[inimino]

In a theoretical sense, Go is certainly solvable, as we can easily construct a Turing machine that plays out every possible move, and provably terminates (given an appropriate ruleset). Practically, the universe will certainly be expected to terminate first.

There may be other ways to solve the game, but we don't know what they are. Because we know it is theoretically solvable, we cannot rule out a practical approach to solving it by some mathematical magic even if we have no idea what that would look like.

By analogy, we can prove many things about infinitely many integers by mathematical induction, but if we didn't have that technique, such proofs might seem impossible.