
 
 Umbhali: Kevin Buzzard Umbhali: Kevin Buzzard Table Of Links Umbala we-Links abstract 
 
 Ukubuyekezwa & Introduction Ukubuyekezwa & Introduction 2. Universal properties 2. I-Universal Properties 3. Products in practice 3. Imikhiqizo emzimbeni 4. Universal properties in algebraic geometry I-Universal Properties ku-Algebraic Geometry 5. The problem with Grothendieck’s use of equality. I-Problem ye-Grothendieck usebenzisa isilinganiso. 6. More on “canonical” maps 6. More on “canonical” amaphepha 7. Canonical isomorphisms in more advanced mathematics 7. Isomorphisms Canonic ku-mathematics engaphezulu 8. Summary And References 8. Ukubuyekezwa kanye nokuxhumana Abstract abstract Thola kanjani umqondo we-equality isetshenziswe kumathemikhali (kuquka Grothendieck), futhi yini umphumela lokhu lapho ukuthatha ukulungiselela imathemikhali. Thola izisombululo ezihlukahlukene ezisebenzayo mayelana nokuningi. Overview Izinto eziningi zemathemathemikhali futhi izakhiwo zihlanganisa ngokuzimela ngokuzimela izakhiwo. Ngokwesibonelo, izibalo ezijwayelekile zihlanganisa indawo ephelele ye-Archimedean, futhi izakhiwo ezininzi ze-algebra (izakhiwo, imikhiqizo ye-tensor, . . .) kanye ne-topology (imikhiqizo ye-spaces ye-topological, ukufinyelela kwesakhiwo se-metric noma i-space e-uniform, . . ) zihlanganisa ngokuzimela ngokuzimela izakhiwo se-universal. Ukuze uthole okwengeziwe, ngithole izakhiwo ezijwayelekile ze-object yemathemikhali kuze ku-isomorphism e-unique. Lokhu kuyinto ucwaningo esihle kakhulu: ukuze uthole isibonelo engaphandle, Kukho kuphela iqembu elifanayo, kuze ku-isomorphism, kodwa le iqembu iye i-automorphisms (isib. I-map ekuthumela isakhiwo ku-quadrate yayo), okwenza ukuthi kuyinto eyingqayizivele kodwa kuphela kuze ku-non-unique isomorphism, futhi ngakho-ke isakhiwo yokuba iqembu le-order 5 ayikho uhlobo lwezakhiwo ukuthi le iphepha kuyinto mayelana. Sishayele manje isakhiwo P okuyinto eyingqayizivele i-object yemathemikhali, kuze ku-isomorphism eyingqayizivele. Ngiyazi ezintathu. Ungathola (ngokuthi kufanele) ukucubungula i-theory ye-objects eyenziwe nge-P usebenzisa kuphela i-assumption P, futhi ungathanda ukuthi kungekho isakhiwo esizayo se-objects (ngaphandle kokufunda ukuthi kukhona isakhiwo esizayo eyenza i-P, ngakho-ke i-theory yakho ayikho emoyeni). Assertion 1. Zonke izindawo ezimbonini ezivumelanisa P kungenziwa ngokuvamile efanayo, ngoba zonke izicelo ezivumelanayo ezivumelanayo ezivumelanayo ngalinye kunezinto ezivumelanayo ngalinye. Assertion 2. Ngaphezulu ngokuvamile, uma unayo izindawo ezimbili ezinezinhlangano izomorphic, lapho lezi izicubu kungenzeka ngokuvamile ukuba zihlanganiswa ekufinyeleleni. Assertion 3. Nakuba sincoma isibonelo. Nakuba kukhona izakhiwo eziningana zezinombolo zayo zayo ezivamile (izigaba ze-Cauchy, izixazululo ze-Dedekind, isivumelwano se-Bourbaki. . . . ), zonke i-analysis esisodwa ethandwa ku-akhawunti yokuqala esifundeni kusungulwa ngokusebenzisa kuphela ama-Achimedean ne-axioms yokusebenza ezivamile. Umathemikhali akuyona ukhiye ukuthi isifinyezo se-number efanele isetshenziselwa - lokhu kuyoba luhlukile. Ngokuba u-Newton, u-Euler no-Gauss basebenzise "izinombolo zayo" ngaphambi kwe-Cauchy, u-Dedekind no-Bourbaki ziye zihlanganisa imodeli Ngibhalisa ngaphezulu mayelana nokufaka ngezinye izikhathi, kodwa kukhona futhi izakhiwo eziningana ze-localization R[1/S] ye-ring ye-commutative ku-subset ye-multiplicative (i-quotient ye-R×S, i-quotient ye-multivariable polynomial ring. . . ) futhi umathemikhali akuyona ngokufanele ukuthi isakhiwo esebenzayo lapho zithole i-R[1/S]; siphinde ukuthi lokhu akufanele. Kule isihloko, ngitholela ukuthi isibambiso yokuqala esitholile, isibambiso se-dangerous, futhi isibambiso se-third ayinempumelelo. Isibambiso se-2 ne-3 kubonakala isetshenziswe eminye izindawo e-algebraic geometry literature – ngokuvamile siza kubhalwe ukusetshenziswa kwe-Grothendieck ye-symbol ye-= ngezinye ububanzi ngemva. Nokho, njengomathemikhali esebenzayo ndiyazi ukuthi kukhona izinto ezingaziwa kakhulu etholakalayo lapha, okuyinto ngingathanda ukucindezeleka. Nangona i-word "canonical" ayikho isixazululo esemthethweni, ama-matematicians akuyona akuyona emangalisayo ekusetshenziswa kwayo kwegama. I-concept ukuthi izinto ezimbili zihlanganisa isomorphic futhi ngoko kunokwenzeka ukuhlangabezana iyinhlangano ebalulekile kakhulu emzimbeni; kuyinto isisombululo esebenzayo yokuhlanganisa, kuncike ukucindezeleka okuhlobene, futhi ku-practice akufinyelela ama-errors e-arguments, okungenani uma umbhali uyazi lokho. Ngaphezu kwalokho, ukwenza imibuzo kanye nokuvumela i-theorems mayelana ne-objects ngokusebenzisa ukwakhiwa okuhlobene kunazo zonke kuyindlela esebenzayo emathemikhali (isibonelo, ukuthatha isisekelo se-vector space ukuze uthole isibuyekezo bese ukubonisa ukuthi isibuyekezo iyatholakala ngaphandle kwe-choice). Okungenani, thina usabela ukulungiselela ezinye izibalo ku-computer (isib. Ukuguqulwa kwezemathemikhali kusuka ku-paper literature ku-language ye-interactive theorem prover - i-computer program that knows the axioms of mathematics and the rules of logic). Ngesikhathi eside, lokhu inqubo kuhlanganise ukulungiselela izinhlelo eziningi ze-arguments. Uma unemibuzo ukucubungula ukuthi akuyona ngokuvamile futhi akakwazi ukucubungula emaphaketheni okungenani-defined ezifana "canonical", noma ukhangela ukuthi izinto ezinguquguqukayo zihlanganisa, ngamanye amaxesha unamathele ukuthi unemibuzo eyengeziwe, noma ngisho ukucubungula kanjani imibuzo ezithile kufanele ifakwe. Ngokuvamile, ngezinye izikhathi, indlela enhle kakhulu ukwenza lokhu umsebenzi kuhlanganise ukuba ukwakha imibuzo emathemikhali ezintsha okuyinto engabikho embhedeni iphepha. Ngicabanga ukuthi imibuzo ayikho, kodwa ngitholile ukuthi imibuzo ezininzi embhedeni akuyona ngokuvamile ephelele kusukela kwegama-formalist. Ngokusho kokuqiniswa kwezemathemikhali malunga ne-geometry ye-algebraic ne-arithmetic ngokusebenzisa i-computer theorem prover, isibonelo, imisebenzi esithatholakala emkhakheni ye-lean prover ([BCM20], [Liv23], [dFF23], [AX23], [Zha23]. . . . ) kanye nemisebenzi esebenzayo e-Isabelle ([BPL21]) kanye ne-Agda ye-cubic ([ZM23]), lezi izinto ziyaqala ukuchithwa. Kwi-section 5 ngezansi nithole isibonelo esibonakalayo se-problems esiyingeni esiyingeni esiyingeni esiyingeni esiyingeni esiyingeni esiyingeni, futhi okuyinto iye y 
 
 Okuzenzakalelayo iyatholakala ku-archiv ngaphansi kwe-license CC BY 4.0 DEED. Okuzenzakalelayo iyatholakala ku-archiv ngaphansi kwe-license CC BY 4.0 DEED. I-Archive ye-Archive

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