Changes

==Origin==

==Origin==

Late Latin ''theorema'', from [[Greek]] ''theōrēma'', from ''theōrein'' to look at, from ''theōros'' [[spectator]], from ''thea'' [[act]] of [[seeing]]

Late Latin ''theorema'', from [[Greek]] ''theōrēma'', from ''theōrein'' to look at, from ''theōros'' [[spectator]], from ''thea'' [[act]] of [[seeing]]

*[http://en.wikipedia.org/wiki/16th_century 1551]

*[https://en.wikipedia.org/wiki/16th_century 1551]

==Definitions==

==Definitions==

*1: a [[formula]], [[proposition]], or [[statement]] in [[mathematics]] or [[logic]] deduced or to be deduced from other formulas or propositions

*1: a [[formula]], [[proposition]], or [[statement]] in [[mathematics]] or [[logic]] deduced or to be deduced from other formulas or propositions

In [[mathematics]], a '''theorem''' is a [[statement]] that has been [[proven]] on the basis of previously established statements, such as other theorems, and previously accepted statements, such as [[axioms]]. The derivation of a theorem is often [[interpreted]] as a [[proof]] of the [[truth]] of the resulting [[expression]], but different deductive systems can yield other interpretations, depending on the [[meanings]] of the derivation rules. The proof of a mathematical theorem is a [[logical]] [[argument]] demonstrating that the [[conclusions]] are a [[necessary]] consequence of the [[hypotheses]], in the sense that if the hypotheses are true then the conclusions must also be true, without any further [[assumptions]]. The concept of a theorem is therefore fundamentally [[deductive]], in [[contrast]] to the notion of a scientific [[theory]], which is [[empirical]].

In [[mathematics]], a '''theorem''' is a [[statement]] that has been [[proven]] on the basis of previously established statements, such as other theorems, and previously accepted statements, such as [[axioms]]. The derivation of a theorem is often [[interpreted]] as a [[proof]] of the [[truth]] of the resulting [[expression]], but different deductive systems can yield other interpretations, depending on the [[meanings]] of the derivation rules. The proof of a mathematical theorem is a [[logical]] [[argument]] demonstrating that the [[conclusions]] are a [[necessary]] consequence of the [[hypotheses]], in the sense that if the hypotheses are true then the conclusions must also be true, without any further [[assumptions]]. The concept of a theorem is therefore fundamentally [[deductive]], in [[contrast]] to the notion of a scientific [[theory]], which is [[empirical]].

Although they can be written in a completely [[symbolic]] form using, for example, [http://en.wikipedia.org/wiki/Propositional_calculus 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 arguments 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]]. [http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem Fermat's Last Theorem] is a particularly well-known example of such a ''theorem''. [http://en.wikipedia.org/wiki/Theorems]

Although they can be written in a completely [[symbolic]] form using, for example, [https://en.wikipedia.org/wiki/Propositional_calculus 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 arguments 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]]. [https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem Fermat's Last Theorem] is a particularly well-known example of such a ''theorem''. [https://en.wikipedia.org/wiki/Theorems]

[[Category: Mathematics]]

[[Category: Mathematics]]