Difference between revisions of "Theorems"
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==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]] | ||
| − | *[ | + | *[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 | ||
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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, [ | + | 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]] | ||
Latest revision as of 02:42, 13 December 2020
Origin
Late Latin theorema, from Greek theōrēma, from theōrein to look at, from theōros spectator, from thea act of seeing
Definitions
- 1: a formula, proposition, or statement in mathematics or logic deduced or to be deduced from other formulas or propositions
- 2: an idea accepted or proposed as a demonstrable truth often as a part of a general theory : proposition <the theorem that the best defense is offense>
- 3: stencil
- 4: a painting produced especially on velvet by the use of stencils for each color
Description
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, 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. Fermat's Last Theorem is a particularly well-known example of such a theorem. [1]
