Engineering is used fairly loosely these days but I insist engineering ends where you have to prove theorems.

That’s a strange delineation. Engineering is essentially about designing a thing, asking whether that thing really does satisfy the desired criteria, and then iterating when it does not. Mathematical models of the world are tremendously useful to this practice—engineers don’t need to guess about many aspect of the real world: they have physics. What they want, more than anything else, is strong evidence that a property holds. Internal validity (proof) and external validity (experiment) are the best evidence that you can get—why would you throw one of those approaches away?