How to Bake Pi

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.

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

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

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.

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

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

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.

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.

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