I think the difference might be this, in undergraduate degrees: * Mathematics is entirely about proving arguments and solving problems within logical systems.

I think the difference might be this, in undergraduate degrees: * Mathematics is entirely about proving arguments and solving problems within logical systems. * The 'hard sciences' (esp. physics) are about working within a theoretical (often mathematical) framework and testing it with experimentation. Contrast this to engineering disciplines, or even medicine, where the focus is on gaining knowledge which you can use to solve problems. It's problem-solving, but doesn't have much of the scientific method about it.