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.
Literally all math you've ever done are "proofs". You just get more rigorous about it.
"Solve the equation $2x+5=0$" really means "prove that there exists a solution to $2x+5=0$ and give an expression for it". Doing math is just doing logical reasoning with certain rules. In higher math, it is more explicitly logical and rigorous but honestly IMO the difference is kind of overstated. Once you get past junior/senior/first year of grad school you stop being so persnickety as well. I recommend https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/
Mathematics is about understanding things. Proofs are one part of this. The bulk of the time is spent asking questions, working out examples or doing calculations, figuring out what others have already done, positing conjectures/hypotheses, and thinking about ideas for why something should or should not be true. There are many famous unsolved problems that mathematicians have spent a lot of time thinking about, so no, we do not spend all of our time writing down proofs. That is just the last, optimal stage of a project (at least for pure mathematicians).
Why you may have this impression: the classes you're taking are likely on well studied topics, and, for efficiency's sake, you're presented with a streamlined version of the material, rather than how it was actually developed. Proofs teach you a way to think about things so you can learn to reason out and make confident deductions on your own, and justify the material you're learning, so they're definitely very important, but not of sole importance, in mathematical training.
Personally, I usually try to spend a lot of time in class explaining motivation and underlying ideas and working out examples, but as a result go through material slower than many of my colleagues. Quite possibly, there's more going on in the courses than just proofs, but depending on the course, they can take up the largest amount of time.
Talking from my own experience (about 3 years of mathematics at university).
I'd like to make few separate points, which somehow connect together, but the connection might be different for different people.
I was (sometimes am) the kind of person, to whom proofs were something boring (at the high school). Somehow, math has that kind of property, that it could be interesting and fun, but then there came some theorem and the proof. This was probably the biggest difference distinguishing math from other subjects.
At the university, this was even more pronounced. At the end of each course there was, among others, the same question regarding the examination, over and over again:
Do we need to learn the proofs?
Certainly, it might be hard to get any different impression, than that university Math is about proofs. But I started to believe, that it's not true. At least not in the way, I think you meant it.
I'd like to give you different point of view. Throughout the years I've got the impression, that each subject (meaning school subject, although it might be generalized (pun intended)) gives you some sort of level, where you are confident, about truthness of what you are learning. E.g. in History class you might use different sources (comparing them etc.), in Chemistry/Physics you might prepare some experiments, which might support some hypothesis. Mathematics somehow differs, at the beginning you have some tools, and you do not question their truthness. And with those tools you construct arguments, which you are able to recognize as true or false.
WIG (what I got)
For me the last years of studying mathematics were about:
- Understanding that once you have a proof of something, it somehow gives you freedom, it is like building another step. The proof helps you to do not worry about it, you can happily rely on something. That is much better, than constantly glance under your feet, whether there is still solid ground.
- It has build up my ability to think abstractly, and still helps me. As one of my professors told me (paraphrasing):
We did not want you to learn Birkhoff's theorem (understand the proof), because we feel, that you need it. We used that to somehow reach the limits of your mind, and help you expand it. So, when you need to understand something new (whether in math or else) you will be able to more likely.
- I did not like proofs, I did not like them, I did not like to write them. Because I've lacked the motivation to do math on my own. But then, while working on my thesis I had to prove something. And I was not a proof I would've known and only copy. Somehow I was very excited, it was great to write something new (well, for me).
- Trying to get around proofs is like crossing out the most interesting tool you have. Sometimes it is very hard to use that tool, but it can pay off.
- Not every proof is a good proof (that does not mean, that it is false). I mean, that you can proof the same things with slightly different words, add a picture or two (to illustrate some facts), and suddenly it becomes clear. And the view "bah, proofs are boring and.." is not there anymore.
- This point is related to the previous one. Many courses look like sequences of Theorem, Proof, Theorem, Proof, Theorem, Proof, $\ldots$. This is due to probably many facts (e.g. not enough time). The one, that scares me most is that some ideas, thoughts we are taught in few lesson were establishing over many (even decades) years. Great mathematicians were having troubles with them.. As an example: we started Mathematical Analysis with 13 axioms, which formed real numbers.
I see, I've drifted away a lot, but I thought that there was not an answer, from, I'd say, a student point of view. Also, many other things could be said, but they intersect with what has been said.
1 I've made many generalizations and simplifications in the following paragraph. Please don't get me wrong about other subject, I like them. I just wanted to stress some points about math.
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.
Einstein, Newton, Euler & 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.