Hilbert's hopes, of course, were dashed by Gödel. However Hilbert's philosophy, named Formalism, became dominant in mathematics.

In this philosophy, mathematics is just a formal game played by formal rules. When we say that something exists, for instance, we are just saying that from some set of axioms, we can prove a statement about existence. It is irrelevant to us whether the axioms are true or the thing "really" exists. All that matters is that we successfully followed the rules of the formal game.