I'm an undergraduate student in mathematics based in Pisa, Italy. My university has always had kind of a reputation for being hard, although I can't compare it to others since it's the only one I've ever attended. I'm doing okay, I'm currently in my third year and I'm almost done with the bigger exams (real analysis I and II, introduction to general and algebraic topology, abstract algebra...). I'm also doing much better (not trying to sound pretentious) than many of my year's students, in fact the vast majority of them quit before the end of the first year and many of the remaining ones just re-attended the first year. The point is, I'm starting to lose a bit of motivation. I like what I'm studying, but I'm certainly not great at it, and I really have no idea what I'm going to do afterwards. I could like becoming an actual mathematician maybe, but I don't think I'd go really far with it since here are a lot of really brilliant students that would outsmart most of the teachers (I'm not kidding, the Scuola Normale Superiore really makes the competition kick in). So the question, briefly, is the following: you average mathematicians, you normal people like me who knew from the start you wouldn't prove the Riemann hypothesis, how did you cope with that feeling? Have you ever felt some kind of inferiority complex? What have you become after graduation? Why? How is the world out there? Does it really need another meh mathematician, or would it benefit more from me just becoming a trashman? Feels a little weird to ask something like this, but this feeling is spread out across most of the students in my mathematics department. A girl I know has just two weeks ago passed the real analysis II and algebra exams (she PASSED THEM!), and is now wondering about whether she should change course and study medicine; that's a bit drastic, me and all of her friends have told her such, but her answer has been "my will to do maths is vastly inferior to the amount of work required in order to keep up with uni". And we're all kinda in that situation, no one excluded. So, the only reply has been "well, you've made it this far. Now it's a bit late." but that's not true. That can't be a motivation for keeping on with maths. I didn't think this would bother me as much as it did, but the fact that everyone has the same feelings as me towards uni has multiplied my anxiety, I don't know why. I also started thinking about my girlfriend's brother: he hasn't even completed high school (he's one year older than me), but he loves programming and has found a secure job with a more than adequate wage in that field. I don't have the same passion towards maths as he has towards IT, as doesn't anyone of my friends at uni. Isn't that weird? The more I study maths, the more I love it and the more I hate studying it. Sorry for the ramble, but being the rational cynic piece of crap that I am it's difficult to put into words my feelings. I know that maths is difficult on its own, I also don't really know what kind of answer I want, I'm just sharing thoughts. I have a lot of friends who are trying the algebra exam for the sixth time, and they're getting very demotivated and kinda make me feel bad that I'm even asking this since I have managed to pass most of the more difficult exams. I don't really know how to express what i'm thinking, so let's just make an example: I had my general and algebraic topology oral exam the other day, and I had to find the fundamental group of the real projective plane. Here's the answer i gave:
If $G$ and $H$ are a right and a left action on a set $X$ such that $(gx)h = g(xh) \ \forall g \in G, h \in H, x \in X$ then $\forall x \in X, \theta_x:G \mapsto H$ which maps $g$ to the only $h: gx=xh$ is a group homomorphism, in particular $H \simeq G/$Ker$(\theta_x) = G/$Stab$_H(x)$. If we consider a topological space $E$ and a group $G$ acting on it, then if such action is compatible with the monodromy of $E/G$, as a corollary $\pi_1(E/G) \simeq G/p_*\pi_1(E)$ where $p_*$ is the induced homomorphism of the covering map $E \mapsto E/G$. As a special case, the fundamental group of the real $n$-dimensional projective space is $\pi_1(S^n/\mathbb{Z}_2)\simeq\mathbb{Z}_2/p_*\pi_1(S^n)$ and since $S^n$ is simply connected $\forall n>1$ the induced subgroup is trivial and therefore the fundamental group of the projective space is $\mathbb{Z}_2$.
Ok, let's recap. I knew what I was talking about, I had studied that, and my professor noticed it (got full marks on the exam btw), but after saying that I just paused for five seconds and realized how MOTHERF***ING COMPLICATED this is. And this is a very basic theorem, not even the proof of SVK (which I fortunately wasn't asked)! Like, explaining this thing to someone not into maths would be quite a long adventure, but it's treated as trivial in this course (as it was by me too, until I had to say it out loud during the exam)! I would never, EVER come up with anything similar, so what hope do I have to become at least a decent mathematician if even such a ""trivial"" theorem is beyond me? Are most mathematicians like me? If so, how do they live with it?
(Sorry for bad grammar)
Might be a bit much to answer the whole question, so I'm going to concentrate on
I enrolled for a Ph.D. in number theory, struggled but did manage to complete it. One examiner's comment was that there was not much in my thesis but it was well written so I deserved to pass. I thought this was absolutely accurate, and it confirmed for me three things which I already knew. (1) By "community" standards I'm a good mathematician. (2) By professional standards I'm not a good mathematician - in fact scarcely a mathematician at all, really. (3) I'm a good writer.
So what did I do? Got an academic position (which I still have 30 years later) and concentrated on teaching, which I love. Not sure I could cope with school teaching, but at the (slightly) higher level of undergraduate teaching, I enjoy having a good understanding of basic post-secondary mathematics, helping students to understand it and trying to help them see how beautiful it is. And writing it elegantly for them. I do occasional very low-level research, not directed, just if I notice something interesting I try to understand the whole problem, write it up carefully, and now and then (very rarely) this turns into a publishable paper.
Also very important is that I'm closely involved with other activities like music and wilderness sports, so that my career (which from many people's point of view would have to be considered spectacularly unsuccessful) is not the only thing in my life.
Good luck, hope this helps.