abstract mathematics might not be directly applicable to real life, but, rather, applicable to something else which is applicable to real life, or via an even longer chain of applications, for example:

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

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.

‘It’s very simple really. You’ve lost something you loved.’

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.

This can seem frustrating (there simply are diagonal highways in Chicago) but the point is to shed light on something rather than to model it precisely.

feeling stupid for not having understood something before just shows that you are now cleverer than you were then.

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.

The key in maths, and with maps, is to find the most appropriate level of abstraction for the given moment.