How are mathematical operators like “plus” and “cos” analyzed?

Question

Consider the mathematical statement

1 + 2 = 3

It is read in English as

One plus two equals three.
One plus two is equal to three.

In English at least, equals is obviously an ordinary verb, but the analysis of "one plus two" isn't obvious. Some other languages have similar constructions; for example:

Spanish

Uno y dos es igual a tres.
Uno más dos es igual a tres.

Notice that unlike the usual "y" ("and"), "es" is the singular conjugation.

Syntactically, plus, minus, times, etc. act a bit like a conjunction, but there are some differences:

• Mathematical operators are an open class, whereas conjunctions are normally considered closed; new operators can be created whenever they are mathematically useful (e.g. "xor", "dot", "cross"). Similarly, "plus", "minus", and "mod"/"modulo" seem to be loanwords from Latin.
• Plural nouns joined by conjunctions are plural, whereas mathematical-operator phrases are always singular ("Cats and mice are animals", vs. "Two cats plus two mice equals two fat cats.")
• Whereas multiple conjuncts joined by the same conjunction usually elide all but the last (e.g. "A, B, or C", "A, B, and C"), this is ungrammatical for mathematical operators ("x plus y plus z", never *"x, y, plus z").

Also, some operators appear to derive from other classes:

• over (division) "pi over 2": preposition
• of (function application) "f of x": preposition
• less (subtraction; synonym of minus) "x less its mean": adjective?
• times (multiplication) "2 times 3": plural noun
• squared, cubed: verbal participles
• to the "e to the x"
• dot, cross (vector operations) "tau equals r cross F": nouns

• many unary operators are derived from nouns:

  • trigonometric and hyperbolic functions: sin, cos, tan, arcsin, ... "sine pi equals zero", "(the) sine of pi equals zero"
  • factorial "four factorial equals twenty-four", "24 is the factorial of 4"
  • root (sqrt) "root two over two", "the square root of two over two"
  • gradient/*del*, div, curl (vector calculus) "div B equals zero"

So what lexical class(es) do mathematical operators belong to, in spoken mathematical usage? I'm interested in how they can be analyzed both in English and in other languages.

Answer 1

Apologies, it seems I don't have enough credit to post a comment, so an "answer" will have to suffice!

I would analyse the elements on either side of a comparative expression such as:

One plus two is equal to three

to be almost like proper nouns referring to abstract mathematical concepts. While the result of the mathematical operation is most certainly non singular in number, we are referring to one, singular, unique, mathematical operation.

If you want to bring algebra into it, replace the left hand side with x and the right hand side with y. Like that, it becomes clear, to me, at least, what the underlying structure of the expression (linguistically speaking) is.

That explains the conjugation...

You may be right that we need another class for mathematical operators, but rather than looking at them as singular words within a larger clause, what happens if we examine them as subclauses in their own right?

It could certainly account for the absence of elision in x + y + z.

Answer 2

Perhaps a bit informal but the introduction to The Princeton Companion to Mathematics includes a rather nice discussion of the grammar of mathematics.