Essay Abstract

Computation is ubiquitous, acting on scales as small as those of particles, up to systems like living creatures. From a mathematical perspective, Nature can be viewed as a computing system. In this context, the laws of physics can be construed as mathematical functions that map inputs onto outputs --- something of the likes of algorithms. A universal computer can store only a limited amount of information because the universe that is physically accessible is finite in space and time. In this essay, I discuss how the concept of predictability changes in light of the computational limit of the Universe. I argue that undecidability emerges naturally in this digital-physicalist view of Nature, and that it severely hinders the prospects for finding an ontic theory of everything.

Author Bio

Rafael Alves Batista is currently a research fellow at Radboud University, in the Netherlands. Previously, he held positions at the University of Oxford and at the University of São Paulo. He received his PhD in Physics from the University of Hamburg.

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"We live in the matter-dominated era, in which the dominant term driving the expansion of the Universe is that of matter, both baryonic and dark." Kroupa is extremely skeptical about the existence of dark matter particles. I say that Milgrom is the Kepler of contemporary cosmology -- on the basis of overwhelming empirical evidence.

What might be wrong with Kroupa's analysis in the following article?

Kroupa, Pavel. "The dark matter crisis: falsification of the current standard model of cosmology." Publications of the Astronomical Society of Australia 29, no. 4 (2012): 395-433.

arXiv preprint

    7 days later

    Dear David,

    Thanks for the comment.

    I think Kroupa's paper is not directly related to the essay. Even if we take his view into account, my arguments are qualitatively valid. Some numerical results will change, though.

    From what I gather from this paper, he is implying that there should be galaxies with no dark matter. They were not known at the time, but they have now been observed. And they are consistent with the standard cosmological model. No need to invoke MOND or anything else.

    I am very open to MOND, but it does lack in evidence. Almost everything can be explained consistently by postulating dark matter. Sure there are problems with the standard cosmological model, but it is in general very successful. The predictions have been confirmed one after the other (some are still missing).

    I am also skeptical about the Kroupa's confidence in the Tully-Fisher relation. It is just an empiracal relation, and many deviations from it have been found.

    Dear David,

    Thanks for the comment.

    Kroupa's criticism of the standard cosmological model is not directly relevant for the point I made in the essay. My arguments hold qualitatively regardless, although some numerical values may change.

    In any case, Kroupa's paper implies the existence of galaxies with no dark matter. They were unknown at the time, but have been recently observed. They are perfectly consistent with the standard cosmological model. No need to invoke MOND or anything else. While I'm very open to modified gravity, at the moment dark matter has been successful in explaining the observations.

    Dear Dr Batista

    You raise absolutely fundamental questions in your essay. However there is a problem with your premise that Nature can be viewed as a computing system, in particular operating with finite numbers. The first problem is that theory requires real numbers in full, e.g. quantum mechanics fall apart without real numbers. So we have an issue what is the relation of quantum mechanics to reality. Another problem is excessive simplification, you write "A free electron can be fully described by its charge, mass, spin. Therefore it can can be easily represented". But there is another level here if one asks why this value of charge and mass and how it interacts with surrounding space. Then it becomes a hellish issue which even after 100 years is still not fully rigorousl solved and it seems to exceed by far reality based on finite numbers. To get a glimpse on it search e.g. 'baez struggles with the continuum'.

      6 days later

      Dear Irek,

      Thanks for your comment.

      You say that "The first problem is that theory requires real numbers in full, e.g. quantum mechanics fall apart without real numbers."

      All the maths work out just fine with the framework of real numbers that we have. It is a good approximation. But I don't see why quantum mechanics would fall apart without it. I'm very familiar with how QM works and I am quite convinced that at the basic level it would be just fine regardless. Could you elaborate on that criticism?

      My main point is that, essentially, 'nothing changes' at the calculation level by supposing finitism. And it's quite easy to show that they don't; if they do, we are not able to know it anyways. In terms of interpretation, many things change.

      "But there is another level here if one asks why this value of charge and mass and how it interacts with surrounding space"

      First, by definition, a free electron is not bound. Therefore, the surroundings don't matter. This is obviously just an example to make things more understandable, it is not relevant for the case I was making.

      I also dedicated a paragraph of the essay to explain the second part of this comment. It relates to the ontology of space-time. If space-time isn't real, only relative position matters, my idea evidently holds. If it is real, the Universe cannot really keep track (in terms of storing information) of the 4-positions of every particle at every moment. Either way, within the finitist framework presented, things can be simplified to the degrees of freedom of particles and the way they interact (which is not different of how things are currently done, by the way).

      Dear Rafael,

      Thank you so much for this essay.

      In the beginning of Epilogue, we would like to ask your opinion on the computer simulation. You cited the Bostrom paper. Therefore, universe is in computer simulation. On this claim, this may be true as you said. On the other hand on philosophy of science, Eric Winsberg pointed out the meaning of the computer simulation in this paper. This seems to be contracted to consider the validity of computer simulation. I also question this again and again to finally reach the random number generation process in computer as reviewed in my essay. What do you think?

      Best wishes,

      Yutaka

        Dear Yutaka,

        I was not aware of this paper by Winsberg.

        I cited Bostrom's simulation argument just to comment on the distinction between living in a Universe that effectively performs computations, and living in a simulation. Those are two different things.

        I will take a look at your essay, thanks!

        Dear Dr Rafael Alves Batista

        Your maximum information content in the Universe is:

        Jmax=(mne/mpr)^[1+ialpha^2log2(mpr/mel)]= 6,387077183705*10^121 bits

        Where, mne, mpr, mel masses of neutron, proton and electron.

        ialpha - inverse fine structure constant.

        Dozens of other formulas give the same result for Jmax.

        Regards,

        Branko

        Dear Rafael (if I may),

        thank you for a brilliant, stimulating and well-argued essay! Our ideas have an unbelievable overlap. I don't know whether you are aware of Gisin's research program on classical indeterminism, based on information theoretic arguments (https://arxiv.org/abs/1803.06824, https://arxiv.org/abs/1909.03697), but there are many elements in common.

        In particular, in [link:arxiv.org/abs/1909.03697[/link] I have put forth the idea that determinism is at adds with two problems: "infinities" and "infinitesimal". In our work we have focused mostly on the latter, while your essay addresses the former, although we also comment on the fact that Lapace's demon cannot exist due to the finiteness of resources in the universe. So we denitely share views of "mathematical physicalism", and I really appreciated your extimate of the largest number, showing how "it relates to the total number of particles in the Universe as well as the number of degrees of freedom of each of them".

        Please, take a moment to have a look at my my essay, because I believe that you may find some elements of common interest therein.

        Thanks again and I really wish you the best for the contest,

        Flavio

        p.s. I truly hope your essay will get more visibility and appreciation than it had so far. The logic of the rating in this contest is really questionable... Anyways, obviously top rating from my side.

          Sorry, the link thing went wrong... This is my paper with Gisin where we discuss the problems of infinities.

          All the best,

          Flavio

          Dear Flavio,

          Thanks for your comment.

          I just read your excellent essay and the overlap is indeed remarkable! We adopt similar approaches and reach similar conclusions.

          I was not aware of your work nor Gisin's. I will read them more carefully.

          At some point I want to write up some ideas of this essay as a paper, since this contest apparently won't give the idea enough visibility. I'm sure I can get some inspiration from your work.

          Best regards,

          Rafael

          hi alves.Great beautiful work on the simulation hypothesis. rated You well. hope you kindly read my take on cognitive input selection bias, on how Brain builds universe here-https://fqxi.org/community/forum/topic/3525.Thanks all the best in the essay contest.

            Dear Michael,

            Thanks for the feedback. Although I have to stress that is more about computation than simulation (I actually argue against the simulation argument towards the end).

            I will take a look at your essay.

            Best regards,

            Rafael

            Hi Rafael,

            Thank you for the wonderful essay! I am totally onboard with your premise that any attempt are realising a universal computing machine is somewhat doomed by the constraint of finiteness. Your conclusion that ``if the Universe is a computer, one cannot harness all of its information capacity (Imax) for computation alone'' because some space is obviously required to store the inputs/outputs!

            I hope you get a chance to take a look at my essay titled 'noisy machines' as we have taken very similar angles to this topic. We have several differences---albeit small---where I argue that indeterminsm is a thermodynamic price which must be paid by our finite machines trying to compute infinite things. However, we both reach the same conclusion!

            Overall, great essay and I really enjoyed it and rated it accordingly!

            Thanks,

            Michael

              Dear Michael,

              Thank you for the comment.

              I haven't yet read your essay thoroughly, but a quick look at it tells me that there is a lot in common. I will read it later.

              Best wishes,

              Rafael

              Dear Rafael,

              you present a bold argument for a radical sort of finitism. You argue your positions well, and present complex concepts in an approachable manner, without sacrificing too much accuracy.

              The initial part of your essay reminded me of Baez' 'Rosetta Stone'-paper, where he develops the equivalence (at a categorical level) between computations and physical processes in great detail (if you haven't read it, I think you might find it enjoyable).

              Another association triggered by your essay is with Jürgen Schmidhuber's 'Algorithmic Theories of Everything'. In particular, you write that there is no reason to really expect that nature should choose less complex realizations over more complex ones---but Schmidhuber argues that there is: assuming the probability distribution from which the history of the universe is sampled to be formally describable, he shows that it is dominated by those universe having a short formal description, i. e. a low Kolmogorov complexity.

              Of course, neither of them shares your finitist commitment. I have to say that, while you argue your case well, this idea sits somewhat uneasily with me. For one, it breaks the quantum mechanical superposition principle---if amplitudes are only, at best, allowed to take on rational values (which one might say has no empirical consequences, the rationals lying dense in the reals after all, and possibly still 'dense enough' with sufficiently large numbers for numerator and denominator), then certain superpositions of states would be normalized to irrational values, i. e. infinite-precision quantities. Those would then have to be 'cast out' from the spectrum.

              The problem also exists in a similar way within quantum field theory, which can be formally written as a continuous infinity of harmonic oscillators at each space-time point.

              Of course, these might turn out to be mere approximations---but it's not immediately clear to me that one can modify the theories appropriately without breaking too much.

              Still, I think it's an idea worth pursuing---and certainly, arguments (such as the one based on Bekenstein-Hawking entropy) that a theory of quantum gravity will only have a finite number of degrees of freedom within a given spacetime volume have been proposed in various ways.

              So, best of luck---both to your program, and in this contest!

              Cheers

              Jochen

                You are vague in your definition of Nmax. You suggest treating the entire universe as a machine. If it is classical, surely Nmax = 10^123. If it is quantum, you get exponentially more branches of the wave function (your 2^10^123), but when you read out the number, it is a classical result - a bit, not a qubit - so you collapse right back down to 10^123. Expressed differently, if this number can be discerned by a Laplacian demon, if the demon lives in real space and performs measurements to get yes-no answers, or sequences of 0s and 1s to express Nmax, then the demon doesn't have access to 2^10^123 bits. If the demon lives in Hilbert space, he/she/it does. But then you are describing a perceived universe different from the one that humans inhabit. A close analogy is black hole entropy, which is equal to the number of BITS it has swallowed, NOT the number of qubits. You cannot inject more information into a black hole than the Hawking entropy by dropping in entangled particles.

                  Dear Jochen,

                  Thanks for the comments.

                  I didn't know Schmidhuber's paper. I will take a look at it.

                  You say:

                  if amplitudes are only, at best, allowed to take on rational values (which one might say has no empirical consequences, the rationals lying dense in the reals after all, and possibly still 'dense enough' with sufficiently large numbers for numerator and denominator), then certain superpositions of states would be normalized to irrational values, i. e. infinite-precision quantities.

                  I was also concerned about this at some point. But in this finitist framework, you can increase the precision by having a larger universe or by waiting longer. Therefore, it's not like one cannot normalise the wave function with, e.g., sqrt(2). It's just that sqrt(2) cannot be fully represented within a finite distance/time, so that values that require Nmax+1 digits of precisions are indeed not accessible at the moment.

                  I acknowledge that the idea has some limitations, and some issues have to be better thought (e.g., if (and how) would physics change without infinities). This essay was just entertaining speculation, but as you said, I believe it is an interesting idea for more serious work.

                  Best wishes,

                  Rafael

                  Dear Paul,

                  Thanks for reading and commenting on the essay.

                  I was indeed vague with Nmax, because I don't want to commit to it. I am aware of this difference between classical and quantum results. But the estimate of Nmax doesn't need to be correct for the overarching thesis of the essay, which is my point (it is NOT to estimate Nmax). As I wrote, it can be just about anything, as long as it is finite. I was going for a conservative estimate (larger Nmax).

                  You do miss one point here, though: I consider everything that could have happened in the Universe to estimate Nmax, since all the outcomes might as well have been "stored". If it can be stored, that determines Nmax. I'm not concerned if it is accessible or not. But you are completely right that a Universal Laplace demon cannot tap onto all the qubits, just the bits. This is a fair criticism and I should've been more clear about it.

                  Best wishes,

                  Rafael

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