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{\bf Some Hints on Mathematical Style}
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David Goss
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Many years ago, just after my degree,
I had the good fortune to be given some hints on
mathematical writing by J.-P. Serre. Through the years I have found myself
trying to repeat this very sound advice to other mathematicians who are
also starting out. Recently, I have been involved in the publishing of a
proceedings volume, as well as being an editor of the Journal of Number
Theory. Many of the papers coming my way are from young authors, and so 
the time seemed right to have a prepared copy of Serre's hints to speed
things along.

Memory can be a fickle thing. So I have asked Serre to help me with this
little undertaking and he has agreed (else I would not have felt confident
enough to start the project). 
He read an early version of these ``hints'' and added a few new ones.

Feel free to use these comments in any way that you would like. If you have
some {\em serious} additions to the list, please let me know of them.

\begin{itemize}
\item
Two basic rules are: 1). {\em Have mercy on the reader}, and 2). {\em
Have mercy on the editor/publisher}. We will illustrate these
 as we move along. 

\item
General Flow of the Paper.
	\begin{itemize}
	\item
	{\bf Definition}: {\em All} basic definitions should be given
	if they are not a standard part of the literature. It is perhaps
	best to err on the side of making life easier on the reader by 
	including a bit too much as opposed to too little (Rule 1).
	\item
	As a very general rule, the definitions should go {\em before}
	the results they are used in (Rule 1).
	\item
	The ``quantifiers'' should always be clear (Rule 1). Some examples to
	avoid:
		\begin{itemize}
		\item
		``We have $f(x)=g(x)$
		($x\in X$).'' What does the parenthesis mean? That
		$f(x)=g(x)$ for {\em all} $x\in X$, or, for {\em some} $x\in
		X$?
		\item
		What does $\displaystyle f_{t,u}(x,y)=O\left(g_{t,u}(x,y)
		\right)$ mean? Very often it means that $t,u,y$ are
		fixed and $x$ is allowed to vary. Quantifier problems
		are especially troublesome with ``big O'' notation.
		\item
		The word ``constant'' is terribly ambiguous. It is
		important to make explicit {\em exactly} which
		variables the constant depends on.
		\end{itemize}
	\item
	{\bf Theorem/Proposition/Lemma/Corollary}: Give clear and
	unambiguous statements of results. These are what other people
	are reading your paper for; so you should ensure that these, at
	least, can be understood by the reader (Rule 1).
		\begin{itemize}
		\item	
		The statement of the Theorem/Proposition/Lemma/Corollary
		should {\bf not} include comments (except for an
		occasional brief remark in parenthesis) or examples.
		\end{itemize}
	\item
	If you use or quote an important result of another person, you should
	give a reference. You should avoid giving the impression that
	such a result is obvious, a generally accepted fact, due to
	you, and so on.
	\item
	{\bf Proof}: A proof should give enough information to make the
	theorem believable {\bf and} leave the reader with the confidence
	(as well as the ability) to fill in details should it be
	necessary (Rule 1).
	\end{itemize}
\item
Other comments:
	\begin{itemize}
	\item
	One should, of course, observe the usual conventions in terms
	of spelling, punctuation, and the other basic elements of style.
	\item
	Long computations that can easily carried out by the reader should
	be avoided. The ideas and results should be given with an invitation to
	the reader to do the calculation should it be desired (Rule 1).
	\item
	The paper should be written in as compact a style as possible. 
	Readers do not like to be subjected to computations that they
	consider obvious nor do journals like to waste pages (Rules 
	1 and 2).
	\item
	One should avoid giving the reader the impression that the
	subject matter can be mastered only with great pain. In fact, this
	is an {\em ideal} way to lose readers (or audiences!).
	\item
	One should avoid using abbreviations like ``w.r.t.'' (with
	respect to), ``iff'' (if and only if)
        and ``w.l.o.g.'' (without loss of generality). They
	simply do not look very nice (Rules 1 and 2).
	\item
	You should strive {\bf not} to begin a sentence with a math symbol. 
	This can confuse the printer (Rule 2).	
	\item
	After the paper is finished, it should be reread (and, perhaps,
	rewritten) from the reader's point of view (Rule 1).
	\item
	A good way to begin is to use a standard classic of
	mathematical exposition (e.g., Bourbaki-Algebra, works
	by Serre or Milnor) as a basic model.
	\end{itemize}
\item
Finally, I quote from the letter Serre wrote commenting on my original 
version: ``It strikes me that mathematical writing is similar to using 
a language. To be understood you have to follow some grammatical rules. 
However, in our case,
nobody has taken the trouble of writing down the grammar; we get it as a
baby does from parents, by imitation of others. Some mathematicians have
a good ear; some not (and some prefer the slangy expressions such
as ``iff''). That's life.''
\end{itemize}
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