In Griffiths, it seems that the conceptual introduction of the magnetic vector potential to electrodynamics was justified based on the fact that the divergence of a curl is zero; so we can define a magnetic field as the curl of another vector A and still maintain consistency with Maxwell's equations. Further, curl-less components could be added to A (introducing the concept of different gauges) and still obtain the same results as well. My question is, basically: was it a purely mathematical justification for introducing the physical concept of the magnetic vector potential? I.e., was it just a question of noticing "Hey, I can make B the curl of another vector!"? So essentially I guess I'm asking: did the physics drive the mathematics or vice versa? My assumption is that the mathematical relationship was noticed first, followed by the introduction of physical concepts, gauges, etc.; but wondering if others may know differently? Thanks.