One of the most influential physicists of all time, John Wheeler, was of the philosophical opinion that physical laws must only be describable in terms of a finite integers [1]. This is an ideal that at least some other physicists hope for. Recently, the "It from Qubit" program in physics [2], attempts to understand reality by reducing everything to computation on qubits, and has led to some interesting insights.

[1] Information, physics, quantum: The search for links, John Wheeler 1990

[[2] The Simon's institute has formally funded such a program, though the movement within physics is much larger https://www.simonsfoundation.org/mathematics-physical-scienc...

Edit: fixed citation 1. And removed reference to infinities, as indicated below.

Do you know if these efforts also only limit the formula describing these Qbit to computable reals?

I think it's a quite interesting crossroad.

Either reals don't exist, and everything is at most turing complete, in which case the universe is detemernistic and predictable, yet there are very hard boundaries to what can be known and proven.

Or reals DO exist in which case the church turing thesis is false, as the universe (with it's operations on reals) is a reified example of a super turing system, in which case we can know a lot more things than in the discrete case, but some of that knowledge can only be grasped intuitively, because the moment you try to articulate it into formulas, it's discretized and thus limited to your sub-universe/brain powerfull turing machine.

It has been claimed that "we can simulate a BQP machine using efficiently computable entries from the set {−1, −4/5 , − 3/5 , 0, 3/5 , 4/5 , 1}" [1].

So any part of reality that you can reduce to an efficient computation on a quantum computer, admits a description using the rationals only. But I might be out of my depth here.

[1] https://arxiv.org/abs/quant-ph/0003035

Reals are a digital model of analog reality. Is just a model enough for your definition of existence?

There are whole books writen about mathematical objects that turned out to not have existed.

Non-computable reals are also perfectly capable of modeling "analog reality" (because for all intents and purposes, reality could be discrete and finitely describable very small scales).

>analog reality.

Quantization would like a word with you.

Continuum is not infinity.

You are correct. I only had distant memories. I have edited.

Eh, because of quantisation you could argue only integers exist ;)

Quantised is not synonymous with discretised. The fact that (for example) energy in a simple harmonic oscillator comes in discrete packets doesn't mean that the kinetic energy of a free particle is. Some quantities in quantum mechanics are as continuous as their classical analogues.

Then you learn about Zeeman effect.

The phrase "mosts [sic] scales of reality" is doing an incredible amount of heavy lifting here, though, since the most reliable mathematics describing three of what appear to be the four fundamental forces of nature are indeed quantized.

It's more accurate to say that only integers appear to be relevant in describing the nature of the nuclear forces and electromagnetism/electroweak force.

And it seems that, at least for the moment, we're headed in a direction of quantization of gravity and a Planck unit of time as well.

Quantum mechanics is not a discrete theory in the sense you seem to be suggesting. There is this idea that is is, which comes from the discrete energy levels it predicts for a simple harmonic oscillator, or the hydrogen atom, but quantities like the momentum, kinetic energy, position etc of a quantum particle are completely allowed to vary continuously.

In some sense (because of the wavefunction) quantum mechanics is less discrete than classical mechanics, a qubit system which would be classicaly described by a single bit is described by a state that sits on a continuous manifold (the Bloch sphere).

I didn't mean to paint it as a discrete theory (and in fact I added the shoutout to the Loop Quantum Gravity stuff at the end in part because I was concerned about this interpretation).

However, across quantum mechanics, there is at least unity on the 'quantum' part. And that's really the point of contention here in terms of integer values, right?

> However, across quantum mechanics, there is at least unity on the 'quantum' part. And that's really the point of contention here in terms of integer values, right?

I don't know what this means, but there is this idea that quantum mechanics leads to integer valued stuff because the first models studied were things like the hydrogen atom and the simple harmonic oscillator which both have a natural integer valued observable (the energy). This is not generally the case, however and in quantum mechanics there are many observables which are not integer valued, for example the kinetic energy of a free particle can be any non-negative real.

To add, perhaps in a very general hand wavy way "quantisation" is something like replacing a commutative algebra A, with a non commutative algebra A_h. I do not necessarily think there is something discrete in this process. In physics functions get sent to operators. States get sent to rays.

It turns out that a lot of interesting physics these operators have physically interesting discrete spectrum.