> Philosophy behind the Napkin approach
> As far as I can tell, higher math for high-school students comes in two flavors:
> • Someone tells you about the hairy ball theorem in the form “you can’t comb the hair on a spherical cat” then doesn’t tell you anything about why it should be true, what it means to actually “comb the hair”, or any of the underlying theory, leaving you with just some vague notion in your head.
> • You take a class and prove every result in full detail, and at some point you stop caring about what the professor is saying.
> Presumably you already know how unsatisfying the first approach is. So the second approach seems to be the default, but I really think there should be some sort of middle ground here. Unlike university, it is not the purpose of this book to train you to solve exercises or write proofs, or prepare you for research in the field. Instead I just want to show you some interesting math. The things that are presented should be memorable and worth caring about. For that reason, proofs that would be included for completeness in any ordinary textbook are often omitted here, unless there is some idea in the proof which I think is worth seeing. In particular, I place a strong emphasis over explaining why a theorem should be true rather than writing down its proof.
And yes, with more mathematical maturity you definitely don't need as much detail. The proofs get terser as you're expected to be able to fill out the more straightforward details yourself.
The style of this textbook does seem to primarily skip the "techniques for evaluating" stuff, on the basis that you just wanted to understand what each branch of mathematics is about and what kinds of theorems it has that might relate to the larger edifice of mathematics.
