I am studying math as an undergraduate, and I have run into a problem that seems to common among students- I enjoy analysis, followed by topology, and find the topics intuitive. However, the road has been a bit bumpy with Abstract Algebra.Unfortunately, I would also like to take higher level classes in Abstract Algebra; because when I do get something (after a lot of hard work), it's very rewarding. I also don't want to feel apprehensive toward topics that involve algebra in the future.
So, while my undergrad Abstract Algebra sequence had some of my favorite classes as an undergrad, it was also frustrating, extremely time consuming, and nerve wracking. I figured that there had to be something wrong with my approach to studying; and I asked a lot of peers and TAs for tips, but didn't find advice that works for me. I try to I have noticed a few key things that are preventing me from processing concepts more quickly:
I seem process information better through symbolic notation, as opposed to words. For example, if I read "2 plus 2 equals 4," I automatically have to translate this into "2 + 2 = 4." Symbolic notation is used a lot in analysis, logic, etc., but most algebra books are very, very wordy. Most of the time, when I read abstract algebra books, I honestly feel like I am reading something in a different language. I feel like I don't know what the words mean conceptually, that I'm not getting the semantics.
Definitions in Algebra tend to be high level from the beginning; as in each definition has a bunch of embedded definitions. I feel like I struggle to see the whole picture because of this, especially when definitions, properties and theorems sometimes take up pages of writing. I don't know how to compress these theorems into pieces that are more manageable, and keep track of the overarching theme of definitions and theorems.
Examples- Often, I only begin to understand definitions when I have an example that I can play with. I really enjoyed learning about modules upon discovering that abelian groups are modules over the integers. This is opposed to other math classes, where I often get a sense of the implications of a definition/theorem before seeing an example.
Conceptualizing structure- At the stage I am at, it seems like Abstract Algebra is all about defining different kinds of structure. The problem is, I often just can't do this, without examples with objects that are "nice," like the integers, complex numbers, fields of characteristic p, etc. Because of this, I tend to think of different structures in algebra as categories of different properties, without really putting these properties together into an intuitive structure. The best analogy I can make is this: if you are a very visual person, and need to see a concrete visual representation of a structure to understand it. Then, you probably translate information into a visual form. But there are some structures that are complex, and very hard to "see" everything in one picture. So, instead, you create a picture that represents each property of the structure, but this means that it is easy to miss how all those pictures work together to build the overall structure. So understanding is lost in translation.
If anyone has had a similar experiences and has found ways to overcome these problems, I would be really grateful for suggestions. I tried to be specific with the problems I am facing. Also, if I need to make improvements to this question, I would also appreciate any advice!
Your experiences are very common. Kudos for raising these concerns. Let me respond to your points, by the number, and not in the order in which you gave them.:)
Point #3 Everyone begins to understand only after acquiring a certain empiricism (studying examples). Mathematics is as experimental as other sciences, opined several mathematicians, including P. Halmos and V. Arnold ("On teaching mathematics"). In his "Lectures on PDE", Arnold says that one's knowledge is nothing more than the collection of examples one has understood thoroughly and completely. So, don't be surprised or alarmed: you are being human.:)
Point #1 This is probably an illusion. Yes, $2+3=5$ reads better in notation than in prose, but slightly more advanced material will reveal the opposite pattern. For example, which definition of continuity of a function $f(x)$ at $x=a$ do you find more palatable (think back to when you first saw this):
the notational definition $$ \forall \epsilon > 0 \; \exists \delta > 0 : |a - x| < \delta \Rightarrow |f(a) - f(x)| < \epsilon, $$ or the prosaic definition: into every neighborhood of $f(a)$ the function $f$ maps some neighborhood of $a$?
As a byproduct, note that the latter definition continues to hold even in the most general setting, when $f$ acts from one topological space to another, hence illustrating how topological spaces are a generalization of the metric ones.
Points ## 2 and 4: I would recommend the following phases (which, unfortunately, may conflict with your university's course offerings, so consider taking academic leave for self-study, it's cheaper:).
First, get a good background on groups (not only what they "are", but also what they are for). Your criterion of having "gotten there": you can clearly connect the concept of a group to: (i) symmetry, and (ii) conservation laws. Literature: First part of "Abel's Theorem in Problems and Solutions" by Alekseev, and the relevant chapters of "Ordinary Differential Equations" by V. Arnol'd. Arnol'd's dictum: there are no other groups except groups of transformations of a set. (Granted, the set can have a complicated structure, and it can be of interest whether the group breaks that structure.)
Second, to make finite groups completely geometric and "unabstract", the first 40 pages of Serre's "Linear representations of finite groups". I said "geometric" and not "visual", because the representation spaces generally have dimension above 4, hence cannot be visualized, but with a good grasp of linear algebra, these two should be identical.:)
Why do we need rings and algebras? One use is to understand the structure of polynomials, and these do arise in analysis (including analysis on manifolds). See page 1 of {https://math.berkeley.edu/~giventh/papers/arn.pdf} A good source on commutative rings: Atiyah & McDonald's "Commutative algebra". Also, Lie algebras arise in geometry (since you already plan to read Arnol'd's "ODE":), you will see those).
You are right to seek physical and technical examples for mathematical concepts. Hope this is helpful, and good luck.