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 brief 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 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)** Whenever possible, try to avoid enumerating equations and formulas; 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 long list of references without providing the reader any guidance about these references.

*Not so good example:* "Recently, there has been a lot of interest in magical theory X [2, 3, 6, 7, 9, 10, 12, 13, 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.

*Somewhat better example:* "Recent progress in magical theory X includes various results towards a classification of all magic wands [2, 3, 10, 34], new insights into the long-term decay of magic energy [12, 13, 16, 19] and several counterexamples which show that the color of a sorcerer's hat is logically independend from the color of their magic wand [6, 7, 9]. A very recent development is the extension of numerous classical results into the realm of super-magics in non-zero characteristics [21, 22, 31, 32]."

**(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 $\sigma_{\operatorname{p}}(A)$.

**(14)** In lenghty arguments, state the goal before giving the argument.

*Bad example:* "For $x \in \mathcal{S}$, the previous inequality implies that ... 10 lines of involved reasoning ... This shows that $x$ is foo."

*Somewhat better example:* "Next we show that each $x \in \mathcal{S}$ is foo, so fix such an $x$. ... 10 lines of involved reasoning ..."

**(15)** When you prove that assertions (i), (ii) and (iii) are equivalent, start the proof of each implication in a new paragraph which begins by stating the implication that will now be proved. Do not write "Now we prove that (ii) implies (iii)", but simply write '(ii) $\Rightarrow$ (iii)' at the beginning of the paragraph. Do this even in cases where the implication is trivial; in such a case the entire paragraph might only consist of the line "'(ii) $\Rightarrow$ (iii)' This implication is clear."

This gives a clear visual structure to the proof which makes it much easier for readers to get an overview of the proof and to readily find a specific argument they are looking for.

Some people might claim that this suggestion is "bad style" or "ugly". Whether or not this claim is correct is irrelevant. Style is important, but readability is more important.

**(16)** When your theorem says "$A$ if and only if $B$", do not use the words "Sufficiency" and "Necessity" to structure the proof. Whenever I read this, I find it annoying that I first have think about which implication is actually meant. Instead, proceed similarly as in point **(15)**: start the proof of each implication in a new paragraph, and begin the paragraph with one of the symbols '$\Rightarrow$' and '$\Leftarrow$'.

As above, some people might claim that this is "bad style" or "ugly". But as above, readability has priority over style.

**(17)** When you define a property which has several equivalent characterizations, use the following guideline to choose which of the equivalent assertions you use as the definition:

The major purpose of definitions is not to be easily checkable or to be easily applicable, but to structure a theory in an intuitive way; good definitions can guide us through a theory. Hence, choose a property for the definition which is most likely to help the reader to understand what is going on. (In some occasions this will indeed be a property that is easy to check or easy to apply; but in some other occasions it won't, and you will have to outsource the equivalent properties that are easy to check or easy to apply to a theorem.)

**(18)** When splitting a proof into several lemmas, do not try to be overly efficient (in terms of logical deduction) when formulating the lemma. When the proof of your theorem requires knowledge of the implication $A \Rightarrow B$, but you can actually show that $A \Leftrightarrow B$, then your lemma should (in many cases) state the equivalence, even if the converse implication is not needed anywhere in your paper.

*Reason 1:* Lemmas are not just proxies for steps in a proof; they also help the reader to get a better understanding of the entire situation, and the question wether the converse implication holds is part of this understanding.

*Reason 2:* While you might need only the implication $A \Rightarrow B$, your reader might need the converse implication. While it might be more or less clear to you that the converse implication holds, it might not be so clear for your reader.

`soft-question`

tag. Feel free to comment on that or delete it, or perhaps pick it up. $\endgroup$shouldbe that the explanation be fully understandable for a long time. On the other hand, on Math Overflow, the quickness of the answer is probably more important. $\endgroup$