Writing Guide
Where To Start
A good place to begin learning to write mathematics is Halmos's well known article:
This article gives a very broad basis for writing in mathematics. It is a classic reference for the writing of mathematics.
It would be worth your time to read this paper before beginning to write mathematics.
Elementary Suggestions
Below I shall give some very elementary suggestions for novice writers of mathematics.
The first advice given to any writer is to read a great deal. The same advice applies to writing in mathematics. Students often make it very far into their studies without ever seriously reading their textbooks. Because of this, their ability to write mathematics is often far behind their ability to solve problems and prove theorems. To be a good writer of mathematics, the best thing you can do is to read mathematics. Try to read many different authors with different styles. This will help develop a style of your own.
When writing mathematics the first and most important thing to remember is that you are trying to convey ideas to your reader. You should write so that the reader will understand everything that you write in a precise yet concise manner. As in any writing, one possible technique you can use, is to read what you have written out loud. This will help you to determine if your writing is conveying your ideas.
Technical Writing
Each theorem must have a proof or have a reference to where the proof can be found. For example, you can say a proof for Theorem 2.3 can be found in [1]. Usually, the proof should follow the theorem but on occasion the proof can precede the theorem if it is presented in the preceding prose. You should make every effort so that your proofs are not only correct but eloquent. An elegant proof is the hallmark of good mathematical writing.
If you use something, it must be defined before it is used. In other words, you should not use the term 'group', for example, before defining what a group is. An exception to this rule is that you can use undefined terms in the abstract when describing what is done in the paper.
When choosing an example, make sure it is an instructive example that illustrates the theorem. It is best to have a paradigm rather than an anomalous example.
Prose should be used to connect lemmas, theorems and corollaries. The paper is written to be read by humans, it is not a program written for computers. Be careful to explain what you are doing.
Whenever an equation or function is important or long it should be displayed. It should also be named if it is going to be referenced. For example, use the \label{} command in latex and this puts a reference like (1) at the end of the line.
Notation and Nomenclature
It is very important to choose your notation carefully. You should use notation that is suggestive of what is happening. As an example, it is better to use h for height and r for radius than to call them x and y. For a more sophisticated example, when we write the action of a group on a set, we usually write it as juxtaposition even though it is a function. This allows the reader to understand what is happening in a natural setting. Moreover, if there is standard notation for something you should use it. You should not use terminology that is inconsistent with the standard names. Using non-standard notation and nomenclature prevents the reader from finding and understanding the foundational results in the literature.
Perhaps the most important aspect of notation and nomenclature is to be consistent throughout the paper. The same notation and nomenclature should be used for an object whenever referring to it. Moreover, if you use letters like f and g to denote functions, then you should continue to use letters nears these in the Latin alphabet to denote functions. If you use a Greek letter to denote a function then you should continue to use Greek letters to denote functions. Similarly, if you use letters late in the alphabet, like x and y for variables then you should continue to use letters late in the alphabet for variables. Usually, these letters are used for real variables and letters like k,m and n are used for integer variables.
Grammar and Punctuation
Follow all of the rules of standard English, for example, all sentences must end with a period. This sounds simplistic but when students write mathematics they often forget that they are writing sentences and forget the period. This occurs frequently when defining piecewise functions or displaying equations. The following are some standard rules when writing mathematics:
1. Never start a sentence with a symbol.
2. Never line up the definitions in a row.
3. Never start a new section with a definition or theorem. These things should always be preceded by prose.
4. Never put anything between a theorem and its proof. Every proof must end with a marker. Classically, this was QED but in modern texts it is usually the box formed by \qed in Tex.
References
Any paper or book that is referenced in the paper must be put in the bibliography. While different journals and publishers have different standards for the bibliography, the important thing when writing for general distribution is to make the entries in the bibliography consistent. When in doubt it is better to reference something rather than omit the reference.
In general, when referencing a theorem you should give the reference where it first appeared. If it is a difficult paper to find or in a language different from what you are writing in, you probably want to give a later reference where it can be more easily found and read. Do not put any paper or book in the references that you do not use in the paper.
Some Examples of Mathematical Writing
Below are some examples of mathematical writing which illustrate the above points.
The first is an expository article by John H. Conway, and Simon Kochen. The second is a research paper by Neil Trudinger and Xu-Jia Wang. Both Conway and Trudinger are recipients of the Steele Prize for Mathematical Writing. The third is a well known paper from the Bulletin. These illustrate different styles and are each written for a different audience in the sense that the first is written for a very general mathematical audience, the second for specialists in a particular area, and the third is for mature mathematicians but not necessarily specialists in the area.
The Strong Free Will Theorem, by John H. Conway and Simon Kochen.
Hessian Measures I, by Neil S. Trudinger and Xu-Jia Wang.
User's Guide to Viscosity Solutions of Second Order Partial Differential Equations, by Michael G. Crandall, Ishii Hitoshi, and Pierre-Louis Lions.
An Example of Good and Bad Mathematical Writing
Here is an excellent example to guide your writing: Writing Mathematics Well