Math overdose? Professional advise how to cope with it [closed]

Question

I'm a PhD student, currently working on my Thesis. Over the years I have many time encountered a problem. Maybe professional mathematicians know what I'm talking about?

When I study a math topic, frequently I reach a stage when I stop understanding anything in this topic anymore. Like when one repeats a word many times over and over it becomes unintelligible and meaningless. I read a book or a series of papers on a topic and everything which I understood before I started actively studying the topic, are now not understandable anymore, even though I understood (or at least I thought I understood) them when I started reading. So I give it up for a while, because continuing this topic is not possible anymore. And then exactly the same thing happens with the next topic. The study is not lost, because when I return to the topic it is much easier and I'm able to progress deeper into it, until I reach a new stage where I have to give it up again.

Has this happened to other professional mathematicians? Or maybe I'm not meant to be a professional mathematician? Any professional advice how to cope with this situation?

Answer 1

I can give three bits of advice. One that has been given by many other people is to take a break. People think that endlessly hitting the books without a break is the mark of a hard working mathematician, and that to do anything else is lazy. This is far from true. Doing so only depletes ones creativity.

The second bit of advice is to form hobbies and interests that may be mathematically related but that give one an opportunity to make connections outside of ones department. Doing a PhD. in a ruthlessly competitive environment can be draining. Getting an opportunity to play pool, MTG, chess, to exercise, play video games, go camping, go hiking or something of the sort may restore a PhD. student's confidence, and also restore one's creativity.

The third bit of advice is not to judge one's worth solely based on one's mathematical achievements. You may say "What else is there to judge oneself by, and why should I stop this judgement if I want to increase my productivity?" The answer is that this judgement does not increase your productivity, in fact it decreases it. When you are highly productive this judgement does not truly increase your productivity. When your productivity is lower, which happens to every mathematician, then this can tank productivity as all elements that further creativity are reduced. Realize that you have innate worth beyond your work, and this will also help to shorten those times when creativity is lacking.

I hope that this is helpful as your question is certainly worthwhile and I suspect that your experience is universal among mathematicians.

Answer 2

I'll try to answer based on a supposition, so I'm sorry if answer won't be particularly well suited for you.

My supposition is that you arrived, maybe multiple times, at a point when your previously acquired knowledge/understanding was pushed under scrutiny and proved, if not plain wrong, at least much improvable. You write “everything which I understood before I started actively studying the topic, are now not understandable anymore”, but isn’t it possible that a finer perception of implications, a more precise realization of what the hypotheses of some key theorem really ask, a broader collection of examples and counterexamples is gradually emerging?

Let me make some examples, to be less vague.

1. In real analysis, one quickly becomes familiar with the concept of limit of a sequence. In the first exercises, one tries to evaluate the limit often without really wondering whether the limit exists or not. Of course the prof was telling “assuming the limit exists” every now and then, but one quickly becomes accustomed to skip mentally this sentence because, hey, it never applies… Then you meet one case in which the limit does not exist but some formal procedure would tell you how to compute it, then another case in which existence is there but is not trivial, then an even subtler one, and so on… You may feel on very fragile ground, but you are just entering the next level of understanding on limits and sequences.
2. In topology, one starts typically playing around with some simple examples of topological spaces and then quickly passes to $\mathbb{R}^n$ with natural topology. A lot of intuition is already built in there. Then you meet the Comb/flea space (https://en.wikipedia.org/wiki/Comb_space), the Alexander sphere (https://en.wikipedia.org/wiki/Alexander_horned_sphere), the long line (https://en.wikipedia.org/wiki/Long_line_(topology)) and so on, and you start hesitating even making the apparently most innocuous assumption on what may or may not happen in general topology. In fact, you are just about to realize how hugely general is the concept of topological space, and why some further requirement is needed in so many applications of topology.
3. When studying elementary topological dynamics, you get typically exposed to examples in which point transitivity and topological transitivity always go together. When you look more carefully (again usually through counter-examples), you realize the differences between the two concepts.

Of course the list can go on endlessly (you may find a lot of related things here: Examples of common false beliefs in mathematics). If I am right, what you should do is probably just relax, maybe break the bad feeling doing something a bit different for a while, and then start rethinking critically to what you have learnt so far.