What is "theorem"

Theorem \The"o*rem\, n. [L. theorema, Gr. ? a sight, speculation, theory, theorem, fr. ? to look at, ? a spectator: cf. F. th['e]or[`e]me. See Theory.]

1. That which is considered and established as a principle; hence, sometimes, a rule.

   Not theories, but theorems (?), the intelligible products of contemplation, intellectual objects in the mind, and of and for the mind exclusively.
   --Coleridge.

   By the theorems, Which your polite and terser gallants practice, I re-refine the court, and civilize Their barbarous natures.
   --Massinger.

2. (Math.) A statement of a principle to be demonstrated.

   Note: A theorem is something to be proved, and is thus distinguished from a problem, which is something to be solved. In analysis, the term is sometimes applied to a rule, especially a rule or statement of relations expressed in a formula or by symbols; as, the binomial theorem; Taylor's theorem. See the Note under Proposition, n., 5.

   Binomial theorem. (Math.) See under Binomial.

   Negative theorem, a theorem which expresses the impossibility of any assertion.

   Particular theorem (Math.), a theorem which extends only to a particular quantity.

   Theorem of Pappus. (Math.) See Centrobaric method, under Centrobaric.

   Universal theorem (Math.), a theorem which extends to any quantity without restriction.

Theorem \The"o*rem\, v. t. To formulate into a theorem.

1550s, from Middle French théorème (16c.) and directly from Late Latin theorema, from Greek theorema "spectacle, sight," in Euclid "proposition to be proved," literally "that which is looked at," from theorein "to look at, behold" (see theory).

n. 1 (context mathematics English) A mathematical statement of some importance that has been proven to be true. Minor theorems are often called ''propositions''. Theorems which are not very interesting in themselves but are an essential part of a bigger theorem's proof are called ''lemmas'' 2 (context mathematics colloquial nonstandard English) A mathematical statement that is expected to be true; as, http://en.wikipedi

1. org/wiki/Fermat's%20Last%20Theorem (as which it was known long before it was proved in the 1990s.) 3 (context logic English) a syntactically correct expression that is deducible from the given axioms of a deductive system v

2. (context transitive English) to formulate into a theorem

1. n. a proposition deducible from basic postulates

2. an idea accepted as a demonstrable truth

In mathematics, a theorem is a statement that has been proved on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.

Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.

Although they can be written in a completely symbolic form, for example, within the propositional calculus, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed. Such arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". These subjective judgments vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.