Some Information on Writing a Mathematical Paper

Note: There is an improved (clearer, better examples) form of this page available here

References on writing

The following two pamphlets contain information on the writing of mathematical papers. There are copies in the library for your reference. Please refer to them.

- Writing Mathematics Well by Leonard Gillman
- How to Write Mathematics by Paul Halmos

Some general guidelines in writing a mathematical paper

1. Organization

a. Crediting sources:

Footnotes are not ordinarily used in writing mathematics. To cite a reference, do it in the following way:

It has been shown that there is a relationship between the number of evaluations per step in the Runge-Kutte methods and the order of the local truncation error [5,p.56].
(Here [5] is the number of the reference as listed in your bibliography.)

The source for a particular statement of a theorem or definitiion is usually cited immediately following the title "Theorem" or "Definition" (before the statement); similarly for examples and proofs. The source is not usually cited for definitions, theorems, etc. that are in common mathematical use (even if they are new to the author of the paper)

b. Numbering systems:

Major statements - theorems, definitions, examples - are numbered consecutively within chapters. The preferred style numbers all statements in order (as distinct from the style in which theorems are numbered separately from definitions, etc.)

i) Number the chapters of your paper as 1,2,3,etc. Then the theorems, definitions, and examples in section one are numbered serially beginning with 1.1,1.2, etc. They would appear as:

1.1 Theorem......
1.2 Definition......
1.3 Example.......

ii) Number any formulas or displays (tables, diagrams) to which you refer. The number may appear in the left- or right-hand margin, but be consistent throughout your paper.

2. Writing Style

a. Do not begin a sentence with a symbol. For example, instead of saying "G is an abelian group," write "The group G is abelian."
b. One generally uses "we" instead of "I" when describing your plan, etc.
c. State a theorem before proving it.
d. State theorems concisely.
e. It is often helpful to provide an example of a theorem before proving it.
f. If a proof is involved, give the underlying idea before plunging into the details.
g. Use the active voice in preference to the passive voice.
h. Use examples liberally.
i. In writing formal mathematics, avoid a mere list of definitions and theorems; provide examples and exposition between the definitions and theorems. As a guide in writing this exposition, consider answering one or more of the following questions:

1) what is the purpose of this definition?
2) what does this theorem or definition really mean?
3) what is the power of this theorem?
4) what is the crux of the proof of the theorem?
5) how is a certain lemma used to prove this theorem?
6) is this theorem a step towards a major result or is it the major result?
7) how does one apply this theorem?
8) why are the hypotheses of this theorem so important?
9) is this theorem the strongest possible result?
10) does the converse of this theorem hold?

3. Text formatting

a. Use standard and accurate notation. For example, the symbol R is commonly understood to represent the reals, C the complex numbers, etc.
b. Theorems, definitions, examples are "displayed" - beginning a new paragraph (not usually indented, but preceded by a blank line), numbered, with the label in bold and underlined
c. Put a long or complicated expression (equations, formulas, etc.) on a line by itself.
d. Leave a double space between symbols and adjacent text.

  e.g. "Let  z  be a complex number" as opposed to "Let z be a complex
   number.
  e.g. "The set  O  is open in  G " as opposed to "The set O is open in G""
  e.g. "The function  f  is differentiable" as opposed to "The function
   f is differentiable."

e. Mark the ends of proofs either by the classic "QED" or some symbol such as .

4. Bibliography - Follow the form below:

The examples which follow show, in order, the formats for a journal article, an article from a book, and a book. Notice that the items are listed in alphabetical order based on the author's last name (underlining may be used in place of italics)

[1] R.P. Boas, Can We Make Mathematics Intelligible?, Amer. Math. Monthly 88(1981), 727-731.
[2] Paul R. Halmos, How to Write Mathematics, in Selecta, Expository Writing, Springer, 1983, 157-186.
[3] John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, 1947.