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.

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

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

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

inaccuracy of the final answer comes from a small percentage possibility that your hypothesis was wrong despite the evidence.

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

to a branch of mathematics called ‘topology’, which studies the shapes of things.

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:

People started realising that those techniques could be used to study other things.