Sharedwww / tables / Notes / goss.texOpen in CoCalc
Author: William A. Stein
1
%Use Latex
2
\documentstyle[]{article}
3
4
\begin{document}
5
\begin{center}
6
{\bf Some Hints on Mathematical Style}
7
\medskip
8
9
David Goss
10
\end{center}
11
\bigskip
12
13
Many years ago, just after my degree,
14
I had the good fortune to be given some hints on
15
mathematical writing by J.-P. Serre. Through the years I have found myself
16
trying to repeat this very sound advice to other mathematicians who are
17
also starting out. Recently, I have been involved in the publishing of a
18
proceedings volume, as well as being an editor of the Journal of Number
19
Theory. Many of the papers coming my way are from young authors; so
20
I have written down these hints in order to speed the process along.
21
22
This is a second (and, most probably, final)
23
version of these ``hints''. I have added comments from
24
a number of mathematicians who read a first version.
25
These hints are presented as a source of ideas on mathematical style. Feel
26
free to use them in any way that you deem useful.
27
28
\begin{itemize}
29
\item
30
Two basic rules are: 1). {\em Have mercy on the reader}, and, 2). {\em
31
Have mercy on the editor/publisher}. We will illustrate these
32
as we move along.
33
34
\item
35
General 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
107
Other comments:
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
188
Some 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
206
Finally, I quote from a letter Serre wrote commenting on my original
207
version: ``It strikes me that mathematical writing is similar to using
208
a language. To be understood you have to follow some grammatical rules.
209
However, in our case,
210
nobody has taken the trouble of writing down the grammar; we get it as a
211
baby does from parents, by imitation of others. Some mathematicians have
212
a good ear; some not (and some prefer the slangy expressions such
213
as ``iff''). That's life.''
214
\end{itemize}
215
\end{document}
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233