The Serial, or Oxford, Comma

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In the sentence

The great historical heroes of applied mathematics include Archimedes, Newton, Euler, and Gauss.

the comma before the “and” is known as a serial comma. Whether or not to include it is a matter of style.

The serial comma is also known as the Oxford comma, because Oxford University Press style rules require it to be present. The Chicago Manual of Style (CMS) requires the serial comma, as does SIAM, which follows the CMS recommendations and explicitly states, in the SIAM Style Manual, “Use the serial comma before the and or or in lists of three or more items.”

Other organizations, such as the New York Times, The Economist, and the University of Oxford, require that the serial comma is used only when necessary to avoid ambiguity. Consider the sentence

Three important techniques in the design of algorithms are bisection, divide and conquer, and recursion.

If the serial comma is omitted the final phrase becomes “are bisection, divide and conquer and recursion”, which will be confusing to anyone who does not know that “divide and conquer” is a technique.

Conversely, the serial comma is sometimes incorrect when it might appear to be optional. In the sentence

The results show that, unlike Algorithm 1, Algorithm 2 and the SVD-based algorithm exhibit forward stable behaviour in all the experiments.

a serial comma must not be put after “Algorithm 2” because the three algorithms do not form a list, so the sentence does not make sense with that extra comma.

Examples such as the last two, where the serial comma either must be used or must not be used, irrespective of style, are relatively infrequent, but they do arise from time to time.

For the last year or two I have been using the serial comma in my papers and books, partly because it is the style of the relevant publishers. In particular, I became accustomed to its use in The Princeton Companion to Applied Mathematics. But I also like the simplicity of the serial comma: I do not have to stop to think whether to use it every time I write a list. For informal writing, such as on this blog, I have not made up my mind which style to use. I think the serial comma would look fussy in the tagline at the top right corner of this page.

In the chapter “Commas the Serial Killer” in his book Making a Point: The Pernickity Story of English Punctuation, David Crystal notes that originally the use of the serial comma was standard, and it was only in the early twentieth century that it started to be avoided, “as part of the trend towards punctuation minimalism”. Interestingly, Crystal uses the serial comma in his book even though the style of his publisher (Profile Books) is to avoid it.

There is a large amount of material on the internet about the serial comma, of which the short post The Oxford, Comma has some good examples of where it is needed, and Wikipedia has a good entry. There is a song “Oxford Comma” by the American rock band Vampire Weekend (thanks to Sam Clark for pointing this out); a video is here, but beware the expletive in the first line of the song. The “comma queen” Mary Norris has produced an excellent video about the serial comma. The serial comma even has its own Twitter account, @IAmOxfordComma.

What better way to support the Oxford comma than by giving up some of your 140 characters for it in a Tweet!

Punctuating Lists

Image courtesy of Stuart Miles at

Lists are common in all forms of writing. The list items can be included within the text or put on separate lines. Separate lines are used in order to draw attention to the items, to ease reading when the items are long or numerous, or to facilitate cross-reference to specific items.

The two main types of lists are enumerated lists, in which the items are numbered or labelled alphabetically, and itemized lists, in which each entry is preceded by a marker such as a bullet symbol. In \LaTeX these are produced by the enumerate and itemize environments, respectively. (The basic enumerate environment allows only numbers as labels, but the enumerate package extends the environment to allow letters, too.)

I have long felt unsure about how to punctuate lists, and especially the sentence that introduces the list. Advice is hard to find in books on English usage and there seems to be no agreed way to do it. Some recent reading, namely of the TUGboat article Makings Lists: A Journey Into Unknown Grammar by James. R. Hunt and the book Making a Point: The Pernickity Story of English Punctuation by David Crystal, has clarified my thinking. I will now explain how I intend to format lists in future.

My approach is based on a key principle, which stems from the fact that in English all sentences must be complete, and not mere fragments. (There is no universally agreed definition of what distinguishes a sentence from a fragment, but a sentence is usually required to contain a verb.) The principle is that

if the labels in a list are removed then what remains should make one or more complete and correctly punctuated sentences.

Here are three examples that are correctly formatted according to this principle.

Example 1. Programming languages from three decades will be compared:

  • C++ (1985),
  • Python (1991), and
  • Julia (2012).

This type of list could readily be collapsed inline into a regular sentence: “Programming languages from three decades will be compared: C++ (1985), Python (1991), and Julia (2012).” The author must judge whether the list form, with its clearer separation of the items, is preferable. Note the “and” after the second item. This is needed, though I think many people would find the example acceptable without it.

Example 2. We used three different algorithms in the experiments. The table reports the performance of

  • Algorithm 3.1 (based on a Taylor series),
  • Algorithm 3.2 (with parameter k = 1), and
  • Algorithm 3.3 (with tolerance 10^{-8}).

The sentence that precedes the list does not end with a colon. If it did, then the principle would be violated.

Example 3. Before printing the paper carry out the following steps.

  1. Run the paper through \LaTeX.
  2. Run the paper through BibTeX.
  3. Repeat steps 1 and 2 until no warnings occur about unresolved references.

In this example each list element is a complete sentence. The numbering of the elements is appropriate because they have a definite order. In some other list in which the ordering is arbitrary one might nevertheless decide to number the elements in order to be able to refer to the elements later on.

The next example does not conform to the principle.

Example 4. We will use three test matrices.

  • rosser: a classic symmetric matrix with close and repeated eigenvalues,
  • hilb: the Hilbert matrix, and
  • pascal: a matrix with elements taken from Pascal’s triangle.

If we remove the list markers then the list items turn into a sentence fragment containing no verb. I suspect few people would object to the formatting of this example. But we can easily modify it to conform.

Example 4 (modified). We will use three test matrices:

  • rosser, a classic symmetric matrix with close and repeated eigenvalues;
  • hilb, the Hilbert matrix; and
  • pascal, a matrix with elements taken from Pascal’s triangle.

Note the use of the semicolons to separate the items, which now contain commas. I much prefer the modified example, although in the past I might have been happy with the original.

Of course, style guidelines for individual publications may override what I have said above. However, my experience is that academic publishers tend not to change authors’ lists unless they are clearly grammatically incorrect. For example the SIAM Style Manual advises copy editors “Retain the author’s list style as long as it’s consistent”.

Finally, I note that the The Princeton Companion to Applied Mathematics follows the principle recommended here, though that is purely a coincidence. I did not think about list punctuation at all while working on the book, leaving it in the hands of copy editor Sam Clark (T&T Productions, London).