Career

Why Writing Well Matters in Mathematics

An argument for why clear mathematical writing is not a luxury but a necessity — with practical advice on how to improve your mathematical prose.

Writing Career

The Undervalued Skill

Among all the skills a mathematician needs — technical mastery, problem-solving ability, creativity, persistence — writing is the most undervalued.

Many students believe that the mathematics speaks for itself. If the proof is correct, what does it matter how it is written?

It matters enormously. A brilliant result buried in impenetrable prose may never be read. A clearly written paper, on the other hand, reaches its audience, influences future work, and stands the test of time.

Why Writing Matters

1. Communication Is the Purpose of a Paper

A mathematical paper is not a private notebook. It is a communication to other mathematicians. If the intended audience cannot follow your argument, the paper has failed — regardless of how correct it is.

The mathematician Jean-Pierre Serre has been described as one of the finest mathematical writers of the 20th century. His papers and books are models of clarity, and their influence is partly attributable to their readability.

"Serre's writing is so clear that when you read him, you feel the mathematics is obvious. It is only when you try to reproduce the argument that you realize how much care went into making it look easy."

— A common observation among algebraists

2. Writing Clarifies Thinking

The act of writing forces you to make your reasoning explicit. Vague intuitions must be sharpened into precise statements. Gaps in your understanding become visible.

Many mathematicians report that they do not fully understand their own results until they have written them up carefully. Writing is not the last step of research — it is part of the research itself.

3. Your Reputation Depends on It

Mathematicians are known by their papers. If your papers are poorly written, your results may be ignored, misunderstood, or attributed to others who explained them more clearly.

Conversely, mathematicians who write well — Serre, Milnor, Atiyah, Halmos — are read and cited disproportionately because their work is accessible.

4. Teaching Requires It

Whether you write lecture notes, textbooks, or problem sets, clear writing is essential for teaching. Students benefit enormously from well-written mathematical exposition.

Principles of Good Mathematical Writing

Clarity Above All

Paul Halmos, who devoted an entire essay to mathematical writing, stated the principle plainly:

"The basic problem in writing mathematics is the same as in writing biology, writing a novel, or writing directions for assembling a harpsichord: the problem is to communicate an idea. To do so, you must have something to say, and you must have someone to say it to, you must organize what you want to say, and you must arrange it in the order you want it said in."

— Paul Halmos, "How to Write Mathematics"

Write for a Specific Reader

Before you begin, identify your intended reader. Is it a specialist in your subfield? A general mathematician? A graduate student? This choice determines your level of detail, notation, and the amount of background you include.

Motivate Before You Formalize

Never begin a section with a bare definition or theorem. First explain informally what is about to happen and why it matters.

Poor:

Definition 4.1. Let $X$ be a topological space. A sheaf $\mathcal{F}$ on $X$ is...

Better:

The key idea in this section is that we can organize local data on a topological space into a global structure called a sheaf. Informally, a sheaf keeps track of functions (or other objects) defined on open sets, along with rules for how they patch together.

Definition 4.1. Let $X$ be a topological space. A sheaf $\mathcal{F}$ on $X$ is...

Use Examples Generously

A well-chosen example is worth a page of abstraction. Examples make definitions concrete, theorems believable, and proofs understandable.

John Milnor's books are masterful in their use of examples. Each abstract concept is immediately illustrated with a concrete instance that reveals its meaning.

Write in Complete Sentences

Mathematical formulas should be part of the sentence structure. Do not write:

Consider $f : X \to Y$ . Continuous. Bijective. Then homeomorphism if $X$ compact and $Y$ Hausdorff.

Write:

Consider a continuous bijection $f : X \to Y$ , where $X$ is compact and $Y$ is Hausdorff. Then $f$ is a homeomorphism.

Introduce Notation Carefully

Every piece of notation should be defined before it is used. Do not assume the reader knows your conventions.

When possible, use standard notation. Non-standard notation forces the reader to maintain a mental dictionary, which consumes cognitive resources that should be directed at understanding the mathematics.

Use Words, Not Just Symbols

A Common Mistake: Writing a proof as a sequence of formulas connected by " $\Rightarrow$ " and " $\therefore$ " symbols, with no English words.

This is not good mathematical writing. Proofs should be written in prose, with equations embedded in sentences and logical connectives expressed in words.

Donald Knuth, in his notes on mathematical writing, emphasizes:

"Good mathematical writing involves a blend of formulas and prose in much the same way that good music involves a blend of melody and harmony."

— Donald Knuth, Mathematical Writing

Practical Advice

Start Writing Early

Do not wait until your results are complete and perfect. Begin writing as soon as you have something to say. Early drafts clarify your thinking and reveal gaps.

Read Good Writing

The best way to improve your writing is to read excellent mathematical exposition. Some recommendations:

Jean-Pierre Serre: Any of his books or lecture notes
John Milnor: Topology from the Differentiable Viewpoint, Morse Theory
Michael Atiyah: Survey papers and essays
Paul Halmos: Finite-Dimensional Vector Spaces, Naive Set Theory
Timothy Gowers: Mathematics: A Very Short Introduction, blog posts

Revise Ruthlessly

Good writing is rewriting. First drafts are always rough. Expect to revise a paper multiple times, each time improving clarity, tightening arguments, and eliminating unnecessary material.

Read Your Writing Aloud

This simple technique catches awkward phrasing, run-on sentences, and unclear passages that are invisible on the page.

Get Feedback

Ask a colleague — ideally one who is not an expert in your specific area — to read your paper. If they have trouble following it, the problem is with the writing, not with the reader.

Learn LaTeX Well

LaTeX is the standard typesetting system for mathematics. Invest time in learning it properly:

Use theorem environments (\begin{theorem}, \begin{proof})
Define macros for frequently used notation
Use \label and \ref for all cross-references
Use BibTeX for bibliography management
Learn the amsmath package thoroughly

Writing at Different Stages

Undergraduate

Focus on writing clear, complete proofs. Every step should be justified. Practice writing solutions to homework problems as if they were being published.

Graduate Student

Begin writing expository material — lecture notes, survey articles, thesis chapters. Learn to motivate your work and place it in context.

Researcher

Develop a writing style that is distinctively your own while remaining clear and accessible. The best mathematical writers have recognizable voices.

The Moral Dimension

There is a moral argument for writing well: you owe it to your readers.

If someone takes the time to read your paper, they deserve clear, careful exposition. Unclear writing wastes the reader's time and energy — a cost that is multiplied across every person who reads the paper.

The mathematical community functions on the assumption that published results can be understood and verified by others. Impenetrable writing undermines this fundamental contract.

A Standard to Aspire To

Write mathematics as if your reader is intelligent, motivated, and busy. Respect their intelligence by not over-explaining trivial points. Respect their time by not under-explaining non-trivial ones. Respect their motivation by showing them why your work matters.

Famous Essays on Mathematical Writing

Several classic essays on mathematical writing are worth reading in full:

Paul Halmos, "How to Write Mathematics" (1970) — The classic essay on mathematical exposition.
Jean-Pierre Serre, "How to Write Mathematics Badly" — An entertaining lecture, available on YouTube, in which Serre describes common mistakes by example.
Donald Knuth, Tracy Larrabee, and Paul Roberts, Mathematical Writing (1989) — Notes from a Stanford course on mathematical communication.
Nicholas Higham, Handbook of Writing for the Mathematical Sciences (1998) — A comprehensive reference covering every aspect of mathematical writing.

Final Thoughts

Writing well is not a distraction from doing mathematics. It is a central part of doing mathematics. The ability to communicate your ideas clearly — in papers, talks, lectures, and conversations — determines how much impact your work will have.

As Halmos wrote:

"The only way to learn to write is to write."

— Paul Halmos

Start now. Write carefully. Revise often. Your future readers — and your future self — will thank you.

References

Paul Halmos, "How to Write Mathematics," L'Enseignement Mathématique, Vol. 16, 1970
Donald Knuth, Tracy Larrabee, and Paul Roberts, Mathematical Writing, MAA Notes, 1989
Nicholas Higham, Handbook of Writing for the Mathematical Sciences, SIAM, 1998
Steven Krantz, A Primer of Mathematical Writing, American Mathematical Society, 2017
Jean-Pierre Serre, How to Write Mathematics Badly (lecture), available on YouTube
William Strunk Jr. and E.B. White, The Elements of Style, Macmillan, 1959 — Not mathematical, but its principles of clarity and brevity apply universally

Back to all posts