{"id":26645,"date":"2014-05-02T14:06:26","date_gmt":"2014-05-02T11:06:26","guid":{"rendered":"https:\/\/bilimvegelecek.com.tr\/?p=26645"},"modified":"2018-09-23T22:41:21","modified_gmt":"2018-09-23T19:41:21","slug":"matematik-ve-estetik","status":"publish","type":"post","link":"https:\/\/bilimvegelecek.com.tr\/index.php\/2014\/05\/02\/matematik-ve-estetik","title":{"rendered":"Matematik ve estetik"},"content":{"rendered":"

\u0130nanc\u0131m matemati\u011fin estetik y\u00f6n\u00fcn\u00fcn g\u00fc\u00e7l\u00fc oldu\u011fu. Baz\u0131lar\u0131 i\u00e7in derin hazlar yaratan matematikte bu hazz\u0131 yaratan estetikten ba\u015fka ne olabilir? Baz\u0131 \u00f6\u011frenciler kendi sezgileriyle matemati\u011fin hazz\u0131n\u0131 ke\u015ffediyor. Bu hazz\u0131 baz\u0131 \u00f6\u011frenciler seziyor ve duyuyorken b\u00fcy\u00fck bir \u00e7o\u011funluk neden duymuyor? Bunun yan\u0131t\u0131 bende \u00e7ok net. \u00d6\u011fretimde matemati\u011fin estetik yap\u0131s\u0131n\u0131 ihmal ediyoruz. Hatta \u00f6\u011fretenler olarak biz de \u00e7ok fark\u0131nda de\u011filiz. Bilinsin ki bu yaz\u0131y\u0131 yazmak i\u00e7in ara\u015ft\u0131r\u0131rken bile yepyeni heyecanlar duydum. Daha \u00f6nce d\u00fc\u015f\u00fcnmedi\u011fim\u2026 Ve \u015fimdi d\u00fc\u015f\u00fcn\u00fcyorum ki; matemati\u011fin her konusu, her matematiksel ispat, her soru \u00e7\u00f6z\u00fcm\u00fc estetik i\u00e7erikli olarak ele al\u0131nabilir.<\/em><\/p>\n

Bilim i\u00e7in ara\u00e7 konumundayken matematik ka\u00e7\u0131n\u0131lmaz olarak daha kat\u0131 bir g\u00f6r\u00fcn\u00fcm al\u0131r. Her ne kadar uygulamada baz\u0131 g\u00fczelliklerden s\u00f6z etsek de \u00f6\u011frenen a\u00e7\u0131s\u0131ndan kat\u0131 bir matematik uygulamas\u0131 i\u00e7indeyiz. Kan\u0131mca matemati\u011fin i\u00e7eri\u011fi bu kat\u0131l\u0131kla s\u0131n\u0131rl\u0131 de\u011fil. Matemati\u011fin gizemli esteti\u011fini uygulamalardan yola \u00e7\u0131karak aralamak, matemati\u011fe kar\u015f\u0131 vefa duygusu olsa gerek. \u00d6\u011frenene de bu estetik yans\u0131mal\u0131\u2026<\/p>\n

\u015eiir gibi<\/strong><\/p>\n

Y\u0131llar \u00f6nce bir s\u0131n\u0131fta matematik dersi i\u015fliyorum. \u0130\u015flenen konu benim aktif olmam\u0131 gerektiriyor. En az\u0131ndan konunun i\u00e7eri\u011fini, ili\u015fkilerini, gere\u011fini kavratmam y\u00f6n\u00fcyle. Her zaman oldu\u011fu gibi, \u00f6n konularla ilgili an\u0131msatmalar, kavray\u0131\u015f i\u00e7in gerekli sezgiyi g\u00fc\u00e7lendirme, merak\u0131 k\u0131\u015fk\u0131rtma \u00e7al\u0131\u015fmalar\u0131 ile giri\u015f yapt\u0131m. Bu \u00f6n \u00e7al\u0131\u015fmalar\u0131n ard\u0131ndan seri bir bi\u00e7imde sezgiler ger\u00e7ekli\u011fe, merak duygusunun doyumu ho\u015flu\u011fa do\u011fru y\u00f6nelmeye ba\u015flad\u0131. \u0130lgi y\u00fcksek. \u00d6\u011frenciler p\u00fcr dikkat ve s\u0131k\u0131nt\u0131s\u0131z dinliyor, soruyor, not al\u0131yor. Bir k\u0131z \u00f6\u011frencim yan\u0131nda oturan arkada\u015f\u0131na \u201cders \u015fiir gibi gidiyor yav\u201d yorumunu yapt\u0131. Duydum. Mutlu oldum elbette. \u0130\u015fler iyi gidiyordu. Ald\u0131\u011f\u0131m onayla vermek istedi\u011fimi vermenin huzuruyla dersi bitirdim. Zil de \u00e7alm\u0131\u015ft\u0131. \u00c7\u0131kt\u0131m. \u00c7\u0131karken de bir ba\u015fka \u00f6\u011frencinin \u201cders nas\u0131l bitti anlamad\u0131m\u201d yorumuyla kar\u015f\u0131la\u015ft\u0131m. Ders ba\u015far\u0131l\u0131yd\u0131.<\/p>\n

\u201c\u015eiir gibi\u201d yorumunu sonradan d\u00fc\u015f\u00fcnd\u00fcm. Anlatt\u0131\u011f\u0131m konuda Escher\u2019in tablolar\u0131n\u0131, alt\u0131n oran\u0131 ya da Fibonacci say\u0131lar\u0131n\u0131 \u00e7a\u011fr\u0131\u015ft\u0131racak \u015feyler yoktu. Ama ders \u015fiir gibiydi! Escher\u2019den, Fibonacci\u2019den \u00f6zellikle s\u00f6z ediyorum. \u00c7\u00fcnk\u00fc matematik ve sanat deyince akla gelen \u00f6rnekler bunlar. Belki bir iki \u015fey daha. Geometrik \u015fekillerin ne kadar estetik oldu\u011fu, m\u00fczik aletlerinin ses titre\u015fiminin oransal ili\u015fkileri gibi basmakal\u0131p \u00f6rnekler\u2026<\/p>\n

Neydi \u201c\u015fiir gibi\u201dnin s\u0131rr\u0131?<\/p>\n

Birincisi haz\u0131rl\u0131klar\u0131m yeterliydi. Konuyu, i\u015fleni\u015f s\u00fcresini, y\u00f6ntemimi, konunun \u00f6n girdilerini iyi planlam\u0131\u015ft\u0131m. Planlama ilkem: Anlatacaklar\u0131m\u0131n anla\u015f\u0131l\u0131r olmas\u0131 ve \u00f6\u011frencilerin ilgisinin canl\u0131 tutulmas\u0131yd\u0131. \u0130kincisi s\u0131n\u0131fa girdi\u011fim andan itibaren \u00f6\u011frencilerin d\u00fc\u015f\u00fcncelerini \u00f6zg\u00fcrce s\u00f6yleyebilece\u011fi ortam\u0131 yaratm\u0131\u015ft\u0131m. Ki bu her ders i\u00e7in gerekliydi. \u00dc\u00e7\u00fcnc\u00fcs\u00fc o anlat\u0131m o s\u0131n\u0131fa \u00f6zg\u00fcyd\u00fc. Ayn\u0131 konu yan s\u0131n\u0131fta hatta ertesi g\u00fcn ayn\u0131 s\u0131n\u0131fta ayn\u0131 \u015fekilde anlat\u0131lamazd\u0131. Bir \u0131rmakta iki kez y\u0131kan\u0131lamayaca\u011f\u0131 gibi. D\u00f6rd\u00fcnc\u00fcs\u00fc kazan\u0131mlar \u00f6ncesi \u00f6n \u00e7al\u0131\u015fma tart\u0131\u015fmal\u0131 sorularla y\u00fcr\u00fct\u00fclm\u00fc\u015ft\u00fc. Yani sezgileri g\u00fc\u00e7lendirmek ve merak\u0131 k\u0131\u015fk\u0131rtmak da tamamd\u0131. Be\u015fincisi belli k\u0131vama gelen ders ortam\u0131nda kazan\u0131mlar ak\u0131c\u0131 bir anlat\u0131mla verilmi\u015fti. Alt\u0131nc\u0131s\u0131 k\u0131sa soru-cevapla kazan\u0131mlar\u0131n ger\u00e7ekle\u015fti\u011fi \u00f6l\u00e7\u00fclm\u00fc\u015ft\u00fc. Yedincisi ve de en \u00f6nemlisi ben konuyu \u201cheyecan\u201d duyarak i\u015flemi\u015ftim.<\/p>\n

Neden bu etkinli\u011fi ayr\u0131nt\u0131l\u0131 bir bi\u00e7imde aktard\u0131m. Elbette \u201cne kadar iyi yap\u0131yorum\u201d demek i\u00e7in de\u011fil. Bu benim i\u015fim. \u0130yi yapmal\u0131y\u0131m. \u00d6\u011frenciye, matemati\u011fe ve yapt\u0131\u011f\u0131 i\u015fe sayg\u0131 duyan her \u00f6\u011fretmen ayn\u0131 \u015feyleri yapar. Amac\u0131m bir\u00e7ok insan\u0131n \u201cben de yapar\u0131m\u201d dedi\u011fi \u00f6\u011fretmenli\u011fin \u201cbildi\u011fini aktarmakla\u201d s\u0131n\u0131rl\u0131 olmad\u0131\u011f\u0131n\u0131 vurgulamak. Ama as\u0131l vurgulamak istedi\u011fim bu da de\u011fil! \u201c\u015eiir gibi\u201d tepkisinin tek ba\u015f\u0131na ses tonuna, dil kullan\u0131m\u0131na, anlat\u0131m ak\u0131c\u0131l\u0131\u011f\u0131na ba\u011fl\u0131 olmad\u0131\u011f\u0131\u2026 Biraz daha ayr\u0131nt\u0131l\u0131 bakal\u0131m: Birincisi hemen her konu ke\u015ffedilmeyi bekleyen gizeme, her zaman \u015fa\u015f\u0131rtabilecek ilgin\u00e7li\u011fe ve g\u00fczelli\u011fe sahiptir. \u0130kincisi bu ke\u015fif eyleminde ortak \u00fcretimin dayan\u0131\u015fmas\u0131, i\u015fbirli\u011fi ve ortak akl\u0131n tekli\u011fi vard\u0131r. \u00dc\u00e7\u00fcnc\u00fcs\u00fc anlama eyleminin hazz\u0131-esteti\u011fi vard\u0131r. Bu haz, duygular\u0131 ok\u015fayan \u00f6yle bir g\u00fczelli\u011fe b\u00fcr\u00fcn\u00fcr ki, \u00f6\u011frenciler zamanlar\u0131n\u0131n bo\u015fa gitmedi\u011fi duygusunu ya\u015far, bilginin kal\u0131c\u0131l\u0131\u011f\u0131 umudu artar.<\/p>\n

Yani tek ba\u015f\u0131na bir matematik dersi bile i\u00e7inde matemati\u011fin kat\u0131l\u0131\u011f\u0131 (!) d\u0131\u015f\u0131nda bir\u00e7ok duygusal \u00f6\u011feler ta\u015f\u0131maktad\u0131r. Gizem, ilgin\u00e7lik, g\u00fczellik, teklik, haz, estetik, kal\u0131c\u0131l\u0131k gibi\u2026 Bu \u00f6\u011feler daha \u00e7ok sanat ya da felsefede ad\u0131 an\u0131lan kavramlard\u0131r. Bir matematik dersinde de yo\u011fun ya\u015fanan bu duyu\u015fsal kavramlarla, matemati\u011fin \u00f6z\u00fc tart\u0131\u015f\u0131ld\u0131\u011f\u0131nda daha yo\u011fun kar\u015f\u0131la\u015faca\u011f\u0131z. Bu nedenle matemati\u011fin estetik yan\u0131n\u0131 sindirmek, bilmek ve \u00f6\u011frenciye yans\u0131tmak ka\u00e7\u0131n\u0131lmaz olsa gerek\u2026<\/p>\n

Ancak o zaman matematik \u00f6\u011fretimi \u201cne i\u015fe yarar\u201d direncinden kurtulur, \u201cke\u015ffetmenin hazz\u0131\u201dna d\u00f6n\u00fc\u015febilir. Arad\u0131\u011f\u0131m\u0131z da bu de\u011fil mi? Anlaml\u0131 ve \u00f6\u011frenilen matematik! Hele de okullarda yapt\u0131\u011f\u0131m\u0131z i\u015fe tek ba\u015f\u0131na \u201c\u00f6\u011fretim\u201d de\u011fil, \u201ce\u011fitim-\u00f6\u011fretim\u201d diyorsak, matemati\u011fin estetik \u00f6zelliklerini nas\u0131l yok sayar\u0131z? Heyecan\u0131n oldu\u011fu yerde estetik vard\u0131r. Uygulamaya bakt\u0131\u011f\u0131m\u0131zda, matematik \u00f6\u011fretirken matemati\u011fin estetik \u00f6zelliklerini kullan\u0131yor muyuz sorusuna verilecek yan\u0131t ne yaz\u0131k ki \u201cevet\u201d olmayacakt\u0131r. En az\u0131ndan bana g\u00f6re \u00f6yle. Okullarda i\u015flenen matematik \u201ckonunun \u00f6\u011fretilmesi\u201d ile s\u0131n\u0131rl\u0131 ele al\u0131nmaktad\u0131r. Kar\u015f\u0131l\u0131\u011f\u0131nda da hakl\u0131 olarak \u00f6\u011frenci \u201cne i\u015fime yarayacak\u201d tepkisi ile kar\u015f\u0131m\u0131za \u00e7\u0131kmaktad\u0131r.<\/p>\n

Kald\u0131 ki sorun sadece \u00fclkemize has bir sorun da de\u011fil. Matematik Sanat\u0131<\/em> kitab\u0131nda Jerry P. King bunu \u015f\u00f6yle ifade ediyor:<\/p>\n

\u201cHepimiz, az da olsa okul matemati\u011finin s\u0131k\u0131nt\u0131s\u0131n\u0131 \u00e7ekmi\u015fizdir. Buna matemati\u011fi \u00f6\u011frenmeye de\u011fer buldu\u011fumuz i\u00e7in, ya da Darwin\u2019in yarat\u0131klar\u0131 gibi ortam elveri\u015fli oldu\u011fu i\u00e7in dayanmad\u0131k. Dayand\u0131k, \u00e7\u00fcnk\u00fc se\u00e7me hakk\u0131m\u0131z yoktu. Uzun bir zaman \u00f6nce birileri matematik bilmenin yararl\u0131 oldu\u011funa ve e\u011fer se\u00e7im bize b\u0131rak\u0131l\u0131rsa onu \u00f6\u011frenmek istemeyece\u011fimize karar vermi\u015fti. Bu y\u00fczden, bizler de bir ortaokul s\u0131n\u0131f\u0131nda, karatahta kar\u015f\u0131s\u0131ndaki sert s\u0131ralara oturmaya zorlanm\u0131\u015ft\u0131k. Bu durumun e\u015fyan\u0131n do\u011fas\u0131 gere\u011fi oldu\u011funa, matemati\u011fin ufak bir az\u0131nl\u0131k d\u0131\u015f\u0131nda kalan insanlar\u0131n sonsuza dek eri\u015fimleri d\u0131\u015f\u0131nda kalaca\u011f\u0131na inanmay\u0131 kabul etmiyorum. Halk\u0131n b\u00fcy\u00fck bir b\u00f6l\u00fcm\u00fcn\u00fcn m\u00fczi\u011fi, resmi, edebiyat\u0131 anlama ve haz duyma yetisine sahip oldu\u011fu; ancak do\u011fu\u015ftan matematik \u00f6z\u00fcrl\u00fc oldu\u011fu d\u00fc\u015f\u00fcncesi bana kendini be\u011fenmi\u015flik, \u00f6z\u00fcr dileyicilik i\u00e7eriyormu\u015f ve d\u00fcped\u00fcz yanl\u0131\u015fm\u0131\u015f gibi geliyor. A\u00e7\u0131k\u00e7a g\u00f6r\u00fcl\u00fcyor ki, matemati\u011fi bilimsel bir ara\u00e7 oldu\u011funu vurgulayarak takdim etmenin en iyi yol oldu\u011fu d\u00fc\u015f\u00fcncesine dayal\u0131 olan g\u00fcn\u00fcm\u00fcz matematik e\u011fitim sistemi ile bu insanlara ula\u015fmay\u0131 ba\u015faramad\u0131k\u2026 Daha ba\u015flang\u0131\u00e7ta, \u00f6\u011frencilerimizi matemati\u011fin, Poincare\u2019nin \u2018bizde bir t\u00fcr estetik duygu geli\u015ftirmeye muktedir olan bu g\u00fczellik ve zarafet niteli\u011fi\u2019 s\u00f6zlerinde belirtilen \u00f6zellikleriyle tan\u0131\u015ft\u0131rmay\u0131 deneyebiliriz.\u201d<\/p>\n

Bu saptamalar \u00fclkemiz matematik\u00e7ilerine de yabanc\u0131 de\u011fil. Yukar\u0131da de\u011findi\u011fimiz saptamalarla da uyum i\u00e7inde. Yazar matemati\u011fin zarafet ve g\u00fczellik \u00f6zelliklerinin ihmal edildi\u011fini d\u00fc\u015f\u00fcn\u00fcyor. Yaz\u0131n\u0131n devam\u0131nda; \u201c\u2026 yeni \u00f6\u011fretim Poincare ve Papert do\u011frultusunda, \u00f6nemli bir \u2018estetik matematik\u2019 b\u00f6l\u00fcm\u00fc i\u00e7ermelidir. Gidi\u015f o y\u00f6nde de\u011fildir. Matematik e\u011fitiminin gelece\u011fi bilgisayarlar, el hesap makineleri ve video g\u00f6sterimleri \u015feklinde, teknolojiye gittik\u00e7e artan bir ba\u011f\u0131ml\u0131l\u0131\u011fa y\u00f6nelmektedir. Amerikal\u0131 \u00f6\u011frencilerin azalan matematik yetenekleri ile teknolojinin okullara sokulmas\u0131 aras\u0131ndaki a\u00e7\u0131k ba\u011f\u0131nt\u0131y\u0131 g\u00f6ren tek ki\u015fi her halde ben de\u011filim\u2026\u201d<\/p>\n

Bu da bizim \u00fczerinde durdu\u011fumuz ve hatta matematik \u00f6\u011fretimini daha da sorunlu hale getirece\u011fini d\u00fc\u015f\u00fcnd\u00fc\u011f\u00fcm\u00fcz bir saptama. \u00c7\u00fcnk\u00fc yeni matematik programlar\u0131 ve y\u00f6ntemleri haz\u0131rlan\u0131rken \u00fclkemizde de \u201cmatematik \u00f6\u011frenme\u201d teknolojiye pas edilmi\u015f durumda. Somuta indirgemeyi teknolojiye kurban ederek\u2026 Matematik \u00f6\u011fretiminde teknolojinin kullan\u0131m\u0131 tart\u0131\u015fmas\u0131n\u0131 sonraya b\u0131rakal\u0131m. Estetik, sanat ve matematik ili\u015fkisini temellendirmeye \u00e7al\u0131\u015fal\u0131m.<\/p>\n

Estetik <\/strong><\/p>\n

Cengiz G\u00fcndo\u011fdu, \u0130nsanc\u0131l Yay\u0131nlar\u0131\u2019ndan \u00e7\u0131kan Estetik Kalk\u0131\u015fma<\/em> adl\u0131 kitab\u0131nda, estetik kalk\u0131\u015fma nedir sorusunu \u015f\u00f6yle yan\u0131tl\u0131yor: \u201c\u0130nsan soyunun d\u00f6rt alanda b\u00fcy\u00fck kalk\u0131\u015fmas\u0131 vard\u0131r. Ta\u015f yontmas\u0131, Pratik Kalk\u0131\u015fmas\u0131\u2019d\u0131r. Att\u0131\u011f\u0131 ta\u015f\u0131n nas\u0131l olup da d\u00fc\u015ft\u00fc\u011f\u00fcn\u00fc bulmas\u0131 Bilimsel Kalk\u0131\u015fmas\u0131\u2019d\u0131r. Yonttu\u011fu ta\u015f\u0131n sap\u0131na g\u00fcl yapmas\u0131 Estetik Kalk\u0131\u015fmas\u0131\u2019d\u0131r. Ya\u015fam\u0131n anlam\u0131n\u0131 aramas\u0131 Felsefi Kalk\u0131\u015fmas\u0131\u2019d\u0131r.<\/p>\n

\u201c\u0130nsan\u0131n kalk\u0131\u015fmas\u0131 geli\u015fig\u00fczellikten kurtulmas\u0131, do\u011fal varl\u0131ktan k\u00fclt\u00fcrel varl\u0131\u011fa d\u00f6n\u00fc\u015fmesidir. Bu d\u00f6n\u00fc\u015f\u00fcm\u00fcn g\u00fczellik yasalar\u0131na uygun olmas\u0131na estetik denir. Estetikten yoksun bir d\u00f6n\u00fc\u015f\u00fcm, d\u00f6n\u00fc\u015f\u00fcm de\u011fil ba\u015fkala\u015f\u0131md\u0131r\u2026<\/p>\n

\u201cEstetik yaln\u0131zca sanatla ilgili de\u011fildir. Estetik, ya\u015fam\u0131n b\u00fct\u00fcn alanlar\u0131n\u0131 kapsar\u2026<\/p>\n

G\u00fcndo\u011fdu bu d\u00f6rt kalk\u0131\u015fmay\u0131 san\u0131r\u0131m bir s\u0131ra g\u00f6zeterek vermemi\u015f. Belki insan ta\u015f yontmaya ba\u015flamadan da g\u00fcl resmi yap\u0131yordu. Hatta ilkel anlamda matematik bile\u2026 Bu nedenle esteti\u011fin ve matemati\u011fin geli\u015fmesinde ko\u015futluklar d\u00fc\u015f\u00fcn\u00fclebilir. Matematiksel geli\u015fimin seyrini Cengiz G\u00fcndo\u011fdu\u2019nun bilimsel ve felsefi kalk\u0131\u015fmas\u0131 ile birlikte ele almak san\u0131r\u0131m en do\u011frusu olacak. Yazar al\u0131nt\u0131n\u0131n devam\u0131nda estetik geli\u015fmenin g\u00fczeli aramas\u0131 ve bulmas\u0131n\u0131 ve bunun da k\u00fc\u00e7\u00fck ya\u015flarda ba\u015flad\u0131\u011f\u0131n\u0131 eklemi\u015f. Bu beni de g\u00fczelliklerin ve niceliksel d\u00fc\u015f\u00fcnmenin k\u00fc\u00e7\u00fck ya\u015flarda geli\u015fti\u011fi ya\u015fanm\u0131\u015fl\u0131klara g\u00f6t\u00fcrd\u00fc. \u0130ki ya\u015fanm\u0131\u015fl\u0131k\u2026<\/p>\n

K\u00fc\u00e7\u00fck o\u011flum anaokuluna gidecek ya\u015flarda. Ailece gezideyiz. Yolun sa\u011f taraf\u0131nda i\u00e7erlikli bir p\u0131nar g\u00f6rd\u00fck. \u00c7evresinde birka\u00e7 a\u011fa\u00e7. Durduk soluk almak i\u00e7in. Elimizi y\u00fcz\u00fcm\u00fcz\u00fc y\u0131kad\u0131k. P\u0131nar\u0131n suyu ile serinledik. Yaz s\u0131ca\u011f\u0131na diren\u00e7. Lafl\u0131yoruz \u00e7evreyi g\u00f6zleyip. O\u011flum biraz hiddetli, \u201csusun bi dakka\u201d dedi. D\u00f6nd\u00fck bakt\u0131k y\u00fcz\u00fcne. A\u00e7\u0131klama bekliyoruz\u2026 Hem\u015fire \u2018sus\u2019u yapar gibi parma\u011f\u0131 duda\u011f\u0131nda sessizce, \u201csusun, sessizli\u011fin sesini dinliyorum\u201d. Sanki trans halinde. Sustuk, dinledik. Kendi ad\u0131ma ben, yan\u0131mda akan p\u0131nar\u0131n \u015f\u0131r\u0131lt\u0131s\u0131n\u0131, esen r\u00fczg\u00e2rla olu\u015fan a\u011fa\u00e7lar\u0131n h\u0131\u015f\u0131rt\u0131s\u0131n\u0131 ve yolun di\u011fer yan\u0131nda akan \u0131rma\u011f\u0131n \u00e7a\u011f\u0131lt\u0131s\u0131n\u0131 duydum. Sessizli\u011fin sesi bu muydu bilmem. Ama k\u00fc\u00e7\u00fck adam duydu\u011fundan emindi\u2026<\/p>\n

Ayn\u0131 y\u0131llar, yine ayn\u0131 k\u00fc\u00e7\u00fck adam. Okula gitmiyor ama saymay\u0131 seviyor, biliyor. Her \u00e7ocuk gibi de alg\u0131lamaya \u00e7al\u0131\u015f\u0131yor. Eve geldi\u011fim bir g\u00fcn kucakla\u015ft\u0131k. Konu\u015furken \u201csen benim bir tanemsin\u201d dedi\u011fimi sonradan an\u0131msad\u0131m. D\u00f6nd\u00fc bana \u00f6nce bir parma\u011f\u0131n\u0131 uzat\u0131p \u201cbaba bir mi \u00e7ok\u201d, sonra be\u015f parma\u011f\u0131n\u0131 uzatt\u0131 \u201cbe\u015f mi \u00e7ok?\u201d Yan\u0131t verdim; \u201celbette be\u015f \u00e7ok\u201d yan\u0131t\u0131 biliyorsun der gibi\u2026 Arkas\u0131ndan yeni bir soru geldi bana; \u201c\u00f6yleyse beni severken neden be\u015f tanemsin demiyorsun?\u201d Ne yan\u0131t verdi\u011fimi an\u0131msam\u0131yorum. \u00d6nemli de de\u011fil. O anda benim \u00e7\u0131kard\u0131\u011f\u0131m sonu\u00e7 \u00f6nemliydi. Onun d\u00fc\u015f\u00fcncesinde, az ile \u00e7ok kavramlar\u0131 ile bir ve be\u015fin niceliksel de\u011ferleri aras\u0131ndaki ili\u015fki kurulmu\u015ftu. Kurulamayan, \u201cbir tanem\u201d deyi\u015finin \u201cbiricikli\u011fi\u201d idi. Cengiz G\u00fcndo\u011fdu\u2019nun dedi\u011fi gibi, ger\u00e7ek ya\u015fam\u0131n bir a\u011fac\u0131, bir adam\u0131, bir evi matematik d\u00fcnyas\u0131nda \u201c1\u201d say\u0131s\u0131na, g\u00fczellik yasalar\u0131na uygun olarak da estetik bir de\u011fere \u201cbiricik\u201d olmaya d\u00f6n\u00fc\u015fm\u00fc\u015ft\u00fc.<\/p>\n

Sessizli\u011fin sesindeki ses ahengini bulmak k\u00fc\u00e7\u00fck adam i\u00e7in sanatsal alg\u0131yd\u0131. \u0130kinci anlat\u0131daki \u201cbir\u201din niceliksel alg\u0131lanmas\u0131nda sorun yoktu. O da t\u00fcm \u00e7ocuklar gibi \u201cbir\u201din, \u201ciki\u201dnin, \u201c\u00fc\u00e7\u201d\u00fcn gizemini sezmi\u015fti. Anlamaya \u00e7al\u0131\u015f\u0131yor, anlad\u0131k\u00e7a anlaman\u0131n co\u015fkusunu ya\u015f\u0131yordu. Ama \u201cbiricik\u201dli\u011fe d\u00f6n\u00fc\u015fen \u201cbir\u201ddeki farkl\u0131la\u015fmay\u0131 hen\u00fcz alg\u0131layamam\u0131\u015ft\u0131. \u201cBir tanem\u201d deyi\u015findeki sevgiyi, g\u00fczelli\u011fi sezdi\u011fi halde. Ona g\u00f6re ya\u015famdaki tek nesnelerin niceliksel ifadesi olan \u201cbir\u201d say\u0131s\u0131 d\u00fc\u015fsel bir nesnesiydi. Ve bir ba\u015fkala\u015f\u0131md\u0131. \u201cBiricik\u201de ge\u00e7mek ise ikinci bir ba\u015fkala\u015f\u0131m. Hem de g\u00fczellemeye evrilen bir ba\u015fkala\u015f\u0131m. San\u0131r\u0131m biraz \u00e7ok gelmi\u015fti ona.<\/p>\n

MATEMAT\u0130K VE SANAT<\/strong><\/h4>\n

\u0130nsan etkinlikleri i\u00e7inde en \u00e7ok tart\u0131\u015f\u0131lan, \u00fczerinde en \u00e7ok yaz\u0131 yaz\u0131lan sanatt\u0131r. Roman\u0131, \u015fiiri, m\u00fczi\u011fi, resmi heykeli\u2026 ile. \u00c7\u00fcnk\u00fc sanat neredeyse t\u00fcm insanlar\u0131n k\u0131y\u0131s\u0131ndan k\u00f6\u015fesinden olsa bile dokunduklar\u0131 bir etkinlik alan\u0131. Gerek uygulay\u0131c\u0131 ve daha \u00e7ok da izleyici olarak. Yemek yapan kad\u0131n mutfa\u011f\u0131nda \u015fark\u0131 m\u0131r\u0131ldan\u0131r. Sesi g\u00fczel olmasa da. Her evin duvar\u0131nda bir resim, bir hal\u0131 dokuma ya da kilim deseni mutlaka vard\u0131r. Be\u011fenilere g\u00f6re de\u011fi\u015fse de. Mahk\u00fbm, ko\u011fu\u015funda c\u00fczdan \u00fczerine boncuk i\u015fler. Af umudu olmasa da. Her berber d\u00fckk\u00e2n\u0131n\u0131n duvar\u0131n\u0131 bir resim s\u00fcsler. Birileri bunlara \u201cbu da resim mi\u201d dese de\u2026 Belki \u00f6yledir de. Ama kimse bana Arde\u015fen\u2019in Tunca\u2019s\u0131nda, da\u011fdan f\u0131\u015fk\u0131ran suyun a\u011fz\u0131na yap\u0131lan iki oluklu p\u0131nara sanat de\u011fildir demesin. Kimin yapt\u0131\u011f\u0131 belli de\u011fil. Ama yapan ayn\u0131 g\u00f6vdeden \u00e7\u0131kan bir \u00e7atal dal bulmu\u015f, \u00e7atal dal\u0131 \u00e7ak\u0131 ya da b\u0131\u00e7akla \u00f6zene bezene oymu\u015f ve suyun \u00e7\u0131kt\u0131\u011f\u0131 a\u011fza yerle\u015ftirmi\u015f. Hi\u00e7bir gereksinime dayanmayan, tamamen yarat\u0131c\u0131l\u0131k eseri, ince ve hatta esprili bir eser\u2026 G\u00f6rd\u00fc\u011f\u00fcn\u00fczde \u00f6nce \u015fa\u015f\u0131r\u0131yorsunuz sonra i\u00e7inizi bir ho\u015fluk kapl\u0131yor. O oluklar\u0131n her birinden akan su \u201ci\u00e7 beni\u201d der gibi davetk\u00e2r\u2026 Be\u011fenilsin ya da be\u011fenilmesin. Bunlar\u0131n her biri insan\u0131n duygu d\u00fcnyas\u0131ndaki g\u00fczellik aray\u0131\u015flar\u0131n\u0131n birer \u00fcr\u00fcn\u00fc. Bunlar\u0131n her birinde se\u00e7icilik de bulursunuz, yarat\u0131c\u0131l\u0131k da estetik de. En yoksul bile evini d\u00fczenlerken bir estetik yaratmak istemez mi? Nak\u0131\u015fl\u0131 tek \u00f6rt\u00fcn\u00fcn yeri evin ba\u015fk\u00f6\u015fesi de\u011fil midir? Ya da tek k\u00f6\u015fesi! Ve de \u00f6yledir ki insan\u0131n g\u00fczellik aray\u0131\u015f\u0131, be\u015fikteki i\u015flemeyle ba\u015flar, mezar ta\u015f\u0131 i\u015flemesine dek s\u00fcrer. Yani sonu\u00e7ta kesiksiz bir i\u00e7 i\u00e7elik vard\u0131r insanla sanat aras\u0131nda. \u0130\u015fte belki de bu i\u00e7 i\u00e7elik nedeniyle sanat deyince bunlar gelmez akla. Daha se\u00e7kinci, daha yayg\u0131n kabul g\u00f6rm\u00fc\u015f eserlerdedir g\u00f6z\u00fcm\u00fcz. Falan ressam\u0131n resmi, falan yontucunun heykeli, filan bestecinin m\u00fczi\u011fi, \u015fu mimar\u0131n eseri\u2026 gibi. Daha akademik bir tav\u0131rd\u0131r bu. Daha evrensel de denilebilir.<\/p>\n

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Nedir o zaman sanat? Nelere sanat eseri diyece\u011fiz? Olduk\u00e7a tart\u0131\u015fmal\u0131\u2026 Bu tart\u0131\u015fmalar bizim i\u00e7eri\u011fimizle ilgili de\u011fil. Ayr\u0131ca ben bu tart\u0131\u015fmaya kat\u0131lacak yetkinlikte de de\u011filim. \u00dczerinde durdu\u011fumuz, sanat konusunda birle\u015filen genel kabuller ve bunlar\u0131n matematikle ba\u011fda\u015f\u0131kl\u0131\u011f\u0131. Sanatla ilgili \u00f6zellikler deyince \u015funlar s\u0131ralan\u0131r: Soyutlama, se\u00e7icilik, yarat\u0131c\u0131l\u0131k, teklik-biriciklik, kal\u0131c\u0131l\u0131k, estetik, \u00e7o\u011fulculuk. Yerine g\u00f6re bunlara ba\u015fka \u00f6zellikler eklenir ya da \u00e7\u0131kar\u0131l\u0131r. Bu \u00f6zelliklerin \u00e7o\u011funlukla birbirini tamamlad\u0131\u011f\u0131 da s\u00f6ylenebilir. \u00d6rne\u011fin sanat eserinin yarat\u0131lmas\u0131 bireyin se\u00e7icili\u011fine, yetene\u011fine ba\u011fl\u0131yken, kal\u0131c\u0131 olmas\u0131n\u0131n \u00e7o\u011funlu\u011fun onay\u0131na, be\u011fenisine ba\u011fl\u0131 olmas\u0131 gibi.<\/p>\n

S\u0131ralad\u0131\u011f\u0131m\u0131z \u00f6zelliklerin matemati\u011fin \u00f6zellikleriyle de b\u00fcy\u00fck \u00f6l\u00e7\u00fcde uyum i\u00e7inde oldu\u011funu g\u00f6rmek matematik\u00e7iler i\u00e7in \u015fa\u015f\u0131rt\u0131c\u0131 de\u011fildir. Ancak genel kan\u0131n\u0131n bu y\u00f6nde olmad\u0131\u011f\u0131 da a\u00e7\u0131kt\u0131r. Bu uyum ger\u00e7ekten var m\u0131d\u0131r? Bana g\u00f6re vard\u0131r. Ama bu uyumu incelemeye ba\u015flamadan \u00f6nce yap\u0131lan yayg\u0131n yanl\u0131\u015f\u0131 yeniden an\u0131msatal\u0131m. Bu yanl\u0131\u015f; matematik ve sanat ili\u015fkisi deyince birilerinin hemen Escher\u2019in tablolar\u0131n\u0131, alt\u0131n oran\u0131, Fibonacci Say\u0131lar\u0131n\u0131 ya da do\u011fadaki \u015fekilleri kan\u0131t olarak sunmas\u0131d\u0131r. Escher matematiksel formlar\u0131 kullanarak e\u015fsiz tablolar yapm\u0131\u015ft\u0131r. Alt\u0131n oran ya da Fibonacci say\u0131lar\u0131 do\u011fada kar\u015f\u0131l\u0131k bulan \u00f6nemli matematiksel kavramlard\u0131r. Estetik de\u011ferleri ve gizemleri \u015fa\u015f\u0131rt\u0131c\u0131d\u0131r. Di\u011fer geometrik \u015fekiller i\u00e7in de benzer \u015feyler s\u00f6ylenebilir. Ama bunlar\u0131 matematikte sanat oldu\u011funun kan\u0131tlar\u0131 bi\u00e7iminde sunmaya itiraz\u0131m\u0131z var. Bu s\u0131ralananlar ancak, ya\u015fam\u0131n her alan\u0131nda olan matemati\u011fin sanatta da g\u00f6zlemlendi\u011finin \u00f6rnekleri olabilir. Benzer \u015fekilde keman tellerinin \u00e7\u0131kard\u0131\u011f\u0131 e\u015fsiz g\u00fczellikteki sesleri tellerin oran\u0131 ile a\u00e7\u0131klay\u0131p \u201ci\u015fte matematik i\u015fte sanat\u201d demek de a\u00e7\u0131klay\u0131c\u0131 de\u011fil. E\u011fer matematikte sanat tart\u0131\u015f\u0131lacaksa ya da matematik ile sanat aras\u0131nda bir ko\u015futluk aranacaksa, bunu matematik ve sanat\u0131n davran\u0131\u015flar\u0131nda ve i\u015fleyi\u015flerinde aramak gerekir. Soyutlama, se\u00e7icilik, gizem, ilgin\u00e7lik-ayr\u0131cal\u0131k, g\u00fczellik, estetik, kal\u0131c\u0131l\u0131k gibi sanat \u00f6zelliklerinin matematikte ne kadar var oldu\u011funu g\u00f6zden ge\u00e7irelim. Geli\u015fim s\u00fcre\u00e7lerini de kar\u015f\u0131la\u015ft\u0131rarak\u2026<\/p>\n

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Geli\u015fim s\u00fcre\u00e7leri paraleldir<\/strong><\/p>\n

Matemati\u011fin ve sanat\u0131n tarihi insanl\u0131k tarihi kadar eskidir. \u00c7\u00fcnk\u00fc her ikisi de insanla\u015fma s\u00fcrecinin yap\u0131 ta\u015flar\u0131d\u0131r. Ma\u011fara resimleri – ta\u015f yontular\u0131 sanat\u0131n, duvar \u00e7izikleri – geometrik fig\u00fcrler matemati\u011fin tarihsel \u00f6nceliklerinin kan\u0131tlar\u0131d\u0131r. En ilkel d\u00f6nemlerde \u00e7izgiler bi\u00e7imindeki \u00e7ocuksu resimler perspektif geli\u015fimi ile nas\u0131l \u00fc\u00e7 boyutlu resimlere evrildiyse, \u00e7entik bi\u00e7imindeki ilk sayma \u00e7abalar\u0131 da sonsuzu ara\u015ft\u0131rmam\u0131za yarayan modern say\u0131lara kadar geli\u015fti. Ayr\u0131nt\u0131l\u0131 incelendi\u011finde her iki seyrin \u015fa\u015f\u0131rt\u0131c\u0131 benzerlikleri g\u00f6r\u00fclebilir.<\/p>\n

Beslenme kaynaklar\u0131 ger\u00e7ek d\u00fcnyad\u0131r<\/strong><\/p>\n

Matemati\u011fin de sanat\u0131n da beslenme kaynaklar\u0131 ger\u00e7ek d\u00fcnyad\u0131r. Bir ba\u015fka deyi\u015fle ger\u00e7ek d\u00fcnyan\u0131n nesneleri ve nesneler aras\u0131ndaki ili\u015fki hem matemati\u011fin hem sanat\u0131n itici g\u00fcc\u00fcd\u00fcr. Sanat\u00e7\u0131 ger\u00e7ek d\u00fcnyadaki bir ta\u015f\u0131 al\u0131r, kendi d\u00fcnyas\u0131na ta\u015f\u0131r, yontar, insan\u0131 ta\u015fa i\u015fler. Ve de onu heykel olarak ger\u00e7ek d\u00fcnyaya geri g\u00f6nderir. Ger\u00e7ek d\u00fcnyada o ta\u015f yoktur art\u0131k. Ta\u015fa i\u015flenen insan da bir g\u00fcn yok olur. Ama heykel ger\u00e7ek d\u00fcnyada ya\u015famaya devam eder.\u00a0\u00a0 Nesnelerin nicelikleriyle ilgilenen matematik\u00e7i de sanat\u00e7\u0131 gibi ger\u00e7ek d\u00fcnyan\u0131n nesnelerine g\u00f6z\u00fcn\u00fc diker. \u00d6rne\u011fin birebir kar\u015f\u0131la\u015ft\u0131r\u0131labilen nesneleri al\u0131r kendi d\u00fcnyas\u0131na. Yeni bir bi\u00e7im verir ve ad\u0131n\u0131 koyar. \u201c\u00dc\u00e7\u201d der \u00f6rne\u011fin. Ger\u00e7ek d\u00fcnyada olmayan, insan akl\u0131n\u0131n geli\u015ftirdi\u011fi bir formdur \u00fc\u00e7. Ya da hareketli bir nesnenin hareketini ele al\u0131r matematik. Onlardan modeller \u00fcretir. Fonksiyon der ad\u0131na, t\u00fcrev der, vekt\u00f6r der. Kendi nesnelerini \u00fcretir. Ger\u00e7ek d\u00fcnyada olmayan nesneler\u2026 Sanat ve matematik ko\u015futlu\u011funun belki de fark\u0131 budur. Heykel bir nedenle yok olabilir. O zaman \u00f6yle bir sanat eseri yoktur art\u0131k. Ama matemati\u011fin \u00fcretimleri ya\u015fam var olduk\u00e7a varl\u0131\u011f\u0131n\u0131 s\u00fcrd\u00fcrmeye devam eder.<\/p>\n

Soyutlama disiplinleridir<\/strong><\/p>\n

Nesneler d\u00fcnyas\u0131nda \u00fc\u00e7 tane olma ili\u015fkisi birebir kar\u015f\u0131la\u015ft\u0131rma sonucudur. \u00dc\u00e7 sandalye – \u00fc\u00e7 insan, \u00fc\u00e7 kedi – \u00fc\u00e7 fare, \u00fc\u00e7 ev – \u00fc\u00e7 a\u011fa\u00e7 – \u00fc\u00e7\u2026 gibi. Nesneler d\u00fcnyas\u0131nda sandalye var, insan, a\u011fa\u00e7, kedi\u2026 var. Ama \u201c\u00fc\u00e7\u201d yok. \u00dc\u00e7 art\u0131k \u201c3\u201d olarak matematik d\u00fcnyas\u0131nda var. O d\u00fcnyada ise sandalye yok, a\u011fa\u00e7 yok, insan yok\u2026 Nesneler d\u00fcnyas\u0131nda \u00fc\u00e7 tane olma ili\u015fkisi ba\u015flang\u0131\u00e7ta matematik d\u00fcnyas\u0131 ile de birebir ili\u015fki i\u00e7indedir.<\/p>\n

Sanat da matematik de se\u00e7icidir<\/strong><\/p>\n

D\u00fcnyada bir\u00e7ok a\u015fk ya\u015fan\u0131r. \u00c7o\u011funlukla da birbirine benzer a\u015fklar. Sanat\u00e7\u0131 a\u015fk\u0131 yazar. Yazd\u0131\u011f\u0131 birbirine benzeyenler de\u011fildir ama\u2026 \u0130lgin\u00e7 oland\u0131r, se\u00e7ilmi\u015f oland\u0131r. Paul ve Virginie\u2019in a\u015fk\u0131d\u0131r yazd\u0131\u011f\u0131 ya da Leyla ile Mecnun\u2019un a\u015fk\u0131. B\u00fclb\u00fcl\u00fcn sesi bestelere konudur. Kargan\u0131n de\u011fil\u2026 Beste yapan o sesi se\u00e7er. \u00c7\u00fcnk\u00fc farkl\u0131d\u0131r, ilgin\u00e7tir. Elbette bu saptamalar da tart\u0131\u015fmal\u0131d\u0131r. D\u00f6neme ba\u011fl\u0131 olarak be\u011fenilerde farkl\u0131l\u0131klar olabilir. Ki\u015filere ve toplumlara ba\u011fl\u0131 olarak da. Se\u00e7ilenin farkl\u0131l\u0131\u011f\u0131 ve ilgin\u00e7li\u011fini tarihsel s\u00fcre\u00e7 ve kal\u0131c\u0131l\u0131\u011f\u0131 ile birlikte d\u00fc\u015f\u00fcnmek gerek. Bir de evrenselli\u011fiyle\u2026<\/p>\n

Matematik de kendi yap\u0131s\u0131 i\u00e7inde ilgin\u00e7 olan\u0131 se\u00e7er. Matematik i\u00e7in cismin d\u00fc\u015fmesi de\u011fil, nas\u0131l d\u00fc\u015ft\u00fc\u011f\u00fc ilgin\u00e7tir. D\u00fc\u015ferken hangi yolu izledi\u011fi, hangi \u015fiddetle d\u00fc\u015ft\u00fc\u011f\u00fc ilgin\u00e7tir. Onu inceler. D\u00fcnyan\u0131n k\u00fcresel olu\u015fu matematik i\u00e7in de\u011fil fizik i\u00e7in ilgin\u00e7tir. K\u00fcreye benzer bir\u00e7ok cisim vard\u0131r ve bunlar di\u011ferlerinden farkl\u0131d\u0131r. Matematik i\u00e7in ilgin\u00e7 olan k\u00fcreye bezer cisimlerin niceliksel \u00f6zellikleridir. Bu nedenle matematik k\u00fcrenin yar\u0131\u00e7ap\u0131, hacmi, alan\u0131, kesitleri ile ilgilenir. D\u00fcnyan\u0131n hacmini veya alan\u0131n\u0131 hesaplamak matematik i\u00e7in olsa olsa matematik \u00f6\u011frenmeyi \u201cilgin\u00e7\u201d hale getirme sorunudur.<\/p>\n

Yarat\u0131c\u0131l\u0131k \u00fcretim bi\u00e7imleridir<\/strong><\/p>\n

Sanat\u00e7\u0131 ger\u00e7ek ya\u015famdan nesneleri se\u00e7erken se\u00e7icidir. Ama se\u00e7ti\u011finin se\u00e7kinli\u011fi ile yetinmez. Ona kendi duygular\u0131n\u0131 katar. Se\u00e7kini daha se\u00e7kin hale getirir. Paul ve Virginie\u2019in a\u015fk\u0131 se\u00e7kindir. Yazar\u0131n alg\u0131s\u0131 daha da se\u00e7kindir. Yazarken duygular\u0131n\u0131 katar a\u015fka. \u00d6yle katar ki, o a\u015fka Paul de \u015fa\u015far, Virginie de. Ger\u00e7ek d\u00fcnyadakinden daha idealdir. Ve de yeni a\u015fklara gebe. Ve onun i\u00e7in \u201csanat\u201d t\u0131r.<\/p>\n

Matematik\u00e7inin se\u00e7ti\u011fi kare herhangi bir d\u00f6rtgene g\u00f6re ilgin\u00e7tir. Bir anlamda da se\u00e7kin. Karenin kaplad\u0131\u011f\u0131 y\u00fczey de ilgin\u00e7tir. Matematik\u00e7i, alan \u00f6l\u00e7\u00fcm\u00fc i\u00e7in bir kurgu ortaya koyar. Onu benzer d\u00f6rtgenler i\u00e7in de uygular. Ortaya att\u0131\u011f\u0131 kurgu kareyi a\u015far, kurama d\u00f6n\u00fc\u015f\u00fcr. O kurgu yeni bir kavramd\u0131r art\u0131k. E\u015fsiz, y\u00fcce ve derde deva\u2026 \u00c7\u00fcnk\u00fc \u201ckuram\u201d t\u00fcm d\u00f6rtgenlere, t\u00fcm \u00e7okgenlere uygundur.<\/p>\n

Yarat\u0131lanlar tektir, g\u00fczeldir, m\u00fckemmeldir<\/strong><\/p>\n

Sanat eserinin m\u00fckemmelli\u011fi onun \u201ce\u015fsiz\u201d olu\u015fundand\u0131r. E\u015fsiz ve bulunmaz oldu\u011fu i\u00e7in tektir. Kurtulu\u015f Sava\u015f\u0131 ile ilgili bir\u00e7ok \u015fiir yaz\u0131lm\u0131\u015ft\u0131r. Ama Naz\u0131m Hikmet\u2019in \u201cKurtulu\u015f Sava\u015f\u0131 Destan\u0131\u201d tektir, biriciktir. Yaz\u0131lanlar\u0131n en co\u015fkulusu, en g\u00fczeli, en m\u00fckemmelidir. \u00c7\u00fcnk\u00fc kamunun be\u011fenisine sunulmu\u015ftur, onaylanm\u0131\u015f, kal\u0131c\u0131 hale gelmi\u015ftir. Heyecan vericidir.<\/p>\n

Matematikte de ayn\u0131 konu bir\u00e7ok matematik\u00e7i taraf\u0131ndan incelenir, kuramlar olu\u015fturulur. Bunlar\u0131n i\u00e7inde en m\u00fckemmeli kabul g\u00f6r\u00fcr. Kabul g\u00f6ren kuram kurala veya teoreme d\u00f6n\u00fc\u015f\u00fcr. Di\u011ferleri elenir. Sanattan farkl\u0131 olarak en basit, en sade, en anla\u015f\u0131l\u0131r ve en \u015fa\u015f\u0131rt\u0131c\u0131 oland\u0131r se\u00e7ilen. Matematiksel m\u00fckemmelli\u011fin temel \u00f6l\u00e7\u00fct\u00fc genel olarak \u201ctutarl\u0131l\u0131k\u201d ve \u201cyararl\u0131l\u0131k\u201dt\u0131r. En tutarl\u0131, en yararl\u0131 olan tektir, \u201ce\u015fsiz\u201ddir. En g\u00fczel olan da odur. Teklik ve g\u00fczellik sanat ve matematik i\u00e7in ortak yand\u0131r. Ama sanatsal g\u00fczellik i\u00e7in m\u00fckemmellik yeterli iken, matematiksel g\u00fczellik i\u00e7in tutarl\u0131l\u0131k ve yararl\u0131l\u0131k aran\u0131r. Biraz daha dolayl\u0131 biraz daha karma\u015f\u0131k\u2026 Belki biraz daha \u00f6zenli. Yararl\u0131l\u0131\u011f\u0131 bir anlamda \u201cciddilik\u201d olarak alan b\u00fcy\u00fck matematik\u00e7i G.H. Hardy, Bir Matematik\u00e7inin Savunmas\u0131<\/em> adl\u0131 kitab\u0131nda; \u201cNas\u0131l ki \u015fiirde bile g\u00fczellik, bir \u00f6l\u00e7\u00fcde, i\u00e7erdi\u011fi fikrin \u00f6nemli olmas\u0131na ba\u011fl\u0131ysa, bir matematik probleminin \u2018g\u00fczelli\u011fi\u2019 de b\u00fcy\u00fck \u00f6l\u00e7\u00fcde, onun ciddi olu\u015funa ba\u011fl\u0131d\u0131r\u2026 G\u00fczellik ilk s\u0131navd\u0131r. \u00c7irkin matematik i\u00e7in d\u00fcnyada yer yoktur\u201d der.<\/p>\n

Hardy\u2019nin \u00e7irkin matematikten kast\u0131 yukar\u0131da s\u00f6z\u00fcn\u00fc etti\u011fimiz elenen kuramlard\u0131r. Elbette yanl\u0131\u015flananlar da elenir. Matematiksel g\u00fczelli\u011fi problemin ciddili\u011finde aramas\u0131 ise, yine s\u00f6z\u00fcn\u00fc etti\u011fimiz tutarl\u0131l\u0131k ve yararl\u0131l\u0131k ilkesine kar\u015f\u0131l\u0131k gelmektedir. Bunun en g\u00fczel \u00f6rneklerinden biri K\u00fcmeler Kuram\u0131\u2019n\u0131n kurucusu olan Cantor\u2019un (1845-1918) sonsuzlukla ilgili \u00e7al\u0131\u015fmas\u0131d\u0131r. \u201cB\u00fct\u00fcn par\u00e7as\u0131ndan b\u00fcy\u00fckt\u00fcr\u201d ilkesini yads\u0131yan bu \u00e7al\u0131\u015fma, basitli\u011fin, sadeli\u011fin, anla\u015f\u0131l\u0131rl\u0131\u011f\u0131n g\u00fczel bir \u00f6rne\u011fidir. Ancak kuram\u0131n kal\u0131c\u0131l\u0131\u011f\u0131 sanatta oldu\u011fu gibi sadece g\u00fczelli\u011fine ba\u011fl\u0131 de\u011fildir. Kal\u0131c\u0131l\u0131\u011f\u0131n nedeni yararl\u0131l\u0131\u011f\u0131 ve tutarl\u0131l\u0131\u011f\u0131d\u0131r. \u00c7\u00fcnk\u00fc Cantor\u2019un sonsuzlukla ilgili kuram\u0131 matematikte bir \u00e7\u0131\u011f\u0131r a\u00e7m\u0131\u015f, analiz konular\u0131n\u0131n geli\u015fmesinin temel ta\u015flar\u0131ndan biri olmu\u015ftur.<\/p>\n

Evrensel etkinliklerdir<\/strong><\/p>\n

Gerek sanat gerekse matematik kitlelerin onay\u0131na a\u00e7\u0131kt\u0131r. Kitlelerin be\u011fenisini kazand\u0131\u011f\u0131 \u00f6l\u00e7\u00fcde sanatt\u0131r ya da matematiktir. O kitle t\u00fcm insanl\u0131kt\u0131r. Yerel motifler i\u00e7erse de sanat\u0131n yurdu yoktur. Sanat\u00e7\u0131n\u0131n da. Beethoven \u00c7in\u2019lidir, Arap\u2019t\u0131r, Avrupa\u2019l\u0131d\u0131r\u2026 Matematik teoremlerinin matematik\u00e7ilerin de yurdu yoktur. O da yery\u00fcz\u00fcnde ya\u015fayan her insan\u0131n be\u011fenisine, hizmetine a\u00e7\u0131kt\u0131r. Yani evrenseldir. Gerek sanat gerek matematik eserler \u00fcretildikleri \u00e7a\u011fa da ait de\u011fildir. \u00c7a\u011f\u0131n ipu\u00e7lar\u0131n\u0131 verse de sonsuzdan gelip sonsuza giden insanl\u0131\u011f\u0131n ortak mallar\u0131d\u0131r. Ma\u011faradaki resim ya da \u00e7etele, kilisedeki ikon, cami avlusundaki tasvir, Pisagor teoremi\u2026 Bunlar g\u00fcn\u00fcm\u00fczde de h\u00e2l\u00e2 taptaze eserlerdir. \u0130nsanl\u0131k var oldu\u011fu s\u00fcrece de kal\u0131c\u0131l\u0131klar\u0131n\u0131 s\u00fcrd\u00fcreceklerdir.<\/p>\n

Matemati\u011fin evrenselli\u011finin sanatta pek olmayan \u00fcst\u00fcnl\u00fc\u011f\u00fc ise, onun \u00f6rg\u00fcs\u00fcnde, b\u00fct\u00fcnselli\u011findedir. Her ne kadar Yunan Matemati\u011fi, Hint Matemati\u011fi, Avrupa ya da \u0130slam Matemati\u011fi denilse de biri di\u011ferinin \u00fcst\u00fcnde y\u00fckselmi\u015ftir. Birini aradan \u00e7ekti\u011finizde, di\u011ferleri bo\u015flukta kal\u0131r.<\/p>\n

Estetik e\u011fitim ara\u00e7lar\u0131d\u0131r<\/strong><\/p>\n

Sanat\u0131n insan\u0131n, insanla\u015fma e\u011fitiminin arac\u0131 oldu\u011fu bilinen ger\u00e7ektir. Her ne kadar bir saray\u0131n mimarisi muktedirin g\u00fcc\u00fcn\u00fc g\u00f6sterse de, ya da zalimin g\u00fcc\u00fcn\u00fc simgeleyen resimler de olsa, yani her zaman \u201cg\u00fcl kokulu\u201d olmasa da, sanat insanla\u015fma \u00e7abas\u0131n\u0131n en g\u00fc\u00e7l\u00fc arac\u0131d\u0131r. \u0130nsanda co\u015fku, mutluluk, zarafet gibi estetik duygular\u0131n\u0131 geli\u015ftirir.<\/p>\n

Bu anlamda matematik de sanattan a\u015fa\u011f\u0131 kalmaz. Matematik insan\u0131n ya\u015fam\u0131 anlama-anlamland\u0131rma yani insanla\u015fma \u00e7abas\u0131n\u0131n en \u00f6nemli ara\u00e7lar\u0131ndand\u0131r. G\u00f6ky\u00fcz\u00fcndeki ku\u015fun bir \u201can\u201ddaki kanat sesini duyars\u0131n matematikle, dokunamad\u0131\u011f\u0131n Merih\u2019le birlikte d\u00f6nersin uzayda, \u0131\u015f\u0131\u011f\u0131n nas\u0131l b\u00fck\u00fcld\u00fc\u011f\u00fcn\u00fc g\u00f6zlersin sonsuzlukta. Bir\u00e7ok matematik probleminin \u00e7\u00f6z\u00fcm\u00fcnde ya da bir teoremin ispat\u0131nda da ya\u015fars\u0131n sanattakine benzer hazz\u0131, co\u015fkuyu, g\u00fczelli\u011fi\u2026<\/p>\n

Yine bir farkla ki sanattan zevk almak \u00e7o\u011funlukla \u00f6zel bir \u00e7aba gerektirmez. \u0130zlemeyi, g\u00f6rmeyi, dinlemeyi bilmek yeter co\u015fku duymak i\u00e7in. Matematik ise biraz \u00e7aba gerektirir. Matemati\u011fi anlama ve \u00f6\u011frenme \u00e7abas\u0131. Sundu\u011fu e\u015fsiz g\u00fczellikleri duyabilmeye de\u011fecek bir \u00e7aba. Bilenler i\u00e7in co\u015fku duyulan, bilmeyenler i\u00e7in korkulan bir etkinliktir matematik\u2026<\/p>\n

\u00d6yleyse yeni bir soru: Korkulan bir etkinlik olmas\u0131 yukar\u0131daki saptamalarla \u00e7eli\u015fmiyor mu? Matematik, sanatsal bir etkinlik ise neden korkulan bir etkinlik olsun? Siz bir \u00f6\u011frencinin, \u201cresim ne i\u015fime yarayacak\u201d veya \u201cm\u00fczik ne i\u015fime yarayacak\u201d dedi\u011fini duydunuz mu? \u0130\u015fte tart\u0131\u015ft\u0131\u011f\u0131m\u0131z bu fiili durum. Bir yanda yararl\u0131l\u0131k yan\u0131yla sistemin \u00f6ne \u00e7\u0131kard\u0131\u011f\u0131 bir anlamda \u201cdayat\u0131lan\u201d matematik. Di\u011fer yanda estetik yan\u0131 ile g\u00fczellikleriyle \u201cg\u00f6rmezlikten gelinen\u201d matematik.<\/p>\n

Yukar\u0131da s\u0131ralad\u0131\u011f\u0131m\u0131z ba\u011fda\u015f\u0131kl\u0131klar artt\u0131r\u0131labilir. \u00d6te yandan itirazlar da olabilir. \u00c7\u00fcnk\u00fc her sanat eseri her insana ayn\u0131 hazz\u0131 vermeyebilir. Anl\u0131k be\u011feniler farkl\u0131 olabilece\u011fi gibi s\u00fcre\u00e7 i\u00e7inde de\u011fi\u015fen alg\u0131lar da \u201csanat eseri\u201d alg\u0131s\u0131nda farkl\u0131la\u015fmaya yol a\u00e7abilir. O zaman \u015f\u00f6yle bir sonu\u00e7 \u00e7\u0131karmak san\u0131r\u0131m do\u011fru olacakt\u0131r: Bir esere d\u00f6nemin ko\u015fullar\u0131n\u0131, anlay\u0131\u015f\u0131n\u0131 alg\u0131layarak bakmak gerek. \u00d6yle ya\u2026 \u00dc\u00e7 boyutlu resme ge\u00e7ilmedi\u011fi d\u00f6nemlerde ma\u011fara duvarlar\u0131na \u00e7izilen \u00e7ocuk\u00e7a resimlere ne demeliyiz? Yandan g\u00f6r\u00fcnen bir resimde at \u00fczerindeki adam\u0131n iki aya\u011f\u0131n\u0131n da g\u00f6r\u00fcnmesine kibirle bakarsak, sanat\u0131n kal\u0131c\u0131l\u0131\u011f\u0131ndan s\u00f6z edebilir miyiz? Matematikte de bug\u00fcnk\u00fc geli\u015fmi\u015flik d\u00fczeyine g\u00f6re ge\u00e7mi\u015fte kullan\u0131lan bir\u00e7ok kabul ve y\u00f6ntemin yetersiz hatta yanl\u0131\u015f oldu\u011funu biliyoruz. \u00d6rne\u011fin Antik Yunan \u00f6ncesi M\u0131s\u0131r\u2019da kenarlar\u0131 a,b,c,d ile g\u00f6sterilen ABCD d\u00f6rtgeninin alan\u0131, A(ABCD) = (a+c) (b+d) \/ 4 form\u00fcl\u00fc ile hesaplan\u0131yordu. Dikd\u00f6rtgen d\u0131\u015f\u0131nda di\u011fer d\u00f6rtgenler i\u00e7in yanl\u0131\u015f olan bu form\u00fcl s\u00fcre\u00e7 i\u00e7inde d\u00fczeldi. Ama o g\u00fcn\u00fcn geli\u015fmi\u015flik d\u00fczeyine g\u00f6re o form\u00fcl \u00f6nemliydi. Aksini d\u00fc\u015f\u00fcnmek \u201ceskiler ne aptalm\u0131\u015f\u201d aptall\u0131\u011f\u0131na d\u00fc\u015fmek olur.<\/p>\n

Matematik ve sanat ili\u015fkisi ile ilgili bir\u00e7ok ara\u015ft\u0131rma ve tart\u0131\u015fma vard\u0131r. Hatta matemati\u011fi do\u011frudan sanat\u0131n bir dal\u0131 olarak tan\u0131mlayan matematik\u00e7iler, bilim felsefecileri de vard\u0131r. Bu tart\u0131\u015fmalar \u00f6nemli ve \u00e7ok yararl\u0131. \u00c7\u00fcnk\u00fc matematik i\u00e7in \u201csanatt\u0131r\u201d diyenlerin de \u201csanat de\u011fildir\u201d diyenlerin de birle\u015fti\u011fi nokta; \u201cmatemati\u011fin sanatsal de\u011ferinin g\u00fc\u00e7l\u00fc\u201d oldu\u011fudur.<\/p>\n

\u00dczerinde durmam\u0131z gereken as\u0131l nokta ise; matematik \u00f6\u011fretiminde matemati\u011fin estetik y\u00f6n\u00fcn\u00fcn nas\u0131l yans\u0131t\u0131laca\u011f\u0131\u2026 Elbette bu yukar\u0131daki maddeler s\u0131ralanarak yap\u0131lamaz. Bu hi\u00e7 ger\u00e7ek\u00e7i de\u011fil. Yap\u0131lmal\u0131 ama nas\u0131l? Nas\u0131l yap\u0131lmal\u0131n\u0131n yan\u0131t\u0131 san\u0131r\u0131m kendi ge\u00e7mi\u015fimizde\u2026 Kendi matematik \u00f6\u011frenme ser\u00fcvenimizde. S\u0131k s\u0131k kendi matematik ser\u00fcvenimi d\u00fc\u015f\u00fcn\u00fcr\u00fcm ve heyecan duyar\u0131m. Bu ser\u00fcven ilk problem \u00e7\u00f6zme y\u0131llar\u0131na dek uzan\u0131r.<\/p>\n

Bir problem \u00e7\u00f6z\u00fcm\u00fc, hele hele uzun u\u011fra\u015f\u0131lardan, uzun i\u015flemlerden sonra sonuca ula\u015fmak \u00e7ocuk ya\u015fta bende ciddi hazlar uyand\u0131r\u0131yordu. Bir futbol ma\u00e7\u0131nda en kritik anda gol atmak gibi. Bin y\u0131ll\u0131k bir heykeli g\u00f6r\u00fcp bin y\u0131l \u00f6nce bu heykelin kanl\u0131 canl\u0131 bir adam oldu\u011funu d\u00fc\u015f\u00fcnmek gibi. Be\u011fendi\u011fim bir k\u0131z\u0131n g\u00f6zlerinin benim \u00fczerimde oldu\u011funu bilmek gibi\u2026 Ka\u00e7 de\u011fi\u015fik kanal, ka\u00e7 de\u011fi\u015fik haz. Ama hepsi insani, hepsi heyecan dolu. Problem \u00e7\u00f6zmenin \u00f6tesindeki matemati\u011fi \u00f6\u011frendik\u00e7e duydu\u011fum haz artt\u0131. Baz\u0131lar\u0131n\u0131 zor \u00f6\u011frensem de\u2026 Havaya att\u0131\u011f\u0131m topun d\u00fc\u015ferken \u00e7izdi\u011fi yolu k\u00e2\u011f\u0131da d\u00f6kmek ilgin\u00e7ti. Atarken h\u0131zl\u0131, y\u00fckseldik\u00e7e yava\u015fl\u0131yor, bir an duruyor sonra y\u00fckselirken yava\u015flamas\u0131n\u0131n tersine h\u0131zlanarak d\u00fc\u015f\u00fcyordu. Simetrik bir yol \u00e7izerek. Ka\u00e7 kez istop oynam\u0131\u015f, matemati\u011fi \u00f6\u011frenene kadar bunu fark edememi\u015ftim. \u0130\u015fte o yolu \u00e7izdim. \u00d6nce ilgin\u00e7 geldi sonra g\u00fczel. Mimar gibi kroki, ressam gibi resim \u00e7iziyordum. Heyecanla\u2026 \u00dc\u00e7 bacakl\u0131 masa yere daha sa\u011flam basarm\u0131\u015f. Denedim ve bakt\u0131m ki ger\u00e7ekten \u00f6yle. \u00d6nce \u015fa\u015f\u0131rd\u0131m sonra heyecan duydum. Ben niye d\u00fc\u015f\u00fcnememi\u015fim? Merih\u2019in benden ne kadar uzakta oldu\u011funu bulabilirmi\u015fim. Hem de Merih\u2019e hi\u00e7 dokunmadan\u2026 Sanki b\u00fcy\u00fcc\u00fc gibiyim. Dudaklar\u0131mda bir g\u00fcl\u00fcmseme, muzaffer. Ve de bir otomobilin bir andaki h\u0131z\u0131n\u0131 da bulabilirmi\u015fim\u2026 Muhte\u015fem\u2026 Otomobil \u015fof\u00f6r\u00fcn\u00fcn bundan haberi bile yok.<\/p>\n

Bir sonraki a\u015fama bunlar\u0131 \u00f6\u011fretmen olarak \u00f6\u011fretmek. Daha do\u011fru deyi\u015fle payla\u015fmak. \u00d6nceki akranlar\u0131mla yani \u00f6\u011frencilerimle. Onlar\u0131n duydu\u011fu \u201cheyecan\u0131\u201d, \u201cgizemi par\u00e7alama\u201d duygusunu, \u201cba\u015farman\u0131n co\u015fkusunu\u201d yeniden ya\u015fayarak. Taptaze kalan heyecanlar. Benden \u00f6nce duyulan, benden sonra duyulacak co\u015fkularla\u2026<\/p>\n

Sonra sorular sorular\u2026 Neden antik \u00e7a\u011f \u00f6ncesine dek gitmi\u015f matematik yapma – matematik \u00f6\u011frenme tutkusu? Salt gereklilik salt yarar i\u00e7in olabilir mi? \u00d6yle olsayd\u0131 hep sanatla birlikte \u00f6\u011frenilir miydi? Neden hep sanatla birlikte an\u0131lm\u0131\u015f? Neden ilk saptanan \u00f6\u011fretim programlar\u0131n\u0131n d\u00f6rt disiplininden de\u011fi\u015fmez \u201cbir\u201di olmu\u015f? Neden suyun bile\u015fenleri bilinmezden, g\u00fcl\u00fcn kokusu ayr\u0131\u015ft\u0131r\u0131lamazdan \u00f6ncesinde varm\u0131\u015f? Hangi matematik, nas\u0131l bir matematikmi\u015f bu? Ne menem \u015feymi\u015f matemati\u011fin esteti\u011fi? Biraz ya\u015fanm\u0131\u015fl\u0131klar, biraz \u00f6rnek, biraz \u00f6neri\u2026<\/p>\n

\u2018\u00d6\u011eRENME\u2019DE MATEMAT\u0130K ESTET\u0130\u011e\u0130<\/strong><\/h4>\n

Ulan \u00d6klid<\/strong><\/p>\n

Teorem kan\u0131tlaman\u0131n hazz\u0131n\u0131 bir ya\u015fanm\u0131\u015fl\u0131kla aktaray\u0131m. Matematik ba\u015far\u0131s\u0131n\u0131n d\u00fc\u015f\u00fck oldu\u011fu bir s\u0131n\u0131fta Metrik Ba\u011f\u0131nt\u0131lar\u0131 i\u015fliyoruz. ABC dik \u00fc\u00e7geninde BC hipoten\u00fcs, AH hipoten\u00fcse ait y\u00fckseklik olmak \u00fczere; |AH|2<\/sup> = |HB|.|HC| (Birinci \u00d6klid) e\u015fitli\u011fini ispatlamalar\u0131n\u0131 istedim. Olduk\u00e7a zay\u0131f olan ve san\u0131r\u0131m bu s\u0131n\u0131fa de\u011fin hi\u00e7bir teorem ispatlamam\u0131\u015f bir \u00f6\u011frencim biraz \u00fcrkek, \u201chocam ben bir \u015feyler yapt\u0131m ama\u2026 bakar m\u0131s\u0131n\u0131z\u201d dedi. Gittim. Kulland\u0131\u011f\u0131 ba\u011f\u0131nt\u0131lar ve g\u00f6sterdi\u011fi benzerlik sonucu do\u011fruydu. Teoremi kan\u0131tlam\u0131\u015ft\u0131. \u201cG\u00fczel, sonucu bulmu\u015fsun\u201d diyerek kutlad\u0131m. \u015ea\u015f\u0131rd\u0131. G\u00f6z\u00fc defterinde co\u015fkuyla, \u201culan \u00d6klid, sen olmasan bu benim teoremim olacakt\u0131\u201d t\u00fcr\u00fc bir tepki verdi. O co\u015fkuyla her rastlad\u0131\u011f\u0131 teoremi kan\u0131tlama \u00e7abas\u0131na girdi. G\u00fcveni artm\u0131\u015ft\u0131.<\/p>\n

\"\"
Elementary school pupil working at desk<\/figcaption><\/figure>\n

Bir kan\u0131t<\/strong><\/p>\n

\u0130ki’nin karek\u00f6k\u00fc \u2018n\u00fcn irrasyonel oldu\u011funun ispat\u0131 matematik\u00e7ilerce bilinir. Bu ispat\u0131n g\u00fczelli\u011fi de\u2026 Teorem ispat\u0131na ge\u00e7meden, a ve b birer tamsay\u0131, b\u00a00 olmak \u00fczere \u201ca\/b\u201d bi\u00e7iminde yaz\u0131lan say\u0131lara rasyonel say\u0131, \u00a0gibi say\u0131lara ise irrasyonel say\u0131 denildi\u011fini an\u0131msatal\u0131m. \u201cSoru\u201dnun nas\u0131l \u201csorun\u201d haline geldi\u011fini de…<\/p>\n

\u00d6teden beri X2<\/sup> = 4 denkleminin \u00e7\u00f6z\u00fcm\u00fcn\u00fcn +2 veya -2 oldu\u011fu kolayl\u0131kla s\u00f6ylenebiliyordu. Hatta X2 <\/sup>= 4\/9 denklemi de X=\u00a02\/3 olarak \u00e7\u00f6z\u00fcl\u00fcyordu. Ama i\u015f X2<\/sup> = 2 denkleminin \u00e7\u00f6z\u00fcm\u00fcne gelince sorun ba\u015fl\u0131yordu. \u00c7\u00fcnk\u00fc karesi 2 olan say\u0131 vard\u0131 ama rasyonel say\u0131 k\u00fcmesinde yoktu. Hatta 2 neyin karesidir sorusunun, say\u0131lara tanr\u0131sal bir g\u00fc\u00e7 atfeden Pisagor Okulu\u2019nda cinayete yol a\u00e7t\u0131\u011f\u0131 bile s\u00f6ylenir\u2026 Sonu\u00e7ta\u2019nin bilinen rasyonel say\u0131 k\u00fcmesinde tan\u0131ml\u0131 olmad\u0131\u011f\u0131 d\u00fc\u015f\u00fcn\u00fcl\u00fcyor. Ama matematikte \u201colmad\u0131\u011f\u0131n\u0131 d\u00fc\u015f\u00fcnmek\u201d yetmedi\u011fi i\u00e7in kan\u0131tlama \u00e7abas\u0131 ba\u015fl\u0131yor\u2026<\/p>\n

Teorem: <\/strong>\u00a0\u00a0irrasyonel say\u0131d\u0131r.<\/p>\n

Kan\u0131t:<\/strong> nin a ile b aralar\u0131nda asal (ortak b\u00f6leni olmayan) ve \u201ca\/b\u201d gibi bir rasyonel say\u0131 oldu\u011funu varsayal\u0131m.<\/p>\n

= a\/b ve a = \u00a0olur. Her iki yan\u0131n karesi al\u0131nd\u0131\u011f\u0131nda; a2 <\/sup>= 2b2 <\/sup>olur ki, 2b2 <\/sup>\u00e7ift say\u0131 oldu\u011fundan onun e\u015fiti olan a2<\/sup> de \u00e7ift say\u0131d\u0131r.<\/p>\n

Karesi \u00e7ift say\u0131 olan say\u0131 da \u00e7ift say\u0131 olaca\u011f\u0131 i\u00e7in \u201ca\u201d \u00e7ift say\u0131d\u0131r. Ve a =2c bi\u00e7iminde yaz\u0131labilir.<\/p>\n

a2 <\/sup>= 2b2 <\/sup>\u2018de \u201ca\u201d yerine \u201c2c\u201d yaz\u0131l\u0131rsa; 4c2<\/sup> =2b2<\/sup> den b2<\/sup> =2c2<\/sup> olur ki, yukar\u0131daki gerek\u00e7e nedeniyle b2<\/sup> ve ba\u011fl\u0131 olarak da \u201cb\u201d \u00e7ift say\u0131 olmak zorundad\u0131r.<\/p>\n

= a\/b varsay\u0131m\u0131nda \u201ca\u201d ve \u201cb\u201d \u00e7ift say\u0131 ise ortak b\u00f6lenleri vard\u0131r ki, a ile b aralar\u0131nda asal olamaz. Aralar\u0131nda asald\u0131r varsay\u0131m\u0131yla \u00e7eli\u015fir. \u00d6yle ise \u00a0rasyonel de\u011fil, irrasyonel say\u0131d\u0131r yarg\u0131s\u0131na var\u0131r\u0131z.<\/p>\n

Bu kan\u0131ttan alaca\u011f\u0131m\u0131z haz bir m\u00fczik par\u00e7as\u0131n\u0131 dinledi\u011fimizde al\u0131nacak hazla ayn\u0131 de\u011fildir. M\u00fczi\u011fi dinler, dinledi\u011finiz anda duygusal hazlar ya\u015far\u0131z. Ancak \u00a0nin irrasyonel oldu\u011funun ispat\u0131 anl\u0131k hazlar yaratmayabilir. \u2019nin y\u0131llarca gizemini korudu\u011fu, insanlar\u0131n ak\u0131llar\u0131n\u0131 i\u015fgal etti\u011fi ve hatta Pisagor Okulu\u2019nda u\u011fruna cinayet i\u015flendi\u011fi bir s\u00fcrecin sonucu oldu\u011fuyla birlikte d\u00fc\u015f\u00fcn\u00fclmelidir. Ard\u0131ndan da insanlar\u0131 uzun s\u00fcre d\u00fc\u015f\u00fcnd\u00fcrm\u00fc\u015f olan sorunun birka\u00e7 ak\u0131l ve kalem darbesiyle \u00e7\u00f6z\u00fclebildi\u011fini g\u00f6rmek gerek. \u0130\u015fte sorun burada. Yukar\u0131daki kan\u0131t kadar \u00f6nemli olan bu ya\u015fanm\u0131\u015fl\u0131k. Bunlar\u0131 bilmek \u00f6\u011frencilerin hakk\u0131. \u0130\u00e7ine duyguyu, co\u015fkuyu katmad\u0131\u011f\u0131n\u0131z hi\u00e7bir bilgi kal\u0131c\u0131 olmaz. \u00dcst\u00fcne \u00fcstl\u00fck matematik bu duygu k\u0131\u015fk\u0131rtmalar\u0131na \u00e7ok m\u00fcsait. Bunu bilmek ve yerine getirmek. Tek sorun bu.<\/p>\n

Bir paradoksun yapt\u0131klar\u0131<\/strong><\/p>\n

Paradokslar genellikle bir \u015feyin, ger\u00e7ek d\u00fcnyadaki davran\u0131\u015f\u0131 ile matematik d\u00fcnyas\u0131ndaki davran\u0131\u015f\u0131 aras\u0131ndaki \u00e7eli\u015fkiler bi\u00e7iminde tan\u0131mlan\u0131r. \u0130lgin\u00e7 oldu\u011fu kadar da \u015fa\u015f\u0131rt\u0131c\u0131d\u0131r. Bu \u0130lgin\u00e7lik gazetecinin k\u00f6pe\u011fi \u0131s\u0131rmas\u0131 gibi bir ilgin\u00e7lik de\u011fil, suyun i\u00e7inde ate\u015f yanmas\u0131 gibi olanaks\u0131z bir ilgin\u00e7liktir. Kar\u015f\u0131m\u0131za \u00e7\u0131k\u0131verdi\u011fi i\u00e7in de\u011fil, d\u00fc\u015f\u00fcnd\u00fck\u00e7e \u015fa\u015fk\u0131nl\u0131\u011f\u0131m\u0131z artar. \u201cYahu d\u00fc\u015f\u00fcnd\u00fck\u00e7e kafay\u0131 yiyece\u011fim\u201d denilen t\u00fcrden bir \u015fa\u015fk\u0131nl\u0131k. Anl\u0131k beklenmediklere de\u011fil derinlikli \u00e7eli\u015fkilere ba\u011fl\u0131\u2026 Herkesin de\u011fil birilerinin buldu\u011fu \u00e7eli\u015fkiler. Birilerinin buldu\u011fu ama herkesin u\u011fra\u015fmaktan kendini alamad\u0131\u011f\u0131\u2026 Sab\u0131r isteyen, derinlikli d\u00fc\u015f\u00fcn\u00fc\u015f isteyen ve e\u011flenceli\u2026 Sonsuzlu\u011fu anlamaya \u00e7al\u0131\u015fmak bunlardan biridir. Ger\u00e7ek d\u00fcnyada oldu\u011fu kadar matematik d\u00fcnyas\u0131nda da\u2026 A\u015fil Paradoksu en yayg\u0131n bilinenlerdendir.<\/p>\n

Soru: <\/strong>A\u015fil ile kaplumba\u011fa yar\u0131\u015facaklar. A\u015fil kaplumba\u011fan\u0131n on kat\u0131 kadar h\u0131zl\u0131 ko\u015fuyor. Biraz adalet i\u00e7in kaplumba\u011fa yar\u0131\u015fa 100 metre \u00f6nden ba\u015fl\u0131yor. A\u015fil kaplumba\u011faya ka\u00e7\u0131nc\u0131 metrede yeti\u015fir?<\/p>\n

\u00c7\u00f6z\u00fcm: <\/strong>G\u00f6r\u00fcn\u00fc\u015fte zor soru de\u011fil. Ama h\u0131z yok, zaman yok. Arada 100 metrelik fark var. Bu fark ad\u0131m ad\u0131m kapanacak. Nas\u0131l ve ka\u00e7 metrede. San\u0131r\u0131m \u015fekil \u00fczerinde d\u00fc\u015f\u00fcnmek en iyisi\u2026<\/p>\n

Aradaki fark yar\u0131\u015f ba\u015flad\u0131\u011f\u0131nda 100 metre. A\u015fil 100 metre ko\u015fup kaplumba\u011fan\u0131n oldu\u011fu yere geldi\u011finde kaplumba\u011fa 10 metre yol al\u0131r. Yani kaplumba\u011fa 10 metre ilerde. A\u015fil 10 metre yol ald\u0131\u011f\u0131nda kaplumba\u011fa 1 metre yol al\u0131r. Bu kez fark yine var ve kaplumba\u011fa 1 metre \u00f6nde\u2026 Sonra fark giderek azal\u0131r; 1\/10 metre, 1\/100 metre, 1\/1000 metre\u2026 Ama kaplumba\u011fa hep \u00f6nde.<\/p>\n

Soru ka\u00e7 metre sonra yeti\u015fir idi. Ama bak\u0131yoruz fark azal\u0131yor ama hi\u00e7 bitmiyor. Bitecek gibi de de\u011fil. Soru bir ba\u015fka bi\u00e7ime d\u00f6n\u00fc\u015ft\u00fc: \u201cA\u015fil kaplumba\u011faya yeti\u015fir mi?\u201d<\/p>\n

Aradaki fark\u0131 bir kez daha g\u00f6zden ge\u00e7irelim:<\/p>\n

100, 10, 1, 1\/10, 1\/100, 1\/1000,\u2026. G\u00f6r\u00fcnen o ki arada hep bir fark olacak ve A\u015fil kaplumba\u011faya asla yeti\u015femeyecek.<\/p>\n

\u0130\u015fte itiraz: \u201cNe demek, ge\u00e7er bile!\u201d Oysa k\u00e2\u011f\u0131t \u00fczerine yapt\u0131\u011f\u0131m\u0131z tart\u0131\u015fma bize ge\u00e7mek bir yana yeti\u015femeyece\u011fini g\u00f6sterdi. \u0130\u015fte paradoks dedi\u011fimiz tam da bu. K\u00e2\u011f\u0131t \u00fczerindeki mant\u0131kl\u0131 tart\u0131\u015fma ger\u00e7ek hayatla \u00e7eli\u015fti. \u201cGe\u00e7er bile\u201d tepkisi, ger\u00e7ek hayat\u0131n yaz\u0131l\u0131 \u00e7izili olana tepkisi.<\/p>\n

Bu tepki bizi; \u201c1\/\u221e = 0\u201d<\/strong> sonucuna g\u00f6t\u00fcr\u00fcr. K\u00e2\u011f\u0131t \u00fczerinde yapt\u0131\u011f\u0131m\u0131z i\u015flemde payda giderek b\u00fcy\u00fcmektedir. Bu de\u011ferin sonsuza de\u011fin b\u00fcy\u00fcyece\u011fini \u00f6ng\u00f6rebiliriz. Yani yukar\u0131daki dizi;<\/p>\n

100, 10, 1, 1\/10, 1\/100, 1\/1000,\u2026., 1\/\u221e <\/strong>bi\u00e7imine d\u00f6n\u00fc\u015fecektir. Di\u011fer yandan ger\u00e7ek hayat bize, A\u015fil\u2019in kaplumba\u011faya yeti\u015fece\u011fini g\u00f6stermektedir. Yeti\u015fme an\u0131nda aradaki fark\u0131n \u201c0\u201d olaca\u011f\u0131n\u0131 biliyoruz. \u00d6yle ise rahatl\u0131kla \u201c1\/\u221e = 0\u201d <\/strong>sonucuna ula\u015f\u0131r\u0131z. Art\u0131k 1 yerine 2,3,4\u2026 gibi hangi do\u011fal say\u0131 gelirse gelsin sonucun \u201c0\u201d a e\u015fit olaca\u011f\u0131 kolayl\u0131kla s\u00f6ylenebilir. Bu sonucun s\u00f6z olarak ifadesi \u201cs\u0131f\u0131rdan farkl\u0131 bir say\u0131n\u0131n sonsuza b\u00f6l\u00fcm\u00fc 0\u2019a e\u015fittir\u201d <\/strong>\u015feklindedir ki matematikte kapal\u0131 bir\u00e7ok kap\u0131n\u0131n a\u00e7\u0131lmas\u0131na yol a\u00e7m\u0131\u015ft\u0131r.<\/p>\n

Bir \u00e7eli\u015fkiden yaratt\u0131\u011f\u0131m\u0131z \u00f6nemli sonucu bir yana b\u0131rakal\u0131m ve as\u0131l soruya ge\u00e7elim: \u201cA\u015fil kaplumba\u011faya ka\u00e7\u0131nc\u0131 metrede yeti\u015fir.\u201d Soruyu bir ba\u015fka bi\u00e7imde \u015f\u00f6yle sorabiliriz: A\u015fil kaplumba\u011faya yeti\u015fti\u011finde ka\u00e7 metre yol alm\u0131\u015ft\u0131r?<\/p>\n

Yan\u0131t\u0131n nas\u0131l bulunaca\u011f\u0131 a\u00e7\u0131k. Aradaki farklar\u0131 toplar\u0131z olur biter. Ki farklar\u0131 yukar\u0131da s\u00f6ylemi\u015ftik. Yap\u0131lacak i\u015flem matematiksel olarak;<\/p>\n

\u201c100 + 10 + 1 + 1\/10 + 1\/100 + 1\/1000 + \u2026.. = ?\u201d i\u015flemi. \u0130\u015fte burada yine sorun var. Toplanacak say\u0131lar s\u0131n\u0131r tan\u0131m\u0131yor. Ald\u0131 ba\u015f\u0131n\u0131 gidiyor\u2026 Gel de topla. \u0130\u015f ba\u015fa d\u00fc\u015ft\u00fc.<\/p>\n

100 + 10 +1 + 1\/10 + 1\/100 + 1\/1000 + \u2026\u2026.. = 111 + 1\/10 + 1\/100 + 1\/1000 + \u2026. e\u015fitli\u011fini yazabiliriz.<\/p>\n

Sorun biraz daha belli oldu. 111 i\u015fin kolay yan\u0131yd\u0131. Ya \u201c1\/10 + 1\/100 + 1\/1000 + \u2026.\u201d toplam\u0131 nas\u0131l yap\u0131lacak?<\/p>\n

\u00d6nce d\u00fczene koymal\u0131 yani bi\u00e7imlendirmeli ve sonra birka\u00e7 ak\u0131ll\u0131 hamle yapmal\u0131y\u0131z. Bu toplama X demekle ba\u015flayal\u0131m.<\/p>\n

X = 1\/10 + 1\/100 + 1\/1000 + 1\/10000 + \u2026. \u00a0Hep 1\/10\u2019un katlar\u0131. 1\/10\u2019dan sonras\u0131n\u0131, \u201c1\/10\u201d parantezine alal\u0131m.<\/p>\n

X = 1\/10 + 1\/10.(1\/10 + 1\/100 + 1\/1000 + \u2026.) olacakt\u0131r. Parantez i\u00e7indeki toplama \u201cX\u201d demi\u015ftik. Yerine koyal\u0131m.<\/p>\n

X = 1\/10 + 1\/10.X olur ve bir oh \u00e7ekeriz. \u201c\u2026..\u201d dan kurtulduk! Art\u0131k i\u015fleme devam:<\/p>\n

X = 1\/10 + X\/10; X \u2013 X\/10 = 1\/10; X.(1 \u2013 1\/10) = 1\/10; 9X\/10 = 1\/10; 9X = 1 ve X = 1\/9<\/strong> olur ki aranan toplam; \u201c111 + 1\/9\u201d dan \u201c1000\/9\u201d <\/strong>olacakt\u0131r.<\/p>\n

\u015eimdi bir soluk almay\u0131 hak ettik. Dural\u0131m ve kazan\u0131mlar\u0131m\u0131za bakal\u0131m.<\/p>\n

Birincisi<\/strong>; g\u00fczel bir ak\u0131l y\u00fcr\u00fctmeyle (g\u00fczel diyorum \u00e7\u00fcnk\u00fc ya\u015fam ger\u00e7e\u011fiyle bir \u00e7eli\u015fkiye son verdik) 1\/\u221e = 0 <\/strong>sonucuna ula\u015ft\u0131k. Bu bizi; 3\/\u221e =0, (3\/5)\/\u221e = 0, \u2026 gibi yani \u201csay\u0131\/sonsuz = 0\u201d <\/strong>kal\u0131c\u0131 genel sonucuna g\u00f6t\u00fcrd\u00fc.<\/p>\n

\u0130kincisi; <\/strong>X = 1\/10 + 1\/100 + 1\/1000 + 1\/10000 + \u2026. \u00a0\u0130\u015fleminde zarif bir hamle ile<\/p>\n

X = 1\/10 + 1\/10.(1\/10 + 1\/100 + 1\/1000 + \u2026.) ve X = 1\/10 + 1\/10.X e\u015fitli\u011fi ile X = (1\/10)\/(1-1\/10) ile devam ettik. Yani say\u0131lar sonsuza da gitse elimizden kurtulamad\u0131.<\/p>\n

Ama matematik\u00e7ilerin iflah olmaz bir yan\u0131 vard\u0131r. S\u00fcrekli soru sorarlar kendilerine. Neden diye, ni\u00e7in diye\u2026 \u201cAcaba\u201d da hep vard\u0131r kafalar\u0131n\u0131n bir yerlerinde. Biri sormasa bir di\u011feri mutlaka sorar: Keramet, toplanacak say\u0131lar\u0131n \u201c1\/10\u201dun kuvvetleri olmas\u0131nda m\u0131? Resmin g\u00fczelli\u011fi a\u011fac\u0131n g\u00fczelli\u011fine ba\u011fl\u0131ym\u0131\u015f gibi!<\/p>\n

Bu sorunun \u00e7\u00f6z\u00fcm\u00fc i\u00e7in matemati\u011fin kulland\u0131\u011f\u0131 yol \u201cgenelleme\u201ddir. Yani her say\u0131 i\u00e7in do\u011fru oldu\u011funu g\u00f6stermek. Bunu g\u00f6sterirsek kerametin \u201c1\/10\u201dda olmad\u0131\u011f\u0131 anla\u015f\u0131lacakt\u0131r. A\u015fa\u011f\u0131daki teoremin ispat\u0131yla matemati\u011fin dozunu biraz daha artt\u0131ral\u0131m.<\/p>\n

Teorem: <\/strong>m\u00a0R, 1\/m + 1\/m2 <\/sup>+ 1\/m3<\/sup> + 1\/m4 <\/sup>+ \u2026 = (1\/m)\/ (1-1\/m) dir.<\/p>\n

\u0130spat: <\/strong>E\u015fitli\u011fin sol yan\u0131na X diyelim.<\/p>\n

X = 1\/m + 1\/m2 <\/sup>+ 1\/m3<\/sup> + 1\/m4 <\/sup>+\u2026 bi\u00e7imine d\u00f6n\u00fc\u015f\u00fcr ve t\u00fcm terimler 1\/m\u2019nin katlar\u0131d\u0131r. Bu kez az\u0131c\u0131k farkl\u0131 davranarak t\u00fcm terimleri \u201c1\/m\u201d ortak \u00e7arpan parantezine alal\u0131m.<\/p>\n

X = (1\/m).( 1 + 1\/m + 1\/m2 <\/sup>+ 1\/m3<\/sup> + 1\/m4 <\/sup>+\u2026) olacakt\u0131r. \u0130kinci parantezde \u201c1\/m + 1\/m2 <\/sup>+ 1\/m3<\/sup> + 1\/m4 <\/sup>+\u2026\u201d yerine de\u011ferini yani X\u2019i yazarsak e\u015fitlik,<\/p>\n

X = (1\/m).(1 + X) durumuna gelir. Devamla X = (1\/m) + (X\/m); X \u2013 (X\/m) = 1\/m olur. Sol yan X ortak \u00e7arpan parantezine al\u0131nd\u0131\u011f\u0131nda;<\/p>\n

X(1 \u2013 1\/m) = 1\/m ve X = (1\/m).(1\/ (1 \u2013 1\/m)<\/strong> ) bulunur ki bulmam\u0131z gereken de buydu.<\/p>\n

X = 1\/9\u2019u bulurken uygulanan zarif i\u015flem, X = (1\/m) . (1\/(1-1\/m)) genellemesine d\u00f6n\u00fc\u015ft\u00fc ki bu genelleme \u00e7ok y\u00f6nl\u00fc kullan\u0131lacak anahtar gibidir. Bu elbette \u00f6nemli. Ama ad\u0131m ad\u0131m, her sat\u0131rda sektirmesiz yapt\u0131\u011f\u0131m\u0131z i\u015flemler daha m\u0131 az \u00f6nemli. Her ad\u0131m bir ahenk i\u00e7ermiyor mu? \u00d6nemli olan bunun fark\u0131nda olmak ve daha da \u00f6nemlisi bunu uygulatt\u0131\u011f\u0131m\u0131z \u00f6\u011frenci bu ahengi ya\u015f\u0131yor mu? Bunun i\u00e7in yeterli zaman ay\u0131r\u0131yor muyuz? Yoksa bu hazz\u0131 ya\u015fatmadan hemen ba\u015fka bir soruya m\u0131 ge\u00e7iyoruz. Sel \u00f6n\u00fcnden k\u00fct\u00fck kapar gibi\u2026 O zaman biz bu ispat\u0131 neden yapt\u0131k. Hani duygu, hani sab\u0131r, hani dikkat? Bu durumda, yanl\u0131\u015f bir nota bir m\u00fczik par\u00e7as\u0131n\u0131 nas\u0131l eser olmaktan \u00e7\u0131kar\u0131rsa, yanl\u0131\u015f bir i\u015flem de matemati\u011fi matematik olmaktan \u00e7\u0131kar\u0131r. Matematik de bir sanat eserinin olu\u015fumundaki gibi \u00f6zen ister, duygu ister, sayg\u0131 ve sab\u0131r ister. Bir ser\u00fcven gibi\u2026 deme hakk\u0131m\u0131z olur mu?<\/p>\n

Sanat\u00e7\u0131 eserini \u201cbe\u011fenilsin\u201d kayg\u0131s\u0131 ile yapmaz. \u0130\u00e7sel bir d\u00fcrt\u00fcd\u00fcr sanat\u00e7\u0131 i\u00e7in eser yaratmak. Be\u011feni, eser ortaya \u00e7\u0131kt\u0131ktan sonraki bir durumdur. Elbette be\u011fenilmenin bir hazz\u0131 vard\u0131r. Matematik\u00e7i de \u201cbe\u011fenilsin\u201d diye matematik yapmaz. Cahit Arf, \u201cba\u015flang\u0131\u00e7ta alk\u0131\u015f almay\u0131 istedim. Ald\u0131m da. Alk\u0131\u015f k\u0131sa s\u00fcrede \u00f6nemini yitirdi\u201d der. Arf\u2019\u0131n bu s\u00f6yleminde apa\u00e7\u0131k olan be\u011feni duygusuyla matematik yapmad\u0131\u011f\u0131d\u0131r. S\u00f6ylemdeki daha gizemli ve bana g\u00f6re daha \u00f6nemli sonu\u00e7 ise matemati\u011fin biteviyeli\u011fi ve k\u0131\u015fk\u0131rt\u0131c\u0131l\u0131\u011f\u0131d\u0131r. Matematikte ula\u015f\u0131lan bir sonu\u00e7 yeni sorulara, o sorular yeni sonu\u00e7lara gebedir. Sorularla, sonu\u00e7larla bezeli bir s\u00fcre\u00e7. S\u00fcrecin kendisidir g\u00fczel olan. Yukar\u0131da \u00f6rnekledi\u011fimiz gibi sorular sonu\u00e7lar\u0131, sonu\u00e7lar sorular\u0131 k\u0131\u015fk\u0131rt\u0131r. A\u00e7may\u0131 bekleyen yeni filizler hep vard\u0131r matematikte. A\u00e7t\u0131k\u00e7a a\u00e7as\u0131 gelir insan\u0131n\u2026<\/p>\n

Biz yukar\u0131daki s\u00fcreci yeni bir soruyla s\u00fcrd\u00fcrelim. Verdi\u011fimiz \u00f6rnekler azalarak sonsuza giden say\u0131larla ilgili idi. Ya artarak sonsuza giden say\u0131larla ilgili benzer sonu\u00e7lara ula\u015fabilir miyiz? Bu kez do\u011frudan genellemeyle bakal\u0131m. Sorumuz s\u0131n\u0131rl\u0131 \u00e7okluklarla ilgili olsun.<\/p>\n

Soru: <\/strong>a + a2 <\/sup>+ a3<\/sup> + a4 <\/sup>+ \u2026\u2026\u2026\u2026+ an <\/sup>toplam\u0131 aR i\u00e7in neye e\u015fittir?<\/p>\n

\u00c7\u00f6z\u00fcm: <\/strong>X = <\/strong>a + a2 <\/sup>+ a3<\/sup> + a4 <\/sup>+ \u2026\u2026\u2026\u2026+ an<\/sup> diyelim. \u00d6nceki \u00f6rneklerde yapt\u0131\u011f\u0131m\u0131za benzer i\u015flemler yapal\u0131m.<\/p>\n

\u00a0\u00a0\u00a0 <\/strong>X = a + a(a + a2 <\/sup>+ a3<\/sup> + a4 <\/sup>+ \u2026\u2026\u2026\u2026+ an-1<\/sup>) olur (I)<\/p>\n

X = <\/strong>a + a2 <\/sup>+ a3<\/sup> + a4 <\/sup>+ \u2026\u2026\u2026 a(n-1)<\/sup> + an<\/sup> den<\/p>\n

X- an<\/sup> = a + a2 <\/sup>+ a3<\/sup> + a4 <\/sup>+ \u2026\u2026\u2026 a(n-1) <\/sup>e\u015fitli\u011fi yaz\u0131labilir. Bu e\u015fitlik, (I) de yerine yaz\u0131l\u0131rsa;<\/p>\n

X = a + a(X \u2013 an<\/sup>) e\u015fitli\u011fi elde edilir. Buradan:<\/p>\n

X = a + a.X \u2013 a.an<\/sup>;\u00a0 a.X \u2013 X = a.an<\/sup> \u2013 a X(a \u2013 1) = a(an<\/sup> \u2013 1) den<\/p>\n

X = a.(an<\/sup> \u2013 1)\/(a \u2013 1<\/strong>) veya X = (an+1<\/sup> \u2013 a)\/(a \u2013 1) <\/strong>elde edilir.<\/p>\n

G\u00f6r\u00fcl\u00fcyor ki aray\u0131\u015f bizi kesin ve yepyeni bir sonuca g\u00f6t\u00fcrd\u00fc. Yeni aray\u0131\u015flarla yeni sonu\u00e7lar; devam\u2026 Bu kez i\u015flemi sonsuza ta\u015f\u0131sak sonsuzu ayd\u0131nlatabilir miyiz?<\/p>\n

Sonsuzu \u0131\u015f\u0131tmak<\/strong><\/p>\n

n\u00a0\u00a0 \u00a0\u221e (n sonsuza gider) oldu\u011funu d\u00fc\u015f\u00fcnelim. Elde etti\u011fimiz ba\u011f\u0131nt\u0131n\u0131n; X = (\u00a0– a )\/ (a \u2013 1) olaca\u011f\u0131 a\u00e7\u0131kt\u0131r. A\u00e7\u0131kt\u0131r da, (\u00a0hatta tek ba\u015f\u0131na (\u00a0ne olacak? Bizden \u00e7ok \u00f6nce de sorulmu\u015f bu soru ve uzun s\u00fcre sorun olmu\u015f. Ve de imdada Cantor yeti\u015fmi\u015f. Hem de \u00e7ok duru, en s\u00fczme ak\u0131l bi\u00e7imiyle\u2026 Soru da \u015f\u00f6yle sorulmu\u015f: \u201cB\u00fct\u00fcn par\u00e7as\u0131ndan b\u00fcy\u00fck m\u00fcd\u00fcr?\u201d Yan\u0131t\u0131 belli gibi: \u201cElbette b\u00fcy\u00fckt\u00fcr.\u201d Bakal\u0131m\u2026<\/p>\n

    \n
  1. Ad\u0131m: <\/strong>\u201cD\u201d 1\u2019den 100\u2019e kadar do\u011fal say\u0131lar\u0131, \u201c\u00c7\u201d, 1\u2019den 100\u2019e kadar \u00e7ift do\u011fal say\u0131lar\u0131 g\u00f6stersin. \u00c7ift do\u011fal say\u0131lar\u0131n, do\u011fal say\u0131lar\u0131n 2 ile \u00e7arp\u0131larak elde edildi\u011fi a\u00e7\u0131kt\u0131r. 2\u00d71 = 2, 2\u00d72 = 4, 2\u00d73 = 6\u2026 gibi. Yani n ve 2n dir. K\u00fcmeler;<\/li>\n<\/ol>\n

    D = {1, 2, 3, 4,\u2026\u2026\u2026\u2026\u2026 99, 100} ve\u00a0 \u00c7 = { 2, 4, 6, 8, \u2026\u2026\u2026\u2026\u2026 98, 100} \u015feklinde yaz\u0131l\u0131r. \u015eimdi bu iki k\u00fcmeyi kar\u015f\u0131la\u015ft\u0131ral\u0131m.<\/p>\n

    D = {1, 2, 3, 4, \u2026\u2026\u00a0\u00a0 49, 50, \u2026.. \u00a099, 100}<\/p>\n

     <\/p>\n

    \u00c7 = { 2, 4, 6, 8, \u2026\u2026\u2026\u2026\u2026 98, 100}<\/p>\n

    G\u00f6r\u00fcld\u00fc\u011f\u00fc gibi \u00c7\u00a0(D k\u00fcmesi \u00c7\u2019yi kapsar) ve D k\u00fcmesinin eleman say\u0131s\u0131n\u0131n \u00c7 k\u00fcmesinin eleman say\u0131s\u0131ndan \u00e7ok oldu\u011fu a\u00e7\u0131k. Yani; b\u00fct\u00fcn par\u00e7as\u0131ndan b\u00fcy\u00fck.<\/strong><\/p>\n

      \n
    1. Ad\u0131m: <\/strong>Her iki k\u00fcmeyi sonlu olmaktan \u00e7\u0131karal\u0131m. Sonsuza y\u00fcr\u00fcs\u00fcnler.<\/li>\n<\/ol>\n

      N= {1, 2, 3, 4, \u2026\u2026. , n, n+1, \u2026\u2026}<\/p>\n

      N\u00c7<\/sub>= { 2, 4, 6, \u2026\u2026. , 2n, 2n+2, \u2026\u2026} Yine \u00e7ift do\u011fal say\u0131lar\u0131n \u201c2n\u201d le elde edildi\u011fi g\u00f6r\u00fcl\u00fcyor.<\/p>\n

      Art\u0131k kar\u015f\u0131la\u015ft\u0131rmak kolay.<\/p>\n

       <\/p>\n

      N= { 1, 2, 3, 4, \u2026\u2026. , n, \u00a0n+1, \u2026\u2026}<\/p>\n\n\n\n\n
      <\/td>\n<\/tr>\n
      <\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      N\u00c7<\/sub>= { 2, 4, 6, \u2026\u2026. , 2n, 2n+2, \u2026\u2026}<\/p>\n

      Bu kadar. Basit bir e\u015fleme. Her do\u011fal say\u0131n\u0131n iki kat\u0131 bir \u00e7ift say\u0131d\u0131r. \u00d6yleyse ne kadar do\u011fal say\u0131 varsa o kadar \u00e7ift do\u011fal say\u0131 vard\u0131r. Kim diyebilir ki, do\u011fal say\u0131lar k\u00fcmesinin alt k\u00fcmesi olan \u00e7ift do\u011fal say\u0131lar\u0131n eleman\u0131 daha azd\u0131r diye? Ya da \u201cb\u00fct\u00fcn par\u00e7adan b\u00fcy\u00fckt\u00fcr\u201d diye\u2026<\/p>\n

      Cantor\u2019un sonsuzlukla ilgili \u00e7al\u0131\u015fmalar\u0131n\u0131 okudu\u011fumda hayran olmu\u015ftum. Sanki bir iki f\u0131r\u00e7a darbesiyle tarihin heykelini yapar gibiydi. Bu kuram sayesinde sonsuzlukla ilgili bellediklerim anlam kazanm\u0131\u015ft\u0131. Elbette Cantor\u2019un \u00e7al\u0131\u015fmas\u0131 bu kadarla kalmam\u0131\u015f. Say\u0131labilir sonsuz, say\u0131lamaz sonsuz gibi yeni kavramlar ve kuramlar geli\u015ftirmi\u015f. Bu kuramlar k\u00fcmeler teorisini ortaya koymu\u015f, \u00f6zellikle fonksiyon analizine yepyeni anlamlar kazand\u0131rm\u0131\u015ft\u0131r. En basit ve yayg\u0131n anlamda da; \u201c\u221e +5\u201d, \u201c\u221e – 5\u201d, \u201c\u221e.5\u201d, \u201c\u221e + \u221e\u201d, \u2026, n\u0131n neden sonsuz oldu\u011fu, \u201c3\/\u221e\u201dun neden \u201c0\u201d oldu\u011fu, \u201c\u221e – \u221e\u201d, \u201c0.\u221e\u201d, \u201c\u221e\/\u221e\u201d, \u201c0\/0\u201d\u0131n neden belirsiz oldu\u011fu bu yolla anlaml\u0131 hale gelmi\u015ftir. Ve ben de \u015funu \u00f6\u011frenmi\u015ftim. Yukar\u0131daki \u00f6zg\u00fcn kar\u015f\u0131la\u015ft\u0131rmay\u0131 yapmadan sonsuzla yap\u0131lan i\u015flemler ve hele de belirsizlik gidermeye \u00e7al\u0131\u015fmak beyhudeydi\u2026<\/p>\n

      \"\"<\/p>\n

      \u00c7ocuktan al haberi<\/strong><\/p>\n

      Yak\u0131n zamanda ya\u015fad\u0131\u011f\u0131m bir olayda da, insan akl\u0131n\u0131n \u00e7eli\u015fkilerinden olan azl\u0131k, \u00e7okluk kavram\u0131n\u0131n bir \u00e7ocuk akl\u0131 i\u00e7in bile ne denli d\u00fc\u015f\u00fcnmeye de\u011fer oldu\u011funu anlad\u0131m. Torunum Sarp 4 ya\u015f\u0131ndan k\u00fc\u00e7\u00fckt\u00fc. Ziyaretlerine gittim. Di\u011fer dedesi zaten yanlar\u0131ndayd\u0131. Nedenini an\u0131msamad\u0131\u011f\u0131m bir anda Sarp iki elini yana a\u00e7\u0131p; \u201cyahu benim ne \u00e7ok dedem oldu\u201d dedi. \u00c7ocuklar\u0131n beklenmedik tepkileri bizleri hep \u015fa\u015f\u0131rt\u0131r. Hep beraber g\u00fcld\u00fck. \u0130ki dede, \u00e7ok dede\u2026 Biraz zaman ge\u00e7ti. Sarp sehpan\u0131n \u00fczerindeki arabalar\u0131yla oynuyor. En az 5-6 araba var. \u201cSarp\u201d dedim. \u201cArabalar\u0131n m\u0131 \u00e7ok, dedelerin mi?\u201d Bu kez o \u015fa\u015f\u0131rd\u0131. Ba\u015f\u0131n\u0131 kald\u0131r\u0131p dedelerine sonra arabalar\u0131na bakt\u0131 bir \u015fey s\u00f6yleyecek gibi oldu. Ama s\u00f6ylemedi. Belki de s\u00f6yleyemedi. G\u00f6zlerime bakarak g\u00fcl\u00fcmsedi, arabalar\u0131yla oynamaya devam etti. San\u0131r\u0131m iki ile s\u0131n\u0131rl\u0131 dede say\u0131s\u0131yla, 6 ile s\u0131n\u0131rl\u0131 olmayan arabalar\u0131n\u0131n say\u0131s\u0131n\u0131 kar\u015f\u0131la\u015ft\u0131rmak ona uygun g\u00f6r\u00fcnmemi\u015fti.<\/p>\n

      An\u0131msad\u0131klar\u0131m beni \u015fimdi de \u00e7ocuklu\u011fumdaki bir ya\u015fanm\u0131\u015fl\u0131\u011fa g\u00f6t\u00fcrd\u00fc. 8-10 ya\u015flar\u0131mda oldu\u011fumu an\u0131ms\u0131yorum. Uzun bir tren yolculu\u011funday\u0131z, Orta Anadolu civar\u0131nda. Ak\u015fam karanl\u0131\u011f\u0131 bast\u0131\u011f\u0131nda kimseye \u2018\u00e7akt\u0131rmadan\u2019 koridora s\u00fcz\u00fcld\u00fcm. Koridor bo\u015f. Zaten bo\u015f olmas\u0131n\u0131 bekliyordum. Akl\u0131ma tak\u0131lan soruya yan\u0131t ar\u0131yorum.<\/p>\n

      \u00d6nceden yan\u0131ma ald\u0131\u011f\u0131m tebe\u015firle yere bir \u00e7izgi \u00e7izdim ve olanca g\u00fcc\u00fcmle havaya do\u011fru s\u0131\u00e7rad\u0131m. Nereye d\u00fc\u015fecektim? Umdu\u011fum gibi olmad\u0131 yine \u00e7izginin \u00fcst\u00fcne d\u00fc\u015ft\u00fcm. Bir daha bir daha denedim. Bu arada birileri \u201cdeli mi bu\u201d demesin diye g\u00f6z\u00fcm sa\u011fda solda. Denemeler de\u011fi\u015fmedi. Hep \u00e7izginin \u00fczerine d\u00fc\u015f\u00fcyordum. Oysa beklentim, ben havada as\u0131l\u0131 iken tren alt\u0131mdan kay\u0131p gitmeliydi. Olmam\u0131\u015ft\u0131. Merak\u0131m\u0131 gideremeden kompart\u0131mana girdim. Bana ayr\u0131lan yere k\u0131vr\u0131l\u0131p uyudum. Sabah uyand\u0131\u011f\u0131mda ortal\u0131k ayd\u0131nlanm\u0131\u015ft\u0131. Camdan d\u0131\u015far\u0131 bakt\u0131m. Ola\u011fan\u00fcst\u00fc bir g\u00fczellik. G\u00f6z alabildi\u011fine sapsar\u0131 ba\u015faklar. Ve de p\u0131r\u0131l p\u0131r\u0131l. G\u00fcne\u015f g\u00f6r\u00fcnm\u00fcyor ama \u0131\u015f\u0131klar\u0131 ba\u015faklar\u0131 yal\u0131yor. Belli ki hafif\u00e7e de bir r\u00fczg\u00e2r esiyor. Ba\u015faklar dalga dalga \u0131\u015f\u0131kl\u0131 ve g\u00f6lgeli. Sanki bir el tar\u0131yor gibi. Uzun s\u00fcre g\u00f6zlerimi ay\u0131rmadan izledim. \u015eu anda bile g\u00f6r\u00fcnt\u00fc belle\u011fimde\u2026 Ve heyecan verici\u2026<\/p>\n

      Ne ili\u015fkisi var s\u00f6ylediklerimin sanat ve matematikle? San\u0131r\u0131m burada ortak yan her ikisindeki ola\u011fan\u00fcst\u00fcl\u00fck. Cantor\u2019un sonsuzlu\u011fu ele almas\u0131nda ve olu\u015fturdu\u011fu kuramlar\u0131n bir\u00e7ok soruna \u00e7\u00f6z\u00fcm getirmesindeki ola\u011fan\u00fcst\u00fcl\u00fck. \u0130nsan soyunun anlama ve anlamland\u0131rma a\u015fk\u0131ndaki ola\u011fan\u00fcst\u00fcl\u00fck. Do\u011fan\u0131n deviniminin insan d\u00fc\u015f\u00fcncesinde yaratt\u0131\u011f\u0131 ola\u011fan\u00fcst\u00fcl\u00fck\u2026 Dali\u2019nin tablosunda at\u0131n (ya da yarat\u0131\u011f\u0131n) aya\u011f\u0131n\u0131n insan boyunun kat kat fazla olmas\u0131ndaki ola\u011fan\u00fcst\u00fcl\u00fck. Ma\u011fara resimlerinde at\u0131n \u00fczerindeki adam\u0131n attan \u00e7ok daha b\u00fcy\u00fck olmas\u0131ndaki ola\u011fan\u00fcst\u00fcl\u00fck\u2026 Dali mi do\u011fru, ma\u011fara ressam\u0131 m\u0131? Her ikisi de mi? Ressam benim g\u00f6rd\u00fc\u011f\u00fcm at\u0131 resmetseydi sanat olur muydu? Matematikte de sanatta da her ikisinde de ak\u0131l ve duygu esteti\u011finin ola\u011fan\u00fcst\u00fcl\u00fc\u011f\u00fc yok mu?<\/p>\n

      D\u00e2hilik \u015fart m\u0131?<\/strong><\/p>\n

      \u0130smihan Yusubov\u2019un Bilim ve Gelecek Kitapl\u0131\u011f\u0131\u2019ndan \u00e7\u0131kan Matematik G\u00fczeldir<\/em> adl\u0131 kitab\u0131nda bir s\u00f6yleyi\u015fi var: \u201cD\u00e2hilerin sa\u00e7malamalar\u0131 da d\u00e2hicedir.\u201d B\u00fcy\u00fck matematik\u00e7i olmak i\u00e7in dahi olmak gerekti\u011fi d\u00fc\u015f\u00fcn\u00fcl\u00fcr. Baz\u0131 matematik\u00e7iler ise farkl\u0131 d\u00fc\u015f\u00fcn\u00fcyor; sab\u0131r, inat ve haz duyma matematik\u00e7i olmak i\u00e7in daha temel gereksinimdir diyorlar. Hangisi daha do\u011fru emin de\u011filim. Ama \u201cmatematik \u00f6\u011frenmek\u201d i\u00e7in sab\u0131r ve haz duygusunun \u00f6ncelikli oldu\u011funa inanm\u0131\u015f\u0131md\u0131r. Dahi olmad\u0131\u011f\u0131n\u0131 bildi\u011fim bir\u00e7ok \u00f6\u011frencinin \u00e7ok ilgin\u00e7 yakla\u015f\u0131mlarla (dahice demiyorum) soru \u00e7\u00f6zd\u00fc\u011f\u00fcn\u00fc hep g\u00f6zlemi\u015fimdir. Beni bile \u015fa\u015f\u0131rtan\u2026<\/p>\n

      Bir lise grubu ile problem \u00e7\u00f6z\u00fcm\u00fcnde denklemleri kullanma \u00e7al\u0131\u015fmas\u0131 yap\u0131yoruz. Bir soru sordum:<\/p>\n

      Ali ile Ay\u015fe\u2019nin paralar\u0131n\u0131n toplam\u0131 600 lira. Ali Ay\u015fe\u2019ye 50 lira verirse, Ay\u015fe\u2019nin paras\u0131 Ali\u2019nin paras\u0131n\u0131n iki kat\u0131 olmaktad\u0131r. Ba\u015flang\u0131\u00e7ta Ali\u2019nin ka\u00e7 lira paras\u0131 vard\u0131? Say\u0131lar bire bir b\u00f6yle olmasa da soru bu kapsamdayd\u0131. Soru lise \u00f6\u011frencisi i\u00e7in zor de\u011fil. Amac\u0131m denklem kullan\u0131m\u0131n\u0131n gere\u011fini vermek. \u00c7\u00f6z\u00fcm i\u00e7in bekledi\u011fim Ali\u2019nin paras\u0131na x, Ay\u015fe\u2019nin paras\u0131na y denmesi ve x + y = 600 ile 2.(x-50) = y+50 denklem sistemiyle soruyu \u00e7\u00f6zmeleri. Ya da Ali\u2019nin paras\u0131na x, Ay\u015fe\u2019nin paras\u0131na 600-x denilerek, 2.(x-50) = 600-x+50 bir bilinmeyenli denklem kurup soruyu \u00e7\u00f6zmeleri. Sorduktan 3-5 saniye sonra bir k\u0131z \u00f6\u011frencim; \u201c250\u201d yan\u0131t\u0131n\u0131 verdi. K\u00e2\u011f\u0131t kalem kullanmam\u0131\u015ft\u0131. Yan\u0131na gittim. Nas\u0131l \u00e7\u00f6zd\u00fc\u011f\u00fcn\u00fc sordum. \u201cHocam para al\u0131\u015fveri\u015fi olsa da olmasa da toplam para de\u011fi\u015fmiyor. Son durumda Ay\u015fe\u2019nin iki Ali\u2019nin bir kat paras\u0131 var. Toplam\u0131 \u00fc\u00e7 kat. 600\u2019\u00fc 3\u2019e b\u00f6ld\u00fcm 200 lira. 200, Ali\u2019nin 50 lira vermi\u015f hali. Demek ki 200+50=250 liras\u0131 varm\u0131\u015f.\u201d<\/p>\n

      Pl\u00e2nlar\u0131m alt \u00fcst oldu. Oysa ben iki ayr\u0131 denklemle iki bilinmeyenli denklemle sorunun \u00e7\u00f6z\u00fcm\u00fcn\u00fc tart\u0131\u015facakt\u0131m. Ayr\u0131ca iki bilinmeyenli yerine bir bilinmeyenli denklemle\u2026 \u00d6\u011frenci deyim yerindeyse beni de soruyu da a\u00e7\u0131k d\u00fc\u015f\u00fcrm\u00fc\u015ft\u00fc. \u00c7\u00f6z\u00fcm\u00fc net, \u00e7\u00f6z\u00fcm tam ve ak\u0131ll\u0131ca. \u00d6\u011frencinin Cahit Arf\u2019\u0131 bilmedi\u011fini sanmam. Ama Cahit Arf\u2019\u0131n \u201cben soru \u00e7\u00f6zerken de\u011fi\u015fmezlerden yola \u00e7\u0131kar\u0131m\u201d s\u00f6ylemini bildi\u011fini sanm\u0131yorum.<\/p>\n

      Bir sorunun birden \u00e7ok \u00e7\u00f6z\u00fcm\u00fc oldu\u011fu gibi bir teoremin de birden \u00e7ok ispat\u0131 olabilir. Baz\u0131 ispatlar olduk\u00e7a ilgin\u00e7tir.<\/p>\n

      Ne g\u00fczel<\/strong><\/p>\n

      Teorem: <\/strong>S\u0131f\u0131rdan farkl\u0131 bir do\u011fal say\u0131 ile tersinin toplam\u0131 en az 2\u2019dir.<\/p>\n

      Teoremi matematik dili ile yazarsak;<\/p>\n

      Teorem: <\/strong>nve nise n + (1\/n) dir.<\/p>\n

      \u0130spat: 1) <\/strong>\u00d6nermenin do\u011frulu\u011funu g\u00f6stermek<\/p>\n

      n + (1\/n) \u00a0ise payda e\u015fitlenerek (n2<\/sup> + 1)\/n \u00a0; n2<\/sup> + 1 \u00a0; n2<\/sup> \u2013 2n + 1 \u00a0dan<\/p>\n

      (n \u2013 1)2 <\/sup>\u00a0elde edilir. Bu \u00f6nermenin \u201cn\u201d in 0\u2019dan b\u00fcy\u00fck b\u00fct\u00fcn de\u011ferleri i\u00e7in do\u011fru oldu\u011fu a\u00e7\u0131kt\u0131r.<\/p>\n

      \u0130spat: 2) <\/strong>nve n\u00a0oldu\u011fu i\u00e7in,<\/p>\n

      n \u00a0n \u2013 \u00a0; \u00a0(n \u2013 1)2 <\/sup>; \u00a0n2 <\/sup>– 2n +1\u00a0 n2<\/sup> + 1 \u00a0ve
      \nn2<\/sup> + 1 \u00a0de her iki yan \u201cn\u201d e b\u00f6l\u00fcn\u00fcrse n + (1\/n) \u00a0elde edilir.<\/p>\n

      Birinci ispat, verilen \u00f6nermenin do\u011fru oldu\u011funu g\u00f6stermek bi\u00e7imindedir. Uygun her teoremin ispat\u0131nda bu yol izlenebilir. \u0130kinci ispat ise burada vurgulamak i\u00e7in yap\u0131lan ispatt\u0131r. \u201cn-1\u201d in 0\u2019dan b\u00fcy\u00fck oldu\u011fu ger\u00e7e\u011finden ba\u015flay\u0131p bir iki sat\u0131rda (elbette do\u011fru matematik i\u015flemlerle) istenilen \u00f6nermeyi elde ettik. Bu ispat\u0131n g\u00fczelli\u011fi, kullan\u0131lacak matematik \u00f6nermeleri do\u011fru se\u00e7mekte ve y\u00f6ntemin sadeli\u011finde.<\/p>\n

      Fazla s\u00f6ze gerek var m\u0131?<\/strong><\/p>\n

      Baz\u0131 sorular\u0131n \u00e7\u00f6z\u00fcm\u00fcnde genel y\u00f6ntemler kullan\u0131ld\u0131\u011f\u0131 gibi \u015fa\u015f\u0131rt\u0131c\u0131 \u00e7\u00f6z\u00fcmler de vard\u0131r. Basit ve kolay akla gelmeyen\u2026<\/p>\n

      Soru: <\/strong>500 tak\u0131m\u0131n kat\u0131ld\u0131\u011f\u0131 tek elemeli \u015fampiyonada, \u015fampiyonu belirlemek i\u00e7in ka\u00e7 ma\u00e7 yap\u0131l\u0131r?<\/p>\n

      \u00c7\u00f6z\u00fcm: <\/strong>Yap\u0131lacak nedirle ba\u015flayal\u0131m. Tak\u0131mlar iki\u015fer iki\u015fer e\u015fle\u015fecek. E\u015flenen tak\u0131mlar ma\u00e7<\/p>\n

      yapacak. Biri mutlaka elenecek. Kalanlar yeniden e\u015flenmeye devam edecek. Son ma\u00e7<\/p>\n

      \u015fampiyonu belirleyecek.<\/p>\n

      Ma\u00e7 say\u0131s\u0131 toplam\u0131: 250 + 125 + 62 + 31 + 16 + 8 + 4 + 2 + 1 = 499<\/p>\n

      Yukar\u0131daki gibi \u00e7izelge haz\u0131rlamasak da \u00e7\u00f6z\u00fcm uzunca. Hep ikiye b\u00f6leceksiniz, \u00e7\u0131karacaks\u0131n\u0131z ve sonunda toplayacaks\u0131n\u0131z. Tak\u0131m say\u0131s\u0131 tek oldu\u011funda bir tak\u0131m\u0131 bekletip uygun durumda ekleyeceksiniz\u2026 \u00c7\u00f6z\u00fcm i\u00e7in ba\u015fka \u00f6neriler de olabilir.<\/p>\n

      Ama bir matematik\u00e7inin \u00e7\u00f6z\u00fcm\u00fc \u015fu: 500 tak\u0131mdan biri \u015fampiyon olacak, 499\u2019u elenecek. Her eleme bir ma\u00e7la olaca\u011f\u0131ndan, ma\u00e7 say\u0131s\u0131 elenen tak\u0131m say\u0131s\u0131na e\u015fittir\u2026 Bu \u00e7\u00f6z\u00fcm\u00fcn ak\u0131lc\u0131l\u0131\u011f\u0131na, inceli\u011fine \u015fapka \u00e7\u0131karmayacak biri olabilir mi?<\/p>\n

      Baz\u0131 aktarmalarla matematik ve sanat ili\u015fkisini vurgulamaya \u00e7al\u0131\u015ft\u0131m. Anla\u015f\u0131lm\u0131\u015ft\u0131r ki inanc\u0131m matemati\u011fin estetik y\u00f6n\u00fcn\u00fcn g\u00fc\u00e7l\u00fc oldu\u011fu. Baz\u0131lar\u0131 i\u00e7in derin hazlar yaratan matematikte bu hazz\u0131 yaratan estetikten ba\u015fka ne olabilir? Baz\u0131 \u00f6\u011frenciler kendi sezgileriyle matemati\u011fin hazz\u0131n\u0131 ke\u015ffediyor. Bu hazz\u0131 baz\u0131 \u00f6\u011frenciler seziyor ve duyuyorken b\u00fcy\u00fck bir \u00e7o\u011funluk neden duymuyor? Bunun yan\u0131t\u0131 bende \u00e7ok net. \u00d6\u011fretimde matemati\u011fin estetik yap\u0131s\u0131n\u0131 ihmal ediyoruz. Hatta \u00f6\u011fretenler olarak biz de \u00e7ok fark\u0131nda de\u011filiz. Bilinsin ki bu b\u00f6l\u00fcm\u00fc yazarken, yazmak i\u00e7in ara\u015ft\u0131r\u0131rken bile yepyeni heyecanlar duydum. Daha \u00f6nce d\u00fc\u015f\u00fcnmedi\u011fim\u2026 Ve \u015fimdi d\u00fc\u015f\u00fcn\u00fcyorum ki; matemati\u011fin her konusu, her matematiksel ispat, her soru \u00e7\u00f6z\u00fcm\u00fc estetik i\u00e7erikli olarak ele al\u0131nabilir.<\/p>\n

       <\/p>\n","protected":false},"excerpt":{"rendered":"

      \u0130nanc\u0131m matemati\u011fin estetik y\u00f6n\u00fcn\u00fcn g\u00fc\u00e7l\u00fc oldu\u011fu. Baz\u0131lar\u0131 i\u00e7in derin hazlar yaratan matematikte bu hazz\u0131 yaratan estetikten ba\u015fka ne olabilir? Baz\u0131 \u00f6\u011frenciler kendi sezgileriyle matemati\u011fin hazz\u0131n\u0131 ke\u015ffediyor. Bu hazz\u0131 baz\u0131 \u00f6\u011frenciler seziyor ve duyuyorken b\u00fcy\u00fck bir \u00e7o\u011funluk neden duymuyor? Bunun yan\u0131t\u0131 bende \u00e7ok net. \u00d6\u011fretimde matemati\u011fin estetik yap\u0131s\u0131n\u0131 ihmal ediyoruz. Hatta \u00f6\u011fretenler olarak biz de \u00e7ok […]<\/p>\n","protected":false},"author":1689,"featured_media":26650,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[160,38,25],"tags":[712,208,695],"class_list":["post-26645","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-123-sayi","category-dergi-sayilari","category-matematik","tag-estetik","tag-matematik","tag-sanat"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t