Relativity of arithmetic as a fundamental symmetry of physics

TL;DR

This paper explores how the freedom in defining arithmetic operations acts as a fundamental symmetry in physics, affecting equations and interpretations in classical, quantum, and relativistic contexts, especially in fractal space-times.

Contribution

It introduces the concept of arithmetic as a symmetry transformation and discusses its implications for physical equations and quantum phenomena.

Findings

Arithmetic realization influences dynamical systems in fractal space-times

Change of arithmetic differs from variable substitution in physics

Implications for Bell theorem and quantum interpretations

Abstract

Arithmetic operations can be defined in various ways, even if one assumes commutativity and associativity of addition and multiplication, and distributivity of multiplication with respect to addition. In consequence, whenever one encounters `plus' or `times' one has certain freedom of interpreting this operation. This leads to some freedom in definitions of derivatives, integrals and, thus, practically all equations occurring in natural sciences. A change of realization of arithmetic, without altering the remaining structures of a given equation, plays the same role as a symmetry transformation. An appropriate construction of arithmetic turns out to be particularly important for dynamical systems in fractal space-times. Simple examples from classical and quantum, relativistic and nonrelativistic physics are discussed, including the eigenvalue problem for a quantum harmonic oscillator. It is explained why the change of arithmetic is not equivalent to the usual change of variables, and why it may have implications for the Bell theorem.