Yea, math is a crazy thing. just "a tool to solve problems" is a wild take. On one extreme, there's an edgy but logical / plausible hypothesis that we live in a universe of mathematical objects, and at the other, math also discovers a lot of questions, the exact opposite of solving problems.
Is there a useful abstraction that doesn't help solve a problem someone has?
You're right, but what I'm saying is that solving the problem isn't necessarily the primary goal and these new abstractions can be valuable in their own right.