The planets orbit because of atmospheric accumulation. There is no space.

Warm regards,

Joe Fisher

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