{"id":47328,"date":"2020-12-01T00:00:18","date_gmt":"2020-11-30T21:00:18","guid":{"rendered":"https:\/\/bilimvegelecek.com.tr\/?p=47328"},"modified":"2020-11-29T22:44:27","modified_gmt":"2020-11-29T19:44:27","slug":"basit-zor-ve-bastan-cikarici","status":"publish","type":"post","link":"https:\/\/bilimvegelecek.com.tr\/index.php\/2020\/12\/01\/basit-zor-ve-bastan-cikarici","title":{"rendered":"Basit, zor ve ba\u015ftan \u00e7\u0131kar\u0131c\u0131\u2026"},"content":{"rendered":"

Evreninin bilinemeyenlerine benzer \u015fekilde matematikte de \u00e7\u00f6z\u00fclemeyen problemlerin oldu\u011funu biliyoruz. G\u00fcn\u00fcm\u00fczde bir\u00e7ok matematik\u00e7i bu b\u00fcy\u00fck problemleri \u00e7\u00f6z\u00fcme kavu\u015fturmak i\u00e7in u\u011fra\u015f\u0131yor.
\n\u00c7\u00f6z\u00fclemeyen problemler matematik\u00e7iler i\u00e7in \u00e7ok caziptir, ama ayn\u0131 zamanda da tehlikelidir, \u00e7\u00fcnk\u00fc aylarca, y\u0131llarca u\u011fra\u015f\u0131p sonu\u00e7 alamama riski vard\u0131r. Elbette, problemle u\u011fra\u015fma s\u00fcrecinde ortaya \u00e7\u0131kan \u201cara sonu\u00e7lar\u0131n\u201d hem matemati\u011fe hem de matematik\u00e7ilere katk\u0131s\u0131 olmaktad\u0131r; fakat bir matematik\u00e7i i\u00e7in akademik d\u00fcnyadan kopmay\u0131 do\u011furabilecek b\u00f6ylesi bir \u00e7aba b\u00fcy\u00fck bir zaman kayb\u0131na yol a\u00e7abilir.
\nBelki de bir\u00e7ok matematik\u00e7i \u201c\u00fcnl\u00fc ve \u00e7etin ceviz\u201d problemlerle u\u011fra\u015fmay\u0131 bu y\u00fczden gereksiz g\u00f6r\u00fcyordur. B\u00fcy\u00fck Alman matematik\u00e7i David Hilbert\u2019in (1862-1943) \u201cNeden Fermat\u2019n\u0131n Son Teoremi\u2019ni kan\u0131tlamak i\u00e7in u\u011fra\u015fmad\u0131n\u0131z?\u201d sorusuna verdi\u011fi yan\u0131t son derece \u00e7arp\u0131c\u0131d\u0131r: \u201cY\u0131llarca s\u00fcrecek yo\u011fun bir \u00e7al\u0131\u015fma sonunda olas\u0131 bir ba\u015far\u0131s\u0131zl\u0131k i\u00e7in harcayacak zaman\u0131m yok\u201d.
\nAnlamas\u0131 kolay, kan\u0131tlamas\u0131 son derece zor problemlerin en ilgin\u00e7 olanlar\u0131ndan birisi ad\u0131n\u0131 Alman matematik\u00e7i Lothar Collatz\u2019dan\u00a0 (1910-1990) alan Collatz san\u0131s\u0131<\/strong> 1937\u2019den bu yana matematik\u00e7ilere meydan okumaya devam ediyor. \u00c7\u00f6z\u00fcm\u00fc yolunda baz\u0131 bilgisayar destekli ad\u0131mlar at\u0131lm\u0131\u015f olsa da ba\u015far\u0131ya ula\u015f\u0131lamad\u0131.
\nTam 82 y\u0131ld\u0131r matematik\u00e7ilere direnen bu problemle matemati\u011fin en b\u00fcy\u00fck \u00f6d\u00fcl\u00fcn\u00fcn ( Fields Madalyas\u0131) sahibi ve ya\u015fayan b\u00fcy\u00fck matematik\u00e7ilerden biri olan Avustralyal\u0131- Amerikal\u0131 matematik\u00e7i Terence Tao\u2019nun da u\u011fra\u015ft\u0131\u011f\u0131n\u0131 biliyoruz. Tao da t\u0131pk\u0131 David Hilbert gibi \u00e7\u00f6z\u00fclemeyen problemlere zaman harcamay\u0131 mesleki bir tehlike olarak g\u00f6rd\u00fc\u011f\u00fcn\u00fc ve \u00fcnlenmi\u015f \u201cb\u00fcy\u00fck sorulara\u201d tak\u0131nt\u0131l\u0131 hale gelerek zaman kaybetmek istemedi\u011fini a\u00e7\u0131klam\u0131\u015ft\u0131. Ancak, tutku al\u0131nan kararlar\u0131 ge\u00e7ersizle\u015ftirdi ve Tao da bir\u00e7ok matematik\u00e7i gibi Collatz san\u0131s\u0131n\u0131n ba\u015ftan \u00e7\u0131kar\u0131c\u0131 cazibesine kap\u0131larak bu probleme \u00f6nceleri y\u0131lda birka\u00e7 g\u00fcn, daha sonra ayda birka\u00e7 g\u00fcn ve sonras\u0131nda daha fazla zaman ay\u0131rarak \u201crakibinin\u201d beyaz bayrak sallamas\u0131 i\u00e7in \u00e7abalad\u0131. Nihai sonuca gidememi\u015f olmas\u0131na kar\u015f\u0131n 2019\u2019da yay\u0131mlad\u0131\u011f\u0131 \u00e7al\u0131\u015fmas\u0131yla problemin \u00e7\u00f6z\u00fcm\u00fc i\u00e7in b\u00fcy\u00fck bir ad\u0131m atm\u0131\u015f oldu.
\nTerence Tao\u2019nun Collatz san\u0131s\u0131 \u00e7al\u0131\u015fmalar\u0131ndan s\u00f6z etmeye ara verip, matemati\u011fin anla\u015f\u0131lmas\u0131 \u00e7ok kolay \u00e7\u00f6z\u00fcm\u00fc bir o kadar zor olan bu davetk\u00e2r problemden habersiz olan okuru daha fazla sab\u0131rs\u0131zland\u0131rmayal\u0131m.
\nCollatz San\u0131s\u0131. <\/strong>1\u2019den b\u00fcy\u00fck herhangi bir do\u011fal say\u0131 se\u00e7elim. Say\u0131 \u00e7iftse ikiye b\u00f6lelim, tekse \u00fc\u00e7le \u00e7arp\u0131p bir ekleyelim. Elde etti\u011fimiz yeni say\u0131ya da tek veya \u00e7ift olmas\u0131na g\u00f6re yine bu i\u015flemlerden birini uygulayal\u0131m. Diyelim 3\u2019\u00fc se\u00e7tik. 3, tek oldu\u011fundan, 3\u2019\u00fc 3\u2019le \u00e7arp\u0131p 1 ekleyelim, 10 elde ettik. 10 \u00e7ift oldu\u011fundan, 10\u2019u ikiye b\u00f6lmeliyiz, 5 elde ettik. 5 tek. Demek ki 5\u2019i \u00fc\u00e7le \u00e7arp\u0131p bir ekleyece\u011fiz. 16 elde ederiz. 16\u2019y\u0131 ikiye b\u00f6lelim, 8 elde ettik. 8 \u00e7ift ikiye b\u00f6lmeliyiz, 4 elde ettik. 4 \u00e7ift, ikiye b\u00f6lmeliyiz, 2 elde ettik. 2 \u00e7ift, 2\u2019ye b\u00f6lmeliyiz, 1 elde ettik. 1\u2019e ula\u015ft\u0131\u011f\u0131m\u0131zda dural\u0131m. Bu i\u015flemleri a\u015fa\u011f\u0131daki gibi ba\u015fka say\u0131larla ba\u015flayarak da yapabiliriz.<\/p>\n

7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.
\n9, 28, 14, 7, \u2026 (7\u2019yle ba\u015flayan yukar\u0131daki dizideki gibi 1\u2019e ula\u015f\u0131r\u0131z.)
\n15, 36, 18, 9, \u2026 (9\u2019la ba\u015flayan yukar\u0131daki dizideki gibi 1\u2019e ula\u015f\u0131r\u0131z.)
\n19, 58, 29, 88, 44, 22, 11, \u2026 (7\u2019yle ba\u015flayan yukar\u0131daki dizideki gibi 1\u2019e ula\u015f\u0131r\u0131z.)
\n29, 88, \u2026 (Yukar\u0131daki dizide 88 var, dolay\u0131s\u0131yla 29 ba\u015flarsak yine 1\u2019e ula\u015f\u0131r\u0131z.)
\n51, 154, 77, 232, 116, 58, 29, \u2026 (29\u2019la ba\u015flayan dizideki gibi 1\u2019e ula\u015f\u0131r\u0131z.)<\/p>\n

\"\"<\/p>\n

Hangi say\u0131yla ba\u015flarsak ba\u015flayal\u0131m bir zaman sonra 1\u2019e ula\u015faca\u011f\u0131z. Peki iyi de, nereden biliyoruz hep 1\u2019e ula\u015faca\u011f\u0131m\u0131z\u0131? Bilmiyoruz. Ama \u00f6yle san\u0131l\u0131yor, \u00e7\u00fcnk\u00fc bir\u00e7ok say\u0131 denenmi\u015f ve hep 1\u2019e ula\u015f\u0131lm\u0131\u015f. Her say\u0131 denenemez elbette. Ama den k\u00fc\u00e7\u00fck her do\u011fal say\u0131 bilgisayar kullan\u0131larak denenmi\u015f ve hep 1\u2019e ula\u015f\u0131lm\u0131\u015f. E\u011fer 1\u2019e ula\u015fmayan bir \u00f6rnek bulmay\u0131 d\u00fc\u015f\u00fcn\u00fcyorsan\u0131z yakla\u015f\u0131k 300 kentilyondan sonraki do\u011fal say\u0131lar\u0131 denemelisiniz! \u00c7ok b\u00fcy\u00fck say\u0131lar i\u00e7in deneme yap\u0131larak 1\u2019e ula\u015f\u0131lm\u0131\u015f olmas\u0131n\u0131n matematiksel hi\u00e7bir de\u011feri yok elbette. Kan\u0131tlamak laz\u0131m! Deneysel ak\u0131l y\u00fcr\u00fctmeyle matematiksel ak\u0131l y\u00fcr\u00fctme aras\u0131ndaki nitelik fark\u0131n\u0131 g\u00f6steren g\u00fczel bir \u00f6rnekle kar\u015f\u0131 kar\u015f\u0131y\u0131z.
\nTahmin edebilece\u011finiz gibi Collatz san\u0131s\u0131 hangi do\u011fal say\u0131yla ba\u015flan\u0131rsa ba\u015flans\u0131n yukar\u0131da verilen i\u015flemleri uygulayarak daima 1\u2019e ula\u015f\u0131laca\u011f\u0131n\u0131 iddia ediyor.
\nMatematik\u00e7iler bu \u00f6nermeyi hen\u00fcz kan\u0131tlayamad\u0131lar. Yakla\u015f\u0131k 50 y\u0131l \u00f6nce \u00fcnl\u00fc Macar matematik\u00e7i Paul Erd\u00f6s (1913-1996), matemati\u011fin hen\u00fcz bu t\u00fcr problemleri \u00e7\u00f6zebilecek d\u00fczeye gelmedi\u011fini s\u00f6ylemi\u015fti. Ama Erd\u00f6s\u2019\u00fcn ke\u015ffetti\u011fi ve \u201cb\u00fcy\u00fck bir matematik\u00e7i olacak\u201d dedi\u011fi Terence Tao ge\u00e7en y\u0131l Collatz san\u0131s\u0131n\u0131 kan\u0131tlamakla u\u011fra\u015ft\u0131\u011f\u0131 d\u00f6nemde bir okurunun yapt\u0131\u011f\u0131 yorumla \u00e7ok \u00f6nemli sonu\u00e7lara ula\u015ft\u0131. Tao\u2019nun blo\u011funa4<\/sup> mesaj atan okuru, Tao\u2019ya Collatz san\u0131s\u0131n\u0131 kan\u0131tlamak yerine bu \u00f6nermeyi \u201cneredeyse\u201d t\u00fcm say\u0131lar i\u00e7in kan\u0131tlamay\u0131 \u00f6nerdi. Tao bu mesaja cevap vermedi ama \u00fczerinde d\u00fc\u015f\u00fcnmeye ba\u015flad\u0131 ve 2019 Eyl\u00fcl\u2019\u00fcnde Collatz san\u0131s\u0131n\u0131n \u201cneredeyse\u201d t\u00fcm say\u0131lar i\u00e7in \u201cneredeyse\u201d do\u011fru oldu\u011funu g\u00f6steren bir kan\u0131t yay\u0131mlad\u0131.2<\/sup> Tao\u2019nun \u00e7al\u0131\u015fmas\u0131n\u0131n temeli Collatz dizilerini s\u0131n\u0131fland\u0131rarak genel bir sonuca gitmeye dayan\u0131yor. Bu \u00e7al\u0131\u015fman\u0131n fikri baz\u0131 \u00f6zel \u00f6rneklemlerden genel sonu\u00e7lar \u00e7\u0131karmaya (anket \u00e7al\u0131\u015fmas\u0131 vb.)\u00a0 benzetilebilir.
\nTao, tam bir kan\u0131t verememi\u015f olsa da, yapt\u0131\u011f\u0131 \u00e7al\u0131\u015fma Collatz san\u0131s\u0131n\u0131n ortaya \u00e7\u0131k\u0131\u015f\u0131ndan bu yana kan\u0131ta giden yolda en parlak geli\u015fme olarak kabul ediliyor.
\nMatematik denizinde Collatz san\u0131s\u0131 gibi dev dalgalarla bo\u011fu\u015fuyor olmak, Paul Erd\u00f6s gibi bir ustan\u0131n umutsuzlu\u011funa ra\u011fmen m\u00fccadeleyi s\u00fcrd\u00fcrmek, \u00e7\u0131k\u0131lan seferden bo\u015f elle d\u00f6nebilmeyi g\u00f6ze almak, insan\u0131n ke\u015ffetmeye, matematiksel kan\u0131ta olan tutkusunun g\u00fczel bir \u00f6rne\u011fi olsa gerek!<\/p>\n

KAYNAKLAR
\n<\/strong>1) https:\/\/en.wikipedia.org\/wiki\/Collatz_conjecture
\n2) https:\/\/arxiv.org\/abs\/1909.03562
\n3) https:\/\/www.quantamagazine.org\/why-mathematicians-still-cant-solve-the-collatz-conjecture-20200922\/
\n4) https:\/\/terrytao.wordpress.com\/2011\/08\/25\/the-collatz-conjecture-littlewood-offord-theory-and-powers-of-2-and-3<\/p>\n","protected":false},"excerpt":{"rendered":"

Evreninin bilinemeyenlerine benzer \u015fekilde matematikte de \u00e7\u00f6z\u00fclemeyen problemlerin oldu\u011funu biliyoruz. G\u00fcn\u00fcm\u00fczde bir\u00e7ok matematik\u00e7i bu b\u00fcy\u00fck problemleri \u00e7\u00f6z\u00fcme kavu\u015fturmak i\u00e7in u\u011fra\u015f\u0131yor. \u00c7\u00f6z\u00fclemeyen problemler matematik\u00e7iler i\u00e7in \u00e7ok caziptir, ama ayn\u0131 zamanda da tehlikelidir, \u00e7\u00fcnk\u00fc aylarca, y\u0131llarca u\u011fra\u015f\u0131p sonu\u00e7 alamama riski vard\u0131r. Elbette, problemle u\u011fra\u015fma s\u00fcrecinde ortaya \u00e7\u0131kan \u201cara sonu\u00e7lar\u0131n\u201d hem matemati\u011fe hem de matematik\u00e7ilere katk\u0131s\u0131 olmaktad\u0131r; fakat […]<\/p>\n","protected":false},"author":375,"featured_media":47330,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7091,38,514,510],"tags":[3562,1008],"class_list":["post-47328","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-200-sayi","category-dergi-sayilari","category-matematik-sohbetleri","category-surekli-bolumler","tag-ali-torun","tag-matematik-sohbetleri"],"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