4/17: The infinities of language

So it turns out language is infinite. Or at least, it's the "infinite use of finite means," according to linguists Wilhelm von Humboldt and Noam Chomsky. Does that mean that we can never completely learn any language, including our native language? Can we ever learn anything completely? And what does language being infinite even mean? 

To illustrate the infinity of language, try this: Think of a sentence, any sentence. It could be in your first language or a language you are learning. Anything that you come up with will be finite—you can count the number of words, characters, letters, syllables, or other type of grammatical unit, and they will add up to a number. Now, try to make the sentence longer. It's easy to do this in English by changing any noun (let's say "the girl"), to a more complex noun (like "the girl's dad," or "the girl's dad's friend"). The same thing can be accomplished in other languages; elements like the possessive in English are recursive, meaning that they can be used over and over again, with no necessary end. In Spanish, we might similarly change la chica to el padre de la chica, or el amigo del padre de la chica. 

With recursion in hand, imagine you could keep expanding your original sentence outward and outward. Every single example you could generate would be finite, because you could count up every word contained in it to a number. But every single example you think up could also be made longer, if only by one or two extra words. Congratulations, you've just proven that there is no longest sentence—so does that make language infinite? 

At this point, we could get into a discussion that would rather resemble mathematics, combing through the differences between infinite and countably infinite and unbounded and more, which is not the goal of this discussion, at least not for right now. (But if you are interested, you can consult papers like this one.) So let us assume that, since there is no longest sentence, the number of possible sentences in a language is, for all intents and purposes, infinite. Does this really matter, when most of this infinite collection of sentences will never be said? The things that actually get said, interpreted, and acted on in natural language are limited, by several factors.

First, we are constrained naturally by time. While repeatedly recursive sentences that would take 200 years to say out loud do exist, they will never be said by any human. And we are also constrained by what is reasonable to say in any given context. While you can say anything you want, at any time, it's most likely that you will only say things that make sense in the context of your current conversation, or situation. Furthermore, our lives are finite, so each human who knows language can only ever say a finite number of things. Even if we add up all language across all human history, it would sum to a massive number, but a number nonetheless, not an infinity.

So, if you're learning a language, its apparent infiniteness shouldn't be much of a problem. All you have to do is learn enough of the language to be able to generate sentences that make sense in conversations you find yourself in. Your own first language is infinite, too, and you know just enough of it to be able to respond to all of the finite contexts that you will experience in your life (At least, that's the hope. Of course, there are times when language feels limited, unable to express the complexity or depth of a feeling, or a situation. More on this in a moment.)

We've just discussed two ways of thinking about the scope of a language—the number of possible sentences, compared to the number of reasonable ones. Even if both sets might be infinite, the latter is much smaller, and we will only encounter a finite portion of it during our lives. But beyond the concrete elements of a sentence, what about the intention behind it, the meaning? Is there an upper bound on the number of things we could say that have meaning? We say things because they have meaning, even if that meaning is as simple as: "I'm not sure what else to say in this conversation, so I just made this small comment to keep things going." Everything you or I say has some intended meaning for the listener, or for ourselves.

But I could also say this sentence: This newsletter is made out of pasta salad. That's not only untrue (as far as I can tell), but it also has no discernible meaning. It doesn't communicate or express anything that has value in our conversation. Maybe, one hundred years in the future, there will be an information-transmitting device that spits out documents or newsletters spelled out using the recipients' favorite food item, and there will be a person with a deep love for pasta salad who has all of his newsletters delivered to him in that medium. At that time, this sentence will have a clear meaning, both to that person and to anyone who understands the pasta-salad-newsletter experience.

And maybe we could even say that by creating this imaginary scenario, I've actually imbued this sentence with a meaning it never had until this exact moment. If we believe this, then we believe that the limits meaning can change over time—which, of course they can. "What's the wifi password?" had no meaning in English 200 years ago, but it's quite meaningful now. Meaning changes not only to accommodate new material circumstances (like wifi), but also because humans constantly use language to purposefully create new meaning: from their imaginations, their poetry, their fantasies. It's this type of change that allows us to create frameworks to respond to those ineffable things, those emotions where language struggles to guide us. 

This possibility, that language has infinite meaning over time, is the most hopeful axis to understand language's apparent infinity, I think. We have the power to alter and imbue language's meaning, to shape our native languages and the languages we learn. By creating new ideas, new sentences, new meaning, we also strive toward the potentially infinite meaning that humans might be able to create, across the finite spans of our lives, together.