Do symmetries "explain" conservation laws? The modern converse Noether theorem vs pragmatism

TL;DR

The paper examines the relationship between symmetries and conservation laws, arguing that pragmatic considerations favor symmetries' primacy despite the lack of strict explanatory direction in Noether's theorems.

Contribution

It highlights the overlooked nature of Noether's converse theorem and discusses pragmatic reasons for prioritizing symmetries over conservation laws in physics and philosophy.

Findings

Noether's direct theorem is more well-known and utilized than its converse.

Practical factors influence the emphasis on symmetries in scientific practice.

The strengthened versions of the converse theorem by Alonso and Olver are significant.

Abstract

Noether's first theorem does not establish a one-way explanatory arrow from symmetries to conservation laws, but such an arrow is widely assumed in discussions of the theorem in the physics and philosophy literature. It is argued here that there are pragmatic reasons for privileging symmetries, even if they do not strictly justify explanatory priority. To this end, some practical factors are adduced as to why Noether's direct theorem seems to be more well-known and exploited than its converse, with special attention being given to the sometimes overlooked nature of Noether's converse result and to its strengthened version due to Luis Mart\'inez Alonso in 1979 and Peter Olver in 1986.