what is the difference of the concept of mathematics as a language and as knowledge
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The language of mathematics is the system used by mathematicians to communicate mathematical ideas among themselves, and is distinct from natural languages in that it aims to communicate abstract, logical ideas with precision and unambiguity.[1][2]
This language consists of a substrate of some natural language (e.g., English), using technical terms and grammatical conventions that are peculiar to mathematical discourse (see mathematical jargon). It is also supplemented by a highly specialized symbolic notation for mathematical formulas.
Similar to natural languages, discourse using the language of mathematics can employ a scala of registers. Research articles in academic journals are sources for detailed theoretical discussions about ideas concerning mathematics and its implications for society.