Paolo Aluffi, author of Algebra: Chapter 0 and professor at Florida State University, sits down to talk about what it actually means to do mathematics, and why most people have never seen the real thing.

00:00 — What does a mathematician actually do?

01:43 — How Algebra: Chapter 0 came to be

03:05 — The three types of students in every math class

07:39 — How Paolo got into mathematics

09:24 — If not math, then music

11:19 — What makes a mathematical structure "natural"?

12:40 — Grothendieck and the bridge between geometry and number theory

16:17 — The rising sea

19:02 — "If mathematics is hard, it's probably wrong"

20:01 — Are we early in the history of mathematics?

20:45 — The four color theorem and mathematical maturity

22:10 — Elliptic curves and the limits of intuition

24:19 — Does the world run on logic?

24:51 — How human intuition solves unsolvable combinatorial problems

29:32 — Is mathematics discovered or invented?

32:50 — Pure math vs applied math — and why Riemann never thought about GPS