What "math" means to people outside of math--including physicists and engineers--is completely different from what "math" means to people inside math. We really ought to have a different word for the two fields. Even "applied math" doesn't cut it, because most applied mathematicians still think like mathematicians, and it ain't like the rest of us.

What mathematicians don't see is that when they rail against the kind of concrete, heuristic definitions that non-mathematicians use for parts of mathematics, is that it isn't just that we don't care that there is a better, more general, more abstract, more justifiable definition out there, it's that we can't understand them so they aren't useful to us, even a little bit. You may as well try explaining Shakespeare to your dog: it'll frustrate you and annoy the dog, who is perfectly happy to use Shakespeare for his intended purpose, which is to say, chewing on the edges of the "Complete Shakespeare" for entertainment.

That's an overstatement, but the gulf between the mathematician's mind-set and the mind-set of physicists and engineers is still far wider than most mathematicians appreciate, and it really would be valuable to have a different word for what physicists and engineers do when we "do math", because it is not "math" the way you mean it.