“Matrix M” or “the matrix M”? English articles and mathematics

Articles are a difficult topic, even for native English speakers, and their usage varies according to the subject and style of a document. In computer science and engineering academic papers, the articles are often omitted, which can look odd or incorrect. However, formal mathematical style often omits many articles. Even for this style, the rules are difficult to articulate and open to interpretation. However, if you ever wondered why it’s “matrix M” instead of “the matrix M,” the general rules of this style are as follows.

  • In phrases such as “update rule h” or “set Q,” the symbol appearing after the noun already makes the noun unique, thus the article “the” is not necessary.
  • For lists of items that take the same article, the article can be omitted from the later terms. Hence, “a cat, dog, and rabbit” is okay, but the list “a cat, a dog, and an owl” needs all the articles for every item.
  • In phrases that use “first,” “second,” or terms like “ith,” e.g., “the ith update rule h,” “the” is needed to be grammatically correct. Logically, the phrase “ith update rule h” is unique, but to be native in style, the “the” must used.

However, these cases differ from ordinary English, so they also make the text more difficult to read. In longer descriptive sections where high level concepts are used, it may be best to keep the article so that the reader can concentrate on the high-level ideas.

  • In phrases where the notation is followed by a restrictive clause, e.g., “the maximum element m_max in set M,” if the article is needed to make the restrictive clause grammatically correct, it should be used.
  • When introducing a new notation, the article “a” can increase readability. In the sentence “The model then outputs a prediction v.” The “a” emphasizes that it is being introduced, so it assists readability. Once the term “prediction v” has been introduced, it does not need the article “the” that would be used in conventional English.

The model then outputs a prediction v. If the class of prediction v matches the true class of the target object, it is called a true positive.

By contrast, if the text is highly mathematical, the article can be omitted according to the rules above, even when a notation is introduced. This is often best because it allows the reader to concentrate on the mathematics. In the example

Let C(r, o) be a circle with radius r centered at point o.

the articles “a” for radius r and point o could be added, but this makes the text more wordy than it needs to be.

Let C(r, o) be a circle with a radius r centered at a point o.

If a low word count is more important than readability, the articles can also be omitted according to the rules above. To delve deeper into this topic, a detailed discussion about mathematical writing can be found in Mathematical Writing by Donald Knuth, Tracy Larrabee, and Paul Roberts.

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