Grammar Mistakes in Math Writing

Question

May be my question is simple. But this question is confused me. In fact, I want to present a lecture and I don't want to have grammar mistakes.

My question: which one is correct?

• A: Let $\mathbb{F}_q$ denote the finite field with $q$ elements. Consider the two elements $\alpha$ and $\beta$ in $\mathbb{F}_q$.

• B: Let $\mathbb{F}_q$ denote the finite field with $q$ elements. Consider two elements $\alpha$ and $\beta$ in $\mathbb{F}_q$.

In other language, I really like to understand how to use the word the in math writing.

Thanks

Edited based on the @user2357112 comment.

Answer 1

A typical statement concerning finite fields would be statement $B$:

Consider two elements $\alpha$ and $\beta$ in $\Bbb F_q$.

Here the two elements are arbitrary. With an article "the two elements" they would be specific elements, such as $0$ and $1$ for example. Usually such a specification would then be given. "Let $\alpha$ and $\beta$ be the elements in $\Bbb F_q$ given by the sum of all squares respectively of all cubes."

Answer 2

The two cases convey different meanings:

• In the first case, it sounds like $\mathbb F_q$ has only two elements, and the two elements are $\alpha$ and $\beta$
• In the second case, $\alpha$ and $\beta$ are two of all possible elements of $\mathbb F_q$ (which may be the same element).

Answer 3

I would not use either form but employ a formal mathematical expression for $\alpha$ and $\beta$ that more precisely defines them. That avoids English entirely as a weak link in conveying your intent. Use slides or a white/blackboard to be explicit about these things when presenting.

English is full of ambiguity (and I'm a native speaker) and for clarity it is best to resort to mathematics proper when defining things.

It's also useful in presentations to have key definitions visible on e.g. a white board so that anyone can refresh their memory at a glance.

Answer 4

As native English speaker, both sentences convey the same meaning: here is a field ${\mathbb F}_q$ and consider (the) two elements $\alpha$ and $\beta$ in there. Another poster has said that "the" conveys the intent that there are only two elements: this isn't true because you introduce the elements by noting that there are $q$ elements in the field. It does however suggest that you want to emphasize these two elements for a reason.

"the" is used in mathematical English in the same way as in non-mathematical English -- which is perhaps a little unfortunate as the way English uses articles (the words "a", "an" and "the") is not always easy to understand. However:

1. "the" can be used to indicate a specific instance of something: "let us consider the unique (up to isomorphism) field of...". Here, we cannot drop "the" because we're talking about an object of some kind, so we need an article to introduce it. We can't use "a(n)" because it's unique, and "a(n)" implies there is more than one

2. "the" can be used to introduce plural objects: "here are the elements of the ring...". "A(n)" introduces singular objects, or count nouns (e.g. "a crowd of people", "a gaggle of geese")

3. we can drop the article ("a", "an" or "the") when talking about an abstract quantity, which reinforces that it is abstract: "We have elements that together form a ring" is equivalent to "We have the elements that together form a ring".

In point 3, the word "the" does subtly change the meaning of the sentence, but not grammatically. It adds a little extra emphasis to the noun (here, "elements") that you might want for a specific reason.

Answer 5

Option A sounds slightly off to me, and I prefer option B. It's a little difficult for me to explain why, but here's my attempt: because $\alpha$ and $\beta$ have not been previously introduced, nor are they in some way uniquely determined.

I looked up the definition of "the", and I found this:

• denoting one or more people or things already mentioned or assumed to be common knowledge.

• used to refer to a person, place, or thing that is unique.

Neither of these criteria are satisfied in this case.