{"id":24020,"date":"2025-01-30T09:36:50","date_gmt":"2025-01-30T08:36:50","guid":{"rendered":"https:\/\/educationalevidence.com\/?p=24020"},"modified":"2025-01-31T09:12:39","modified_gmt":"2025-01-31T08:12:39","slug":"quines-son-les-obres-dart-de-les-matematiques","status":"publish","type":"post","link":"https:\/\/educationalevidence.com\/ca\/quines-son-les-obres-dart-de-les-matematiques\/","title":{"rendered":"Quines s\u00f3n les obres d\u2019art de les matem\u00e0tiques?"},"content":{"rendered":"

Quines s\u00f3n les obres d\u2019art de les matem\u00e0tiques?<\/h1>\n

Les demostracions s\u00f3n seq\u00fc\u00e8ncies l\u00f2giques de pensament, \u00e9s a dir, s\u00f3n el nexe amb la filosofia i, m\u00e9s important, s\u00f3n el nexe de les matem\u00e0tiques amb les humanitats<\/p>\n

\"\"
Leonhard Euler donaria la primera demostraci\u00f3 formal del teorema de Fermat el 1736. \/ Viquip\u00e8dia<\/a><\/figcaption><\/figure>\n

Llic\u00e8ncia Creative Commons<\/a><\/p>\n

\"\"<\/p>\n

 <\/p>\n

Francesc Gass\u00f3 Minguet<\/strong><\/a><\/span><\/p>\n

 <\/p>\n

L\u2019any 1990 William Dunham va publicar el llibre \u201cJourney through genius: The Great Theorems of Mathematics\u201d. Aquest llibre parteix de la premissa que quasi totes les disciplines art\u00edstiques tenen catalogades les seves obres mestres<\/strong> i que aquestes poden ser contemplades en museus. Les humanitats tamb\u00e9 tenen clar els llibres o assaigs que cal prioritzar per entendre millor la condici\u00f3 humana i, sigui en museus, biblioteques o exposicions se\u2019n fa divulgaci\u00f3. Podria continuar fent paral\u00b7lelismes amb la m\u00fasica, l\u2019arquitectura, etc. Per a la persona que visita un d\u2019aquests espais \u00e9s \u00fatil saber que per entendre b\u00e9 aquella disciplina ha d\u2019estudiar aquelles obres d\u2019art. Per\u00f2 a les matem\u00e0tiques els hi manca aquesta tria <\/strong>de les seves obres mestres.<\/p>\n

S\u00ed que existeixen museus<\/strong> de les matem\u00e0tiques, per\u00f2 acostumen a ser espais on es vol sembrar la llavor per l\u2019inter\u00e8s en les matem\u00e0tiques i la l\u00f2gica. Molts dels recursos exposats van destinats als infants i adolescents, com si a un museu pict\u00f2ric el que hi trob\u00e9ssim fossin pinzells, llen\u00e7os i altres eines que ens permetessin comen\u00e7ar a pintar un retrat. Aquesta tipologia de museus \u00e9s necess\u00e0ria perqu\u00e8 en part es busca contrarestar les opinions de moltes persones que tenen influ\u00e8ncia als infants i mostren amb orgull les seves mancances matem\u00e0tiques, quan en qualsevol altra disciplina intentarien que la seva ignor\u00e0ncia pass\u00e9s desapercebuda.<\/p>\n

Per tant, als museus de matem\u00e0tiques hi manca el que Dunham va proposar<\/strong> en aquest llibre tan exquisit: una tria dels grans matem\u00e0tics i dels seus teoremes i demostracions.<\/p>\n

La tria que ell feu permet con\u00e8ixer d\u2019una forma contextualitzada en la hist\u00f2ria com les idees de les ments m\u00e9s brillants de les matem\u00e0tiques han modelat la nostra comprensi\u00f3 del m\u00f3n. Mostra l\u2019aut\u00e8ntica bellesa intr\u00ednseca de les matem\u00e0tiques: les demostracions. Cal l\u00f2gica i precisi\u00f3 per a partir d\u2019uns axiomes ben definits per Euclides<\/strong> deduir-ne qualsevol dels teoremes que coneixem avui dia. I s\u00ed, tamb\u00e9 cal molta bellesa.<\/p>\n

La demostraci\u00f3 com a obra d\u2019art<\/em>. En la Gr\u00e8cia cl\u00e0ssica van entendre que no era suficient en enunciar all\u00f2 que semblava intu\u00eftiu, calia comprovar-ho, \u00e9s a dir, calia deduir-ho pas a pas a partir d\u2019all\u00f2 que era segur, que era cert.<\/p>\n

Per exemple, van demostrar que la diagonal d\u2019un quadrat de costat la unitat no \u00e9s un nombre racional en contra de la perfecci\u00f3 del m\u00f3n \u201ccommensurable\u201d tal com ells l\u2019entenien. En aquesta demostraci\u00f3 s\u2019utilitza la reducci\u00f3 a l\u2019absurd, m\u00e8tode que consisteix a suposar una hip\u00f2tesi i a partir de raonaments l\u00f2gics que s\u2019encadenen s\u2019arriba a un absurd, per tant, l\u2019error es troba en la hip\u00f2tesi. Aquest resultat supos\u00e0 un trasbals pels pitag\u00f2rics que atorgaven la supremacia als enters i com a c\u00e0stig llen\u00e7aren a la mar a Hippasus com a c\u00e0stig<\/strong>. No sempre la humanitat ha estat preparada per a rebre les demostracions que revoquen les idees preconcebudes, o potser no ho ha estat mai? Arquimedes<\/strong> utilitz\u00e0 iteracions per aproximar el nombre pi, tot inventant una nova forma de buscar nombres. Euler<\/strong> feu servir el m\u00e8tode d\u2019inducci\u00f3 per demostrar el petit Teorema de Fermat<\/strong>. Cantor<\/strong>, gr\u00e0cies a les bijeccions, comprov\u00e0 que hi ha tants nombres naturals com racionals, per\u00f2 que els nombres reals s\u00f3n d\u2019un ordre d\u2019infinit superior, demostraci\u00f3 que ens permet entretenir-nos al joc de l\u2019hotel d\u2019infinites habitacions.<\/p>\n

Totes aquestes demostracions, i les que no he enumerat, ens aporten m\u00e8todes de raonament, formes l\u00f2giques d\u2019encadenar deduccions per a les quals moltes persones no s\u2019han entrenat i que la humanitat va trigar segles a idear-les i aplicar-les.<\/p>\n

Aix\u00ed doncs, les demostracions s\u00f3n seq\u00fc\u00e8ncies l\u00f2giques de pensament, \u00e9s a dir, s\u00f3n el nexe amb la filosofia i, m\u00e9s important, s\u00f3n el nexe de les matem\u00e0tiques amb les humanitats.<\/p>\n

En un moment hist\u00f2ric en qu\u00e8 certs discursos populistes que no aguantarien ni un raonament per reducci\u00f3 a l\u2019absurd guanyen adeptes, no val la pena donar la import\u00e0ncia que es mereix a les demostracions matem\u00e0tiques? No val la pena que tothom pugui gaudir d\u2019aquestes obres d\u2019art matem\u00e0tiques que no s\u00f3n m\u00e9s que una altra pe\u00e7a imprescindible del pensament universal?<\/p>\n

I no oblidem el m\u00e9s important: Les grans demostracions matem\u00e0tiques s\u00f3n belles per elles mateixes i ens obliguen a q\u00fcestionar-nos algunes de les nostres intu\u00efcions<\/strong>.<\/p>\n

\n

Font:\u00a0educational EVIDENCE<\/strong><\/p>\n

Drets:\u00a0Creative Commons<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

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