# How to improve writing mathematics?

My first language is not English. How can I improve my mathematical writing. I feel like the only things I can write down are numbers and equations. Is there any good suggestion for improving writing, especially for mathematical writing (math-philosophy)?

• Since your native tongue is not english, i feel the best advice is i could give is: read, read as much as you can -of similar literature- in english ... – Konstantinos Kanakoglou Sep 10 '17 at 21:03
• However, i think this question would better fit math.stackexchange – Konstantinos Kanakoglou Sep 10 '17 at 21:04
• this question is similar mathoverflow.net/questions/1243/how-to-write-math-well – j.c. Sep 10 '17 at 22:06
• Thank you all for your useful advice. I am very appreciated for that. Especially, I would like to thank to j.c. for pointing out the similar question's link, where I also get a lot of useful information. – Shaoyun Yi Sep 10 '17 at 22:38
• My own advice: read anything by Serre, whose English is clearer, more elegant, and more transparent than that of most anglophones. – Lubin Sep 10 '17 at 23:52

I want to highlight two tools for learning: imitation and practice.

Read a lot of mathematics. You will find that some texts are easier to follow than others. What makes you like a text? What texts do you like most? When you write, try to write as your favorite author would. If you keep on writing mathematics long enough, you will find your own voice, but imitation is a necessary first step.

Write a lot. You say you can write numbers and equations. What are they about? Explain. It doesn't have to be perfect, but explain it in your own words. Tell a story about your calculation. What would you say out loud to explain your work to a fellow student? Write that down. Try to make a habit out of making explained calculations that anyone could read.

Lastly, if you don't know how to write something, ask for help. Composing good mathematical prose is not trivial, and learning it is an important part of any degree in mathematics. You are entitled for help with it, not only with your calculations.

If I understand correctly, your problem is in writing relatively simple and short things. Writing and structuring a thesis, a paper, or other extended piece of work is a story I will exclude here.

• Thank you so much, Dr. Ilmavirta. I am very appreciated for your considerable answer for my question and I cannot agree with you more. So I need to write down the mathematics as much as I can (not just reading or doing calculations). Just like you say, at beginning I can follow the favorite way, then I may have my own way later. Thanks again. – Shaoyun Yi Sep 11 '17 at 0:03
• +1 Joined (finally) to upvote tis answer. – Ethan Bolker Sep 12 '17 at 19:42
• Also, writing technique for a thesis, a paper or something seems more like a matter without certain language. – Lwins Oct 1 '17 at 17:27

At the early stages of my mathematical career, I found the following (little) book very useful.

Trzeciak, Jerzy, Writing mathematical papers in English. A practical guide, Zürich: European Mathematical Society Publishing House (ISBN 3-03719-014-0/pbk). 49 p. (2005). ZBL1077.00008.

Halmos, Paul R. "How to Write Mathematics." L'Enseignement Mathématique 16.2 (1970): 123-152. (PDF download.)

Idiosyncratic, but a classic. I especially enjoyed the section, "Think about the alphabet," and his discussion of "frozen" letters:

"many readers would feel offended if $n$ were used for a complex number, $\epsilon$ for a positive integer, and $z$ for a topological space. (A mathematician's nightmare is a sequence $n_\epsilon$ that tends to $0$ as $\epsilon$ becomes infinite.)"

• I have probably already mentioned somewhere on this site Hugh Montgomery's thought-experiment about writing a paper where, whenever you need a new symbol, you just use the first letter of the alphabet that you haven't used yet. "The Riemann $a$-function, defined by $a(b)=\sum_cc^{-b},\dots$". He concluded such a paper would be unreadable. – Gerry Myerson Sep 10 '17 at 22:13
• @GerryMyerson: Ha! Imagine how different it would feel if we used $\alpha{-}\beta$ arguments in calculus rather than $\epsilon{-}\delta$ arguments."Why do we use ϵ and δ?" – Joseph O'Rourke Sep 10 '17 at 22:19
• @ShaoyunYi: Re: "Dr.": In MO, we are all students. :-) – Joseph O'Rourke Sep 11 '17 at 0:32
1. Read what you have written out loud. This way you will hear most of the things that could go wrong.

If it sounds awkward, change it.

If it bores you, say something more interesting or cut it down.

If it confuses you, look for a clearer presentation.

I find that to be good advice for writing in any language.

1. It's also useful to do this as the central step with a machine translator. For instance, to write text in Spanish, I will:

• a) Write a draft in English;
• b) Run Google Translate on it from English to Spanish;
• c) Read the result out loud, and edit so that it sounds good to me in Spanish;
• d) Run Google Translate on the result from Spanish to English;
• e) Check if anything in this English version shows something ungrammatical or unintended in Spanish, and if so, go back to c.

This works best with substantial revisions in the middle step -- substantial enough for me that I feel a physical strain in my cheeks from enunciating all the Spanish sounds. But it saves me from lots of errors of getting genders wrong, and it comes up with better words than I would think of on my own.

Anyway, if you've done enough reading in English to have a good ear for it, you may like the results from this technique.

Here(数学写作漫谈) is some tips by wwl(李文威) in Chinese.