No form of mathematics applies to everything. And no matter how you define a mathematical system, there will always be valid statements within it which are true but which can't be proved within that mathematical system. Inconsistent systems are uninteresting. But even a consistent system may not yield practical results. A mathematical system is an artificial construct which may or may not be applicable to a real system.
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.