Of universal computers, infinities, and demons by Rafael Alves Batista

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

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.

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 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.