How to write Mathematics

Jan Mandel
University of Colorado at Denver

January 2005

When the mathematician would solve a difficult problem, he first frees the equation of all incumbrances, and reduces it to its simplest terms.  -  Henry D. Thoreau

These simple rules apply to all forms of mathematical writing  -  from homeworks to exams to journal papers and books.

How to make Mathematics pleasant to read - things to do

  1. When read aloud, the text and formulas must form complete English sentences. If you get lost, say aloud what you mean and write it down.
  2. Every mathematical statement must be complete and meaningful. Avoid fragments.
  3. If a statement is something you want to prove or something you assume temporarily, e.g., to discuss possible cases or to get a contradiction, say so clearly. Otherwise, anything you put down must be a true statement that follows from your up front assumptions.
  4. There must be sufficient detail to verify your argument. If you do not have the details, you have no way of knowing if what you wrote is correct or not. If you are not sure, even slightly, about something, work out the details on the side with utmost honesty, going as deep as necessary. Decide later how much detail to include. Keep the level of detail uniform.
  5. Do not write irrelevant things just to fill paper and show you know something.
  6. Your argument should flow well. Make the reading easy. Logical and intuitive notation matters.
  7. Keep in mind what the problem is to make sure you are not doing something else. Many problems are solved and proofs done simply by understanding what is what.

How to make Mathematics unintelligible - things NOT to do

  1. Put down a lot of fragments of formulas with little or no text. They will figure it out if they are willing to spend the time.
  2. Never say what your plan is and just arrive at the result in the smallest space possible. As long as it is logically correct, who cares if it provides no insight into what is going on?
  3. Hide crucial information in the middle of a lot of irrelevant or well-known things and never in a displayed equation. They will have fun searching your paper over and over for the first time that new concept was mentioned, but they will sort it out eventually.
  4. Introduce a lot of new, counterintuitive notation. Maximize the number of subscripts and other decoration. Make similar symbols stand for completely different things. New notation is important for your identity and fame as a Mathematician. Never mind everyone else has been using a different notation and nobody will understand anything when casually browsing through your paper.
  5. Never drop any hints or reminders. If someone does not have the ability to look behind two corners  and the perfect memory to remember all your definitions, notation, and theorems from many pages before, they are not worthy reading what you wrote.
  6. Establish and follow conventions in your notation to lull the reader into complacency, then break them occasionally. This will mislead the casual reader every time.
  7. If you follow as many of these rules as possible, your writing will be mathematically correct yet well defended. It will really show how smart you are. If someone is not willing to contemplate in detail every single line you wrote and its precise consequences for every other line, or just does not have the time to burn, they have no business understanding the Mathematics you wrote.
  8. Mathematics and programming are similar. The above rules how not to write are inspired by the article How to Write Unmaintainable Code .