How to write Mathematics
University of Colorado at Denver
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
- 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.
- Every mathematical statement must be complete and meaningful.
- If a statement is something you want to prove or something you
assume temporarily, e.g., to discuss possible cases or to get a
say so clearly. Otherwise, anything you put down must be a true
statement that follows from your up front assumptions.
- There must be sufficient detail to verify your argument. If you
have the details, you have no way of knowing if what you wrote is
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.
- Do not write irrelevant things just to fill paper and
show you know something.
- Your argument should flow well. Make the reading easy. Logical
and intuitive notation matters.
- 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
- 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.
- 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?
- 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.
- 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.
- 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.
- 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.
- 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.
Mathematics and programming are similar. The above rules how not to write are
inspired by the article
How to Write Unmaintainable Code