Mathematical structure of nature

For a while now, the correspondence between a mathematical structure and the way nature behaves has puzzled and amazed me. In other words, Why is it that I can write down equations on a sheets of paper and predict something about a natural object?
Newtonian Mechanics had a certain mathematical structure and Maxwell's Electrodynamics had a certain mathematical structure, and these two structures couldn't co-exist. Just by trying make the equations co-variant, we were able to get great insights into the way nature worked. We could get testable predictions which we were ofcourse verified by experiment.
Quantum Mechanics brought in another mathematical structure which takes into account the relationship between the observer and the observed, and gives us the nature of a measurement.

There may be domains where experimental physics cannot yet reach. There may be some underlying physical phenomena which cannot yet be detected by experiments. To probe into those domains, as of now, we may only have mathematical tools at our disposal.
Here is a paragraph from a book on Quantum Mechanics by Bohm ( Ref 3) that I think is worth mentioning:

" Physicists believe that there is something in nature, or in each restricted domain of it, that may be "understood"; that there is a structure in nature. To "understand" means to bring this structure into congruence with some structure in our mind, with a structure of thought objects, with a structure that has been created by our minds. For physics, this structure of thought objects is a mathematical structure. So to understand part of physical nature means to map its structure on a mathematical structure. To obtain a physical theory, then, means to obtain a mathematical image of a physical system. 
For the domains of quantum physics the mathematical structures are algebras of linear operators in linear spaces. The discovery of this, the fundamental properties of the algebra, and the other basic assumptions of quantum mechanics was a very difficult process. " 

I just hope we have the capability to recognise the true underlying structure of nature( however naive this may sound).


P.s. The references are given in the column on the right.