The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas. Much of this is common English, but with a specific non-obvious meaning when used in a mathematical sense.
Some phrases, like "in general", appear below in more than one section.
Philosophy of mathematics
—The proof that there are infinitely many prime numbers.
—The proof of the irrationality of the square root of two.
Although ultimately every mathematical argument must meet a high standard of precision, mathematicians use descriptive but informal statements to discuss recurring themes or concepts with unwieldy formal statements. Note that many of the terms are completely rigorous in context.