Question

Book Recommendations for Non-mathematician Curious About Advanced Topics

Answer 1

I think the book you're looking for is *The Princeton Companion to Mathematics* by Timothy Gowers, which pretty much serves as an encyclopaedia for the entirety of pure mathematics as it stands now. There is also a companion book which does the same for Applied Mathematics - *The Princeton Companion to Applied Mathematics* by Nicholas Higham.

I would also recommend the books by Keith Devlin, John Stillwell and Ian Stewart which cover various mathematical fields - though not as comprehensively as *The Princeton Companion.*

Answer 2

i think you'd be best watching channels such as 3 blue 1 brown and numberphile. youtube is dominating content in the space of mathematics-for-non-mathematicians. mathematicians don't tend to write for general audiences and when they do, it tends to be too loose and flowery or too technical to be suited for the audience. i think visual media is just superior for this sort of stuff

Answer 3

If I were you, I would invest time understanding the foundations:

- Infimum & supremum
- Continuity & limits
- surjective and injective function
- Bezout's identity, congruent numbers
- Linear maps, linear dependence/indepence, dimension, basis
- Basic set theory notation

Definitions, proofs and propositions: this is the holy trinity of math. (Also known as blood, sweat and tears.)

Answer 4

"Proof" has entered the chat.

What you want is to pick up is a first book on Number Theory or Graph Theory. It will be a new topic for you, but will be presented in a way that's aimed at people who are ready to "level up" their math education. After you've done that, and have a sense of the joy of proof, you'll be ready to return to calculus and such with a new brain.

Answer 5

Nahin's When Least is Best

Answer 6

The Road to Reality by Roger Penrose