Most of these work if you use them properly. To use it you have to understand the symbols, remember the rule and know when to apply it. Mathematics The study of ideas like number, quantity, arrangement, shape and space. In the Middle Ages, music was an important part of pure mathematics. Mathematical Models Mathematics provides us with models of external reality.
Pure mathematics has no necessary relation to external reality, it is a thought form.
Applied mathematics uses some of these thought forms as models of external reality. A model has a clearer shape than the real thing. It does not have all the confusing detail of the external reality and it allows us to manipulate an idea of the reality in our minds. Looking at your watch you calculate that the train is due in London in two hours. You think that means the train is only travelling at fifty miles an hour.