Hmm, I'm about twelve years late to the party - anyway, since the post has just popped up at the front page, here's a list of suggestions that I try to follow when writing mathematics, mainly because it reflects my preferences when I *read* mathematics. 

(I'll phrase the suggestions in imperative mode, since I try to keep telling them to myself when I write. The enumeration is there to facilitate reference to the single suggestions in potential comments; it does not indicate importance.)

1) Do not write down things how you think they should be written down; write things down how you would like to read them if the author were someone else.

2) Length of mathematical writing can be measured (a) in numbers of words, lines and pages, and (b) in the amount of time a reader needs to read and understand what you have written and to get all information relevant to them while doing so. Try to write as short as possible, but measure length exclusively in terms of (b).

3) Do not introduce unnecessary notation. Example: I've seen books introduce an extra symbol for the imaginary line, which everyone who does not use the book on a regular basis has to look up; this is pointless, $i\mathbb{R}$ works just as well.

4) Apply the first paragraph of [Scott Carter's answer](https://mathoverflow.net/a/1257/102946) also to equations or formulas you're referring to. Not so good example: "Assume that (3) and (6) hold. Then the equation $Au = f$ has a unique solution $u$, and $u$ satisfies (11)." Better example: "Assume that the integrability condition (3) holds and that the operator $A$ satisfies the ellipticity estimate (6). Then the equation $Au = f$ has a unique solution $u$, and $u$ satisfies the regularity property (11)."

5) Try to avoid enumerating equations and formulas whenever possible; this will force you to be more disciplined about the structure of your text. This applies in particular, but not exclusively, to proofs.

6) Try to avoid acronyms for mathematical properties. Most definitely avoid using half a dozen lengthy acronyms for various mathematical properties.

7) Do not assume that readers will read your entire paper (they won't). Results should be self-contained wherever possible.

8) Do not, under any circumstances, scatter various assumptions for a theorem throughout the text, when the theorem is worded in a way which makes it impossible to note from the theorem alone that these assumptions apply.

9) When citing a theorem, include the theorem number. If the paper you cite is very short and has only five pages and two theorems, include the theorem number anyway.

10) Do not use phrases of the following type in your introduction (or anywhere else): "Recently, there has been a lot of interest in magical theory X [2, 3, 6, 7, 9, 10, 12, 12, 16, 19, 21, 22, 31, 32, 34]." Such lists of references are useless, since nobody will look up all those papers without any more precise indication of what to find there.

11) When using a result you cite, try to indicate at least briefly what the result says. Not so good example: "... and hence, [2, Theorem 3.11] implies the claim." Somewhat better example: "... and hence, the open mapping theorem [2, Theorem 3.11] implies the claim."

12) Try to encode mathematical properties in words rather than in notation. Not so good example: "*Preliminaries.* Within $\mathbb{C}^{d \times d}$, we denote the set of all positively semi-definite matrices by $\mathcal{PS}$, the set of all positively definite matrices by $\mathcal{P}$, and the group of all invertible matrices by $\mathcal{GL}_n(\mathbb{C}^d)$. ... 5 pages later ... *Theorem.* We have $\mathcal{PS} \cap \mathcal{GL}_n(\mathbb{C}^d) = \mathcal{P}$."

13) Try to use notation that is self-explanatory. Example: If $\sigma(A)$ denotes the spectrum of a linear operator $A$, denote its point spectrum by $\sigma_{\operatorname{pnt}}(A)$ rather than by $\sigma_{\operatorname{p}}(A)$.

**Background information (in order to assess the validity of my suggestions):** I've been told by a non-empty set $S_1$ of mathematicians that my mathematical writing is quite good. I've also been told by a non-empty set $S_2$ (which is disjoint from $S_1$, and smaller in cardinality) of mathematicians that my mathematical writing is quite bad. Clearly, I won't comment on who is right...

**Disclaimer:** I'm quite sure that I have violated almost all my suggestions multiple times in the past.