Writing Mathematics

If you google “mathematical writing advice”, you'll find plenty of excellent resources. Many of them, though, focus on style and tend to be fairly detailed. This note, intended primarily for students writing a master's thesis or their first paper, instead aims to provide short, targeted advice on the writing process and a mental model of the reader. As with all advice, I advise you not to take it too seriously.

The writing process

• Manage your deadline: Discuss your timeline with your supervisor early. Allow time to write, receive feedback, and revise, and expect unexpected delays. (If relevant, account for the time needed for your thesis examiners or referees to review your manuscript.)
• Writing is part of research: Filling in details can be hard work, often revealing gaps and mistakes, and sometimes leading to new discoveries. This is natural and to be expected.
• Fix the scope and audience: Decide who the intended audience is, and, in light of that, what should be included or omitted. Discuss this with your supervisor early.
• Write non-linearly, starting from the body: Start from the body, rather than the introduction or abstract. You are allowed to write the sections in any order, and to work backwards from a key result to its supporting lemmas. Leave the introduction and final connecting prose until the structure is stable.
• Protect your momentum: Writing need not be slow and laborious. Start by trying to produce the bulk of the text, and polish and revise later. Do not interrupt your flow – it is useful to postpone tasks such as looking for references, writing certain proofs, or perfecting arguments. When doing so, leave a small, searchable note, for example using a \todo macro containing a few words on how to complete the proof, if you know how, or the arXiv identifier and proposition number while the reference is still open.
• Read cover-to-cover at the end: Check mathematics and structure as you go. Only when you have a complete draft, read it from beginning to end, making sure it tells a coherent story and looking for gaps or inconsistencies.

The manuscript

Keep two facts about the reader in mind.

• Readers read selectively and non-linearly: They typically read the title and abstract, and skim the introduction. If interested, they may read other parts selectively. They're often interested in side results rather than the main one. (This is less true for a master's thesis than a paper.)
• The reader is not in your head: They do not have direct access to the motivation and overall strategy that you have in mind.

In light of these, here are some principles.

• Make the logical order linear: Definitions should appear before they are used. Proofs should only depend on previous results. (This doesn't apply to the introduction.)
• Explain the strategy ahead at every scale: It is often difficult to follow the broader strategy of a proof, a section or the entire manuscript, and a brief explanation goes a long way. The introduction should explain the overall strategy; a section should first outline its contents; a long proof should start with its method (e.g., “we verify the conditions of Proposition 2.7”, or “we argue by induction on ...”).
• Use environments and connecting prose: Definitions, theorems, proofs, examples, etc. should be in a clear environment. Use paragraphs between them to narrate the story and explain the transition, rather than presenting a wall of environments.
• Exposition order is not discovery order: Present the results in the order in which they are easiest to comprehend, rather than the order in which you discovered them. Remember that the parts that were difficult for you to figure out won't necessarily be difficult to understand, and vice versa.

Writing style and LaTeX

As I mentioned above, numerous excellent style guides already cover this subject well. Nevertheless, here are a few recommendations and common pitfalls I see often.

• State the main results early: The introduction should state the main results early, clearly displayed in theorem environments, and without requiring long preliminaries. Make sure to include a cross-reference to their statements in the body.
• Explain ideas, but don't over-philosophize: Explaining ideas and motivation significantly improves readability; overdoing it is off-putting.
• Use display mode: Use \[ \] for long or important expressions (or an align* environment for multiline expressions).
• Use cross-references: Give clear references such as “by Lemma 3.2”. I'd recommend using \usepackage[capitalise,nameinlink]{cleveref} in the preamble, then e.g. \begin{lemma}\label{uniqueness-lemma} and later \cref{uniqueness-lemma}.
• Use a single numbering sequence: Ensure that your environment numbering is linear, e.g. Definition 2.1, Lemma 2.2, Theorem 2.3, Lemma 2.4, ... To that end, you need to have a single counter, e.g. [theorem] in the following snippet

  \newtheorem{theorem}{Theorem}[section]
  \newtheorem{proposition}[theorem]{Proposition}
  \newtheorem{lemma}[theorem]{Lemma}
  \newtheorem{corollary}[theorem]{Corollary}
  
  \theoremstyle{definition}
  \newtheorem{definition}[theorem]{Definition}
  \newtheorem{example}[theorem]{Example}

• Use a paper you like: If there's a paper whose format you like, you may wish to download its TeX source, which is normally available on arXiv, and use it as a template.