Is university math all about proofs? 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...
Yes, mathematicians do a lot more besides writing proofs!  While some already pointed out that “they also have lunch” I assume that you are more interested into the intellectual processes involved in the mathematical activity.

When I¹ examine my intellectual processes, I can see that some exercise my intuition and some exercise my rationality.  My intuition tells me “there is something there” and with my rationality I can organise intuitions, see causal relations between facts and all the like. I am not, by far, versed in that area of philosophy, and I am confident that many people investigated this and wrote beautiful books on that topic. But for the purpose of this answer, I kindly bid the right of being superficial and approximative.

Mathematics is all about the interaction of intuition and rationality. As I study mathematics, my intuition leads me to genuine astonishment and makes me wonder about mathematical phenomenons I experience:


“could the same obstruction be at work in these two seemingly unrelated problems?”
“what is the geometric meaning of that algorithm?”
“where do the solutions of that problem live?”
“why do this topologist and this algebraist run into the same problem?”
“is there a formalism turning this gibberish into a straight computation?”


Producing all these questions is the work of my intuition – and this is a pretty common experience for mathematicians. Sometimes, my intuition is on a big day and can even give me a glimpse at the answer of the question!  This is where my rationality comes into play, and helps me to reconnect the new intuitive idea to the body of knowledge which I am already familiar with.  And this reconnection is what we call a proof, and this is how I can tell other mathematicians how they can approach and tame the intuition I had.

Think of mathematics as a journey in a place where you cannot take pictures of what you see (your intuitions) but you still can draw a “treasure map” to teach others how they can go the same place where you had been (rationality at work).

As a conclusion, proofs are the most visible aspects of mathematics, but as in many other activities, there is much more that one can experience than one can see. If mathematics were all about proofs, then mathematicians would be like computers which randomly explore the space of proved or provable statements by combining known facts together, but this is not the case, as this method would hardly produce anything interesting.

¹ I write this at the first person to provide a vivid and entertaining picture, with the implicit assumption that the case I describe is the common case.
Einstein, Newton, Euler &amp; Hawkins to name but a few of the greats all solved problems, found solutions and created new way's of seeing the world, looking at problems and adding an explanation to all things. Not just maths, but the universe and many other wonders.

Most of it was born in their head, an idea, a theory.  The theory's all have to be proved, and sadly approved, before they are accepted.

Mathematical proofs are the answer to scientific experimentation.  If it can be repeatedly shown to have a consistent outcome, you have proved what started as a theoretical solution.