It depends what you mean by "solved." In a rigorous, pure math sense, no. But so what? Math is only about itself and its own esoteric rules, not about practical reality. In a scientific sense, it is proven in the same way we know there are no unicorns: there isn't a scrap of evidence they exist, nor is there any reason to believe they should.

We use math as a framework for reasoning about physical reality. When doing this we assume that the logic of our math holds for physical reality and that we can use it to make useful predictions. If a mathematical theory has held up so-far then that is evidence that it will continue to. It's not a guarantee, but it is evidence.