Geometry is an example, or rather geometries, for there are many. Euclidian geometry, the granddaddy of them all, starts as does any mathematical systems with a small set of axioms. An axiom is a statement which is assumed to be true. Finally, someone decided to try reductio ad absurdum , a new mathematical concept. If you work with an assumption and from it you prove a false conclusion, then the assumption is false.
Spherical geometry has a large number of other results which are different.
For instance, in plane geometry, the sum of the angles of a triangle are always a constant, but that's not the case in spherical geometry, where the sum can range from 180 to 540 degrees. Well, they both are. If you try to use Euclidian geometry to navigate, you'll get lost. Spherical geometry gives you the right answer. This is geometry done on a specific curved surface shaped a bit like a saddle.