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

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.

• @prash: How is that at all related? That question is about semantics. I'm asking about syntax. – Mechanical snail Oct 22 '12 at 2:43
• If 'plus' is a conjunction for mathematicians, 'less', 'times' and 'over' are also. They have to be the same part of speech. – Peter Shor Oct 22 '12 at 14:41
• My suspicion is that in English, most of these operators can convincingly be argued to be conjunctions, prepositions, marginal prepositions, and adjectives depending on the situation and the arguer. Personally I think that in mathematical speech, there are covert categories overloading regular parts of speech. – Mark Beadles Oct 23 '12 at 2:11
• This is a great question! I remember coming up against it myself years ago, possibly when working on a machine translation project when I was pos-tagging words in my dictionary or from my corpus. – hippietrail Nov 1 '12 at 16:17
• if you listen to advanced math lectures you might come to the conclusion that you need completely new categories for grammar altogether ;] "take double u - tau bar lower z to the y" – Anno2001 Nov 3 '12 at 20:14

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

Because mathematics is almost a language unto itself, it need not use exactly the same grammatical categories as the discourse language surrounding it. Mathematics itself has "parts of speech" corresponding to the "arity" (valency, rank, adicity, or degree) of an operator:

• Prefix unary operators (root x, cos x, sin x, div B)
• Postfix unary operators (x squared, x cubed, x factorial)
• Infix binary operators (x plus y, x minus y, x times y, x over y, x xor y, x mod y, y to the [power of] x, x equals y, x is greater than y)

However, each of the examples you gave does map fairly cleanly onto one or two English parts of speech.

Prefix unary operators behave like noun-genitive constructions. For example, "cos τ/4 = 1" the cosine of tau over four equals one. This is clearest to see with root, elided from square root of.

Postfix unary operators are either participles (as you mentioned) or genitive-noun constructions (2 factorial or 3 factorial like Ford parts or Nissan parts).

Infix binary operators behave like adpositions, the same part of speech as the English words over and of that you mentioned. One might analyze cross as if it had elided from crossed with in the same way that inside as a preposition (inside the trailer). In this way, plus behaves as if it were with the addition of: one, with the addition of one, equals two. This model solves the problem of "multiple conjuncts joined by the same conjunction": the book in the bookcase in the room in the house, not *the book, the bookcase, the room, in the house.

Virtually all English adpositions are prepositions, but times often acts as a postposition: three times five or five three times. Consider the title of the song "Three Times a Lady" by the Commodores: (three times) a lady, not *three (times a lady).

The exception to this is comparison operators, which turn two non-boolean values into a boolean value. These act as copulas (linking verbs, the same category as is): two plus three equals five, six is less than nine. They agree with a singular subject because an object in mathematics is a single thing even if its value exceeds one. I model these by treating the subject as an understood appositive the number, as in the number six is less than the number nine.

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`.