\u00d6\u011frencilik hayat\u0131m boyunca sadece iki ders kitab\u0131n\u0131n ilgimi \u00e7ekti\u011fini hat\u0131rl\u0131yorum. Biri insan \u00fcreme sisteminin anlat\u0131ld\u0131\u011f\u0131 biyoloji kitab\u0131yd\u0131 (!), di\u011feriyse lise son s\u0131n\u0131fta okutulan matematik kitab\u0131. Bu kitab\u0131 saklam\u0131\u015f\u0131m, ge\u00e7enlerde elime ge\u00e7ti, sayfalar\u0131n\u0131n \u00fczerine \u201cBu teoremi bir kez daha ispatla, \u00f6nemli\u201d, \u201cAral\u0131\u011fa bak, s\u00fcrekli mi?\u201d, \u201cG\u00fczel bir \u00f6rnek\u201d gibi bir\u00e7ok not yazm\u0131\u015f\u0131m. 40 y\u0131l \u00f6ncesinden s\u00f6z ediyorum, ama aylarca bu kitapla hemhal oldu\u011fumu \u00e7ok iyi an\u0131ms\u0131yorum. \u0130yi bir kitapt\u0131, adeta konu\u015furdum onunla!<\/p>\n
Ku\u015fkusuz, g\u00fcn\u00fcm\u00fczde de internet ortam\u0131n\u0131n olanaklar\u0131na ra\u011fmen ders kitaplar\u0131 hem \u00f6\u011frenci hem de \u00f6\u011fretmen i\u00e7in h\u00e2l\u00e2 olduk\u00e7a \u00f6nemli.<\/p>\n
Bir lise matematik ders kitab\u0131 nas\u0131l olmal\u0131? Bu soruya \u00e7ok farkl\u0131 yan\u0131tlar verilebilir, ama isterseniz \u00f6nce \u015fu soruyu soral\u0131m: \u201c\u0130yi\u201d bir matematik ders kitab\u0131n\u0131n \u00f6\u011frenci \u00fczerinde nas\u0131l bir etkisi olabilir? Benim yan\u0131t\u0131m k\u0131saca \u015f\u00f6yle: Kitapta matematiksel d\u00fc\u015f\u00fcncenin ipu\u00e7lar\u0131 verilerek, tan\u0131m ve kan\u0131t hakk\u0131yla ele al\u0131nabilirse, \u00f6\u011frenci matemati\u011fin ezberlenmesi gereken kurallar ve y\u00f6ntemlerden olu\u015fmad\u0131\u011f\u0131n\u0131 anlayacakt\u0131r. Ayr\u0131ca, ele al\u0131nan kavramlar\u0131n tarihsel arka plan\u0131na dikkat \u00e7ekilirse, \u00f6\u011frenci \u00f6\u011frenmeye \u00e7al\u0131\u015ft\u0131\u011f\u0131 kavramlar\u0131n ke\u015fif s\u00fcreci hakk\u0131nda bilgi sahibi olaca\u011f\u0131ndan soyutlaman\u0131n zorluklar\u0131n\u0131 a\u015fmas\u0131 kolayla\u015facakt\u0131r.<\/p>\n
\u00dclkemizde yay\u0131mlanm\u0131\u015f, yukar\u0131daki nitelikleri ta\u015f\u0131yan az say\u0131daki matematik ders kitaplar\u0131 aras\u0131nda bence en \u00f6nemli olanlar\u0131 Ali Nesin\u2019in fen liseleri i\u00e7in yazm\u0131\u015f oldu\u011fu kitaplard\u0131r. Yukar\u0131daki, \u201cBir lise matematik ders kitab\u0131 nas\u0131l olmal\u0131?\u201d sorusunun cevab\u0131n\u0131 merak eden okurun bu kitaplar\u0131 incelemesini \u00f6neririm. Bu yaz\u0131da K\u00fcmeler Kuram\u0131 I<\/em>, Do\u011fal Say\u0131lar Yap\u0131s\u0131, Tamsay\u0131lar Yap\u0131s\u0131 <\/em>isimli kitaplardan se\u00e7ti\u011fim birka\u00e7 kan\u0131t, kavram ve tarihi bilgiyi payla\u015fmak istiyorum.<\/p>\n Usta matematik\u00e7i yazd\u0131\u011f\u0131 \u00f6ns\u00f6zde \u201cBen bir yalanc\u0131y\u0131m\u201d paradoksuna g\u00f6nderme yaparak, \u201cParadoks gibi gelecek ama matematik anlat\u0131rken bir liseliye illa ki yalan s\u00f6ylenmeli\u201d c\u00fcmlesini kurarak yalan\u0131 asgariye indirmeye \u00e7al\u0131\u015ft\u0131\u011f\u0131n\u0131 ve hi\u00e7 olmazsa yalans\u0131z dolans\u0131z matematik i\u00e7in ba\u015fvurulmas\u0131 gereken kaynaklar\u0131 g\u00f6sterdi\u011fini ifade ediyor. Burada s\u00f6z\u00fc edilen yalan, \u201clise matemati\u011fiyle\u201d akademik d\u00fczeydeki matematik aras\u0131ndaki farktan kaynaklan\u0131yor.<\/p>\n Yalan ve matematiksel d\u00fcr\u00fcstl\u00fck <\/strong><\/p>\n K\u00fcmeler Kuram\u0131 I <\/em>kitab\u0131n\u0131n ilk c\u00fcmlesi \u015f\u00f6yle: \u201cBirtak\u0131m nesnelerden olu\u015fan topluklara matematikte k\u00fcme <\/em><\/strong>ad\u0131 verilir.\u201d Asl\u0131nda bu tan\u0131mlaman\u0131n do\u011fru olmad\u0131\u011f\u0131 kitab\u0131n 69\u2019uncu sayfas\u0131nda a\u00e7\u0131klan\u0131p, \u00f6\u011feleri t\u00fcm <\/strong>k\u00fcmelerden olu\u015fan bir k\u00fcmenin olamayaca\u011f\u0131n\u0131n kan\u0131t\u0131 yap\u0131l\u0131yor ve \u015fu notu okuyoruz: \u201c[\u2026] Demek ki herhangi bir k\u00fcme t\u00fcm k\u00fcmeleri \u00f6\u011fe olarak i\u00e7eremez, mutlaka bir k\u00fcme d\u0131\u015f\u0131nda kalmak zorunda. Buradan da \u00f6\u011feleri t\u00fcm k\u00fcmeler olan bir k\u00fcmenin olamayaca\u011f\u0131 \u00e7\u0131kar. Bu asl\u0131nda bu kitab\u0131n en ba\u015f\u0131nda verdi\u011fimiz k\u00fcme kavram\u0131yla \u00e7eli\u015fir, \u00e7\u00fcnk\u00fc kitab\u0131n ba\u015f\u0131nda bir k\u00fcmeyi herhangi bir topluluk olarak tan\u0131mlam\u0131\u015ft\u0131k, dolay\u0131s\u0131yla verdi\u011fimiz k\u00fcme tan\u0131m\u0131na g\u00f6re t\u00fcm k\u00fcmelerden olu\u015fan bir k\u00fcme olmas\u0131 laz\u0131m. Bu i\u015fte tam olarak Russell Paradoksu<\/em><\/strong>\u2019dur. Sorun, kitab\u0131n en ba\u015f\u0131nda verdi\u011fimiz k\u00fcme tan\u0131m\u0131nda. O tan\u0131m do\u011fru olamaz. Okura yalan s\u00f6yledik!\u201d<\/p>\n Bir\u00e7ok lise ders kitab\u0131nda k\u00fcme <\/em>\u201c\u0130yi tan\u0131mlanm\u0131\u015f nesneler toplulu\u011fudur\u201d diye tan\u0131mlan\u0131r, belki bir lise \u00f6\u011frencisi i\u00e7in bu a\u00e7\u0131klaman\u0131n yeterli olaca\u011f\u0131 d\u00fc\u015f\u00fcn\u00fclebilir, ama asl\u0131nda bu c\u00fcmlenin do\u011fru olmad\u0131\u011f\u0131n\u0131n a\u00e7\u0131klanmas\u0131n\u0131n sa\u011fl\u0131kl\u0131 bir matematik \u00f6\u011fretimi i\u00e7in gerekli oldu\u011fu inanc\u0131nday\u0131m. \u00d6rne\u011fin, Milli E\u011fitim Bakanl\u0131\u011f\u0131\u2019nca yay\u0131mlanan 9\u2019uncu s\u0131n\u0131f matematik ders kitab\u0131nda (Kaynak 4) Alman matematik\u00e7i Georg Cantor\u2019un 1878 y\u0131l\u0131nda yay\u0131mlad\u0131\u011f\u0131 makalesine at\u0131fta bulunarak \u201cK\u00fcme, iyi tan\u0131mlanm\u0131\u015f birbirinden farkl\u0131 nesneler toplulu\u011fudur\u201d c\u00fcmlesi kurularak k\u00fcmenin bu tan\u0131m\u0131 (!) nihai bir bilgi olarak verilmi\u015f. Oysa ger\u00e7ek bu de\u011fil! K\u00fcmeler Kuram\u0131 I<\/em>\u2019e g\u00f6z atal\u0131m: \u201cCantor\u2019un (matematiksel oldu\u011funu hi\u00e7 iddia etmedi\u011fimiz) k\u00fcme tan\u0131m\u0131 \u015f\u00f6yleydi: \u2018K\u00fcme, alg\u0131m\u0131z\u0131n ya da d\u00fc\u015f\u00fcncemizin a\u00e7\u0131k ve net nesnelerinden olu\u015fan bir topluluktur\u2019. Oysa g\u00fcn\u00fcm\u00fcz\u00fcn matemati\u011finde k\u00fcme <\/em>kavram\u0131 ve bir k\u00fcmenin \u00f6\u011fesi olmak <\/em>ili\u015fkisi matematiksel olarak tan\u0131mlanmadan kabul edilmesi gereken kavramlar olarak kabul edilir. Matemati\u011fin t\u00fcm kavramlar\u0131 bu iki kavrama dayand\u0131r\u0131larak tan\u0131mlanabilir.\u201d<\/p>\n Matematiksel d\u00fc\u015f\u00fcnmenin ipu\u00e7lar\u0131\u2026 <\/strong><\/p>\n Bir\u00e7ok \u00f6\u011frencide yayg\u0131n olan g\u00f6r\u00fc\u015f, matematiksel tan\u0131m ve kavramlar\u0131n bir \u201cTanr\u0131 buyru\u011fu\u201d gibi ortaya \u00e7\u0131kt\u0131\u011f\u0131 do\u011frultusundad\u0131r. Oysa biliyoruz ki matematiksel yap\u0131 matematik\u00e7iler taraf\u0131ndan matemati\u011fin i\u015fine nas\u0131l geliyorsa, yani sonraki ad\u0131m ve s\u00fcre\u00e7lerde tutars\u0131zl\u0131k yaratmayacak bir \u015fekilde in\u015fa edilmi\u015ftir. \u00d6rne\u011fin y\u0131llard\u0131r, 0! = 1 e\u015fitli\u011fine itiraz eden \u00f6\u011frenciler \u201c\u0130lk s\u0131f\u0131r do\u011fal say\u0131n\u0131n \u00e7arp\u0131m\u0131 neden s\u0131f\u0131r de\u011fil hocam?\u201d diye sorup bu e\u015fitli\u011fin kan\u0131tlanmas\u0131n\u0131 istemi\u015flerdir ve onlar\u0131 \u201cMatematiksel yap\u0131n\u0131n tutarl\u0131l\u0131\u011f\u0131 i\u00e7in 0! = 1 olarak tan\u0131mlanm\u0131\u015ft\u0131r\u201d cevab\u0131 pek tatmin etmez, \u00e7\u00fcnk\u00fc matematiksel yap\u0131n\u0131n in\u015fas\u0131 hakk\u0131nda yeterince bilgi ve g\u00f6r\u00fc\u015fe sahip de\u011fildirler. Ve elbette, bu eksiklik onlar\u0131n su\u00e7u de\u011fil, matematik \u00f6\u011fretimindeki sorunlardan kaynaklan\u0131r. \u201cMatematikte 0! = 1 neden bir tan\u0131m olarak kabul edilmi\u015ftir?\u201d sorusunu Ali Nesin Tamsay\u0131lar Yap\u0131s\u0131 <\/em>kitab\u0131nda matematiksel yap\u0131n\u0131n in\u015fas\u0131na ait ipu\u00e7lar\u0131 vererek \u015f\u00f6yle yan\u0131tl\u0131yor:<\/p>\n \u201c[\u2026] 34, d\u00f6rt tane 3\u2019\u00fc \u00e7arpmak anlam\u0131na gelir. 31, bir tane 3\u2019\u00fc \u00e7arpmak anlam\u0131na gelir. Dolay\u0131s\u0131yla 30, s\u0131f\u0131r tane 3\u2019\u00fc \u00e7arpmak anlam\u0131na gelir. Size s\u0131f\u0131r tane 3 veriyorum, hadi \u00e7arp\u0131n bu 3\u2019leri! \u00c7arpamazs\u0131n\u0131z tabii, \u00e7arpacak 3 yok \u00e7\u00fcnk\u00fc. Tan\u0131m gere\u011fi, s\u0131f\u0131r tane say\u0131n\u0131n \u00e7arp\u0131m\u0131n\u0131n sonucu 1\u2019dir. Ba\u015fka bir nedenden de\u011fil, tan\u0131mdan! Bu y\u00fczden 30 = 1 olur. 5! de be\u015f tane say\u0131y\u0131 \u00e7arpmak demektir:<\/p>\n 5! = 1 \u00d7 2 \u00d7 3 \u00d7 4 \u00d7 5<\/p>\n 3! i\u00e7in 1\u2019den 3\u2019e kadar \u00fc\u00e7 tane say\u0131 \u00e7arp\u0131l\u0131r. 0! i\u00e7in 0 tane, yani hi\u00e7 tane say\u0131 \u00e7arp\u0131lmal\u0131. Bir \u00f6nceki paragrafta hi\u00e7 tane say\u0131n\u0131n \u00e7arp\u0131m\u0131n\u0131 1 olarak tan\u0131mlam\u0131\u015ft\u0131k. Demek ki 0! = 1 olmal\u0131.\u201d<\/p>\n K\u00fcmeler Kuram\u0131 I<\/em>\u2019de okuru olduk\u00e7a \u015fa\u015f\u0131rtacak, ama matematiksel d\u00fc\u015f\u00fcnmenin ipu\u00e7lar\u0131n\u0131n verildi\u011fine inand\u0131\u011f\u0131m ilgin\u00e7 bir teorem kan\u0131tlan\u0131yor:<\/p>\n \u201cTeorem. <\/strong>Bo\u015fk\u00fcmenin her \u00f6\u011fesi 1\u2019e e\u015fittir! Kan\u0131t\u0131n p\u00fcf noktas\u0131 bo\u015fk\u00fcmenin hi\u00e7 \u00f6\u011fe i\u00e7ermemesidir. Tan\u0131m gere\u011fi hi\u00e7 \u00f6\u011fe i\u00e7ermeyen bo\u015f k\u00fcmenin her \u00f6\u011fesi 1\u2019e e\u015fittir. Bunu kan\u0131tlayal\u0131m. Diyelim ki sav\u0131m\u0131z yanl\u0131\u015f, yani diyelim bo\u015f k\u00fcmenin her \u00f6\u011fesi 1\u2019e e\u015fit de\u011fil\u2026 O zaman bo\u015fk\u00fcmede 1\u2019e e\u015fit olmayan bir \u00f6\u011fe vard\u0131r. Ama hani bo\u015fk\u00fcmede hi\u00e7 \u00f6\u011fe yoktu? Hi\u00e7 \u00f6\u011fesi olmayan bo\u015fk\u00fcmede 1\u2019e e\u015fit olmayan bir \u00f6\u011fe olabilir mi? Elbette olamaz. Demek ki bo\u015fk\u00fcmenin her \u00f6\u011fesi 1\u2019e e\u015fittir!<\/p>\n Dikkat edin bo\u015fk\u00fcmede 1\u2019e e\u015fit olan bir \u00f6\u011fe var demiyoruz. Bo\u015fk\u00fcmede \u00f6\u011fe olmad\u0131\u011f\u0131n\u0131 biliyoruz tabii. Sadece bo\u015fk\u00fcmenin her <\/strong>\u00f6\u011fesinin 1\u2019e e\u015fit oldu\u011funu s\u00f6yl\u00fcyoruz\u2026 Arada fark var\u2026<\/p>\n [\u2026] Bo\u015fk\u00fcmenin her \u00f6\u011fesi istedi\u011fimiz t\u00fcm \u00f6zellikleri sa\u011flar. Bo\u015fk\u00fcmenin her \u00f6\u011fesi sar\u0131d\u0131r, ye\u015fildir, uzundur ve ayn\u0131 zamanda k\u0131sad\u0131r da. Hi\u00e7 \u00f6\u011fesi olmayan bo\u015fk\u00fcmenin t\u00fcm \u00f6\u011feleri t\u00fcm \u00f6zellikleri ve e\u015fitlikleri sa\u011flar. Bunu bo\u015fk\u00fcmenin hi\u00e7 \u00f6\u011fesi olmamas\u0131na bor\u00e7luyuz.\u201d<\/p>\n Yukar\u0131da kan\u0131tlanan teorem, matematikte sezgilerimizin ihanetine u\u011frayabilece\u011fimizin g\u00fczel bir \u00f6rne\u011fini g\u00f6steriyor. \u00d6te yandan, \u201cbo\u015fk\u00fcmenin t\u00fcm \u00f6\u011feleri t\u00fcm \u00f6zellikleri ve e\u015fitlikleri sa\u011flar\u201d sonucunun matemati\u011fin daha ileriki b\u00f6l\u00fcmlerinde (bo\u015fk\u00fcme ba\u011f\u0131nt\u0131s\u0131n\u0131n \u00f6zellikleri gibi) \u00f6nemli bir yeri oldu\u011funu d\u00fc\u015f\u00fcn\u00fcrsek matematiksel tutarl\u0131\u011f\u0131n ne oldu\u011funu daha iyi anlayabiliriz. Daha do\u011frusunu s\u00f6ylemek gerekirse matemati\u011fi b\u00f6ylesi kitaplardan \u00f6\u011frenen bir lise \u00f6\u011frencisi matematiksel ufkuyla matemati\u011fin \u00f6z\u00fcn\u00fc daha iyi kavrayacakt\u0131r.<\/p>\n Matematik tarihi <\/strong><\/p>\n Matematik \u00f6\u011fretiminde matematik tarihinin yeri ve gereklili\u011fi tart\u0131\u015f\u0131lmaz bir ger\u00e7ek. Art\u0131k eskisi gibi de\u011fil, bir\u00e7ok matematik ders kitab\u0131 matematik tarihiyle ilgili bilgilere yer veriyor, fakat bu bilgilerin bir\u00e7o\u011fu maalesef y\u00fczeysel ve ele al\u0131nan kavramlardan kopuk oldu\u011fundan \u00f6\u011frencilerin ilgisini \u00e7ekemiyor. Ali Nesin kitaplar\u0131nda \u201csadece matemati\u011fi de\u011fil, matemati\u011fin ke\u015fif s\u00fcrecini de anlatmal\u0131, hatta belki de teoriden \u00f6d\u00fcn verip ke\u015fif s\u00fcrecini \u00f6ne \u00e7\u0131karmal\u0131\u201d diyerek okuru adeta bir matematik tarihi \u015f\u00f6lenine davet ediyor. \u00d6yle eklektik ve salt tarihsel bilgilere dayanarak de\u011fil, matemati\u011fin insanl\u0131k tarihindeki geli\u015fim s\u00fcrecini kitapta i\u015flenen konu ba\u011flam\u0131nda \u00f6\u011frenebilme \u015fans\u0131na sahip olabiliyorsunuz.<\/p>\n K\u00fcmeler Kuram\u0131 I<\/em>\u2019den k\u00fcmelerin k\u0131sa tarihinin anlat\u0131ld\u0131\u011f\u0131 bir b\u00f6l\u00fcm\u00fc payla\u015fmak istiyorum.<\/p>\n \u201cVar oldu\u011fundan beri insano\u011flu ve insank\u0131z\u0131 matematikte olmasa da g\u00fcnl\u00fck ya\u015fam\u0131nda k\u00fcme kavram\u0131yla a\u015finayd\u0131 elbette, \u00f6rne\u011fin koyun s\u00fcr\u00fcs\u00fc, bu\u011fday tarlas\u0131, kabile, bir sepet yumurta gibi s\u00f6zler k\u00fcme fikrinin \u00e7e\u015fitli tezah\u00fcrleridir. Ancak matematiksel anlamda k\u00fcmenin olduk\u00e7a yak\u0131n bir ge\u00e7mi\u015fi vard\u0131r. K\u00fcmelerden a\u00e7\u0131k a\u00e7\u0131k ilk kez 1847\u2019de \u015fimdiki \u00c7ek Cumhuriyeti\u2019nin ba\u015fkenti Prag\u2019da ya\u015fam\u0131\u015f olan matematik\u00e7i Bolzano (1781-1848) s\u00f6z etmi\u015ftir. O zamanlar sonsuz say\u0131da \u00f6\u011fesi olan k\u00fcmelerin \u00e7eli\u015fki i\u00e7erece\u011finden, yani matematikte \u00e7eli\u015fkiye yol a\u00e7aca\u011f\u0131ndan korkuluyordu. \u00d6rne\u011fin do\u011fal say\u0131larla \u00e7ift do\u011fal say\u0131lar\u0131n<\/p>\n 0 \u2194 0<\/p>\n 1 \u2194 2<\/p>\n 2 \u2194 4<\/p>\n 3 \u2194 6<\/p>\n \u2026<\/p>\n bi\u00e7iminde n <\/em>\u2194 2n <\/em>kural\u0131yla e\u015fle\u015ftirilebilmesi biliminsanlar\u0131n\u0131 korkutuyordu (\u00f6rne\u011fin Galileo\u2019nun \u00f6d\u00fc patlam\u0131\u015ft\u0131) ne de olsa \u00e7ift do\u011fal say\u0131lar k\u00fcmesinde do\u011fal say\u0131lar k\u00fcmesinden daha az say\u0131 olmal\u0131yd\u0131, yar\u0131s\u0131 kadar! 19. y\u00fczy\u0131l\u0131n sonlar\u0131na do\u011fru, yani bundan 100 k\u00fcsur y\u0131l \u00f6nce Alman matematik\u00e7i Georg Cantor (1845-1918) sonsuz k\u00fcmelerin \u00fcstlerine \u00fcstlerine gitmi\u015f, onlar\u0131 anlamaya \u00e7al\u0131\u015fm\u0131\u015f, \u00f6rne\u011fin do\u011fal say\u0131lar ile \u00e7ift do\u011fal say\u0131lar k\u00fcmesinin (yukar\u0131daki e\u015flemeden dolay\u0131) ayn\u0131 say\u0131da \u00f6\u011fesi oldu\u011funa h\u00fckmetmi\u015f ve bug\u00fcnk\u00fc matematiksel anlam\u0131na \u00e7ok yak\u0131n bir k\u00fcmeler kuram\u0131n\u0131 matematik camias\u0131n\u0131n a\u011f\u0131r bask\u0131lar\u0131na kar\u015f\u0131 koyarak neredeyse tek ba\u015f\u0131na geli\u015ftirmi\u015ftir. Nitekim zaman\u0131n bir\u00e7ok \u00fcnl\u00fc matematik\u00e7isi k\u00fcmeler kuram\u0131n\u0131 gereksiz bir u\u011fra\u015f olarak g\u00f6rm\u00fc\u015ft\u00fc. K\u00fcmeler kuram\u0131yla gen\u00e7lik ger\u00e7ek matematikten uzakla\u015ft\u0131r\u0131l\u0131p, gereksiz ve eften p\u00fcften d\u00fc\u015f\u00fcncelere y\u00f6nlendiriliyor diye d\u00fc\u015f\u00fcn\u00fcl\u00fcyordu. K\u00fcmeler kuram\u0131n\u0131 ciddiye alanlar matemati\u011fi getirisi olmayan bir alana s\u00fcr\u00fcklemekle su\u00e7lan\u0131yordu. O \u00e7a\u011f\u0131n en \u00fcnl\u00fc ve en etkin iki matematik\u00e7isi Alman David Hilbert ve Frans\u0131z Henri Poincar\u00e9 matemati\u011fin \u00f6z\u00fcyle ilgili bu m\u00fccadelede ayr\u0131 cephelerde yer alm\u0131\u015ft\u0131r. M\u00fccadeleyi Hilbert kazanm\u0131\u015ft\u0131r; bug\u00fcn k\u00fcmeler kuram\u0131 matematikte merkezi bir konuma gelmi\u015ftir. Matemati\u011fin klasik dallar\u0131n\u0131n bir\u00e7ok \u00f6nemli problemi \u00e7\u00f6z\u00fcm\u00fcn\u00fc k\u00fcmeler kuram\u0131nda bulmu\u015ftur.<\/p>\n K\u00fcmeler kuram\u0131 varl\u0131\u011f\u0131n\u0131 sonsuz k\u00fcmelere bor\u00e7ludur. Cantor\u2019dan \u00f6nce sonsuz k\u00fcmelerin varl\u0131\u011f\u0131ndan ku\u015fkulan\u0131l\u0131yordu, ku\u015fkulanmayanlar da sonsuz k\u00fcmlerle i\u015flem yapmaya \u00e7ekiniyorlard\u0131 ya da iki sonsuz k\u00fcme aras\u0131nda matematiksel olarak (\u00f6\u011feleri d\u0131\u015f\u0131nda) bir fark g\u00f6remiyorlard\u0131. Cantor 1874\u2019te yay\u0131mlad\u0131\u011f\u0131 bir makalesinde iki sonsuz k\u00fcme aras\u0131nda (\u00f6\u011fe say\u0131s\u0131 a\u00e7\u0131s\u0131ndan mesela) derin farklar olabilece\u011fini g\u00f6stererek matematikte bir \u00e7\u0131\u011f\u0131r a\u00e7m\u0131\u015ft\u0131r. K\u00fcmeler kuram\u0131n\u0131n geli\u015fimi on y\u0131llar boyunca matematik d\u00fcnyas\u0131nda matematiksel ve felsefi d\u00fczeyde sert tart\u0131\u015fmalara, hatta bilimsel kavgalara neden olmu\u015ftur.\u201d<\/p>\n Ali Nesin\u2019in lise ders kitaplar\u0131ndan se\u00e7tiklerimden olu\u015fan bu yaz\u0131 \u201cBir lise matematik ders kitab\u0131 nas\u0131l olmal\u0131?\u201d sorusuna yan\u0131t olabildi mi? Bilemiyorum, ama merakl\u0131 okurun ve ilgili lise \u00f6\u011frencilerinin yukar\u0131daki sorunun yan\u0131t\u0131n\u0131 bu kitaplarda bulaca\u011f\u0131na eminim. Ayr\u0131ca, okullarda yap\u0131lan matemati\u011fin y\u00fcrek par\u00e7alay\u0131c\u0131 halini d\u00fc\u015f\u00fcn\u00fcrsek b\u00f6ylesi matematik ders kitaplar\u0131na ve bu do\u011frultudaki bir m\u00fcfredat program\u0131na ne denli ihtiyac\u0131m\u0131z\u0131n oldu\u011fu a\u00e7\u0131k\u00e7a g\u00f6r\u00fcl\u00fcr san\u0131r\u0131m.<\/p>\n Kaynaklar <\/strong><\/p>\n 1) Ali Nesin, K\u00fcmeler Kuram\u0131 I<\/em>, Nesin Yay\u0131nc\u0131l\u0131k, 2017.<\/p>\n 2) Ali Nesin, Do\u011fal Say\u0131lar Yap\u0131s\u0131<\/em>, Nesin Yay\u0131nc\u0131l\u0131k, 2017.<\/p>\n 3) Ali Nesin, Tamsay\u0131lar Yap\u0131s\u0131<\/em>, Nesin Yay\u0131nc\u0131l\u0131k, 2017.<\/p>\n 4) Mehmet Mavi\u015f, G\u00fcray G\u00fcl, Himmet Solakl\u0131o\u011flu, Hakan Tarku, Fatih Bulut, Mahmut G\u00f6k\u015fen, Matematik 9 Ders Kitab\u0131<\/em>, MEB, 2017.<\/p>\n","protected":false},"excerpt":{"rendered":" \u00d6\u011frencilik hayat\u0131m boyunca sadece iki ders kitab\u0131n\u0131n ilgimi \u00e7ekti\u011fini hat\u0131rl\u0131yorum. Biri insan \u00fcreme sisteminin anlat\u0131ld\u0131\u011f\u0131 biyoloji kitab\u0131yd\u0131 (!), di\u011feriyse lise son s\u0131n\u0131fta okutulan matematik kitab\u0131. Bu kitab\u0131 saklam\u0131\u015f\u0131m, ge\u00e7enlerde elime ge\u00e7ti, sayfalar\u0131n\u0131n \u00fczerine \u201cBu teoremi bir kez daha ispatla, \u00f6nemli\u201d, \u201cAral\u0131\u011fa bak, s\u00fcrekli mi?\u201d, \u201cG\u00fczel bir \u00f6rnek\u201d gibi bir\u00e7ok not yazm\u0131\u015f\u0131m. 40 y\u0131l \u00f6ncesinden s\u00f6z […]<\/p>\n","protected":false},"author":375,"featured_media":26065,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[3343,25,514],"tags":[3363,3362,208],"class_list":["post-26064","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-172-sayi","category-matematik","category-matematik-sohbetleri","tag-ali-nesin","tag-kumeler","tag-matematik"],"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