For me, it became clear that we had crossed some horrible line between sanity and madness when journalists started laughing during news stories. Leaks of stories of hallucinatory misbehavior, treason, criminality, and just plain evil have been coming out of the Trump government so fast that it’s become surreal. Potential reasonable reactions include despair, and humor. Taking the second option, the New York Times web site recently published a satire piece called The White House Leak Template for Journalists. You click on various and sundry choices…
…and it generates a little news story about a leaked Trump administration scandal for you.
Scroll to the bottom of this page, and you’ll find screen shots of the whole thing. It’s sadly hilarious, but behind the hilarity is an important point about how language works.
One of the things that’s interesting about language is that every human language (what we call in my line of work “natural” languages, as opposed to computer languages) is capable of saying an infinite number of things. “Infinite” is a big claim, and you’re right to be skeptical about it. So, let me just show you that with even a very small amount of knowledge of a language, you can say an enormous number of things–much more than you might ever have thought–and as you’ll see at the end of the post, this is a fact that has important implications for the many people reading this blog who are trying to learn a second language.
Let’s suppose that you know how to say a simple declarative sentence in some language or another–my dog ate my shoes. You’ve got a subject, a verb, and an object. Suppose that you know 10 nouns and 10 verbs. You can now say the following number of sentences:
10 nouns * 10 verbs * 9 nouns = 900 sentences
Why only 9 nouns in the object position? Because I’m assuming that you won’t use the same noun for the object as you did for the subject. So, whichever noun you pick for the subject, you now have nine choices left for the object, rather than the 10 that you started with.
Let’s suppose that you have a language–like French or Spanish–that inflects all verbs differently for singular versus plural subjects. Let’s also suppose that in our calculation above, we included only singular forms of the verbs. Add the plural form of the nouns and the plural form of the verbs, and now you have the following additional sentences:
10 plural nouns * 10 verbs * (9 plural nouns plus 9 singular nouns) = 1800 sentences
To recap: 900 sentences if you only know the singulars, plus another 1800 if you add the plurals, so you’ve got 2700 sentences that you can say.
Note: this post relies heavily on a branch of math called combinatorics. I stink at combinatorics, so please be kind! Corrections are welcome in the Comments section.
To this point, we’ve only been using nouns and verbs. Let’s add a new kind of word: and. Even if we didn’t know the plural forms of the verbs, and lets us say a truly remarkable number of sentences with just our 10 singular nouns and our 10 singular verbs. Recall how many simple declarative sentences we could say with just 10 nouns and 10 verbs:
10 nouns * 10 verbs * 9 nouns = 900 sentences
Once you’ve picked a noun for the subject, you have 9 nouns left for your object, leaving eight unused nouns. Suppose that you’re going to use and in your object: you have 9 possibilities for the first noun (since you used 1 for the subject) , and 8 possibilities for the second one (since you used one for the subject, and you’ve already used one in the object). So, with and, you have the following number of possibilities:
10 nouns * 10 verbs * (9 nouns + and + 8 nouns) = 1700 sentences
…and if you’re keeping track, that’s 900 + 1800 + 1700 sentences, or 4,400 sentences.
Of course, we’re not done with and yet–since you’ve learnt to use the plural forms of verbs, you can use and in the subject, too. The calculation of the number of sentences that you can make with and in the subject (but just a single noun in the object) is similar to what we just did:
(10 nouns + and + 9 nouns) * 10 verbs * 8 nouns = 1520 sentences
…getting us to 5,920 sentences.
Of course, you can have two nouns in the subject and two nouns in the object, as well–you can do the math. What’s cooler is that you can use and to join together two sentences, too. Let’s take the “formula” that gave us the smallest number of sentences: singular subject, singular verb, singular object. Remember how we calculated the number of sentences that we could make with only 10 nouns and 10 verbs:
10 nouns * 10 verbs * 9 nouns = 900 sentences
How many sentences can you make by joining two sentences together with and? The possible assumptions are numerous. Can you repeat the subject? Why not? (Dogs chase cats and dogs chase balls.) Can you repeat the object? Why not? (Dogs chase cats and children chase cats.) Certainly those are weird, though, so let’s estimate that maybe 10% of our possibilities aren’t going to be OK, and just calculate from the numbers that we used for the simple declarative sentences. That gives us this:
10 nouns * 10 verbs * 9 nouns and 10 nouns * 10 verbs * 9 nouns = 1800 sentences; subtract 10% of that for the ones that repeat too much and you still went from 900 sentences to 1620 sentences with just one additional word.
…in other words: as soon as you throw and into the mix at the level of sentences, you double the number of sentences that you can make. (The last time we tried to total how many sentences we could make, we had 5,920. Double that with and, subtract 10% for the sentences that repeat too much, and you have 10,656 sentences.)
What happens if you add or to your armamentarium? You just doubled the number of sentences that you can make again. How about throwing but in there? You just doubled it again. (We’re around 40,000 sentences right now, even with our 10% adjustment for repeated things.) Add one more tense and you just…well, it just got really, really big. And let’s review what you know–it’s very little:
- 10 nouns, singular and plural
- 10 verbs, singular and plural
- two tenses
For those of us who are as math-challenged as I am: that’s 23 words and two tenses to give you around 40,000 sentences. Throw in some adjectives… Learn how to turn a simple declarative sentence into a question… Learn a few names… Learn to say he, she, and it… Add because…
Now, I know what you’re thinking: I know a hell of a lot more than 10 nouns and verbs in French, but it sure doesn’t feel like I know how to say very many things. Remember, though: as we discussed recently, you can get a surprisingly long way on a pretty small amount of a language. This is a skill that you can develop with practice: think about simple ways to communicate your wants and needs, and I bet you’ll come up with creative ways to work around your lack of knowledge of a language.
A technical excursus: recursion
When we got into and, we touched on an important mechanism of language that leads to the fact that every human language is capable of saying an infinite number of things. Called recursion, it has a specific definition in mathematical formalism that you can find here; for our point of view, it means that some things in language that we care about, such as sentences, can be made up of other things of the same type. For example, we used recursion when we made the sentence Dogs chase cats and dogs chase balls out of two sentences: dogs chase cats and dogs chase balls. We could also use recursion to make noun phrases (the groups of words that make up the subjects and objects in our examples): the noun phrase my dog and your cat is made up of the noun phrase my dog and the noun phrase your cat. In principle, is there any limit to this? No, actually. You would die before you could say an infinitely long sentence, and even if you could live long enough to hear one, by the time you got to the end you would most likely have forgotten the beginning. But, that doesn’t change the fact that the language, by virtue of having this fundamental property of recursion, can produce an infinite number of things to say.
If no one could ever say an infinitely long sentence, who cares about understanding how and why languages can produce the things? For one thing, infinity is a pretty big deal, and if you’re dealing with a system of any sort that’s capable of infinity, then if you want to be able to understand how it works, you need to understand that aspect of it. I believe it was Chomsky (who in many ways was a horrible thing to have happen to linguistics) who made the analogy that just because no marathon runner can run forever doesn’t mean that it’s not useful and important to understand the physiological mechanisms that let them do it.
You made it this far? Great! Your reward is the New York Times Leak Template. Read it and laugh–then go subscribe to a newspaper. Keeping journalism alive is essential to getting the traitors that are currently running our federal government out of the White House. Feeling geeky? Calculate how many news stories about Trumpworld scandals this would generate–and ask yourself if that would be enough…