The Concept of Existence in Mathematics

• Autores: Pablo M. Jacovkis
• Localización: Metatheoria, ISSN-e 1853-2330, ISSN 1853-2322, Vol. 7, Nº. Extra 2, 2017 (Ejemplar dedicado a: Special Issue - Tribute to Mario Bunge), págs. 17-23
• Idioma: inglés
• DOI: 10.48160/18532330me7.149
• Títulos paralelos:
  • El concepto de existencia en matemáticas
• Enlaces
  • Texto completo
• Resumen
  • español

    Afirmamos que, desde un punto de vista pragmático, los matemáticos tratan los objetos matemáticos como si fueran reales. Si una teoría es consistente, los teoremas se descubren (a veces con análisis no necesariamente diferentes de los aplicados en ciencias naturales) y las demostraciones se inventan; la tecnología moderna no puede existir sin aceptar la ley del tercero excluido; una demostración constructiva puede suministrar nuevas ideas o métodos pero, desde el punto de vista matemático, una demostración no constructiva es tan sólida como una constructiva. En consecuencia, ningún matemático, puro o aplicado, prescinde del axioma de elección; por otra parte, aunque según se acepte o no la hipótesis del continuo pueden aparecer distintos teoremas y objetos, no existe –al menos hasta ahora– ningún teorema importante aplicable al mundo real que dependa de aceptar o no dicha hipótesis. Los objetos matemáticos construidos por matemáticos aplicados son a menudo tan útiles como los objetos físicos, incluso aquellos objetos que fueron creados mediante métodos computacionales o probabilísticos. 

  • English

    We assert that, from a pragmatic point of view, mathematicians treat mathematical objects as if they were real. If a theory is consistent, theorems are discovered (sometimes with analyses not necessarily different from those applied in sciences) and proofs are invented; modern technology cannot exist without accepting the law of excluded middle; a constructive proof may provide new ideas or methods but, from a mathematical point of view, a non-constructive proof is as sound as a constructive one. Accordingly, no mathematician, pure or applied, gets by without the axiom of choice; on the other hand, although different theorems and objects may appear depending on the acceptance or not of the continuum hypothesis, no important theorem applicable to the real world exists – at least until now – which depends on accepting or not this hypothesis. Mathematical objects built by applied mathematicians are often as useful as physical objects, even those objects created via computer-assisted or probabilistic methods. 

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