17

I am a young PhD-student who just started doing real research in mathematics. Of course, this starts with learning stuff which has already been investigated. If I try to understand for instance a paper someone wrote, I find it difficult to focus on the main aspects and ideas of the paper. Most of the time, I get stuck on some elementary details, which I had not thought about until I read them in that paper. Convincing myself that this detail is true is a matter of a few minutes, but the time adds up, so I end up losing quite a lot of time trying to understand details not directly related to the problem, but occurring in the solution of the problem. But if I skip this detail, I have the feeling that I did not really understand what the author is doing in the paper. On the other side, if I think about every detail, I lose my focus on the main aspects of the topic, which is not great either.

Conversely, if I work on my own problems, I tend to get stuck on minor details as well. Unfortunately, my advisor is not a good help in this regard.

My question is therefore the following, which can be also asked in other areas of research, not only in mathematics:

How does one find the right balance between focussing on the main aspects of a topic and the details, both in actual research and in trying to understand works of other people?

6

In mathematics, being able to read papers is a skill on its own, and learning how to do it properly is part of “mathematical maturity." Most everyone runs into this sort of problem at first, and with practice, you get better at seeing the forest through the trees.

One concrete strategy is to skip the small steps and to assume the details in them are true. You can verify those details after you understand the main ideas. Of course, you still need the skill of telling apart the small details from the important ideas. Same goes for research -- you can make a big plan of attack and assume the details will work out later, and then if the big plan succeeds, you can go back and do the detailed calculations carefully. Here, what is needed is intuition about what "should be true if worked out." But such intuitions take time and practice to develop.

4

I think your concern is very reasonable, especially in the context of the usual paranoia-inducing undergrad and beginning graduate curriculum, where, somehow, every detail matters as much as every other, and any gaffe or gap is as "wrong" as any other. In real life, of course, things are nothing like that.

Similarly, analogous to real life, or... in real life, the point is to get to the end, whatever debts or doubts are incurred on the way, and often to meet some schedule, whether internally or externally imposed. Indeed, quite often one finds that the whole conclusion is not sufficiently interesting (as far as can be ascertained on this pass...) to look at or worry about the details. So much the better, then, to have not invested time in them.

As @LevReyzin already noted, being able to sift out the critical details (especially on just one pass or so) is a skill that is acquired, and is very important to professional function. In particular, the palpable fact that novices are not good at this is completely unsurprising, and is perhaps their chief difficulty. The problem cannot be quickly solved, not by willpower nor by "tricks"... although noting times and dates can help you track your own intransigence. :)

I do seriously recommend analogies to very practical things, like going to the grocery to get food. Sure, there are details that come up along the way, some things that may be hard to understand, present other issues, or add to your to-do list, but things have to reach a substantial thresh-hold to actually prevent "getting food". True, lack of transportation, for example, may provide a direct obstacle, but many public issues that are substantial in their own right do not, in fact, fatally obstruct getting groceries... So it would be an error to allow such things to do so.

To believe that this is a reasonable analogy requires believing certain things about (most) mathematical papers, namely, that most of it is "just the usual"... so the fact that it appears anomalous or disturbing or ineffable to a novice _means_nothing_. The default assumption should be that 90% or more of a paper is completely routine... whether or not one can personally vouch for this.

Yes, making notes of what one hasn't understood, at least as a to-do-when-there's-time list, is useful. But one does not make progress as a professional by "waiting" until one has understood all the background... if only because such an "understanding" is often misguided or even erroneous if one has not seen the things _in_use_. Thus, it is all the more important to "skip ahead"... to understand the things one is skipping.

2

As Lev said, as you get more experienced, this will resolve naturally. However, I have a couple of ideas for you to speed things up a bit.

  • When you attend a talk someone is giving about his research, you can't stop and smell the roses. The presenter is plowing ahead, and you can't go back to the previous slide and think carefully about how that equation was derived. If you have not had this experience often, then please attend some more talks, to give yourself this experience of letting things wash over you.

    Once you are feeling more comfortable with this type of experience, you can use it as a mental image for what you want some of your paper-reading experiences to be like.

    Of course, not all your paper-reading experiences need to be like this.

  • Join, or organize, a journal club for students, where each week someone presents a paper he read and liked. As you are preparing for your turn, you'll want to skim through a nice stack of articles to narrow the field. When you've got a short list, then you can read for more detail, in preparation for making your final choice, and for preparing your presentation.
  • There's a technique from psychology you can try if you're still having trouble: Pick up a paper you are interested in reading. On a page of a notebook, write today's date on the first line. Start reading. When you get to the first equation that's not immediately obvious to you, write down a number between 0 and 7 to the right of the date, on the same line. 0 means you have NO urge to verify that equation for your own satisfaction. 7 means you have the MAXIMUM urge to verify. Now keep reading, without verifying. After an interval, write down your new measure of your urge. Keep reading. Keep taking your "pulse" every so often. You might see your number increase initially, and then start to decline. You may stop measuring when it has stabilized at zero.

    If continuing to read the paper is very painful, you can set it aside and do something else, such as listen to music.

    You can aim to do this exercise once or twice a day. Probably after one or two weeks you'll find that you're able to choose when you want to verify a step and when you are okay with skimming that bit, reading ahead, and possibly coming back later to verify.

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