> Language can exist entirely independently from senses and cognition.
Helen Keller begs to disagree. Language and cognition were clearly linked for her.
> It wasn't until April 5, 1887, when Anne took Helen to an old pump house, that Helen finally understood that everything has a name. Sullivan put Helen’s hand under the stream and began spelling “w-a-t-e-r” into her palm, first slowly, then more quickly.
> Keller later wrote in her autobiography, “As the cool stream gushed over one hand she spelled into the other the word water, first slowly, then rapidly. I stood still, my whole attention fixed upon the motions of her fingers. Suddenly I felt a misty consciousness as of something forgotten–-a thrill of returning thought; and somehow the mystery of language was revealed to me. I knew then that ‘w-a-t-e-r’ meant the wonderful cool something that was flowing over my hand. That living word awakened my soul, gave it light, hope, joy, set it free! There were barriers still, it is true, but barriers that could in time be swept away.”
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).