Exposition as science: some ideas for how to make progress
In a previous post, I argued for treating exposition as a scientific field of inquiry as the best main strategy to use to get people to create more great exposition. However, I did not really explain what “exposition as science” means in detail. So in this post, I give more details, in particular by pointing to examples of things I consider progress and some other ideas on how to make further progress.
This list is pretty disorganized, I think mostly because my own thoughts are still disorganized. Some of the items in the list aren’t even of the same “type” as others. I hope that I (or someone else!) will come up with a better typology/categorization/”roadmap” of things to do in this space in the future.
(The sections in this post can mostly be read independently. There are a few places where I refer to different sections.)
Take a genre of exposition (such as “solutions manual”), try to make the best possible version of that thing, and see what techniques you discover or lessons you learn
In March 2020, I started a blog where I would write up solutions to Terence Tao’s Analysis I, an undergraduate-level real analysis textbook. I had a bunch of motivations going into this project, but one of them was something like “Why do solutions manuals suck so much? Shouldn’t it be easy to improve on the state of the art and create ‘the greatest solutions manual ever’?” I finished blogging solutions to about 1⁄3 of the book’s exercises before deprioritizing the project, so I did not get to create the “the greatest solutions manual ever”, but I’d like to think that filtering to solutions manuals with at least 50 solutions, the average quality of solutions is very high. In terms of quality of the math exposition itself, I still think it’s good but not so good that you can’t find the same level or above in a bunch of places.
I think I came up with a bunch of “techniques” to make solutions manuals much better, including:
organizing the solutions manual as a blog, so that people can find each exercise via internet search
having a comments section for each exercise so people can ask questions or submit alternative proofs
including a variety of different proofs if the exercise can be done in multiple ways
standardizing on some sections, including most importantly a “How to think about the exercise” section where I try to say something about how I came up with the solution, rather than just writing out the proof
paying to get an independent review, to give the audience some idea of the quality of the solutions
None of the above ideas are difficult to come up with or implement. But they just weren’t really a thing that was done in the genre of solutions manuals (I can think of a few exceptions for some of the points above).
So by working on this project, I got the sense that people don’t really care to make really good solutions manuals. (They are also probably written for professors or TAs to assist in grading problem sets, so aren’t even reaching the audience that would most benefit.)
The solutions manual genre is just the one I decided to explore. I think there are a lot of other genres of exposition that one could similarly attempt to “be the best at” (some other genres: textbook, recorded lecture, math explainer video, discovery fiction (see below for more), multiple choice quiz, spaced repetition prompts, stream-of-consciousness transcript of how one thinks about a problem). I think one would similarly discover a bunch of techniques and maybe learn how to be better at exposition in general.
Discover techniques for improving a specific part of exposition (such as how to give better examples in math)
Over on the Learning Subwiki I created a page about examples in math, where I broke down examples into four categories: obvious examples, surprising non-examples, surprising examples, and obvious non-examples. (I won’t reiterate what I wrote on that page, but understanding what I wrote there is important for what follows.)
In my experience, most math textbooks don’t give examples of all four kinds when they give examples. They instead mostly stick to the obvious examples, and sometimes give surprising non-examples. I think most textbook authors don’t even have these four concepts in mind when they write down examples for their textbooks. In other words, they don’t think to themselves “Should I give all four kinds of examples? I won’t give an obvious non-example since I know my audience is quite mathematically mature, but I will give the other three”; instead, they probably just think “Oh yeah, I should give an example” and then move on. I think this kind of lack of reflectivity/lack of having the right concept, as well as just not giving all four kinds of examples more commonly, makes mathematical exposition noticeably more confusing to students.
I just gave an example of a way to improve examples in math. Are there other ways to improve examples? Are there ways to improve theorem statements? proofs? What are some analogous components in non-mathematical exposition, and how do we improve those components? I have the intuition that there’s a bunch of progress that can be made by simply asking these obvious questions.
One may object that finding examples of various kinds is already something that good students do, and that students ought to be doing this more, not the expositors. I like to instead turn this around and ask: Can we find more things that good learners already do, and then automate them, or explicitly spell them out, for even not-good learners to do or benefit from? In this framing, the four kinds of math examples is just one principle I have discovered. What are others?
There are various analogies here (and elsewhere) to various other kinds of writing like mathematical proofs, computer programs, and literature. When writing proofs, one has techniques such as proof by cases and “without loss of generality”; when writing computer programs, one has different kinds of looping structure, ideas like modularity, different programming paradigms, etc.; in literature, one has tons of techniques. What are the analogous techniques in exposition?
Find perspective shifts in the subject matter itself
Sometimes, people come up with different ways of looking at an existing discipline, and this makes it possible to produce exposition that was not possible before. Two examples I am familiar with are:
Sheldon Axler’s Linear Algebra Done Right explains linear algebra by delaying the mention of determinants as much as possible. This requires reworking proofs of many results to avoid the use of determinants.
Noson Yanofsky’s paper “A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points” (and Lawvere’s prior work, which is less accessible) ties together many of the diagonalization/self-referential results one encounters in computability theory and logic, explaining how “they are all really the same thing”.
And a third example that also seems worth mentioning:
Tristan Needham’s Visual Complex Analysis explains complex analysis using a lot of really good pictures. I have only read a little bit of this book but it seems good.
I am sure there are tons of other examples in both math and other fields. Searching for more examples might be useful, but by far the more important task is to generate more examples. Are there topics that seem just a bit too difficult to understand? Is there a proof one keeps forgetting? Might there be a way to look at the whole subfield differently, to make it a lot more “natural” to understand and remember?
There is an interesting question here of whether this task is for expositors or for “researchers in the field itself”. Maybe it is for both? Maybe a collaboration is needed? Maybe there isn’t much of a difference between exposition and research? These are just some low-effort obvious questions I have thought of.
In some sense, the kind of perspective shift described here is the highest order bit in an expository piece: you could be doing all the little “techniques” right but still be missing something big that makes the material much harder to understand.
As a sub-category under perspective shifts, there is notation. Adam Shimi pointed me to this page which catalogs examples of ways in which notation can influence how one thinks about a subject.
Invent new genres or styles of exposition (such as “discovery fiction”)
Discovery fiction is a style of exposition where one fabricates a story of how someone might have discovered something. When done well, it gives readers a good idea of where something comes from, why it works the way it does, etc. It can also teach a little bit of mental habits one needs to discover new ideas for oneself.
Creating more discovery fiction, and figuring out how to make really good discovery fiction would be good. See the section “Take a genre of exposition …” above for more. This section, however, is instead about creating entirely new genres on the level of “discovery fiction” or “solutions manual” or “textbook” or “interactive video” or “an essay with spaced repetition prompts interspersed”. I don’t know how to do this; I have only kind of done it once. I don’t think anyone knows how to do this reliably. But it seems worth figuring out.
Invent psychological techniques to improve exposition (such as “write as if you are writing a children’s picture book about a topic”)
Jessica Taylor’s “Writing children’s picture books” and Eliezer Yudkowsky’s earlier “Explainers Shoot High. Aim Low!” both describe a similar technique one can use to improve exposition: to imagine explaining to someone with much less background than one’s actual audience. I suspect there are other such “psychological” techniques one can use. What I mean by “psychological” technique is that instead of acting like a checklist or a local “move” one can use (such as with the four kinds of math examples I gave earlier in the post), it is more of a global “mindset” or “mood” change that causes a bunch of other changes to the way one explains something.
Port pedagogical platforms to other fields and see what happens
Programming has a lot of interesting tools for learning, such as LeetCode, Exercism, Scrimba, and probably a lot more. The video games community has “let’s play” videos, speedruns, and probably more. What happens if we port these ideas over to other fields? What would a LeetCode for math look like? What would a Scrimba for piano look like? etc.
Create more “playbooks” or “outlines” of expository pieces, instead of fully fleshed out pieces
I believe “playbooks” or scratch outlines of how to explain a topic are just as important as fully fleshed out expository pieces. Many people write textbooks on real analysis with slightly different strategies. But they will all have to explain certain bits in the same way. There’s a lot of duplicate effort going on. I think it would be much better if people could first verbally argue this out a bit, do some adversarial collaboration or something, write “outlines” of the book and use it to argue more, test stuff in classes, and then write up the book.
Go through textbooks to spot patterns or techniques the authors are (possibly unconsciously) using
There is already a lot of variation in quality of textbooks. Our eventual goal is to produce exposition of such high quality that no examples currently exist. But that is a hard goal, so one idea is to go through both good and bad textbooks and see what works and what doesn’t. I did this once with Terence Tao’s Analysis I.
More small scale experiments and prototypes
It is easy to think that because the eventual goal is to produce explanations that people can easily understand, that all outputs of the field must be easy for everyone to understand. However, just as experts in other fields produce work only intended to be consumed by experts, I think in an “exposition as science” field it is also good/necessary to have prototypes and such that only people who are familiar with exposition can understand or appreciate.
Create more things like X and see what happens
Some values of X for which I’d be interested in seeing the result:
Extremely detailed study guides like the Teach Yourself Logic guide. Most study guides are quite barebones; I would like to see more things like Teach Yourself Logic instead of more textbooks on the exact same subject.
Articles on the Tricki.
The Tricki itself.
Tim Gowers’s blog posts and old web pages.
Create an explicit model of the reader’s understanding, and then try to write so as to steer the reader’s understanding to some desired state
I have the intuition that many explanations fail because they don’t model the reader’s understanding in sufficient detail (e.g. they don’t account for reader’s working memory, or the fact that there’s a nearby misunderstanding, or reader engagement level/where the reader’s eyes glaze over). Better writers have a good feel for what the reader is thinking, and then try to anticipate reader questions or preemptively combat misunderstanding. But I think these better writers mostly do this very informally/intuitively. Can we do better via explicit reasoning? In other words, can we explicitly model the reader’s state of mind as they read, and then use that model to write better explanations? Can we write so as to backchain from some desired state of ‘reader fully comprehends material’, and thereby make writing explanations as simple as solving a maze? I don’t know the answers to any of these questions, but I am curious to try to answer them.
Thanks to Adam Shimi for helpful discussion and comments on a draft of this post. Thanks also to Vipul Naik for reading a draft of this post and reminding me to include one thing.
Thanks to Adam Shimi for reminding me about literary techniques.
I came up with the idea of “live math videos” but didn’t follow through to actually creating any myself. And then Tim Gowers came up with the same idea (almost surely independently) and actually went ahead and made a bunch of videos.
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This post was apparently translated to Chinese, and there is some discussion there. I can’t quite tell if it’s actually humans writing the comments (and Chrome’s translation is just not very good) or if the content and discussion is all AI-generated.