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).

    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

      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.

      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.

      Hi Edwin--

      Thank you very much for your comments. I'm genuinely thrilled that you found the essay to be both entertaining and useful. To be honest, when I wrote it, I wasn't quite sure whether I was hitting the right "notes", as it were.

      Very best regards to you,

      Bill.

      Hi--

      Thank you for your comments. I will make sure to look at your essay. I'm glad we can agree that there is no physical paradox associated with Hilbert's Hotel.

      Best regards,

      Bill.

      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.

      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?

        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.

        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

          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.

          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

                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.

                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: mathematical, physical, and metaphysical. In my opinion, "metaphysical infinity" concerns speculative statements about Nature which invoke various sorts of infinite characteristics. For example, our current cosmological theories seem to be telling us that spatial sections may be physically infinite. OK, fine, that's a fair point of view. However, I consider it to be a scientifically-informed metaphysical statement about Reality. It is a metaphysical statement because I don't see anyway to make scientific statements on objects, etc., that are, in principle, always unobservable, either directly or indirectly.

                I do not reject statements concerning metaphysical infinity. I just don't consider them to be scientific statements. The purpose of my essay was to gently suggest that physics does not need physical infinity. I conjecture that everything that we need to do, physically, can be accomplished by relying on an appropriate-sized Bravo. I make no criticism of either mathematical infinity or metaphysical infinity. Both can be extremely useful within their fields.

                Finally, I really don't have an ontological dog in this fight. As a physicist, I started out believing in physical infinity. I came to question the concept of physical infinity only after reading papers by "Team Ellis", considering carefully what Dedekind was telling us, and by taking a hard look at the subject of "super tasking" (the notion that an infinite number of acts, etc., can be accomplished in finite time and so on). I concluded that physical infinity was neither necessary nor useful for doing physics. However, someday, some team of physicists may show how a phenomenon can only be explained by physical infinity. In such a case, I would admit defeat and move on. It wouldn't the first time that Nature threw physics a curveball.

                Best regards,

                Bill.

                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.

                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

                  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.

                  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