TITLE: Should an undergrad accept that some things don't make sense, or study the foundation of mathematics to resolve this?
QUESTION [90 upvotes]: I'm a second year math student. And I've the following problem. 
When I prepare myself for an exam, I can distinguish two phases. First I'm mainly  interested in whatever is necessary to pass the exam. This means that I do not always read the theory too intensely, and I'm more focused on finding the "tricks" that I need to solve the problems.
Second phase: After some time, I feel like I've found most of the tricks, and at this point, I usually spend more time in understanding the theory. Just because, I know I will pass the exam, and I think it is fun to understand the theory better.
However, sometimes when I really try to understand the theory, I feel like understanding the foundation of something, doesn't make me feel I understand it better, it makes me only feel like I never really understood it. And this kind of frustrates me, and I'm not sure if it is even smart to try to understand a concept in that way as an undergraduate student.
For example, I'm studying complex analysis. I skipped the formal definition of complex numbers a little bit, and I just began making calculation and showing if mappings are conformal/holomorphic etc.  Now I'm back trying to understand the construction of the complex numbers as pairs of real numbers etc. My brain is getting more and more confused the more I think about it. 
My brains goes like: "If you define $\mathbb{C}$ as pairs of real numbers with complex addition and complex multiplication in the usual way. Okay that makes sense, we clearly don't have $\mathbb{R}$ as a subset of $\mathbb{C}$, but we can construct a bijection $f:\mathbb{R} \to \{(x,y):y=0\}:x\mapsto (x,0)$ that preserves every single property you want real numbers to have. So in that sense there is no difference between $\mathbb{R}$ and $\{(x,y):y=0\}$, and if we would define the real numbers as $\{(x,y):y=0\}$, then we have $\mathbb{R} \subset \mathbb{C}$. "Euphoric" Two seconds later, but wait... now I basicly define $\mathbb{R}$ as $\mathbb{R} \times 0$.... this doesn't make sense.... "Frustration" (just to give an example, I wouldn't post the whole list of things about theory of complex analysis that confuses me)
Sometimes I wonder if it there is any use of trying to understand mathematical concepts in this way as an undergraduate student. Should I just accept that it doesn't totally make sense and go on with solving problems?

REPLY [2 votes]: I think you have to be practical and make a balance.  Obviously the more you  understand bases, the easier it is to remember techniques.  All that said, things would crawl to a halt if you had to build every basis before learning a technique.  Should we teach 5  year olds to add or should we teach them some theory about intervals?  
It is a practical, human trait to learn iteratively.  To learn simple models first.  It actually helps you to understand relativity to have learned Newtonian mechanics first.  And there is a lot of power in math techniques also.  A lot of use.  And a lot of explanation of natural science.
So my PRACTICAL advice is to NAIL the techniques, tricks, etc.  A+ level.  And then with extra time (or later with harder courses) come back and prove some theorems.
Even for a research mathematician, life is not just inspecting other people's proofs.  Manipulative skill and puzzle-solving ability will help you to discover new math.  And it will make it easier to read papers.