Suggestions for writing mathematics well. Many mathematicians know little about writing well, and some, unfortunately, don't even care. The suggestions here are aimed at those who want others to be able to read their work easily. First, see "Some Hints on Mathematical Style", by David Goss (https://www.cnsr.ictas.vt.edu/mentoring/Goss.pdf; formerly http://www.math.ohio-state.edu/~goss/hint.pdf, but this link no longer works) and "Some Remarks on Writing Mathematical Proofs" by John M. Lee (https://sites.math.washington.edu/~lee/Writing/writing-proofs.pdf). I have only a little to add to this--for now, just a few topics. Summary Formulas should (almost) always be separated by words. Give explanations, definitions, etc. before they are used. Give specific references, for example including theorem numbers. Hypotheses of results should be included in their statements. No run-on sentences, no "we have that", no missing "let", etc. No "For all $0 < a < b$": this means "for every $0$ such that ..."; similar mistakes. "$n$ points" not "$n$-points" (just like "seven animals", not "seven-animals"). Quantify over the correct variable: for all $n$, not for all $x_n$; choose $x$, not $f (x)$, etc. No reasoning in statements of theorems etc. Number consecutively throughout. First, Goss rightly complains about the all too common constructions "$f (x) > 0$, $x \in X$" and "$f (x) > 0$ ($x \in X$)", but I think there is an even more important reason than the one he gives for avoiding them. That is the general principle: Formulas should (almost) always be separated by words. The point is to make it visually clear on the page where formulas begin and end. Mathematics is hard enough to read that everything the author can do to help the reader is valuable. Having to look carefully for where to separate something into several formulas imposes an extra burden on the reader (even if the reader can't explicitly point to it). (Note: Lee does observe this point, but it is important enough to repeat, with more examples.) The examples above are corrected to read: $f (x) > 0$ for all $x \in X$ or $f (x) > 0$ for $x \in X$. Other examples: For every $x \in X$, $f (x) > 0$ is corrected to read For every $x \in X$, we have $f (x) > 0$. The phrase Let $x \in X, y \in Y$ is corrected to read Let $x \in X$ and let $y \in Y$. The phrase Let $x \in X$, $y \in Y$, and $z \in Z$ is corrected to read Let $x \in X$, let $y \in Y$, and let $z \in Z$. (Occasionally one gets really stuck, but this is rare.) Second, give explanations, definitions, reasons, etc. before they are used rather than afterwards. I have on a number of occasions read a chain of calculations, gotten stuck somewhere, and only afterwards discovered that the author is using some major theorem or some later calculation to justify the step at which I got stuck. This is extremely annoying. Thus, $A \cong B \cong C \cong D$, where the second step follows from Theorem 3.2. is corrected to Using Theorem 3.2 at the second step, we have $A \cong B \cong C \cong D$. This example is relatively mild; it is much worse if additional justification is needed at the third, fourth, and sixth steps of a calculation of seven steps spread out over a multiline display. The phrase Let $A = C (X)$, where $X$ is a compact Hausdorff space. is corrected to Let $X$ be a compact Hausdorff space. Set $A = C (X)$. Even worse is something like: Set $b = f (a)$, where $f \colon [0, 1] \to [0, 1]$ satisfies $f (0) = 0$ and f (t) = 1 for $t \geq \epsilon$, where $\epsilon > 0$ is chosen so that ... Third, give specific references. Don't write "It follows from [17] that every nuclear C*-algebra is ..."; instead, write, "It follows from Theorem 2.3 of [17] that every nuclear C*-algebra is ...". The first version is especially bad if the paper is long or if the theorem is somewhat tangential to the main results of the paper, and can be forgiven if the paper is short and the result being cited is the main result of the paper. I have seen two absolutely awful examples of this sort of thing. One was a reference simply to a paper [17] which happened to be a 100 page Memoir of the AMS; the result used was not even stated separately as a lemma but rather buried in the middle of a remark on page 61 or so. Nor did the citing paper give the statement of the result used. If I had not known ahead of time where it was, I would never ever have found it in the paper. (And the only reason I did know is that the result cited was one I had used for the same purpose in an earlier paper of mine.) The other awful example was a reference saying it followed from Section 3 of [17] that something was true. There was nothing quite like the claimed statement in Section 3 of [17]; as far as I can tell, one was supposed to modify the line of reasoning spread out over several results in Section 3 of [17] to get the claimed statement, but the author of the paper I was reading didn't want to be bothered to actually write down how to do this, preferring to leave a substantial task to every reader of his paper who wanted to figure out how he proved the claimed statement. (It would be legitimate to write something like the following: "The methods of Section 3 of [17] can be used to prove that ... . We omit the details." At least, this way, the reader knows he will have to understand of Section 3 of [17], and do some work, rather than being led to expect a theorem in Section 3 of [17] whose statement easily implies what is wanted.) Fourth, hypotheses of results should be included in their statements. Standing hypotheses stated elsewhere are dangerous. (This is implicitly contained in one very general sentence in Goss.) If you write an interesting paper, other people will cite it, often in the form, "It follows from Lemma 4.5 of [your paper] that ...". Now suppose a third person reads the second paper and wants to know more about this step. Quite possibly the first thing that person will look at in your paper is Lemma 4.5. If there are hypotheses for Lemma 4.5 that are not in the statement of the lemma, such a reader may well not find out about them, believe Lemma 4.5 is stronger than it really is, and then possibly use it incorrectly. If you have a (complicated, or even not so complicated) standing hypothesis for many results, you can put it in its own numbered item, maybe "4.1 Standing hypotheses", and in the statement of Lemma 4.5 write, "Assume the hypotheses in 4.1". On the other hand, if your standing hypothesis is something like "All C*-algebras in this paper are separable", then it costs very little, and protects the reader, to simply put, "Let $A$ be a separable C*-algebra" everywhere you would otherwise have put, "Let $A$ be a C*-algebra". As with other things, some versions of this are worse than others. A hypothesis in the text immediately before Lemma 4.5 is somewhat likely to be seen. (But why put it before Lemma 4.5 rather than in the statement of Lemma 4.5?) A standing hypothesis stated early in the introduction has a reasonable chance of being seen, but could certainly be overlooked. (Will the person following up on the citation of Lemma 4.5 read any of the introduction? Maybe not.) Somewhat worse is a standing hypothesis introduced at the beginning of the section containing Lemma 4.5. Still worse is a standing hypothesis introduced somewhere in the middle of the section containing Lemma 4.5, or at the beginning of an earlier section. I have even seen the following truly awful thing: the standing hypothesis is introduced somewhere in the middle of one of the earlier sections. The same considerations apply to notation used in a nonstandard way. They also apply to some extent to new notation introduced in the paper, but less strongly, since somebody reading Lemma 4.5 and finding unexplained notation will at least know to look elsewhere in the paper for its meaning (provided the notation doesn't already have some other generally accepted meaning). For this reason, one need not put a reference to the notation everywhere it is used, but one should still put references to it reasonably often. Fifth, here are a few examples of grammatical mistakes and ugly phrases which one should avoid. Run-on sentences: "We have $x > 0$, therefore $x^3 > 0$." is an example of a run-on sentence: two sentences strung together with a comma. The following versions are correct: "We have $x > 0$. Therefore $x^3 > 0$." "We have $x > 0$; therefore $x^3 > 0$." A very commonly seen wrong example is: "All nuclear C*-algebras are exact, see [17]." This needs to be rewritten as "All nuclear C*-algebras are exact. See [17]." or as "All nuclear C*-algebras are exact; see [17].". (In some other languages, run-on sentences are allowed. But not in English.) "We have that": This phrase is ugly and probably grammatically wrong. Many places I see it, one should simply delete "that". Thus, "We have that $x > 0$." is corrected to read "We have $x > 0$.". Other places people want to use this construction need more rewriting to avoid. Missing "let": "Let $\varepsilon > 0$, $A$ a C*-algebra, $a \in A$" is corrected to read: "Let $\varepsilon > 0$, let $A$ be a C*-algebra, and let $a \in A$". Don't omit "let", "be", or "and". "For all $0 < a < b$": This phrase means, "for all (objects called) $0$ which satisfy $0 < a < b$", which is not what is intended. If $b$ is given, it must be "for all $a$ such that $0 < a < b$", or, here, "for all $a \in (0, b)$". If neither $a$ nor $b$ is given, then write, "for all $a$ and $b$ such that $0 < a < b$". Generally, if you have "for all" or "there is", that phrase applies to the immediate next symbol. Thus, "for all $a < b$" says, among other things, that the reader is supposed to already know what $b$ is. Another example: If $x$ is given, you can't write, "there exists $x \in U$ such that ..." or "there exists a neighborhood $x \in U$ such that ...". It must be, "there exists a neighborhood $U$ of $x$ such that ...", or (again a special situation; the analog of this doesn't always exist) "there exists a neighborhood $U \ni x$ such that ..." (with backwards "$\in$"). No hyphens with numbers. The phrases, "let $x_1, x_2, \ldots, x_n$ be $n$ points in $X$" and "let $x_1, x_2, \ldots, x_n$ be $n$ distinct points in $X$" are correct as written; adding a hyphen after "$n$" gives the wrong versions, "let $x_1, x_2, \ldots, x_n$ be $n$-points in $X$" and "let $x_1, x_2, \ldots, x_n$ be $n$-distinct points in $X$". For comparison with ordinary language, "I saw seven animals" and "I saw seven large animals" are correct, but "I saw seven-animals" and "I saw seven-large animals" are wrong. Makes sure to quantify over the correct variable. If $f \colon X \to Y$ and $C \subset Y$, you can't introduce $x \in X$ by saying "Let $f (x) \in C$." It is similarly incorrect to write "Every $x_n$" is positive." when you mean "$x_n$ is positive for every $n$." I see this done a lot. Often this construction is harmless, but occasionally it leads to disaster. For example, I have seen false proofs that $f (A) \cap f (B) \subset f (A \cap B)$ in which incorrect quantification of this kind was the main error. Reasoning should not appear in the _statement_ of a lemma, theorem, proposition, corollary, or definition. (Goss mentions this briefly.) Don't write: Theorem 4.5. Let $A$ be a unital C*-algebra. Suppose ... . Then $A$ has a faithful tracial state, and is therefore stably finite. This is corrected to read: Theorem 4.5. Let $A$ be a unital C*-algebra. Suppose ... . Then: (1) The algebra $A$ has a faithful tracial state. (2) The algebra $A$ is stably finite. Or: Theorem 4.5. Let $A$ be a unital C*-algebra. Suppose ... . Then $A$ has a faithful tracial state. Proof. ... QED. Corollary 4.6. Let $A$ be as in Theorem 4.5. Then $A$ is stably finite. It is, however, standard to have reasoning in remarks and examples. Use one numbering system throughout. Thus: Definition 2.1. Lemma 2.2. Corollary 2.3. Lemma 2.4 Lemma 2.5 Definition 2.6 ... Theorem 2.14. The following numbering scheme seems more logical, but in practice makes it very hard to find things in the paper (especially the theorem): Definition 2.1. Lemma 2.1. Corollary 2.1. Lemma 2.2 Lemma 2.3 Definition 2.2 ... Theorem 2.1. (I prefer to write "2.14 Theorem" instead of "Theorem 2.14", but this seems not to be common.)