> Being able to create systems by writing specifications and having the computer figure out how to execute them was basically the point of fifth generation programming languages.
Yeah, but in maths you can specify anything, including things that the computer is unlikely to figure out how to execute if it's possible at all. Programming languages of every generation are very useful, as is mathematics, even though they're not the same thing.
> More relevant today, you can execute other "specification" languages like Coq and Idris because they support things outside the narrow feature set of specification usecases.
Coq and Idris are very different in the way they're typically used, and I'd say TLA+ is much closer to Coq than to Idris (and is probably more popular than the two combined), but to "execute" anything the specification needs to be at a certain level that's detailed enough to produce a program, and oftentimes that is very much not what you want.
It would be extremely useful to have a language that you could describe the various properties of a car and it would compile your specification into the design of a car (you would need to give it sufficient detail as there are choices to be made). But it would also be extremely useful to have a language that could be used to learn certain things about a car -- say, it's braking distance -- without specifying it in sufficient detail to actually build one. That is what maths is good for -- describing things at arbitrary levels of detail to answer relevant questions.
For example, you may have a 5 MLOC distributed system, and you want to know if a certain kind of failure may lead to data loss. You could use TLA+ to describe just the relevant details to answer the question in, say, 200 lines of formulas. That you cannot compile those formulas into a working 5 MLOC piece of software is not a downside of mathematics, but rather the point.
> TLA+ isn't executable and doesn't look like an imperative language because the authors don't want it to be
And because it's not a language for programming but a language for mathematics, so it looks and feels pretty close to plain mathematics (only it's formal), as that's the obvious choice for writing mathematics.
