Well, it's not exactly engineering, but physics often uses quite informal math. For a pretty modern example, the Dirac delta "function" was used long before it was formally described; and I have heard it said that even today String theory uses some math that is not fully formalized - though I can't say I know what specifically, so I may be wrong. Newton expressed calculus in terms of inifinitesimals (the dx notation was simply an infinitesimal delta x, not merely notation for derivation), which were not not formalized until much later, after they had already fallen out of favor and had been replaced in formal math by the delta-epsilon limits-based constructions.

Edit: one better example from modern physics - the path integral formulation, used in both string theory and other areas of QM/QFT, is not fully formalized and formally proven to work. Also, a more concrete example of a widely used but actually still unproven conjecture in string theory is the famous AdS/CFT correspondence.