How can mathematical knowledge be independent of experience and at the same time turns out to be such a fruitful and necessary instrument for elaborating theoretical tools for experimental sciences?

Is mathematical knowledge independent from an existence that is assigned by our mental perceptions (our physical reality) ?

Mathematics is built from definitions and axioms. The conclusions which you can draw from these definitions and axioms would not change in some type of different reality.

Math is not independent of our physical reality.

It is a compliment to it.

Math is just another way to describe reality.

Geometry and trig are 2 dimentional worlds.

Calculas is a 3D world.

Diff eq is about 3D objects that move and interact.

Fractals decribe an infinately irregular surface.

Mathematics is a language. It is a form of expression that develops over time. It is independant from it's subject, expressing its subject with symbols and numbers to convey a relationship, without using words. Language itself is abstract and universal, taking many forms.