Infinite worlds beyond language

All English sentences can be expressed with a finite number of characters (the alphabet, the space, punctuation, maybe a few other things). Thus we can assign a number to English sentences. “The King of France is Bald” might be 17217245, for example.

Notably, the number of possible sentences is going to be countably infinite.

Now right away this is very interesting. Because a sentence can (among other functions) describe the world. Every way the world can be, that we can describe using English can thus be put into a one-to-one correspondence with the counting numbers.

To see why this is extraordinary, consider that there are unfathomably more irrational numbers than counting numbers. See the proof here. Thus there are unfathomably more numbers than can be described in English.

Furthermore, for any language with a finite number of symbols, there will be numbers that language cannot describe. Almost all numbers, in fact.

Let’s make the following bold conjecture:

Anything that can be expressed in a language with a finite number of symbols can be expressed in English (or pretty much any other natural language) given enough (but finite) characters.

If this is correct, then all numbers that can be talked about in any language can be talked about in English, and all the numbers that can’t be talked about in English are dark to all languages. They cannot be talked about at all in any possible language- at least not specifically- one can refer to the general category and prove its existence. Call the set of all numbers that can’t be specifically referred to the dark numbers(1).

Consequently to there being dark numbers, many, perhaps in some sense, most, ways the world can be (e.g. X being some dark number of meters away from Y) cannot be talked about specifically in language. We can indicate that these possible states of affairs outside language exist, but not name them in particular.

What a wonderful world, eh.

But what about context?

I made a mistake, not so much in the analysis, but in the simplification. I underestimated my readers. I thought I could get away with not talking about the contextual element of language here to keep things simple. Wrong! Isaac and N0st have pointed out, correctly, that when we factor in the contextual aspect of language, the potential richness increases.

This is true, but not, I think, in any way that’s going to get us beyond the limits of language. We can refer to the distance between X & Y, and if the universe is continuous, there’s a good chance, depending on the exact formulation of the laws of nature, that this distance will be a dark number. But there’s no way we can know this, or talk about, so to speak, the quantity itself. If we try to measure it, we’ll approximate it to some rational number, or in a few cases (circumference of a circle), to some irrational number. In some sense that’s hard to pin down exactly, the number is still outside language.

Now beyond numbers, what about the general thought that language is merely countably infinite, which seems so scarce compared to the potential richness of the world? Does contextuality in all its forms- definite articles, indexicals, pointing- get us out of this? No, I don’t think so.

First note that, within a given context, there’s only a countably infinite number of things I can say. Next note that, within a context, we can ‘context jump’- “Imagine I am in such-an-such a context when you interpret the next thing I’m going to say.” Thus context is no way out. There are a few more subtleties here that I’d have to deal with in a proper treatment, and I’m happy to talk about them in the comments, but I think this effectively takes us back to square one.

Footnotes:

(1): The term dark numbers was my coinage, but it seems like, upon googling it, some other guys might have also coined it, albeit using a very different argument to me. Not quite sure what this guy is saying, but maybe it’s something similar? Also, what about this guy? I think, perhaps surprisingly, they all mean something similar- unnameable numbers.

Anyway, take everything I’m saying with a grain of salt, I’m not a mathematician or anything, I’m just screwing around.