Abstract The physical phenomenon of symmetries and their physical–mathematical conceptualisation traverse the history of science, particularly in geometry, physics and mathematics. In order to understand and describe the laws of nature (in certain situations, e.g. status, relationships, applicability, fields etc.), scientists, historians and philosophers persistently worked on these subjects, particularly on the significance of gauge symmetry and its correlated mathematical operations (theorems, transformations, conserved quantities, violation, breaking, etc.). One of the main purposes is to understand when and how a violation - a breaking of symmetry - is performed. For the sake of brevity and taking into account the physics–mathematics relationship between Physics and Mathematics in the history of physics/mathematics, this paper concisely discusses the intellectual role played by the Physics and Mathematics relationships in the work by James Clerk Maxwell, Emmy Noether and Albert Einstein.