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Section 5.12 Writing for Mathematics

Contrary to the popular belief that mathematics and writing are opposites, strong writing skills will make you a better mathematician. Writing for mathematics is all about communicating complex ideas with precision and concision. Initially new genres like proofs may be intimidating, but practice, your professors, and this section can help you learn the craft of writing in the field of Mathematics.

## Genres.

“*You shouldn’t be afraid to do proofs.*”

The fundamental genre of writing in mathematics is the proof. In a proof, a mathematician walks through the logical steps from a known statement to prove another statement. Proofs should be devoid of complexity beyond the logical steps required to prove the statement. They range in length from just a few lines to several pages or longer, depending on the complexity of the problem.

The other genre more common in upper division mathematics classes is a paper using mathematics to solve real-world problems. This genre involves more expository writing, as the problem must be used to write a mathematical model, and then the results of the model must be contextualized with respect to the original problem. They are generally 5-7 pages in length, including graphics.

## Purpose.

The purpose of writing in mathematics is to work through difficult ideas and communicate them clearly. Writing helps develop and deepen thinking. Proofs are applications of complex concepts and methods learned in class; in order to write them successfully, you have to explore and understand those ideas, a process that facilitates learning.

In the professional mathematical world, a proof is also something more, a way of demonstrating (proving!) previously unknown knowledge to the academic community. While student proofs are not usually new discoveries, they require the student to practice the same kinds of thinking and logical moves required for more advanced academic work.

## Valued Characteristics.

“*Erdős (a famous mathematician) said that God has a book with all the proofs in it… and when you wrote a nice proof he would say, ‘oh, that’s from the book.’*”

Precision and concision are the most highly valued characteristics in mathematical writing. Precision means that the piece of writing is logically correct. You cannot write a good proof without correct logic. Concision means that the ideas are communicated efficiently, without anything unnecessary included. For example, if you’re using a theorem from the book, cite it instead of writing the entire thing out again. Avoid unnecessary explanations like “now I will show that X is 3…” instead, simply show it. Concision also means knowing your audience and not including details they would already know. For most proofs, the expected audience is another student doing reasonably well in your class, who wouldn’t need to see every basic algebraic step you took. If, for example, you already covered solving systems of equations in class, there’s no need to list every step you used to solve a system. You can simply tell the reader what technique you used and what the results were. Concision, along with mathematical elegance, also means using big theorems that move the proof along in fewer steps. A good proof does its work as efficiently as possible, without descending into detailed computations where they aren’t needed.

To write concisely, also keep in mind that a proof is the logic itself, not the story of how you worked it out. X did not become three when you figured it out; it always was three, and you simply discovered that fact. In the process of writing a proof, you will likely struggle down many dead ends, but the final project should only include the most efficient logical steps.

## Evidence.

Pre-existing theorems and definitions are important starting evidence, which are generally cited from the textbook. Mathematical equations, incorporated into the text in complete sentences, also provide evidence. In more applied projects, working out specific cases or examples of a theorem can also provide important explanation. Longer projects might also cite mathematical literature, usually books and occasionally journal articles.

## Conventions and Tips.

A proof is a piece of writing in paragraph form. This means that even when mathematical equations are used, they go within complete sentences with proper punctuation.

When possible, don’t start sentences with numbers or symbols.

Proofs are written in the present tense using the first person plural (“we”). Write as though you are walking through the logical steps with the reader.

Choose notation carefully and define variables, parameters, and constants clearly when they are introduced.

While expressions are used in text, don’t use shorthand like replacing the words “less than” with the “<” symbol.

Be thoughtful when inserting graphics: use consistent designs that convey contents clearly, and be sure to reference them in the text.

Mathematical writing is generally done in LaTeX, a document markup language. This makes proper formatting of equations in text much more elegant.

Most professors return incomplete or incorrect proofs for revisions. This is an important part of the learning process; be comfortable with working through revisions several times.

When writing about equations or formulas, remember that they exist and are not evolving or springing into being. Refer to them as factual entities that simply *are*.

Writing in mathematics classes is generally done individually, while professionals may work solo, in pairs or small groups, and infrequently in large groups

Many students find starting proofs challenging but, as with any piece of writing, it’s important to just start and not worry too much about being right immediately. Try strategies recommended by your professors, like starting with definitions of important terms or working backwards.

## Additional Resources.

A

very useful guide^{ 1 } to proof writing is available from Dr. John Lee at the University of Washington. Your professor may also have written something along these lines.

Mathematical Writing^{ 2 } by Donald Knuth, Tracy Larrabee, and Paul Roberts is a somewhat more in-depth booklet on writing in the discipline (based on a course taught at Stanford), available free online.

A Primer of Mathematical Writing by Stephen Krantz is a great book for students interested in diving deeper into the topic.

`sites.math.washington.edu/~lee/Writing/writing-proofs.pdf`

`jmlr.csail.mit.edu/reviewing-papers/knuth_mathematical_writing.pdf`