Well, the liar paradox dates to antiquity, the halting problem has been well known for 70+ years, etc., and yet.... it turns out we can still use logic and software to solve lots of practical problems anyway, simply by avoiding self-referential contradictions and infinite loops.

Arithmetic seems to have a number (so to speak) of practical applications as well.

Perhaps it's a bit like theory vs. practice, or maybe mathematics vs. engineering.