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
(http://www.math.ohio-state.edu/~goss/hint.pdf). I have only a
little to add to this--for now, just a few topics.
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).
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, 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.