In section 5.3 from the book Book of Proof by Hammack (3rd edition, this link is to the author's website), the author outlines 12 mathematical writing guidelines to help the young mathematician with writing better proofs.

Those guidelines, with their examples are as follows:

  1. Begin each sentence with a word, not a mathematical symbol:

    Wrong: $A$ is a subset of $B$.

    Correct: The set $A$ is a subset of $B$.

  2. End each sentence with a period, even when the sentence ends with a mathematical symbol or expression:

    Wrong: Euler proved that $\sum_{k=1}^\infty\frac{1}{k^s}=\prod_{p\in P}\frac{1}{1-\frac{1}{p^s}}$

    Correct: Euler proved that $\sum_{k=1}^\infty\frac{1}{k^s}=\prod_{p\in P}\frac{1}{1-\frac{1}{p^s}}$.

  3. Separate mathematical symbols and expressions with words:

    Wrong: Because $x^2-1=0$, $x=1$ or $x=-1$.

    Correct: Because $x^2-1=0$, it follows that $x=1$ or $x=-1$.

  4. Avoid misuse of symbols:

    Wrong: The empty set is a $\subseteq$ of every set.

    Correct: The empty set is a subset of every set.

  5. Avoid using unnecessary symbols:

    Wrong: No set $X$ has negative cardinality.

    Correct: No set has negative cardinality.

  6. Use first person plural:

    Use the words "we" and "us" rather than "I," "you" or "me."

  7. Use the active voice:

    Wrong: The value $x=3$ is obtained through division of both sides by $5$.

    Correct: Dividing both sides by $5$, we get $x=3$.

  8. Explain each new symbol:

    Wrong: Since $a\mid b$, it follows that $b=ac$.

    Correct: Since $a\mid b$, it follows that $b=ac$ for some integer $c$.

  9. Watch out for "it":

    Wrong: Since $X\subseteq Y$, and $0<|X|$, we see that it is not empty.

    Correct: Since $X\subseteq Y$, and $0<|X|$, we see that $Y$ is not empty.

  10. Since, because, as, for, so:

    The following statements all mean that $P$ is true (or assumed to be true) and as a consequence $Q$ is true also:

    • $Q$ since $P$
    • $Q$ because $P$
    • $Q$, as $P$
    • $Q$, for $P$
    • $P$, so $Q$
    • Since $P$, $Q$
    • Because $P$, $Q$
    • As $P$, $Q$
  11. Thus, hence, therefore, consequently:

    These adverbs precede a statement that follows logically from previous sentences or clauses:

    Wrong: Therefore $2k+1$.

    Correct: Therefore $a=2k+1$.

  12. Clarity is the gold standard of mathematical writing:

    If you think breaking a rule makes your writing clearer, then break the rule.

Are there any other rules or personal experiences that lead to writing a better proof?

  • 2
    $\begingroup$ Your last 3 examples for guideline 10 violate guideline 3, so wouldn't it be better to leave them out? And "$Q$, for $P$" entails using "for" to mean "because", which is a rather poetic usage which the reader won't expect. $\endgroup$ – Rosie F Jul 16 at 18:57
  • $\begingroup$ I have voted to close this as being too opinion based. What makes for good mathematical writing is fairly personal, and may depend quite a lot on one's advisor or the editor of the journal where one has submitted work (for example, with respect to (6), I would argue that pronouns should be avoided entirely). $\endgroup$ – Xander Henderson Jul 29 at 14:14

Short answer in case there's somebody who hasn't seen this before: Halmos, Paul R. "How to write mathematics." Enseign. Math 16.2 (1970): 123-152.

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Please have a look at Nicholas Higham's book Handbook of Writing for the Mathematical Sciences.

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