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

TL;DR

This paper challenges the physical relevance of real numbers in classical mechanics, proposing a finite-information, non-deterministic alternative that aligns with quantum theory and reexamines the nature of determinism and realism in physics.

Contribution

It introduces a finite-information classical mechanics that is empirically equivalent to traditional mechanics but non-deterministic, and reinterprets the role of real numbers and determinism in physics.

Findings

Real numbers contain infinite information, which is physically implausible.

A finite-information classical mechanics can replicate classical mechanics' predictions.

Both the alternative classical mechanics and quantum theory can be made deterministic with additional variables.

Abstract

It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can't contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is ``random numbers'', as their series of bits are truly random. I propose an alternative classical mechanics, which is empirically equivalent to classical mechanics, but uses only finite-information numbers. This alternative classical mechanics is non-deterministic, despite the use of deterministic equations, in a way similar to quantum theory. Interestingly, both alternative classical mechanics and quantum theories can be supplemented by additional variables in such a way that the supplemented theory is deterministic. Most physicists straightforwardly supplement classical theory with real numbers to which they attribute physical existence, while most physicists reject Bohmian mechanics as supplemented quantum theory, arguing that Bohmian positions have no physical reality.