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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 nonsense: Also ''general abstract nonsense'' or ''generalized abstract nonsense'', a 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. ; canonical: A reference to a standard or choice-free presentation of some mathematical object. The term ''canonical'' is also used more informally, meaning roughly "standard" or "classic". For example, one might say that Euclid's proof's is the "canonical proof" of the infinitude of primes. ; deep: A result is called "deep" if its proof requires concepts and methods that are advanced beyond the concepts needed to formulate the result. The prime number theorem, proved with techniques from complex analysis, was thought to be a deep result until elementary proofs were found. The fact that π is irrational is a deep result because it requires considerable development of real analysis to prove, even though it can be stated in terms of simple number theory and geometry. ; elegant: Also ''beautiful''; an 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 providing a technique of proof which is either particularly simple, or captures the intuition or imagination as to why the result it proves is true. 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. ; elementary: A proof or result is called "elementary" if it requires only basic concepts and methods, in contrast to so-called deep results. The concept of "elementary proof" is used specifically in number theory, where it usually refers to a proof that does not use methods from complex analysis. ; folklore : A result is called "folklore" if it is non-obvious, has not been published, and yet is generally known among the specialists in a field. Usually, it is unknown who first obtained the result. If the result is important, it may eventually find its way into the textbooks, whereupon it ceases to be folklore. ; natural: Similar to "canonical" but more specific, this term makes reference to a description (almost exclusively in the context of transformations) which holds independently of any choices. Though long used informally, this term has found a formal definition in category theory. ; pathological: An object behaves pathologically (or, somewhat more broadly used, in a ''degenerated'' way) if it fails to conform to the generic behavior of such objects, fails to satisfy certain regularity properties (depending on context), or simply disobeys mathematical intuition. These can be and often are contradictory requirements. Sometimes the term is more pointed, referring to an object which is specifically and artificially exhibited as a counterexample to these properties. :Note for that latter quote that as the differentiable functions are meagre in the space of continuous functions, as Banach found out in 1931, differentiable functions are colloquially speaking a rare exception among the continuous ones. Thus it can hardly be defended any-more to call non-differentiable continuous functions pathological. ; rigor (rigour): Mathematics strives to establish its results using indisputable logic rather than informal descriptive argument. Rigor is the use of such logic in a proof. ; well-behaved: An object is well-behaved (in contrast with being ''pathological'') if it ''does'' satisfy the prevailing regularity properties, or sometimes if it conforms to intuition (but intuition often suggests the opposite behavior as well). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「List of mathematical jargon」の詳細全文を読む スポンサード リンク
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