Writing Mathematics

These guidelines apply to every proof-writing course I teach; each of those courses is writing-intensive, and its syllabus points here. Standard professional standards for technical writing (mathematical writing) apply. The style is mathematical prose written for a human reader. An otherwise “correct” argument presented as a disorganized or unintelligible pile of symbols is not a solution: it will be returned without grading and earn no credit.

The standard

Work that ignores the following is returned ungraded and scores zero.

  • Write in sentences. Every line —including a line of algebra— belongs to a sentence that has a verb and can be read aloud as English. “=” is the verb equals; a display is a clause, not a bullet point.
  • Every letter or symbol must have a clear meaning and scope before it is used; e.g.: “let n be a positive integer”; “let G be a group and let H be a subgroup of G”, etc. An undeclared symbol is an undefined term.
  • State what you are proving, in words, before you prove it, and mark clearly where the proof ends (the text “Q.E.D.” or “QED”, or a less conspicuous box “◼︎”, are common conventions).
  • Justify each step. Name the definition, hypothesis or theorem that licenses it. “Clearly” and “obviously” are not justifications.
  • Put quantifiers in words, in the right order. “For every ε>0 there exists δ>0 such that […]”; “there exists δ>0 such that for every ε>0 […] carry quite different meaning: using the wrong one is a mathematical mistake, not a typo.
  • Using logical symbols (e.g., “∀”, “∃”, “⇒” and “∴”) as “shorthand” is unacceptable. Their proper use is restricted to extremely narrow settings such as discussing syntactic formulas, or writing purely symbolic proofs —likely never outside of courses in logic and foundations.
  • Hand in proofread, complete work. Unproofread work — including unedited AI output, with its telltale artifacts —scores zero under the AI policy in the syllabus.

Handwriting is acceptable as long as it is legible. What is not acceptable is a draft turned in as finished work.

Learning to write mathematics

Read these in this order. The first is three pages long and is the single highest-return thing on this list.

  1. Francis Su, Some Guidelines for Good Mathematical Writing (3 pp., free) — PDF. Read it before your first problem set, and again before your second.
  2. Mathematical writing advice: Do’s and Don’ts — on your own course’s Class-only files page (encrypted file; the passphrase for that course is in its Canvas announcements).
  3. Michael Hutchings, Introduction to Mathematical Arguments (27 pp., free) — PDF. The mechanics: what a proof of “if P then Q” actually has to contain, how to handle quantifiers, and the errors everyone makes first.
  4. Donald Knuth, Tracy Larrabee and Paul Roberts, Mathematical Writing (free) — PDF. The extensive reference (119 pp.). Read only §1: twenty-seven numbered rules in pages 1–6 (pages 3–8 of the PDF) —the rest is a treasure trove of advice as you grow academically.
  5. Purdue OWL — owl.purdue.edu. Ordinary English grammar, punctuation, sentence structure and mechanics, with a good section for ESL writers.

Get human help. The UTSA Writing Center is free, takes appointments seven days a week (also online), and will work on a mathematics assignment with you —bring the actual problem set. Book appointments at utsa.mywconline.com. Also: take advantage of office hours (bring a draft, not a blank page).

Typesetting with LaTeX

Although LaTeX is not required, essentially it is the world standard for mathematical typesetting; the hour it costs to learn the basics pays for itself many times over.

Start here. The easiest way to get started with LaTeX is at Overleaf —it runs in the browser, so there is nothing to install and nothing to configure. Work through Learn LaTeX in 30 minutes; that is genuinely enough to typeset your first problem set.

Keep these four within reach:

  • George Grätzer, Text and Math into LaTeX, 6th ed. (Springer, 2024): the standard reference, and the one to own as a UTSA student. UTSA Libraries e-book (free to UTSA users). FYI: The author (G. Grätzer) also makes the older 4th edition, More Math into LaTeX (older, but not obsolete) freely available for download: ResearchGate

  • AMS, Short Math Guide for LaTeX (~20 pp., free) — PDF. The lookup table: symbols, alignment environments, matrices, spacing.
  • Detexify — detexify.kirelabs.org. Draw a symbol with the mouse, get its command. Use this instead of searching for the name of a squiggle.
  • Tobias Oetiker et al., The Not So Short Introduction to LaTeX (free) — PDF. The full beginner’s tour, past the 30-minute tutorial.

A myriad expert solutions (Q&A) are available online at Stack Exchange (SE) tex.stackexchange.com. You can even post a tricky question at SE if you are really stuck.

Six habits worth having from day one

  • The first line of your document should be \documentclass[...]{amsart} in most cases (where ... stands for a suitable comma-separated list of options, e.g. 12pt, letterpaper). Do not use \documentclass{article}: the amsart style has far better modern defaults.
  • Load universally useful packages: \usepackage{amsmath,amssymb,amsthm}, and use their environments: theorem, lemma, definition (declared with \newtheorem) and proof, which prints the ∎ for you.
  • Use \(...\) for inline mathematics and \[...\] for a display. Never double-dollar signs $$...$$ displayed equations: amsmath deprecates it; instead, read the next bullet item. (It is acceptable to use single dollar-sign delimiters $...$ for inline equations instead of \(...\).)
  • Line up multi-line calculations in one align environment rather than stacking separate displays. Each step then sits under the last, and the = signs line up.
  • Use the named operators — \sin, \log, \gcd, \deg, \ker, \dim — and declare your own with \DeclareMathOperator. Do not write sin(x) in math mode; it typesets as the product of three variables s, i, n.
  • Write words in mathematics with \text{...}, spacing with \quad, and divisibility with \mid rather than a bare |.
  • Compile after every few lines, and fix the first error before reading the rest —a single missing brace produces a cascade of misleading ones.

Do not spend your time fighting the layout: no manual page breaks, no hand-tuned vertical spacing, no fiddling with margins. LaTeX’s defaults were designed by people who knew what they were doing, and every hour spent overriding them is an hour not spent on the mathematics.