%Use Latex1\documentstyle[]{article}23\begin{document}4\begin{center}5{\bf Some Hints on Mathematical Style}6\medskip78David Goss9\end{center}10\bigskip1112Many years ago, just after my degree,13I had the good fortune to be given some hints on14mathematical writing by J.-P. Serre. Through the years I have found myself15trying to repeat this very sound advice to other mathematicians who are16also starting out. Recently, I have been involved in the publishing of a17proceedings volume, as well as being an editor of the Journal of Number18Theory. Many of the papers coming my way are from young authors; so19I have written down these hints in order to speed the process along.2021This is a second (and, most probably, final)22version of these ``hints''. I have added comments from23a number of mathematicians who read a first version.24These hints are presented as a source of ideas on mathematical style. Feel25free to use them in any way that you deem useful.2627\begin{itemize}28\item29Two basic rules are: 1). {\em Have mercy on the reader}, and, 2). {\em30Have mercy on the editor/publisher}. We will illustrate these31as we move along.3233\item34General Flow of the Paper.35\begin{itemize}36\item37{\bf Definition}: {\em All} basic definitions should be given38if they are not a standard part of the literature. It is perhaps39best to err on the side of making life easier on the reader by40including a bit too much as opposed to too little (Rule 1).41\begin{itemize}42\item43Some redundancy should be built into the paper so that44one or two misprints cannot destroy the understandability.45The analogy is with ``error-correcting codes''46which allow transmission of information through noisy47and defective channels.48\end{itemize}4950\item51As a very general rule, the definitions should go {\em before}52the results that they are used in (Rule 1).53\item54The ``quantifiers'' should always be clear (Rule 1). Some examples to55avoid:56\begin{itemize}57\item58``We have $f(x)=g(x)$59($x\in X$).'' What does the parenthesis mean? That60$f(x)=g(x)$ for {\em all} $x\in X$, or, for {\em some} $x\in61X$?62\item63What does ``$\displaystyle f_{t,u}(x,y)=O\left(g_{t,u}(x,y)64\right)$'' mean? Very often it means that $t,u,y$ are65fixed and $x$ is allowed to vary. Quantifier problems66are especially troublesome with ``big O'' notation.67\item68The word ``constant'' is terribly ambiguous. It is69important to make explicit {\em exactly} which70variables the constant depends on.71\end{itemize}72\item73{\bf Theorem/Proposition/Lemma/Corollary}: Give clear and74unambiguous statements of results. These are what other people75are reading your paper for; so you should ensure that these, at76least, can be understood by the reader (Rule 1).77\begin{itemize}78\item79The statement of the Theorem/Proposition/Lemma/Corollary80should {\bf not} include comments (except for an81occasional brief remark in parenthesis) or examples.82\end{itemize}83\item84If you use or quote an important result of another person, you should85give a reference. You should avoid giving the impression that86such a result is obvious, a generally accepted fact, due to87you, and so on.88\begin{itemize}89\item90A reference to a book should always give the page!91\item92Try to avoid using ``by the proof of'' when the proof93is in the paper and the statements can be rewritten94to be {\em directly} quoted.95\item96A ``well-known'' result that is {\em not} in the literature97should be proved if needed (Rule 1).98\end{itemize}99\item100{\bf Proof}: A proof should give enough information to make the101theorem believable {\bf and} leave the reader with the confidence102(as well as the ability) to fill in details should it be103necessary (Rule 1).104\end{itemize}105\item106Other comments:107\begin{itemize}108\item109One should, of course, observe the usual conventions in terms110of spelling, punctuation, and the other basic elements of style.111Use complete sentences, with subject, {\em verb},112and complement (Rule 1).113\begin{itemize}114\item115A verb should {\bf not} be replaced by a symbol. It is116bad to write: ``... if $x=2$, $y=3$, $z=4$'' meaning117``... if $x=2$ and $y=3$, then $z$ is equal to 4''.118\item119It is also bad to write: ``... we prove $\zeta_{\bf Q}(2n)120\in \pi^{2n}\bf Q$'' instead of: ``... we prove {\em that}121$\displaystyle \frac{\zeta_{\bf Q}(2n)}{\pi^{2n}}$ belongs122to $\bf Q$'' (or ``is rational'').123\end{itemize}124\item125Use the present -- not the past -- form.126\begin{itemize}127\item128As an example of bad writing, we have: ``We have proved129that $f(x)$ was equal to $g(x)$...''. This is corrected to:130``We have proved that $f(x)$ is equal to $g(x)$...''.131\end{itemize}132\item133Long computations that can easily be carried out by the reader should134be avoided. The ideas and results should be given with an invitation to135the reader to do the calculation should it be desired (Rule 1).136\begin{itemize}137\item138The {\em exception} to this rule is when the long139computation is an {\em essential} part of the argument.140In this case, it should be given in full (Rule 1).141\end{itemize}142\item143One should avoid giving the reader the impression that the144subject matter can be mastered only with great pain. In fact, this145is an {\em ideal} way to lose readers (or audiences!).146\item147One should avoid using abbreviations like ``w.r.t.'' (with148respect to), ``iff'' (if and only if),149and ``w.l.o.g.'' (without loss of generality). They150simply do not look very nice (and ``iff'' is offensive! --151Rules 1 and 2).152\item153You should {\bf not} begin a sentence with a math symbol.154This can confuse the printer as well as the reader (Rules 1 and 2).155\begin{itemize}156\item157As a example of such bad writing, we have: ``... we want to158prove the continuity of $f(x)=2\cos^2 x\cdot\sin x$. $\cos x$159being continuous....''. This is corrected to:160``...$f(x)=2\cos^2 x\cdot \sin x$. Since $\cos x$ is161continuous...''.162\end{itemize}163\item164If your paper raises a natural question, and you don't know the165answer, by all means {\em say so}! This may turn out to be166more interesting than the theorems that you prove.167\begin{itemize}168\item169Conversely, refrain from making ``conjectures'' too170hastily. Use instead the words ``question'' or ``problem''.171Remember that a good ``question'' should be answerable by172``yes'' or ``no''. To ask ``under what conditions does A173hold'' is not a question worth printing.174\end{itemize}175\item176It is often helpful to begin a new section of the paper with177a summary of the general setting.178\item179After the paper is finished, it should be reread (and, perhaps,180rewritten) from the reader's point of view (Rule 1).181\item182A good way to begin is to use a standard classic of183mathematical exposition (e.g., Bourbaki-Algebra, works184by Serre or Milnor) as a basic model.185\end{itemize}186\item187Some further sources to look at:188\begin{itemize}189\item190P. Halmos: {\it How to write mathematics}, Enseign. Math.,191{\bf 16}, (1970), 123--152.192\item193W. Strunk, Jr., \& E. B. White: {\it The Elements of Style},194Macmillan Paperbacks Edition, (1962).195\item196D. Knuth et al.: {\it Mathematical Writing}, MAA Notes \#14, (1989).197\item198Some conventions on citations and pronouns may be found in:199S. Zucker: {\it Variation of a mixed Hodge structure II},200Inventiones Math. 80, (1985), p. 545.201\end{itemize}202203204\item205Finally, I quote from a letter Serre wrote commenting on my original206version: ``It strikes me that mathematical writing is similar to using207a language. To be understood you have to follow some grammatical rules.208However, in our case,209nobody has taken the trouble of writing down the grammar; we get it as a210baby does from parents, by imitation of others. Some mathematicians have211a good ear; some not (and some prefer the slangy expressions such212as ``iff''). That's life.''213\end{itemize}214\end{document}215216217218219220221222223224225226227228229230231232233