The Liar's Paradox is still considered an "unsolved problem in philosophy"? I don't see why it's considered a big problem that we're able to define things that can neither be sorted into the "true" bucket nor the "false" bucket. If you could derive a paradox from, say, the Peano axioms, then that would indeed be problematic, but as it is, why is the fact that you can say "This sentence is false" any more problematic than the fact that you can say "let X = ¬X" without all of logic imploding?

Math is the art of constructing tautologies complicated enough to be useful. I don't think it's any mark against it that you can use the same language to describe things that are neither useful nor tautologous.