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 abstract nonsenseA tongue-in-cheek reference to category theory, using which one can employ arguments that establish a (possibly concrete) result without reference to any specifics of the present problem. For that reason, it's also known as general abstract nonsense or generalized abstract nonsense. canonicalA reference to a standard or choice-free presentation of some mathematical object (e.g., canonical map, canonical form, or canonical ordering). The same term can also be used more informally to refer to something "standard" or "classic". For example, one might say that Euclid's proof is the "canonical proof" of the infinitude of primes. deepA result is called "deep" if its proof requires concepts and methods that are advanced beyond the concepts needed to formulate the result. For example, the prime number theorem — originally proved using techniques of complex analysis — was once thought to be a deep result until elementary proofs were found. On the other hand, the fact that π is irrational is usually known to be a deep result, because it requires a considerable development of real analysis before the proof can be established — even though the claim itself can be stated in terms of simple number theory and geometry. elegantAn aesthetic term referring to the ability of an idea to provide insight into mathematics, whether by unifying disparate fields, introducing a new perspective on a single field, or by providing a technique of proof which is either particularly simple, or which captures the intuition or imagination as to why the result it proves is true. In some occasions, the term "beautiful" can also be used to the same effect, though Gian-Carlo Rota distinguished between elegance of presentation and beauty of concept, saying that for example, some topics could be written about elegantly although the mathematical content is not beautiful, and some theorems or proofs are beautiful but may be written about inelegantly. elementaryA proof or