# Where to break paragraphs in a proof?

I have some rules of thumb about writing research papers that I can actually articulate. For example, leave all definitions as late as possible (but not later!), so the reader won't fear "Do I need to understand this definition for this theorem?" nor wonder "Did I understand that theorem, insofar as it didn't seem to use the previous definition?"

One I don't have, though, is where to include paragraph breaks in proofs. If I follow the usual maxim that each paragraph should be about a single idea, then interpreted variously, this leads to sentence-long or page-long paragraphs.

More specifically, do you think of "It remains only to show some big thing" as an announcement with which to begin a new paragraph, or as a capstone with which to end the previous one?

Do you have a maxim describing where to put paragraph breaks in proofs, that doesn't lead to ones that are too short or too long?

• I would just say: do it in the way you think it looks nice in the particular case. -- I wonder whether you think there is a rule which gives always good results(?) – Stefan Kohl Sep 14 '13 at 19:30
• I can't think of any good reason to use different rules for paragraph breaking in a proof than in any other kind of writing. – Mariano Suárez-Álvarez Sep 14 '13 at 19:59
• @MarianoSuárez-Alvarez I'm not sure it would be okay in "any other kind of writing" to say "and the rest follows from Lemma 4 combined with Proposition 1". There is a separate syntax and cultural expectation from math writing. – Vidit Nanda Sep 14 '13 at 20:01
• I cannot see in what way that connects to what I said, really —for one thing, using that phrase or not in any piece of writing has nothing to do with paragraph breaking! – Mariano Suárez-Álvarez Sep 14 '13 at 20:04
• No, but insofar as there is a separate syntax, it's reasonable to think it might include separate rules for paragraph breaking. – Allen Knutson Sep 14 '13 at 21:07

Here is a useful quote from Zinsser's "On writing well", a timeless guide to all such matters:

Keep your paragraphs short. Writing is visual—it catches the eye before it has a chance to catch the brain. Short paragraphs put air around what you write and make it look inviting, whereas a long chunk of type can discourage a reader from even starting to read.

But don't go berserk. A succession of tiny paragraphs is as annoying as a paragraph that's too long. I'm thinking of all those midget paragraphs—verbless wonders—written by modern journalists trying to make their articles quick 'n' easy. Actually they make the reader's job harder by chopping up a natural train of thought.

• I'd love to imagine that a verbless paragraph would never get past a referee... – Mariano Suárez-Álvarez Sep 14 '13 at 20:19

I'm not sure what type of answer to give here, except don't be tasteless, and do what looks right. The obvious thing to do, if an idea is spanning multiple paragraphs, is to make it a separate proposition and cite it from within the larger argument. I've found how-to-write-readable-computer-code type guides rather useful in this regard.

The harder problem appears to be when to group separate ideas into a larger paragraph, for which it seems difficult to consistently defend any universal prescriptivist stance. Here's an example (yes, I understand that it is a bit of a low-blow to bring up something you wrote a long time ago): it has often confounded me why the proof of Lemma 7 on page 18 of your honeycomb paper spans three paragraphs. On the other hand, I was really glad that Lemma 8 was broken into various pieces, otherwise it would have taken me much longer to parse.

Maybe the golden rule should apply: would I want to read this in someone else's paper? Personally, I find it much more psychologically satisfying to see "only one more thing remains" at the end of a paragraph rather than at the beginning of the next one.

• I invite others who want to use examples of what not to do, to make use of my papers in this way! Looking at lemma 7 again, I still feel like I would make it three paragraphs today, and am annoyed that I can't articulate why. (Hence the question!) – Allen Knutson Sep 14 '13 at 21:09
• I would have used one :-) «Fix a region $\Omega$ in $D(h)$, let $x$ be a vertex of $\tau$ lying in the closure of $\Omega$ (so that its image $h(x)$ is in the support of the diagram of $h$) and denote $\sigma$ the connected component of $x$ in the subtinkertoy of $\tau$ consisting of those vertices mapped by to $h(x)$ by $h$. We can check that the component $\sigma$ is a honeycomb tinkertoy in its own right ---the only non-trivial observation needed is that all six vertices of a hexagon map to $h(x)$ as soon as four do--- and ...» – Mariano Suárez-Álvarez Sep 14 '13 at 22:54
• «... it follows from this and some chasing down of definitions that the region bounded by $D(\sigma)$ is precisely $\Omega$. The claim is therefore a consequence of Lemma 4.» – Mariano Suárez-Álvarez Sep 14 '13 at 22:55
• Without having read the rest of the paper (which might be an advantage or a disadvantage here), I think I see an articulable motivation for the division of the proof of Lemma 7 into three paragraphs. Something called a tinkertoy appears at the beginning of the second paragraph and disappears (from the text, though not from the reader's thought) at the end of that paragraph. So I think the motivation for the paragraph breaks may well have been to separate the tinkertoy material from the rest of the proof. – Andreas Blass Sep 15 '13 at 0:23

While writing, I usually read aloud the text: I like that what I write be not very different from what I would say. Since paragraphs are separated by pauses, which are usually quite obvious when talking, a bad paragraph break or a missing one tends to make itself quite obvious in this way.

This does not constitute a rule, of course, for paragraph breaking, but is a useful way to probe them.

Good paragraph breaking is like pornography, in a way.

Regarding your specific example: without context, I don't think we can decide if a «It remains only to show some big thing» is better at the end of a paragraph or at the beginning. If you have a paragraph whose point is to explain why to obtain the desired result A it is enough to prove B, then it may well end with a «It remains only to show that B holds.» On the other hand, if that point was done earlier in the argument, and since then time has been spent on doing some other menial task, it would be nice to start the paragraph that begins the remaining task with an announcement that «It remains only to show that B holds» if only to be nice to the reader who might have by now forgotten the big lines of our reasoning.

• I search for the specific phrase "It remains only to show" in JSTOR (restricting to JAMS, TAMS, PAMS and the Annals) and sampling randomly I could not in fact find any occurrence at the end or at the beginning of a paragraph :-) – Mariano Suárez-Álvarez Sep 14 '13 at 21:34