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Most people throughout their career encounter at least one paper that seems especially daunting to them.

I'm interested in real stories of how you successfully overcame that to extract the knowledge you were interested in and how long it took. Please name the paper and provide details of what was putting you off and of the process you used.

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    $\begingroup$ The best answer will depend on the reason for the dauntingness: Is the book/paper daunting because it is long? because it is written in a foreign language or notation? because it is written badly? or because the underlying mathematics is difficult? $\endgroup$ Feb 2 at 23:03
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    $\begingroup$ This is dumb advice but it sometimes works: just read the theorems and definitions, don't read the proofs. Then try to prove the theorems yourself. If that doesn't work, go back to the paper and figure out what you are missing. Often enough, the paper is much easier to read once get that extra motivation. $\endgroup$ Feb 3 at 4:08

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Just sit down and start chewing on it.

Don't think: I should finish this within X days/weeks/years.

Read it sentence by sentence, pausing whenever you encounter anything you do not understand, and then apply all means at your disposal to understand it (including asking others, letting your subconscious do its work, etc.). Do not set any time limits for this, but don't give up on it either, but whenever you are stuck and wrestling with something say to yourself: this is precisely why I have embarked on this enterprise. It is only when we are stuck that we have a real opportunity to learn something.


I am sorry to say that I never took this advice myself, while I was still in academia. I have mainly applied it when reading texts in ancient Greek, forcing myself to understand every line, and every odd grammatical detail that didn't quite make sense to me. This was purely a hobby for me. And this is key for me: there shouldn't be any additional pressure other than the pure drive to know and understand. When you are in academia, this is very hard to achieve, maybe even impossible.

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    $\begingroup$ I think that it is also important to be willing not to read things sentence by sentence, but to skip results, proofs, sections, … without guilt. I have known graduate students who could not make serious progress because they would not even try to read Section 2 of a paper until they understood every detail in Section 1. This is in some sense admirable, but in anoher sense it means that you never have a chance to build up an intuition that can be assisted by later rigour, and are forced instead always to build up the rigour first and hope that the intuition comes along with it. $\endgroup$
    – LSpice
    Feb 1 at 16:20
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    $\begingroup$ It is not always obvious when it's time to move on. For me the emphasis should be on "go slower" rather than "keep moving". But sometimes you should just forge ahead, even when there are still some question marks. $\endgroup$
    – R.P.
    Feb 1 at 16:31
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    $\begingroup$ Re, perhaps I may say this way why I am wary of leaning too hard into this perspective: reading too far ahead is self correcting, in the sense that, if you try to prove a generalisation of a theorem in Section 4.3, you'll soon realise that you need to read the proof of that lemma in Section 2.1 more carefully. But not reading far enough ahead is not self correcting; you'll never realise that the offhand remark that crystallises everything is just on the next page, because you haven't read it! Still, I certainly agree on "go slow." $\endgroup$
    – LSpice
    Feb 1 at 16:36
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    $\begingroup$ @R.P. I think it also depends on the nature of the text. Sometimes I will read sacred texts from religions other than my own, with the same motivations as you. In that case, I usually don't skip passages. But in mathematics, even if all you're trying to do is understand the paper, reading it linearly is not always the optimal approach, even for a well-written paper. The order in which the material is presented in the paper is likely a logical order, which need not coincide with the best order in which to read it for understanding. $\endgroup$ Feb 2 at 23:55
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    $\begingroup$ I adopted R.P.'s mentality during grad school, and while it is psychically very calming and makes you resistant to being "daunted", in practice, one should probably mix in some of LSpice's advice, or else it's just not a very efficient use of time. $\endgroup$
    – Will Chen
    Feb 2 at 23:56
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I can think of two distinct reasons why a particular paper or book might seem daunting. The first is the sheer size. The second is that I don't understand what's going on.

If the only issue is sheer size, then I think the "just do it" advice that others have given is sound.

If the problem is that I don't understand what's going on, then my advice is perhaps contrarian:

Do everything in your power to not read the paper.

By this I don't mean that you should procrastinate. I mean that you should first ask yourself whether you really need to understand the paper. Is the knowledge you want available from some other, more accessible source? Does it suffice to extract a small piece from the paper that you really need while ignoring the rest? If the paper seems incomprehensible, then there's a decent chance that it's not optimally written, and that your difficulties are at least partially the author's fault. Before forcing yourself through something that is not well written, you had better be darn sure that the task is unavoidable.

Suppose after all that, you still find that your research goals are forcing you to understand this paper, because it contains essential material that is available nowhere else. If the paper is really that important, then chances are there are other people who have studied it carefully and have said useful things about it. So start by reading as much commentary as you can get your hands on (and talking to experts, if they are willing to give you their time). This should help you get some sense of the overall structure of the argument. Perhaps some parts are simply computational and there's not much for you to do other than verify that the argument is formally correct. But there may be parts which are driven by some key ideas that aren't articulated explicitly in the paper itself, but that have been pointed out by commentators. In these cases, it's important to know what the guiding ideas are when you're struggling through the argument.

As a concrete example, I remember really wanting to understand Razborov and Rudich's paper, Natural proofs. On my first read, I found the paper baffling, because the main result clearly had major conceptual content, but when I looked at the proof, the bulk of it seemed to be some kind of technical argument which I could follow line by line, but left me no wiser as to how such a powerful conceptual conclusion could be drawn. It was only when I read "around" the proof that I came to realize that the conceptual content could be captured in a very simple (albeit hand-waving) argument: if you had a property $\mathscr{P}$ that supposedly an $\mathsf{NP}$-hard function has but no polytime function has, and $\mathscr{P}$ is efficiently computable and is true of a random function with non-negligible probability, then you would have on your hands a powerful cryptanalytic weapon that could distinguish cryptographic functions from truly random functions. This was the conceptual core, and the technical argument that had baffled me was simply a careful verification that this hand-waving argument can be made precise.

The Razborov–Rudich paper is perhaps not the world's best example of a "daunting" paper, because it's well written and even the technically hardest parts are not that complicated, but I think it illustrates the sort of thing that can make a paper seem daunting, namely an argument that seems baffling, and that does not seem to reward repeated careful reading. In such cases, a patient "just do it" approach is not likely to get you very far. It's important to get some high-level understanding of what is going on.

In the absence of external commentary, you may have to generate your own commentary. The way I typically tackle this is to start by formulating a relatively weak, but still nontrivial, result $X$ that the paper implies. Without the results of the paper, I can't see how to prove $X$ with the tools I know. So how does the paper surmount the difficulties? What do I get stuck on when I try to prove $X$ myself, and where in the paper is that sticking point addressed or bypassed? Asking yourself such questions and trying to answer them helps you read with a sense of purpose, which is always better than simply slogging through an argument line by line.

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  • $\begingroup$ I feel yours answer are practically useful and in the spirit of research in action. $\endgroup$
    – Akira
    Feb 2 at 22:31
  • $\begingroup$ I believe what Timothy Chow writes, only I think there is also a very important gray area between a) (wisely) avoiding reading certain papers that are Not Appropriate to you at the moment for whatever reasons and b) happily plowing ahead with a good paper for this person at this moment: This gray area consists of the papers that lie outside your current comfort zone, yet not so far outside that you would immediately reject reading them. I once spent all summer (1971) reading "Groups of homotopy spheres" — well outside my comfort zone at the time — and never looked back. $\endgroup$ Feb 2 at 22:48
  • $\begingroup$ Razborov and Rudich, I just looked. The kind of paper where hand-waving arguments are dressed with impeccable technical proofs to arrive at a "rigorous" result which depends on a series of conjectures about NP. Haha, avoid, indeed. In fact, a "daunting" paper will often turn out to be something like this, so kudos to @Timothy Chow for "Do everything in your power to not read the paper." $\endgroup$
    – Kphysics
    Feb 3 at 8:56
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Sometimes you just have to admit that a reading project is too ambitious for you at the moment, and set it aside. After building more maturity and skill in surrounding topics you may have a better go at reading the difficult paper.

Conversely it is also important to know when you do have enough background/skill and it is time to start reading.

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    $\begingroup$ But it's important to have given it a serious go first before setting it aside! I think I'm not the only one who has a problem with preëmptively setting things aside because their reputation makes me think that they're too hard to read, when actually they are, if not easy, then at least accessible to mortals. $\endgroup$
    – LSpice
    Feb 1 at 16:16
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    $\begingroup$ @LSpice Ah yes, agreed that actually giving it a go at some point, even if you think you’re not prepared is a good idea. At least it will point out areas you’re lacking in, or you may discover that you actually are capable of doing it after all. $\endgroup$
    – Nate River
    Feb 1 at 16:18
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    $\begingroup$ Very sorry, I seem simply to have blindly skipped over the sentence "Conversely it is also important to know when you do have enough background/skill and it is time to start reading.", which says better what my comment presumed to add. $\endgroup$
    – LSpice
    Feb 1 at 16:22
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    $\begingroup$ @LSpice Ah, I added that in later so maybe that’s why you missed it. A lot of people I think are guilty of being stuck on the “I need more background” phase, so I think your advice of giving it a serious go anyway at some point is a very good idea. $\endgroup$
    – Nate River
    Feb 1 at 16:23
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I, like I think many people who are called upon to read them, have found SGA3, and the œuvre of Bruhat–Tits, both quite daunting. In both cases, the remedy has been, with encouragement from wiser colleagues (thanks, Brian Conrad, Sean Cotner, and Stephen DeBacker, as well as anyone else I've forgotten!), just to sit down and read them, and find that they are by and large beautifully written. For SGA3, this is helped immensely by the lovely réedition by Gille and Polo. For Bruhat and Tits, J.-K. Yu's introduction Bruhat–Tits theory and buildings was a wonderful roadmap.

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My experience with this changed a lot when I transitioned from graduate student to more experienced researcher. In earlier days, I followed the advice of starting at the beginning and working systematically through a big work. An example for me was Wall's book Surgery on Compact Manifolds, which is daunting and hard, with some chapters particularly rough going. I didn't succeed in reading it front to back, but I'm not alone. (In the introduction to the second edition, Ranicki wrote "…I always had it with me on my visits home, and once my mother asked me: ‘Haven’t you finished reading it yet?’".)

In more recent years, time constraints meant that this approach was harder to achieve, and usually I had a better idea of what I was looking for. So I would read what I needed to get started, and then go back repeatedly to fill in details (or even major points) as needed for further progress. A prime example for me was the series of papers, Spectral Asymmetry and Riemannian Geometry I–III of Atiyah–Patodi–Singer, which are daunting for the sheer amount of ideas they contain. I think I wrote 3 papers related to the $\eta$-invariant (from those papers) before I finally read the full proof of the main theorem.

For a great description of how to learn in this somewhat circuitous manner, I suggest reading the introductory portion of a beautiful article in the AMS Bulletin, "Harish-Chandra and his work", by Rebecca Herb. The author describes how she was led into his difficult work by her advisor, and found a thesis problem by starting with a single page and ‘working backward and forward’. Of course, it helps to have a good advisor!

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    $\begingroup$ Your second paragraph is similar to the answer I would give: the older one gets, the better one learns to skim... $\endgroup$ Feb 2 at 22:53
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Just do it.

Do it with a purpose

Having a solid intent behind reading a paper or fully understanding its content absolutely helps motivate your reading. Also, it provides the proper lens to reading that paper. If you need to understand the intricate details then it will take more time, but you know you only need to read to understand the core part, not any fluff language that isn't giving you the relevant info.

If ya just need to understand the gist then ya don't need to read as deep. Always can start here and reread if ya need more depth. Same applies to needing only certain sections of the text.

Do it a bit at a time

Tricking yourself into doing it for longer by doing it for 2 or 5 minutes and when your brain thinks or wants to do something else, treat yourself to that something else after 2 or 5 minutes more. Repeat until you're not making progress.

Of course this enters the lengthy discussion of optimizing how to focus and do tasks. Could try mindfulness meditation, or yoga. Tons of things to try to figure out what works best for you.

Also, if you find you need to read something else to better understand the current work, then prioritize that prior knowledge first. Same can go for practicing skills or really understanding key concepts.

Do it with accountability

Hold yourself accountable by working on it for some set amount of time a day or week. Tracking the time you commit to an activity can be useful when to encourage you to do more if you only did a bit. Or reading it for some time before throwing on a video or playing or worse, social media scrolling. High dopamine activities will make ya want more of them so do em later in day and do the reading earlier in the day. Sleep will reset your brain.

Do it with a friend

Share your suffering. :] This is accountability + a friend to talk the content and concepts over with. You could take a rubber duck approach where they didn't read it but will listen and ask thoughtful questions if being more responsive than an actual rubber duck. If your friend or colleague also read it then discussing can help you both understand it.

Similarly, can read as if you were learning to teach someone else the concept. Works great esp. When targeting non experts.

Do it with more of your senses

Definitely taste the book. Don't forget to do that. I consume a book for breakfast daily.

When in undergrad or grad classes and I was studying, I would read aloud to slow down and fully comprehend the text, Sometimes along with writing out the key points in note form. Same goes for attempting to make visuals or trying out the thing I was learning with hands on experience. Basically do all the learning styles. 😆

Personal Experience

I have had multiple papers and books be daunting to me, esp. throughout the literature review for my ongoing paper. These have covered a wide range of intertwining fields, including, probabilistic and algorithmic statistics, math, machine learning theory, & computability theory. In this case the problem and breadth was daunting. That's okay because I wanted to learn this and do this.

After all that I found papers to be "daunting" as in "I don't want to decipher the jargon to understand the intricacies from this niche field that is incredibly new and not as well formalized as the rest" (looking at you, algorithmic statistics). This is an important difference from feeling that I couldn't understand the text. I had the latter feeling more in studying during undergrad and early graduate school. The more you learn, the more you find overlapping patterns you can leverage and also realize the more others don't know and that's alright because you wouldn't want to force people to have to do all that reading you had to do.

Though you certainly come to appreciate writers who respect your time. Btw, philosophy doesn't tend to be those kind of writers, in my experience. Physicists, now they get to the point.

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  1. Richard Feynman mentioned a paper that was very hard for him to read. I can't remember if it was in his bio or letters volume (I think the former, have both). He told his sister that he just couldn't handle it...was too hard. And she counseled him to just sit down and push through it. And he did...and once he'd done it, said it wasn't that bad. I know that it sounds minor, but I can just imagine how he felt.

  2. One minor idea to add (not listed yet). I have found in the physical sciences, that pulling every reference, and printing it and skimming/reading them, can be very helpful. I wouldn't say that I get everything when I do that. After all the paper was already tough for me. And the volume of content has increased. However, I still find things that way...unexpected easy explanation, new gems, etc. Sometimes I even pull all the refs to the refs (one layer down of turtles). You really do start to get a gestalt that way. For example, with the one layer down of turtles, you'll start to see repeated key refs in the field. Also...practically, I find that killing trees and filing them in subject folders is a key aspect of this. You'll never get it to work, by online skimming. But if you kill the trees, it somehow works in a way that skimming the physical shelves of a library works. (Also, having them in a file cabinet, means you can go back to them in other contexts, in a way that you would never do if electronic...and more easily hidden/forgotten.)

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