This is a longer-form response to a tweet from my friend Steve:

Are there things (concepts, feelings, thoughts, materials) that actually cannot be described in words? Or are some things just hard to describe but could be described with enough patience & vocabulary?

— steveflanagan.eth (@flevestanagan) June 16, 2020

In other words, indescribability:

I think in a strictly formal, theory-sense:

- Let’s define a “description” as a finite sequence of morphemes.
- Any human language necessarily has a finite number of morphemes. Because every morpheme must be used at least once over time.
- … thus it follows that the set of all
*descriptions*as we defined it is*countably infinite*.

On the other hand, I think it’s pretty reasonable to believe that the set of all things that we may ever want to describe is *uncountably infinite*. Which implies that this set is infinitely larger than the number of possible linguistic descriptions we could ever create.

Perhaps if there were a hypothetical language with an infinite number of morphemes/symbols?

## What does this mean, practically?

I think a practical interpretation is that there are infinitely more variations and nuances and kinds of experiences available to us as humans, than there exist symbols that we have time to make up to represent those things. I think the best we can do as a result is to provide reasonably good “effective descriptions” whose margins of error overlap and disappear over time. In theoretical-physics parlance, we may call this a perturbative theory of descriptions.

I think the fact that language and vernacular evolve constantly over time is a consequence / proof of this big discrepancy between the set of human experiences and the set of possible composition of symbols we have available to us in languages.

## The limit theorem of descriptions

These are two related questions:

- Are there indescribable things? (I think, yes.)
- Are there objects for which there exist no descriptions whose margin of error is acceptably low?

(2) is less about the nature of descriptions per se and more a question of whether there exist objects that we may want to describe, whose margin of error does not vanish as we craft ever-more-precise linguistic descriptions, i.e. things that defy precise description.

We’re basically asking whether there exist objects in human experience whose subjective experience cannot be approximated arbitrarily closely. In other words, are there objects, for which there is a nonzero lower bound to the margin of error of descriptions? I think this sounds very close to the epsilon-delta formulation of limits. These would be “indescribable” in the sense of question (2) above.

So, what kinds of things don’t have well-defined limits in mathematics? A list:

- Things that exhibit asymptotic behavior against a finite abscissa
- Things that oscillate rather than settling in on a value, or whose left- and right-limit values differ

I’m not sure what the correct analogy for (1) would be, but a good analogy for (2) I think is objects whose experience is different every time, or different for every person. These things don’t have well-defined descriptions because they don’t have well-defined values to begin with. The object itself is mercurial in time and between people, and in that way defies a single static description.

Fear, love, and other subjective things come to mind as examples.