Math 5070 Spring 2005

Handout 4

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.
  1. When read aloud, the text and formulas must form complete English sentences. If you get lost, say aloud what you mean, then 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. Write what your plan is.  It will also help you focus on what to do.
  5. 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. Keep the level of detail uniform.
  6. If you are not sure about something, even slightly, work out the details on a different sheet of paper just for yourself, with utmost honesty, going as deep as necessary. Decide later how much detail to include.
  7. Do not write irrelevant things just to fill paper and show you know something.
  8. Your argument should flow well. Make the reading easy. Logical and intuitive notation matters.
  9. 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.
  10. The state of mind when you are inventing a solution is completely different from the mode of work when you are writing the solution down and verifying it. Learn how to go back and forth between the two.