Mathematical Writing

Forgive me, but I’m often overjoyed to provide feedback on student writing as a math professor. Who knew?! Below are some of my favorite pieces of advice about writing mathematics, and I hope you will find them useful.


Paul Halmos wrote How to Write Mathematics (original available here), and these are his key points:

  1. Say something. To have something to say is by far the most important ingredient of good exposition.
  2. Speak to someone. Ask yourself who it is that you want to reach.
  3. Organize. Arrange the material so as to minimize the resistance and maximize the insight of the reader.
  4. Use consistent notation. The letters (or symbols) that you use to denote the concepts that you’ll discuss are worthy of thought and careful design.
  5. Write in spirals. Write the first section, write the second section, rewrite the first section, rewrite the second section, write the third section, rewrite the first section, rewrite the second section, rewrite the third section, write the fourth section, and so on.
  6. Watch your language. Good English style implies correct grammar, correct choice of words, correct punctuation, and common sense.
  7. Be honest. Smooth the reader’s way, anticipating difficulties and forestalling them. Aim for clarity, not pedantry; understanding, not fuss.
  8. Remove the irrelevant. Irrelevant assumptions, incorrect emphasis, or even the absence of correct emphasis can wreak havoc.
  9. Use words correctly. Think about and use with care the small words of common sense and intuitive logic, and the specifically mathematical words (technical terms) that can have a profound effect on mathematical meaning.
  10. Resist symbols. The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it.

The Math Association of America is now hosting an entire webpage devoted to Mathematical Communication. This looks like it will become a fantastic resource going forward.


Over at Harvey Mudd College, Francis Su has posted his Guidelines for Good Mathematical Writing. This seems to be part of a wider campus-wide effort in communication. In this post, Rachel Levy explains why Every Math Major Should Take a Public-Speaking Course.

Francis Su also has an article Some Guidelines for Good Mathematical Writing in the August/September 2015 MAA Focus Newsmagazine (a subscription or MAA membership might be required).


Annalisa Crannell at Franklin & Marshall College offers A Guide to Writing in Mathematics Classes for undergraduate math students, but the advice is useful for a much wider audience.

  1. Why Should You Have To Write Papers In A Math Class?
  2. How is Mathematical Writing Different?
  3. Following the Checklist
  4. Good Phrases to Use in Math Papers
  5. Helpful Hints for the Computer
  6. Other Sources of Help

I’ve also been told from several people to consider Serre’s How to Write Mathematics Badly. I found the video, but had a tough time watching it through to the end. Maybe Serre should have taken the public speaking course at Harvey Mudd College?


 

Dave Richeson from Dickinson College writes about The Nuts and Bolts of Writing Mathematics at his Division by Zero blog. He also has a very nice Checklist for Editing Mathematical Writing, and an example of a poorly written proof for a class exercise.

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