Matematik\u2019in ne menem bir \u015fey oldu\u011funa dair pek \u00e7ok yaz\u0131 mevcut bu blogda. An\u0131msamak i\u00e7in; t\u00fcm\u00fcyle hayal \u00fcr\u00fcn\u00fc oldu\u011fundan s\u00f6z etmi\u015ftik. \u0130kincileyin; Say\u0131 Sistemi (\u2018Number Theory\u2019) eksikli ve geli\u015ftirilmeye muhta\u00e7 (\u2018immature\u2019) demi\u015ftik. \u00d6rne\u011fin, her yerde rastlad\u0131\u011f\u0131m\u0131z \u221e say\u0131 de\u011filken say\u0131 say\u0131lm\u0131\u015ft\u0131r. Ayr\u0131ca, \u00c7ift Tam Say\u0131lar\u0131n \u221e\u2019u ile Tek Tam Say\u0131lar\u0131n \u221e\u2019u t\u0131pa t\u0131p ayn\u0131 imi\u015fcesine de\u011ferlendirilmektedir. \u00d6rne\u011fin, 1\/3=0,333\u2026 ve 3(1\/3)=1 iken, 3(0,333\u2026)\u2019\u00fcn ka\u00e7a e\u015fit oldu\u011fu konusunda tam ve kesin ve herkesin onay(lay)aca\u011f\u0131 bir de\u011fer ileri s\u00fcr\u00fclememi\u015ftir. Zaten, Matematik\u2019ten =\u2019i \u00e7\u0131kar\u0131n geriye anlams\u0131z bir tak\u0131m simgeler y\u0131\u011f\u0131n\u0131 kal\u0131r. B\u00f6yle Matematik olur mu, b\u00f6yle Matematik mi olur? Sentetik Geometri\u2019de nokta, \u00e7izgi ve di\u011fer \u015fekiller tan\u0131ms\u0131zd\u0131r. \u00d6rne\u011fin, hi\u00e7 kimse nokta olu\u015fturamaz, \u00e7izgi \u00e7izemez hele \u00e7ember, elips vb. \u00e7izemez. Analitik Geometri (Kartezyen Geometri) birbirine dik veya dar (\u00f6tesinde geni\u015f) a\u00e7\u0131l\u0131 iki \u00e7izginin kesi\u015fimiyle ba\u015flar. Gelgelelim, \u00e7izgilerin \u00e7izilebildi\u011fi varsay\u0131m\u0131 alt\u0131nda bile dik a\u00e7\u0131 veya di\u011fer t\u00fcrden a\u00e7\u0131 yaratabilmenin olanaks\u0131zl\u0131\u011f\u0131 a\u00e7\u0131kt\u0131r. (x\u00b2 + y\u00b2 = R\u00b2 yazabilirsiniz. Bunda hi\u00e7bir beis yok. Ama bu denkleme uyan \u00e7izgiyi yani \u00e7emberi \u00e7izemezsiniz, ne ka\u011f\u0131t \u00fczerine ne de ekran \u00fczerinde.) Trigonometri\u2019de ise, 0, 30 ve 90 derecelerin ve bunlara 180 derecenin tam katlar\u0131 eklenerek elde edilecek a\u00e7\u0131lar\u0131n sin\u00fcsleri d\u0131\u015f\u0131nda hi\u00e7bir a\u00e7\u0131n\u0131n, \u00f6rne\u011fin on yedi bu\u00e7uk derecenin sin\u00fcs\u00fcn\u00fc biliyor de\u011filiz. Keza 0, 60 ve 90 derecelerin ve bunlara 180 derecenin katlar\u0131 eklenerek elde edilecek a\u00e7\u0131lar\u0131n kosin\u00fcsleri d\u0131\u015f\u0131nda hi\u00e7bir a\u00e7\u0131n\u0131n, \u00f6rne\u011fin on yedi bu\u00e7uk derecenin kosin\u00fcs\u00fcn\u00fc biliyor de\u011filiz. Bu durum da hayli kar\u0131\u015f\u0131kl\u0131\u011fa yol a\u00e7maktad\u0131r. \u00d6rne\u011fin 30 derecenin sin\u00fcs\u00fc \u00bd\u2019dir ama \u00bd\u2019nin arcsin\u00fcs\u00fc 30 derece de\u011fildir. (#2)
\nMatematik\u2019te daha pek \u00e7ok eksiklik, yanl\u0131\u015fl\u0131k bulup yanl\u0131\u015fl\u0131klar listesini alabildi\u011fine uzatmak m\u00fcmk\u00fcnd\u00fcr. (#3) Ama Matematikselle\u015ftiri\u015f de\u011fil (Bu da yanl\u0131\u015f bir ifadedir.) Matematikle\u015ftiri\u015f gayet \u00f6nemli bir konudur Fizik Bilgisi yani Bilim Bilgisi i\u00e7in.
\nHemen altta, Gelileo\u2019nun o \u00fcnl\u00fc denkleminin alt\u0131nda yatan Matematikselle\u015ftiri\u015f ile devam edece\u011fiz.<\/p>\n
EK 1 ==><\/p>\n
\u0130lkin; herhangi bir Fizik olgusuna ili\u015fkin herhangi bir Matematik denklemi gibi, Galileo\u2019nun denklemi de bir aldatmacad\u0131r. \u00c7\u00fcnk\u00fc, herhangi bir Fizik olgusuna ili\u015fkin herhangi bir Matematik denklemine t\u0131pat\u0131p uygun davranan nesne yoktur, bulmak m\u00fcmk\u00fcn de\u011fildir. Hi\u00e7bir nesne, hi\u00e7bir denkleme t\u0131pat\u0131p uygun davranmaz. Ancak, \u015fu ya da bu \u00f6l\u00e7\u00fcde yakla\u015f\u0131k olarak uygunluk g\u00f6sterir. (#4)<\/p>\n
Buradaki temel sorunlardan biri, Fizik Bilgisi\u2019nin sadece noktasal tanecikler i\u00e7in ge\u00e7erli olu\u015fudur. Noktasal tanecik \u015fudur: k\u00fctlesi var ama eni, boyu, derinli\u011fi yok! Oysa, az yukar\u0131da tart\u0131\u015ft\u0131\u011f\u0131m\u0131z gibi, evrende nokta yok, evrende nokta \u00fcretilemez, ancak \u201cVar.\u201d diye hay\u00e2l edilebilir. Demek ki, i\u015fin hen\u00fcz en ba\u015f\u0131nda bile yakla\u015f\u0131kl\u0131kl\u0131\u011f\u0131 kabul etmek zorunday\u0131z. \u00d6rne\u011fin, G\u00fcne\u015f Sistemi b\u00f6lgesinde D\u00fcnya\u2019y\u0131 da G\u00fcne\u015f\u2019i de nokta par\u00e7ac\u0131k olarak var say\u0131p (eskilerin deyimiyle, tel\u00e2kki edip) Kepler Yasalar\u0131n\u0131 yahut Newton\u2019un K\u00fctle\u00e7ekim ba\u011f\u0131nt\u0131s\u0131n\u0131 de\u011ferlendirmek m\u00fcmk\u00fcn olabilir.<\/p>\n
Konuyla ilintili temel sorunlardan bir ba\u015fkas\u0131, evren Fizi\u011finde herhangi \u00e7ok say\u0131daki par\u00e7ac\u0131k anda\u015f ve kar\u015f\u0131l\u0131kl\u0131 (\u2018mutual\u2019) olarak etkile\u015firken Fizik Bilgisi\u2019nin sadece iki (noktasal) par\u00e7ac\u0131\u011f\u0131n etkile\u015fmesini \u00e7\u00f6zebilmi\u015f olmas\u0131d\u0131r. \u00d6rne\u011fin, G\u00fcne\u015f_D\u00fcnya, D\u00fcnya_Ay etkile\u015fmelerini tek tek ve ayr\u0131 ayr\u0131 de\u011ferlendirebiliyoruz ama G\u00fcne\u015f_D\u00fcnya_Ay \u00fc\u00e7l\u00fcs\u00fc i\u00e7in kapal\u0131, analitik ba\u011f\u0131nt\u0131 \u00fcretemeyip, N\u00fcmerik Hesaplama Y\u00f6ntemleri kullanarak yakla\u015f\u0131k sonu\u00e7lar elde edebiliyoruz. Ho\u015f, ikili etkile\u015fimlerde de tam de\u011fil, noktasal par\u00e7ac\u0131k yakla\u015f\u0131kl\u0131\u011f\u0131na mecbur oldu\u011fumuzdan ancak yakla\u015f\u0131k sonu\u00e7lar elde edebiliyoruz.<\/p>\n
Galileo\u2019nun denklemini ayr\u0131nt\u0131s\u0131yla irdelemeye ge\u00e7mezden evvel, bu denklemin zaman parametresi i\u00e7erdi\u011fini ve zaman parametresi i\u00e7eren ilk Fizik denklemi oldu\u011funu notlayal\u0131m. \u0130kinci notumuz da, \u015fu olsun: Denklem, denklik demektir; e\u015fit i\u015faretinin iki yan\u0131n\u0131n denkli\u011fini ifade eder. Dolay\u0131s\u0131 ile, her denklem bir ifadedir (\u2018expression\u2019). <==EK 1 sonu<\/p>\n
<\/p>\n
EK 2 ==><\/p>\n
Birinci EK b\u00f6l\u00fcm\u00fcnde, (#3) say\u0131l\u0131 dip notu ile de\u011finilen Matematik\u2019in hay\u00e2l \u00fcr\u00fcn\u00fc oldu\u011fu konusundaki fikrimize bir kar\u015f\u0131 \u00e7\u0131k\u0131\u015f gerek\u00e7esi de \u2018evrenin Matematik ile a\u00e7\u0131klanabiliyor\u2019 oldu\u011fu \u00f6nermesidir. Biraz daha ileri gidenler, evrenin Matematik\u2019e uydu\u011funu iddia etmektedir.<\/p>\n
Okumakta oldu\u011funuz bu yaz\u0131, ni\u00e7in \u2018evrenin Matematik ile a\u00e7\u0131klanabiliyor\u2019 oldu\u011funa ili\u015fkindir zaten. Ama, evrenin Matematik\u2019e uydu\u011fu fikri k\u00fclliyen yanl\u0131\u015ft\u0131r. \u00c7\u00fcnk\u00fc, en az\u0131ndan, Matematik insan yap\u0131s\u0131 iken evren de\u011fildir. Dahas\u0131, evrenin ne menem bir \u015fey oldu\u011funu da bilmiyoruz, bil_E_miyoruz. Kald\u0131 ki, insan yap\u0131s\u0131 oldu\u011fundand\u0131r, Matematik de gayet g\u00fcd\u00fckt\u00fcr; t\u0131pk\u0131 insan\u0131n evren kar\u015f\u0131s\u0131nda g\u00fcd\u00fck oldu\u011fu gibi. \u00d6nceki yaz\u0131lar\u0131m\u0131zda de\u011findi idik; sadece 5 tane g\u00fcd\u00fck alg\u0131\/duyu organ\u0131 ile ne kadar\u0131n\u0131 tan\u0131y\u0131p anlayabilip kavrayabileceksiniz (\u2018comprehension\u2019 (#5)) ki evrenin? \u0130nsanl\u0131k zek\u00e2s\u0131 \u00e7ok mu ileri ki, her t\u00fcrl\u00fc alg\u0131\/duyu organ\u0131 zay\u0131fl\u0131\u011f\u0131n\u0131 giderecek kertede geli\u015fkin alet edevat yapabilsin. Yapm\u0131\u015f m\u0131 ki?<\/p>\n
B\u00fct\u00fcn bunlarla birlikte, Matematikle\u015ftiri\u015f insanl\u0131\u011f\u0131n Fizik Bilgisi elde etmesi i\u00e7in en \u00f6nemli i\u015ftir. Olmazsa olmaz kertesindedir. Yani, Matematikle\u015ftiri\u015f olmaks\u0131z\u0131n Matematikle\u015ftiri\u015f yapmaks\u0131z\u0131n Fizik Bilgisi edinilemez. <==EK 2 sonu<\/p>\n
<\/p>\n
EK 3 ==><\/p>\n
Matematikle\u015ftiri\u015f, herhangi bir olay\u0131n do\u011frudan do\u011fruya nicelikler (\u201cquantity\u201d) cinsinden ifade edilmesiyle ba\u015flar. Yani, \u201c\u015eu oldu.\u201d, \u201c\u015e\u00f6yle oldu.\u201d, \u201cBu oldu.\u201d, \u201cB\u00f6yle oldu.\u201d gibi nitel (\u201cquality\u201d) ifadeler matematikle\u015ftiri\u015f d\u0131\u015f\u0131d\u0131r.<\/p>\n
Herhangi bir olaya ait nicelikler ise, ancak ve ancak \u00f6l\u00e7\u00fcm yoluyla elde edilebilir. Demek ki, matematikle\u015ftiri\u015f \u00f6l\u00e7\u00fcm ile ba\u015flar. \u00d6l\u00e7\u00fcm olmaks\u0131z\u0131n nicelik eldesi, nicelle\u015ftiri\u015f (\u201cquantification\u201d); nicelik olmaks\u0131z\u0131n da matematikle\u015ftiri\u015f yap\u0131lamaz, m\u00fcmk\u00fcn de\u011fildir.<\/p>\n
Her \u00f6l\u00e7\u00fcm\u00fcn g\u00fcvenilirli\u011fi de do\u011frudan do\u011fruya kullan\u0131lan \u00f6l\u00e7me ayg\u0131tlar\u0131n\u0131n duyarl\u0131l\u0131k (hassal\u0131k, hassayiyet) inceli\u011fine ba\u011fl\u0131d\u0131r.<\/p>\n
Galileo\u2019nun 1564-1642 d\u00fcnyas\u0131nda ise, sarka\u00e7l\u0131 saat yoktu (#6); zemberekli saatler de hem tek t\u00fck hem de herkesin edinip Galileo deneylerini s\u0131namakta kullanamayaca\u011f\u0131 kerte pahal\u0131 idi. Uzakl\u0131k \u00f6l\u00e7\u00fcmleri de en iyisi bir milimetre hassal\u0131\u011fa sahip ayg\u0131tlarla yap\u0131labiliyordu. Su saati denilen ve bir deli\u011finden akan su miktar\u0131n\u0131n tart\u0131lmas\u0131 yoluyla saptanan zaman s\u00fcreleri de hayli kaba \u00f6l\u00e7\u00fcmler sonucunda elde edilebiliyordu. \u0130\u015fte Galileo\u2019nun denkleminin alt\u0131nda yatan nicelikler ve ilgili ba\u011f\u0131nt\u0131lar, b\u00f6ylesi kaba \u00f6l\u00e7\u00fcmlere dayan\u0131yordu.<\/p>\n
Hem zaman ve konum \u00f6l\u00e7\u00fcmlerini anda\u015f olarak yapacaks\u0131n\u0131z, hem de serbest b\u0131rak\u0131l\u0131\u015ftan sonra git gide h\u0131zlanan nesnelerin devinim \u00f6zelliklerini saptayacaks\u0131n\u0131z. A\u00e7\u0131kt\u0131r ki, hi\u00e7 de kolay bir i\u015f de\u011fil.<\/p>\n
Bir ba\u015fka g\u00fc\u00e7l\u00fck de D\u00fcnya\u2019n\u0131n \u00e7ekim kuvvetinden kaynaklanan do\u011fal y\u00fcksek h\u0131zlan\u0131\u015f idi. Galileo bu sorunu e\u011fik d\u00fczlem kullanarak yapay yoldan d\u00fc\u015f\u00fcrd\u00fc. Nokta par\u00e7ac\u0131k niyetine de iki \u00fc\u00e7 santim \u00e7apl\u0131 bronz g\u00fclleler kulland\u0131.<\/p>\n
Yerle (yatayla) yap\u0131lm\u0131\u015f de\u011fi\u015fik e\u011fim a\u00e7\u0131lar\u0131na sahip d\u00fczlemlerin tepesinden b\u0131rakt\u0131\u011f\u0131 bronz k\u00fcrelerin konumunu \u00f6l\u00e7erken, devinim ba\u015flad\u0131\u011f\u0131nda a\u00e7t\u0131\u011f\u0131 su kab\u0131ndaki deli\u011fi devinim bitti\u011finde kapayarak bir ba\u015fka kaba d\u00f6k\u00fclm\u00fc\u015f suyu tartarak aradan ge\u00e7en zaman\u0131 da \u00f6l\u00e7\u00fcyordu. (#7)<\/p>\n
\u0130\u015fte o muhte\u015fem Galilei Denklemi, b\u00f6yle bir (kelimenin tam anlam\u0131yla) bilimsel \u00e7aban\u0131n \u00fcr\u00fcn\u00fcyd\u00fc. <==EK 3 sonu<\/p>\n
<\/p>\n
(#1) https:\/\/www.merriam-webster.com\/dictionary\/mathematization : reduction to mathematical form. (#6) Sarka\u00e7l\u0131 saati Galileo icat etmi\u015ftir.<\/p>\n (#7) Reconstructing Galileo\u2019s Inclined Plane Experiments for Teaching Purposes, Falk RIESS, Peter HEERING, Dennis NAWRATH<\/p>\n https:\/\/www.researchgate.net\/publication\/254334167_Reconstructing_Galileo’s_Inclined_Plane_Experiments_for_Teaching_Purposes\/link\/0a85e53cc1ebc17a57000000\/download?_tp=eyJjb250ZXh0Ijp7ImZpcnN0UGFnZSI6InB1YmxpY2F0aW9uIiwicGFnZSI6InB1YmxpY2F0aW9uIn19<\/p>\n","protected":false},"excerpt":{"rendered":" Matematik\u2019in ne menem bir \u015fey oldu\u011funa dair pek \u00e7ok yaz\u0131 mevcut bu blogda. An\u0131msamak i\u00e7in; t\u00fcm\u00fcyle hayal \u00fcr\u00fcn\u00fc oldu\u011fundan s\u00f6z etmi\u015ftik. \u0130kincileyin; Say\u0131 Sistemi (\u2018Number Theory\u2019) eksikli ve geli\u015ftirilmeye muhta\u00e7 (\u2018immature\u2019) demi\u015ftik. \u00d6rne\u011fin, her yerde rastlad\u0131\u011f\u0131m\u0131z \u221e say\u0131 de\u011filken say\u0131 say\u0131lm\u0131\u015ft\u0131r. Ayr\u0131ca, \u00c7ift Tam Say\u0131lar\u0131n \u221e\u2019u ile Tek Tam Say\u0131lar\u0131n \u221e\u2019u t\u0131pa t\u0131p ayn\u0131 imi\u015fcesine […]<\/p>\n","protected":false},"author":1425,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"categories":[1],"tags":[],"class_list":["post-1743","post","type-post","status-publish","format-standard","hentry","category-genel"],"_links":{"self":[{"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/posts\/1743","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/users\/1425"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/comments?post=1743"}],"version-history":[{"count":0,"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/posts\/1743\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/media?parent=1743"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/categories?post=1743"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/tags?post=1743"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n(#2) Bu konuda Google AI yanl\u0131\u015f bilgi vermektedir, Google AI\u2019nin verdi\u011fi bilgi yanl\u0131\u015ft\u0131r. Ayr\u0131ca, https:\/\/blog.metu.edu.tr\/caglart\/2026\/06\/30\/ben-mi-matematik-bilmiyorum-su-ispat-mi-yanlis-baslikli-yazidaki-hata\/ adresteki yanl\u0131\u015fl\u0131k \u015fudur: arctan(\u221e)=tan-1(\u221e)=\u03c0 ifadeleri yanl\u0131\u015ft\u0131r.
\n(#3) Bir de diyorlar ki, “\u2013Diyelim ki, ‘Matematik hay\u00e2l \u00fcr\u00fcn\u00fcd\u00fcr.’ Peki, bu durumda herkes mi ayn\u0131 hay\u00e2li g\u00f6rmekte? Zira, Matematik herkes i\u00e7in ayn\u0131! De\u011fil mi?” Evet! Matematik herkes i\u00e7in ayn\u0131. Ama bu durum, Matematik’in hay\u00e2l \u00fcr\u00fcn\u00fc ve bir kurallar silsilesi oldu\u011fu ger\u00e7e\u011fini de\u011fi\u015ftirmez ki. Bak\u0131n\u0131z, mesel\u00e2, Satran\u00e7 da Tavla da herkes i\u00e7in ayn\u0131, bir kurallar silsilesi olup hay\u00e2l \u00fcr\u00fcn\u00fcd\u00fcr! De\u011fil mi?
\n(#4) \u015eunca zamand\u0131r Galileo Galilei ile yarenlik eden bir ki\u015fi olarak, b\u00fct\u00fcn bunlar\u0131 Galileo’nun da elbet bildi\u011fini rahatl\u0131kla s\u00f6yleyebilirim.\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(#5) https:\/\/blog.metu.edu.tr\/caglart\/2024\/09\/<\/a><\/p>\n