> In other words, where a mid-wit would step through the code (and be right), a high-brow might deems the problem impossible because it fits to a generic class of which this particular instance is trivial.

In this situation, “mid-wit” might best apply to the person who studies theoretical computer science, but never manages to understand when this result is applicable: these people will fall into the second category. But see my comment at the end about whether this terminology is really appropriate.

Someone who quotes the undecidability of HALT in reference to the question of whether that specific program halts is probably not a “high-brow” (whatever that means). It’s a perfect example of “a little knowledge being a dangerous thing”: those who actually understand the theory will know when it applies; it’s those who don’t have a good internalised model of the theory, but who have a vague recollection of something along the lines “it’s impossible to tell if programs halt or not”, who are going to get it wrong.

As a side note, I don’t think “mid-wit” vs. “high-brow” is a good distinction here: whether or not you remember (or were taught) about the halting problem is not a matter of intelligence, and I suspect most people who get it wrong aren’t doing so because they never understood theoretical computer science, but rather because they haven’t been practicing academia, but rather the craft/art of software engineering, and have just forgotten it.