>We walked down the path to the well-house, attracted by the fragrance of the honeysuckle with which it was covered. Some one was drawing water and my teacher placed my hand under the spout. 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 left the well-house eager to learn. Everything had a name, and each name gave birth to a new thought. As we returned to the house every object which I touched seemed to quiver with life. That was because I saw everything with the strange, new sight that had come to me.

-Helen Keller, The Story of My Life

See http://www.slate.com/blogs/lexicon_valley/2013/10/16/piraha_...

'Among Pirahã's many peculiarities is an almost complete lack of numeracy, an extremely rare linguistic trait of which there are only a few documented cases. The language contains no words at all for discrete numbers and only three that approximate some notion of quantity—hói, a "small size or amount," hoí, a "somewhat larger size or amount," and baágiso, which can mean either to "cause to come together" or "a bunch."'

...

'We can only count so fast, after all, but the Pirahã appear not to be counting at all—because, well, how could they? Instead, they're employing what Everett calls an "analog estimation strategy," which works well for a few items but breaks down beyond that.'

In the exact same way that mathematical notation expands cognition of math. By giving us a way to agree on and define ideas in a way that allows us to discuss, think about, and develop further, related ideas. As we grow, we construct a massive scaffold of ideas built on other ideas.

Some of the accounts of Pirahã subjects are really interesting: IIRC when shown two groups of (for example) 4 and 7 objects and asked to describe the difference between them, some people didn't care to answer or said they were the same. Others were puzzled; they could tell there was a difference, but couldn't articulate what it was. And you need to be able to articulate/describe that difference before you can go on to trigonometry.

Naming things makes them easier to distinguish, see the Wharfian Hypothesis [1] (which has been validated experimentally).

For example, if you don't have different words for blue and green (which is the case for some languages), then it's harder (slower) to distinguish distinct color samples that are blue and green than if you had words for the different colors.

If you're an Inuit, and you have words for different kinds of snow, it's going to be easier to disambiguate them, as you have a mental anchor (the name) for each concept.

In programming terms, I think there's a meaningful analogy with heap vs. dictionary lookups, with words allowing more efficient retrieval of unrelated concepts.

[1] https://en.wikipedia.org/wiki/Linguistic_relativity