When is one thing equal to some other thing

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1、Whenisonethingequaltosomeotherthing?BarryMazurJune12,2007InmemoryofSaundersMacLane1TheawkwardnessofequalityOnecan’tdomathematicsformorethantenminuteswithoutgrappling,insomewayorother,withtheslipperynotionofequality.Slippery,becausethewayinwhichobjectsarepresentedtoushardlyever,perhapsnever,immediate

2、lytellsus–withoutfurthercommentary–whentwoofthemaretobeconsideredequal.Weevenseethis,forexample,ifwetrytodefinerealnumbersasdecimals,andthenhavetomentionaliaseslike20=19.999...,afactnotunknowntothemerchantswhopricetheiritems$19.99.Theheartandsoulofmuchmathematicsconsistsofthefactthatthe“same”objectca

3、nbepresentedtousindifferentways.Evenifwearefacedwiththesimple-seemingtaskof“giving”alargenumber,thereisnowayofdoingthiswithoutalso,atthesametime,“giving”aheftyamountofextrastructurethatcomesasaresultofthewaywepindown—orthewaywepresent—ourlargenumber.Ifwewriteournumberas1729weare,sottovoce,offeringapre

4、ferredwayof“computingit”(addonethousandtosevenhundredstotwotenstonine).Ifwepresentitas1+123wearerecommendinganothermodeofcomputation,andifwepinitdown—asRamanujuandid—asthefirstnumberexpressibleasasumoftwocubesintwodifferentways,wearebeinglessspecificabouthowtocomputeournumber,buthaveunderscoredacharact

5、erizingpropertyofitwithinasubtlediophantinearena.Theissueof“presentation”sometimescomesupasasmallpedagogicalhurdle—nomorethanapebbleintheroad,perhaps,butitisthere–whenoneteachesyoungpeopletheideaofcongruencemodN.Howshouldwethinkof1,2,3,...mod691?Aretheseciphersjustmembersofanewnumbersystemthathappen

6、stohavesimilarnotationassomeofourintegers?Arewetothinkofthemasequivalence1classesofintegers,wheretheequivalencerelationiscongruencemod691?Orarewehappytodealwiththemasthegoodoldintegers,butsubjectedtothatequivalencerelation?Theeventualanswer,ofcourse,is:allthreeways—havingtheflexibilitytoadjustourview

7、pointtotheneedsofthemomentisthekey.Butthatmaybetoostiffadoseofflexibilitytoimposeonourstudentsallatonce.Todefinethemathematicalobjectsweintendtostudy,weoften—perhapsalways—firstmakeitunderstood,moreofteni