Indeterminism in Physics, Classical Chaos and Bohmian Mechanics. Are Real Numbers Really Real?

There's an interesting paper[1] that argues that real numbers aren't physical, for the precise reasons you stated. That is, only the subset of real numbers that contain a finite amount of information (like constructive real numbers) are physically meaningful. A consequence of this is that classical physics is not really deterministic: this is because, in general (ie. including chaotic systems), the evolution of a system depends on a set of initial conditions that are specified by full real numbers, impossible to measure with finite precision. So, the use of real numbers is hiding the indeterminism in the initial conditions, much like the function in this article encodes a dataset in a single parameter. [1]: https://arxiv.org/abs/1803.06824

We should also mention Nicolas Gisin. One of his mantras is Time is real; Real numbers are not. He conceives of real numbers resulting from processes (approximations, relaxations, computable calculations) that unfold over time. So there is a Heisenbergish uncertainty principle of precision and elapsed time. Selected papers: Indeterminism in Physics, Classical Chaos and Bohmian Mechanics. Are Real Numbers Really Real? https://arxiv.org/pdf/1803.06824 Real Numbers are the Hidden Variables of Classical Mechanics https://philarchive.org/rec/GISRNA Time Really Passes, Science Can’t Deny That https://arxiv.org/pdf/1602.01497v1 Popular articles: Real numbers don’t cut it in the real world, this physicist argues https://www.sciencenews.org/article/real-numbers-physics-fre...

If it really was continuous so that physical quantities were real numbers as defined in mathematics, then it is in contradiction to maximal information density. Because almost all real numbers contain infinite amount of information. Full argument is elaborated here "Indeterminism in Physics, Classical Chaos and Bohmian Mechanics. Are Real Numbers Really Real? by Nicolas Gisin": https://arxiv.org/abs/1803.06824

I been juggling the following three papers for a bit and trying to brush my maths up. I don't know why I feel a hunch between them, but I do know that the hunch is to do with this problem. https://arxiv.org/pdf/1803.06824.pdf https://arxiv.org/pdf/1306.0533.pdf https://arxiv.org/pdf/1204.2779.pdf I will almost certainly be wrong about whatever it is. However I may have more updates on how wrong and what it is I am wrong about, in maybe a decade or so ;)

Ah, okay, so it boils down to the question whether real numbers are physically realizable. There is a debate about that in the philosophy of physics and mathematics. AFAIK, according to the current state of physics real numbers cannot be realized within a finite amount of space, at least Gisin [1] recently argued that way and I've seen other arguments for this view before [2,3,4,5:167-170]. I also remember some argument based on thermodynamics but cannot find the reference. [1] https://arxiv.org/pdf/1803.06824.pdf [2] https://arxiv.org/abs/math/0411418 [3] https://news.ycombinator.com/item?id=14080024 [4] https://ptolemy.berkeley.edu/~eal/pnerd/blog/are-real-number... [5] https://www.sciencedirect.com/science/article/pii/S030439750...

It is, this idea is explored a bit in this paper: https://arxiv.org/abs/1803.06824. It also argues that using real numbers for physical quantities makes classical mechanics misleadingly deterministic, that is, they're hiding information that is impossible to measure in the initial conditions.