I did a BSc in Theoretical Physics, meaning that a lot of my time was spent deriving equations, making hand-wavy arguments, and arriving at solutions with a distinct lack of rigour.
I'm now doing an MSc course which involves maths more in the way of a Maths BSc.
I was explaining to a friend who I did physics with that we didn't really do proper maths in the sense that, despite being very difficult at times, it's really nothing like the maths that you would do an a maths degree, and I proceeded to say that "maths is all about proofs, we never really did proofs", to which he replied: "We derived equations, isn't that a proof?"
I didn't really know how to respond. Is there a formally defined difference between a proof and a derivation? Are they even different?
When I think of derivations I think of the sort of unrigorous maths you get in physics, and when I think of proofs I think of the rigorous maths you get in maths.