Indeterministic finite-precision physics and intuitionistic mathematics

TL;DR

This paper explores the implications of modeling physical quantities with finite precision rather than real numbers, examining indeterminism in classical mechanics and the challenges of formalizing finite precision mathematically.

Contribution

It analyzes alternative finite-precision models in classical mechanics and discusses the limitations of intuitionistic mathematics for formalizing such models.

Findings

Finite-precision models suggest natural indeterminism in classical mechanics.

Intuitionistic mathematics faces conceptual contradictions when formalizing finite precision.

A classical-mathematics-inspired approach to finite precision is proposed as an alternative.

Abstract

In recent publications in physics and mathematics, concerns have been raised about the use of real numbers to describe quantities in physics, and in particular about the usual assumption that physical quantities are infinitely precise. In this thesis, we discuss some motivations for dropping this assumption, which we believe partly arises from the usual point-based approach to the mathematical continuum. We focus on the case of classical mechanics specifically, but the ideas could be extended to other theories as well. We analyse the alternative theory of classical mechanics presented by Gisin and Del Santo, which suggests that physical quantities can equivalently be thought of as being only determined up to finite precision at each point in time, and that doing so naturally leads to indeterminism. Next, we investigate whether we can use intuitionistic mathematics to mathematically express the idea of finite precision of quantities, arriving at the cautious conclusion that, as far as we can see, such attempts are thwarted by conceptual contradictions. Finally, we outline another approach to formalising finite-precision quantities in classical mechanics, which is inspired by the intuitionistic approach to the continuum but uses classical mathematics.