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There is an easy counter model. The universe is _literally_ a real number machine, i.e. even a particular point is only describable to us in the approximate. The laws of physics get to be finite, but you can't talk about a particular location or place in time without infinite information, or you talk about it in the approximate.

Am I correctly summarizing this statement, for my own understanding?

“All values in the cosmos are integers. Decimals exist to help humans make comparisons between objects.”

I mean, they seem to be saying the exact opposite of that based on the usual way people talk about the "reals".

No. They're talking about real numbers; even decimal numbers (rationals) are isomorphic to integers. The reals are different.

You could put all the human knowledge in a single real number and still have infinite room left.

There exists an abstract mathematical mapping, sure. However, finding the spot in the real number where that's the case may require infinite storage (you need to store the pointer) or infinite time (you start searching through the number to find what you want). It also has a halting problem where you don't actually know how you'll know when you find that data.

> The universe is _literally_ a real number machine

I am wary of this. Just because you have a good predictive model based on math, and the math can be made rigorous using ZFC / Cauchy sequences / infinite decimals / etc., doesn't mean that any of those rigorous mathematical concepts are literally running the real universe.

Personally I don't think there's any reason to believe that any numbers, even zero and one, have any existence outside of being concepts in our heads.

I don't get what you say in the last paragraph. Isn't the mapping of zero and one to nothing and something kind of the easiest thing (e.g. one photon or no photon)?

For the first paragraph it is my understanding that the universe is discrete/quantizied and it is not running on 'real numbers'.

> I don't get what you say in the last paragraph. Isn't the mapping of zero and one to nothing and something kind of the easiest thing (e.g. one photon or no photon)?

You can map and mapping is useful, but the map lives in your head and isn't the territory.

> For the first paragraph it is my understanding that the universe is discrete/quantizied and it is not running on 'real numbers'.

This is a widely believed folk myth among software engineers, but it isn't supported by evidence. There are some speculative theories that try to solve some problems in physics by quantizing space, but right now they are no more empirically supported than other speculations.

Worth a read:

1. "Russell’s No Man’s Land" [0]

2. "The hollow universe of modern physics" [1]

[0] https://edwardfeser.blogspot.com/2020/08/russells-no-mans-la...

[1] https://edwardfeser.blogspot.com/2022/05/the-hollow-universe...

Mh, but I thought it was widely held that there are things such as the Planck constant which constitute bounds on divisibility. There rather seems to be no indication to believe conversely that the universe works on real numbers. Which makes at least intuitive sense that there isn't anything which contains infinite information.

There is such a thing as the Planck constant. It doesn't constitute a bound on the divisibility of space. It is a length scale below which we do not expect our current physical theories to make good predictions. See https://www.fnal.gov/pub/today/archive/archive_2013/today13-...

> Which makes at least intuitive sense that there isn't anything which contains infinite information.

There are no real numbers outside of human beings' minds. The information in a number, real or otherwise, is a theoretical mathematical concept from human minds for which there is no evidence in nature.

There is no definitive evidence that nature confirms to either continuous or discrete math at the smallest scales, but for now, if you look at theoretical physics papers, you will see integrals and derivatives, not sums and differences.

>There is an easy counter model. The universe is _literally_ a real number machine, i.e. even a particular point is only describable to us in the approximate.

There is actually some debate about that, and for good reason. There are even a couple of practicing physicists looking at how predictions would differ assuming the universe has finite precision, and if I recall correctly the results from that investigation are non-trivial.

>Then everything humanity will ever reach or know has Kolmogorov complexity no larger than this X.

Only if it is deterministic, which certainly seems not very plausible. (You could say everything is a probability distribution, but to me it seems to get extremely complex after that, collapse?)

>Because we already know the laws of physics are relatively simple, I would put my bets on the initial conditions having most of the entropy.

Just because the laws are simply expressible does not imply they are simple. A variational minimization problem can be very easy to state (certainly that seems a good candidate for a "true" law of nature), but it can easily be as hard as a PDE!

It’s also possible that our __models__ are simple, but the underlying reality is not, so as long as there exists some error after sufficient evaluations it is safe to say that our model’s complexity is less than the universe’s.

Yes, I agree. It is definitely possible that the underlying laws are extremely complex and there is absolutely no simple way to state them.

But what I think is really interesting that this extreme complexity can only be found in extremely large and small scales. E.g. Newtons laws are phenomenal at predicting the universe at "human scales". The systems which can arise in Newtonian physics can be extremely complex, but their descriptions are usually very simple. Deviations from Newtonian predictions either are very small or arise only at large/small scales.

> Newtonian physics can be extremely complex, but their descriptions are usually very simple

IIRC Constructor theory builds on this, in a way, it defines some very basic and simple building blocks out of which the rest of the laws that govern our universe.

https://en.wikipedia.org/wiki/Constructor_theory

> Only if it is deterministic, which certainly seems not very plausible.

A collapsing quantum state is so far the only proposed process which would result in a nondeterministic universe. The problem is that absolutely nothing is known about it, and everything around it is fishy.

I personally think that the concept of the universal wavefunction makes more sense.

The point about determinism is correct, but since wavefunction evolution is unitary, the universe is fully deterministic. Only if you take some interpretation that there's real physical process of measurement which we don't understand and is indeterministic you can get randomness.

But maybe if you subscribe to multiple worlds interpretation then the hardest part might be specifying which universe you're in, which would require number of bits proportional to the time since the big bang.

> The point about determinism is correct, but since wavefunction evolution is unitary, the universe is fully deterministic.

A problem: if you consider the contents of the universe (and the precise state of the contents at a given snapshot in time) to be part of "the universe", you have the hard problem of consciousness to tend with (in that humans are a part of the universe, and they rearrange its contents on a regular basis).

> Only if you take some interpretation that there's real physical process of measurement which we don't understand and is indeterministic you can get randomness.

Randomness is not the only means to achieve a non-deterministic universe, there is also the possibility of non-deterministic forces being in play, like human activity.

Of course, one can assert that human cognition is deterministic (no other option may even be available in some cases!), but science strongly suggests that simply asserting something to be true does not necessarily cause it to be true (though it does often cause it to appear to be true to pre-trained instances of human consciousness, a phenomenon which science has well demonstrated).

Abstractly(!), humans seem to be in a very similar cognitive state as they were when they ran on sub-perceptual religious axioms.

https://ell.stackexchange.com/questions/155593/meaning-of-th...

Either that, or reality is inherently quantum and the entropy of an evolving unitary wave function (well, the entropy of any given amplitude) basically goes to infinity.

> Surely not everything is so incredibly compressible, is it?

I like your line of thinking and I'm actually quite excited about this recent shift in physics to a more information-theoretic line of inquiry.

Is it fair to state a corollary of your question this way? Given a complete description of some hypothesized universe/multiverse generator (essentially the next layer down that we are actively trying to discover), the Kolmogorov complexity (or some variation of K-Complexity) of this description must be greater than the K-complexity of any generated universe.

In my view, it would be very weird if this were not true. And (to me) it would be even weirder if this K-complexity was some large number but still less than infinity.

You could have an incredibly simple universe generator that just produces universes with random laws of physics and random initial conditions. From our human point of view that's indistinguishable from a highly sophisticated universe generator, since we can only observe the successes (or rather only one, that by our definition is a success)

> My intuition is that if you take X to be too small, the general "nothing's special about the universe and it's all extremely typical" argument and you must surely reach a contradiction. Surely not everything is so incredibly compressible, is it?

This sounds similar to saying, "This photograph can be expressed in this small number of bits, therefore it's not special." Why does one imply the other?

We don’t know that the universe is discrete, hence a description in a finite number of bits, and Kolmogorov complexity, may be inapplicable.

This line of reasoning assumes that all physical processes are deterministic. If they are not (as is the current mainstream belief) then I don’t think that line of reasoning would stand. Instead, the current state is the result of that initial set of inputs, and countless other nob-deterministic dice rolls.

Also keep in mind that everything assumes the equations are Lipschitz in order to guarantee uniqueness of the solution given specific initial conditions. But this is too optimistic.

y’=y^(1/3)

with y(0) = 0

has an infinite number of solutions.

It gets far worse if it is not an ODE, but a PDE, there a uniqueness criteria is somewhere between "for this PDE in this extremely special case with these extra assumptions ..." up to "?????".

The universe appears chaotic, in that outcomes have sensitive dependence on its initial conditions. Even if all laws of physics are deterministic.

So for there to be a wide range of possible things happening from the initial starting point (including you and I having this conversation right now), IMO it follows that a complete description of the initial conditions would have to be absurdly (infinitely?) large to cause the universe to replay to this exact moment if you started it all over again.

In other words, the size of X would have to be extremely large… so I think I agree with you.

(Edited once I realized that myself and OP are likely in agreement.)

> Then everything humanity will ever reach or know has Kolmogorov complexity no larger than this X

No. Information complexity does not obey conservation laws. The initial rule for generating the Universe is simple, but as time goes on information complexity only increases exponentially. It's a singularity of information complexity and we are at the center of it.

(This is the real reason, by the way, why aliens don't exist and Fermi's "paradox" isn't really.)

What do you mean by "information complexity"?

Quantity of information absolutely does follow conservation laws, at least in quantum mechanics.

Same thing as the Kolmogorov complexity the original poster mentioned. It corresponds to entropy somehow, so I don't think quantum mechanics is on topic here.

Does this assume the/a universe is deterministic?