Have a look at our Knot Exhibition, which aims to explain how mathematics gets into knots. It explains some of the methods used: representation, classification, invariants, analogies, laws and applications. The applications come after one has developed the necessary concepts and methods, and may also be motivated by such potential applications.

Mathematics develops rigorous language for expression, proof, analogy, verification, falsification, calculation. As an example, the language of abstract group theory was developed by many pure mathematicians, and was found necessary to determine all the 230 crystallographic groups, and also to develop the theory of quarks.

The abstract language of mathematics is really about analogy. Much modern pure mathematics is about describing abstract structures, relating them, often via category theory, and the difficult task of describing their interaction. For this, numbers are not enough.