[beiller]

Explained better above. There are more irrational numbers, almost guaranteeing any number come across in nature is irrational. Interesting thought since I think one thing that makes irrational numbers is there is no function for them. So it's kind of a cheeky way to say no math formulas can ever describe the real world since all the numbers are irrational.

[dagw]

There are more irrational numbers, almost guaranteeing any number come across in nature is irrational

There is nothing that says that the distribution between rational and irrational numbers that show up in nature is the same as the distribution in our construction of the real numbers.

[adastra22]

You’re missing the point: physics has shown the world to be discretized. Almost every number you go out and measure IS ACTUALLY AN INTEGER. In that sense the real number system doesn’t exist. It’s super useful, yes, but it’s an abstraction away from reality.

[2snakes]

I thought that "infinities of infinities" ala Cantor exist though. I agree with you that reality being quantized means everything in physical reality can be normalized to integers.

[eru]

> physics has shown the world to be discretized.

It's a bit more complicated than that.

[adastra22]

I am a physicist, and I believe what I said is accurate. Although yes I am (1) oversimplifying, and (2) assuming that some form of quantum gravity is correct.

[eru]

To expand a bit with some examples: the energy a photon in a standing wave in a specific cavity can have is quantised. But a photon out in space can have any old energy it wants to. (Of course, a given energy level will correspond to a specific wave length etc.)

Similar, an electron in a single isolated atom has specific quantised energy levels. But if you look at the electrons in a hunk of copper, they are essentially free to absorb and emit energy in almost arbitrary amounts.

An even stronger example is time: as far as I am aware, time is not quantised in any of our accepted theories.

There's some reasonable speculation that ultimately everything is quantised at the Planck scale, including time. But that's just a very reasonable hunch, not something that 'physics has shown'. (And you already point out that trying to marry quantum mechanics with general relativity is a hot mess.)

I agree that 'quantisation all the way down' is the way to bet. But that's just speculation.

(But I strongly disagree with your claim that physics is build on integers. Yes, it might be discretised, but there are plenty of discrete structures that are not integers. Look at a Rubik's cube for a simple example. On top of that: almost any real world measurement is better described by a probability distribution than by single number, be that an integer or otherwise.)