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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; so
I have written down these hints in order to speed the process along.

This is a second (and, most probably, final)
version of these ``hints.'' I have added comments from
a number of mathematicians who read a first version.
These hints are presented as a source of ideas on mathematical style. Feel
free to use them in any way that you deem useful.

\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).
		\begin{itemize}
		\item
		Some redundancy should be built into the paper so that
		one or two misprints cannot destroy the understandability. 
		The analogy is with ``error-correcting codes'' 
		which allow transmission of information through noisy
		and defective channels.
		\end{itemize}
		
	\item
	As a very general rule, the definitions should go {\em before}
	the results that 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. 
		\begin{itemize}
		\item
		A reference to a book should always give the page!
		\item
		Try to avoid using ``by the proof of'' when the proof
		is in the paper and the statements can be rewritten
		to be {\em directly} quoted.
		\item
		A ``well-known'' result that is {\em not} in the literature
		should be proved if needed (Rule 1).
		\end{itemize}
	\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.
	Use complete sentences, with subject, {\em verb},
	and complement (Rule 1).
		\begin{itemize}
		\item
		A verb should {\bf not} be replaced by a symbol. It is 
		bad to write: ``... if $x=2$, $y=3$, $z=4$'' meaning
		``... if $x=2$ and $y=3$, then $z$ is equal to $4$.''
		\item
		It is also bad to write: ``... we prove $\zeta_{\bf Q}(2n)
		\in \pi^{2n}\bf Q$'' instead of: ``... we prove {\em that}
		$\displaystyle \frac{\zeta_{\bf Q}(2n)}{\pi^{2n}}$ belongs
		to $\bf Q$'' (or ``is rational'').
		\end{itemize}
	\item
	Use the present --  not the past -- form.
		\begin{itemize}
		\item
		As an example of bad writing, we have: ``We have proved
		that $f(x)$ was equal to $g(x)$...''. This is corrected to:
		``We have proved that $f(x)$ is equal to $g(x)$...''.
		\end{itemize}
	\item
	Long computations that can easily be 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).
		\begin{itemize}
		\item
		The {\em exception} to this rule is when the long
		computation is an {\em essential} part of the argument.
		In this case, it should be given in full (Rule 1).
		\end{itemize}
	\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 (and ``iff'' is offensive! --
 	Rules 1 and 2).
	\item
	You should {\bf not} begin a sentence with a math symbol. 
	This can confuse the printer as well as the reader (Rules 1 and 2).
		\begin{itemize}
		\item
		As a example of such bad writing, we have: ``... we want to
		prove the continuity of $f(x)=2\cos^2 x\cdot\sin x$.  $\cos x$
		being continuous....''. This is corrected to: 
		``...$f(x)=2\cos^2 x\cdot \sin x$. Since $\cos x$ is
		continuous...''.
		\end{itemize}
	\item
	If your paper raises a natural question, and you don't know the
	answer, by all means {\em say so}! This may turn out to be
	more interesting than the theorems that you prove.
		\begin{itemize}
		\item
		Conversely, refrain from making ``conjectures'' too
		hastily. Use instead the words ``question'' or ``problem.''
		Remember that a good ``question'' should be answerable by
		``yes'' or ``no.'' To ask ``under what conditions does A
		hold'' is not a question worth printing.
		\end{itemize}
	\item
	It is often helpful to begin a new section of the paper with
	a summary of the general setting.
	\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
Some further sources to look at:
	\begin{itemize}
	\item
	P. Halmos: {\it How to write mathematics}, Enseign. Math.
	{\bf 16} (1970), 123--152.
	\item
	 W. Strunk, Jr., \& E. B. White: {\it The Elements of Style},
	Macmillan Paperbacks Edition, (1962).
	\item
	D. Knuth et al.: {\it Mathematical Writing}, MAA Notes \#14 (1989).
	\item
	Some conventions on citations and pronouns may be found in:
	S. Zucker: {\it Variation of a mixed Hodge structure II},
	Invent. Math. {\bf 80} (1985), p. 545.
	\end{itemize}

	
\item
Finally, I quote from a 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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