WRITING MATHEMATICS Tevian Dray2014 (a revised version of a 1998 essay)

Writing mathematics well requires a blend of mathematical knowledge with traditional writing skills such as spelling, grammar, and usage.

Overview

The following overview is adapted from the instructions for a writing course based on non-Euclidean geometry, but can easily be adapted to other contexts.

• Choose a topic
For instance, pick a non-Euclidean geometry you find interesting. Is there some aspect of it which was discussed briefly somewhere without much detail? Is there some way of changing the rules which intrigues you?
Once you have tentatively chosen a topic, write a few sentences explaining it. If you are creating your own model, describe exactly what it is. If there's something missing from a proof, or from the coverage of a topic in a textbook, or whatever, describe what's missing.

Why is your topic interesting? Why is it important? Write down a few bullet points that address these questions.

• Make an outline
Now that you have chosen the topic, you should know at least in principle what geometric model(s) you will be working with. The next step is to decide what questions to ask about it. So make up a list of questions about your model. Does it need a distance function? Do you plan to determine what corresponds to circles?
Select several of these questions (1 is too few; 10 is too many) which you hope to answer while writing your essay. Divide them into appropriate categories. Now you're ready for the outline: Start with an introduction, end with a summary/discussion/conclusion, and put the various (categories of) problems in the middle. Briefly describe each part. Incorporate the list of bullet points you made above.

• Do the math
Solve the problems. This is the fun part!

• Make a rough draft
Write up what you did. You need to include enough detail so that people can understand it. Most calculations should be given explicitly. Lots of figures (with suitable captions/descriptions) are a big help. But you also need to include enough words so that people can understand it; theorems and proofs may be appropriate, but are certainly not sufficient.

• Rewrite as needed
Be a perfectionist. Fix your math mistakes. Fix your grammar mistakes. Fix your spelling mistakes. Make sure your logic is sound. Make sure your reader will know at each stage what you're doing. Perhaps some reminders are needed: “Now we will solve the Dray conjecture” or “We therefore see that the Dray conjecture is false”.

Ground Rules

Again, the following ground rules are adapted from the instructions for a writing course based on non-Euclidean geometry, but can easily be adapted to other contexts.

• You must do some math
A discussion of the history of non-Euclidean geometry is not appropriate. A comparison of different (historical) versions of neutral geometry might be.

• Your work must be original
This does not necessarily mean that you must do something nobody's ever thought of before, although you'll certainly get brownie points if that is the case. You do need to work through the math yourself, and present the results in your own words. And you should carefully distinguish your own contribution from previous efforts, indicating that your work is a simplification/generalization/application of previous work (which should be explicitly cited), as the case may be.

• References must be cited
You may use whatever references you can find which might be appropriate. But you must give appropriate credit. A direct quote, for instance the statement of a postulate or a theorem, should be clearly labeled as such. A figure which appears elsewhere must be so labeled. It is not appropriate to make minor changes in text, or to redraw a figure, without giving a proper reference; this is plagiarism. By all means paraphrase an argument you find elsewhere. But give credit to the author. And don't fill up the entire essay this way; that's a book report.
Your references should appear separately at the end of your essay, with a section heading such as References or Bibliography. Full publication data must be given, including title, author(s), publisher, and year. Page numbers may be given if appropriate.

Your essay should be easy to read in another sense: Use a word processor! Get that new printer cartridge you've been thinking about! Use section headings. Indent your paragraphs. Don't run lengthy equations into the text—display them neatly on separate lines. (You may hand-write equations if you can not type special symbols.)

• Figures
By all means include lots of figures! These can appear in the text or on separate pages at the end, and may be hand-drawn. Each one should have a label, such as Figure 1, as well as a caption. You should describe each figure in the text in enough detail so the reader can figure out why it's there.

• Mathematical content
It's a good idea to get the math right!

Technical Details

• Use complete sentences! Always!
• Don't copy the problem; restate it in your own words. (A reasonable goal is to be able to pick up your written work five years from now and still understand it—without the use of any additional references.)
• An abstract is useful, summarizing the main conclusions in a few sentences.
• Lengthy derivations or proofs that would interrupt the flow of the narrative can be included as appendices.
• Your introduction is a good place for a background paragraph or two that discusses previous work (with citations to appropriate sources).
Equations
• Short equations, such as the statement that $y=x^2$, should normally be inline, that is, contained within a sentence as though they were words.
• Longer equations, such as $$\int\limits_0^\infty e^{-x^2} \,dx = \frac{\sqrt\pi}{2} , \label{test}$$ should be displayed, that is, set off from the rest of the paragraph. This also applies to particularly important equations, such as $$E = mc^2 .$$
• All equations should be grammatically correct parts of sentences, even if they are displayed.
• Displayed equations should be, well, equations; they require an equals sign or some other mathematical verb, and can not be merely an isolated mathematical expression.
• It is therefore an error to display a long sequence of computations as separate equations, unconnected by words. Such equations should be connected by short phrases, such as “so that”, etc.
• All displayed equations should be numbered, so that they can be referred to (“see Equation (\ref{test})”)—not only by you, but by others who read your work. (Some journal styles do not permit the numbering of equations you don't actually reference yourself.)
Figures
• Unlike equations, figures are not part of the flow of the text.
• Figures can be displayed as floats, either at the top (or bottom) of the page, above (or below) all other text.
• Figures can also be displayed at the left or right margin in the middle of a page, with text flowing around them.
• Finally, figures can be collected onto separate pages at the end of a manuscript.
• All figures should have captions, and labels such as “Figure 1”.
• All figures should be referred to explicitly in the text (“see Figure 1”), and explained in words.
• Tables are treated similarly to figures.
Miscellaneous
• There is a difference between a hyphen (“-”), used between words, a dash (“—”), used within sentences, a minus sign (“−”), used in equations, and a short dash (“–“), used in numerical ranges. (Short dashes are called “en-dashes”, long dashes are “em-dashes”. The spacing around minus signs (“$3-2$”) is different from that used with en-dashes (“2–3”).)
• Avoid the use of pronouns such as “it”, “this”, “that” wherever possible. Good mathematics writing requires precision; don't make your reader guess what these words refer to.
• Similarly, avoid the use of “fluff”, such as starting a sentence with “Note that”.
• Mathematical precision also requires you to select the correct verb. For example, integrals are evaluated (not solved).
• Many mathematical terms also have common meanings. Avoid using such terms unless you intend them in their mathematical sense. For example, a problem is complicated (not complex).

Examples

Below are some online definitions of Euclidean geometry. These definitions are taken out of context, and are not intended to be mathematical in the first place, so it is a bit unfair to critique them according to our standards for good mathematical writing. We do so nonetheless.

Wikipedia
• Euclidean geometry is a mathematical system attributed to Euclid, which he described in his textbook on geometry: the Elements.
• Euclidean geometry consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these.
• Euclidean geometry is an axiomatic system, in which all theorems ("true statements") are derived from a small number of axioms.
Encyclopedia Brittanica
• Euclidean geometry is the plane and solid geometry commonly taught in secondary schools.
• Euclidean geometry is the study of plane and solid figures on the basis of axioms and theorems employed by the Greek mathematician Euclid.
Oswego (NY) City School District Regents Exam Prep Center
• Euclidean Geometry is the study of flat space.
• Euclidean Geometry (the high school geometry we all know and love) is the study of geometry based on definitions, undefined terms (point, line and plane).
• Euclidean Geometry (the high school geometry we all know and love) is the study of geometry based on definitions, undefined terms (point, line and plane) and the assumptions of the mathematician Euclid.
• All of these definitions except possibly the last are descriptive, in that they don't actually tell us what Euclidean geometry is, only what (some of) its properties are. That may be good prose, but it is not good mathematics.
• Three of the definitions refer to Euclid's work; an explicit citation should be given. (When citing something like Euclid's Elements, good practice is to cite both the original and a more accessible modern translation; in this case, the latter could be a reference to an online copy.)
• These definitions contain too many undefined terms. What is a “mathematical system”? What does “consists in assuming” mean?
• These definitions are imprecise. What does it mean to “study” geometry?
• The first Oswego definition is incorrect! The non-Euclidean geometry used in special relativity is also flat. (It is important to know the audience for whom you are writing. Yes, precision matters, but so does communication. Mathematicians have an unfortunate tendency to sacrifice clarity of exposition for precision. That style may be acceptable when addressing experts, but not usually otherwise. In this example, an important consideration would be whether the intended audience would know enough to be confused by this error.)
• In the last two definitions, the parenthetical clause about high school geometry is fluff.
• Finally, note the two different spellings “Euclidean geometry” and “Euclidean Geometry”. Each usage is correct, but carries a slightly different meaning.

An online sample of good mathematical writing in the context of non-Euclidean geometry can be found here; a PDF version can be found here. (This manuscript was later published as Pi Mu Epsilon Journal 11, 78–96 (2000).)

Rubric

Below are some of the criteria that can be used to assess mathematical writing.

Content
• Correct computation.
• Correct justification.
• Use of more than one method.
• Use of multiple representations.
• Making conjectures and/or generalizations.