[Q_is_4_Quantum]

Agree with this - the jury is still out on "reality" or otherwise of a wavefunction, and people expend their "real" research efforts differently according to their opinion! Saying "well everything is just mathematics" is a bit like advocating solipsism: people will roll their eyes and stop inviting you to cocktail parties, its a boring position.

Another historical example would be electric and magnetic field lines. My impression is that even Faraday originally had doubts about their reality; others certainly did. They could have just been forever considered a useful mathematical construct. At some point it became clear that thinking of them as really existing, permeating space, and having physical properties akin to those of accepted real stuff (momentum etc) was more useful. Many years after GR was formulated there were arguments about similar mathematical objects (dynamical components of a tensor, or "gravitational waves") should be considered real or not. The story is that Feynman convinced many with a simple thought experiment about how they transmit energy.