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1、TheHeartofMathematicsAuthor(s):P.R.HalmosSource:TheAmericanMathematicalMonthly,Vol.87,No.7(Aug.-Sep.,1980),pp.519-524Publishedby:MathematicalAssociationofAmericaStableURL:http://www.jstor.org/stable/2321415Accessed:03/06/201005:31YouruseoftheJSTORarchiveindicatesyouracceptanceofJSTOR'sTer
2、msandConditionsofUse,availableathttp://www.jstor.org/page/info/about/policies/terms.jsp.JSTOR'sTermsandConditionsofUseprovides,inpart,thatunlessyouhaveobtainedpriorpermission,youmaynotdownloadanentireissueofajournalormultiplecopiesofarticles,andyoumayusecontentintheJSTORarchiveonlyforyour
3、personal,non-commercialuse.Pleasecontactthepublisherregardinganyfurtheruseofthiswork.Publishercontactinformationmaybeobtainedathttp://www.jstor.org/action/showPublisher?publisherCode=maa.EachcopyofanypartofaJSTORtransmissionmustcontainthesamecopyrightnoticethatappearsonthescreenorprintedp
4、ageofsuchtransmission.JSTORisanot-for-profitservicethathelpsscholars,researchers,andstudentsdiscover,use,andbuilduponawiderangeofcontentinatrusteddigitalarchive.Weuseinformationtechnologyandtoolstoincreaseproductivityandfacilitatenewformsofscholarship.FormoreinformationaboutJSTOR,pleaseco
5、ntactsupport@jstor.org.MathematicalAssociationofAmericaiscollaboratingwithJSTORtodigitize,preserveandextendaccesstoTheAmericanMathematicalMonthly.http://www.jstor.orgTHEHEARTOFMATHEMATICSP.R.HALMOSIntroduction.Whatdoesmathematicsreallyconsistof?Axioms(suchastheparallelpos-tulate)?Theorems
6、(suchasthefundamentaltheoremofalgebra)?Proofs(suchasGodel'sproofofundecidability)?Concepts(suchassetsandclasses)?Definitions(suchastheMengerdefinitionofdimension)?Theories(suchascategorytheory)?Formulas(suchasCauchy'sintegralformula)?Methods(suchasthemethodofsuccessiveapproximations)?Math
7、ematicscouldsurelynotexistwithouttheseingredients;theyareallessential.Itisneverthelessatenablepointofviewthatnoneofthemisattheheartofthesubject,thatthemathematician'smainreasonforexistenceistosolveproblems,andthat,therefore,whatmathematicsreallyconsistsofisproblemsa
