This is the preface to a set of notes I'm writing for a seminar this semester. It will be a fun and bumpy ride !
Reading research papers in mathematical disciplines takes training. A student’s natural tendency is to read a paper from beginning to end in a depth-first manner. Every time they get stuck at a concept, they branch down another path, and then another, to the point where even getting through an abstract for a paper can take days and is exhausting.
Needless to say, this is not the best way to read technical material of any kind, mathematical or otherwise.
I have spent many hours explaining to my students how they should be reading papers, and if you’re reading this and you have students of your own, you’ve probably done the same. While I don’t claim to have the perfect method for reading any given paper, I’ve learnt over the years how to quickly skim a paper, flying over it and scanning the terrain as it were. I’ve learnt when to make a deep dive all the way to ground level, and when to resurface. And above all, I’ve learnt to distinguish the different reasons for reading technical material in the first place, and that each mode requires a different treatment.
I’ve looked all over the web for good material to give to students to learn how to read papers in (theoretical) computer science. I’ve found a few short articles here and there: S. Keshav’s note on reading systems papers, Michael Mitzenmacher’s helpful document, a page by Jason Eisner, and the many blog posts by Terry Tao, to name a few. There are also a number of good short articles on reading mathematics that I’ve drawn some inspiration from.
But none of these capture the essence of what I try to achieve in my reading and what I want my students to be able to do. It’s a kind of Thurstonian ideal of understanding: where the proof falls away and what you’re left with is a deep conceptual appreciation of how the parts of a proof (or a theorem, or a set of definitions) fit together, and what they really mean.
Moreover, if one is to teach the art of reading, it is important not just to describe the goal, but to find a way to get there: with exercises, little practices, and muscle-strengthening activities. This is after all the difference between teaching and showing.
That is my motivation for this collection of notes. It is my attempt to demystify and systematize a set of tools for reading papers. It will not be complete, and it will likely not resemble how you read papers. But hopefully it provides one way of getting to the promised land.