E.g. the word “carbohydrate”, or it’s equivalent in different languages, describes a concept; but the concept itself exists independently of any language construct that describes it. Similarly, in math, a concept e.g. “prime number” is independent of the many ways and many languages you can use to describe it. It just is.
It's less clear in informal languages that concepts exist apart from the language to describe them. This is one of the great difficulties in translation; different languages conceptualize the world in different (and sometimes incompatible) ways. For example, Latin has a variety of words for relationships of power that don't convey cleanly into English. The Greek word λόγος also doesn't cleanly render in English. Similarly for much of the Tao Te Ching. To what extent our language shapes our conceptualization of the world is currently an open question, but it seems like it's at least some.
These seem like extreme cases that just take more words to translate due to limited overlap of shared experiences. If it never snows in Hawaii, the Hawaiian language may not have a word for it. But snow is a real thing apart from language. I think what is meant by informal language is more the structural rules, which ultimately seem to just be arbitrary patterns of behavior people fell into, instead of sometime deeper.
Those rules too exhibit cross-language conceptual shear. For example, English doesn't really have an equivalent of Latin's subjunctive mood, and our clause boundary system is way weaker. We use punctuation marks in a way that the Romans didn't (they used punctuation words in a much more systematic way than we do). Greek has a separate grammatical number for pairs of things as opposed to our singular and plural. Those rules, in addition to the individual words (perhaps) bespeak a different way of conceptualizing the world.
It actually does snow in Hawaii. People on the big island have skis and snowboards for the few times it snows on top of their mountains. It also snows on Haleakala in Maui which is pretty fun to think about.
The concept of "a number that cannot be written as a product of two other numbers" just is. It might happen in some setting that none exist, but the general concept most certainly just is (and they can be shown to exist among the natural numbers if you insist - but they do not exist in e.g. Z_2 - the binary field).
