Paul Halmos was a mathematician who among other things was recognized by his excellent written exposition.

His article on how to write mathematics left a lasting impression on me, and this year I decided to read it again, and capture some notes for my future reference.

Although the word mathematics is in the title, some ideas presented apply to all kinds of writing, which makes them potentially useful to anyone wanting to improve their writing.

How to write mathematics

Say something and be brief

Saying a lot about nothing is worse than having nothing to say. If you have something to say, keep it concise. Do not introduce too many ideas, it will be harder to remember the main one.

Write for your audience

Writing for a specific audience makes things simpler because you can make big assumptions about their shared knowledge.

Writing for a general audience is difficult because any assumption you make will likely be wrong.


Capture your ideas and establish the connections between them. This could include going over all of your notes and creating references where necessary.

Make it self-contained

Include all the necessary pre-requisites so that the readers won’t have to look up external sources. In some cases, this also means to remove certain concepts in order to keep the size of the text manageable.

Create an outline

A good start is to outline the text in three sections, sometimes referred as the Aristotelian triptych: Say what you’re going to say, say it, say what you said.

Say what you’re going to say

Prepare your audience for what’s coming. In this section you can motivate the topic by providing some historical background in a way that shows the connections between the ideas of previous thinkers and the ones that will be discussed later in your text.

Say it

Make your text as self-contained as possible, do not interrupt the reader’s flow by making references to ideas outside the current text.

Provide examples and draw straightforward connections to the key concepts. Counter-examples can also help you delimit the scope.

Say what you said

Remind your audience about the takeaways. Make them short and memorable.


Organizing the material is about drawing the interconnections between ideas, arranging is about choosing the order in which they will be laid out.

The spiral plan

Write the first section, then write the second section, go back and read the first section. You will often find things that can be fixed or improved, now you can rewrite the first section.

Continue writing the next section, then go back and review the previous ones, rewrite what needs to be rewritten, repeat this process until the text is good enough.

Linearize dependencies

Arrange the sections in a linear order, this makes for a better reading experience since it’s not necessary to jump back and forth.


Start writing. Write bit by bit. Focus on the process, write a little each time. Focusing on the end goal can increase anxiety and lead to procrastination.

Continuously review what you’ve written using the spiral plan to find more connections and possible improvements.

Stop. Do not write more than what’s necessary to communicate your message. If you follow the spiral plan you will have plenty of opportunities to find gaps, mistakes or other things to fix.

Re-write and re-write

When you find something that can be improved, or that should be fixed, try rewriting it instead of just editing it. Rewriting gives you the opportunity to express an idea in a totally different — and hopefully better — way.

Mechanical details matter

Check your work for mistakes, orthographic problems, style issues, inconsistent use of notation or terminology. Also, try to use punctuation correctly.

Too many mistakes can distract your audience from what you are trying to say.

  1. Paul Halmos article in Wikipedia
  2. Dieudonne, Jean, Menahem M. Schiffer and Paul R. Halmos. “How to Write Mathematics.” (1973).
  3. Google search: “L’Enseignement Mathématique, Vol.16 (1970) seals halmos how to write math”