Someone over at Metafilter had a great question: What syntactic category are mathematical operands? (Their username is notsnot, in case this needs to go in a dissertation someday.)
Let’s start with something like
Three plus four is seven.
For now, we’ll say the numerals are NPs. In a construction NP <word> NP, the <word> could be various things: a verb, a conjunction, a preposition. We can immediately rule out verbs, since plus and its friends (minus, times, over, etc.) are not conjugated.
We should also look at non-mathematical sentences, like
Determination plus luck means victory.
Let’s do some syntax. There are some standard though fallible tests for prepositions. For instance, they can usually be modified by right:
He fell right in the river.
She lives right down the street.
Go to the cave right in the forest.
*Determination right plus luck means victory.
*Three right plus four is seven.
Prepositional phrases (PP) can often be fronted:
Up the hill she walked.
*Plus four three is seven.
You can front a PP and replace the NP with an interrogative, or front just the questioned element:
Sam is the king of England. Seven is three plus four.
Of what is Sam the king? *Plus what is seven three?
What is Sam king of? *What is seven three plus?
PPs allow gapping:
Sam is king of England, and Joe, of France.
*Seven is three plus four, and eight, plus five.
These tests aren’t definitive, but plus is failing every one of them. A better match might be conjunctions. Plus, like and, can link NPs or sentences, and can be used multiple times:
Bill and Anne and Rahesh came. Two plus two plus one make five.
Bill came, and Anne left. Bill came, plus Anne left.
On the other hand, this transformation sure doesn’t work:
Sam is a king and Sam is a dancer. >> Sam is a king and a dancer.
X is 4, plus X is sin θ >> *X is 4 plus sin θ.
It looks like the construction S, plus S isn’t really the same plus as in two plus two. And other operators don’t allow it at all:
*Bill left, minus Anne stayed.
*Bill left, over Anne stayed.
A bigger problem is that English sentences allow literally infinite amounts of inserted material.
Sam and Alice are nobles.
Sam and Alice are fine, just nobles.
Sam and Alice are still nobles.
Sam, prince of Florin, and Alice, duchess of Guilder, are nobles of Sylvania.
Sam and possibly Alice are, as of Tuesday, nobles.
How much of this can we do with mathematical expressions?
Three plus four is seven.
Three plus lovely four is seven.
Three plus four is still seven.
Three, square root of nine, plus four, half of eight, are seven.
Three and possibly four are, as of Tuesday, seven.
These are not impossible, but at best they sound jocular. The additions are not math; they’re intrusions of ordinary English.
There’s also the complication that mathematical plus and minus can be unary: you can say Minus three plus four is one. You can have a conjunction beginning a sentence (And the Lord said to Moses…), but that’s not how minus is working here; it’s obviously a modifier for three.
Not to belabor the obvious, but many of the basic things we can do with a sentence don’t really work in mathematics. You can’t really put a mathematical expression in the past tense, or use the present perfect, or use pronouns, or passivize, or insert a relative clause, or nominalize, or cleft, or topicalize.
And all this is looking at a very basic expression that probably did arise out of normal syntax. It’s even harder to apply our notions of normal English syntax to something like
x equals minus b plus or minus square root of b squared minus four a c over two a.
e to the i n equals cosine of n plus i times sine of n.
I’ve gone into this much detail to convince you (and myself) that ordinary English syntax doesn’t really explain mathematical expressions. I hope my conclusion doesn’t shock or appall you: mathematical expressions don’t follow English syntactic rules; they follow mathematical rules.
Now, maybe you could shoehorn the quadratic formula or Euler’s formula into the syntactic framework of your choice. I will bet you, however, that you’ll end up with a pile of very idiosyncratic special rules and special syntactic categories, and a bunch of ad hoc exclusions of normal English rules.
And there’s an alternative formulation that would end up far simpler than that: mathematical expressions have their own cross-linguistic syntax, based on their written form, and languages have conventions on how to say them aloud.
I don’t think this is terribly surprising… it’s like discovering that the Russian of Tolstoy’s War and Peace contains a number of passages which are written in the Roman alphabet and don’t follow ordinary Russian syntax. Is this a revolutionary discovery about weird undercurrents of Russian? No, it’s just that Tolstoy included quite a bit of French in the text. Similarly, English sentences can have embedded mathematics.
Still, I hadn’t thought about it this way, and I find it interesting that a pretty ordinary part of English turns out to be, well, not really English at all.
Now, for historical and practical reasons, there’s a certain overlap, especially with basic arithmetic. People undoubtedly said “Two and two are four” (or “twá and twá sind féower”) long before international mathematics was formalized. So these behave more like ordinary English than the quadratic formula does.
Plus, the conventions for speaking math out loud were, of course, invented by speakers of the language out of existing (or newly borrowed) words, and follow ordinary language conventions– where possible. So you can read cos (2θ) as “cosine of two times theta”. On the other hand you can just read it as “cos two theta”, which probably has no non-math analogue in English.
(I should add that programmers are very familiar with the idea that math expressions have a particular syntax. They don’t bother with linguistic categories at all; they define their own, such as operators, variables, constants, functions, and statements.)