TITLE: Is university math all about proofs?
QUESTION [34 upvotes]: Do mathematicians do anything else beside writing proofs? It seems like all the "upper-division" math here are about proving something rather than solving for something i.e. instead solving for $x^2 = -1$, prove that solution to $x^2$ is something adic, on some sort of space, satisfy some vexity, generalizable to some blah...

REPLY [2 votes]: It's about not taking theorems for granted.
I'm not a mathematician, but it was quite a realization for me when it hit me that a lot of the things we "found" in high school were... complete magic at that point. For example, the Rational Root Theorem and Descartes' Rule of Signs... did you ever wonder why those work? Or did you--like me--just think "Cool!" when someone gave you those theorems, without ever giving them a second thought as to why they should work? How would you discover these theorems for the first time?
Or even, let's go back earlier: at some point or another, maybe in middle school, someone told you that the shortest path between two points is a line, and you probably took this as "postulate" and never gave it a second thought. But why? Is it really something you can't prove? Well, maybe not within a geometric framework, but it turns out that with the help of differential equations, you can solve (yes, solve!) for the curve that minimizes the distance between two points in a vector space. The solution is much more complicated than what you'd ever expect coming out of high school, but it is quite enlightening, and it is as much of a solution as it is a proof. The two go hand in hand.