From what I know, this happened in a large part due to the Three Body Problem. Once it was proved that there was no closed form solution in terms of elementary functions, Poincare suggested that we instead try to see if solutions existed at all, and whether they were unique. If so, we could at least hope for numerical solutions. I think this was partly responsible for the development of "analysis situs", i.e. topology. [1]
This approach was fruitful in yielding concrete techniques - we now have efficient techniques like the fast multipole method by Rokhlin and Greengard, to numerically tackle the three body problem. Without existence and uniqueness results, there is no basis for the correctness of these algorithms.
This approach was fruitful in yielding concrete techniques - we now have efficient techniques like the fast multipole method by Rokhlin and Greengard, to numerically tackle the three body problem. Without existence and uniqueness results, there is no basis for the correctness of these algorithms.
[1] https://en.wikipedia.org/wiki/Poincar%C3%A9_and_the_Three-Bo...