Writing About Mathematics,
             adapted, with permission, from
                 SIAM News, June 1996

                  by Gilbert Strang

This is my first attempt to write about writing. There is a
good SIAM book on this subject by Nick Higham, and this article adds
only a few suggestions. Some of the suggestions are almost tricks,
and I hope they will help. Others go deeper because writing is very
hard work---if you seriously want to be read and understood. Maybe
that is the point of this article.

In all these suggestions, I assume that the mathematical ideas have
been thought through---there is something to say. But the *saying*
is distinct from the *thinking*. To get the equations and proofs
correct is absolutely not the final step. You have to move beyond
the research, where the goal is insight. In writing, the goals are
less abstract and more human:

1. Make the paper *interesting*.
2. Connect to what the reader already knows.
3. Find a good way to start.
4. Keep some parts simple---as simple as speech.

Some writing is only for the record. I am not speaking about
that style at all. This article is about communication to a reader,
not storage in a journal. If you want the attention of readers, you
have to give them *your* attention. What do they already know, and
how does the new part fit in? This comes naturally with speech, so
I often say the words aloud. That tends to keep the sentences

The enemy of good writing is not lack of space. It is lack of time
and energy and patience. Sometimes the ideas flow easily into words,
sometimes they get stuck and won't move. I try to write when I feel
psychologically strong (these words are written in the early
morning; I am impressed by anyone who can keep writing after
dinner). Above all, remember the reader.

        A Few Tricks

Creating a string of equations is easy, and no good.
It's much better to express your idea twice, and in two ways--
first in words and then in the equation.

Innocent bad example:
The angle between two subspaces is inf  cos^-1 |<u,v>|/ ||u|| ||v||.

Better to have said it in words too:

The angle between two subspaces is the smallest angle between
vectors in the subspaces:  inf ....

Tell the reader what the damn equation means! That person has
other things to do. Given half a chance, he or she will stop reading
your article. You must grab the reader's attention and hold on. If
the point is not clearly made, and decoding the notation requires an
effort, phooey on the rest (unless you are proving Fermat's Last

The notation maps ideas into symbols. If the notation is good, the
reader won't keep searching back to the start of the paper.
Just add the words the reader needs:

Poor: Analogous to (3.1.5), if we take the union of the bases
in (3.1.12), N=1,2,..., we arrive at (3.1.14).

Better: We can include the following simple proof of (3.1.14).
Best: Just say it: We can quickly show that T is bounded.
Another useful device has been hiding in these paragraphs (and
in this sentence). I think it is called anthropomorphism.
An inanimate and abstract concept is assigned human properties.
It hides in a paragraph.  It gets
stuck and won't move. This is nonsense, of course. An idea can't
do such things, it just sits there. I can't stop doing this....

Humanization is a simple way to write more actively. It definitely
adds life to the paper.

         The First Paragraph

The first words are a signal to the reader, green or red. Work
on those important words. Read them aloud, their rhythm is important.
A mixture of long and short sentences will help. You
absolutely must find an interesting way to start.

With a specific example, I can try to show in detail
how the pieces might fit. The nearest example is the beginning of this
article. May I ask you to look back at the first paragraph? We can
analyze it and improve it.

The first sentence has nine words and no commas. It introduces the
subject: "to write about writing." In a sense it also introduces the
author: "This is my first attempt." The first-person adjective *my*
is informal and personal (not always appropriate). The important point is
that you can not only read those nine words, *you can hear them*.
Living speech is extra powerful--- it is impossible to
keep up such strength in writing.

In that opening paragraph, notice the word *suggestions*. The
repetition of that word connects everything. *A few suggestions* are
separated into *some of the suggestions* and *others*. It is like
multiresolution, moving from the broad topic (writing) to the
specific contents (suggestions) and then to the details.

The last sentence of the paragraph is suitably short, but too weak.
A purist would complain about the vague reference to *that*. Better to
repeat the key idea: "Maybe serious work is
the point of this article." Now I have to think again about that
gentle word *maybe*.

At this rate we won't finish! This isn't a poem, it is just a
little article about writing. Every word counts, but the
first words count the most. A typical mathematics paper will change
gear, as it goes from introduction to exposition. For this main
part, where the research is described, I add only one new thought:

1. Equations need a phrase to explain them.
2. A single word can recall the notation.
3. It is all right to pretend that ideas are alive.
4. A list that is numbered and indented gives the eye a break.

A list of more than four items is stretching it. I will end with
with two very small suggestions, which you can safely ignore

          Two Grumbles

"The components y_i, i = 1,2,...,n, are all positive." I bet
that i = 1,2,...,n is obvious from the context. Stating it just
interferes with the reader's thinking.

I don't think that L^2 and a Besov space should be defined in the
same paper. If L^2 is not already an old friend, there is no point
whatsoever to introducing B (sub s, super p,k).
I believe that definitions of L^2 should be
forbidden in research papers.

I can't end with grumbles. Maybe
unhappiness is good for novelists, but mathematics is basically
optimistic. We are writing for friends, and they really want to
understand. *Talk to them as you write*!

Gilbert Strang, a professor of mathematics at MIT, is the author of six textbooks, most recently Wavelets and Filter Banks. He has also written (and spoken) about wavelets for SIAM; his 1989 paper in SIAM Review is often held up as a model of expository writing.

[more explanation]

Cameron Laird makes this article available as one of a series [give reference] on expository writing/claird@phaseit.net