This was a handout that I made for my Discrete Mathematics class. At our college this course is the gateway to the mathematics major and is the students’ introduction to writing mathematical arguments. Here is a pdf version of the text shown below.
The nuts and bolts of writing mathematics
“Mathematics must be written so that it is impossible to misunderstand, not merely so that it is possible to understand.” ~ Herman Rubin
“You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.” ~ C. F. Gauss
“I would not have made this so long except that I do not have the leisure to make it shorter.” ~ B. Pascal
“Why does Milnor get to prove all the easy theorems?” ~ Anonymous comment about Fields Medal winner John Milnor, who is known for his short, elegant, and deep proofs.
1. The writing process. Use the same writing process that you would for any writing project.
a. Prewriting. This is the most mathematical step of the process. Often this step takes place on scratch paper. Figure out the mathematics: test conjectures, work out examples, try various proof techniques, etc.
b. Writing. When you understand the mathematics it is time to write the first draft. The draft may have extraneous information, be missing information, be written in the wrong order, contain some minor mathematical errors, etc.
c. Revising. Once you have a first draft, go back and revise the writing. Focus on large changes such as adding, removing, rearranging, and replacing. Fix any mathematical errors.
d. Editing/proofreading. At this stage you must attend to the fine details. Fix any problems with spelling, grammar, word choice, punctuation, etc. Make sure all of the mathematics is typeset correctly.
e. Publishing. Make the final changes so that you can submit your work. You may need to fit it to a style guide (get the margins correct, add a title page, etc.), convert it to a certain file type, or print it.
2. Your ultimate goal. Your aim is to succinctly and convincingly communicate a mathematical argument to the reader.
3. This is writing. When writing mathematics you must obey all of the usual writing rules—spelling, grammar, punctuation, sentence structure, etc.
4. Not a creative writing exercise. Avoid flowery, imprecise, descriptive, and vague language.
5. Know your audience. The way you write a proof should depend on who will read it. In a mathematics course you should write so that your classmates can understand your proof—it should not be written for the professor or for high school students.
6. Find your voice. Mathematics is always written in first person plural (we, us, our), not first person singular (I, me, my) or third person singular (one). Unlike most sciences, mathematicians do not use the passive voice. Mathematics is usually written in the present tense.
- Yes: “Using the quadratic formula we find the root of
to be 2.”
- No: “Using the quadratic formula the root of
- No: “Using the quadratic formula one finds the root of
- No: “Using the quadratic formula I found the root of
7. Symbols vs. words. You should read the mathematical symbols as words. For example, in the following sentence, the two “>” symbols are verbs.
- Since
, it follows that
.
However, symbols should not replace words in a sentence.
- If all three angles of a triangle are =, then it is equilateral.
You may want to read your proof aloud (including the equations)—this is a great way to find sentence fragments.
8. Too many symbols/too many words. It takes some practice to determine when to write the mathematics in words, and when to write in symbols.
- Too many words: The integral of
from zero to one is four.
- Too many symbols: We say that
if
,
s.t.
whenever
.
9. The devil is in the details. If the reader is mathematically sophisticated, you can often omit certain details, especially straightforward calculations. The work on your scrap paper does not always have to appear in your proof.
- Too many details: Suppose
. Factoring the polynomial we obtain
. Setting both sides equal to zero we have
and
. Solving both equations we find that
or
.
- Better: Suppose
- Another example that is appropriate for a post-calculus course: The derivative of
is
.
You should always include the details for any technique that you have just learned in your class.
10. Can I have an example of that? No. Do not put examples in the middle of your proof to illustrate your argument. The proof should consist only of your well-crafted logical argument.
11. Let it flow. Use the following words and phrases to help your writing flow: therefore, thus, so, hence, consequently, accordingly, it follows that, we see that, from this we obtain, moreover, provided that, notice that, note that, recall, since, because. Special tip: if you want to use the word “get”, use the word “obtain” instead. It always sounds better.
12. State your intentions. If you are about to make a multi-sentence mini-argument within your proof, you may want to announce your plan at the start.
- Our first objective is to show that no maximum occurs in the region…
13. 
14. Be concise. A longer proof does not guarantee clearer writing. Often short, clear arguments are better than long, winding descriptions.
15. Reread what you have written. Always do this! (And, as much as it may pain you, rewrite it if necessary.)
16. Practice, practice, practice. As with any worthwhile endeavor, you must practice in order to improve.
David Richeson: Division by Zero 
These are great guidelines! I think I’ll use most of them in my high school geometry class.
I was told by one of my professors, Dr. Stopple, never to use “obviously” or “clearly.”
If something is obvious or clear, we need not say it. If that same thing is then not obvious, or unclear, we should not introduce it as “obvious.” (obviously)
These are dangerous words. I’m not totally opposed to them, but they are definitely misused. I think that sometimes it is an ego thing (the author wants to make him/herself look smart) or a laziness thing (couldn’t be bothered writing the details). There are times, though, when something needs to be stated for the sake of the proof, but it is too trivial to waste space on proving it. In that case it is probably OK to say “clearly.”
I’m sure we’ve all had the experience of reading a sentence that starts with “clearly,” pondering it for a long while, then realizing that, yes, it was clear :-)
I like these. I haven’t taught any proof-oriented classes in a long time, and may never need this for students, but you clarify some of it for me for my own mathematical writing.
>Special tip: if you want to use the word “get”, use the word “obtain” instead. It always sounds better.
I do disagree with this one, though. I mostly write for students, and have always tried to use simple language. (as simple as possible, and no simpler, who said that?) I like ‘get’ better, although if I had to say it a few times, I might also use obtain for variety’s sake.
I really like that you occassionally repost a blog post like this. It makes a lot of sense since this type of information does not go out of date.
I remember when you first posted this one and I forwarded the link to my theoretical mathematics PhD student son (now a PhD). Now I realize that I was one of your early followers so I just went back and read the first 7 posts done prior to finding your blog (I’m the Banach Obama guy).
I strongly recommend to any new reader to go back and review earlier blogs … there are some real gems back there. And to Dr Richeson, keep reposting particularly useful posts like this one.
Thanks, David! Congratulations to your son.
I still love the Banach Obama t-shirt idea.
I also like ‘get’.
I’d like more discussion of the placement of examples than what you said in #9. I write for students, and I want to make sure they see why we’re going to head in a particular direction before we do so.
I’m unhappy with the order of topics in the conventional calculus textbooks, and have been writing a few things for my students. I expect to be doing more of this as time goes on. So I especially appreciate this discussion of how to write mathematics.
I lean toward the fanciful, though. Have you read Math Girls?
I’d point to #4: know your audience. I’m teaching the students to write like mathematicians, and “proofs by example” are frowned upon in rigorous mathematical writing. However, if the audience is different, it may be appropriate to pause in the proof to illustrate with an example. I have the same response to the fanciful language question: know your audience.
Reminds me of the good advice of Lyn Dupré’s marvelous book for Computer Scientists called Bugs in Writing http://www.amazon.com/BUGS-Writing-Revised-Guide-Debugging/dp/020137921X . Although all of the examples are from CS, my wife, a professional writer, keeps my copy on her shelves, because almost everything applies to writing in general. Dupré used to edit for Addison-Wesley before AW was bought out, even before LaTeX was widely used.
I’ve never seen that book. Thanks for the tip.