You wrote:
"Personally my impression is that work in description logics, databases and such has been held back by a focus on mathematics."
and
"highly descriptive representations"
These two remarks were very popular during the hype period of 'artificial intelligence' (AI). The dream was that we could avoid math, use your
"highly descriptive representations"
and some magic dust, and, presto, get great results. We didn't.
Instead, it remains: To build something solid, we need solid specifications of (1) what the thing is to do and (2) what we have to work with and then actually design the thing so that we are fairly sure from the design that we will get (1).
E.g., to build a bridge across the Golden Gate of SF, we still need solid specifications of (1) what the bridge is to do and (2) steel, concrete, the bedrock, etc. Otherwise you won't want to drive across the bridge.
Well (1) is a 'specification', and so far we still need one. AI, etc. just wanted to say, I won't have a specification and, instead, will know I like it when I see it. Doesn't work well enough.
When my team and I gave a paper at an AAAI IAAI conference, nearly all the good work was from solid, traditional engineering with careful design as I have outlined.
For problems such as you mention,
"Captain Kirk is a person is the Star Trek universe"
a specification is difficult to write. It remains, for this difficulty, the only hope we have is math, but, yes, the math for such a specification is difficult and likely needs some advanced prerequisites and maybe some original work.
Possible in some seemingly challenging cases? Yes. Easy? No. Doable for all such problems now? No.
Without the math, with a lot of effort, some human 'domain expertise', some heuristics, a lot of fitting ('machine learning') to a lot of empirical data, a lot of testing and revision, we can build software that can do well, say, playing chess or answering questions on TV.
Still, you wouldn't want to drive across a bridge built that way. If the bridge had stood for a year, then maybe you'd try it. But you wouldn't want to be a test pilot for an airplane designed that way.
If constructing solid math specifications is too difficult and just "I will know I like it when I see it" is too sloppy, then we could use some new 'paradigms'. Here's a quarter, and it's worth more than a pair of dimes!