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 mathematici

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. I don't think that's true, at least in areas where physicists or engineers are actually going to be trying to use the relevant math. I think the difference is mostly in norms of written presentation. In physics, the concrete motivating example is often seen as an important thing to present; in math, it seems more typical to polish your definitions until they're as minimal and general as can be and start the presentation there. This often obscures the more concrete examples, but I've found plenty of times that if you read or talk to mathematicians in informal settings they can be pretty good about the examples that motivate the ideas.