The devil's guide to organizing a paper

Hard-to-scan labels

``Some authors number their theorems from 1 to n, their definitions from 1 to k, their lemmas from 1 to p, their corollaries from 1 to r---each item having its own numbering system,'' Krantz writes on page 19. ``Do not laugh: this describes the default in LaTeX. As a reader, I find this method maddening; the upshot is that I can never find anything.''

``Some authors have been browbeaten by non-mathematical editors into placing the number attached to a definition, lemma, or theorem to the right of this heading,'' Steenrod writes on page 8. ``This makes it difficult to locate a desired reference number by scanning pages, especially so when the author numbers lemmas separately from theorems. Actually it would be most convenient to have the numbers appear in the left margins, but this requires exceptionally expensive setting of type. The next best procedure is to use boldface numbers close to the left margin. Authors must deal firmly with editors who complain of the ugliness of the boldface splotches running down the page.''

Expense is no longer an issue. Steenrod would have been pleased to see that, for example, the Handbook of Applied Cryptography uses boldface numbers in the left margin:

     3.3    The RSA problem
       3.28 Definition ...
       3.29 Remark ...
       3.30 Fact ...
     3.4    The quadratic residuosity problem
       3.31 Definition ...
       3.32 Remark ...
       3.33 Fact ...
     3.5    Computing square roots in Zn

The numbers are much easier to scan than numbers embedded in the text. They would have been even easier to scan if the section numbers had used the same monotonic numbering system as the paragraph numbers.

The most effective numbering system I've seen has two levels: one number for the section and one number for the item within the section. Section 1 has items 1.1, 1.2, 1.3, etc.; section 2 has items 2.1, 2.2, 2.3, etc.; section 3 has items 3.1, 3.2, 3.3, etc.; and so on through the document. The occasional citations to portions of a document larger than a section are handled as, e.g., ``Sections 18-25.''

If you want the reader to suffer, don't use easy-to-scan labels! Here's what to do instead:

Bury your labels inside the text. Make sure that they don't leap off the printed page. An italic number will be much harder to spot than a boldface number.
Introduce several more levels of numbering. For example, you can incorporate volume numbers, part numbers, chapter numbers, and subchapter numbers, so that a typical theorem is labelled I.C.3.b.2.4 instead of 58.4. Claim that you've chosen labels to make the structure of your document clear to the reader.
Omit as much information as possible in local references. For example, when you're talking about Theorem I.C.3.b.2.4 inside Section I.C.3.b.2, refer to it as Theorem 4. Many people flipping through pages will miss the section boundary and will think you're talking about Theorem 4 in a different section.