# Self-learning mathematics – help needed!

First, I apologise for the nebulous nature of my title but I can’t adequately explain myself concisely.

I am about to start an MSc in pure maths after a fairly shaky undergraduate degree. I am very passionate about maths but I have several problems with self-learning which I can’t seem to cure (even after 4 years of trying hard!) and they are extremely detrimental to my progress. I would be delighted to receive any advice for this (admittedly very particular) set of circumstances or to hear from anyone who is somewhat similar in nature but has found a way through. For brevity’s sake I will just list the problems I have:

1. I seem to be obsessed with understanding “the basics”; that is, whenever trying to learn something new, say a first course on modular forms, I go back and try to systematize and relearn all the undergraduate complex analysis I did, which, in turn, leads me to go back and relearn all the undergraduate real analysis I did, which, in turn, takes me back to some naive set theory etc. This really slows progress (pretty much to a stop) but it just feels so wrong not to understand the basics first.

2. Often, when I learn a new subject, I have several different treatments of it at my disposal, each of which, naturally, takes its own perspective on the material. As such, I proceed to read the corresponding chapters of each the 5 or 6 (say) expositions and try to order the material optimally or put it in the most general setting (i.e. in categorical language). This is often very difficult (and, again, very time-consuming) and so I use LaTeX in an attempt to make “re-ordering without re-writing” easier but then I get a bit obsessed with the layout and formatting…

3. I am very pedantic about proofs. For example, I really dislike proofs which appeal to geometry (in the naive sense i.e. plane geometry) because there seem to be so many special cases to check and texts always seem to dismiss such things as obvious or relegate them to exercises. Also, “tedious” details such as “this map is obviously continuous” or “we can clearly assume wlog…” really bog me down. I find that, often, to give a careful proof of such things can be quite intricate.

There are definitely a few more points/clarifications I could make but this post is quite lengthy as it is and so I’ll stop there for now.

As I say, any help would be greatly appreciated – I have asked many people and read many articles over the past years to try to overcome these difficulties but I just can’t crack it.

Many thanks!

What you’re talking about seems much less like a mathematical or academic complaint than a psychological one. Here’s what I read in your post:

• You seem to be insecure about your understanding of higher-level topics, so you continuously and obsessively revisit lower-level topics, despite that this is probably not necessary: really, if you got into a program for an M.Sc in Mathematics, you probably don’t need to re-read books on naive set theory.
• You seem to be obsessively pedantic about details. This is the way most beginning mathematicians start out — making sure that all their proofs are definitely watertight — and as they progress, they allow themselves a little more leeway in the rigor of their proofs: certain statements just become obvious and don’t feel worth the time to prove. Now, you are clearly not a beginning student, so this is a fairly atypical behavior.
• The two points above, in combination, more or less waste a great deal of time for you, and paralyze your learning.

This actually reads like a textbook case of a particular type of procrastination to me. In particular, your pedantic attention to detail (even regarding comparatively unimportant aspects, like the layout and formatting of your notes) is commensurate with perfectionist behavior. Indeed, to me this reads like perfectionist procrastinator-type behavior: you try to perfect every aspect of the less important tasks (taking notes, revisiting the most elementary set theory, etc.) and then don’t have the time or energy for the more important tasks, like studying complex analysis.

This is a very subtle and dangerous form of procrastination, because you mentally trick yourself into thinking that you’re doing important work (re-reading DeMorgan’s Laws, making your notes pretty) when you’re really not. You’re doing things you already know how to do (like proving elementary results or typing up some $\LaTeX$), which is a ‘safe’, comfortable activity, whereas the work that you actually should be doing is more of a challenge, which you are avoiding with this form of procrastination.

This form of procrastination is sometimes accompanied by some intellectual insecurity or anxiety.

I do not hold any formal qualifications in the field of psychology, so you should take my judgement with a grain of salt. However, I suggest the following two coping strategies:

1. When you find yourself doing something that seems somewhat unnecessary, like looking back at elementary set theory: ask yourself if what you’re currently doing is really necessary for your learning. If the answer is not a clear-cut “yes”, then stop and get back to your initial task. Use self-control.
2. See a psychologist or counselor about this problem. Issues like this are fairly commonplace, so you’ll be in good hands.

Good Luck!