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1、PhilosophyofMathematics:StructureandOntologySTEWARTSHAPIROOXFORDUNIVERSITYPRESSPHILOSOPHYofMATHEMATICSThispageintentionallyleftblankPHILOSOPHYofMATHEMATICSStructureandOntologySTEWARTSHAPIRO13OxfordNewYorkAthensAucklandBangkokBogotáBuenosAiresCalcuttaCapeTownChenn
2、aiDaresSalaamDehliFlorenceHongKongIstanbulKarachiKualaLumpurMadridMelbourneMexicoCityMumbaiNairobiParisSãoPauloSingaporeTaipeiTokyoTorontoWarsawandassociatedcompaniesinBerlinIbadanCopyright©1997byStewartShapiroFirstpublishedin1997byOxfordUniversityPress,Inc.198Ma
3、disonAvenue,NewYork,NewYork10016FirstissuedasanbyOxfordUniversityPresspaperback,2000OxfordisaregisteredtrademarkofOxfordUniversityPress.Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,electroni
4、c,mechanical,photocopying,recordingorotherwise,withoutthepriorpermissionofOxfordUniversityPress.LibraryofCongressCataloging-in-PublicationDataShapiro,Stewart,1951–Philosophyofmathematics:structureandontology/StewartShapiro.p.cm.Includesbibliographicalreferencesan
5、dindex.ISBN0-19-509452-2;0-19-513930-5(pbk)1.Mathematics—Philosophy.I.Title.QA8.4.S5331997510'.1—dc2096-31722135798642PrintedintheUnitedStatesofAmericaonacid-freepaperForBeverlyThispageintentionallyleftblankNumbers...areknownonlybytheirlaws,thelawsofarithmetic,so
6、thatanyconstructsobeyingthoselaws—certainsets,forinstance—areeligible...explicationsofnumber.Setsinturnareknownonlybytheirlaws,thelawsofsettheory...arith-meticisallthereistonumber...thereisnosay-ingabsolutelywhatthenumbersare;thereisonlyarithmetic.Quine[1969,44–4
7、5]Ifintheconsiderationofasimplyinfinitesystem...setinorderbyatransformation...weentirelyneglectthespecialcharacteroftheelements;sim-plyretainingtheirdistinguishabilityandtakingintoaccountonlytherelationstooneanotherinwhichtheyareplacedbytheorder-settingtrans-form
8、ation...,thenaretheseelementscallednaturalnumbersorordinalnumbersorsimplynumbers.Dedekind[1888,§73]ThispageintentionallyleftblankContentsIntroduction3PARTIPERS
