Writing Mathematics Properly (for the youngest ones)

Writing a paper is part of doing research. This page collects the advice and the reading I give to the students who work under my supervision.

The Mistakes You Should Avoid

Here is a list containing some of the mistakes I often notice in the first paper drafts of young researchers, with pointers to people who have written about each of them at length.

Structure

  1. No real introduction. The introduction should contain background, history, motivation, the statement of your results, and a road map of the paper. It is also where you convince the reader that the paper is worth reading. (Tao, Use the introduction to “sell” the key points of your paper.)
  2. Results stated inaccurately. Claim exactly what you proved, no more and no less. (Tao, Describe the results accurately.)
  3. One long argument instead of lemmas. A proof that runs for pages should be cut into lemmas with clear statements, each proved once and used many times. (Tao, Create lemmas.)
  4. Important statements buried inside proofs. Many readers skip the proofs. Each milestone of the paper deserves its own clearly stated proposition or theorem, placed before the technical details, and anything the rest of the paper depends on must stand on its own in the text. (Tao, Organise the paper.)
  5. Writing up the path of discovery. The reader wants the shortest correct route to the result, not the story of how you found it. (Rogness, Writing tips.)

Mathematics and Logic

  1. Notation introduced on the fly, or notation that changes meaning. Define every symbol before you use it, and use it for one thing only. (Tao, Use good notation; Miller, Notes on Writing Mathematics.)
  2. “It is easy to see”, “clearly”, “obviously”. These words usually cover a gap. If the step is easy, show it in one line; if it is not, prove it. (J. M. Lee, Some Remarks on Writing Mathematical Proofs.)
  3. Quantifiers in the wrong order or left implicit. Say “for every” and “there exists” in words, in the right order, and avoid the ambiguous “any”. (Milne, Tips for authors; J. M. Lee, Some Remarks on Writing Mathematical Proofs.)
  4. Too much detail, or too little. Routine verifications can be summarized; the steps that carry the argument cannot. (Tao, Give appropriate amounts of detail.)

Language

  1. A sentence that starts with a symbol. A symbol has no capital letter, so the reader cannot see where the sentence begins. Put a word in front of it. (Khovanova, Common Mistakes; Knuth, Larrabee, and Roberts, Mathematical Writing.)
  2. Symbols where words belong. Logical symbols such as \(\forall\), \(\exists\) and \(\Rightarrow\) do not belong in prose, and a formula is not a sentence. (Tao, Take advantage of the English language; Conrad, Advice on Mathematical Writing.)
  3. Contractions and abbreviations. “Don’t”, “we’ll” and “WLOG” are for notes, not for a paper. (Higham, Handbook of Writing for the Mathematical Sciences.)
  4. Words used loosely. “So that” is not “such that”, “verify” is not “satisfy”, and “any” can mean “every” or “some”. (Milne, Tips for authors.)
  5. Displayed formulas without punctuation. A displayed equation is part of a sentence and takes the comma or period that the sentence needs. (Knuth, Larrabee, and Roberts, Mathematical Writing; Higham, Handbook of Writing for the Mathematical Sciences.)

Typography and LaTeX

  1. Every equation numbered. Number only the equations you refer to. (Khovanova, Common Mistakes; Knuth, Larrabee, and Roberts, Mathematical Writing.)
  2. Hard-coded numbers. Refer to theorems, equations and sections with \label and \ref, never by typing the number. (Higham, Handbook of Writing for the Mathematical Sciences.)
  3. Typographic details left to chance. Quotation marks, dashes, \ldots versus \cdots, bracket sizes, and multi-letter operators in upright type all have rules; learn them once. (Knuth, Larrabee, and Roberts, Mathematical Writing; K. P. Lee, A Guide to Writing Mathematics; Higham, Handbook of Writing for the Mathematical Sciences.)

References and Acknowledgments

  1. Sloppy references. Cite the published version when there is one, check every name and title, and do not copy a misspelling from someone else’s bibliography. (Milne, Tips for authors.)
  2. No acknowledgments. Thank the people and the programs that helped you. (Khovanova, Common Mistakes; Higham, Handbook of Writing for the Mathematical Sciences.)

Before You Send a Draft

  1. Sending a first draft. Spell-check, proofread, and read the whole paper as a stranger would before anyone else sees it. (Tao, Proofread and double-check your paper before submission and Submit a final draft, not a first draft.)