I said language can exist independently, not that all language exists independently.

"one plus one equals two" can be understood and worked with without ever feeling water over your hand. It is a priori knowledge (see Hume's fork for an explanation).

You have to understand that the richness of language linked to cognition is due to your experience with that part of language and resulting romantization of it. It doesn't mean that it is a core defining feature of language, even though it feels that way (and as touching as that anecdote is).