## December 19, 2017

### On Writing Short Papers

#### Posted by Mike Shulman

In the old days, when mathematics journals were all published on paper, there were hard budgetary constraints on the number of pages available in any issue, so long papers were naturally a much harder sell than short ones. But now that the primary means of dissemination of papers is electronic, this should no longer be the case. So journals that still impose draconian page constraints (I’m looking at you, CS conference proceedings), or reject papers because they are too long, are just a holdover from the past, an annoyance to be put up with until they die out.

At least, that’s what I used to believe.

Recently I’ve become quite enamored with the idea of writing short papers. The reason is quite simple: I prefer to read short papers! Every day seems to have fewer productive hours in it than the last, and an 80-page paper just doesn’t make me want to spend the time.

Many of my papers are quite long. Recently I made a list of all my papers and their page counts, and 13 of them are 70 pages or more. Some of those papers really do have to be that long: getting to the punchline just requires that much setup. Others are long because of the “encyclopedic impulse”, trying to include all potentially-important facts about the concept being studied. I still think there is a place for papers of this length; one might argue that they occupy a sort of novella ground in between short papers and research monographs. And so I’m still glad that the Internet is making it easier for some journals, at least, to publish long papers.

However, I’ve noticed that some of the papers that I’m proudest of are among the shortest I’ve ever written. A short paper, like a short story, doesn’t waste any time: it sets up exactly what’s needed and gets right to the point. A short paper generally has just one idea to get across, and does it as efficiently as possible. And a short paper doesn’t require an immense time committment of the reader, hence is more likely to get read — and more likely to get refereed quickly.

With all that in mind — in addition to the draconian page constraints imposed by some journals and conference proceedings — I’ve been thinking more recently about how to write short papers. I think probably anyone who’s tried can testify that it’s not as easy as it may sound! Here are a few tricks that I’ve come up with so far:

• As memorably enjoined by Strunk and White, omit needless words. Actually, my own experience suggests that they should instead have said “remove needless words”. I find it much easier to recognize the needlessness of words after I’ve written them. Usually when I come back to a paragraph I’ve written a day or two later, I can shorten it by a line or two without reducing the content, and make it read better in the bargain.

• Suppress the encyclopedic impulse. A paper doesn’t need to explore every possible ramification of its definitions. In fact, it may be better if it doesn’t: open problems help get other people excited about a subject. There’s a balance here, of course. Fortunately, the encyclopedic impulse now has other outlets, like writing on the nLab. (Writing on Wikipedia would probably fall afoul of the No Original Research policy, but the nLab encourages original research.)

• Write, then delete. Blaise Pascal is often quoted as saying “I made this letter longer than usual because I have had not time to make it shorter.” Often I find that the way to write a short paper is to first write a long paper and then remove things from it. It doesn’t have to happen in two discrete stages: I can be removing things as I’m writing other things. But I often find that it’s not until I’ve written something out reasonably carefully that I can be sure that it can be omitted, or that I can see how best to condense it into a two-sentence remark. This strategy also helps deal with the encyclopedic impulse: just writing something down helps stop it clamoring in my head, even if that something never makes it into the actual paper.

• Omit or shorten proofs. This is the most controversial. Of course a math paper needs proofs, but omitting or only sketching some proofs that can be reconstructed by the reader can help streamline it and make it easier to read. On the other hand, it’s quite frustrating when the author of a paper thinks a proof is straightforward, but it doesn’t seem so straightforward to the reader. It would be great if proofs could be “hidden” in the default version of a paper, making it shorter, but with a button for the reader to click that says “expand this proof”. Maybe there could be even multiple levels of expansion: first a telegraphic version that gives the basic ideas, then a more detailed version with all the nitty-gritty. Unfortunately, while something like this is possible when writing math on a wiki, all “real” journals that I’m aware of publish papers as PDFs which don’t have any feature like this. But there are workarounds; for instance, in computer science (at least, among the computer scientists I hang out with) it seems to be not uncommon to publish short versions of papers, with most proofs omitted, in conference proceedings that have very stringent page limits, but to also make a longer version containing the proofs available on a personal web site. Also, if the proofs are formalized in a computer proof assistant, then the burden of absolute rigor is lifted from the written paper.

What other tricks are there for writing short papers?

Posted at December 19, 2017 11:24 PM UTC

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### Re: On Writing Short Papers

Not that I write many articles, short or long, but I’d say a big one might be: let the reader in on the most illustrative example(s) you have, the one(s) which really guided your thinking – and let that hammer home the point.

Ignoring this is one of the things that many category theorists do in articles that makes them unnecessarily hard to absorb. Instead they focus on making their propositions very general, but then the real force of the point often gets buried. Oftentimes I feel I could have figured out the generality for myself, once I have really grasped the actual guiding example(s).

Or at least they could write, “Using a similar proof, one may readily establish the following more general result” or words to that effect.

Posted by: Todd Trimble on December 20, 2017 2:04 PM | Permalink | Reply to this

### Re: On Writing Short Papers

That’s good advice. However, it’s not clear to me that it will always result in shortening the paper. I would expect that sometimes, seriously discussing the motivating example (though still the right thing to do) would make for a somewhat longer paper than just hitting the maximum generality directly.

Posted by: Mike Shulman on December 21, 2017 10:45 AM | Permalink | Reply to this

### Re: On Writing Short Papers

What of the associated concern that motivating, illustrative examples haven’t even been developed? One may just know that a certain construction is the way things should go.

I note in the recent An M5-Brane Model (which, by the way, is a form of the higher gauge theory that so concerned the Café in its early days)

the ease with which generalizations are made using category theory makes development of the general theory more appealing than the rather cumbersome process of working out specific examples. On closer inspection, and with some guidance from string theory, it is however not too hard to find examples.

Can a desire for pithy brevity work against the needed elaboration of detailed cases?

Posted by: David Corfield on December 22, 2017 8:25 AM | Permalink | Reply to this

### Re: On Writing Short Papers

However, I’ve noticed that some of the papers that I’m proudest of are among the shortest I’ve ever written.

I’d love to know what these papers are!

Best, Thomas

Posted by: Thomas Hunter on December 20, 2017 2:47 PM | Permalink | Reply to this

### Re: On Writing Short Papers

I was thinking of this one (17 pages), this one (24 pages, with Kate Ponto — arguably an instance of Todd’s comment), and this one (10 pages, with Dan Licata — perhaps artificially compressed since it was in a CS conference proceedings, but would probably have been pretty short anyway.)

Posted by: Mike Shulman on December 21, 2017 10:43 AM | Permalink | Reply to this

### Re: On Writing Short Papers

Of the technical things I’ve written that I’m fairly happy with, one is fairly long and another is quite short. My PhD thesis set the length record in my department by about a factor of three, partly because I found the literature I built upon to do a really bad job of explaining things. So, rather than being able to refer my (hypothetical) readers to good textbooks or review articles elsewhere, I had to construct pedagogical treatments that I could be satisfied with. But, now that it’s done with, I’m happy I did it.

On the other extreme, last year I wrote a paper about certain geometrical structures arising in quantum information theory that was only two pages long. I kept that one as short as possible, because I liked the challenge of that, and because any frills would distract from the oddness of the result. (The journal version ended up just a smidge longer, by a few sentences.)

I generally find it easier to write and then cut. This is unlike a friend of mine who is currently finishing their PhD thesis; for them, producing the words in the first place has been like pulling teeth. Which approach works best is, I think, going to be a very personal thing, and the only way to find it out for oneself is to experiment, remembering to not feel bad if a technique happens to fail.

And when I cut material, I save it elsewhere. Good ideas are worth revisiting! The journal edition of a paper might be crisp and succinct, but perhaps the more meandering treatment in an earlier draft, with a few byways and a couple descriptions of false starts, is still useful for something else. It could be adapted into a lecture, or a book chapter, or a more freewheeling genre of writing, like a series of blog posts — in another context, perhaps those false starts could illuminate the process of exploration. Or, perhaps the rough version included some unfinished thoughts — ideas that didn’t quite come to fruition, but which still seem as well-motivated as they did to start with — and those can be the nuclei of new projects. When brainstorming potential collaborations with colleagues or research problems for students, it helps to have material like that on hand.

So, I’d say, be as encyclopedic as you feel like! But, after indulging that impulse, be willing to trim the results back.

Posted by: Blake Stacey on December 20, 2017 5:29 PM | Permalink | Reply to this

### Re: On Writing Short Papers

Wow, two pages including the bibliography! I have a dream of someday writing a paper that’s under 10 pages, but I doubt I’ll ever achieve 2. I think some subjects are probably more amenable to short papers than others.

Which approach works best is, I think, going to be a very personal thing

That’s true.

And when I cut material, I save it elsewhere.

Yes, so do I, in git revision history if nowhere else. (-:

Posted by: Mike Shulman on December 20, 2017 6:26 PM | Permalink | Reply to this

### Re: On Writing Short Papers

I too have been trying to figure out how to write short papers. One way is to write more papers. If you have an idea for a paper that’s about two or more things, maybe it would be better as two or more separate papers, each more tightly focused.

Miles Davis, famous for his laconic wit, has some other useful advice. John Coltrane was in his band in the 50’s, and they made amazing music. But Trane was fond of long, elaborate solos. Miles wanted them short and to the point. Coltrane said, “I don’t know how to stop.” Miles replied: “Try taking the fucking horn out of your mouth.”

Posted by: John Baez on December 20, 2017 10:57 PM | Permalink | Reply to this

### Re: On Writing Short Papers

That’s another good suggestion!

One might worry that splitting papers up too much would run the risk of appearing to chase minimum publishable units. But I don’t think this is likely to be a real problem; if the real reason for splitting a paper up is that it would actually be more readable as two separate papers, then that should be evident to readers.

On the other hand, even if a paper should clearly be split up, it’s not always clear how to split it up.

Posted by: Mike Shulman on December 21, 2017 10:28 AM | Permalink | Reply to this

### Re: On Writing Short Papers

Physical Review has the concept of supplementary material.

It is excellent for this style - write a shorter, punchier paper and include full proofs, extra diagrams or data, some side notes, etc in the Supplementary Material, and you get the best of boat worlds.

Posted by: Andy Ferris on December 21, 2017 11:15 AM | Permalink | Reply to this

### Re: On Writing Short Papers

Can you hyperlink from the paper to the supplementary material?

Of course one can get a similar effect by simply putting proofs in an appendix, to which it’s easy to hyperlink. But then the length of the proofs is counted as part of the length of the paper.

Posted by: Mike Shulman on December 21, 2017 12:08 PM | Permalink | Reply to this

### Re: On Writing Short Papers

I wonder if papers are the most efficient way to get an idea across. In computer science I usually prefer to look at a slide deck if it is available. You can get the main ideas in a fraction of the time. Even more efficient is having somebody explain the idea to you in person. I wonder if the difference is inherent to the medium, or whether it’s just the way people write papers right now.

Posted by: Jules Jacobs on December 21, 2017 12:34 PM | Permalink | Reply to this

### Re: On Writing Short Papers

In his essay On proof and progress in mathematics, Thurston wrote:

Mathematical knowledge can be transmitted amazingly fast within a subfield. When a significant theorem is proved, it often (but not always) happens that the solution can be communicated in a matter of minutes from one person to another within the subfield. The same proof would be communicated and generally understood in an hour talk to members of the subfield. It would be the subject of a 15- or 20-page paper, which could be read and understood in a few hours or perhaps days by members of the subfield.

Why is there such a big expansion from the informal discussion to the talk to the paper? One-on-one, people use wide channels of communication that go far beyond formal mathematical language. They use gestures, they draw pictures and diagrams, they make sound effects and use body language. Communication is more likely to be two-way, so that people can concentrate on what needs the most attention. With these channels of communication, they are in a much better position to convey what’s going on, not just in their logical and linguistic facilities, but in their other mental facilities as well.

In talks, people are more inhibited and more formal. Mathematical audiences are often not very good at asking the questions that are on most people’s minds, and speakers often have an unrealistic preset outline that inhibits them from addressing questions even when they are asked.

In papers, people are still more formal. Writers translate their ideas into symbols and logic, and readers try to translate back.

Personally, I often find that planning a talk about a paper forces me to think more deeply about what its essential ideas are and how to present them, sometimes to the extent that I wish I could rewrite the paper. And of course slides are forced to be short and to omit details, so the only thing you can convey is the idea. In a good slides talk the idea does get conveyed, and if you’re especially lucky you can get the idea from just the slides without hearing the talk.

I think the linear nature of a paper, and the requirement that it contain sufficient background, details, citations, and so on, does make for a fundamental difference. But I can imagine a more interactive sort of “research product” with a “top level” that looks kind of like slides (but designed to be read alone rather than accompanied by a talk), with “zooming in” links where you can read more about the background, more about the literature, details of the proofs, details of the examples, whatever you want. Almost like, oh I don’t know, a set of wiki pages. (-:O

On the other hand, I also like the linear nature of a paper, because like it or not, time is linear, people do and read things in some sequence, and writing a linear paper allows the author to choose a particular sequence to present things in, and thereby “tell a good story.” Interactive video games and choose-your-own-adventure books are all very well, but they don’t replace ordinary novels and movies. Reading about things on a wiki is a fine experience, but it’s not the same as reading a paper that’s been thoughtfully constructed and written; it’s easy to get lost in a train of hyperlinks. This is, perhaps, why I would prefer to think of detailed proofs as “expandable” at the point where they appear in the paper, like a knowl, rather than as links to separate pages, papers, or supplementary material.

Posted by: Mike Shulman on December 21, 2017 1:01 PM | Permalink | Reply to this

### Re: On Writing Short Papers

I wonder if papers will become less formal if computer checked proofs become more widespread. Papers perform the dual role of providing ironclad proofs and communicating the idea. If the former can be checked by a computer then perhaps the body of the paper can focus on the idea. You see a bit of this in computer science: instead of providing a (boring) soundness proof in the paper, some papers simply refer to a formal proof in Coq.

Posted by: Jules Jacobs on December 21, 2017 3:27 PM | Permalink | Reply to this

### Re: On Writing Short Papers

I think that’s likely. It’s a balancing act, but having a formalization does mean that the human-readable version of a paper has somewhat fewer jobs to do.

Posted by: Mike Shulman on December 22, 2017 3:21 AM | Permalink | Reply to this

### Re: On Writing Short Papers

A few months ago a few homotopy theory colleagues were daydreaming about the features that we’d want to see in a new electronic journal, in an imaginary world in which we all had the time to launch such a thing.

Something like you’ve suggested - a feature that would let you write a single linear narrative with expanding and contracting levels of detail - would be at the top of my list.

I’m trying very hard at the moment to write a “short” semi-general audience paper on $(\infty,1)$-category theory and am finding at each juncture where I have an idea about how to make the paper shorter the change simultaneously makes it better and worse.

Posted by: Emily Riehl on December 22, 2017 9:42 PM | Permalink | Reply to this

### Re: On Writing Short Papers

Now I want to know what other features you all came up with!

Posted by: Mike Shulman on December 23, 2017 5:53 AM | Permalink | Reply to this

### Re: On Writing Short Papers

“Maybe there could be even multiple levels of expansion: first a telegraphic version that gives the basic ideas, then a more detailed version with all the nitty-gritty.”

(Actually … this doesn’t just apply to math papers, I prefer to digest information in general in multiple increasingly detailed passes. For example, when reading a poet, some people like to take a couple especially famous poems and dive deep into the details of those, then later explore the rest of their oeuvre if those poems sufficiently captivated them. I typically do the opposite and read a large number of the poet’s poems to get a general idea for how they think, then dive deep into the ones that interested me most. Back to math papers, having multiple levels of expansion would make getting the general idea(s) of the paper before diving into details much easier/swifter.)

Posted by: Trent on December 22, 2017 6:11 AM | Permalink | Reply to this

### Re: On Writing Short Papers

I am currently a civil servant. One of the imperatives in the civil service is to keep advice to elected politicians as short as possible while remaining clear and incisive (we expect them to be read at the end of a long day). One technique is to draft the advice in PowerPoint rather than Word. PowerPoint forces you to make pages containing exactly one step in your argument with about half a dozen really short sentences or fewer, longer ones. The sequence of pages must make a single overarching narrative. The medium forces you to focus on brevity and clarity of presentation, while the ability to easily rearrange the order of the slides encourages you to consider how to segue from one idea to the next.

Posted by: Roger Joseph Witte on December 23, 2017 3:51 PM | Permalink | Reply to this

### Re: On Writing Short Papers

A rule of thumb I use (in physics, admittedly) is that if I want to emphasize a certain result, the paper is as short as possible; if I want to emphasize a technique, the paper should be encyclopedic and self-contained. Obviously results versus techniques isn’t a sharp dichotomy, but I find this to be pretty useful as a guideline.

Posted by: Dan on December 23, 2017 6:35 PM | Permalink | Reply to this

### Re: On Writing Short Papers

I’m curious what your reasoning for that distinction is. If one were going to make a distinction along that axis (which I don’t immediately see any reason to do), I could just as easily see it going the other way: if you’re proving a result, you might want to explore all ramifications of that result to make the paper a good reference, whereas if you’re introducing a technique, you might want to make it short and sweet to encourage as many people as possible to read it and apply the technique in their own work.

Also, what if you want to emphasize a definition? Or do physicists not do that? (-:

Posted by: Mike Shulman on December 23, 2017 9:26 PM | Permalink | Reply to this

### Re: On Writing Short Papers

It’s largely personal taste in terms of what I, myself, like to read. The rationale is basically: if I am trying to use someone else’s technique, then I prefer to have access to a long, detailed paper, with examples, steps filled in, etc., basically a detailed user’s manual. Whereas with a result, I just want to digest the core set of ideas as easily as possible, which in my case means skipping most of the nitty-gritty steps.

A merger of these that I love is when people write a handful of detailed papers and then summarize the main results in a short letter. I also love people with blogs who write summaries like this!

As for definitions, that’s an interesting question that I’ve never had to deal with ;)

Posted by: Dan on December 24, 2017 9:58 PM | Permalink | Reply to this

### Re: On Writing Short Papers

Interesting! Hearing you say that makes me want even more to be able to write an “expandable” paper, because different readers may want different things from the same paper. Even, the same reader at different times may want different things; it’s happened several times to me that when I first read a paper I’m mainly interested in the results and so I skip over details, but then a long time later I realize that I want to do something similar and so I go back and try to understand the details of the techniques.

Posted by: Mike Shulman on December 25, 2017 11:45 PM | Permalink | Reply to this

### Re: On Writing Short Papers

Then there’s the game of reducing it to 140 characters.

Posted by: Tom Leinster on December 25, 2017 6:13 PM | Permalink | Reply to this

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