I heartily recommend Mathematics Made Difficult by Carl Linderholm.

https://en.wikipedia.org/wiki/Mathematics_Made_Difficult?wpr...

Imagine an introduction to math in that style.

Fun aside: I'd love to see the reaction of an applied-math class, like an engineering class, in which the instructor after having introduced the course, says, "let's go through some math prequisites for this class" and essentially starts reciting something equivalent to Principia Mathematica (without mentioning what he's trying to prove, meaning in a bottom-up fashion),

- "If A is true, and B is true, then C is true"

- "We know A is true, therefore if B is true, C must be true"

- "B is true if and only if D is true and E is true"

- ...

- Therefore C is true.

- (on and on and on for a whole hour).

Do that to any pure math class and they will similarly walk out. You need to know where something abstract like this is going to follow at all.

> You need to know where something abstract like this is going to follow at all.

Indeed, and do remind your audience from time to time about where you're going. Otherwise, your whole line of reasoning might become useless to someone who lost track of the point, or just didn't hear or grasp it the first time you said it.