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Author: William A. Stein
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6{\bf Some Hints on Mathematical Style}
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8
9David Goss
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13Many years ago, just after my degree,
14I had the good fortune to be given some hints on
15mathematical writing by J.-P. Serre. Through the years I have found myself
16trying to repeat this very sound advice to other mathematicians who are
17also starting out. Recently, I have been involved in the publishing of a
18proceedings volume, as well as being an editor of the Journal of Number
19Theory. Many of the papers coming my way are from young authors; so
20I have written down these hints in order to speed the process along.
21
22This is a second (and, most probably, final)
23version of these hints''. I have added comments from
24a number of mathematicians who read a first version.
25These hints are presented as a source of ideas on mathematical style. Feel
26free to use them in any way that you deem useful.
27
28\begin{itemize}
29\item
30Two basic rules are: 1). {\em Have mercy on the reader}, and, 2). {\em
31Have mercy on the editor/publisher}. We will illustrate these
32as we move along.
33
34\item
35General Flow of the Paper.
36	\begin{itemize}
37	\item
38	{\bf Definition}: {\em All} basic definitions should be given
39	if they are not a standard part of the literature. It is perhaps
40	best to err on the side of making life easier on the reader by
41	including a bit too much as opposed to too little (Rule 1).
42		\begin{itemize}
43		\item
44		Some redundancy should be built into the paper so that
45		one or two misprints cannot destroy the understandability.
46		The analogy is with error-correcting codes''
47		which allow transmission of information through noisy
48		and defective channels.
49		\end{itemize}
50
51	\item
52	As a very general rule, the definitions should go {\em before}
53	the results that they are used in (Rule 1).
54	\item
55	The quantifiers'' should always be clear (Rule 1). Some examples to
56	avoid:
57		\begin{itemize}
58		\item
59		We have $f(x)=g(x)$
60		($x\in X$).'' What does the parenthesis mean? That
61		$f(x)=g(x)$ for {\em all} $x\in X$, or, for {\em some} $x\in 62 X$?
63		\item
64		What does $\displaystyle f_{t,u}(x,y)=O\left(g_{t,u}(x,y) 65 \right)$'' mean? Very often it means that $t,u,y$ are
66		fixed and $x$ is allowed to vary. Quantifier problems
67		are especially troublesome with big O'' notation.
68		\item
69		The word constant'' is terribly ambiguous. It is
70		important to make explicit {\em exactly} which
71		variables the constant depends on.
72		\end{itemize}
73	\item
74	{\bf Theorem/Proposition/Lemma/Corollary}: Give clear and
75	unambiguous statements of results. These are what other people
76	are reading your paper for; so you should ensure that these, at
77	least, can be understood by the reader (Rule 1).
78		\begin{itemize}
79		\item
80		The statement of the Theorem/Proposition/Lemma/Corollary
81		should {\bf not} include comments (except for an
82		occasional brief remark in parenthesis) or examples.
83		\end{itemize}
84	\item
85	If you use or quote an important result of another person, you should
86	give a reference. You should avoid giving the impression that
87	such a result is obvious, a generally accepted fact, due to
88	you, and so on.
89		\begin{itemize}
90		\item
91		A reference to a book should always give the page!
92		\item
93		Try to avoid using by the proof of'' when the proof
94		is in the paper and the statements can be rewritten
95		to be {\em directly} quoted.
96		\item
97		A well-known'' result that is {\em not} in the literature
98		should be proved if needed (Rule 1).
99		\end{itemize}
100	\item
101	{\bf Proof}: A proof should give enough information to make the
102	theorem believable {\bf and} leave the reader with the confidence
103	(as well as the ability) to fill in details should it be
104	necessary (Rule 1).
105	\end{itemize}
106\item
108	\begin{itemize}
109	\item
110	One should, of course, observe the usual conventions in terms
111	of spelling, punctuation, and the other basic elements of style.
112	Use complete sentences, with subject, {\em verb},
113	and complement (Rule 1).
114		\begin{itemize}
115		\item
116		A verb should {\bf not} be replaced by a symbol. It is
117		bad to write: ... if $x=2$, $y=3$, $z=4$'' meaning
118		... if $x=2$ and $y=3$, then $z$ is equal to 4''.
119		\item
120		It is also bad to write: ... we prove $\zeta_{\bf Q}(2n) 121 \in \pi^{2n}\bf Q$'' instead of: ... we prove {\em that}
122		$\displaystyle \frac{\zeta_{\bf Q}(2n)}{\pi^{2n}}$ belongs
123		to $\bf Q$'' (or is rational'').
124		\end{itemize}
125	\item
126	Use the present --  not the past -- form.
127		\begin{itemize}
128		\item
129		As an example of bad writing, we have: We have proved
130		that $f(x)$ was equal to $g(x)$...''. This is corrected to:
131		We have proved that $f(x)$ is equal to $g(x)$...''.
132		\end{itemize}
133	\item
134	Long computations that can easily be carried out by the reader should
135	be avoided. The ideas and results should be given with an invitation to
136	the reader to do the calculation should it be desired (Rule 1).
137		\begin{itemize}
138		\item
139		The {\em exception} to this rule is when the long
140		computation is an {\em essential} part of the argument.
141		In this case, it should be given in full (Rule 1).
142		\end{itemize}
143	\item
144	One should avoid giving the reader the impression that the
145	subject matter can be mastered only with great pain. In fact, this
146	is an {\em ideal} way to lose readers (or audiences!).
147	\item
148	One should avoid using abbreviations like w.r.t.'' (with
149	respect to), iff'' (if and only if),
150        and w.l.o.g.'' (without loss of generality). They
151	simply do not look very nice (and iff'' is offensive! --
152 	Rules 1 and 2).
153	\item
154	You should {\bf not} begin a sentence with a math symbol.
155	This can confuse the printer as well as the reader (Rules 1 and 2).
156		\begin{itemize}
157		\item
158		As a example of such bad writing, we have: ... we want to
159		prove the continuity of $f(x)=2\cos^2 x\cdot\sin x$.  $\cos x$
160		being continuous....''. This is corrected to:
161		...$f(x)=2\cos^2 x\cdot \sin x$. Since $\cos x$ is
162		continuous...''.
163		\end{itemize}
164	\item
165	If your paper raises a natural question, and you don't know the
166	answer, by all means {\em say so}! This may turn out to be
167	more interesting than the theorems that you prove.
168		\begin{itemize}
169		\item
170		Conversely, refrain from making conjectures'' too
171		hastily. Use instead the words question'' or problem''.
172		Remember that a good question'' should be answerable by
173		yes'' or no''. To ask under what conditions does A
174		hold'' is not a question worth printing.
175		\end{itemize}
176	\item
177	It is often helpful to begin a new section of the paper with
178	a summary of the general setting.
179	\item
180	After the paper is finished, it should be reread (and, perhaps,
181	rewritten) from the reader's point of view (Rule 1).
182	\item
183	A good way to begin is to use a standard classic of
184	mathematical exposition (e.g., Bourbaki-Algebra, works
185	by Serre or Milnor) as a basic model.
186	\end{itemize}
187\item
188Some further sources to look at:
189	\begin{itemize}
190	\item
191	P. Halmos: {\it How to write mathematics}, Enseign. Math.,
192	{\bf 16}, (1970), 123--152.
193	\item
194	 W. Strunk, Jr., \& E. B. White: {\it The Elements of Style},
195	Macmillan Paperbacks Edition, (1962).
196	\item
197	D. Knuth et al.: {\it Mathematical Writing}, MAA Notes \#14, (1989).
198	\item
199	Some conventions on citations and pronouns may be found in:
200	S. Zucker: {\it Variation of a mixed Hodge structure II},
201	Inventiones Math. 80, (1985), p. 545.
202	\end{itemize}
203
204
205\item
206Finally, I quote from a letter Serre wrote commenting on my original
207version: It strikes me that mathematical writing is similar to using
208a language. To be understood you have to follow some grammatical rules.
209However, in our case,
210nobody has taken the trouble of writing down the grammar; we get it as a
211baby does from parents, by imitation of others. Some mathematicians have
212a good ear; some not (and some prefer the slangy expressions such
213as iff''). That's life.''
214\end{itemize}
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