{"id":9734,"date":"2026-07-25T20:21:08","date_gmt":"2026-07-25T11:21:08","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9734"},"modified":"2026-07-26T23:12:27","modified_gmt":"2026-07-26T14:12:27","slug":"math-abstraction-01-essence","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/mathematical-abstraction\/math-abstraction-01-essence\/","title":{"rendered":"\uc218\ud559\uc801 \ucd94\uc0c1\ud654\uc758 \uae30\ubcf8 \uac1c\ub150"},"content":{"rendered":"<p><!-- \uc218\ud559\uc801 \ucd94\uc0c1\ud654 1\ud3b8: \ucd94\uc0c1\ud654\ub780 \ubb34\uc5c7\uc778\uac00 --><\/p>\n<div class=\"math-abs-series\">\n<p>\uc77c\uc0c1\uc5b4\uc5d0\uc11c \u2018\ucd94\uc0c1\uc801\u2019\uc774\ub77c\ub294 \ub9d0\uc740 \ub300\uccb4\ub85c \uae0d\uc815\uc801\uc778 \uc758\ubbf8\ub85c \uc4f0\uc774\uc9c0 \uc54a\ub294\ub2e4. \uc124\uba85\uc774 \ucd94\uc0c1\uc801\uc774\ub77c\ub294 \ub9d0\uc740 \ub300\uac1c \ubaa8\ud638\ud574\uc11c \uc54c\uc544\ub4e3\uae30 \uc5b4\ub835\ub2e4\ub294 \ubd88\ud3c9\uc744 \ub73b\ud558\uba70, \uc73c\ub808 \uad6c\uccb4\uc801\uc778 \uc608\ub97c \ub4e4\uc5b4 \ub2ec\ub77c\ub294 \uc694\uccad\uc774 \ub4a4\ub530\ub978\ub2e4. <!-- \uadf8\ub9bc\uc744 \ub450\uace0 \ucd94\uc0c1\uc801\uc774\ub77c\uace0 \ud560 \ub54c\ub3c4 \ube44\uc2b7\ud558\ub2e4. \ubb34\uc5c7\uc744 \uadf8\ub838\ub294\uc9c0 \ud55c\ub208\uc5d0 \ud30c\uc545\ud558\uae30 \uc5b4\ub835\ub2e4\ub294 \ub73b\uc774 \ub2f4\uaca8 \uc788\ub2e4. --><\/p>\n<p>\uc218\ud559\uc5d0\uc11c\ub294 \uc0ac\uc815\uc774 \ub2e4\ub974\ub2e4. \ucd94\uc0c1\ub300\uc218\ud559\uc758 \uad70, \ucd94\uc0c1 \ubca1\ud130\uacf5\uac04, \uc704\uc0c1\uacf5\uac04\uc758 \uc815\uc758\ub294 \uc5c4\ubc00\ud558\uace0 \uba85\ub8cc\ud55c \ubb38\uc7a5\uc73c\ub85c \uae30\uc220\ub41c\ub2e4. \uacf5\ub9ac\ub294 \ub300\uac1c \ud2b9\uc815 \ub300\uc0c1 \ud558\ub098\ub97c \uc720\uc77c\ud558\uac8c \uc815\ud558\ub294 \uac83\uc774 \uc544\ub2c8\ub77c, \uc5b4\ub5a4 \uad6c\uc870\uac00 \uadf8 \uac1c\ub150\uc758 \uc0ac\ub840\uac00 \ub418\uae30 \uc704\ud574 \ub9cc\uc871\ud574\uc57c \ud560 \uc870\uac74\uc744 \uba85\uc2dc\ud55c\ub2e4. \uac19\uc740 \uacf5\ub9ac\uacc4\ub97c \ub9cc\uc871\ud558\ub294 \uc11c\ub85c \ub2e4\ub978 \uad6c\uc870\ub4e4\uc774 \uc874\uc7ac\ud560 \uc218 \uc788\uc9c0\ub9cc, \ubb34\uc5c7\uc744 \uac00\uc815\ud558\uace0 \ubb34\uc5c7\uc744 \uc99d\uba85\ud558\ub294\uc9c0\ub294 \ubd84\uba85\ud574\uc9c4\ub2e4. \uc218\ud559\uc790\uac00 \uc5b4\ub5a4 \uac1c\ub150\uc744 \ucd94\uc0c1\ud654\ud588\ub2e4\uace0 \ub9d0\ud560 \ub54c, \uadf8\uac83\uc740 \uac1c\ub150\uc744 \ubaa8\ud638\ud558\uac8c \ub9cc\ub4e4\uc5c8\ub2e4\ub294 \ub73b\uc774 \uc544\ub2c8\ub77c \uad00\ub828\ub41c \uad6c\uc870\uc640 \uc131\uc9c8\uc744 \uba85\ud655\ud788 \ub4dc\ub7ec\ub0c8\ub2e4\ub294 \ub73b\uc774\ub2e4. \ub3d9\uc77c\ud55c \ub2e8\uc5b4\uac00 \uc77c\uc0c1\uc5d0\uc11c\ub294 \ubaa8\ud638\ud568\uc744, \uc218\ud559\uc5d0\uc11c\ub294 \uc815\uad50\ud55c \uac1c\ub150 \ud615\uc131\uc744 \uac00\ub9ac\ud0a4\ub294 \uc148\uc774\ub2e4.<\/p>\n<p>\uc774 \uc5f0\uc7ac\ub294 \uc774\ub7ec\ud55c \ucc28\uc774\uc810\uc5d0\uc11c \ucd9c\ubc1c\ud55c\ub2e4. \uc218\ud559\uc801 \ucd94\uc0c1\ud654\ub780 \uc5b4\ub5a0\ud55c \u2018\uc870\uc791\u2019\uc774\uba70, \uc774 \uc870\uc791\uc740 \uc5b4\ub5bb\uac8c \uc218\ud559\uc744 \uc5c4\ubc00\ud558\uac8c \ub9cc\ub4dc\ub294\uac00? \ub098\uc544\uac00 \uc218\ud559\uc790\ub4e4\uc740 \uc5b8\uc81c\ubd80\ud130 \uc790\uc2e0\uc774 \uc774\ub7ec\ud55c \uc870\uc791\uc744 \uc218\ud589\ud558\uace0 \uc788\uc74c\uc744 \uc790\uac01\ud588\ub294\uac00? \uc774 \uae00\uc5d0\uc11c\ub294 \ucd94\uc0c1\ud654\uc758 \ub73b\uc744 \uc815\uc758\ud558\uace0, \uadf8 \uc870\uc791\uc774 \uc2e4\uc81c\ub85c \uc218\ud589\ub418\ub294 \ubc29\uc2dd \uc911 \ud558\ub098\ub97c \uc218\ud559\uc758 \uc815\ub9ac\uc640 \uc99d\uba85\uc744 \ud1b5\ud574 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<h4>\ucd94\uc0c1\uacfc \uc0ac\uc0c1<\/h4>\n<p>\uc6b0\uc120 \ub2e8\uc5b4\uc758 \uc5b4\uc6d0\ubd80\ud130 \uc0b4\ud3b4\ubcf4\uc790. \uc601\uc5b4 \ub2e8\uc5b4 \u2018abstract\u2019\ub294 \ub77c\ud2f4\uc5b4 \u2018abstractus\u2019(\ub3d9\uc0ac \u2018abstrahere\u2019\uc758 \uacfc\uac70\ubd84\uc0ac)\uc5d0\uc11c \uc654\uc73c\uba70, \u2018abstrahere\u2019\ub294 \u2018~\ub85c\ubd80\ud130 \ub5a8\uc5b4\uc838\u2019\ub97c \ub73b\ud558\ub294 \u2018ab-\/abs-\u2019\uc640 \u2018\ub04c\ub2e4\u2019\ub97c \ub73b\ud558\ub294 \u2018trahere\u2019\uc758 \uacb0\ud569\uc774\ub2e4<a class=\"math-abs-series-ref\" title=\"Merriam-Webster. (n.d.). Abstract. In Merriam-Webster.com Dictionary. Retrieved July 25, 2026.\" href=\"#ref-MerriamWebsterND\">[MerriamWebsterND]<\/a>. \uc544\ub9ac\uc2a4\ud1a0\ud154\ub808\uc2a4\ub294 \uc218\ud559\uc801 \ub300\uc0c1\uc744 \ub17c\ud558\uba74\uc11c \u2018\uc81c\uac70\uc5d0 \uc758\ud574\u2019 \ub610\ub294 \u2018\uc81c\uac70 \uc18d\uc5d0\uc11c\u2019\ub97c \ub73b\ud558\ub294 \u2018\uc544\ud30c\uc774\ub808\uc2dc\uc2a4(aphairesis)\u2019 \uacc4\uc5f4\uc758 \ud45c\ud604\uc744 \uc0ac\uc6a9\ud588\ub2e4<a class=\"math-abs-series-ref\" title=\"Mendell, H. (2016). Aristotle and mathematics. In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Fall 2016 ed.).\" href=\"#ref-Mendell2016\">[Mendell2016]<\/a>. \uadf8\uc758 \uc124\uba85\uc5d0\uc11c \uc218\ud559\uc790\ub294 \uac10\uac01\uc801 \uc0ac\ubb3c\uacfc \ubcc4\ub3c4\ub85c \uc874\uc7ac\ud558\ub294 \uc218\ub098 \ub3c4\ud615\uc744 \ub9cc\ub4e4\uc5b4 \ub0b4\ub294 \uac83\uc774 \uc544\ub2c8\ub77c, \uac10\uac01\uc801 \uc0ac\ubb3c\uc744 \uadf8\uac83\uc758 \uac10\uac01\uc801, \uc6b4\ub3d9\uc801 \uce21\uba74\uc774 \uc544\ub2cc \uc591\uacfc \uc5f0\uc18d\uc131\uc758 \uce21\uba74\uc5d0\uc11c \uace0\ucc30\ud558\uba70 \ubb34\uac8c, \uc5f4, \uc6b4\ub3d9 \ub4f1\uc758 \uc131\uc9c8\uc744 \uace0\ub824\uc5d0\uc11c \uc81c\uc678\ud55c\ub2e4<a class=\"math-abs-series-ref\" title=\"Aristotle. (1998). The Metaphysics (H. Lawson-Tancred, Trans.). Penguin Classics.\" href=\"#ref-Aristotle1998\">[Aristotle1998]<\/a>. \uc774\ub294 \ud6c4\ub300 \uc11c\uc591 \ucca0\ud559\uc758 \ucd94\uc0c1\ub860\uc5d0 \ud070 \uc601\ud5a5\uc744 \uc900 \uace0\uc804\uc801 \ucd9c\ubc1c\uc810\uc774\uba70, \uc774\uc5d0 \ub300\ud55c \ubcf8\uaca9\uc801\uc778 \ub17c\uc758\ub294 <a href=\"..\/math-abstraction-02-implicit-era\/\">2\ubd80<\/a>\uc5d0\uc11c \ub2e4\ub8ec\ub2e4.<\/p>\n<p>\ud55c\uc790\uc5b4\ub3c4 \uc774\uc640 \ube44\uc2b7\ud55c \uc774\ubbf8\uc9c0\ub97c \ub2f4\uace0 \uc788\ub2e4. \ucd94\uc0c1(\u62bd\u8c61)\uc740 \uc5ec\ub7ec \ub300\uc0c1\uc774\ub098 \ud45c\uc0c1\uc5d0\uc11c \uacf5\ud1b5\ub41c \uc131\uc9c8\uc744 \ubf51\uc544\ub0b4\uc5b4 \uac1c\ub150\uc744 \uad6c\uc131\ud558\ub294 \uc791\uc6a9\uc744 \ub73b\ud558\uba70<a class=\"math-abs-series-ref\" title=\"Kotobank. (n.d.). \u300c\u62bd\u8c61\u300d.\" href=\"#ref-KotobankChusho\">[KotobankChusho]<\/a>, \ucca0\ud559\uc5d0\uc11c \uc774\uc640 \uc9dd\uc744 \uc774\ub8e8\uc5b4 \uc4f0\uc774\ub294 \uc0ac\uc0c1(\u6368\u8c61)\uc740 \uadf8 \uacfc\uc815\uc5d0\uc11c \ub2e4\ub978 \uc131\uc9c8\uc774\ub098 \uce21\uba74\uc744 \ubc84\ub9ac\uace0 \uace0\ub824\ud558\uc9c0 \uc54a\ub294 \uc77c\uc744 \uac00\ub9ac\ud0a8\ub2e4<a class=\"math-abs-series-ref\" title=\"Kotobank. (n.d.). \u300c\u6368\u8c61\u300d.\" href=\"#ref-KotobankShasho\">[KotobankShasho]<\/a>. \uc774 \ub450 \ud45c\ud604\uc740 \ubcc4\uac1c\uc758 \uc870\uc791\uc774\ub77c\uae30\ubcf4\ub2e4 \ud558\ub098\uc758 \uc120\ud0dd \uacfc\uc815\uc5d0\uc11c \uc11c\ub85c \ub9de\ubb3c\ub9ac\ub294 \ub450 \uce21\uba74\uc73c\ub85c \ubcfc \uc218 \uc788\ub2e4. \uce60\ud310\uc5d0 \uadf8\ub9b0 \uc0bc\uac01\ud615\uc5d0\ub294 \ud06c\uae30\uc640 \uc704\uce58\uac00 \uc815\ud574\uc838 \uc788\uace0, \uc120\uc758 \uad75\uae30\uc640 \ubd84\ud544\uc758 \uc0c9\uc0c1\ub3c4 \uc874\uc7ac\ud55c\ub2e4. \uc5ec\uae30\uc11c \u2018\uc138 \uc120\ubd84\uc73c\ub85c \ub458\ub7ec\uc2f8\uc778 \ub3c4\ud615\u2019\uc774\ub77c\ub294 \uc131\uc9c8\uc5d0 \uc8fc\ubaa9\ud558\uace0 \ub098\uba38\uc9c0\ub97c \uace0\ub824\uc5d0\uc11c \uc81c\uc678\ud560 \ub54c \uc77c\ubc18\uc801\uc778 \u2018\uc0bc\uac01\ud615\u2019\uc758 \uac1c\ub150\uc73c\ub85c \ub098\uc544\uac08 \uc218 \uc788\ub2e4. \ubb34\uc5c7\uc744 \ucde8\ud560\uc9c0 \uc815\ud558\ub294 \uc77c\uc740 \uace7 \ubb34\uc5c7\uc744 \uace0\ub824\ud558\uc9c0 \uc54a\uc744\uc9c0 \uc815\ud558\ub294 \uc77c\uacfc \uc5f0\uacb0\ub418\uba70, \uc774 \uc810\uc740 \uc774 \uc5f0\uc7ac \uc804\ubc18\uc5d0 \uac78\uccd0 \uc790\uc8fc \ub4f1\uc7a5\ud560 \uac83\uc774\ub2e4.<\/p>\n<h4>\ucd94\uc0c1\ud654\uc758 \uc138 \uac00\uc9c0 \ubc29\uc2dd<\/h4>\n<p>\uc774 \uc5f0\uc7ac\uc5d0\uc11c <span class=\"defined\">\ucd94\uc0c1\ud654<\/span>(abstraction)\ub780 \ub2e4\uc74c\uacfc \uac19\uc740 \uc870\uc791\uc744 \uc758\ubbf8\ud55c\ub2e4. \ud558\ub098 \ub610\ub294 \uc5ec\ub7ec \ub300\uc0c1\uc5d0\uc11c \ud2b9\uc815\ud55c \uc131\uc9c8\ub9cc\uc744 \ucd94\ucd9c\ud558\uace0 \ub098\uba38\uc9c0 \uc131\uc9c8\uc740 \ubc30\uc81c\ud558\uc5ec, \uc624\uc9c1 \ucd94\ucd9c\ub41c \uc131\uc9c8\ub85c\ub9cc \uaddc\uc815\ub418\ub294 \uc0c8\ub85c\uc6b4 \uac1c\ub150\uc774\ub098 \ub300\uc0c1\uc744 \ud615\uc131\ud558\ub294 \uc870\uc791\uc774\ub2e4. \uc218\ud559\uc5d0\uc11c \uc774 \uc870\uc791\uc740 \ud06c\uac8c \uc138 \uac00\uc9c0 \ubc29\uc2dd\uc73c\ub85c \uc218\ud589\ub41c\ub2e4.<\/p>\n<ul>\n<li>\uccab \ubc88\uc9f8\ub294 <span class=\"defined\">\u2018\uac1c\ub150 \ud615\uc131\u2019\uc73c\ub85c\uc11c\uc758 \ucd94\uc0c1\ud654<\/span>\uc774\ub2e4. \uc5ec\ub7ec \uc0ac\ub840\uc5d0\uc11c \uacf5\ud1b5\ub41c \uc131\uc9c8\uc744 \ucd94\ucd9c\ud558\uc5ec \ud558\ub098\uc758 \uac1c\ub150\uc73c\ub85c \ubb36\ub294 \ubc29\uc2dd\uc774\ub2e4. \uc0ac\uacfc \ub2e4\uc12f \uac1c, \uc591 \ub2e4\uc12f \ub9c8\ub9ac, \uc190\uac00\ub77d \ub2e4\uc12f \uac1c\uc5d0\uc11c \uc9c0\uce6d\ud558\ub294 \uad6c\uccb4\uc801\uc778 \ub300\uc0c1\uc744 \uace0\ub824\uc5d0\uc11c \uc81c\uc678\ud558\uba74 \u2018\uc218 5\u2019\ub77c\ub294 \uac1c\ub150\ub9cc \ub0a8\ub294\ub2e4. \uc774\ub294 \uc778\ub958\uac00 \uc218\ud589\ud574 \uc628 \uac00\uc7a5 \uc6d0\ucd08\uc801\uc778 \ud615\ud0dc\uc758 \ucd94\uc0c1\ud654 \uac00\uc6b4\ub370 \ud558\ub098\uc774\uba70, \uc218\ud559\uad50\uc721 \uc5f0\uad6c\uc5d0\uc11c\ub3c4 \uc911\uc694\ud55c \uad00\uc810\uc73c\ub85c \ub2e4\ub8e8\uc5b4\uc9c4\ub2e4<a class=\"math-abs-series-ref\" title=\"Mitchelmore, M., &amp; White, P. (2007). Abstraction in mathematics learning. Mathematics Education Research Journal, 19, 1\u20139.\" href=\"#ref-MitchelmoreWhite2007\">[MitchelmoreWhite2007]<\/a>. \uc774 \ubc29\uc2dd\uc758 \uc5ed\uc0ac\uc801 \ubc30\uacbd\uc740 <a href=\"..\/math-abstraction-02-implicit-era\/\">2\ubd80<\/a>\uc5d0\uc11c, \uc774\uc640 \uad00\ub828\ub41c \ud559\uc2b5\uc758 \ubb38\uc81c\ub294 <a href=\"..\/math-abstraction-09-cognitive-process\/\">9\ubd80<\/a>\uc5d0\uc11c \ub2e4\ub8e8\uae30\ub85c \ud55c\ub2e4.<\/li>\n<li>\ub450 \ubc88\uc9f8\ub294 <span class=\"defined\">\u2018\ub3d9\uce58\ub958\ub97c \uc0ac\uc6a9\ud55c \uc815\uc758\u2019<\/span>\uc774\ub2e4. \ub300\uc0c1\ub4e4 \uc0ac\uc774\uc5d0 \uc131\ub9bd\ud558\ub294 \uad00\uacc4\uc5d0\uc11c \ucd9c\ubc1c\ud558\uc5ec, \uadf8 \uad00\uacc4\ub85c \ubb36\uc778 \ubd80\ub958 \ud558\ub098\ud558\ub098\ub97c \uc0c8\ub85c\uc6b4 \ub300\uc0c1\uc73c\ub85c \ucde8\uae09\ud558\ub294 \ubc29\uc2dd\uc774\ub2e4. \uc774 \ubc29\uc2dd\uc740 \ubcf8\ubb38\uc758 \ud6c4\ubc18\ubd80\uc5d0\uc11c \uc790\uc138\ud788 \uc0b4\ud3b4\ubcfc \uac83\uc774\ub2e4.<\/li>\n<li>\uc138 \ubc88\uc9f8\ub294 <span class=\"defined\">\u2018\uacf5\ub9ac\ud654\u2019\ub97c \uc0ac\uc6a9\ud55c \ucd94\uc0c1\ud654<\/span>\uc774\ub2e4. \ub300\uc0c1\uc774 \ub9cc\uc871\uc2dc\ucf1c\uc57c \ud560 \uc131\uc9c8\uc758 \ubaa9\ub85d, \uc989 \uacf5\ub9ac(axiom)\ub97c \uc81c\uc2dc\ud558\uace0, \uadf8 \uacf5\ub9ac\uacc4\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubaa8\ub4e0 \uc218\ud559\uc801 \ub300\uc0c1\uc744 \ud3ec\uad04\ud558\uc5ec \uc815\uc758\ud558\ub294 \ubc29\uc2dd\uc774\ub2e4. \uad70, \ubca1\ud130\uacf5\uac04, \uc704\uc0c1\uacf5\uac04 \ub4f1\uc774 \uc774\ub7ec\ud55c \ubc29\uc2dd\uc73c\ub85c \uc815\uc758\ub41c\ub2e4. \uacf5\ub9ac\ud654\ub294 19\uc138\uae30 \ub9d0, \ud2b9\ud788 \ud790\ubca0\ub974\ud2b8\uac00 1899\ub144\uc5d0 \uc81c\uc2dc\ud55c \uae30\ud558\ud559\uc758 \uacf5\ub9ac\ud654\ub97c \uac70\uce58\uba70 \ud604\ub300\uc801\uc778 \ud615\ud0dc\ub97c \uac16\ucd94\uc5c8\uace0, 20\uc138\uae30 \uc218\ud559\uc758 \uc8fc\uc694 \ubc29\ubc95\ub860\uc774 \ub418\uc5c8\ub2e4<a class=\"math-abs-series-ref\" title=\"Zach, R. (2023). Hilbert's program. In E. N. Zalta &amp; U. Nodelman (Eds.), The Stanford Encyclopedia of Philosophy.\" href=\"#ref-Zach2023\">[Zach2023]<\/a>. \uc790\uc138\ud55c \ub0b4\uc6a9\uc740 <a href=\"..\/math-abstraction-04-axiomatic-method\/\">4\ubd80<\/a>\uc5d0\uc11c \uc0c1\uc138\ud788 \uc0b4\ud3b4\ubcfc \uac83\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc774 \uc138 \uac00\uc9c0 \ubc29\uc2dd\uc740 \uc11c\ub85c \uae34\ubc00\ud558\uac8c \uc5f0\uacb0\ub418\uc5b4 \uc788\ub2e4. \uccab \ubc88\uc9f8\uac00 \ucd94\uc0c1\ud654\uc5d0 \ub300\ud55c \uc9c1\uad00\uc801\uc778 \uc11c\uc220\uc774\ub77c\uba74, \ub450 \ubc88\uc9f8\uc640 \uc138 \ubc88\uc9f8\ub294 \uc774\ub97c \uc218\ud559\uc801\uc73c\ub85c \ub354 \uc815\uad50\ud558\uac8c \ub2e4\ub4ec\uc740 \uae30\ubc95\uc774\ub2e4.<\/p>\n<h4>\uc77c\ubc18\ud654, \uc774\uc0c1\ud654, \ud615\uc2dd\ud654<\/h4>\n<p>\ucd94\uc0c1\ud654\uc640 \ud63c\ub3d9\ud558\uae30 \uc26c\uc6b4 \uba87 \uac00\uc9c0 \uac1c\ub150\uc774 \uc788\ub2e4. <!-- \ub2e4\ub9cc \uc774 \uc6a9\uc5b4\ub4e4\uc758 \uacbd\uacc4\uc640 \uc6a9\ubc95\uc740 \ubb38\ud5cc\uacfc \ub9e5\ub77d\uc5d0 \ub530\ub77c \ub2ec\ub77c\uc9c8 \uc218 \uc788\uc73c\ubbc0\ub85c, \uc774 \uc5f0\uc7ac\uc5d0\uc11c\ub294 \ubd84\uc11d\uc744 \uc704\ud574 \ub2e4\uc74c\uacfc \uac19\uc774 \uc791\uc5c5\uc801\uc73c\ub85c \uad6c\ubd84\ud55c\ub2e4. --><\/p>\n<ul>\n<li>\uccab\uc9f8, <span class=\"defined\">\uc77c\ubc18\ud654<\/span>(generalization)\ub294 \uba85\uc81c\ub098 \uc815\ub9ac\uc758 \uc801\uc6a9 \ubc94\uc704\ub97c \ub354 \ub113\uc740 \ub300\uc0c1\uad70\uc73c\ub85c \ud655\uc7a5\ud558\ub294 \uacfc\uc815\uc774\ub77c\uace0 \ud558\uc790. \uc608\ub97c \ub4e4\uc5b4 \ud3c9\uba74 \\(\\mathbb R^2\\)\uc758 \ubca1\ud130\uc5d0 \ub300\ud558\uc5ec \uc99d\uba85\ub41c \uc815\ub9ac\ub97c \\(n\\)\ucc28\uc6d0 \uacf5\uac04 \\(\\mathbb R^n\\)\uc758 \ubca1\ud130\ub85c \ud655\uc7a5\ud558\ub294 \uac83\uc740 \uc77c\ubc18\ud654\uc758 \uc804\ud615\uc801\uc778 \uc0ac\ub840\ub2e4. \ud55c\ud3b8 \\(\\mathbb R^n\\)\uc5d0\uc11c \uc784\uc758\uc758 \uccb4 \uc704\uc758 \ucd94\uc0c1 \ubca1\ud130\uacf5\uac04\uc73c\ub85c \ub118\uc5b4\uac08 \ub54c\ub294 \uc88c\ud45c\ub77c\ub294 \uad6c\uccb4\uc801\uc778 \ud45c\ud604\uc744 \ubc30\uc81c\ud558\uace0 \ub367\uc148\uacfc \uc2a4\uce7c\ub77c\ubc30\uc758 \uacf5\ub9ac\uc801 \uad6c\uc870\uc5d0 \uc8fc\ubaa9\ud55c\ub2e4. \uadf8 \uacb0\uacfc \ub2e4\ud56d\uc2dd, \uc5f0\uc18d\ud568\uc218, \ub3d9\ucc28 \uc120\ud615 \ubbf8\ubd84\ubc29\uc815\uc2dd\uc758 \ud574\uacf5\uac04\ucc98\ub7fc \ucc98\uc74c\ubd80\ud130 \uc88c\ud45c\ub85c \uc81c\uc2dc\ub418\uc9c0 \uc54a\uc740 \ub300\uc0c1\ub3c4 \ubca1\ud130\uacf5\uac04\uc73c\ub85c \ub2e4\ub8f0 \uc218 \uc788\ub2e4. <!-- \uc774 \ud6c4\uc790\uc758 \uc804\uac1c\ub294 \ucd94\uc0c1\ud654\uc778 \ub3d9\uc2dc\uc5d0 \uc801\uc6a9 \ubc94\uc704\ub97c \ub113\ud78c\ub2e4\ub294 \ub73b\uc5d0\uc11c\ub294 \uc77c\ubc18\ud654\ub77c\uace0\ub3c4 \ubd80\ub97c \uc218 \uc788\uc73c\ubbc0\ub85c, \ub450 \uc6a9\uc5b4\uac00 \uc5b8\uc81c\ub098 \ubc30\ud0c0\uc801\uc73c\ub85c \uac08\ub9ac\ub294 \uac83\uc740 \uc544\ub2c8\ub2e4. --> \uc774 \uc5f0\uc7ac\uc5d0\uc11c\ub294 \uc8fc\ub85c \u2018\uc801\uc6a9 \ubc94\uc704\uc758 \ud655\uc7a5\u2019\uacfc \u2018\ud45c\ud604\uc5d0\uc11c \uad6c\uc870\ub97c \ubd84\ub9ac\ud558\ub294 \uc77c\u2019 \uac00\uc6b4\ub370 \uc5b4\ub290 \uce21\uba74\uc744 \uac15\uc870\ud558\ub294\uc9c0\uc5d0 \ub530\ub77c \ub450 \ub9d0\uc744 \uad6c\ubd84\ud574 \uc0ac\uc6a9\ud560 \uac83\uc774\ub2e4. \ubca1\ud130\uacf5\uac04\uc5d0 \uad00\ud55c \uc790\uc138\ud55c \uc774\uc57c\uae30\ub294 <a href=\"..\/math-abstraction-06-vector-spaces\/\">6\ubd80<\/a>\uc5d0\uc11c \uc0b4\ud3b4\ubcf8\ub2e4.<\/li>\n<li>\ub458\uc9f8, <span class=\"defined\">\uc774\uc0c1\ud654<\/span>(idealization)\ub294 \ud604\uc2e4\uc758 \ubcf5\uc7a1\ud55c \ub300\uc0c1\uc744 \ub2e4\ub8e8\uae30 \uc704\ud574 \uc2e4\uc81c \ub300\uc0c1\uacfc \uc758\ub3c4\uc801\uc73c\ub85c \uc5b4\uae0b\ub098\ub294 \uac00\uc815\uc744 \ub3c4\uc785\ud558\ub294 \uacfc\uc815\uc774\ub77c\uace0 \ud558\uc790. \ubb3c\ub9ac\ud559\uc5d0\uc11c \uac00\uc815\ud558\ub294 \ub9c8\ucc30 \uc5c6\ub294 \ud3c9\uba74, \ubd80\ud53c\uac00 \uc5c6\ub294 \uc9c8\uc810, \uc804\uae30 \uc800\ud56d\uc774 \uc5c6\ub294 \ub3c4\uc120 \ub4f1\uc774 \uc774\uc5d0 \uc18d\ud55c\ub2e4. \ud55c \uc804\ud615\uc801\uc778 \uad6c\ubd84\uc5d0 \ub530\ub974\uba74 \ucd94\uc0c1\ud654\ub294 \ub300\uc0c1\uc758 \uc138\ubd80 \uce21\uba74\uc744 \uace0\ub824\uc5d0\uc11c \uc81c\uc678\ud558\ub294 \ub370 \ucd08\uc810\uc744 \ub450\uace0, \uc774\uc0c1\ud654\ub294 \uc2e4\uc81c\uc640 \ub9de\uc9c0 \uc54a\ub294 \uc131\uc9c8\uc744 \ub3c4\uc785\ud558\ub294 \ub370 \ucd08\uc810\uc744 \ub454\ub2e4. <!-- \ub2e4\ub9cc \uc5b4\ub5a4 \ubaa8\ub378\uc744 \ucd94\uc0c1\ud654\ub85c \ubcfc\uc9c0 \uc774\uc0c1\ud654\ub85c \ubcfc\uc9c0\ub294 \ub9e5\ub77d\uc5d0 \ub530\ub77c \ub2ec\ub77c\uc9c8 \uc218 \uc788\uc73c\uba70, \u2018\ucd94\uc0c1\ud654\ub294 \uc5b8\uc81c\ub098 \uadf8\ub300\ub85c \ucc38\uc774\uace0 \uc774\uc0c1\ud654\ub294 \uc5b8\uc81c\ub098 \uadfc\uc0ac\uc801\uc73c\ub85c\ub9cc \ucc38\uc774\ub2e4\u2019\ub77c\uace0 \ub2e8\uc21c\ud558\uac8c \ub098\ub204\uae30\ub294 \uc5b4\ub835\ub2e4<a class=\"math-abs-series-ref\" title=\"Levy, A. (2021). Idealization and abstraction: Refining the distinction. Synthese, 198(Suppl. 24), 5855\u20135872.\" href=\"#ref-Levy2021\">[Levy2021]<\/a>. --><\/li>\n<li>\uc14b\uc9f8, <span class=\"defined\">\ud615\uc2dd\ud654<\/span>(formalization)\ub294 \uba85\uc81c\uc640 \ucd94\ub860\uc744 \uba85\uc2dc\uc801\uc778 \ud615\uc2dd\uc5b8\uc5b4, \uacf5\ub9ac, \ucd94\ub860 \uaddc\uce59\uc73c\ub85c \ud45c\ud604\ud558\uc5ec \uac01 \uc99d\uba85 \ub2e8\uacc4\uac00 \uc815\ud574\uc9c4 \uaddc\uce59\uc744 \ub530\ub974\ub294\uc9c0\ub97c \uad6c\ubb38\uc801\uc73c\ub85c \uac80\uc0ac\ud560 \uc218 \uc788\uac8c \ud558\ub294 \uacfc\uc815\uc774\ub2e4<a class=\"math-abs-series-ref\" title=\"Shapiro, S., &amp; Kouri Kissel, T. (2026). Classical logic. In E. N. Zalta &amp; U. Nodelman (Eds.), The Stanford Encyclopedia of Philosophy.\" href=\"#ref-ShapiroKouriKissel2026\">[ShapiroKouriKissel2026]<\/a>. <!-- \uc774\ub294 \ucd94\ub860\uc744 \ub2e8\uc21c\ud788 \u2018\uae30\uacc4\uc801\uc778 \uae30\ud638 \uc870\uc791\uc73c\ub85c \ud658\uc6d0\ud55c\ub2e4\u2019\ub294 \uc8fc\uc7a5\ubcf4\ub2e4 \uc81c\ud55c\uc801\uc774\uace0 \uc815\ud655\ud55c \ub73b\uc774\ub2e4. --> \ud615\uc2dd\ud654\ub294 \uacf5\ub9ac\ud654\uc640 \ubc00\uc811\ud558\uac8c \uc5f0\uad00\ub418\uc5b4 \uc788\uc73c\ub098 \ub3d9\uc77c\ud55c \uac1c\ub150\uc740 \uc544\ub2c8\ub2e4. \uc774 \ub450 \uac1c\ub150\uc758 \uc5c4\ubc00\ud55c \uad6c\ubd84\uc740 \ud5a5\ud6c4 \uacf5\ub9ac\uc801 \ubc29\ubc95\uc744 \ub2e4\ub8f0 \ub54c \ub2e4\uc2dc \ub17c\uc758\ud560 \uc608\uc815\uc774\uba70, \ucd94\uc0c1\ud654\ub97c \ud3ec\ud568\ud55c \ub124 \uac00\uc9c0 \uac1c\ub150\uc758 \uccb4\uacc4\uc801\uc778 \uc815\ub9ac\ub294 \uc5f0\uc7ac\uc758 \ub9c8\uc9c0\ub9c9 \uae00\uc778 <a href=\"..\/math-abstraction-11-topography\/\">11\ubd80<\/a>\uc5d0\uc11c \ub9c8\ubb34\ub9ac\ud560 \uac83\uc774\ub2e4.<\/li>\n<\/ul>\n<h4>\ub3d9\uce58\ub958\ub97c \uc0ac\uc6a9\ud55c \uc815\uc758<\/h4>\n<p>\uc774\uc81c \ub450 \ubc88\uc9f8 \ubc29\uc2dd\uc778 \u2018\ub3d9\uce58\ub958\ub97c \uc0ac\uc6a9\ud55c \uc815\uc758\u2019\ub97c \uc790\uc138\ud788 \uc0b4\ud3b4\ubcf4\uc790. \ucd9c\ubc1c\uc810\uc73c\ub85c \ub2e4\uc74c\uacfc \uac19\uc740 \uc9c8\ubb38\uc744 \ub358\uc838 \ubcfc \uc218 \uc788\ub2e4. \u201c\uc9c1\uc120\uc758 \ubc29\ud5a5\uc774\ub780 \ubb34\uc5c7\uc778\uac00?\u201d<\/p>\n<p>\uc9c1\uc120\uc758 \ubc29\ud5a5\uc740 \uae30\uc900\ucd95\uc5d0 \ub300\ud55c \uac01\uc73c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\uc9c0\ub9cc, \uadf8 \uac01\ub3c4 \uac12\uc740 \uc5b4\ub5a4 \uae30\uc900\ucd95\uc744 \uc120\ud0dd\ud558\ub290\ub0d0\uc5d0 \uc758\uc874\ud55c\ub2e4. \ub610\ud55c \ud654\uc0b4\ud45c\ub294 \ubc29\ud5a5\uc744 \ub098\ud0c0\ub0b4\ub294 \ud558\ub098\uc758 \uad6c\uccb4\uc801\uc778 \ub300\ud45c\ubb3c\uc774\uc9c0 \ubc29\ud5a5 \uadf8 \uc790\uccb4\ub294 \uc544\ub2c8\ub2e4. \ud504\ub808\uac8c(Gottlob Frege)\ub294 \u300e\uc0b0\uc220\uc758 \uae30\ucd08\u300f(<i>Die Grundlagen der Arithmetik<\/i>)\uc5d0\uc11c \ubc29\ud5a5\uc758 \uace0\ub9bd\ub41c \u2018\ubcf8\uc9c8\u2019\uc744 \uba3c\uc800 \uc815\uc758\ud558\ub294 \ub300\uc2e0, \ub450 \uc9c1\uc120\uc774 \uc5b8\uc81c \uac19\uc740 \ubc29\ud5a5\uc744 \uac16\ub294\uc9c0\ub97c \ucd9c\ubc1c\uc810\uc73c\ub85c \uc0bc\uc558\ub2e4<a class=\"math-abs-series-ref\" title=\"Frege, G. (1884). Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung \u00fcber den Begriff der Zahl. W. Koebner. Internet Archive.\" href=\"#ref-Frege1884\">[Frege1884]<\/a>. \uc9c1\uc120 \\(a\\)\uc758 \ubc29\ud5a5\uc744 \\(\\mathrm{dir}(a)\\)\ub77c\uace0 \uc4f0\uba74,<br \/>\n\\[<br \/>\n\\mathrm{dir}(a)=\\mathrm{dir}(b)\\quad\\Longleftrightarrow\\quad a\\parallel b<br \/>\n\\]<br \/>\n\ub77c\ub294 \ub3d9\uc77c\uc131 \uc870\uac74\uc744 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \uc774\ud558\uc5d0\uc11c\ub294 \\(\\mathcal L\\)\uc744 \uace0\uc815\ub41c \uc720\ud074\ub9ac\ub4dc \ud3c9\uba74\uc5d0 \ub193\uc778 \uc9c1\uc120\ub4e4\uc758 \uc9d1\ud569\uc774\ub77c \ud558\uace0, \\(a\\parallel b\\)\ub294 \ub450 \uc9c1\uc120\uc774 \uc77c\uce58\ud558\uac70\ub098 \ud1b5\uc0c1\uc801\uc778 \uc758\ubbf8\uc5d0\uc11c \ud3c9\ud589\ud558\ub2e4\ub294 \ub73b\uc73c\ub85c \uc0ac\uc6a9\ud55c\ub2e4. \ub2e8, \uc5ec\uae30\uc11c \uc5b4\ub5a4 \uc9c1\uc120\uc740 \ud56d\uc0c1 \uc790\uae30 \uc790\uc2e0\uacfc \ud3c9\ud589\ud558\ub2e4\uace0 \uc57d\uc18d\ud55c\ub2e4. \uc774 \uc870\uac74\uc774 \uc804\uc81c\ub418\uc5b4\uc57c \ud6c4\uc220\ud560 \u2018\ubc18\uc0ac\uc131\u2019\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\ud504\ub808\uac8c\ub294 \uba3c\uc800 \uc704 \uc30d\uc870\uac74\ubb38\uc744 \u2018\ubc29\ud5a5\u2019\uc774\ub77c\ub294 \ud45c\ud604\uc758 <span class=\"defined\">\ubb38\ub9e5\uc801 \uc815\uc758<\/span>(contextual definition)\ub85c \uc0bc\uc744 \uc218 \uc788\ub294\uc9c0 \uac80\ud1a0\ud588\ub2e4<a class=\"math-abs-series-ref\" title=\"Frege, G. (1884). Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung \u00fcber den Begriff der Zahl. W. Koebner. Internet Archive.\" href=\"#ref-Frege1884\">[Frege1884]<\/a>. \uc774 \uc6d0\ub9ac\ub294 \ub450 \ubc29\ud5a5\uc774 \uc5b8\uc81c \uac19\uc740\uc9c0\ub294 \uc54c\ub824 \uc8fc\uc9c0\ub9cc, \ubc29\ud5a5 \ud45c\ud604\uacfc \uc804\ud600 \ub2e4\ub978 \uc885\ub958\uc758 \ud45c\ud604 \uc0ac\uc774\uc758 \ub3d9\uc77c\uc131\uae4c\uc9c0 \uacb0\uc815\ud558\uc9c0\ub294 \ubabb\ud55c\ub2e4. \uc608\ucee8\ub300 \u2018\uc9c0\uad6c\ucd95\uc758 \ubc29\ud5a5\uc774 \uc601\uad6d\uacfc \uac19\uc740\uac00\u2019\uc640 \uac19\uc740 \ubb38\uc7a5\uc740 \uc774 \uc6d0\ub9ac\ub9cc\uc73c\ub85c \ucc38\uc778\uc9c0 \uac70\uc9d3\uc778\uc9c0 \ud310\uc815\ub418\uc9c0 \uc54a\ub294\ub2e4. \ud504\ub808\uac8c\ub294 \uc774\ub7ec\ud55c \ubb38\uc81c\uac00 \uc815\uc758\uc758 \ubd88\uc644\uc804\uc131\uc744 \ub4dc\ub7ec\ub0b8\ub2e4\uace0 \ubcf4\uc558\ub2e4<a class=\"math-abs-series-ref\" title=\"Frege, G. (1884). Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung \u00fcber den Begriff der Zahl. W. Koebner. Internet Archive.\" href=\"#ref-Frege1884\">[Frege1884]<\/a>.<\/p>\n<p>\uc774\uc5d0 \ud504\ub808\uac8c\ub294 \uc9c1\uc120 \\(a\\)\uc758 \ubc29\ud5a5\uc744 \u2018\\(a\\)\uc640 \ud3c9\ud589\ud55c \uc9c1\uc120\u2019\uc774\ub77c\ub294 <span class=\"defined\">\uac1c\ub150\uc758 \uc678\uc5f0<\/span>\uc73c\ub85c \uba85\uc2dc\uc801\uc73c\ub85c \uc815\uc758\ud588\ub2e4<a class=\"math-abs-series-ref\" title=\"Frege, G. (1884). Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung \u00fcber den Begriff der Zahl. W. Koebner. Internet Archive.\" href=\"#ref-Frege1884\">[Frege1884]<\/a>. \uc774\ub97c \ud604\ub300 \uc9d1\ud569\ub860\uc758 \uc5b8\uc5b4\ub85c \ub098\ud0c0\ub0b4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[<br \/>\n\\mathrm{dir}(a):=[a]_{\\parallel}=\\{b\\in\\mathcal L:b\\parallel a\\}.<br \/>\n\\]<br \/>\n\uc774\uc81c \ubc29\ud5a5\uc740 \\(\\mathcal L\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc73c\ub85c \uc8fc\uc5b4\uc9c0\ubbc0\ub85c, \ub450 \ubc29\ud5a5\uc758 \ub3d9\uc77c\uc131\uc740 \uc9d1\ud569\uc758 \ub3d9\uc77c\uc131\uc73c\ub85c \ud310\uc815\ud560 \uc218 \uc788\ub2e4. <!-- \ub2e4\ub9cc \ud504\ub808\uac8c \uc790\uc2e0\uc758 \u2018\uac1c\ub150\uc758 \uc678\uc5f0\u2019\uacfc \ud604\ub300 \uc9d1\ud569\ub860\uc758 \uc9d1\ud569\uc744 \uc5ed\uc0ac\uc801\uc73c\ub85c \uadf8\ub300\ub85c \ub3d9\uc77c\uc2dc\ud574\uc11c\ub294 \uc548 \ub41c\ub2e4. \uc704 \uc2dd\uc740 \ud504\ub808\uac8c\uc758 \uc81c\uc548\uc744 \uc624\ub298\ub0a0\uc758 \ubaab\uc9d1\ud569 \uc5b8\uc5b4\ub85c \uc7ac\uad6c\uc131\ud55c \ud45c\ud604\uc774\ub2e4. --><\/p>\n<p>\uc5ec\uae30\uc11c\ub294 \ub450 \uce35\uc704\ub97c \uad6c\ubd84\ud560 \ud544\uc694\uac00 \uc788\ub2e4. \uc77c\ubc18\uc801\uc73c\ub85c<br \/>\n\\[<br \/>\nA(x)=A(y)\\quad\\Longleftrightarrow\\quad x\\sim y<br \/>\n\\]<br \/>\n\uc640 \uac19\uc774 \uc0c8 \ud45c\ud604 \\(A(x)\\)\uc758 \ub3d9\uc77c\uc131 \uc870\uac74\uc744 \uc81c\uc2dc\ud558\ub294 \uc6d0\ub9ac\ub97c <span class=\"defined\">\ucd94\uc0c1\ud654 \uc6d0\ub9ac<\/span>(abstraction principle) \ub610\ub294 <span class=\"defined\">\ucd94\uc0c1\uc5d0 \uc758\ud55c \uc815\uc758<\/span>(definition by abstraction)\ub77c\uace0 \ubd80\ub978\ub2e4. \ubc18\uba74 \\(A(x)=[x]_{\\sim}\\)\ub85c \ub450\uc5b4 \uc0c8 \ub300\uc0c1\uc744 \uc2e4\uc81c \ub3d9\uce58\ub958\ub85c \uad6c\ud604\ud558\ub294 \uac83\uc740 \ud604\ub300 \uc9d1\ud569\ub860\uc758 <span class=\"defined\">\ubaab\uc9d1\ud569 \uad6c\uc131<\/span>\uc774\ub2e4. \uc774 \ub458\uc740 \ubc00\uc811\ud558\uac8c \uc5f0\uacb0\ub418\uc9c0\ub9cc \ub17c\ub9ac\uc801\uc73c\ub85c \uac19\uc740 \ub2e8\uacc4\ub294 \uc544\ub2c8\uba70, \u2018definition by abstraction\u2019\uc774\ub77c\ub294 \uc804\ubb38 \uc6a9\uc5b4\ub294 \ud504\ub808\uac8c \uc774\ud6c4 19\uc138\uae30 \ub9d0 \ud398\uc544\ub178 \ud559\ud30c\uc758 \ub17c\uc758\ub97c \uac70\uce58\uba70 \uc815\ucc29\ud588\ub2e4<a class=\"math-abs-series-ref\" title=\"Mancosu, P. (2016). Abstraction and Infinity. Oxford University Press. Publisher page.\" href=\"#ref-Mancosu2016\">[Mancosu2016]<\/a>.<\/p>\n<p>\ud504\ub808\uac8c\uac00 \ubc29\ud5a5\uc758 \uc0ac\ub840\ub97c \uac80\ud1a0\ud55c \uadfc\ubcf8\uc801\uc778 \ubaa9\uc801\uc740 <span class=\"defined\">\uae30\uc218<\/span>(cardinal number)\ub97c \uc5c4\ubc00\ud788 \uc815\uc758\ud558\ub294 \ub370 \uc788\uc5c8\ub2e4. \ub450 \uac1c\ub150 \\(F\\)\uc640 \\(G\\)\uc5d0 \uc18d\ud558\ub294 \ub300\uc0c1\ub4e4 \uc0ac\uc774\uc5d0 \uc77c\ub300\uc77c \ub300\uc751\uc774 \uc874\uc7ac\ud560 \ub54c \ub450 \uac1c\ub150\uc774 \ub3d9\uc218(equinumerous)\ub77c\uace0 \ud558\uace0 \\(F\\approx G\\)\ub85c \uc4f0\uba74,<br \/>\n\\[<br \/>\n\\#F=\\#G\\quad\\Longleftrightarrow\\quad F\\approx G<br \/>\n\\]<br \/>\n\ub77c\ub294 \ucd94\uc0c1\ud654 \uc6d0\ub9ac\ub97c \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ub098 \uc774 \uc6d0\ub9ac\ub9cc\uc73c\ub85c\ub294 \uc218\uac00 \uc544\ub2cc \ub300\uc0c1\uacfc \uc218 \uc0ac\uc774\uc758 \ub3d9\uc77c\uc131 \ubb38\uc81c\uac00 \ub0a8\ub294\ub2e4. \uadf8\ub798\uc11c \ud504\ub808\uac8c\ub294 \\(F\\)\uc5d0 \uc18d\ud558\ub294 \uae30\uc218\ub97c \u2018\\(F\\)\uc640 \ub3d9\uc218\uc778 \uac1c\ub150\u2019\uc774\ub77c\ub294 \uac1c\ub150\uc758 \uc678\uc5f0\uc73c\ub85c \uba85\uc2dc\uc801\uc73c\ub85c \uc815\uc758\ud588\ub2e4<a class=\"math-abs-series-ref\" title=\"Frege, G. (1884). Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung \u00fcber den Begriff der Zahl. W. Koebner. Internet Archive.\" href=\"#ref-Frege1884\">[Frege1884]<\/a>. \uc774\ub7ec\ud55c \ub17c\ub9ac\uc8fc\uc758\uc801 \uc2dc\ub3c4\uac00 \uc5b4\ub5a4 \uacb0\ub9d0\uc744 \ub9fa\uc5c8\ub294\uc9c0\ub294 \uc5f0\uc7ac\uc758 \ub9c8\uc9c0\ub9c9 \uae00\uc778 <a href=\"..\/math-abstraction-11-topography\/\">11\ubd80<\/a>\uc5d0\uc11c \uc0c1\uc138\ud788 \ub2e4\ub8f0 \uac83\uc774\ub2e4.<\/p>\n<h4>\ub3d9\uce58\uad00\uacc4\uc640 \ubd84\ud560<\/h4>\n<p>\uc774\uc81c \uc55e\uc11c \ubc29\ud5a5\uc758 \uc608\uc5d0\uc11c \uc218\ud589\ud55c \uc870\uc791\uc744 \uc218\ud559\uc801\uc73c\ub85c \uc77c\ubc18\ud654\ud574 \ubcf4\uc790. \uc774\uac83\uc744 \uc704\ud574 \uac00\uc7a5 \uba3c\uc800 \ud544\uc694\ud55c \uac83\uc740 \ub300\uc0c1\ub4e4\uc744 \u2018\uac19\ub2e4\uace0 \uac04\uc8fc\ud558\uae30\u2019 \uc704\ud574 \uad00\uacc4\uac00 \ub9cc\uc871\uc2dc\ucf1c\uc57c \ud560 \uc870\uac74\ub4e4\uc744 \uba85\ud655\ud788 \uaddc\uc815\ud558\ub294 \uc77c\uc774\ub2e4.<\/p>\n<p>\uc5c4\ubc00\ud788 \ub9d0\ud574 \uc9d1\ud569 \\(X\\) \uc704\uc758 \uc774\ud56d\uad00\uacc4 \\(R\\)\uc740 \uacf1\uc9d1\ud569 \\(X\\times X\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uba70, \\(x\\mathrel{R}y\\)\ub77c\ub294 \ud45c\uae30\ub294 \uc21c\uc11c\uc30d \\((x,y)\\)\uac00 \\(R\\)\uc758 \uc6d0\uc18c\ub77c\ub294 \ub73b\uc774\ub2e4. \uc9d1\ud569 \\(X\\) \uc704\uc758 \uad00\uacc4 \\(\\sim\\)\uac00 \ub2e4\uc74c \uc138 \uac00\uc9c0 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0ac \ub54c, \uc774 \uad00\uacc4\ub97c \\(X\\) \uc0c1\uc758 <span class=\"defined\">\ub3d9\uce58\uad00\uacc4<\/span>(equivalence relation)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<ul>\n<li>\uc784\uc758\uc758 \uc6d0\uc18c \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\sim x\\)\uc774\ub2e4. (<span class=\"defined\">\ubc18\uc0ac\uc131<\/span>)<\/li>\n<li>\\(x\\sim y\\)\uc774\uba74 \\(y\\sim x\\)\uc774\ub2e4. (<span class=\"defined\">\ub300\uce6d\uc131<\/span>)<\/li>\n<li>\\(x\\sim y\\)\uc774\uace0 \\(y\\sim z\\)\uc774\uba74 \\(x\\sim z\\)\uc774\ub2e4. (<span class=\"defined\">\ucd94\uc774\uc131<\/span>)<\/li>\n<\/ul>\n<p>\uc774 \uc870\uac74\ub4e4\uc740 \uc77c\uc0c1\uc5d0\uc11c \uc6b0\ub9ac\uac00 \u2018\uac19\ub2e4\u2019\ub77c\ub294 \uac1c\ub150\uc5d0 \uc554\ubb35\uc801\uc73c\ub85c \uc694\uad6c\ud558\ub294 \uc131\uc9c8\ub4e4\uc744 \uc218\ud559\uc801\uc73c\ub85c \ud615\uc2dd\ud654\ud55c \uac83\uc774\ub2e4. \uc55e\uc11c \uc815\ud55c \uc57d\uc18d \uc544\ub798\uc5d0\uc11c, \uace0\uc815\ub41c \uc720\ud074\ub9ac\ub4dc \ud3c9\uba74\uc758 \uc9c1\uc120\ub4e4 \uc0ac\uc774\uc758 \u2018\ud3c9\ud589\u2019\uc740 \ub3d9\uce58\uad00\uacc4\ub2e4. \ub450 \ud559\uc0dd\uc774 \u2018\uac19\uc740 \ud559\ub144\uc5d0 \uc18d\ud55c\ub2e4\u2019\ub77c\ub294 \uad00\uacc4\ub098, \uc815\uc218\ub860\uc5d0\uc11c \u2018\ub450 \uc815\uc218\ub97c 6\uc73c\ub85c \ub098\ub208 \ub098\uba38\uc9c0\uac00 \uac19\ub2e4(\ud569\ub3d9\uc774\ub2e4)\u2019\ub77c\ub294 \uad00\uacc4 \uc5ed\uc2dc \ub3d9\uce58\uad00\uacc4\uc5d0 \ud574\ub2f9\ud55c\ub2e4. \ubc18\uba74 \uc2e4\uc218\uc758 \ub300\uc18c \uad00\uacc4\uc778 \u201c\\(x\\)\uac00 \\(y\\)\ubcf4\ub2e4 \uc791\uac70\ub098 \uac19\ub2e4(\\(x\\le y\\))\u201d\ub294 \ubc18\uc0ac\uc131\uacfc \ucd94\uc774\uc131\uc744 \ub9cc\uc871\uc2dc\ud0a4\uc9c0\ub9cc \ub300\uce6d\uc131\uc740 \ub9cc\uc871\uc2dc\ud0a4\uc9c0 \uc54a\uc73c\ubbc0\ub85c \ub3d9\uce58\uad00\uacc4\uac00 \uc544\ub2c8\ub2e4.<\/p>\n<p>\uc9d1\ud569 \\(X\\) \uc0c1\uc5d0 \ub3d9\uce58\uad00\uacc4 \\(\\sim\\)\uac00 \uc8fc\uc5b4\uc9c8 \ub54c, \ud2b9\uc815 \uc6d0\uc18c \\(x\\in X\\)\uc640 \ub3d9\uce58\uc778 \ubaa8\ub4e0 \uc6d0\uc18c\ub4e4\uc744 \ubaa8\uc740 \ubd80\ubd84\uc9d1\ud569<br \/>\n\\[ [x]=\\{y\\in X:x\\sim y\\} \\]<br \/>\n\uc744 \\(x\\)\ub97c \ub300\ud45c\uc6d0\uc73c\ub85c \ud558\ub294 <span class=\"defined\">\ub3d9\uce58\ub958<\/span>(equivalence class)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989, \uad00\uacc4 \\(\\sim\\) \ud558\uc5d0\uc11c \\(x\\)\uc640 \u2018\uac19\ub2e4\u2019\uace0 \ucde8\uae09\ub418\ub294 \ub300\uc0c1 \uc804\uccb4\uc758 \uc9d1\ud569\uc774\ub2e4. \uadf8\ub9ac\uace0 \uc774\ub7ec\ud55c \ub3d9\uce58\ub958\ub4e4\uc744 \uc6d0\uc18c\ub85c \uac00\uc9c0\ub294 \uc0c8\ub85c\uc6b4 \uc9d1\ud569<br \/>\n\\[ X\/{\\sim}=\\{[x]:x\\in X\\} \\]<br \/>\n\uc744 \\(X\\)\uc758 \\(\\sim\\)\uc5d0 \ub300\ud55c <span class=\"defined\">\ubaab\uc9d1\ud569<\/span>(quotient set) \ub610\ub294 <span class=\"defined\">\uc0c1\uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\uc774\uc640 \ubcc4\uac1c\ub85c, \uc9d1\ud569 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569\ub4e4\uc744 \ubaa8\uc544\ub193\uc740 \uc871(family) \\(\\mathcal P\\)\uac00 \ub2e4\uc74c \uc138 \uac00\uc9c0 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0ac \ub54c, \\(\\mathcal P\\)\ub97c \\(X\\)\uc758 <span class=\"defined\">\ubd84\ud560<\/span>(partition)\uc774\ub77c\uace0 \uc815\uc758\ud55c\ub2e4. \uc9c1\uad00\uc801\uc778 \uc774\ud574\ub97c \ub3d5\uae30 \uc704\ud574 \\(\\mathcal P\\)\uc5d0 \uc18d\ud55c \uac01\uac01\uc758 \ubd80\ubd84\uc9d1\ud569\uc744 \ud558\ub098\uc758 \u2018\uc870\uac01(block)\u2019\uc774\ub77c\uace0 \ubd80\ub974\uae30\ub85c \ud558\uc790.<\/p>\n<ul>\n<li>\uac01 \uc870\uac01\uc740 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\ub2e4.<\/li>\n<li>\uc11c\ub85c \ub2e4\ub978 \ub450 \uc870\uac01\uc740 \uc11c\ub85c\uc18c\uc774\ub2e4.<\/li>\n<li>\uc870\uac01 \uc804\uccb4\uc758 \ud569\uc9d1\ud569\uc740 \\(X\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc774 \ubd84\ud560\uc758 \uc815\uc758\ub97c \uc0ac\uc6a9\ud558\uc5ec \uace7\ubc14\ub85c \uc720\ub3c4\ud560 \uc218 \uc788\ub294 \uc911\uc694\ud55c \uc131\uc9c8\uc774 \uc788\ub2e4. \ubc14\ub85c \uc9d1\ud569 \\(X\\)\uc758 \uc784\uc758\uc758 \uc6d0\uc18c\ub294 \u2018\uc815\ud655\ud788 \ub2e8 \ud558\ub098\uc758\u2019 \uc870\uac01\uc5d0\ub9cc \uc18d\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc774\ub2e4. \uc804\uccb4 \ud569\uc9d1\ud569\uc774 \\(X\\)\uac00 \ub418\uc5b4\uc57c \ud558\ubbc0\ub85c \ubaa8\ub4e0 \uc6d0\uc18c\ub294 \uc801\uc5b4\ub3c4 \ud558\ub098\uc758 \uc870\uac01\uc5d0 \uc18d\ud574\uc57c \ud558\uba70, \uc11c\ub85c \ub2e4\ub978 \uc870\uac01\ub4e4\uc740 \uc11c\ub85c\uc18c\uc5ec\uc57c \ud558\ubbc0\ub85c \uc5b4\ub5a4 \uc6d0\uc18c\ub3c4 \ub450 \uac1c \uc774\uc0c1\uc758 \uc870\uac01\uc5d0 \ub3d9\uc2dc\uc5d0 \uc18d\ud560 \uc218 \uc5c6\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n<p>\ud45c\uba74\uc801\uc778 \uc815\uc758\ub9cc \ub193\uace0 \ubcf4\uba74 \ub3d9\uce58\uad00\uacc4\uc640 \ubd84\ud560\uc740 \uc11c\ub85c \ubb34\uad00\ud55c \uac1c\ub150\ucc98\ub7fc \ubcf4\uc778\ub2e4. \ub3d9\uce58\uad00\uacc4\uac00 \ub450 \ub300\uc0c1 \uc0ac\uc774\uc758 \uad00\uacc4\uc131\uc744 \ud310\ubcc4\ud558\ub294 \uae30\uc900\uc774\ub77c\uba74, \ubd84\ud560\uc740 \uac70\uc2dc\uc801\uc73c\ub85c \uc9d1\ud569 \uc804\uccb4\ub97c \uc5ec\ub7ec \uad6c\uc5ed\uc73c\ub85c \ub098\ub204\ub294 \uad6c\uc870\uc801 \ubc29\uc2dd\uc774\uae30 \ub54c\ubb38\uc774\ub2e4. \uadf8\ub7ec\ub098 \ub2e4\uc74c \uc815\ub9ac\ub294 \uc774 \ub450 \uac1c\ub150\uc774 \uc218\ud559\uc801\uc73c\ub85c \ud558\ub098\uac00 \ub2e4\ub978 \ud558\ub098\ub97c \uc644\ubcbd\ud788 \uacb0\uc815\ud568\uc744 \ubcf4\uc5ec\uc900\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p>\n<span class=\"theorem\">\uc815\ub9ac: \ub3d9\uce58\uad00\uacc4\uc640 \ubd84\ud560\uc758 \ub300\uc751<\/span><\/p>\n<p>\uc9d1\ud569 \\(X\\) \uc0c1\uc758 \ub3d9\uce58\uad00\uacc4 \\(\\sim\\)\uac00 \uc8fc\uc5b4\uc9c0\uba74, \uadf8\uc5d0 \ub530\ub978 \ubaab\uc9d1\ud569 \\(X\/{\\sim}\\)\ub294 \\(X\\)\uc758 \ubd84\ud560\uc774 \ub41c\ub2e4. \uc5ed\uc73c\ub85c, \uc9d1\ud569 \\(X\\)\uc758 \uc784\uc758\uc758 \ubd84\ud560 \\(\\mathcal P\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \u201c\\(x\\)\uc640 \\(y\\)\uac00 \ub3d9\uc77c\ud55c \uc870\uac01\uc5d0 \uc18d\ud55c\ub2e4\u201d\ub77c\uace0 \uc815\uc758\ub41c \uad00\uacc4\ub294 \\(X\\) \uc0c1\uc758 \ub3d9\uce58\uad00\uacc4\uac00 \ub41c\ub2e4. \ub098\uc544\uac00 \uc774 \ub450 \ub300\uc751\uc740 \uc11c\ub85c \uc5ed(inverse)\uc758 \uad00\uacc4\uc5d0 \uc788\ub2e4. \uc989, \uc9d1\ud569 \\(X\\) \uc0c1\uc758 \ub3d9\uce58\uad00\uacc4 \uc804\uccb4\uc758 \uc9d1\ud569\uacfc \\(X\\)\uc758 \ubd84\ud560 \uc804\uccb4\uc758 \uc9d1\ud569 \uc0ac\uc774\uc5d0\ub294 \uc790\uc5f0\uc2a4\ub7ec\uc6b4 \uc77c\ub300\uc77c \ub300\uc751(\uc804\ub2e8\uc0ac)\uc774 \uc874\uc7ac\ud55c\ub2e4.\n<\/p>\n<\/div>\n<p>\uc99d\uba85\uc740 \uc138 \ub2e8\uacc4\ub85c \ub098\ub204\uc5b4 \uc9c4\ud589\ud55c\ub2e4. \uccab\uc9f8, \ub3d9\uce58\uad00\uacc4\ub85c\ubd80\ud130 \uc5bb\uc740 \ubaab\uc9d1\ud569\uc774 \uc2e4\uc81c\ub85c \ubd84\ud560\uc758 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294\uc9c0 \ud655\uc778\ud55c\ub2e4. \uad00\uacc4\uc758 \ubc18\uc0ac\uc131\uc5d0 \uc758\ud574 \uc784\uc758\uc758 \uc6d0\uc18c \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\sim x\\)\uac00 \uc131\ub9bd\ud558\ubbc0\ub85c \\(x\\in[x]\\)\uc774\ub2e4. \ub530\ub77c\uc11c \uc5b4\ub5a4 \ub3d9\uce58\ub958\ub3c4 \uacf5\uc9d1\ud569\uc774 \ub420 \uc218 \uc5c6\uc73c\uba70, \ubaa8\ub4e0 \ub3d9\uce58\ub958\ub4e4\uc744 \ud569\uc9d1\ud569\ud558\uba74 \uc804\uccb4\uc9d1\ud569 \\(X\\)\uac00 \ub428\uc740 \uc790\uba85\ud558\ub2e4. \ub0a8\uc740 \uac83\uc740 \uc11c\ub85c \ub2e4\ub978 \ub450 \ub3d9\uce58\ub958\uac00 \ud56d\uc0c1 \uc11c\ub85c\uc18c\uc784\uc744 \ubcf4\uc774\uba74 \ub41c\ub2e4. \uc774\ub97c \uc704\ud574 \ub300\uc6b0\uba85\uc81c, \uc989 \u201c\ub450 \ub3d9\uce58\ub958 \\([x]\\)\uc640 \\([y]\\)\uac00 \uacf5\ud1b5 \uc6d0\uc18c\ub97c \uac00\uc9c0\uba74 \uc0ac\uc2e4\uc0c1 \\([x]=[y]\\)\uc774\ub2e4\u201d\ub77c\ub294 \uba85\uc81c\ub97c \uc99d\uba85\ud558\uc790. \uc5b4\ub5a4 \uc6d0\uc18c \\(z\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(z\\in[x]\\cap[y]\\)\ub77c\uace0 \uac00\uc815\ud574 \ubcf4\uc790. \ub3d9\uce58\ub958\uc758 \uc815\uc758\uc5d0 \uc758\ud574 \\(x\\sim z\\)\uc774\uace0 \ub3d9\uc2dc\uc5d0 \\(y\\sim z\\)\uc774\ub2e4. \ub300\uce6d\uc131\uc5d0 \uc758\ud574 \\(z\\sim y\\)\uac00 \uc131\ub9bd\ud558\uace0, \ub2e4\uc2dc \ucd94\uc774\uc131\uc744 \uc801\uc6a9\ud558\uba74 \\(x\\sim z\\)\uc640 \\(z\\sim y\\)\ub85c\ubd80\ud130 \\(x\\sim y\\)\ub97c \uc5bb\ub294\ub2e4. \uc774\uc81c \\([y]\\subset[x]\\)\uc784\uc744 \ubcf4\uc774\uae30 \uc704\ud574 \uc784\uc758\uc758 \\(w\\in[y]\\)\ub97c \ud0dd\ud558\uc790. \uc815\uc758\uc0c1 \\(y\\sim w\\)\uac00 \uc131\ub9bd\ud558\ub294\ub370, \uc55e\uc11c \uc5bb\uc740 \\(x\\sim y\\)\uc5d0 \ucd94\uc774\uc131\uc744 \uc801\uc6a9\ud558\uba74 \\(x\\sim w\\)\uac00 \ub3c4\ucd9c\ub418\ubbc0\ub85c \\(w\\in[x]\\)\uac00 \ub41c\ub2e4. \ub530\ub77c\uc11c \\([y]\\subset[x]\\)\uc774\ub2e4. \ub300\uce6d\uc131\uc5d0 \uc758\ud574 \\(y\\sim x\\) \uc5ed\uc2dc \uc131\ub9bd\ud558\ubbc0\ub85c, \\(x\\)\uc640 \\(y\\)\uc758 \uc5ed\ud560\uc744 \ubc14\uafb8\uc5b4 \ub3d9\uc77c\ud55c \ub17c\uc99d\uc744 \uc804\uac1c\ud558\uba74 \\([x]\\subset[y]\\)\ub3c4 \ub3c4\ucd9c\ub41c\ub2e4. \uacb0\uacfc\uc801\uc73c\ub85c \\([x]=[y]\\)\uac00 \uc99d\uba85\ub418\uc5c8\ub2e4.<\/p>\n<p>\ub458\uc9f8, \uc5ed\ubc29\ud5a5\uc758 \uad6c\uc131\uc744 \ud655\uc778\ud55c\ub2e4. \uc784\uc758\uc758 \ubd84\ud560 \\(\\mathcal P\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \u201c\ub450 \uc6d0\uc18c \\(x, y\\)\ub97c \ubaa8\ub450 \ud3ec\ud568\ud558\ub294 \uc870\uac01 \\(A\\in\\mathcal P\\)\uac00 \uc874\uc7ac\ud558\uba74 \\(x\\approx y\\)\uc774\ub2e4\u201d\ub77c\uace0 \uc0c8\ub85c\uc6b4 \uc774\ud56d\uad00\uacc4 \\(\\approx\\)\ub97c \uc815\uc758\ud558\uc790. \uc774 \uad00\uacc4\uac00 \ub3d9\uce58\uad00\uacc4\uc758 \uc138 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294\uc9c0 \uc0b4\ud3b4\ubcf4\uc544\uc57c \ud55c\ub2e4. \ubd84\ud560\uc758 \ud2b9\uc131\uc0c1 \uac01 \uc6d0\uc18c \\(x\\)\ub294 \ubc18\ub4dc\uc2dc \uc5b4\ub5a4 \uc720\uc77c\ud55c \uc870\uac01\uc5d0 \ud3ec\ud568\ub418\ubbc0\ub85c, \uadf8 \uc870\uac01 \uc548\uc5d0\ub294 \ub2f9\uc5f0\ud788 \\(x\\) \uc790\uc2e0\ub3c4 \ud3ec\ud568\ub41c\ub2e4. \ub530\ub77c\uc11c \\(x\\approx x\\)\uac00 \uc131\ub9bd\ud558\uc5ec \ubc18\uc0ac\uc131\uc744 \ub9cc\uc871\ud55c\ub2e4. \ub300\uce6d\uc131 \uc5ed\uc2dc \uc815\uc758 \uc790\uccb4\uac00 \\(x\\)\uc640 \\(y\\)\uc5d0 \ub300\ud574 \ub300\uce6d\uc801\uc774\ubbc0\ub85c \uc790\uba85\ud558\uac8c \uc131\ub9bd\ud55c\ub2e4. \ucd94\uc774\uc131\uc758 \uacbd\uc6b0, \\(x\\approx y\\)\uc774\uace0 \\(y\\approx z\\)\ub77c\uace0 \uac00\uc815\ud574 \ubcf4\uc790. \uc815\uc758\uc5d0 \ub530\ub77c \\(x, y\\in A\\)\ub97c \ub9cc\uc871\ud558\ub294 \uc870\uac01 \\(A\\)\uc640 \\(y, z\\in B\\)\ub97c \ub9cc\uc871\ud558\ub294 \uc870\uac01 \\(B\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c \uc6d0\uc18c \\(y\\)\uac00 \\(A\\)\uc640 \\(B\\)\uc758 \uad50\uc9d1\ud569\uc5d0 \uc18d\ud558\uac8c \ub418\ubbc0\ub85c \ub450 \uc870\uac01 \\(A, B\\)\ub294 \uc11c\ub85c\uc18c\uac00 \uc544\ub2c8\ub2e4. \ubd84\ud560\uc758 \uc870\uac74\uc5d0\uc11c \uc11c\ub85c \ub2e4\ub978 \ub450 \uc870\uac01\uc740 \ubc18\ub4dc\uc2dc \uc11c\ub85c\uc18c\uc5ec\uc57c \ud558\ubbc0\ub85c, \uacb0\ub860\uc801\uc73c\ub85c \\(A=B\\)\uc77c \uc218\ubc16\uc5d0 \uc5c6\ub2e4. \ub530\ub77c\uc11c \\(x\\)\uc640 \\(z\\)\ub294 \ubaa8\ub450 \uac19\uc740 \uc870\uac01 \\(A\\)\uc5d0 \uc18d\ud558\uac8c \ub418\uc5b4 \\(x\\approx z\\)\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\uc14b\uc9f8, \uc774 \ub450 \uc0dd\uc131 \uacfc\uc815\uc774 \uc11c\ub85c \uc5ed\uc758 \uad00\uacc4\uc784\uc744 \uc785\uc99d\ud55c\ub2e4. \uba3c\uc800 \uc5b4\ub5a4 \ub3d9\uce58\uad00\uacc4 \\(\\sim\\)\uc5d0\uc11c \ucd9c\ubc1c\ud558\uc5ec \ubd84\ud560\uc778 \ubaab\uc9d1\ud569 \\(X\/{\\sim}\\)\ub97c \uad6c\uc131\ud558\uace0, \uc774 \ubd84\ud560\ub85c\ubd80\ud130 \uc55e\uc11c \uc815\uc758\ud55c \ubc29\uc2dd\ub300\ub85c \uc0c8\ub85c\uc6b4 \uad00\uacc4 \\(\\approx\\)\ub97c \uc5bb\uc5c8\ub2e4\uace0 \ud558\uc790. \uc815\uc758\uc0c1 \\(x\\approx y\\)\ub77c\ub294 \uac83\uc740 \ub450 \uc6d0\uc18c\ub97c \ub3d9\uc2dc\uc5d0 \uac00\uc9c0\ub294 \uc5b4\ub5a4 \ub3d9\uce58\ub958\uac00 \uc874\uc7ac\ud55c\ub2e4\ub294 \ub73b\uc774\ub2e4. \ub9cc\uc57d \uc6d0\ub798\uc758 \uad00\uacc4\uc5d0\uc11c \\(x\\sim y\\)\uc600\ub2e4\uba74 \ub450 \uc6d0\uc18c\ub294 \ubaa8\ub450 \ub3d9\uce58\ub958 \\([x]\\)\uc5d0 \uc18d\ud558\ubbc0\ub85c \uc790\uc5f0\uc2a4\ub7fd\uac8c \\(x\\approx y\\)\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc5ed\uc73c\ub85c \\(x\\approx y\\)\ub77c\uba74 \uc5b4\ub5a4 \uc6d0\uc18c \\(z\\)\ub97c \ub300\ud45c\uc6d0\uc73c\ub85c \ud558\ub294 \ub3d9\uce58\ub958\uc5d0 \ub300\ud574 \\(x, y\\in[z]\\)\uac00 \ub41c\ub2e4\ub294 \uc758\ubbf8\uc774\ubbc0\ub85c, \\(z\\sim x\\)\uc774\uace0 \\(z\\sim y\\)\uc774\ub2e4. \ub300\uce6d\uc131\uacfc \ucd94\uc774\uc131\uc744 \uc801\uc6a9\ud558\uba74 \uacb0\uacfc\uc801\uc73c\ub85c \\(x\\sim y\\)\ub97c \uc5bb\ub294\ub2e4. \uc989, \uc0c8\ub85c \uc720\ub3c4\ub41c \uad00\uacc4 \\(\\approx\\)\ub294 \uc6d0\ub798\uc758 \ub3d9\uce58\uad00\uacc4 \\(\\sim\\)\uc640 \uc644\ubcbd\ud788 \uc77c\uce58\ud55c\ub2e4.<\/p>\n<p>\ubc18\ub300 \ubc29\ud5a5\ub3c4 \ub9c8\ucc2c\uac00\uc9c0\ub2e4. \ud2b9\uc815\ud55c \ubd84\ud560 \\(\\mathcal P\\)\uc5d0\uc11c \ucd9c\ubc1c\ud558\uc5ec \uad00\uacc4 \\(\\approx\\)\ub97c \uc815\uc758\ud588\uc744 \ub54c, \uc774 \uad00\uacc4\ub85c\ubd80\ud130 \ud30c\uc0dd\ub41c \ubaab\uc9d1\ud569 \\(X\/{\\approx}\\)\uac00 \uc6d0\ub798\uc758 \ubd84\ud560 \\(\\mathcal P\\)\uc640 \uc77c\uce58\ud568\uc744 \ubcf4\uc774\uba74 \ub41c\ub2e4. (\uc774\ud558\uc758 \ub17c\uc99d\uc5d0\uc11c \\([x]\\)\ub294 \uc720\ub3c4\ub41c \uad00\uacc4 \\(\\approx\\)\uc5d0 \ub300\ud55c \ub3d9\uce58\ub958\ub97c \uc758\ubbf8\ud55c\ub2e4.) \uc784\uc758\uc758 \uc6d0\uc18c \\(x\\in X\\)\uc5d0 \ub300\ud574 \\(x\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc720\uc77c\ud55c \uc870\uac01\uc744 \\(A\\in\\mathcal P\\)\ub77c\uace0 \ud558\uc790. \uc5b4\ub5a4 \uc6d0\uc18c \\(y\\)\uac00 \\(A\\)\uc5d0 \uc18d\ud55c\ub2e4\uba74, \uc870\uac01 \\(A\\)\ub294 \\(x\\)\uc640 \\(y\\)\ub97c \ubaa8\ub450 \uc6d0\uc18c\ub85c \uac00\uc9c0\ubbc0\ub85c \\(x\\approx y\\)\uac00 \ub418\uace0, \uc774\uac83\uc740 \uace7 \\(y\\in[x]\\)\uc784\uc744 \ub73b\ud55c\ub2e4. \uc5ed\uc73c\ub85c \\(y\\in[x]\\)\ub77c\uba74 \\(x\\approx y\\)\uc774\ubbc0\ub85c \\(x\\)\uc640 \\(y\\)\ub97c \ub3d9\uc2dc\uc5d0 \uc6d0\uc18c\ub85c \uac00\uc9c0\ub294 \uc5b4\ub5a4 \uc870\uac01 \\(B\\)\uac00 \uc874\uc7ac\ud574\uc57c \ud55c\ub2e4. \uc774\ub54c \uc6d0\uc18c \\(x\\)\uac00 \uc870\uac01 \\(A\\)\uc640 \\(B\\)\uc5d0 \ubaa8\ub450 \uc18d\ud558\uac8c \ub418\uc5b4 \ub450 \uc870\uac01\uc758 \uad50\uc9d1\ud569\uc774 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uac8c \ub418\ubbc0\ub85c \ubd84\ud560\uc758 \uc870\uac74\uc5d0 \uc758\ud574 \\(A=B\\)\uac00 \uc131\ub9bd\ud558\uace0, \uacb0\uacfc\uc801\uc73c\ub85c \\(y\\in A\\)\uac00 \ub41c\ub2e4. \uc774\uac83\uc740 \uace7 \\([x]=A\\)\uc784\uc744 \uc758\ubbf8\ud55c\ub2e4. \uc989, \uc720\ub3c4\ub41c \ub3d9\uce58\uad00\uacc4\uc5d0 \ub300\ud55c \uac01\uac01\uc758 \ub3d9\uce58\ub958\ub294 \uc6d0\ub798 \ubd84\ud560\uc758 \uc870\uac01\uacfc \uc815\ud655\ud788 \uc77c\uce58\ud558\uba70, \ubaa8\ub4e0 \uc870\uac01\uc740 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\ubbc0\ub85c \uc5b4\ub5a4 \ub300\ud45c\uc6d0\uc744 \uc7a1\uc544\ub3c4 \uadf8 \ub3d9\uce58\ub958\ub85c \ud45c\ud604\ub420 \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c \ub3c4\ucd9c\ub41c \ubaab\uc9d1\ud569 \\(X\/{\\approx}\\)\ub294 \uc6d0\ub798\uc758 \ubd84\ud560 \\(\\mathcal P\\)\uc640 \ub3d9\uc77c\ud55c \uc9d1\ud569\uc871\uc774 \ub41c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<p>\uc774 \uc815\ub9ac\uac00 \ubcf4\uc5ec\uc8fc\ub294 \uac83\uc740 \uc9d1\ud569 \uc704\uc5d0 \ub3d9\uce58\uad00\uacc4\ub97c \ubd80\uc5ec\ud558\ub294 \uc77c\uacfc \uadf8 \uc9d1\ud569\uc744 \ubd84\ud560\ud558\ub294 \uc77c\uc774 \uac19\uc740 \uc815\ubcf4\ub97c \uc11c\ub85c \ub2e4\ub978 \ubc29\uc2dd\uc73c\ub85c \ud45c\ud604\ud55c\ub2e4\ub294 \uc810\uc774\ub2e4. \ub3d9\uce58\uad00\uacc4 \\(\\sim\\)\uac00 \uc8fc\uc5b4\uc9c0\uba74 \ud45c\uc900 \ubaab\uc0ac\uc0c1<br \/>\n\\[<br \/>\n\\pi:X\\longrightarrow X\/{\\sim},\\qquad \\pi(x)=[x]<br \/>\n\\]<br \/>\n\uc744 \uc815\uc758\ud560 \uc218 \uc788\uace0,<br \/>\n\\[<br \/>\n\\pi(x)=\\pi(y)\\quad\\Longleftrightarrow\\quad x\\sim y<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uac01 \ub3d9\uce58\ub958\ub294 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc778 \ub3d9\uc2dc\uc5d0 \ubaab\uc9d1\ud569 \\(X\/{\\sim}\\)\uc758 \ud55c \uc6d0\uc18c\uc774\ubbc0\ub85c, \ubc30\uacbd \uc9d1\ud569\ub860 \uc548\uc5d0\uc11c\ub294 \uc774\ub97c \uc0c8\ub85c\uc6b4 \uc218\ud559\uc801 \ub300\uc0c1\uc73c\ub85c \ucde8\uae09\ud560 \uc218 \uc788\ub2e4<a class=\"math-abs-series-ref\" title=\"Halmos, P. R. (1960). Naive Set Theory. Van Nostrand. Internet Archive.\" href=\"#ref-Halmos1960\">[Halmos1960]<\/a>. <!-- \ub2e4\ub9cc \ub300\uc751 \uc815\ub9ac\ub9cc\uc73c\ub85c \ucd94\uc0c1\uc801 \ub300\uc0c1\uc774 \ub17c\ub9ac\uc801\uc73c\ub85c \u2018\ucc3d\uc870\ub41c\ub2e4\u2019\uace0 \ub9d0\ud560 \uc218\ub294 \uc5c6\ub2e4. \ub3d9\uce58\ub958\uc640 \ubaab\uc9d1\ud569\uc758 \uc874\uc7ac\ub294 \uc0ac\uc6a9 \uc911\uc778 \uc9d1\ud569\ub860\uc758 \uc6d0\ub9ac\ub4e4\uc5d0 \uae30\ub300\uae30 \ub54c\ubb38\uc774\ub2e4. \uc55e\uc11c \ubcf8 \ubc29\ud5a5\uc758 \uc0ac\ub840\ub3c4 \ud504\ub808\uac8c \uc790\uc2e0\uc758 \uac1c\ub150\uc758 \uc678\uc5f0\uc5d0 \uc758\ud55c \uc815\uc758<a class=\"math-abs-series-ref\" title=\"Frege, G. (1884). Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung \u00fcber den Begriff der Zahl. W. Koebner. Internet Archive.\" href=\"#ref-Frege1884\">[Frege1884]<\/a>\uc640, \uadf8\uac83\uc744 \ud604\ub300\uc801\uc73c\ub85c \ub098\ud0c0\ub0b8 \ubaab\uc9d1\ud569 \uad6c\uc131\uc744 \uad6c\ubd84\ud574\uc11c \uc774\ud574\ud574\uc57c \ud55c\ub2e4. \uc774 \uae00\uc5d0\uc11c \ub9d0\ud558\ub294 \u2018\ub3d9\uce58\ub958\ub97c \uc0ac\uc6a9\ud55c \ucd94\uc0c1\ud654\u2019\ub294 \uad00\uacc4 \\(\\sim\\)\uac00 \uad6c\ubcc4\ud558\uc9c0 \uc54a\ub294 \ub300\uc0c1\ub4e4\uc744 \ud558\ub098\uc758 \ub3d9\uce58\ub958\ub85c \ubb36\uace0, \uadf8 \ub3d9\uce58\ub958\ub97c \ubaab\uc9d1\ud569\uc758 \uc6d0\uc18c\ub85c \ub2e4\ub8e8\ub294 \uad6c\uc131\uc774\ub2e4. --><\/p>\n<h4>\uc815\uc218\uc758 \uad6c\uc131<\/h4>\n<p>\uc774\uc81c \ub3d9\uce58\ub958\ub97c \uc0ac\uc6a9\ud55c \uc815\uc758\uc758 \uad6c\uccb4\uc801\uc778 \uc608\ub85c\uc11c, \uc0c8\ub85c\uc6b4 \uc218 \uccb4\uacc4\ub97c \uad6c\uc131\ud558\ub294 \uacfc\uc815\uc744 \uc0b4\ud3b4\ubcf4\uc790<a class=\"math-abs-series-ref\" title=\"Tao, T. (2022). Analysis I (4th ed.). Springer.\" href=\"#ref-Tao2022\">[Tao2022]<\/a>. \uc790\uc5f0\uc218 \uc804\uccb4\uc758 \uc9d1\ud569\uc744 \\(\\mathbb N=\\{0,\\,1,\\,2,\\,\\ldots\\}\\)\uc774\ub77c\uace0 \ud558\uc790. \\(\\mathbb N\\)\uc740 \ub367\uc148\uacfc \uacf1\uc148 \uc5f0\uc0b0\uc5d0 \ub300\ud558\uc5ec \ub2eb\ud600 \uc788\uc9c0\ub9cc, \ube84\uc148\uc5d0 \ub300\ud574\uc11c\ub294 \ub2eb\ud600 \uc788\uc9c0 \uc54a\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \ubc29\uc815\uc2dd \\(x+5=3\\)\uc740 \\(\\mathbb N\\) \ub0b4\uc5d0 \ud574\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc6b0\ub9ac\ub294 \uc784\uc758\uc758 \ube84\uc148\uc774 \ud56d\uc0c1 \uac00\ub2a5\ud558\ub3c4\ub85d \uc790\uc5f0\uc218 \uccb4\uacc4\ub97c \ud655\uc7a5\ud558\uc5ec \uc815\uc218 \uccb4\uacc4\ub97c \uad6c\uc131\ud558\uace0\uc790 \ud55c\ub2e4.<\/p>\n<p>\ud575\uc2ec \uc544\uc774\ub514\uc5b4\ub294 \uc74c\uc758 \uc815\uc218\ub97c \ub450 \uc790\uc5f0\uc218\uc758 \u2018\ucc28(difference)\u2019\ub85c \uac04\uc8fc\ud558\ub294 \uac83\uc774\ub2e4. \uc989, \uc790\uc5f0\uc218\uc758 \uc21c\uc11c\uc30d \\((3,\\,5)\\)\ub97c \u201c3\uc5d0\uc11c 5\ub97c \ube80 \uac12\u201d\uc73c\ub85c \uac04\uc8fc\ud558\ub294 \uac83\uc774\ub2e4. \uadf8\ub7f0\ub370 \\((3,\\,5)\\)\ubfd0\ub9cc \uc544\ub2c8\ub77c \\((4,\\,6)\\), \\((0,\\,2)\\) \ub4f1 \ubb34\uc218\ud788 \ub9ce\uc740 \uc21c\uc11c\uc30d\uc774 \ubaa8\ub450 \uc218\ud559\uc801\uc73c\ub85c \ub3d9\uc77c\ud55c \uac12(\uc989, \\(-2\\))\uc744 \ub098\ud0c0\ub0b4\uc57c \ud55c\ub2e4. \uadf8\ub807\ub2e4\uba74 \ub450 \uc21c\uc11c\uc30d\uc774 \uac19\uc740 \uac12\uc744 \ub098\ud0c0\ub0b4\uae30 \uc704\ud55c \ud615\uc2dd\uc801 \uc870\uac74\uc740 \ubb34\uc5c7\uc77c\uae4c? \uc9c1\uad00\uc801\uc73c\ub85c \\(a-b=c-d\\)\uac00 \uc131\ub9bd\ud574\uc57c \ud558\ub294\ub370, \uc774 \ub4f1\uc2dd\uc740 \\(a+d=b+c\\)\uc640 \ub3d9\uce58\uc774\ub2e4. \uc8fc\ubaa9\ud560 \uc810\uc740 \ud6c4\uc790\uc758 \ub4f1\uc2dd\uc5d0\ub294 \ube84\uc148 \uc5f0\uc0b0\uc774 \ub4f1\uc7a5\ud558\uc9c0 \uc54a\ub294\ub2e4\ub294 \uac83\uc774\ub2e4. \uc989, \uc544\uc9c1 \uc815\uc758\ub418\uc9c0 \uc54a\uc740 \ube84\uc148\uc744 \ubc30\uc81c\ud558\uace0 \uc624\uc9c1 \uc790\uc5f0\uc218 \uccb4\uacc4\uc5d0 \uc774\ubbf8 \uc874\uc7ac\ud558\ub294 \u2018\ub367\uc148\u2019\ub9cc\uc73c\ub85c \ub450 \uc21c\uc11c\uc30d \uc0ac\uc774\uc758 \ub3d9\uce58 \uc870\uac74\uc744 \ud45c\ud604\ud574 \ub0b8 \uac83\uc774\ub2e4. \uc774\ub97c \ubc14\ud0d5\uc73c\ub85c \uacf1\uc9d1\ud569 \\(\\mathbb N\\times\\mathbb N\\) \uc0c1\uc5d0 \ub2e4\uc74c\uacfc \uac19\uc740 \uc774\ud56d\uad00\uacc4 \\(\\sim\\)\ub97c \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[ (a,\\,b)\\sim(c,\\,d)\\ \\Longleftrightarrow\\ a+d=b+c \\]\n<\/p>\n<p>\uc774\ub807\uac8c \uc815\uc758\ub41c \uad00\uacc4\ub294 \ub3d9\uce58\uad00\uacc4\uac00 \ub41c\ub2e4. \ubc18\uc0ac\uc131(\\(a+b=b+a\\))\uacfc \ub300\uce6d\uc131\uc740 (\uc790\uc5f0\uc218 \ubc94\uc704\uc5d0\uc11c) \ub367\uc148\uc758 \uad50\ud658\ubc95\uce59\uc5d0 \uc758\ud574 \uc790\uba85\ud558\uac8c \uc131\ub9bd\ud55c\ub2e4. \ucd94\uc774\uc131\uc744 \uc99d\uba85\ud574 \ubcf4\uc790. \\((a,\\,b)\\sim(c,\\,d)\\)\uc774\uace0 \\((c,\\,d)\\sim(e,\\,f)\\)\ub77c\uace0 \uac00\uc815\ud558\uba74, \uc815\uc758\uc5d0 \uc758\ud574 \\(a+d=b+c\\)\uc774\uace0 \\(c+f=d+e\\)\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc774 \ub450 \ub4f1\uc2dd\uc758 \uc591\ubcc0\uc744 \uac01\uac01 \ub354\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[ a+d+c+f=b+c+d+e \\]<br \/>\n\uc790\uc5f0\uc218\uc758 \ub367\uc148\uc740 \uc18c\uac70\ubc95\uce59(cancellation law)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ubbc0\ub85c, \uc591\ubcc0\uc5d0\uc11c \uacf5\ud1b5\ub41c \\(c+d\\)\ub97c \uc18c\uac70\ud558\uba74 \\(a+f=b+e\\)\uac00 \ub0a8\ub294\ub2e4. \uc774\uac83\uc740 \uace7 \\((a,\\,b)\\sim(e,\\,f)\\)\uc784\uc744 \ub73b\ud558\ubbc0\ub85c \ucd94\uc774\uc131\uc774 \uc99d\uba85\ub418\uc5c8\ub2e4.<\/p>\n<p>\uc774 \ub3d9\uce58\uad00\uacc4\ub97c \ubc14\ud0d5\uc73c\ub85c, \uc815\uc218 \uc804\uccb4\uc758 \uc9d1\ud569\uc744 \uace7 \uc21c\uc11c\uc30d\ub4e4\uc758 \ubaab\uc9d1\ud569<br \/>\n\\[ \\mathbb Z=(\\mathbb N\\times\\mathbb N)\/{\\sim} \\]<br \/>\n\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \uc790\uc5f0\uc218 \uc21c\uc11c\uc30d\ub4e4\uc758 \ub3d9\uce58\ub958 \ud558\ub098\uac00 \uc815\uc218 \ud558\ub098\ub97c \ub098\ud0c0\ub0b4\ub294 \uac83\uc774\ub2e4. \ub098\uc544\uac00 \uc815\uc218\uc758 \ub367\uc148 \uc5f0\uc0b0\uc740 \uac01 \ub3d9\uce58\ub958\uc5d0\uc11c \ub300\ud45c\uc6d0\uc744 \ucd94\ucd9c\ud558\uc5ec \uc131\ubd84\ubcc4\ub85c \ub354\ud558\ub294 \ubc29\uc2dd\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[ [(a,\\,b)]+[(c,\\,d)]=[(a+c,\\,b+d)] \\]<br \/>\n\uc5ec\uae30\uc11c \ub17c\ub9ac\uc801\uc73c\ub85c \ubc18\ub4dc\uc2dc \uc9da\uace0 \ub118\uc5b4\uac00\uc57c \ud560 \uc810\uc774 \uc788\ub2e4. \uc88c\ubcc0\uc740 \u2018\ub3d9\uce58\ub958 \uc9d1\ud569 \uc0ac\uc774\uc758 \uc5f0\uc0b0\u2019\uc778 \ubc18\uba74, \uc6b0\ubcc0\uc740 \uc784\uc758\ub85c \uc120\ud0dd\ub41c \u2018\ub300\ud45c\uc6d0\ub4e4 \uc0ac\uc774\uc758 \uc5f0\uc0b0\u2019\uc758 \uacb0\uacfc\uc5d0 \uc758\ud574 \uacb0\uc815\ub41c\ub2e4\ub294 \uc0ac\uc2e4\uc774\ub2e4. \uc989, \ub3d9\uc77c\ud55c \ub3d9\uce58\ub958 \ub0b4\uc5d0\uc11c \ub2e4\ub978 \ub300\ud45c\uc6d0\uc744 \uc120\ud0dd\ud558\uc5ec \uacc4\uc0b0\ud558\ub354\ub77c\ub3c4 \ucd5c\uc885 \uacb0\uacfc\ubb3c\uc774 \ud56d\uc0c1 \ub3d9\uc77c\ud55c \ub3d9\uce58\ub958\uc5d0 \uc18d\ud558\ub294\uc9c0\ub97c \uc99d\uba85\ud574\uc57c\ub9cc \uc774 \uc5f0\uc0b0\uc774 \uc218\ud559\uc801 \ubaa8\uc21c \uc5c6\uc774 \uc815\uc758\ub41c\ub2e4. \uc774 \uac80\uc99d \uc808\ucc28\ub97c \ub9c8\uce5c \uc5f0\uc0b0\uc744 \uc77c\uceec\uc5b4 <span class=\"defined\">\uc798 \uc815\uc758\ub41c<\/span>(well-defined) \uc5f0\uc0b0\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. <!-- \uad6c\uccb4\uc801\uc778 \uc99d\uba85 \uacfc\uc815\uc740 \ub300\uc218\uc801\uc778 \ub2e8\uc21c \uacc4\uc0b0\uc774\ubbc0\ub85c \uc5ec\uae30\uc11c\ub294 \uc0dd\ub7b5\ud55c\ub2e4. --><!-- [\uc989, \\((a,\\,b)\\sim(a',\\,b')\\)\uc774\uace0 \\((c,\\,d)\\sim(c',\\,d')\\)\uc77c \ub54c \ud56d\uc0c1 \\((a+c,\\,b+d)\\sim(a'+c',\\,b'+d')\\)\uac00 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uba74 \ub41c\ub2e4. \uc815\uc218\uc758 \uacf1\uc148 \uc5ed\uc2dc \\([(a,\\,b)]\\cdot[(c,\\,d)]=[(ac+bd,\\,ad+bc)]\\)\ub85c \uc815\uc758\ud558\uba70, \ub367\uc148\uacfc \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc5f0\uc0b0\uc774 \uc798 \uc815\uc758\ub428\uc744 \uc99d\uba85\ud574\uc57c \ud55c\ub2e4.] \uc694\ucee8\ub300 \ub3d9\uce58\ub958\ub97c \uc774\uc6a9\ud574 \uc0c8\ub85c\uc6b4 \uc218\ud559\uc801 \ub300\uc0c1\uc744 \uc815\uc758\ud558\uace0 \uc5f0\uc0b0\uc744 \ubd80\uc5ec\ud560 \ub54c\ub9c8\ub2e4 \uc774 \u2018\uc798 \uc815\uc758\ub428(well-definedness)\u2019\uc5d0 \ub300\ud55c \uac80\uc99d\uc740 \ud544\uc5f0\uc801\uc73c\ub85c \ub530\ub77c\uc628\ub2e4. --><\/p>\n<p>\uc774\uc81c \uae30\uc874\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc744 \uc0c8\ub85c\uc6b4 \uc815\uc218\uacc4\uc758 \ub3d9\uce58\ub958 \\([(n,\\,0)]\\)\uacfc \uad6c\uc870\uc801\uc73c\ub85c \ub3d9\uc77c\uc2dc(identify)\ud558\uba74, \uc6d0\ub798\uc758 \uc9d1\ud569 \\(\\mathbb N\\)\uc744 \uc790\uc5f0\uc2a4\ub7fd\uac8c \ud655\uc7a5\ub41c \uc9d1\ud569 \\(\\mathbb Z\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc73c\ub85c \uac04\uc8fc\ud560 \uc218 \uc788\ub2e4. \ub354 \ub098\uc544\uac00 \uc0c8\ub86d\uac8c \uc815\uc758\ub41c \uc784\uc758\uc758 \ub3d9\uce58\ub958 \\([(a,\\,b)]\\)\uc5d0 \ub300\ud558\uc5ec,<br \/>\n\\[ [(a,\\,b)]+[(b,\\,a)]=[(a+b,\\,a+b)]=[(0,\\,0)] \\]<br \/>\n\uc774 \uc131\ub9bd\ud558\ubbc0\ub85c \ubaa8\ub4e0 \uc815\uc218\ub294 \ub367\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0\uc744 \uac00\uc9c0\uac8c \ub41c\ub2e4. [\\((a+b,\\,a+b)\\sim(0,\\,0)\\)\uc784\uc740 \ub3d9\uce58\uad00\uacc4\uc758 \uc815\uc758\uc5d0 \uc758\ud574 \uc989\uac01\uc801\uc73c\ub85c \ud655\uc778\ub41c\ub2e4. \\([(0,\\,0)]\\)\uc740 \uc815\uc218\uc758 \ub367\uc148\uc5d0 \ub300\ud55c \ud56d\ub4f1\uc6d0, \uc989 \\(0\\)\uc5d0 \ud574\ub2f9\ud55c\ub2e4.] \uacb0\ub860\uc801\uc73c\ub85c, \uae30\uc874\uc758 \uc790\uc5f0\uc218 \uccb4\uacc4 \\(\\mathbb N\\)\uc5d0\uc11c\ub294 \ud574\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub358 \ubc29\uc815\uc2dd \\(x+5=3\\)\uc774, \uc0c8\ub86d\uac8c \ud655\uc7a5\ub41c \uc815\uc218 \uccb4\uacc4 \\(\\mathbb Z\\)\uc5d0\uc11c\ub294 \\([(3,\\,5)]\\)\ub77c\ub294 \uba85\ud655\ud55c \ud574\ub97c \uac16\uac8c \ub418\uba70, \uc774 \ub3d9\uce58\ub958\uac00 \ubc14\ub85c \uc6b0\ub9ac\uac00 \uc77c\uc0c1\uc801\uc73c\ub85c \ubd80\ub974\ub294 \uc74c\uc758 \uc815\uc218 \u2018\\(-2\\)\u2019\uc758 \uc218\ud559\uc801 \uc2e4\uccb4\uc774\ub2e4.<\/p>\n<p>\uc774\ub7ec\ud55c \ubaab\uc9d1\ud569 \uae30\ubc95\uc740 \ub2e4\ub978 \uc218 \uccb4\uacc4\ub97c \uad6c\uc131\ud560 \ub54c\ub3c4 \uc0ac\uc6a9\ub41c\ub2e4. \uc608\ucee8\ub300 \uc784\uc758\uc758 \uc815\uc218 \\(p\\)\uc640 \\(0\\)\uc774 \uc544\ub2cc \uc815\uc218 \\(q\\)\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc21c\uc11c\uc30d\ub4e4\uc758 \uc9d1\ud569 \uc704\uc5d0<br \/>\n\\[ (p,\\,q)\\sim(r,\\,s)\\ \\Longleftrightarrow\\ ps=qr \\]<br \/>\n\uc774\ub77c\ub294 \uad00\uacc4\ub97c \ubd80\uc5ec\ud558\uba74 \ub3d9\uce58\ub958 \\([(p,\\,q)]\\)\ub97c \uc720\ub9ac\uc218 \\(p\/q\\)\ub85c \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \uc6b0\ub9ac\uac00 \ubd84\uc218\ub97c \uc57d\ubd84\ud558\uac70\ub098 \ud1b5\ubd84\ud558\ub294 \uacfc\uc815\uc740 \ub3d9\uc77c\ud55c \ub3d9\uce58\ub958 \uc548\uc5d0\uc11c \ub2e4\ub978 \ub300\ud45c\uc6d0\uc744 \ud0dd\ud558\ub294 \uc870\uc791\uc73c\ub85c \uc774\ud574\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\uc2e4\uc218\uc758 \ud55c \uac00\uc9c0 \uad6c\uc131\uc740 \uc720\ub9ac\uc218 \ucf54\uc2dc \uc218\uc5f4 \uc804\uccb4\uc758 \uc9d1\ud569 \\(\\mathcal C\\)\uc5d0\uc11c \ucd9c\ubc1c\ud55c\ub2e4. \ub450 \uc218\uc5f4 \\((x_n)\\)\uacfc \\((y_n)\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n(x_n)\\sim(y_n)\\quad\\Longleftrightarrow\\quad \\lim_{n\\to\\infty}(x_n-y_n)=0<br \/>\n\\]<br \/>\n\uc73c\ub85c \uc815\uc758\ud558\uace0 \ubaab\uc9d1\ud569 \\(\\mathcal C\/{\\sim}\\)\ub97c \ub9cc\ub4e0 \ub4a4, \ud56d\ubcc4 \ub367\uc148\uacfc \uacf1\uc148 \ubc0f \uc801\uc808\ud55c \uc21c\uc11c\ub97c \uc815\uc758\ud558\uba74 \uc644\ube44\uc21c\uc11c\uccb4\uc778 \uc2e4\uc218\uccb4\uc758 \ud55c \uad6c\uc131\uc744 \uc5bb\ub294\ub2e4<a class=\"math-abs-series-ref\" title=\"Tao, T. (2022). Analysis I (4th ed.). Springer.\" href=\"#ref-Tao2022\">[Tao2022]<\/a>. 19\uc138\uae30\uc5d0\ub294 \uce78\ud1a0\uc5b4\uac00 1872\ub144\uc5d0 \uc774\uc640 \uad00\ub828\ub41c \u2018\uae30\ubcf8\uc218\uc5f4\u2019 \uad6c\uc131\ubc95\uc744 \ubc1c\ud45c\ud588\uc73c\uba70, \uba54\ub808\uc640 \ud558\uc774\ub124\uc758 \uad00\ub828 \uc791\uc5c5 \ubc0f \ub370\ub370\ud0a8\ud2b8 \uc808\ub2e8\uc744 \uc774\uc6a9\ud55c \ub300\uc548\uc801 \uad6c\uc131\ub3c4 \uc788\uc5c8\ub2e4<a class=\"math-abs-series-ref\" title=\"Weiss, I. (2015). The real numbers\u2014A survey of constructions. Rocky Mountain Journal of Mathematics, 45(3), 737\u2013762.\" href=\"#ref-Weiss2015\">[Weiss2015]<\/a>. <!-- \ub530\ub77c\uc11c \ubc29\ud5a5, \uc815\uc218, \uc720\ub9ac\uc218, \uc2e4\uc218\ub294 \ubaa8\ub450 \ubaab\uc9d1\ud569 \uae30\ubc95\uc73c\ub85c \uad6c\uc131\ud558\uac70\ub098 \ud45c\ud604\ud560 \uc218 \uc788\ub294 \uc911\uc694\ud55c \uc0ac\ub840\ub4e4\uc774\uc9c0\ub9cc, \uc774\uac83\uc774 \uac01\uac01\uc758 \ub300\uc0c1\uc744 \uc815\uc758\ud558\ub294 \uc720\uc77c\ud55c \ubc29\ubc95\uc774\ub77c\ub294 \ub73b\uc740 \uc544\ub2c8\ub2e4. --><\/p>\n<p>\uc774 \uc5f0\uc7ac\uc5d0\uc11c\ub294 \uc218\ud559\uc0ac \uc804\ubc18\uc5d0 \uac78\uccd0 \ucd94\uc0c1\ud654\uac00 \uc2e4\uc81c\ub85c \uc218\ud589\ub418\ub294 \uad6c\uccb4\uc801\uc778 \ubc29\uc2dd\ub4e4\uc744 \ud558\ub098\uc529 \uba85\uba85\ud558\uba70 \uc0b4\ud3b4\ubcf4\uace0, <a href=\"..\/math-abstraction-11-topography\/\">11\ubd80<\/a>\uc5d0\uc11c \uadf8 \ubc29\uc2dd\ub4e4\uc744 \uc694\uc57d\ud558\uc5ec \ub3cc\uc544\ubcfc \uc608\uc815\uc774\ub2e4. \u2018\ub3d9\uce58\ub958\ub97c \uc0ac\uc6a9\ud55c \uc815\uc758\u2019, \uc989 \ubb34\ud615\uc758 \uad00\uacc4\ub97c \ud3ec\ucc29\ud558\uc5ec \uad6c\uccb4\uc801\uc778 \ub300\uc0c1\uc73c\ub85c \uad73\ud600\ub0b4\ub294 \uc870\uc791\uc774 \uadf8 \uccab \ubc88\uc9f8 \ucd94\uc0c1\ud654 \ubc29\uc2dd\uc774\ub2e4.<\/p>\n<h4>\ucd94\uc0c1\ud654\uc758 \uc5ed\uc124<\/h4>\n<p>\uc9c0\uae08\uae4c\uc9c0 \ucd94\uc0c1\ud654\ub780 \ub300\uc0c1\uc5d0\uc11c \ud2b9\uc815\ud55c \uc131\uc9c8\uc5d0 \uc8fc\ubaa9\ud558\uace0 \ub098\uba38\uc9c0\ub97c \uace0\ub824\uc5d0\uc11c \uc81c\uc678\ud558\uc5ec \uc0c8\ub85c\uc6b4 \uac1c\ub150\uc774\ub098 \ub300\uc0c1\uc744 \uaddc\uc815\ud558\ub294 \uc870\uc791\uc774\ub77c\uace0 \uc815\uc758\ud558\uc600\ub2e4. \ub610\ud55c \uc774 \uc870\uc791\uc774 \uc218\ud589\ub418\ub294 \uc138 \uac00\uc9c0 \ubc29\uc2dd\uc744 \ubd84\ub958\ud558\uace0, \uccab \ubc88\uc9f8 \uc0c1\uc138 \uc0ac\ub840\ub85c \ucd94\uc0c1\ud654 \uc6d0\ub9ac\uc640 \ubaab\uc9d1\ud569 \uad6c\uc131\uc744 \uad6c\ubd84\ud558\uc5ec \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \ub3d9\uce58\uad00\uacc4\uc640 \ubd84\ud560\uc758 \ub300\uc751 \uc815\ub9ac\ub294 \ubaab\uc9d1\ud569\uc774 \uc5b4\ub5bb\uac8c \uc791\ub3d9\ud558\ub294\uc9c0\ub97c \uc124\uba85\ud558\uba70, \ubc29\ud5a5\uacfc \uc815\uc218\uc758 \uc0ac\ub840\ub294 \ub3d9\uc77c\uc131 \uc870\uac74\uc744 \uc81c\uc2dc\ud558\ub294 \ub2e8\uacc4\uc640 \ub3d9\uce58\ub958\ub97c \uc0ac\uc6a9\ud574 \ub300\uc0c1\uc744 \uba85\uc2dc\uc801\uc73c\ub85c \uad6c\uc131\ud558\ub294 \ub2e8\uacc4\uac00 \uc5b4\ub5bb\uac8c \uc5f0\uacb0\ub418\ub294\uc9c0\ub97c \ubcf4\uc5ec\uc900\ub2e4. \uc774\ub294 \ud504\ub808\uac8c\uac00 \ubc29\ud5a5\uacfc \uc218\ub97c \ub17c\ud558\uba74\uc11c \ubb38\ub9e5\uc801 \ub3d9\uc77c\uc131 \uc6d0\ub9ac\uc640 \uac1c\ub150\uc758 \uc678\uc5f0\uc5d0 \uc758\ud55c \uba85\uc2dc\uc801 \uc815\uc758\ub97c \uad6c\ubd84\ud55c \ud750\ub984\uacfc\ub3c4 \ub9de\ub2ff\uc544 \uc788\ub2e4<a class=\"math-abs-series-ref\" title=\"Frege, G. (1884). Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung \u00fcber den Begriff der Zahl. W. Koebner. Internet Archive.\" href=\"#ref-Frege1884\">[Frege1884]<\/a>.<\/p>\n<p>\uc774 \uae00\uc758 \uc11c\ub450\uc5d0\uc11c, \u2018\uc218\ud559\uc801 \ucd94\uc0c1\ud654\u2019\uac00 \uc77c\uc0c1\uc5d0\uc11c \uc0ac\uc6a9\ud558\ub294 \u2018\ucd94\uc0c1\uc801\u2019\uc774\ub77c\ub294 \ub9d0\uacfc \ub2ec\ub9ac \uac1c\ub150\uc744 \uc5c4\ubc00\ud558\uace0 \uba85\ub8cc\ud558\uac8c \ub2e4\ub4ec\ub294 \ub370 \uc0ac\uc6a9\ub41c\ub2e4\uace0 \ud558\uc600\ub2e4. \ubcf8\ubb38\uc5d0\uc11c \uc0b4\ud3b4\ubcf8 \ubaab\uc9d1\ud569 \uad6c\uc131\uc740 \uadf8 \ucc28\uc774\ub97c \uc798 \ubcf4\uc5ec\uc900\ub2e4. \ub450 \uc21c\uc11c\uc30d \\((3,\\,5)\\)\uc640 \\((4,\\,6)\\) \uc0ac\uc774\uc5d0\ub294 \uc5b4\ub5a4 \uc790\uc5f0\uc218\ub97c \ub300\ud45c\uc6d0\uc73c\ub85c \uc37c\ub294\uac00 \ud558\ub294 \ucc28\uc774\uac00 \uc788\uc9c0\ub9cc, \uc815\uc218 \uad6c\uc131\uc5d0\uc11c\ub294 \\(a+d=b+c\\)\ub77c\ub294 \uad00\uacc4\uac00 \uadf8 \ucc28\uc774\ub97c \ubb34\uc2dc\ud558\uace0 \ub450 \uc21c\uc11c\uc30d\uc744 \uac19\uc740 \ub3d9\uce58\ub958\uc5d0 \ub123\ub294\ub2e4. \uc5ec\uae30\uc11c \ubb34\uc5c7\uc744 \uad6c\ubcc4\ud558\uace0 \ubb34\uc5c7\uc744 \uac19\ub2e4\uace0 \ubcfc \uac83\uc778\uc9c0\ub97c \uc815\ud558\ub294 \uac83\uc740 \uad6c\uccb4\uc801\uc73c\ub85c \uc120\ud0dd\ud55c \uad00\uacc4 \\(\\sim\\)\uc774\ub2e4. \ubc18\uc0ac\uc131, \ub300\uce6d\uc131, \ucd94\uc774\uc131\uc740 \uc120\ud0dd\ub41c \uad00\uacc4\uac00 \uc9d1\ud569\uc744 \uc11c\ub85c \uacb9\uce58\uc9c0 \uc54a\ub294 \ub3d9\uce58\ub958\ub4e4\ub85c \uc77c\uad00\ub418\uac8c \ubd84\ud560\ud558\ub3c4\ub85d \ubcf4\uc7a5\ud558\ub294 \uc870\uac74\uc774\ub2e4. \uc218\ud559\uc801 \ucd94\uc0c1\ud654\uc758 \uba85\ub8cc\ud568\uc740 \uace0\ub824\uc5d0\uc11c \uc81c\uc678\ud560 \ucc28\uc774\uc640 \ubcf4\uc874\ud560 \uad6c\uc870\ub97c \uc774\ucc98\ub7fc \uba85\uc2dc\uc801\uc778 \uc815\uc758\uc640 \uc870\uac74\uc73c\ub85c \uc81c\uc2dc\ud558\ub294 \ub370\uc11c \ub098\uc628\ub2e4.<\/p>\n<p>\uc544\uc9c1 \ud55c \uac00\uc9c0 \uc758\ubb38\uc774 \ub0a8\uc544 \uc788\ub2e4. \uc218\ud559\uc758 \uc5ed\uc0ac\uc5d0\uc11c \ucd94\uc0c1\ud654\uc758 \uc870\uc791\uc740 \uc5b8\uc81c\ubd80\ud130, \uc5b4\ub5a4 \ubaa8\uc2b5\uc73c\ub85c \ub098\ud0c0\ub0ac\uc744\uae4c? \uc55e\uc11c \uc0b4\ud3b4\ubcf8 \uac83\uacfc \uac19\uc740 \ucd94\uc0c1\ud654 \uc6d0\ub9ac\uc640 \ubaab\uad6c\uc131\uc758 \uc0ac\ub840\ub4e4\uc740 \uc624\ub798\uc804\ubd80\ud130 \ud615\uc131\ub418\uc5b4 \uc654\uc9c0\ub9cc, \uadf8 \ub17c\ub9ac\uc801 \ud615\uc2dd\uacfc \u2018\ucd94\uc0c1\uc5d0 \uc758\ud55c \uc815\uc758\u2019\ub77c\ub294 \uc804\ubb38 \uc6a9\uc5b4\ub294 19\uc138\uae30 \ub9d0\uc5d0 \uc774\ub974\ub7ec \uccb4\uacc4\uc801\uc73c\ub85c \ubd84\uc11d\ub418\uae30 \uc2dc\uc791\ud588\ub2e4<a class=\"math-abs-series-ref\" title=\"Mancosu, P. (2016). Abstraction and Infinity. Oxford University Press. Publisher page.\" href=\"#ref-Mancosu2016\">[Mancosu2016]<\/a>. <!-- \uadf8\ub807\ub2e4\uace0 \uadf8 \uc774\uc804\uc758 \uc218\ud559\uc790\ub4e4\uc774 \uc790\uc2e0\ub4e4\uc758 \uac1c\ub150 \ud615\uc131 \ubc29\uc2dd\uc744 \uc804\ud600 \uc790\uac01\ud558\uc9c0 \ubabb\ud588\ub2e4\uace0 \ub2e8\uc815\ud560 \uc218\ub294 \uc5c6\ub2e4. --> <a href=\"..\/math-abstraction-02-implicit-era\/\">2\ubd80<\/a>\uc5d0\uc11c\ub294 \uba54\uc18c\ud3ec\ud0c0\ubbf8\uc544\uc758 \uc148\ub3cc\uacfc \uc218 \ud45c\uae30\uc758 \ubc1c\ub2ec\uc774 \uc218 \uac1c\ub150\uc758 \ud615\uc131\uc5d0 \uc5b4\ub5bb\uac8c \uad00\uc5ec\ud588\ub294\uc9c0, \uadf8\ub9ac\uace0 \u2018\uad6c\uccb4\uc801\uc778 \uc148\ub3cc\uc5d0\uc11c \uc21c\uc218\ud55c \uc218\uac00 \ub2e8\uc120\uc801\uc73c\ub85c \ubd84\ub9ac\ub418\uc5c8\ub2e4\u2019\ub294 \uc124\uba85\uc774 \uc65c \ub17c\uc7c1\uc801\uc778\uc9c0\ub97c \ud568\uaed8 \uc0b4\ud3b4\ubcf8\ub2e4<a class=\"math-abs-series-ref\" title=\"Schmandt-Besserat, D. (1984). Before numerals. Visible Language, 18(1), 48\u201360.\" href=\"#ref-SchmandtBesserat1984\">[SchmandtBesserat1984]<\/a><a class=\"math-abs-series-ref\" title=\"Overmann, K. A. (2018). Updating the \u201cabstract\u2013concrete\u201d distinction in ancient Near Eastern numbers. Cuneiform Digital Library Journal, 2018(1), 1\u201322.\" href=\"#ref-Overmann2018\">[Overmann2018]<\/a>. \ub610\ud55c \uace0\ub300 \uadf8\ub9ac\uc2a4\uc5d0\uc11c \ubb38\uc790\ub85c \ud45c\uc2dc\ud55c \ub3c4\ud615, \uc815\ud615\ud654\ub41c \uc5b8\uc5b4, \uc99d\uba85 \uad00\ud589\uc774 \uc5f0\uc5ed\uc801 \uc218\ud559\uc758 \ud615\uc131\uc5d0 \uc5b4\ub5bb\uac8c \uae30\uc5ec\ud588\ub294\uc9c0\ub3c4 \uac80\ud1a0\ud560 \uac83\uc774\ub2e4<a class=\"math-abs-series-ref\" title=\"Netz, R. (1999). The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History. Cambridge University Press.\" href=\"#ref-Netz1999\">[Netz1999]<\/a>.<\/p>\n<h4>\ucc38\uace0\ubb38\ud5cc<\/h4>\n<ul>\n<li id=\"ref-Aristotle1998\">[Aristotle1998] Aristotle. (1998). <i>The Metaphysics<\/i> (H. Lawson-Tancred, Trans.). Penguin Classics.<\/li>\n<li id=\"ref-Frege1884\">[Frege1884] Frege, G. (1884). <i>Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung \u00fcber den Begriff der Zahl<\/i>. W. Koebner. <a href=\"https:\/\/archive.org\/details\/bub_gb_pjzrPbOS73UC\">Internet Archive<\/a>.<\/li>\n<li id=\"ref-Halmos1960\">[Halmos1960] Halmos, P. R. (1960). <i>Naive Set Theory<\/i>. Van Nostrand. <a href=\"https:\/\/archive.org\/details\/naivesettheory00halm\">Internet Archive<\/a>.<\/li>\n<li id=\"ref-KotobankChusho\">[KotobankChusho] Kotobank. (n.d.). \u300c\u62bd\u8c61\u300d. <a href=\"https:\/\/kotobank.jp\/word\/%E6%8A%BD%E8%B1%A1-97265\">https:\/\/kotobank.jp\/word\/\u62bd\u8c61-97265<\/a>.<\/li>\n<li id=\"ref-KotobankShasho\">[KotobankShasho] Kotobank. (n.d.). \u300c\u6368\u8c61\u300d. <a href=\"https:\/\/kotobank.jp\/word\/%E6%8D%A8%E8%B1%A1-525201\">https:\/\/kotobank.jp\/word\/\u6368\u8c61-525201<\/a>.<\/li>\n<li id=\"ref-Levy2021\">[Levy2021] Levy, A. (2021). Idealization and abstraction: Refining the distinction. <i>Synthese<\/i>, 198(Suppl. 24), 5855\u20135872. <a href=\"https:\/\/doi.org\/10.1007\/s11229-018-1721-z\">https:\/\/doi.org\/10.1007\/s11229-018-1721-z<\/a>.<\/li>\n<li id=\"ref-Mancosu2016\">[Mancosu2016] Mancosu, P. (2016). <i>Abstraction and Infinity<\/i>. Oxford University Press. <a href=\"https:\/\/global.oup.com\/academic\/product\/abstraction-and-infinity-9780198746829\">Publisher page<\/a>.<\/li>\n<li id=\"ref-Mendell2016\">[Mendell2016] Mendell, H. (2016). Aristotle and mathematics. In E. N. Zalta (Ed.), <i>The Stanford Encyclopedia of Philosophy<\/i> (Fall 2016 ed.). <a href=\"https:\/\/plato.stanford.edu\/archives\/fall2016\/entries\/aristotle-mathematics\/\">https:\/\/plato.stanford.edu\/archives\/fall2016\/entries\/aristotle-mathematics\/<\/a>.<\/li>\n<li id=\"ref-MerriamWebsterND\">[MerriamWebsterND] Merriam-Webster. (n.d.). Abstract. In <i>Merriam-Webster.com Dictionary<\/i>. Retrieved July 25, 2026, from <a href=\"https:\/\/www.merriam-webster.com\/dictionary\/abstract\">https:\/\/www.merriam-webster.com\/dictionary\/abstract<\/a>.<\/li>\n<li id=\"ref-MitchelmoreWhite2007\">[MitchelmoreWhite2007] Mitchelmore, M., &amp; White, P. (2007). Abstraction in mathematics learning. <i>Mathematics Education Research Journal<\/i>, 19, 1\u20139. <a href=\"https:\/\/doi.org\/10.1007\/BF03217452\">https:\/\/doi.org\/10.1007\/BF03217452<\/a>.<\/li>\n<li id=\"ref-Netz1999\">[Netz1999] Netz, R. (1999). <i>The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History<\/i>. Cambridge University Press. <a href=\"https:\/\/doi.org\/10.1017\/CBO9780511543296\">https:\/\/doi.org\/10.1017\/CBO9780511543296<\/a>.<\/li>\n<li id=\"ref-Overmann2018\">[Overmann2018] Overmann, K. A. (2018). Updating the \u201cabstract\u2013concrete\u201d distinction in ancient Near Eastern numbers. <i>Cuneiform Digital Library Journal<\/i>, 2018(1), 1\u201322. <a href=\"https:\/\/cdli.earth\/articles\/cdlj\/2018-1\">https:\/\/cdli.earth\/articles\/cdlj\/2018-1<\/a>.<\/li>\n<li id=\"ref-SchmandtBesserat1984\">[SchmandtBesserat1984] Schmandt-Besserat, D. (1984). Before numerals. <i>Visible Language<\/i>, 18(1), 48\u201360. <a href=\"https:\/\/journals.uc.edu\/index.php\/vl\/article\/view\/5376\">https:\/\/journals.uc.edu\/index.php\/vl\/article\/view\/5376<\/a>.<\/li>\n<li id=\"ref-ShapiroKouriKissel2026\">[ShapiroKouriKissel2026] Shapiro, S., &amp; Kouri Kissel, T. (2026). Classical logic. In E. N. Zalta &amp; U. Nodelman (Eds.), <i>The Stanford Encyclopedia of Philosophy<\/i>. <a href=\"https:\/\/plato.stanford.edu\/entries\/logic-classical\/\">https:\/\/plato.stanford.edu\/entries\/logic-classical\/<\/a>.<\/li>\n<li id=\"ref-Tao2022\">[Tao2022] Tao, T. (2022). <i>Analysis I<\/i> (4th ed.). Springer. <a href=\"https:\/\/doi.org\/10.1007\/978-981-19-7261-4\">https:\/\/doi.org\/10.1007\/978-981-19-7261-4<\/a>.<\/li>\n<li id=\"ref-Weiss2015\">[Weiss2015] Weiss, I. (2015). The real numbers\u2014A survey of constructions. <i>Rocky Mountain Journal of Mathematics<\/i>, 45(3), 737\u2013762. <a href=\"https:\/\/doi.org\/10.1216\/RMJ-2015-45-3-737\">https:\/\/doi.org\/10.1216\/RMJ-2015-45-3-737<\/a>.<\/li>\n<li id=\"ref-Zach2023\">[Zach2023] Zach, R. (2023). Hilbert\u2019s program. In E. N. Zalta &amp; U. Nodelman (Eds.), <i>The Stanford Encyclopedia of Philosophy<\/i>. <a href=\"https:\/\/plato.stanford.edu\/entries\/hilbert-program\/\">https:\/\/plato.stanford.edu\/entries\/hilbert-program\/<\/a>.<\/li>\n<\/ul>\n<h4>\uc800\uc791\uad8c<\/h4>\n<p>\uc774\uc2ac\ube44, 2026. designeralice\uff20daum.net.<\/p>\n<div class=\"math-abs-series-contents\">\n<p class=\"math-abs-series-menu-title\"><a href=\"..\/\">\uc218\ud559\uc801 \ucd94\uc0c1\ud654<\/a><\/p>\n<ol class=\"math-abs-series-menu-list\">\n        <!-- \ud604\uc7ac \ud398\uc774\uc9c0\uc5d0 \ud574\ub2f9\ud558\ub294 li \ud0dc\uadf8\uc5d0 class=\"math-abs-series-current-page\" \uc18d\uc131 \ucd94\uac00 --><\/p>\n<li class=\"math-abs-series-current-page\"><a href=\"..\/math-abstraction-01-essence\/\">\uc218\ud559\uc801 \ucd94\uc0c1\ud654\uc758 \uae30\ubcf8 \uac1c\ub150<\/a><\/li>\n<li><a href=\"..\/math-abstraction-02-implicit-era\/\">\ucd08\uae30 \uc218\ud559\uc5d0\uc11c\uc758 \uc554\ubb35\uc801 \ucd94\uc0c1\ud654<\/a><\/li>\n<li><a href=\"..\/math-abstraction-03-algebraic-symbols\/\">\ub300\uc218\uc801 \uae30\ud638\uc640 \uc5f0\uc0b0 \ubc95\uce59<\/a><\/li>\n<li><a href=\"..\/math-abstraction-04-axiomatic-method\/\">\uacf5\ub9ac\uc801 \ubc29\ubc95\uc5d0 \uc758\ud55c \ucd94\uc0c1\ud654<\/a><\/li>\n<li><a href=\"..\/math-abstraction-05-mathematical-structures\/\">\uc218\ud559\uc801 \uad6c\uc870\uc640 \ubc94\uc8fc\ub860<\/a><\/li>\n<li><a href=\"..\/math-abstraction-06-vector-spaces\/\">\ubca1\ud130\uacf5\uac04\uacfc \uc120\ud615 \uad6c\uc870<\/a><\/li>\n<li><a href=\"..\/math-abstraction-07-topological-spaces\/\">\uac70\ub9ac\uacf5\uac04\uacfc \uc704\uc0c1\uacf5\uac04<\/a><\/li>\n<li><a href=\"..\/math-abstraction-08-measure-spaces\/\">\uce21\ub3c4\uacf5\uac04\uacfc \ud655\ub960\uacf5\uac04<\/a><\/li>\n<li><a href=\"..\/math-abstraction-09-cognitive-process\/\">\uc218\ud559\uc801 \ucd94\uc0c1\ud654\uc758 \uc778\uc9c0\uc640 \ud559\uc2b5<\/a><\/li>\n<li><a href=\"..\/math-abstraction-10-optimal-generality\/\">\ud604\ub300 \uc218\ud559 \uc5f0\uad6c\uc5d0\uc11c\uc758 \ucd94\uc0c1\ud654<\/a><\/li>\n<li><a href=\"..\/math-abstraction-11-topography\/\">\uc218\ud559\uc801 \ucd94\uc0c1\ud654\uc758 \ubc29\ubc95\uacfc \ucca0\ud559<\/a><\/li>\n<\/ol>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\uc77c\uc0c1\uc5b4\uc5d0\uc11c \u2018\ucd94\uc0c1\uc801\u2019\uc774\ub77c\ub294 \ub9d0\uc740 \ub300\uccb4\ub85c \uae0d\uc815\uc801\uc778 \uc758\ubbf8\ub85c \uc4f0\uc774\uc9c0 \uc54a\ub294\ub2e4. \uc124\uba85\uc774 \ucd94\uc0c1\uc801\uc774\ub77c\ub294 \ub9d0\uc740 \ub300\uac1c \ubaa8\ud638\ud574\uc11c \uc54c\uc544\ub4e3\uae30 \uc5b4\ub835\ub2e4\ub294 \ubd88\ud3c9\uc744 \ub73b\ud558\uba70, \uc73c\ub808 \uad6c\uccb4\uc801\uc778 \uc608\ub97c \ub4e4\uc5b4 \ub2ec\ub77c\ub294 \uc694\uccad\uc774 \ub4a4\ub530\ub978\ub2e4. \uc218\ud559\uc5d0\uc11c\ub294 \uc0ac\uc815\uc774 \ub2e4\ub974\ub2e4. \ucd94\uc0c1\ub300\uc218\ud559\uc758 \uad70, \ucd94\uc0c1 \ubca1\ud130\uacf5\uac04, \uc704\uc0c1\uacf5\uac04\uc758 \uc815\uc758\ub294 \uc5c4\ubc00\ud558\uace0 \uba85\ub8cc\ud55c \ubb38\uc7a5\uc73c\ub85c \uae30\uc220\ub41c\ub2e4. \uacf5\ub9ac\ub294 \ub300\uac1c \ud2b9\uc815 \ub300\uc0c1 \ud558\ub098\ub97c \uc720\uc77c\ud558\uac8c \uc815\ud558\ub294 \uac83\uc774 \uc544\ub2c8\ub77c, \uc5b4\ub5a4 \uad6c\uc870\uac00 \uadf8 \uac1c\ub150\uc758 \uc0ac\ub840\uac00 \ub418\uae30 \uc704\ud574 \ub9cc\uc871\ud574\uc57c \ud560 \uc870\uac74\uc744 \uba85\uc2dc\ud55c\ub2e4. \uac19\uc740 \uacf5\ub9ac\uacc4\ub97c \ub9cc\uc871\ud558\ub294 \uc11c\ub85c \ub2e4\ub978 \uad6c\uc870\ub4e4\uc774 \uc874\uc7ac\ud560&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9726,"menu_order":100,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9734","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9734","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=9734"}],"version-history":[{"count":30,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9734\/revisions"}],"predecessor-version":[{"id":9994,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9734\/revisions\/9994"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9726"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=9734"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}