maths is defined by the techniques it uses to study things,

In maths the technique we use is logic. We only want to use sheer logical reasoning. Not experiments, not physical evidence, not blind faith or hope or democracy or violence. Just logic. So what are the things we study? We study anything that obeys the rules of logic.

Mathematics is the study of anything that obeys the rules of logic, using the rules of logic.

That’s why it’s so important when teaching maths to understand the student’s way of thinking, and point out what was wrong with their logic, not just what was wrong with their final answer.

At the heart of maths is the desire to understand things rather than just know them.

The inaccuracy of your final answer will now come from the information you threw away when you performed the initial abstraction.

First you have to turn the reality into an abstraction. Then you do your logical reasoning in the abstract world. Then finally you have to turn that back into reality again. Different people are good at different parts of this process. But really the key part is being able to move back and forth between the abstract and the real.

Mathematics is different. The first step is the same – you start with a hypothesis that you think might be true for some reason. But instead of testing it rigorously using evidence, you test it rigorously using logic.

My friend’s wisdom was a type of abstraction, abstracting heartbreak down to its very essence. Abstraction can appear to take you further and further away from reality, but really you’re getting closer and closer to the heart of the matter. To get to the heart, you have to strip away clothes and skin and flesh and bone.