Believe it or not, that monster actually makes sense.1 It’s a sentence diagram of mathematical notation. This has probably never been done before by anyone ever.
Non-math people sometimes joke about how they quit math when it started to be more letters than numbers. Well, math uses symbols for words and phrases, then sentences like this one. This is as much as I can personally speak for, but I have it on good authority that since it’s all written in notation anyway, professionals can sometimes read actual math research papers in their specialty that aren’t even in their language. This diagram is my attempt to prove that one particular lump of notation is also a grammatically correct English sentence.
(Sure, “5-2=3” is a grammatically correct English sentence (for that matter, so is “5-2=2”), but those are boring simple sentences. This one’s more interesting.)
You might argue that it isn’t English because it doesn’t look English, or because it isn’t using the Roman alphabet. Well, what language is “Dôs moi pâ stô, kaì tàn gân kīnā́sō” 2 written in? It’s Greek, but it doesn’t look like it. It is in fact the same sentence as “δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω.” 2 Just because it doesn’t look English doesn’t mean it’s not.
Here’s what my sentence looks like in math notation:
“∀ε>0, Ǝδ>0 s.t. ∀p,q∈X dX(p,q)<δ ⇒ dY(f(p),f(q))<ε”
And in words:
“For all epsilon greater than zero, there exists some delta greater than zero such that, for all points p and q in metric space X, if the distance in X between p and q is less than delta then the distance in Y between f(p) and f(q) is less than epsilon.”
(Epsilon and delta are small positive numbers, f is a function taking points from the space X to the space Y, so that p and q are points in X and f(p) and f(q) are the points in Y where it dropped them off. Distance in Y and distance in X might be measured differently. It’s like Narnia. The sentence itself is a definition. It doesn’t make sense to say whether it’s true without specifying what function fX->Y we’re talking about. It’s true about a particular f iff that function is continuous.)
This sentence is a definition of a particular mathematical concept, and it’s a concept that they might be teaching your kids in school! If you’re a concerned parent, check your child’s textbooks for any reference to “continuity,” then throw this blog post at the superintendent and demand that they remove all age-inappropriate materials from math curriculums. Do you want your kids to be subjected to math like this? This must be stopped before it’s too late!
I took the definition in words, diagrammed that, and then substituted the symbols back in for the words. This worked almost perfectly, but there were a few slight differences.
For one example (actually TWO examples!), take the phrase, “For all epsilon greater than zero…” Here, we’ve got a preposition (“for”), with the noun “epsilon” as the object of the preposition. Epsilon is modified by the adjective “all,” and by the adjective “greater.” “Greater” is itself modified by the prepositional phrase “than zero.”
Did that make sense? We’re talking about the noun epsilon, described as being greater than zero, and specifying that we’re talking about all possible epsilons greater than zero.
In math, the symbol “∀” means “for all.” 3 So instead of having the preposition “for” and the adjective “all,” it’s combined into one word. It functions as a preposition, so that’s where I put it on the diagram. (“For all” as a preposition isn’t normal, but in math it is. Math: Not even once.)
Something similar is happening with “greater than zero.” Ordinarily, “greater” is an adjective (describing epsilon), and then the prepositional phrase “than zero” modifies the word “greater” by describing what it’s greater than. When written in symbols, there’s the one symbol “>” that means “greater than.” Like “∀,” it functions as a single word, a preposition. In this case, the object of the preposition is the noun “zero.”
As you can see, the mathematical symbols are a specialized vocabulary allowing math people to say things like this more quickly, more aesthetically, and more efficiently. I’m guessing that grammatical efficiency wasn’t what they were thinking of when they started using “>” as a preposition, but it’s a nice side-effect.
On the other hand, if it’s about efficiency, then why in the name of the golden rectangle do we not have a symbol for “∀ε>0 Ǝδ>0 s.t.”? When you keep having to write that repeatedly for a bunch of different definitions, you really wish you could abbreviate it, but you can’t because it’ll change the meaning. This seems like the perfect case for where there should be a single symbol meaning “for all epsilon greater than zero there exists a delta greater than zero such that the following statement is true….” But there isn’t and so that string of symbols gets repeated over and over and over. OK, done griping.
Back to grammar. The big horizontal arrow means “implies,” in the mathematical/logical sense of “if the statement in the subject is true then the statement in the predicate is true, if it isn’t then no comment.” It functions as a verb.
The symbol “∈” means “is an element of,” but can also just be read as “in.” (Short for “which is an element of…” That is grammatically not the same thing as a preposition at all, but it sure acts like one.) That’s the way it’s used every time it shows up here. It’s similar to “<” and “>,” where it can be a either preposition or a verb. In this sentence, “>” is a preposition both times it appears and “<” is a verb both times. That wasn’t planned, but it makes things simpler. (And remember not to confuse ε with ∈, because despite my handwriting those are totally different things.)
Diagramming the “such that” clause was beyond my powers, so I called in backup. My sister Rachel (artist’s rendition) once taught me all the nerd stuff I knew and is roughly analogous to my Jedi Master. She said it’s an adverbial subordinate clause. It’s kind of ambiguous because the verb is “exists,” so any description of what kind of existence could be either adjectival or adverbial, but I think she’s probably right.
The big slanted dotted line represents the fact that we’re dealing with a subordinate clause. (It was a straight vertical line until my editor corrected me. And good thing she did, too, because obviously that would have been unforgivably terrible.) It’s drawn from the main verb down to the verb of the subordinate clause. The main verb here is “Ǝ.” 5 (The subject (delta) exists.) That other clause functions as a big long adverb. It’s talking about the qualifications on that existence.
The phrase “such that” functions as a conjunction that connects the clauses while showing which is more important. The abbreviation “s.t.” for “such that” is not standard as far as I know, or at least it never received the Official Math Seal of Officialness, but I learned it with that abbreviation, it’s fairly commonly used among the math people that I know, and it was either use that or leave the subordinating conjunction off the diagram entirely.
The small vertical dotted lines just represent “and.” The distance between points p and q, for instance, where both points are objects of the preposition “between.” Although, that preposition itself doesn’t show up in the sentence ∀ε>0 Ǝδ>0 s.t. ∀p,q∈X: dX(p,q)<δ ⇒ dY(f(p),f(q))<ε, so there’s no symbol for it on the diagram. I could have put the commas where the conjunctions go, but that would have been weird.
In conclusion, this worked. The sentence “∀ε>0 Ǝδ>0 s.t. ∀p,q∈X: dX(p,q)<δ ⇒ dY(f(p),f(q))<ε” is in fact a grammatically correct English sentence, even if it uses a specialized vocabulary and notation. And mathematicians can have whole conversations like that. It. Is. Awesome.
1At least, it should. If I messed up, correct me and I’ll credit you on the rewrite.
2“Give me somewhere to stand and I will move the earth.” Google Translate doesn’t recognize this, but I got it from Wikipedia, which I trust more. If anyone who actually knows Greek can verify this, please do.
3Sort of. It depends on context and grammar, so if you quote the Three Musketeers as saying “All for One and One ∀” then you’re wrong.4
4 Let P(a,b) mean “a is for b.” (In this case, “is for” means something along the lines of “a acts for the benefit of b,” or “a is on b’s side,” or “a will stick a pointy piece of metal into anyone who threatens b.” Astonishingly, there isn’t a mathematical symbol for this, hence the arbitrary symbol “P(a,b).”) Then, the motto is “Ǝy:∀x:P(x,y) & Ǝx:∀y:P(x,y).” (“There exists at least one person such that all are for that one, and there exists at least one person who is for all.”) What the Musketeers really want to say is ∀x,y:P(x,y), “Everyone is for everyone else,” which you will notice is not the same thing as what they actually said.
5 First person to explain why it obviously must actually be a preposition wins a cookie. The cookie is currently unclaimed.