Math is an ideal, a purity, useful for studying and hypothesizing messy physical realities. Good math is crucial to making controlled observations and discoveries. In the absence of actual physical access to a problem, it can be guessed at, and even pinpointed using mathematical models.
Slideshow:
Mathematics studies perfect objects – the Platonic ideal – but often the proofs can be messy and confusing.
The technological shift in electronics from analog to digital was paralleled by a paradigm shift in mathematics, with discrete and combinatorial maths replacing traditional continuous analytical maths. Continuous objects are now converted to discrete representations for processing on digital computers.
Mathematical abstractions never look like the real thing, but they are often as powerful, and are frequently more conceptually illuminating, in an abstract way.
It’s a simple matter to tie a complex knot, or set a combination or code. It’s much harder to untie the knot or crack the code.
Ever since Pythagorus and likely before, mathematicians have seen the world as being composed of numbers, which has led to great insights.