From: Subject: Diario El Sur: Consejos para comprender el infinito Date: Fri, 12 Jan 2007 15:43:55 -0300 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----=_NextPart_000_01FF_01C73660.7773B320" X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.3028 This is a multi-part message in MIME format. ------=_NextPart_000_01FF_01C73660.7773B320 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Content-Location: http://www.elsur.cl/edicion_hoy/secciones/ver_rep.php?id=2763&dia=1151208000 Diario El Sur: Consejos para comprender el = infinito
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> PORTADA<= /A>=20 - REPORTAJES
Concepci=F3n, = Chile, domingo=20 25 de junio de 2006
=
Consejos para=20 comprender el infinito =
  • Numerosos=20 libros intentan explicar el m=E1s = complejo y=20 abstracto concepto de las matem=E1ticas, = presente=20 adem=E1s en nuestras vidas diarias. = Podemos saber=20 por referencias que, por ejemplo, el = Universo no=20 tiene bordes, es decir, no tiene fin. = Sin=20 embargo, pocos seres humanos son capaces = de=20 entender esa idea.=20


Hugo Jara=20 = Goldenberg

      El diccionario de la RAE = define al=20 infinito como aquello =93que no tiene ni = puede=20 tener fin ni t=E9rmino=94. Se trata de = una acepci=F3n=20 que nos deja satisfechos, pero = =BFsignifica eso=20 que comprendemos realmente ese concepto? = Por=20 supuesto que no, ya que esta idea, al = igual que=20 algunas otras que utilizamos = habitualmente,=20 poseen un nivel de complejidad tal que = su cabal=20 entendimiento queda m=E1s all=E1 de = nuestra=20 capacidad cognitiva. Acostumbramos a = hablar de=20 infinito cuando nos referimos a = magnitudes que=20 se proyectan m=E1s all=E1 de los = extremos de lo muy=20 grande y lo muy peque=F1o, lo que nos = indica que=20 este concepto est=E1 =EDntimamente = asociado al=20 razonamiento matem=E1tico, el que junto = con el=20 lenguaje constituye un aspecto = fundamental de=20 nuestro andamiaje=20 = intelectual.
     =20 Cuando el ser humano comenz=F3 a = desarrollar el=20 pensamiento num=E9rico, ya tuvo = dificultades para=20 asimilar plenamente las grandes cifras. = Algunas=20 tribus abor=EDgenes australianas = actuales poseen=20 en su lenguaje vocablos s=F3lo para las = cantidades=20 =93uno=94 y =93dos=94, cualquier cifra = superior lo=20 asocian a la categor=EDa de muchos. Esto = nos=20 recuerda que en los inicios de nuestra=20 civilizaci=F3n ocurri=F3 una situaci=F3n = similar, ya=20 que si pesquisamos el origen de la = palabra=20 =93tres=94 veremos que est=E1 = emparentada con el=20 concepto de =93trans=94, que significa = =93m=E1s=20 = all=E1=94.
      Sin=20 embargo, gracias al desarrollo de las=20 matem=E1ticas hemos aprendido a manejar = con=20 soltura cualquier cifra, e incluso = creamos un=20 sistema especial de notaci=F3n para los = grandes=20 n=FAmeros. El recordado astr=F3nomo y = divulgador=20 cient=EDfico, Carl Sagan, acu=F1=F3 el = t=E9rmino =93miles=20 de millones=94 (que incluso fue el = t=EDtulo de uno=20 de sus libros), para referirse a = aquellas=20 inmensas cantidades que utilizamos con = tanta=20 propiedad, pero cuya plena comprensi=F3n = escapa a=20 nuestra capacidad de asimilaci=F3n = intelectual.=20 Cuando abordamos las grandes magnitudes=20 num=E9ricas, existe un l=EDmite hasta el = cual=20 podemos acceder sin mayor dificultad. = Pero=20 cuando lo traspasamos, nos vemos = obligados a=20 buscar ayuda en conceptos matem=E1ticos = abstractos=20 y en modernas herramientas = inform=E1ticas, ya que=20 s=F3lo con ellas podemos explorar todos = los=20 rincones de nuestra realidad, desde lo=20 inconmensurablemente peque=F1o hasta lo=20 infinitamente=20 = grande.
     
    &nbs= p;El=20 c=E1lculo=20 = infinitesimal
     
   &nbs= p; =20 Desde los tiempos de la Grecia cl=E1sica = se ha=20 especulado sobre el verdadero sentido = del=20 infinito. Anaximandro introdujo el = t=E9rmino=20 =93apeiron=94 para referirse a =93lo = indefinido=94, =93sin=20 l=EDmite=94 o =93lo ilimitado=94. = Despu=E9s de mucho=20 divagar sobre esta idea tan = embriagadora, los=20 fil=F3sofos hel=E9nicos acabaron = rechaz=E1ndola por=20 considerarla contraria al sentido = com=FAn. A=20 partir de ellos, los principales = pensadores y=20 cient=EDficos de todas las =E9pocas se = enfrentaron=20 al desaf=EDo de develar el secreto de = este=20 concepto que induc=EDa a tantas = paradojas. Galileo=20 Galilei lleg=F3 a decir que su = comportamiento era=20 tan contradictorio que lo mejor era = evitarlo. Y=20 Newton trat=F3 de eludir a los = infinitesimales en=20 la explicaci=F3n de su nuevo m=E9todo = del=20 = c=E1lculo.
      Fue=20 necesario esperar hasta fines del siglo = XIX,=20 cuando el matem=E1tico Georg Cantor = elabor=F3 una=20 aritm=E9tica del infinito, para que = finalmente se=20 esparciera la luz en la confusi=F3n. El = infinito=20 no es un n=FAmero muy grande, es s=F3lo = una cualidad=20 asociada a lo indeterminable. Cantor = explicaba=20 que el gran problema era que el infinito = necesitaba de una matem=E1tica = diferente, y era su=20 combinaci=F3n con los c=E1lculos = cotidianos lo que=20 llevaba a situaciones parad=F3jicas. = Defini=F3=20 varias categor=EDas de infinitud, a las = que=20 represent=F3 mediante la primera letra = del=20 alfabeto hebreo: el aleph. La = domesticaci=F3n=20 matem=E1tica del infinito fue una = condici=F3n que=20 ayud=F3 a preparar el terreno a la = f=EDsica moderna,=20 que a partir de los primeros a=F1os del = siglo XX=20 transform=F3 para siempre la forma como = entendemos=20 el tiempo, el espacio y la=20 = materia.
     =20 Trat=E1ndose de un concepto tan = perturbador, no=20 resulta extra=F1o que sea un tema = recurrente en la=20 literatura, principalmente de la ciencia = ficci=F3n, pero tambi=E9n ha sido = utilizado por=20 autores universales, como Jorge Luis = Borges,=20 quien incluso titul=F3 a unos de sus = cuentos m=E1s=20 famosos como =93El Aleph=94, ese = m=E1gico ente=20 cantoriano que permite que todo el = Universo est=E9=20 contenido en cada una de sus=20 = part=EDculas.
     =20 Pero quienes mejor partido sacan al = infinito son=20 los escritores que se dedican a la = divulgaci=F3n=20 de la ciencia, despu=E9s de todo se = trata de un=20 concepto que surge de su territorio. De = entre=20 los muchos autores y t=EDtulos que es = posible=20 encontrar en las librer=EDas y = bibliotecas se=20 puede destacar a =93G=F6del, Escher, = Bach, un eterno=20 y gr=E1cil bucle=94, de Douglas R. = Hofstadter. Aqu=ED=20 el infinito se analiza como el = fen=F3meno que=20 surge de la auto referencia (ciclos auto = contenidos), estudiando la creaci=F3n de = un=20 matem=E1tico, un artista gr=E1fico y un = m=FAsico.=20 Otras obras interesantes son =93De = aqu=ED al=20 infinito=94, de Ian Stewart. O =93La = historia=20 definitiva del Infinito=94, de Richard = Morris,=20 quien enfoca su trabajo con una visi=F3n = m=E1s=20 f=EDsica que matem=E1tica. Y el =FAltimo = texto=20 aparecido en los escaparates es = =93Comprendiendo=20 el infinito=94 (Fondo de cultura = econ=F3mica, 2005),=20 de Shaughan Lavine, un cient=EDfico que = intenta=20 explicar lo indeterminado desde la = experiencia=20 de la cotidianidad finita y ef=EDmera, = en una=20 mezcla que posee dosis precisas de = n=FAmeros,=20 historia, filosof=EDa y=20 = l=F3gica.
      Como=20 vemos, existen muchas alternativas = interesantes=20 para acceder a una mejor comprensi=F3n = de uno de=20 los conceptos m=E1s inalcanzables y = parad=F3jicos a=20 los que se ha enfrentado el intelecto = humano.=20



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=A9 2007 Todos los derechos reservados = para Diario El=20 Sur S.A.
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All rights = reserved. Coded by: Taewook Kang (txkang.REMOVETHIS@hotmail.com) Web Site: http://txkang.com Script featured on Dynamic Drive (http://www.dynamicdrive.com) =09 Please retain this copyright notice in the script. License is granted to user to reuse this code on=20 their own website if, and only if,=20 this entire copyright notice is included. --------------------------------------------------------------*/ //Specify affected tags. 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try { if ( self != top ) referrer = top.document.referrer; } catch(e) { referrer = document.referrer; } return referrer; } ------=_NextPart_000_01FF_01C73660.7773B320 Content-Type: application/octet-stream Content-Transfer-Encoding: quoted-printable Content-Location: http://www.elsur.cl/certifica/certifica_n.js /*=20 Copyright 2004 - Certifica.com=20 $Id: certifica.js,v 1.12 2004/10/27 20:12:21 leus Exp $ */ DEFAULT_PIVOT_NAME =3D 'cert_Pivot'; DEFAULT_REDIRECT_TIME =3D 3000; DEFAULT_PERIODIC_REDIRECT_TIME =3D 60000; function cert_qVal(sValue)=20 { var pos =3D String(document.location).indexOf('?'); if (pos !=3D -1) { var query =3D String(document.location).substring(pos+1); var vars =3D query.split("&"); for (var i=3D0; i < vars.length; i++) { var pair =3D vars[i].split("=3D"); if (pair[0] =3D=3D sValue) return pair[1]; } =20 } return null; =20 } function cert_getReferrer() { var referrer =3D document.referrer; if (self.cert_getReferrer14) return cert_getReferrer14(); /*@cc_on @if(@_jscript_version >=3D 5 ) try {=20 if ( self !=3D top ) referrer =3D top.document.referrer; } catch(e) {}; @end @*/ return referrer; } function cert_getTagCertifica(iSiteId, sPath, sAppend)=20 { var size, colors, referrer, url; size =3D colors =3D referrer =3D 'otro'; var o =3D cert_qVal('url_origen'); if (o !=3D null && o !=3D '') referrer =3D o; else=20 referrer =3D escape(cert_getReferrer()); if ( window.screen.width ) size =3D window.screen.width; if ( window.screen.colorDepth ) colors =3D window.screen.colorDepth; else if ( window.screen.pixelDepth ) colors =3D = window.screen.pixelDepth; url =3D=20 'http://hits.e.cl/cert/hit.dll?sitio_id=3D' + iSiteId + = '&path=3D' + sPath + '&referer=3D' + referrer + '&size=3D' + size + '&colors=3D' + = colors; url +=3D '&java=3D' + navigator.javaEnabled(); if (sAppend) url +=3D sAppend; return url; =20 } /* Efectua un hit en certifica usando una imagen pivote. */ function cert_registerHit(iSiteId, sPath, sPivotName)=20 { var sAppend =3D '&cert_cachebuster=3D' + (1 + Math.floor = (Math.random() * 10000)); if ( !sPivotName ) sPivotName =3D DEFAULT_PIVOT_NAME; if ( document.images ) if ( document.images[sPivotName] ) document.images[sPivotName].src =3D = cert_getTagCertifica(iSiteId, sPath, sAppend); } /* Efecta una redirecci n marcando la ruta de salida */ function cert_registerHitAndRedirect( sURL, iSiteId, sPath, sPivotName ) = { cert_registerHit( iSiteId, sPath, sPivotName ); setTimeout( "location.href =3D '" + sURL + "'", DEFAULT_REDIRECT_TIME = ); } /* Abre una nueva ventana, marcando el hit */ function cert_registerHitAndOpenWindow( sURL, iSiteId, sPath, = sPivotName, sName, sFeatures, bReplace ) { cert_registerHit( iSiteId, sPath, sPivotName ); if (!sName) sName =3D 'Downloads'; if (!sFeatures) sFeatures =3D = 'toolbar=3Dno,location=3Dno,directories=3Dno,status=3Dyes,menubar=3Dno, = scrollbars=3Dno,resizable=3Dno,width=3D425,height=3D510,screenX=3D20,scre= enY=3D20'; window.open( sURL,=20 sName,=20 sFeatures,=20 bReplace=20 ); return false; } /* Marca el hit y reemplaza/abre una URL en el frame 'sName' */ function cert_registerHitAndReplaceOtherFrame( sURL, sName, iSiteId, = sPath, sPivotName )=20 { cert_registerHitAndOpenWindow( sURL, iSiteId, sPath, sPivotName, = sName, 0, true ); 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