On the independence problem of Newton's first law

TL;DR

This paper offers a formal mathematical explanation for why Newton's first law is necessary, based on Euclidean geometry, and reviews previous explanations for the law's independence.

Contribution

It introduces a novel formal explanation linking Euclidean geometry to the necessity of Newton's first law and reviews existing explanations comprehensively.

Findings

The definitions of Euclidean geometry imply the need for Newton's first law.

Previous explanations are often fragmented and lack a formal basis.

The paper supports the formal explanation with evidence.

Abstract

Newton's laws of motion pose an apparent problem, sometimes referred to as "the independence problem": the first law seems to be a simple consequence of the second law, raising the question of why it was included as a separate law. Numerous answers to this question have been proposed in the literature. The main contribution of this paper is a novel answer which we call "the formal explanation." Unlike previous accounts it relies on mathematical formalism and argues that the definitions of Euclidean geometry necessitate the inclusion of the first law. We provide evidence in support of this claim. A second contribution is a comprehensive review of previously suggested explanations, which so far have often been treated in a fragmented manner, and a discussion of the plausibility of the various answers.