Commonly Accepted Principles of Writing  Mathematics Well

1.  General Remarks on Mathematical Content and Style

     a.  Always be precise and mathematically correct, yet concise.   For more details, suggestions, and examples, see the guidelines for Mathematics students at MIT: Writing a Math Phase Two Paper .
     b.  If using the first person, use "we" instead of "I".
     c.  State  a theorem (or proposition, lemma, corollary) before proving it.   State the theorem concisely, completely, and correctly.  (Hypotheses and conclusion).
     d.   Do not begin a sentence with a symbol.
     e.  Introduce a theorem appropriately before proving.   If appropriate, give an example of the use a theorem before  giving the proof.
     f.   If a complicated proof is involved, it may be helpful to give an outline of the method of the proof prior to the body of the proof.
     g. Use the active voice instead of passive voice where possible.
     h.  Elaborate results with examples, applications, and diagrams.
     k.  Make sure the level of the writing is appropriate to the target audience.
     i.   For formal mathematics (such as a paper for a course or a publication), do not give just lists of   definitions and theorems.   Elaborate (as suggested above above) with examples and exposition between the definitions and theorems.
     k.  Do not just list a sequence of equations or inequalities as the body of a proof.  Guide the reader with transitional phrases.

2.  Guidelines for the Establishing the Content of Expository Mathematical Text:

          1) What is the purpose of this theorem, result, algorithmic method, or definition?
          2) What does this theorem or definition really mean?
          3) Why is this result or definition important?
          4) What is the central idea  in the proof of the theorem?
          5) How are other theorems or lemmas used to prove this theorem?
          6) Is this theorem a step towards a major result or is it the major result?
          7) How does one apply this theorem? What is an example of a mathematical object satisfying (or not satisfying) the definition or result?
          8)  Why are the hypotheses of this theorem so important? (How will the theorem fail if the hypotheses are not satisfied.)
          9)  Is this theorem the strongest possible result?  Does it have any important corollaries?
          10) Does the converse of this theorem hold?  If not, why (provide an example perhaps).
 

2. Guidelines for Appropriate Formatting of Mathematical Text.

     a. Use standard and accurate notation. For example, the symbol R is commonly  understood to represent the set of real numbers;  C the complex numbers (although in handwritten format one normally uses what's called "blackboard bold" to write the symbols for the more common sets of numbers - I'll demonstrate in class); x, y,z real variables; u, v vectors;  a, b, c real constants;  k, n, m integers;   f, g, h  functions,  p, q for polynomials.   Note how symbols are used in your mathematical texts and refer to them if you have doubts.
     b. Theorems, definitions, examples are offset from the rest of the text. Start a new paragraph and normally indent the theorem, precede by a blank line. If numbering your results or naming them, they should be bold faced.   Be consistent in the style you choose.
    c. Put a long or complicated expression (equations, formulas, etc.) on a line by itself -- normally centered and offset by blank lines.  If you will need to refer to it later, you should use a consistent numbering scheme.   If your paper is divided into numbered sections, it is helpful to incorporate these into your numbering scheme.
     d. Leave a space between symbols and adjacent text.   (Some suggest a double space.)  Also use appropriate, conventional  formatting of mathematics ( such as italicizing variables).  Example:

                             Let  z = 2 + 4i.

     e. If you mark the ends of proofs use accepted notation such as QED.   Alternatively, appropriately format the proof so the beginning and end of the proof are easily identified.  Whatever style you use, be consistent.
    f.  Do not use a symbol instead of a verbal description unless it is necessary (and helpful) for later reference.
    g.  Define every symbol before it is used.
    h.  Use different symbols for different objects.
    i.  Do not use abbreviation symbols such as =, >, in text  (as opposed to within formulas).

Appropriate:
Whenever the least upper bound is equal to the maximum of the set ...
Inappropriate:
Whenever the least upper bound =  the maximum of the set ..
   j.  Use mathematical terms correctly.  Commonly confused are functions, values of functions, equations, and expressions.
   k.  Do not write equality ( = ) unless you mean equality!
 
Important Note:  One of the best ways to improve your style of writing mathematics is to READ well written mathematics.  Pay attention to the wording and style of writing in your mathematics texts.  Note the customary formatting.  Refer to these when making decisions about the content and style of your mathematical writing.

Web sites with help for writing mathematics:
Guide to Writing
Some thoughts on writing mathematics
Writing Mathematics
How to Use the Equation Editor for Microsoft Word
Writing a Math Phase Two Paper

Assignment  Using the handout and the suggestions above, look at the sample Word document which contains a number of violations of accepted mathematical writing style. Find as many as you can and mark appropriate corrections.  Then edit the document in Word incorporating your suggestions. Save under a new file name and submit to me via email.  You may work with a partner if you wish, and submit your work jointly.