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dbpedia-owl:abstract
  • A les matemàtiques hi ha diverses demostracions amb contradiccions òbvies. Tot i que les demostracions són errònies, els errors són subtils. Aquestes fal·làcies són considerades simples curiositats, però poden ser utilitzades per il·lustrar la importància del rigor en aquesta àrea.
  • En matemáticas, hay múltiples demostraciones matemáticas de contradicciones obvias. A pesar de que las demostraciones son erróneas, los errores son sutiles, y la mayor parte de las veces, intencionados. Estas falacias se consideran normalmente meras curiosidades, pero pueden ser usadas para ilustrar la importancia del rigor en esta área.La mayoría de estas demostraciones dependen de variantes del mismo error. El error consiste en usar una función f que no es biyectiva, para observar que f(x) = f(y) para ciertas x e y, concluyendo (erróneamente) que por tanto x = y. La división por cero es un caso particular: la función f es x → x × 0, y el paso erróneo es comenzar con x × 0 = y × 0 y con ello concluir que x = y.
  • In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept of mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof: a mistake in a proof leads to an invalid proof just in the same way, but in the best-known examples of mathematical fallacies, there is some concealment in the presentation of the proof. For example, the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way. Therefore these fallacies, for pedagogic reasons, usually take the form of spurious proofs of obvious contradictions. Although the proofs are flawed, the errors, usually by design, are comparatively subtle, or designed to show that certain steps are conditional, and should not be applied in the cases that are the exceptions to the rules.The traditional way of presenting a mathematical fallacy is to give an invalid step of deduction mixed in with valid steps, so that the meaning of fallacy is here slightly different from the logical fallacy. The latter applies normally to a form of argument that is not a genuine rule of logic, where the problematic mathematical step is typically a correct rule applied with a tacit wrong assumption. Beyond pedagogy, the resolution of a fallacy can lead to deeper insights into a subject (such as the introduction of Pasch's axiom of Euclidean geometry). Pseudaria, an ancient lost book of false proofs, is attributed to Euclid.Mathematical fallacies exist in many branches of mathematics. In elementary algebra, typical examples may involve a step where division by zero is performed, where a root is incorrectly extracted or, more generally, where different values of a multiple valued function are equated. Well-known fallacies also exist in elementary Euclidean geometry and calculus.
  • In matematica, un sofisma algebrico è una dimostrazione o un ragionamento matematico contenente un errore, che porta quindi ad un risultato errato o contraddittorio. Usualmente questi sofismi sono utilizzati a scopo didattico, per dimostrare l'importanza del rigore nelle dimostrazioni matematiche; per questo motivo, gli errori presenti sono in generale molto sottili e difficili da rilevare (relativamente al pubblico cui sono destinati) ma alla fine il ragionamento presenta conclusioni evidentemente erronee. La storia della matematica registra comunque numerosi casi di ragionamenti erronei dovuti a matematici importanti. Di seguito vengono riportati alcuni esempi classici di sofismi algebrici, suddivisi in base alla tipologia dell'errore che viene introdotto.
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  • A les matemàtiques hi ha diverses demostracions amb contradiccions òbvies. Tot i que les demostracions són errònies, els errors són subtils. Aquestes fal·làcies són considerades simples curiositats, però poden ser utilitzades per il·lustrar la importància del rigor en aquesta àrea.
  • In matematica, un sofisma algebrico è una dimostrazione o un ragionamento matematico contenente un errore, che porta quindi ad un risultato errato o contraddittorio.
  • In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept of mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof: a mistake in a proof leads to an invalid proof just in the same way, but in the best-known examples of mathematical fallacies, there is some concealment in the presentation of the proof.
  • En matemáticas, hay múltiples demostraciones matemáticas de contradicciones obvias. A pesar de que las demostraciones son erróneas, los errores son sutiles, y la mayor parte de las veces, intencionados. Estas falacias se consideran normalmente meras curiosidades, pero pueden ser usadas para ilustrar la importancia del rigor en esta área.La mayoría de estas demostraciones dependen de variantes del mismo error.
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  • Pseudo-démonstration d'égalité entre nombres
  • Demostració invàlida
  • Demostración inválida
  • Mathematical fallacy
  • Prova inválida
  • Sofisma algebrico
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