[Math] Where to break paragraphs in a proof

Question

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?

Answer 1

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.