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Are there any rules about when to say exist and when to say exists in mathematics? For example, both these sentences appear in a book of mine:
There exist αi in I such that xn = Σ αi xi.
There exists s'' in S such that s''(s'm – sm') = 0.
I'm a bit confused about when to use exist and when exists.
Exists and exist follow the ordinary convention for verbs: one is singular and the other is plural. Where mathematical usage differs from ordinary usage is in the way singular and plural are indicated in the subject that follows, and an implied “for all” later if the subject is plural.
Exists is singular:
There exists s'' in S such that s''(s'm – sm') = 0.
Or, spelled out more explicitly:
There exists a number s'' in the set S, such that s''(s'm – sm') = 0.
Exist is plural:
There exist αi in I such that xn = Σ αi xi.
This is where the mathematical usage differs from ordinary usage. Spelled out explicitly, this would be:
There exist numbers α1, α2, α3, etc., which are elements of the set I, such that for each subscript, if we refer to the subscript as i, then xn = Σ αi xi.
The convention is tricky for a beginner to understand because it depends on your knowing that i is commonly used as a variable that will stand for multiple subscripts. The plural form of “exist” is actually a helpful clue. However, you must understand the implied “for each” later in the sentence. It’s implied by the fact that the sentence uses i to stand for multiple values.
I would say some author would like to use "there exist" as a direct translation of the existential quantifier. In this case, like in your case, the author does not bother to fit his writing in a plain English context. Note that if your readers are all mathematicians or people with pure mathematics background, then you need not worry about such a usage would confuse your reader.
Nevertheless, I prefer to write, for example: "For every real number ϵ there is a real number δ such that ..." or "There are integers x, y, z such that x2 + y2 = z2 " or "There is a triple of integers $x, y, z$ such that x2 + y2 = z2.
The case that interests you is not the only case. For instance, we can easily find that some author would write "Given ϵ > 0, ..." instead of "Given an ϵ > 0".
A beautiful mathematical writing should be like the author is talking and walking with the reader.
If you are interested, please be referred to How to write mathematics
