How to avoid getting "lost in notation"

Question

Having just finished my first (and wonderful) year of undergraduate math studies, there is a certain "skill" I find myself still struggling with, as if I've made no progress regarding it since my first lecture in my first course.

As the title subjects, I'm referring to being able to keep track of the different notations, scripts, symbols assigned "on the fly" to note specific objects, either during lecture, or when I find mydelf reading a mathematical text. It's important to emphasize that I'm not referring to the difficulty of understanding formal / rigorous proofs, but simply to that of keeping track of all the different objects currently under discussion.

I'll provide two examples to illustrate what I mean more concretely:

1. This is a direct construction of a completion for some metric space. I find myself easily getting lost between all of the possible combinations of ($[x] \ \hat{x} \ x \ x_n \ \langle x \rangle $), even though these notations are used to refer to completely different things. This confusion persists even after I know the proof good enough to write it down and explain it myself.
2. This simple exercise, that involves showing that if a locally compact subset $A$ of some topological $T_2$ space $X$, is dense ($A$ is dense in $X$), than $A$ is open. Even though after finishing my topology course, the exercise itself seems easy, I find it overwhelmingly hard to read the short answers in the link provided. The hardest part: remembering what $V, U, O, D, A, X$ all stand for, and transitioning every single moment form treating (for example) U as a subset of spaces, $D$, or $A$, or $X$.

Now the main frustration is that I can easily find myself investing a great deal of time when reading material. Alternatively, during lecture I can find myself totally lost.

Question:

What can one do improve his ability to quickly "digest" patterns, notation, values, etc. (In other words - to avoid the difficulty I've just described)? I would especially appreciate those who've had the same struggles and found a way to deal with it.

A couple of remarks:

• I know that a lot (maybe most) of people seem to be completely unaffected by this kind of difficulty, but I'm also pretty certain that a lot of students are familiar with this problem. Thus - I find it relevant to post as question on this site.
• I've found the following question relevant, but not the same: This one explicitly states similar challenges (among other things) regarding notation, but answers seem to focus on how mathematicians write in order to make their ideas easier to read.

Answer 1

I think this is something almost everyone learning some sort of abstract math goes through.

I think what you should focus more on, is looking at what those symbols define, and what mathematical object those symbols are a placeholder for.

The only advice I can give is to simply push through this stage, it gets better trust me. There'll come a time down the road when you'll be looking at symbols and seeing the underlying concepts instead of just some Greek letter.

I know this seems like some very vague and even weird answer, so to concretely make sense of what I'm saying. Suppose we had the following passage from your favorite topology book:

"Let $X$ be a topological space, let $x, y \in X$ be points in $X$, and let $U$ be a neighborhood of $x$, and let $V$ be a neighborhood of $y$ with $U \cap V = \emptyset$"

What any topologist would do in this situation is either draw a diagram or form a mental image, similar to the one below, to keep track of whats being said

Once you have a clean grasp of what those symbols define, instead of $X$ I could put a symbol of a baby panda, it wouldn't make a difference though because I'd still be talking about the same topological space.

Specifically for Topology, it's incredibly helpful to draw pictures to avoid getting lost behind symbols.

Answer 2

Working your way through arguments cluttered with notation is indeed difficult.

Some thoughts:

Sometimes authors don't help. I find the uniqueness proof in your first link quite hard to read. I'd try to write it with more words and fewer symbols, better differentiated.

The writer in the second link did follow some helpful conventions. The set $D$ is dense, so that's a good name for it. Often $U$ is used for a neighborhood; if you need a second neighborhood then $V$ is a natural choice.

When I see complicated summations written with sigmas and subscripts I often write them out on scratch paper with ellipses between explicitly expanded first few and last or typical terms.

If you're searching the internet for proofs you might look at several in order to pick one you find comprehensible.

Remember all this when you start writing your own mathematics.

Answer 3

I understand your problem as being one of memorizing notation on the fly. Thankfully, there are tricks you can pratice to help you do that.

First and foremost, make sure you know how to name the symbols properly: $\hat x$ is "x-hat", $\mathcal U$ is "U-cal", $[x]$ is "x-in-bracket" or whatever. When you read a symbol, you need to consistenly read its long name in your head, as if you were speaking out loud.

The second trick takes a bit more practice and experience writing complex proofs yourself. You must associate to each symbol a feeling. This "feeling" is usually an image, but it can be anything related to your senses. Everytime you see the symbol, not only do you read its name in your head, but you also take a look at it in the wild. For example, when I see $\mathcal U$ refering to an open set, I imagine the curves of the letter as representing the boundary of the set. The image you associate to the symbol can be context dependent and doesn't have to be mathematically meaningfull. You can even associate to the symbol the muscular feeling of writing it with a pen.

Next, you pair up each symbol's image with your intuition of the concept it represents. You picture the two together in your head, so that both will be linked in your memory.

There's no shortcut to this process, although it can become automatic. You can't just see a symbol alongside a definition and remember what it refers to afterwards. You need to picture the two things together and internalize their relationship in some way.

Answer 4

Like anything else, practice. It's important tho not to get discouraged. You have to accept this is part of the process of learning anything new, and when you find yourself in that place were your head hurts, don't run from it, because that is where you want to be. That is where learning takes place. Instead of wasting 15 free minutes on facebook, browse some problems on here. Just try and make sense of what the problem is asking, make your brain hurt, but make it hurt on things you can figure out.

It's also really helpful to bookmark some problems that you struggled with at one point, so you can go back later and see the progress you've made. Just being aware that you have made progress, both feels really good, and helps you to stay positive as you struggle forward.

Also, this isn't unique to you, its something everyone has to go through. Math truly is a language. Just because it's in english (well most of it, lol) doesn't mean you should be able to instantly understand it. You wouldn't expect yourself to understand Spanish or French or some other foreign language right away, would you? Give yourself that same freedom with math.