Well...I think about Godel's theorem as the mathematical implementation of Kant's ideas about the limits of reason based upon human experience, i.e. there are some truths which are inaccessible because time and space are preconditions of all human experience.

Not to dwell on arguments about whether or not time and space actually exist independently of human experience, what Kant was getting at is that the way in which humans experience the world limits our ability to draw conclusions to a particular subset of all truths.

If Godel's theorem is true, then from a Kantian perspective, mathematics no longer enjoys a uniquely privileged place in regards to human rationality. That's pretty important philosophically - at least to some people.

Positivists may take a different view.