{"id":807,"date":"2011-04-03T16:58:31","date_gmt":"2011-04-03T14:58:31","guid":{"rendered":"http:\/\/simplexportal.ro\/?p=807"},"modified":"2013-09-12T15:27:59","modified_gmt":"2013-09-12T13:27:59","slug":"a-matematikai-indukcio-modszere","status":"publish","type":"post","link":"https:\/\/simplexportal.ro\/?p=807","title":{"rendered":"A matematikai indukci\u00f3 m\u00f3dszere"},"content":{"rendered":"
\n
\"\"<\/div>\n
Beleolvasok<\/a><\/div>\n<\/div>\n

“Ne higgy el semmit, de csak azt
\nk\u00e9rdezd, amit k\u00e9rdezni \u00e9rdemes.”<\/p>\n

El\u0151sz\u00f3<\/p>\n

Minden tud\u00e1sunk felt\u00e9telez\u00e9sekb\u0151l indul ki \u00e9s a matematik\u00e1n k\u00edv\u00fcli tud\u00e1sunk nagyr\u00e9szt ezekb\u0151l is \u00e1ll. A felt\u00e9telez\u00e9sek k\u00f6zt nagy k\u00fcl\u00f6nbs\u00e9gek \u00e9szlelhet\u0151ek. Vannak elfogadhat\u00f3, megb\u00edzhat\u00f3 felt\u00e9telez\u00e9sek (pl. a fizika \u00e1ltal\u00e1nos t\u00f6rv\u00e9nyeiben megnyilv\u00e1nul\u00f3ak) \u00e9s vannak nem megb\u00edzhat\u00f3 \u00e9s el nem fogadhat\u00f3 felt\u00e9telez\u00e9sek, amilyenekkel (pl. a napi sajt\u00f3t b\u00f6ng\u00e9szve) minduntalan tal\u00e1lkozhatunk.
\nAlapvet\u0151en k\u00e9tfajta okoskod\u00e1ssal tal\u00e1lkozunk: bizony\u00edt\u00f3 okoskod\u00e1ssal, ezzel biztos\u00edtjuk be matematikai tud\u00e1sunkat \u00e9s a plauzibilis okoskod\u00e1ssal, ezzel t\u00e1masztjuk al\u00e1 sejt\u00e9seinket, ilyenek a fizikus indukt\u00edv bizony\u00edt\u00e9kai, a t\u00f6rt\u00e9n\u00e9sz \u00edr\u00e1sos leletei, a k\u00f6zgazd\u00e1sz statisztikus eredm\u00e9nyei. A bizony\u00edt\u00f3 okoskod\u00e1s \u00e1titatja a tudom\u00e1nyokat, de \u00f6nmag\u00e1ban nem k\u00e9pes \u00fajat mondani a benn\u00fcnket k\u00f6r\u00fclvev\u0151 vil\u00e1gr\u00f3l. Minden \u00faj ismeret megszerz\u00e9se mag\u00e1ban foglalja a plauzibilis okoskod\u00e1st.
\nA matematika kit\u0171n\u0151 alkalmat k\u00edn\u00e1l a bizony\u00edt\u00f3 okoskod\u00e1s elsaj\u00e1t\u00edt\u00e1s\u00e1ra, de bizony\u00edt\u00e1s nincs el\u0151zetes tal\u00e1lgat\u00e1s n\u00e9lk\u00fcl. Ez\u00e9rt matematik\u00e1t m\u0171velni csak az k\u00e9pes, aki a tal\u00e1lgat\u00e1s m\u0171v\u00e9szet\u00e9t is elsaj\u00e1t\u00edtja. Hiszen a matematika nem puszt\u00e1n bizony\u00edt\u00f3 tudom\u00e1ny. B\u00e1r a legt\u00f6bb matematikai k\u00f6zlem\u00e9ny bizony\u00edt\u00f3 jelleg\u0171, \u00e1ll\u00edt\u00e1sokat \u00e9s bizony\u00edt\u00e1sokat tartalmaz, ezen eredm\u00e9nyek l\u00e9trej\u00f6tte hasonl\u00f3 minden egy\u00e9b tud\u00e1s sz\u00fclet\u00e9s\u00e9hez: plauzibilis okoskod\u00e1ssal, tal\u00e1lgat\u00e1ssal, megfigyel\u00e9sekb\u0151l, anal\u00f3gi\u00e1k k\u00f6vet\u00e9s\u00e9b\u0151l j\u00f6n l\u00e9tre. \u00c9ppen ez\u00e9rt a plauzibilis okoskod\u00e1s a bizony\u00edt\u00f3hoz hasonl\u00f3 fontoss\u00e1ggal b\u00edr.
\nAz indukt\u00edv okoskod\u00e1s a plauzibilis okoskod\u00e1s egy k\u00fcl\u00f6nleges esete. Az indukci\u00f3 alapj\u00e1t mindig k\u00fcl\u00f6nb\u00f6z\u0151 \u00e9szrev\u00e9telek k\u00e9pezik, ezeket al\u00e1t\u00e1maszt\u00f3 \u00e9szrev\u00e9telek er\u0151s\u00edtik meg, mindazon\u00e1ltal indukt\u00edv okoskod\u00e1sunk m\u00e9g mindig hib\u00e1snak bizonyulhat.
\nA matematikai indukci\u00f3 ezt a hibalehet\u0151s\u00e9get k\u00fcsz\u00f6b\u00f6li ki \u00e9s b\u00e1r alapj\u00e1t indukt\u00edv, plauzibilis okoskod\u00e1s k\u00e9pezi (ezzel jutunk el az \u00e1ll\u00edt\u00e1saink megfogalmaz\u00e1s\u00e1hoz), az indukci\u00f3s bizony\u00edt\u00e1s m\u00e1r bizony\u00edt\u00f3 okoskod\u00e1s. A matematikai indukci\u00f3 a tapasztalatokb\u00f3l kiindul\u00f3 tudom\u00e1nyos m\u00f3dszer.<\/p>\n","protected":false},"excerpt":{"rendered":"

Beleolvasok “Ne higgy el semmit, de csak azt k\u00e9rdezd, amit k\u00e9rdezni \u00e9rdemes.” El\u0151sz\u00f3 Minden tud\u00e1sunk felt\u00e9telez\u00e9sekb\u0151l indul ki \u00e9s a matematik\u00e1n k\u00edv\u00fcli tud\u00e1sunk nagyr\u00e9szt ezekb\u0151l is \u00e1ll. A felt\u00e9telez\u00e9sek k\u00f6zt nagy k\u00fcl\u00f6nbs\u00e9gek \u00e9szlelhet\u0151ek. Vannak elfogadhat\u00f3, megb\u00edzhat\u00f3 felt\u00e9telez\u00e9sek (pl. a fizika \u00e1ltal\u00e1nos t\u00f6rv\u00e9nyeiben megnyilv\u00e1nul\u00f3ak) \u00e9s vannak nem megb\u00edzhat\u00f3 \u00e9s el nem fogadhat\u00f3 felt\u00e9telez\u00e9sek, amilyenekkel (pl. a […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13],"tags":[20],"class_list":["post-807","post","type-post","status-publish","format-standard","hentry","category-fokozati","tag-szilagyi-judit"],"_links":{"self":[{"href":"https:\/\/simplexportal.ro\/index.php?rest_route=\/wp\/v2\/posts\/807","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/simplexportal.ro\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/simplexportal.ro\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/simplexportal.ro\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/simplexportal.ro\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=807"}],"version-history":[{"count":4,"href":"https:\/\/simplexportal.ro\/index.php?rest_route=\/wp\/v2\/posts\/807\/revisions"}],"predecessor-version":[{"id":1053,"href":"https:\/\/simplexportal.ro\/index.php?rest_route=\/wp\/v2\/posts\/807\/revisions\/1053"}],"wp:attachment":[{"href":"https:\/\/simplexportal.ro\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=807"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/simplexportal.ro\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=807"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/simplexportal.ro\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=807"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}