【罗素】数学哲学导论 214页

INTRODUCTIONTOMATHEMATICALPHILOSOPHYBertrandRussellDOVERPUBLICATIONS,INC.NewYork\nBibliographicalNoteThisDoveredition,firstpublishedin1993,isanunabridgedandunalteredrepublicationofthesecondedition(1920)oftheworkfirstpublishedin1919byGeorgeAllen&Unwin,Ltd.,London,andTheMacmillanCo.,NewYork.LibraryofCongressCatahging-in-PublicationDataRussell,Bertrand,r87z-rg7o.Introductiontomathematicalphilosophy/BertrandRussell.P.cm.Originallypublished:znded.London:G.AllenandUnwin;NewYork:Macmillan,r9r9.Includesinden.ISBNo-486-27724-0(pbk.)r.Mathematics-Philosophy.I.Tifle.QA8.4.R87r993b5ro'.r-dczog3-2r477CIPManufacturedintheUnitedStatesofAmericaDoverPublications.Inc..31East2ndStreet,Mineola,N.Y.11501\nPREFACE('Introductionr"Tslsbookisintendedessentiallyasananddoesnotaimatgivinganexhaustivediscussionoftheproblemswithwhichitdeals.Itseemeddesirabletoserforthcertainresults,hithertoonlyavailabletothosewhohavemasteredlogicalsymbolism,inaformofferingtheminimumofdifficultytothebeginner.Theutmostendeavourhasbeenmadetoavoiddogmatismonsuchquestionsasarestillopentoseriousdoubt,andthisendeavourhastosomeextentdominatedthechoiceoftopicsconsidered.Thebeginningsofmathematicallogicarelessdefinitelyknownthanitslaterportions,butareofatleastequalphilosophicalinterest.Muchofwhatissetforthinthefollowingchaptersisnotproperlytobecalled"philosophy,"thoughthemattersconcernedwereincludedinphilosophysolongasnosatisfactoryscienceofthemexisted.Thenatureofinfinityandcontinuity,forexample,belongedinformerdaystophilosophl,butbelongsnowtomathematics.Mathematicalphilosoplt!,inthestrictsense,cannot,perhaps,beheldtoincludesuchdefinitescientificresultsashavebeenobtainedinthisregion;thephilosophyofmathematicswillnaturallybeex-pectedtodealwithquestionsonthefrontierofknowledge,astowhichcomparativecertaintyisnotyetattained.Butspeculationonsuchquestionsishardlylikelytobefruitfulunlessthemorescientificpartsoftheprinciplesofmathematicsareknown.Abookdealingwithtfiosepartsfray,therefore,claimtobeanintroductiontomathematicalphilosophy,thoughitcanhardlyclaim,exceptwhereitstepsoutsideitsprovince,tobeactuallydealingwithapaftofphilosophy.Itdoesdeal,Y\nviIntroductiontoMathematicalPhilosoplt'yhowever,dtrabodyofknowledgewhich,tothosewhoacceptit,appearstoinvalidatemuchtraditionalphilosoPhT,andevenagooddealofwhatiscurrentinthePresentd^y.InthiswaY,aswellasbyitsbearingonstillunsolvedproblems,mathematicallogicisrelevanttophilosophy.Forthisreason'aswellasonaccountoftheintrinsicimportanceofthesubject,somePurPosemaybeservedbyasuccinctaccountofthemainresultsofmathematicallogicinaformrequiringneitheraknowledgeofmathematicsnoranaptitudeformathematicalsymbolism.Here,however,aselsewhere,themethodismoreimportantthantheresults,fromthepointofviewoffurtherresearch;andthemethodcannotwellbeexplainedwithintheframeworkofsuchabookasthefollowing.Itistobehopedthatsomereadersmaybesuficientlyinterestedtoadvancetoastudyofthemethodbywhichmathematicallogiccanbemadehelpfulininvestigatingthetraditionalproblemsofphilosophy.ButthatisatopicwithwhichthefollowingPageshavenotattemptedtodeal.BERTRANDRUSSELL.\nEDITOR'SNOTETuosnwho,relyingonthedistinctionbetweenMathematicalPhilosophyandthePhilosophyofMathematics,tlinkthatthisbookisoutofplaceinthepresentLibrary,maybereferredtowhattheauthorhimselfsaysonthisheadinthePreface.Itisnotnecessarytoagreewithwhathetheresuggestsastothereadjustmentofthefieldofphilosophybythetransferencefromittornathematicsofsuchproblemsasthoseofclass,continuity,infinity,inordertoperceivethebearingofthedefinitionsanddiscussionsthatfollowontheworkof3'traditionalphilosophy."Ifphilosopherscannotconsenttorelegatethecriticismofthesecategoriestoanyofthespecialsciences,itisessential,atanyrate,thattheyshouldknowtheprecisemeaningthatthescienceofmathematics,inwhichtheseconceptsplaysolargeapart,assignstothem.If,ontheotherhand,therebemathematicianstowhomthesedefinitionsanddiscussion$seemtobeanelabora-tionandcomplicationofthesimple,itmaybewelltoremindthemfromthesideofphilosophythathere,aselsewhere,apparentsimplicitymayconcealacomplexitywhichitisthebusinessofsomebody,whetherphilosopherormathematician,ot,liketheauthorofthisvolume,bothinone,tounravel.vli\nCONTENTSPACEPREFACEvEDITOR,SNOTEViiI.THESERIESOFNATURALNUMBERS....I2.DEFINITIONOFNUMBER..II3.FINITUDBANDMATHEMATICALINDUCTION.20+.THEDEFTNITIONOFORDER295.KINDSOFRELATIONS,.+26.STMTLARTTYOFRELATIONS...527.RATToNAL,REAL,ANDcoMpLExNUMBERS638,INFTNTTECARDTNALNUMBERS..,779-TNFINITESERTEsANDoRDINALS....89ro.LIMITSANDCONTINUITY..97II.LIMTTSANDCONTTNUITYOFFUNCTTONS..rO712.SELECTIONSANDTHEMULTIPLICATIVEAXrOM..ll7t3.THEAXrOMOFTNFTNITYANDLOGTCALTYPESl3rr+.TNCoMPATTBILTTYANDTHETHEORYOFDEDUCTION.14+r5.PROPOSITTONALFUNCTTONS..I55r6.DESCRTPTToNS16717.cLAssEs.rrgrr8.MATTTEMATTCSANDLOGIC,.19+INDEX...ir.r..2O7vru\nIntroductiontoMathematicalPhilosophyCHAPTBRITIIESERIEgOFNATURALNUMBERSMenrruATrcsisastudywhich,whenwestartfromitsmostfamiliarportions,maybepursuedineitheroftwooppositedirections.Themorefamiliardirectionisconstructive,towardsgraduallyincreasingcomplexity:fromintegerstofractions,realnumbers,complexnumbers;fromadditionandmulti-plicationtodifferentiationandintegration,andontohighermathematics.Theotherdirection,whichislessfamiliar,proceeds,byanalysing,togreaterandgreaterabstractnessandlogicalsimplicityIinsteadofaskingwhatcanbedefinedanddeducedfromwhaiisassumedtobeginwith,weaskinsteadwhatmoregeneralideasandprinciplescanbefound,intermsofwhichwhatwasourstarting-pointcanbedefinedordeduced.Itisthefactofpursuingthisoppositedirectionthatcharacterisesmathematicalphilosophyasopposedtoordinarymathematics.Butitshouldbeunderstoodthatthedistinctionisone,notinthesubjectmatter,butinthestateofmindoftheinvestigator.EarlyGreekgeometers,passingfromtheempiricalrulesofEgyptianland-surveyingtothegeneralpropositionsbywhichthoseruleswerefoundtobejustifiable,andthencetoEuclid'saxiomsandpostulates,wereengagedinmathematicalphilos-oph/,accordingtotheabovedefinition;butwhenoncetheaxiomsandpostulateshadbeenreached,theirdeductiveemploy-ment,aswefinditinEuclid,belongedtomathematicsinthe\nzIntroductiontoMathematicalPhilosoplt1ordinarysense.Thedistinctionbetweenmathematicsandmathematicalphilosophyisonewhichdependsupontheinterestinspiringtheresearch,anduponthestagewhichtheresearchhasreached;notuponthepropositionswithwhichtheresearchisconcerned.Wemaystaterthesamedistinctioninanotherway.Themostobviousandeasythingsinmathematicsarenotthosethatcomelogicallyatthebeginning;theyarethingsthat,fromthepointofviewoflogicaldeduction,comesomewhereinthemiddle.Justastheeasiestbodiestoseearethosethatareneitherverynearnorveryfar,neitherverysmallnorverygreat,sotheeasiestconceptionstograsparethosethatareneitherverycomplexnorverysimple(using"simple"inalogicalsense).Andasweneedtwosortsofinstruments,thetelescopeandthemicroscope,fortheenlargementofourvisualpowers,soweneedtwosortsofinstrumentsfortheenlargementofourlogicalpowers,onetotakeusforwardtothehighermathematics,theothertotakeusbackwardtothelogicalfoundationsofthethingsthatweareinclinedtotakeforgrantedinmathematics.Weshallfrndthatbyanalysingourordinarymathematicalnotionsweacquirefreshinsight,newpowers,andthemeansofreachingwholenewmathematicalsubjectsbyadoptingfreshlinesofadvanceafterourbackwardjourney.Itisthepurposeofthisbooktoexplainmathematicalphilos-ophysimplyanduntechnically,withoutenlarginguponthoseportionswhicharesodoubtfulordifficultthatanelementarytreatmentisscarcelypossible.AfulltreatmentwillbefoundinPrincipiaMathematica;Lthetreatmentinthepresentvolumeisintendedmerelyasanintroduction.Totheaverageeducatedpersonofthepresentd^y,theobviousstarting-pointofmathematicswouldbetheseriesofwholenumbers,,,,,3,4,etc.rCambridgeUniversityPress,vol.i.,rgto;vol.ii.,rgrr;vol.iii.,r9r3.ByWhiteheadandRussell.\nTheSeriesofNaturalNunbers3Probablyonlyapersonwithsomemathematicalknowledgewouldthinkofbeginningwithoinsteadofwithr,butwewillpresumethisdegreeofknowledge;wewilltakeasourstarting-pointtheseries:o,r,z,3,...lt,rtfr,"..anditisthisseriesthatweshallmeanwhenwespeakofthettgeriesofnaturalnumbers."Itisonlyatahighstageofcivilisationthatwecouldtakethisseriesasourstarting-point.Itmusthaverequiredmanyagestodiscoverthatabraceofpheasantsandacoupleofdayswerebothinstancesofthenumberz:thedegreeofabstractioninvolvedisfarfromeasy.Andthediscoverythatrisanumbermusthavebeendifficult.Asforo,itisaveryrecentaddition;theGreeksandRomanshadnosuchdigit.Ifwehadbeenembarkinguponmathematicalphilosophyinearlierdays,weshouldhavehadtostartwithsomethinglessabstractthantheseriesofnaturalnumbers,whichweshouldreachasastageonourbackwardjourney.Whenthelogicalfoundationsofmathe-maticshavegrownmorefamiliar,weshallbeabletostartfurtherback,atwhatisnowalatestageinouranalysis.Butforthemomentthenaturalnumbersseemtorepresentwhatiseasiestandmostfamiliarinmathematics.Butthoughfamiliar,theyarenotunderstood.Veryfewpeoplearepreparedwithadefinitionofwhatismeantbyttttnumberr"or"or"orr.t'Itisnotverydifficulttoseethat,startingfromo,anyotherofthenaturalnumberscanbereachedbyrepeatedadditionsofr,butweshallhavetodefinewhatttwemeanbyaddingrr"andwhatwemeanby"repeated."Thesequestionsarebynomeanseasy.Itwasbelieveduntilrecentlythatsome,atleast,ofthesefirstnotionsofarithmeticmustbeacceptedastoosimpleandprimitivetobedefined.Sincealltermsthataredefinedaredefinedbymeansofotherterms,itisclearthathumanknowledgemustalwaysbecontenttoacceptsometermsasintelligiblewithoutdefinition,inorder\n+fntroductiontoMathenaticalPhilosophytohaveastarting-pointforitsdefinitions.Itisnotclearthattheremustbetermswhicharcincapablcof.definition:itispossiblethat,howeverfarbackwegoindefining,wealwaysmightgofurtherstill.Ontheotherhand,itisalsopossiblethat,whenanalysishasbeenpushedfarenough,wecanreachtermsthatreallyaresimple,andthereforelogicallyincapableofthesortofdefinitionthatconsistsinanalysing.Thisisaquestionwhichitisnotnecessaryforustodecide;forourpurposesitissufficienttoobservethat,sincehumanPowersarefinite,thedefinitionsknowntou$mustalwaysbeginsome-where,withtermsundefinedforthemoment,thoughperhapsnotpermanently.Alltraditionalpuremathematics,includinganalyticalgeom-etry,mayberegardedasconsistingwhollyofpropositionsaboutthenaturalnumbers.Thatistosay,thetermswhichoccurcanbedefinedbymeansofthenaturalnumbers,andthepropositionscanbededucedfromtheProPertiesofthenaturalnumbers-withtheaddition,ineachcase'oftheideasandpropositionsofpurelogic.Thatalltraditionalpuremathematicscanbederivedfromthenaturalnumbersisaf.airlyrecentdiscovery,thoughithadlongbeensuspected.Pythagoras,whobelievedthatnotonlymathematics,buteverythingelsecouldbededucedfromnumbers,wasthediscovererofthemostseriousobstaclein'(thewayofwhatiscalledthearithmetising"ofmathematics.ItwasPythagoraswhodiscoveredtheexistenceofincom-mensurables,and,inparticular,theincommensurabilityofthesideofasquareandthediagonal.Ifthelengthofthesideisrinch,thenumberofinchesinthediagonalisthesquarerootoLz,whichappearednottobeanumberatall.Theproblemthusraisedwassolvedonlyinourownday,andwasonlysolvedcompletelybythehelpofthereductionofarithmetictologic,whichwillbeexplainedinfollowingchapters.ForthePresent,weshalltakeforgrantedthearithmetisationofmathematics,thoughthiswasaf.eatoftheverygreatestimportance.\nTheSeriesofNaturalNunbersjHavingreducedalltraditionalpuremathematicstothetheoryofthenaturalnumbers,thenextstepinlogicalanalysiswastoreducethistheoryitselftothesmallestsetofpremissesandundefinedtermsfromwhichitcouldbederived.ThisworkwasaccomplishedbyPeano.Heshowedt"hattheentiretheoryofthenaturalnumberscouldbederivedfromthreeprimitiveideasandfiveprimitivepropositionsinadditiontothoseofpurelogic.Thesethreeideasandfivepropositionsthusbecame,asitwere,hostagesforthewholeoftraditionalpuremathe-matics.Iftheycouldbedefinedandprovedintermsofothers,socouldallpuremathematics.Theirlogical"weight,"ifonemayusesuchanexpression,isequaltothatofthewholeseriesofsciencesthathavebeendeducedfromthetheoryofthenaturalnumbers;thetruthofthiswholeseriesisassuredifthetruthofthefiveprimitivepropositionsisguaranteed,provided,ofcourse,thatthereisnothingerroneousinthepurelylogicalapparatuswhichisalsoinvolved.Theworkofanalysingmathe-maticsisextraordinarilyfacilitatedbythisworkofPeano's.ThethreeprimitiveideasinPeano'sarithmeticare:o,number,successor.ttBy"Euccessorhemeansthenextnumberinthenaturalorder.Thatistosay,thesuccessorofoisr,thesuccessoroftttttisz,andsoon.Bynumberhemeans,inthisconnection,theclassofthenaturalnumbers.rHeisnotassumingthatweknowallthemembersofthisclass,butonlythatweknowwhatwemeanwhenwesaythatthisorthatisanumber,justttasweknowwhatwemeanwhenwesayJonesisamanrttthoughwedonotknowallmenindividually.ThefiveprimitivepropositionswhichPeanoassumesare:(l)oisanumber.(z)Thesuccessorofanynumberisanumber.G)Notwonumbershavethesamesuccessor.1Weshalluse"number"inthissenseinthepresentehaptor..A.Itonwardsthewordwillbeusedinamoregeneralsense.\nfntoductiontoMatltematicalPhilosoplry(+)oisnotthesuccessorofanynumber.G)Arypropertywhichbelongstoo,andalsotothesuccessorofeverynumberwhichhastheproperty,belongstoallnumbers.Thelastoftheseistheprincipleofmathematicalinduction.Weshallhavemuchtosayconcerningmathematicalinductioninthesequel;forthepresent,weareconcernedwithitonlyasitoccursinPeano'sanalysisofarithmetic.Letusconsiderbrieflythekindofwayinwhichthetheoryofthenaturalnumbersresultsfromthesethreeideasandfivettpropositions.Tobeginwith,wedefinerasthesuccessorofor"ttzasthesuccessorofrr"andsoon.WecanobviouslygoDnaslongaswelikewiththesedefinitions,since,invirtueof(z),everynumberthatwereachwillhaveasuccessor,and,invirtueofb),thiscannotbeanyofthenumbersalreadydefined,because,ifitwere,twodifierentnumberswouldhavethesamesuccessor;andinvirtueof(+)noneofthenumberswereachintheseriesofsuccessorscanbeo.Thustheseriesofsuccessorsgivesusanendlessseriesofcontinuallynewnumbers.Invirtueof(S)allnumberscomeinthisseries,whichbeginswithoandtravelsonthroughsuccessivesuccessors:for(a)obelongstothisseries,and(b)if.anumberabelongstoit,sodoesitssuccessor,whence,bymathematicalinduction,everynumberbelongstotheseries.Supposewewishtodefinethesumoftwonumbers.Takinganynumberm,wedefinen+oastn,andm*(n*t)asthesuccessorof.m*nInvirtueof(S)thisgivesadefinitionofthesumof.mandn,whatevernumberzmaybe.Similarlywecandefinetheproductofanytwonumbers.ThereadercaneasilyconvincehimselfthatanyordinaryelementaryProPositionofarithmeticcanbeprovedbymeansofourfivepremisses,andifhehasanydifficultyhecanfindtheproofinPeano.ItistimenowtoturntotheconsiderationswhichmalceitnecessarytoadvancebeyondthestandpointofPeano,who\nTheSniesofNaturalNumbers7representsthelastperfectionofthe"arithmetisation"of'nmathematics,tothatofFrege,whofirstsucceededin"logicisingmathematics,e.e.inreducingtologicthearithmeticalnotionswhichhispredecessorshadshowntobesufficientformathematics.Weshallnot,inthischapter,actuallygiveFrege'sdefinitionofnumberandofparticularnumbers,butweshallgivesomeofthereasonswhyPeano'streatmentislessfinalthanitappearstobe'Inthefirstplace,Peano'sthreeprimitiveideas-namely,"or"tttttt-21gnumberrt'andsuccessslcapableofaninfinitenumberofdifierentinterpretations,allofwhichwillsatisfythefiveprimitivepropositions.Wewillgivesomeexamples.(r)Let$o"betakentomeantoo,andlett6numberttbetakentomeanthenumbersfromrooonwardintheseriesofnaturalnumbers.Thenallourprimitivepropositionsaresatisfied,eventhefourth,for,thoughrooisthesuccessorofttgg,ggisnotanumber"inthesensewhichwearenowgivingtotheword"number."Itisobviousthatanynumbermaybesubstitutedforroointhisexample.$tt(z)Leto"haveitsusualmeaning,butletnumber"ttmeanwhatweusuallycallevennumbersrttandletthe"successor"ofanumberbewhatresultsfromaddingtwotoit.Then3(!"willstandforthenumbertwo,3c2"willstandttttforthenumberfour,andsoon;theseriesofnumbersnowwillbeo,two,four,six,eight...AllPeano'sfivepremissesaresatisfiedstill.c(tt(3)Leto"meanthenumberone,let"numbermeanthesetrrt,lrt,1t6r...t6ttttandletsuccessormeanhalf.ttThenallPeanotsfiveaxiomswillbetrueofthisset.Itisclearthatsuchexamplesmightbemultipliedindefinitely.Infact,givenanyseriestCO,fr1,t?CZ,frV...tCne...\nfntroductiontoMathematicalPhilosophywhichisendless,containsnorepetitions,hasabeginning,andhasnotermsthatcannotbereachedfromthebeginninginafinitenumberofsteps,wehaveasetoftermsverifyingPeano'saxioms.Thisiseasilyseen,thoughtheformalproofissome-t(ttttwhatlong.Leto"meanro,letnumbermeanthewhole.t'?'cetofterms,andletthesuccessorofftnmeanfin+t,Then('o(t)isanumberr"i,e,roisamemberoftheset.t'The(r)successorofanynumberisanumberrtti.e.takinganytermxnintheset,*r,*,isalsointheset,tt(l)Notwonumbershavethesamesuccessorrt'i,c,iIx^andxnaretwodifierentmembersoftheset,r,r+1andxn*,aredifferent;thisresultsfromthefactthat(byhypothesis)therearenorepetitionsintheset.(+)"oisnotthesuccessorofanynumberrt'i.e.noterminthesetcomesbefore*0.fi)Thisbecomes:Anypropertywhichbelongstor0,andbelongstoxtt+rprovideditbelongsto#n,belongstoallthex's.Thisfollowsfromthecorrespondingpropertyfornumberg.Aseriesoftheformt(g1lC1;1C22nn,inwhichthereisafirstterm,asuccessortoeachterm(sothatthereisnolastterm),norepetitions,andeverytermcanbereachedfromthestartinafinitenumberofsteps,iscalledaprogression.Progressionsareofgreatimportanceintheprinci-plesofmathematics.Aswehavejustseen,everyprogressionverifiesPeano'sfiveaxioms.Itcanbeproved,conversely,thateveryserieswhichverifiesPeano'sfiveaxiomsisapro-gression.Hencethesefiveaxiomsmaybeusedtodefinethettttclassofprogressions:progressions"atethoseserieswhichverifythesefiveaxioms."Aryprogressionmaybetakenasthebasisofpuremathematics:wemaygivethename6(ot't'tttoitsfirstterm,thenamenumbertothewholesetofitsttterms,andthenamesuccessor"tothenextintheprogression.Theprogressionneednotbecomposedofnumbers:itmaybe\nTlteSeriesofNaturalNumbers9composedofpointsinspace,ormomentsoftime,oranycithertermsofwhichthereisaninfinitesupply.Eachdifierentprogressionwillgiverisetoadifierentinterpretationofallthepropositionsoftraditionalpuremathematics;allthesepossibleinterpretationswillbeequallytrue.InPeano'ssystemthereisnothingtoenableustodistinguishbetweenthesedifierentinterpretationsofhisprimitiveideas.Itisassumedthatweknowwhatismeantby"orttandthatweshallnotsupposethatthissymbolmeansrooorCleopatra'sNeedleoranyoftheotherthingsthatitmightmean.t(t'ttttsuccessorThispoint,thato"andnumberand"cannotbedefinedbymeansofPeano'sfiveaxioms,butmustbeindependentlyunderstood,isimportant.Wewantournumbersnotmerelytoverifymathematicalformula,buttoapplyintherightwaytocommonobjects.Wewanttohave'3tenfingersandtwoeyesandonenose.Asysteminwhichr"Kmeantloo,and2"meanttot,andsoon,mightbeallrightforpuremathematics,butwouldnotsuitdailylife.WewantKo"andttnumber"andttsuccessor"tohavemeaningswhichwillgiveustherightallowanceoffingersandeyesandnoses.Wehavealreadysomeknowledge(thoughnotsufficientlyttttarticulateoranalytic)ofwhatwemeanbyrand"2"ar,.dsoon,andouruseofnumbersinarithmeticmustconformtothisknowledge.WecannotsecurethatthisshallbethecasebyPeano'smethod;allthatwecando,ifweadopthismethod,istosayttweknowwhatwemeanbytotandtnumbertandtsuccessorrtthoughwecannotexplainwhatwemeanintermsofothersimplerconcepts."Itisquitelegitimatetosaythiswhenwemust,andatsornepointweallmust;butitistheobjectofmathematicalphilosophytoputoffsayingitaslongaspossible.Bythelogicaltheoryofarithmeticweareabletoputitoffforaverylongtime.Itmightbesuggestedthat,insteadofsettingoP"o"andttttttnumberandsuccessor"astermsofwhichweknowthemeaningalthoughwecannotdefinethem,wemightletthem\nrofnnoductiont0MatlzernaticalPhilosophystandforanythreetermsthatverifyPeano'sfiveaxioms.Theywillthennolongerbetermswhichhaveameaningthatisdefinitet'variablesr"thoughundefined:theywillbetermsconcerningwhichwemakecertainhypotheses,namely,thosestatedinthefiveaxioms,butwhichareotherwiseundetermined.Ifweadoptthisplan,ourtheoremswillnotbeprovedconcerninganascer-tttainedsetoftermscalledthenaturalnumbersr"butconcerningallsetsoftermshavingcertainproperties.Suchaprocedureisnotfallacious;indeedforcertainpurposesitrePresentsavaluablegeneralisation.Butfromtwopointsofviewitfailstogiveanadequatebasisforarithmetic.Inthefirstplace,itdoesnotenableustoknowwhetherthereareanysetsoftermsverifyingPeano'saxioms;itdoesnotevengivethefaintestsuggestionofanywayofdiscoveringwhethertherearesuchsets.Inthesecondplace,asalreadyobserved,wewantournumberstobesuchascanbeusedforcountingcommonobjects,andthisrequiresthatournumbersshouldhaveadefinitemeaning,notmerelythattheyshouldhavecertainformalproperties.Thisdefinitemeaningisdefinedbythelogicaltheoryofarithmetic.\nCHAPTERIIDEFINITIONOFNUMBERTunquestion'3Whatisanumber?"isonewhichhasbeenoftenasked,buthasonlybeencorrectlyansweredinourowntime.TheanswerwasgivenbyFregein1884,inhisGrundlagcnderAritbmetih.LAlthoughthisbookisquiteshort,notdifficult,andoftheveryhighestimportance,itattractedalmostnoattention,andthedefinitionofnumberwhichitcontainsre-mainedpracticallyunknownuntilitwasrediscoveredbythepresentauthorinr9or.Inseekingadefinitionofnumber,thefirstthingtobeclearaboutiswhatwemaycallthegrammarofourinquiry.Manyphilosophers,whenattemptingtodefinenumber,arereallysettingtoworktodefineplurality,whichisquiteadifferentthing.Nuffiberiswhatischaracteristicofnumbers,asrnaniswhatischaracteristicofmen.Apluralityisnotaninstanceofnumber,butofsomeparticularnumber.Atrioofmen,forexample,isaninstanceofthenumber3,andthenumber3isaninstanceofnumber;butthetrioisnotaninstanceofnumber.Thispointmayseemelementaryandscarcelyworthmentioning;yetithasprovedtoosubtleforthephilosophers,withfewexceptions.Aparticularnumberisnotidenticalwithanycollectionoftermshavingthatnumber:thenumber3isnotidenticalwithIThesameanswerisgivenmorefullyaadwithmoredevelopmentinlliisGrundgasctzaduArithmetih,vol.i.,1893.\ntzInrodactiontoMathcmaticalPhilosophythetrioconsistingofBrown,Jones,andRobinson.Thenumber3issomethingwhichalltrioshaveincommon,andwhichdis-tinguishesthemfromothercollections.Anumberissomethingthatcharacterisescertaincollections,namely,thosethathavethatnumber.ttInsteadofspeakingofacollectionrttweshallasanrlerpeakttt'get.ttofaclassrttorsometimecaOtherwordgusedinttttmathematicsforthesamethingareaggregate"andmani-fold."Weshallhavemuchtosaylateronaboutclasses.Forthepresent,wewillsayaslittleaspossible.Buttherearesomeremarksthatmustbemadeimmediately.Aclassorcollectionmaybedefinedintwowaysthatatfirstsightseemquitedistinct.Wemayenumerateitsmembers,aswhenwesa/r"ThecollectionImeanisBrown,Jones,andRobinson."Orwemaymentionadefiningproperty,aswhentt66wespeakofmankind"ortheinhabitantsofLondon."Thet'exten-definitionwhichenumeratesiscalledadefinitionbysion,"andtheonewhichmentionsadefiningpropertyiscalledadefinitionby"intension."Ofthesetwokindsofdefinition,theonebyintensionislogicallymorefundamental.Thisisshownbytwoconsiderationst(l)thattheextensionaldefini-tioncanalwaysbereducedtoanintensionalone;(z)thattheintensionaloneoftencannoteventheoreticallybereducedtotheextensionalone.Eachofthesepointeneedrawordoferplanation.(r)Brown,Jones,andRobinsonallofthempossessacertainpropertywhichispossessedbynothingelseinthewholeuniverse,namely,thepropertyofbeingeitherBrownorJonesorRobinson.ThispropertycanbeusedtogiveadefinitionbyintensionoftheclassconsistingofBrownandJonesandRobinson.Con-sidersuchaformulaas"xisBrownor*isJonesorrisRobinson."Thisformulawillbetrueforjustthreer's,namely,BrownandJonesandRobinson.Inthisrespectitresemblesacubicequa-tionwithitsthreeroots.Itmaybetakenasassigningapropertycomrnontothemembersofthedassconsistingofthesethree\nDefnitionofNumber13men,andpeculiartothem.Asimilartreatmentcanobvioualybeappliedtoanyotherclassgiveninextension.(z)Itisobviousthatinpracticewecanoftenknowagreatdealaboutaclasswithoutbeingabletoenumerateitsmembers.Noonemancouldactuallyenumerateallmen,orevenalltheinhabitantsofLondon,yetagreatdealisknownabouteachoftheseclasses.Thisisenoughtoshowthatdefinitionbyextensionisnotntccssnrytoknowledgeaboutaclass.Butwhenwecometoconsiderinfiniteclasses,wefindthatenumerationisnoteventheoreticallypossibleforbeingswhoonlyliveforafinitetime.Wecannotenumerateallthenaturalnumbers:th.yareo,t,z,3,andsoot.Atsomepointwemustcontentourselvegwithttandsoon.ttWecannotenumerateallfractionsorallirrationalnumbers,orallofanyotherinfinitecollection.Thusourknow-ledgeinregardtoallsuchcollectionscanonlybederivedfromadefinitionbyintension.Theseremarksarerelevant,whenweareseekingthedefinitionofnumber,inthreedifferentways.Inthefirstplace,numbersthemselvesformaninfinitecollection,andcannotthereforebedefinedbyenumeration.Inthesecondplace,thecollectionshavingagivennumberoftermsthemselvespresumablyformaninfinitecollection:itistobepresumed,forexample,thatthereareaninfinitecollectionoftriosintheworld,forifthiswerenotthecasethetotalnumberofthingsintheworldwouldbefinite,which,thoughpossible,seemsunlikely.Inthethirdttplace,wewishtodefinenumber"insuchawaytiliatinfinitenumbersmaybepossible;thuswemustbeabletospeakofthenumberoftermsinaninfinitecollection,andsuchacollectionmustbedefinedbyintension,i.e.by^propertycommontoallitsmembersandpeculiartothem.Formanypurposee,aclassandadefiningcharacteristicofitarepracticallyinterchangeable.Thevitaldifferencebetweenthetwoconsistsinthefactthatthereisonlyoneclasshavingagivensetofmembers,whereastherearealwaysmanydifierentcharacteristicsbywhichagivenclassmaybedefined.Men\nr+fnnoductiontoMathematicalPhilasoplt1maybedefinedasfeatherlessbipeds,orasrationalanimals,or(morecorrectly)bythetraitsbywhichSwiftdelineatestheYahoos.Itisthisfactthatadefiningcharacteristicisneveruniquewhichmakesclassesuseful;otherwisewecouldbecontentwiththepropertiescommonandpeculiartotheirmembers.lAnyoneofthesepropertiescanbeusedinplaceoftheclasswheneveruniquenessisnotimportant.Returningnowtothedefinitionofnumber,itiscleart"hatnumberisawayofbringingtogethercertaincollections,namely,thosethathaveagivennumberofterms.Wecansupposeallcouplesinonebundle,alltriosinanother,andsoon.Inthiswayweobtainvariousbundlesofcollections,eachbundleconsistingofallthecollectionsthathaveacertainnumberofterms.Eachbundleisaclasswhosemembersarecollections,i.e.classes;thuseachisaclassofclasses.Thebundlecon-sistingofallcouples,forexample,isaclassofclasses:eachcoupleisaclasswithtwomembers,andthewholebundleofcouplesisaclasswithaninfinitenumberofmembers,eachofwhichisaclassoftwomembers.Howshallwedecidewhethertwocollectionsaretobelongtothesamebundle?Theanswerthatsuggestsitselfis:"Findouthowmanymemberseachhas,andputtheminthesamebundleiftheyhavethesamenumberofmembers."Butthispresupposesthatwehavedefinednumbers,andthatweknowhowtodiscoverhowmanytermsacollectionhas.Wearesousedtotheoperationofcountingthatsuchapresuppositionmighteasilypassunnoticed.Infact,however,counting,thoughfamiliar,islogicallyaverycomplexoperation;more-overitisonlyavailable,asameansofdiscoveringhowmanytermsacollectionhas,whenthecollectionisfinite.Ourdefirti-tionofnumbermustnotassumeinadvancethatallnumbersarefinite;andwecannotinanycase,withoutaviciouscircle,1Aswillbeexplainedlater,classesmayberegardedaslogicalfictions,manufacturedoutofdefiningcharacteristics.Butforthepresentitwillsimplifyourexpositiontotreatclassesasiftheywerereal.\nDefnitionofNunberrsusecountingtodefinenumbers,becausenumbersareusedincounting.Weneed,therefore,someothermethodofdecidingwhentwocollectionshavethesamenumberofterms.Inactualfact,itissimplerlogicallytofindoutwhethertwocollectionshavethesamenumberoftermsthanitistodefinewhatthatnumberis.Anillustration$'illmakethisclear.Iftherewerenopolygamyorpolyandryanywhereintheworld,itisclearthatthenumberofhusbandslivingatanymomentwouldbeexactlythesameasthenumberofwives.Wedonotneedacensustoassureusofthis,nordoweneedtoknowwhatistheactualnumberofhusbandsandofwives.Weknowthenumbermustbethesameinbothcollections,becauseeachhusbandhasonewifeandeachwifehasonehusband.Therelationofhusbandandwifeiswhatiscalled"one-one."Arelationissaidtobe"one-one"when,if.xhastherelationinquestiontolrloothertermr'hasthesamerelationtoy,andrdoesnothavethesamerelationtoanytermy'otherthany.Whenonlythefirstofthesetwoconditionsisfulfilled,t'therelationiscalled"one-many;whenonlythesecondisfulfilled,itiscalled"many-one."Itshouldbeobservedthatthenumberrisnotusedinthesedefinitions.lnChristiancountries,therelationofhusbandtowifeisone-one;inMahometancountriesitisone-many;inTibetitismany-one.Therelationoffathertosonisone-many;thatofsontofatherismany-one,butthatofeldestsontofatherisone-one.If.nisanynumber,therelationof.ntoz{risone-one;soistherelationofntoznorto3n.Whenweareconsideringonlypositivenumbers,therelationof.nton2isone-one;butwhennegativenumbersareadmitted,itbecomestwo-one,sincenand-nhavethesamesquare.Theseinstancesshouldsufficetomakeclearthenotionsofone-one,one-many,andmany-onerelations,whichpl^yagreatpartintheprinci-plesofmathematics,notonlyinrelationtothedefinitionofnumbers,butinmanyotherconnections.ttTwoclassesaresaidtobesimilar'nwhenthereisaone-one\n16[ntnaductiontoMath'ematicalPltilosopltyrelationwhichcorrelatesthetermsoftheoneclasseachwithonetermoftheotherclass,inthesamemannerinwhichtherelationofmarriagecorrelateshusbandswithwives.Afewpreliminarydefinitionswillhelpustostatethisdefinitionmoreprecisely.Theclassofthosetermsthathaveagivenrelationtosomethingorotheriscalledthedomainof'thatrelation:thusfathersarethedomainoftherelationoffathertochild,husbandsarethedomainoftherelationofhusbandtowife,wivesarethedomainoftherelationofwifetohusband,andhusbandsandwivestogetherarethedomainoftherelationofmarriage.Therelationofwifetohusbandiscalledtheconaerseoftherelationofhusbandtowife.Similarlylessistheconverseof.greater,lateristheconverseof.eailier,andsoon.Generally,theconverseofagivenrelationisthatrelationwhichholdsbetweenyandxwheneverthegivenrelationholdsbetweenxaridy.Theconaersedomainof.arelationisthedomainofitsconv€rse:thustheclassofwivesistheconversedomainoftherelationofhusbandtowife.Wemaynowstateourdefinitionofsimilarityasfollows:-Oneclassissaidtobe"sirnilar"toAnotberwhentbereisaone-onercIationofwbicbthconeclassisthedomain,ubiletheatberistbeconoersedomain.Itiseasytoprove(r)thateveryclassissimilartoitself,(z)thatifaclassoissimilartoaclassB,thenBissimilartocb(3)thatifaissimilartopandBtoltthenoissimilartoy.Arelationissaidtoberef.exivewhenitpossessesthefirstoftheseproperties,syfnmetricalwhenitpossessesthesecond,andtransi-tivewhenitpossessesthethird.Itisobviousthatarelationwhichissymmetricalandtransitivemustbereflexivethroughoutitsdomain.RelationswhichPossesstheseProPertiesareanimportantkind,anditisworthwhiletonotethatsimilarityisoneofthiskindofrelations.Itisobvioustocommonsensethattwofiniteclasseshavethesamenumberoftermsiftheyaresimilar,butnototherwise.Theactofcountingconsistsinestablishingaone-onecorrelation\nDefnitionofNunberrybetweenthesetofobjectscountedandthenaturalnumbers(excludingo)thatareusedupintheprocess.Accordinglycommonsenseconcludesthatthereareasmanyobjectsinthesettobecountedastherearenumbersuptothelastnumberusedinthecounting.Andwealsoknowthat,solongasweconfineourselvestofinitenumbers,therearejustanumbersfromrupton.Henceitfollowsthatthelastnumberusedincountingacollectionisthenumberoftermsinthecollection,providedthecollectionisfinite.Butthisresult,besidesbeingonlyapplicabletofinitecollections,dependsuponandassumesthefactthattwoclasseswhicharesimilarhavethesamenumberofterms;forwhatwedowhenwecount(r"y)roobjectsistoshowthatthesetoftheseobjectsissimilartothesetofnumbersrtoro.Thenotionofsimilarityislogicallypresupposedintheoperationofcounting,andislogicallysimplerthoughlessfamiliar.Incounting,itisnecessarytotaketheobjectscountedinacertainorder,asfirst,second,third,etc.,butorderisnotoftheessenceofnumber:itisanirrelevantaddition,anun-necessarycomplicationfromthelogicalpointofview.Thenotionofsimilaritydoesnotdemandanorder:forexample,wesawthatthenumberofhusbandsisthesameasthenumberofwives,withouthavingtoestablishanorderofprecedenceamongthem.Thenotionofsimilarityalsodoesnotrequirethattheclasseswhicharesimilarshouldbefinite.Take,forexample,thenaturalnumbers(excludingo)ontheonehand,andthefractionswhichhaverfortheirnumeratorontheotherhand:itisobviousthatwecancorrelatezwithf,3with$,andsoon,thusprovingthatthetwoclassesaresimilar.Wemaythususethenotionof"similarity"todecidewhentwocollectionsaretobelongtothesamebundle,inthesenseinwhichwewereaskingthisquestionearlierinthischapter.Wewanttomakeonebundlecontainingtheclassthathasnomembers:thiswillbeforthenumbero.Thenwewantabundleofalltheclassesthathaveonemember:thiswillbeforthenumberr.Then,forthenumber2,wewantabundleconsisting\nr8fnnoductiontoMathematicalPlzilosoplt1ofallcouples;thenoneofalltrios;andsoon.Givenanycollec-tion,wecandefinethebundleitistobelongtoasbeingtheclassofallthosecollectionsthatare"similar"toit.Itisveryeasytoseethatif(forexample)acollectionhasthreemembers,theclassofallthosecollectionsthataresimilartoitwillbetheclassoftrios.Andwhatevernumberoftermsacollectionmayhave,thosecollectionsthatare"similar"toitwillhavethesamenumberofterms.Wemaytakethisasadef.nitionof"havingthesamenumberofterms."Itisobviousthatitgivesresultsconformabletousagesolongasweconfineourselvestofinitecollections.Sofarwehavenotsuggestedanythingintheslightestdegreeparadoxical.Butwhenwecometotheactualdefinitionofnumberswecannotavoidwhatmustatfirstsightseemaparadox,thoughthisimpressionwillsoonwearoff.Wenaturallythinkthattheclassofcouples(forexample)issomethingdifierentfromthenumberz.Butthereisnodoubtabouttheclassofcouples:itisindubitableandnotdifficulttodefine,whereasthenumberz,inanyothersense,isametaphysicalentityaboutwhichwecanneverfeelsurethatitexistsorthatwehavetrackeditdown.Itisthereforemoreprudenttocontentourselveswiththeclassofcouples,whichwearesureof,thantohuntforaproblematicalnumber2whichmustalwaysremainelusive.Accordinglywesetupthefollo*itgdefinition:-Thenumberofaclassistbeclassofallthoseclassesthataresimilartoit.Thusthenumberofacouplewillbetheclassofallcouples.Infact,theclassofallcouplesvnllbethenumberz,accordingtoourdefinition.Attheexpenseofalittleoddity,thisdefinitionsecuresdefinitenessandindubitableness;anditisnotdifficulttoprovethatnumberssodefinedhavealltheProPertiesthatweexpectnumberstohave.Wemaynowgoontodefinenumbersingeneralasanyoneofthebundlesintowhichsimilaritycollectsclasses.Anumberwillbeasetofclassessuchasthatanytwoaresimilartoeach\nDefnirionofNrmberr9other,andnoneoutsidethesetaresimilartoanyinsidetheset.Inotherwords,anumber(ingeneral)isanycollectionwhichisthenumberofoneofitsmembers;or,moresimplystill:Anumberisanythingwltichisthenumberofso?neclass.Suchadefinitionhasaverbalappearanceofbeingcircular,butinfactitisnot.Wedefine"thenumberofagivenclass"withoutusingthenotionofnumberingeneral;thereforewemaydefinenumberingeneralintermsof"thenumberofagivenclass"withoutcommittinganylogicalerror.Definitionsofthissortareinfactverycommon.Theclassoffathers,forexample,wouldhavetobedefinedbyfirstdefiningwhatitistobethefatherofsomebody;thentheclassoffatherswillbeallthosewhoaresomebody'sfather.Similarlyifwewanttodefinesquarenumbers(r"y),wemustfirstdefinewhatwemeanbysayingthatonenumberisthesquareofanother,andthendefinesquarenumbersasthosethatarethesquaresofothernumbers.Thiskindofprocedureisverycommon,anditisimportanttorealisethatitislegitimateandevenoftennecessary.Wehavenowgivenadefinitionofnumberswhichwillserveforfinitecollections.Itremainstobeseenhowitwillserveforinfinitecollections.Butfirstwemustdecidewhatwemeanby"finite"and"infinitert'whichcannotbedonewithinthelimitsofthepresentchapter.\nCHAPTERIIIFINITUDEANDMATHEMATICALINDUCTIONTnnseriesofnaturalnumbers,aswesawinChapterI.,canallbedefinedifweknowwhatwemeanbythethreeterms"or".,,,numberrt,andsuccessor.rtButwemaygoastePfarther:wecandefineallthenaturalnumbersifweknowwhatwemeanby,,oDand,,successor.,tItwillhelpustounderstandthedifierencebetweenfiniteandinfinitetoseehowthiscanbedone,andwhythemethodbywhichitisdonecannotbeextended((beyondthefinite.Wewillnotyetconsiderhowo))and"suc-cessor"aretobedefined:wewillforthemomentassumethatweknowwhatthesetermsmean'andshowhowthenceallothernaturalnumberscanbeobtained.Itiseasytoseethatwecanreachanyassignednumber,sayc(tt3O'OOO.wefirStdefinet,)asthesuccessorofor"thenwe(cttdefine2t'astheSuccessorofIr"andsoon.Inthecaseofanassignednumber,suchas3O2ooO,theproofthatwecanreachitbyproceedingstepbystepinthisfashionmaybemade,ifwehavethepatience,byactualexperiment:wecangoonuntilweactuallyarriveat3o,ooo.Butalthoughthemethodofexperimentisavailableforeachparticularnaturalnumber'itisnotavailableforprovingthegeneralpropositionthatallsuchnumberscanbereachedinthiswaY,i.e.byproceedingfromostepbystepfromeachnumbertoitssuccessor.Isthereanyotherwaybywhichthiscanbeproved?Letusconsiderthequestiontheotherwayround.Whatare(6ttthenumbersthatcanbereached,giventhetermsOand\nFinitudeandMathematicalInductionztttttsuccessorIIsthereanywaybywhichwecandefinethewholeclassofsuchnumbers?Wereachl,asthesuccessorofo;2,asthesuccessorofr;3,asthesuccessorofzIandsoon.Itisthis"andsoon"thatwewishtoreplacebysomethinglessvagueandindefinite.Wemightbetemptedtosaythat"andsoon"meansthattheprocessofproceedingtothesuccessormayberepeatedanyf,nitenumberoftimes;buttheproblemuponwhichweareengagedistheproblemofdefining..finitenumberrt'andthereforewemustnotusethisnotioninourdefini-tion.Ourdefinitionmustnotassumethatweknowwhatafinitenumberis.Thekeytoourproblemliesinmatltematicalinduction.Itwillberememberedthat,inChapterI.,thiswasthefifthofthefiveprimitivepropositionswhichwelaiddownaboutthenaturalnumbers.Itstatedthatanypropertywhichbelongstoo,andtothesuccessorofanynumberwhichhastheproperty,belongstoallthenaturalnumbers.Thiswasthenpresentedasaprinciple,butweshallnowadoptitasadefinition.Itisnotdifficulttoseethatthetermsobeyingitarethesameasthenumbersthatcanbereachedfromobysuccessivestepsfromn€xttonext,butasthepointisimportantwewillsetforththematterinsomedetail.Weshalldowelltobeginwithsomedefinitions,whichwillbeusefulinotherconnectionsalso.Apropertyissaidtobe"hereditary"irthenatural-numberseriesif,wheneveritbelongstoanumbern,italsobelongston*r,thesuccessorofa.Similarlyaclassissaidtobe"heredi-))taryif,whenevetnisamemberoftheclass,soisn+r.Itiseasytosee,thoughwearenotyetsupposedtoknow,thattosayapropertyishereditaryisequivalenttosayingthatitbelongstoallthenaturalnumbersnotlessthansomeoneofthem,a.g.itmustbelongtoallthatarenotlessthanroo,orallthatarelessthanrooo,oritmaybethatitbelongstoallthatarenotlessthano,i.e.toallwithoutexception.Apropertyissaidtobe"induetive"whenitisahereditary\nzzInroductiontaMatlzewaticalPhilosophypropertywhichbelongstoo.Similarlyaclassis"inductive"whenitisahereditaryclassofwhichoisamember.Givenahereditaryclassofwhichoisamember,itfollowsthatrisamemberofit,becauseahereditaryclasscontainsthetuccessorsofitsmembers,andristhesuccessorofo.Similarly,givenahereditaryclassofwhichrisamember,itfollowsthatzisamemberofit;andsoon.Thuswecanproveby^step-by-stepprocedurethatanyassignednaturalnumber,say3orooo,isamemberofeveryinductiveclass.Wewilldefinethe"posterity"ofagivennaturalnumberwithrespecttotherelation"immediatepredecessor"(whichtttt)istheconverseofsuccessorallthosetermsthatbelong"ttoeveryhereditaryclasstowhichthegivennumberbelongs.Itisagaineasytoseethattheposterityofanaturalnumbercon-sistsofitselfandallgreaternaturalnumbers;butthisalsowedonotyetofficiallyknow.Bytheabovedefinitions,theposterityofowillconsistofthosetermswhichbelongtoeveryinductiveclass.Itisnownotdifficulttomakeitobviousthattheposterityofoisthesamesetasthosetermsthatcanbereachedfromobysuccessivestepsfromnexttonext.For,inthefirstplace,obelongstoboththesesets(inthesenseinwhichwehavedefinedourterms)Iinthesecondplace,ifabelongstobothsets,sodoesn+r.Itistobeobservedthatwearedealingherewiththekindofmatterthatdoesnotadmitofpreciseproof,namely,thecomparisonofarelativelyvagueideawitharelativelypreciset'thoseone.Thenotionoftermsthatcanbereachedfromobysuccessivestepsfromnexttonext"isvague,thoughitseemsasifitconveyedadefinitemeaning;ontheotherhand,"theposterityofo"ispreciseandexplicitjustwheretheotherideaishazy,Itmaybetakenasgivingwhatwernea.nltomeanwhenwespokeofthetermsthatcanbereachedfromobysuccesgivestePs.Wenowlaydownthefollo*iogdefinition:-the"naturalnumbers"arctheposterityofowitbrespccttotlte\nFinitudeandMatlrematicalfnduction23rclation"immediatepredecessor"(whichistheconverseoft'successortt).WehavethusarrivedatadefinitionofoneofPeano'sthreeprimitiveideasintermsoftheothertwo.Asaresultofthisdefinition,twoofhisprimitivepropositions-namely,rheoneassertingthatoisanumberandtheoneassertingmathematicalinduction-becomeunnecessary,sincetheyresultfromthedefini-tion.Theoneassertingthatthesuciessorofanaturalnumberisanaturalnumberisonlyneededintheweakenedformt'everynaturalnumberhasasuccessor."Wecan,ofcourse,easilydefine((o)'and,,successor,,bymeansofthedefinitionofnumberingeneralwhichwearrivedatinchapterII.Thenumberoisthenumberoftermsinaclasswhichhasnomembers,i.e.intheclasswhichiscalledthe(Gnull-class."Bythegeneraldefinitionofnumber,thenumberoftermsinthenull-classisthesetofallclassessimilartothenull-class,i.e.(asiseasilyproved)thesetconsistingofthenull-classallalone,i.e.theclasswhoseonlymemberisthenull-class.(Thisisnotidenticalwiththenull-class:ithasonemember,namely,thenull-class,whereasthenull-classitselfhasnomembers.Aclasswhichhasonememberisneveridenticalwiththatonemember,asweshallexplainwhenwecometothetheoryofclasses.)Thuswehavethefollowingpurelylogicaldefinition:-oistlteclasswhoseonlymemberisthenull-class.Itremainstodefinettsuccessor."Givenanynumbera,letobeaclasswhichhasnmembers,andletrbeatermwhichisnotamemberofo.Thentheclassconsistingofowithraddedonwillhaven+rmembers.Thuswehavethefollowingdefinition:-TltesuccessoroftltcnumberoJtermsintlteclasss,istltenumberoftermsintheclassconsistingofatogetherwithx,wherexisanyterrnnotbelongingtotbeclass.Certainnicetiesarerequiredtomakethisdefinitionperfecr,buttheyneednotconcernus.lItwillberememberedthatwe1SeePfincipiaMathcnatica,vol.ii.r11e,\n2+[ntroductiontoMatltematicalPltilosoplt1havealreadygiven(inChapterII.)alogicaldefinitionofthenumberoftermsinaclass,namely,wedefineditasthesetofallclassesthataresimilartothegivenclass.WehavethusreducedPeano'sthreeprimitiveideastoideasoflogic:wehavegivendefinitionsofthemwhichmakethemdefinite,nolongercapableofaninfinityofdifierentmeanings,astheywerewhentheywereonlydeterminatetotheextentofobeyingPeano'sfiveaxioms.WehaveremovedthemfromthefundamentalapparatusoftermsthatmustbemerelyaPPre-hended,andhavethusincreasedthedeductivearticulationofmathematics.Asregardsthefiveprimitivepropositions,wehavealreadysucceededinmakingtwoofthemdemonstrablebyourdefinitionof"naturalnumber."Howstandsitwiththeremainingthree?Itisveryeasytoprovethatoisnotthesuccessorofanynumber,andthatthesuccessorofanynumberisanumber'Butthereisadifficultyabouttheremainingprimitiveproposition,namely,t'notwonumbershavethesamesuccessot."Thedifficultydoesnotariseunlessthetotalnumberofindividualsintheuniverseisfinite;forgiventwonumbersmanda,neitherofwhichisthetotalnumberofindividualsintheuniverse,itiseasytoprovethatwecannothavern+r-nf.runlesswehaverrr:n,.Butletussupposethatthetotalnumberofindividualsintheuniversewere(say)ro;thentherewouldbenoclassofrrindividuals,andthenumberrrwouldbethenull-class.Sowouldthenumbertz.Thusweshouldhavetr:rzithereforethesuccessorofrowouldbethesameasthesuccessorofrr,althoughrowouldnotbethesameasrr.Thusweshouldhavetwodifierentnumberswiththesamesuccessor.Thisfailureofthethirdaxiomcannotarise,however,ifthenumberofindi-vidualsintheworldisnotfinite.Weshallreturntothistopicatalaterstage.lAssumingthatthenumberofindividualsintheuniverseisnotfinite,wehavenowsucceedednotonlyindefiningPeano'sI$geChapterXIII\nI;initudeandMatltenaticalfnduction2sthreeprimitiveideas,butinseeinghowtoprovehisfiveprimitivepropositions,bymeansofprimitiveideasandpropositionsbelong-ingtologic.Itfollowsthatallpuremathematics,insofarasitisdeduciblefromthetheoryofthenaturalnumbers,isonlyaprolongationoflogic.Theextensionofthisresulttothosemodernbranchesofmathematicswhicharenotdeduciblefromthetheoryofthenaturalnumbersoffersnodifficultyofprinciple,aswehaveshownelsewhere.lTheprocessofmathematicalinduction,bymeansofwhichwedefinedthenaturalnumbers,iscapableofgeneralisation.Wedefinedthenaturalnumbersasthe"posterity"ofowithrespecttotherelationofanumbertoitsimmediatesuccessor.IfwecallthisrelationN,anynumbermwtllhavethisrelationtorn+r.Apropertyis"hereditarywithrespecttoNr"orsimply"N-hereditaryr"if,wheneverthepropertybelongstoanumberm,italsobelongstorn*r,i,e.tothenumbertowhichmhastherelationN.Andanumbernvnllbesaidtobelongtothe"posterity"ofmwithrespecttotherelationNifnhaseveryN-hereditarypropertybelongingtorn.ThesedefinitionscanallbeappliedtoanyotherrelationjustaswellastoN.ThusifRisanyrelationwhateverrw€canlaydownthefollowingdefinitions'2-Apropertyiscalled"R-hereditary"when,ifitbelongstoatermr,andrhastherelationRtoy,thenitbelongstoy.AclassisR-hereditarywhenitsdefiningpropertyisR-hereditary.Atermrissaidtobean"R-ancestor"ofthetermyifyhaseveryR-hereditarypropertythatxhas,providedrisatermwhichhastherelationRtosomethingortowhichsomethinghastherelationR.(Thisisonlytoexcludetrivialcases.)1Forgeometry,insofarasitisnotpurelyanalytical,seePrinciplesofMathematics,partvi.;forrationaldynamics,ibid.,partvii.?Thesedefinitions,andthegeneralisedtheoryofinduction,areduetoFrege,andwerepublishedsolongagoasr87gintrisBegrffischrift.Inspiteofthegreatvalueofthiswork,Iwas,Ibelieve,thefirstpersonwhoeverreadit-morethantwentyyearsafteritspublication.\n26IntrodactiontoMathematicelPhihsophl'6TheR-posterity"ofxisallthetermsofwhichIisanR-ancestor.Wehaveframedtheabovedefinitionssothatifatermistheancestorofanythingitisitsownancestorandbelongstoitsownposterity.Thisismerelyforconvenience.ItwillbeobservedthatifwetakeforRtherelation"parent,"ttttttancestor"andposteritywillhavetheusualmeanings,exceptthatapersonwillbeincludedamonghisownancestorst'andposterity.Itis,ofcourse,obviousatoncethatancestor"t'parentr"mustbecapableofdefinitionintermsofbutuntilFregedevelopedhisgeneralisedtheoryofinduction,noonecouldttt'('havedefinedancestorpreciselyintermsofparent."Abriefconsiderationofthispointwillservetoshowtheimportanceofthetheory.Apersonconfrontedforthefirsttimewiththettttttproblemofdefiningancestor"intermsofparentwouldnaturallysaythatAisanancestorofZif.,betweenAandZ,thereareacertainnumberofpeople,B,C,.,ofwhomBisachildofA,eachisaparentofthenext,untilthelast,whoisaparentof.Z.Butthisdefinitionisnotadequateunlessweaddthatthenumberofintermediatetermsistobefinite.Take,forexample,suchaseriesasthefollowing:--r,-t,-1,-*'t'l'*'r'Herewehavefirstaseriesofnegativefractionswithnoend,andthenaseriesofpositivefractionswithnobeginning.Shallwesaythat,inthisseries,-litanance$torof$IItwillbesoaccordingtothebeginner'sdefinitionsuggestedabove,butitwillnotbesoaccordingtoanydefinitionwhichwillgivethekindofideathatwewishtodefine.ForthispurPose,itisessentialthatthenumberofintermediariesshouldbefinite.ttBut,aswesaw,finite"istobedefinedbymeansofmathe-maticalinduction,anditissimplertodefinetheancestralrelationgenerallyatoncethantodefineitfirstonlyforthecaseoftherelationof.nton*r,andthenextendittoothercases.Here,asconetantlyelsewhere,generalityfromthefirst,thoughitmay\nFinindcandMartemaicalInduction27requiremorethoughtatthestart,willbefoundinthelongruntoeconomisethoughtandincreaselogicalpower.'ofTheusemathematicalinductionindemonstrationswas,inthepast,somethingofamystery.Thereseemednoreason-abledoubtthatitwasavalidmethodofproof,butnoonequiteknewwhyitwasvalid.Somebelievedittobereallyacaseofinduction,inthesenseinwhichthatwordisusedinlogic.Poincar6rconsideredittobeaprincipleoftheutmostimport-ance,bymeansofwhichaninfinitenumberofsyllogismscouldbecondensedintooneargument.Wenowknowthatallsuchviewsaremistaken,andthatmathematicalinductionisadefinition,notaprinciple.Therearesomenumberstowhichitcanbeapplied,andthereareothers(asweshallseeinChapterVIII.)('towhichitcannotbeapplied.Wedef.nethenaturalnumbers"asthosetowhichproofsbymathematicalinductioncanbeapplied,i.e.asthosethatpossessallinductiveproperties.Itfollowsthatsuchproofscanbeappliedtothenaturalnumbers,notinvirtueofanymysteriousintuitionoraxiomorprinciple,butasapurelyverbalproposition.If"quadrupeds"aredefinedasanimalshavingfourlegs,itwillfollowthatanimalsthathavefourlegsarequadrupeds;andthecaseofnumbersthatobeymathematicalinductionisexactlysimilar.Weshallusethephrase"inductivenumbers"tomeanthesamesetaswehavehithertospokenofasthe6tnaturalnumbers."Thephrase"inductivenumbers"ispreferableasaffordingareminderthatthedefi.nitionofthissetofnumbersisobtainedfrommathematicalinduction.Mathematicalinductionaffords,morethananythingelse,theessentialcharacteristicbywhichthefiniteisdistinguishedfromtheinfinite.Theprincipleofmathematicalinduction'(mightbestatedpopularlyinsomesuchformaswhatcanbeinferredfromnexttonextcanbeinferredfromfirsttolast.ttThisistruewhenthenumberofintermediatestepsbetweenfirstandlastigfinite,nototherwise.AnyonewhohasevertSaicncaandMathod,chap.iv.\n28fntoductiontoMathematicalPhilosopltywatchedagoodstrainbeginningtomovewillhavenoticedhowtheimpulseiscommunicatedwithajerkfromeachtrucktothenext,untilatlasteventhehindmosttruckisinmotion.Whenthetrainisverylong,itisaverylongtimebeforethelasttruckmoves.Ifthetrainwereinfinitelylong,therewouldbeaninfinitesuccessionofjerks,andthetimewouldnevercomewhenthewholetrainwouldbeinmotion.Nevertheless,iftherewereaseriesoftrucksnolongerthantheseriesofinductivenumbers(which,asweshallsee,isaninstanceofthesmallestofinfinites),everytruckwouldbegintomovesoonerorlater'engineifthepersevered,thoughtherewouldalwaysbeothertrucksfurtherbackwhichhadnotyetbeguntomove.Thisimagewillhelptoelucidatetheargumentfromnexttonext,anditsconnectionwithfinitude.Whenwecometoinfinitenumbers,whereargurnentsfrommathematicalinductionwillbenolongervalid,thepropertiesofsuchnumberswillhelptomakeclear,bycontrast,thealmostunconscioususethatismadeofmathematicalinductionwherefinitenumbersareconcerned.\nCHAPTERIVTHEDEFINITIONOFORDERWrhavenowcarriedouranalysisoftheseriesofnaturalnumberstothepointwherewehaveobtainedlogicaldefinitionsofthemembersofthisseries,ofthewholeclassofitsmembers,andoftherelationofanumbertoitsimmediatesuccessor.Wemustnowconsidertheserialcharacterofthenaturalnumbersintheordero,r,2,3,Weordinarilythinkofthenum-bersasinthisorder,anditisanessentialpartoftheworkccofanalysingourdatatoseekadefinitionof"order"orseries"inlogicalterms.Thenotionoforderisonewhichhasenormousimportanceinmathematics.Notonlytheintegers,butalsorationalfrac-tionsandallrealnumSershaveanorderofmagnitude,andthisisessentialtomostoftheirmathematicalproperties.Theorderofpointsonalineisessentialtogeometry;soistheslightlymorecomplicatedorderoflinesthroughapointinaplane,orofplanesthroughaline.Dimensions,ingeometryrareadevelopmentoforder.Theconceptionof.alimal,whichunderliesallhighermathematics,isaserialconception.Therearepartsofmathematicswhichdonotdependuponthenotionoforder,buttheyareveryfewincomparisonwiththepartsinwhichthisnotionisinvolved.Inseekingadefinitionoforder,thefirstthingtorealiseisthatnosetoftermshasjustoneordertotheexclusionofothers.Asetoftermshasalltheordersofwhichitiscapable.Some-timesoneorderissomuchmorefamiliarandnaturaltoour29\n30fntnductionbMaizemaicalPltilonphythoughtsthatweareinclinedtoregarditastbeorderofthatsetofterms;butthisisamistake.Thenanrralnumbers-ttortheinductive"numbers,asweshallalsocallthem-occurtousmostreadilyinorderofmagnitude;buttheyarecapableofaninfinitenumberofotherarrangements.Wemight,forexample,considerfirstalltheoddnumbersandthenalltheevennumbers;orfirstt,thenalltheevennumbers,thenalltheoddmultiplesof3,thenallthemultiplesof5butnotof2or3,thenallthemultiplesof7butnotofzor3or5,andsoonthroughthewholeseriesofprimes.Whenwesaythatwettttarrangethenumbersinthesevariousorders,thatisaninaccurateexpression:whatwereallydoistoturnourattentiontocertainrelationsbetweenthenaturalnumbers,whichthem-selvesgeneratesuch-and-suchanarrangement.Wecannottttmorearrangethenaturalnumbersthanwecanthesta'rryheavens;butjustaswemaynoticeamongthefixedstarseithertheirorderofbrightnessortheirdistributioninthesky,sotherearevariousrelationsamongnumberswhichmaybeobserved,andwhichgiverisetovariousdifierentordersamongnumbers,allequallylegitimate.Andwhatistrueofnurnbersisequallytrueofpointsonalineorofthemomentsoftime:oneorderismorefamiliar,butothersareequallyvalid.Wemight,forexample,takefirst,onaline,allthepointsthathaveintegralco-ordinates,thenallthosethathavenon-integralrationalco-ordinates,thenallthosethathavealgebraicnon-rationalco-ordinates,andsoon,throughanysetofcomplica-tionsweplease.Theresultingorderwillbeonewhichthepointsofthelinecertainlyhave,whetherwechoosetonoticeitornot;theonlythingthatisarbitraryaboutthevariousordersofasetoftermsisourattention,forthetermsthemselveshavealwaysalltheordersofwhichtheyarecapable.Oneimportantresultofthisconsiderationisthatwemustnotlookforthedefinitionoforderinthenatureofthesetoftermstobeordered,sinceonesetoftermshasmanyorders.Theorderli,es,notintheclassofterms,butinarelationamong\nThcDefnitionofOrder3rthemembersoftheclass,inrespectofwhichsomeappearasearlierandsomeaslater.Thefactthataclassmayhavemanyordersisduetothefactthattherecanbemanyrelationsholdingamongthemembersofonesingleclass.Whatpropertiesmustarelationhaveinordertogiverisetoanorder?Theessentialcharacteristicsofarelationwhichistogiverisetoordermaybediscoveredbyconsideringthatinrespectofsucharelationwemustbeabletosay,ofanytwotermsintheclasswhichistobeordered,thatone"precedes"andtheother"follows."Now,inorderthatwemaybeabletousethesewordsinthewayinwhichweshouldnaturallyunderstandthem,werequirethattheorderingrelationshouldhavethreeproperties.-(t)Ifrpreced.ry,ymustnotalsoprecede*.Thisisanobviouscharacteristicofthekindofrelationsthatleadtoseries.If*islessthan!,!isnotalsolessthanr.Ifrisearlierintimethan!,!isnotalsoearlierthanr.Ifxistotheleftof!,!isnottotheleftof,x.Ontheotherhand,relationswhichdonotgiverisetoseriesoftendonothavethisproperty.Ifrisabrotherorsisterofy,yisabrotherorsisterof*.If*isofthesameheightasltyisofthesameheightasr.Ifrisofadifferentheightfrom!,!isofadifferentheightfromr.Inalltlresecases,whentherelationholdsbetweenxandy,italsoholdsbetweenyandx.Butwithserialrelationssuchathingcannothappen.Arelationhavingthisfirstpropertyiscalledasymmetrical.(z)Ifrprecedesyandyprecedesz,#mustprecedez.Thismaybeillustratedbythesameinstancesasbefore:Iess,eailier,Itftof,Butasinstancesofrelationswhichdonothavethispropertyonlytwoofourpreviousthreeinstanceswillserve.If*isbrotherorsisterofy,andyotz,frmaynotbebrotherorsisterofz,sincerandzmaybethesameperson.Thesameappliestodifferenceofheight,butnottosamenessofheight,whichhasoursecondpropertybutnotourfirst.Therelation53fatherr"ontheotherhand,haeourfirstpropertybutnot\n32fntroductiontoMathematicalPltilosopltyoursecond.Arelationhavingoursecondpropertyiscalledtransitive.(l)Givenanytwotermsoftheclasswhichistobeordered,theremustbeonewhichprecedesandtheotherwhichfollows.Forexample,ofanytwointegers,orfractionsrorrealnumbers,oneissmallerandtheothergreater;butofanytwocomplexnumbersthisisnottrue.Ofanytwomomentsintime,onemustbeearlierthantheother;butofevents,whichmaybesimultaneous,thiscannotbesaid.Oftwopointsonaline,onemustbetotheleftoftheother.Arelationhavingthisthirdpropertyiscalledconnected.WhenarelationPossessesthesethreePloPerties,itisofthesorttogiverisetoanorderamongthetermsbetweenwhichitholds;andwhereveranorderexists,somerelationhavingthesethreepropertiescanbefoundgeneratingit.Beforeillustratingthisthesis,wewillintroduceafewdefinitions.(I)Arelationissaidtobeanaliorelative,lortobecontainedinorimptydirtersity,ifnotermhasthisrelationtoitself.ttgreaterrt'ttdifierentinsizert'ttbrotherr"Thus,forexampl",t'husbandrt'ttfatherttarealiorelatives;but"equalr"t'bornttofthesameparentsr"dearfriend"arenot.(z)Thesquareofarelationisthatrelationwhichholdsbetweentwotermsrandzwhenthereisanintermediatetermysuchthatthegivenrelationholdsbetweenxandyandbetween6(yandz.Thuspaternalgrandfather"isthesquareof"fatherr"((greaterby,ttisthesquareofttgreaterbyr,"andsoon.$)Thedomainofarelationconsistsofallthosetermsthathavetherelationtosomethingorother,andtheconaersedomainconsistsofallthosetermstowhichsomethingorotherhastherelation.Thesewordshavebeenalreadydefined,butarerecalledhereforthesakeofthefollowingdefinition:-(4)Thef,etdofarelationconsistsofitsdomainandconversedomaintogether.ITh'istermisduetoC.S.Peirce.\nTlreDefnitionofOrder33(5)Onerelationissaidtocontainorbeimpliedbyanotherifitholdswhenevertheotherholds.Itwillbeseenthatanasymmetricalrclationisthesamethingasarelationwhosesquareisanaliorelative.Itoftenhappensthatarelationisanaliorelativewithoutbeingasymmetrical,thoughanasymmetricalrelationisalwaysanaliorelative.Forttttexample,spouseisanaliorelative,butissymmetrical,sinceifristhespouseof!,!isthespouseof#.Butamongtransitiaerelations,allaliorelativesareasymmetricalaswellasoiceoersa,Fromthedefinitionsitwillbeseenthatatransitiverelationisonewhichisimpliedbyitssquare,or,aswealsosay,"con-(stains"itssquare.Thusancestor"istransitive,becausettt'anancestortsancestorisanancestor;butfatherisnottransitive,becauseafather'sfatherisnotafather.Atransitivealiorelativeisonewhichcontainsitssquareandiscontainedindiversityiorrwhatcomestothesamething,onewhosesquareimpliesbothitanddiversity-because,whenarelationistransitive,asymmetryisequivalenttobeinganaliorelative.Arelationisconnectedwhen,givenanytwodifferenttermsofitsfield,therelationholdsbetweenthefirstandthesecondorbetweenthesecondandthefirst(notexcludingthepossibilitythatbothmayhappen,thoughbothcannothappeniftherelationisasymmetrical).ttItwillbeseenthattherelationancestorr"forexample,isanaliorelativeandtransitive,butnotconnected;itisbecauseitisnotconnectedthatitdoesnotsufficetoarrangethehumanraceinaseries.ttTherelationlessthanorequaltor"amongnumbers,istransitiveandconnected,butnotasymmetricaloranaliorelative.t(Therelationgreaterorless"amongnumbersisanalio-relativeandisconnected,butisnottransitive,forif*isgreaterorlessthany,andyisgreaterorlessthanz,itmayhappenthatxandzarethesamenumber.Thusthethreepropertiesofbeing(t)analiorelative,(z)\n3+IntroductiontoMathernaticalPlzilosophytransitive,and(3)connected,aremutuallyindependent,sincearelationmayhaveanytwowithouthavingthethird.Wenowlaydownthefollowingdefinition:-Arelationisserialwhenitisanaliorelative,transitive,andconnectedIor,whatisequivalent,whenitisasymmetrical,transitive,andconnected.Aseriesisthesamethingasaserialrelation.Itmighthavebeenthoughtthataseriesshouldbethefieldofaserialrelation,nottheserialrelationitself,Butthiswouldbeanerror.Forexample,Ir213itr312i213rti2rrr3i3rrr2i312rlaresixdifferentserieswhichallhavethesamefield.Ifthefieldweretheseries,therecouldonlybeoneserieswithagivenfield.Whatdistinguishestheabovesixseriesissimplythedifferentorderingrelationsinthesixcases.Giventheorderingrelation,thefieldandtheorderarebothdeterminate.Thustheorderingrelationmaybetakentobetheseries,butthefieldcannotbesotaken.Givenanyserialrelation,sayP,weshallsaythat,inrespectofthisrelation,*"precedes"yif.xhastherelationPtoy,whichweshallwrite"*Py"forshort.ThethreecharacteristicswhichPmusthaveinordertobeserialare:(t)WemustneverhavexPx,i.e.notermmustprecedeitself.(z)P,mustimplyP,i.e.if*precedesyandyprecedesz,#mustprecedez.6)IfrandyaretwodifierenttermsinthefieldofP,weshallhaverPyorlPli.,i.e.oneofthetwomustprecedetlreother.Thereadercaneasilyconvincehimselfthat,wherethesethreepropertiesarefoundinanorderingrelation,thecharacteristicsweexpectofserieswillalsobefound,andviceversa.Wearethereforejustifiedintakingtheaboveasadefinitionoforder\nTheDefnirionofOrder3jorseries.Anditwillbeobservedthatthedefinitioniseffectedinpurelylogicalterms.Althoughatransitiveasymmetricalconnectedrelationalwaysexistswhereverthereisaseries,itisnotalwaystherelationwhichwouldmostnaturallyberegardedasgeneratingtheseries.Thenatural-numberseriesmayserveasanillustration.Therelationweassumedinconsideringthenaturalnumberswastherelationofimmediatesuccession,d.e.therelationbetweenconsecutiveintegers.Thisrelationisasymmetrical,butnottransitiveorconnected.Wecan,however,derivefromit,6sbythemethodofmathematicalinduction,theancestral"relationwhichweconsideredintheprecedingchapter.Thist'relationwillbethesameas"lessthanorequaltoamonginductiveintegers.Forpurposesofgeneratingtheseriesoft'lessnaturalnumbers,w€wanttherelationthanrttexcluding"equalto.ttThisistherelationof.mtoawhenmisanancestorofabutnotidenticalwithn,or(whatcomestothesamething)whenthesuccessorofmisanancestorof.ninthesenseinwhichanumberisitsownancestor.Thatistosay,weshalllaydownthefollowingdefinition:-Aninductivenumbermissaidtobelessthananothernumberzwhennpossesseseveryhereditarypropertypossessedbythesuccessorof.m.It,iseasytosee,andnotdifficulttoprove,thattherelationt'lessthanrt'sodefined,isasymmetrical,transitive,andcon-nected,andhastheinductivenumbersforitsfield.Thusbymeansofthisrelationtheinductivenumbersacquireanorderttinthesenseinwhichwedefinedthetermorderr"andthisordert'istheso-called"naturalorder,ororderofmagnitude.Thegenerationofseriesbymeansofrelationsmoreorlessresemblingthatofzton+risverycommon.TheseriesoftheKingsofEngland,forexample,isgeneratedbyrelationsofeachtohissuccessor.Thisisprobablytheeasiestway,whereitisapplicable,ofconceivingthegenerationofaseries.Inthismethodwepassonfromeachtermtothenext,aslongasthere\n36fnnoductiontoMathernaticalPlrilosoplt1isanext,orbacktotheonebefore,aelongasthereisonebefore.Thismethodalwaysrequiresthegeneralisedformofmathe-ttmaticalinductioninordertoenableustodefineearlier"and66t't'properlaterinaseriessogenerated.Ontheanalogyoffractionsr"letusgivethename"properposterityofxwithrespecttoR"totheclassofthosetermsthatbelongtotheR-posterityofsometermtowhichxhastherelationR,inthesensewhich((wegavebeforetoposterityr"whichincludesaterminitsownposterity.Revertingtothefundamentaldefinitions,wefindthat('theproperposterity"^^ybedefinedasfollows.-t'properTheposterity"ofrwithrespecttoRconsistsofalltermsthatpossesseveryR-hereditarypropertypossessedbyeverytermtowhichrhastherelationR.ItistobeobservedthatthisdefinitionhastobesoframedastobeapplicablenotonlywhenthereisonlyonetermtowhichxhastherelationR,butalsoincases(ase.g.thatoffatherandchild)wheretheremaybemanytermstowhichxhastherelationR.Wedefinefurther:Aterm*isa"properancestor"ofywithrespecttoRifybelongstotheproperposterityofrwithrespecttoR.Weshallspeakforshortof"R-posterity"and"R-ancestors"whenthesetermsseemmoreconvenient.RevertingnowtothegenerationofseriesbytherelationRbetweenconsecutiveterms,weseethat,ifthismethodistobepossible,therelation"properR-ancestor"mustbeanaliorela-tive,transitive,andconnected.UnderwhatcircumstanceswillthisoccurIItwillalwaysbetransitive:nomatterwhatsortttttttofrelationRmayb",R-ancestorandproperR-ancestor"arealwaysbothtransitive.Butitisonlyundercertaincircum-stancesthatitwillbeanaliorelativeorconnected.Consider,forexample,therelationtoone'sleft-handneighbouratarounddinner-tableatwhichtherearetwelvepeople.IfwecallthisrelationR,theproperR-posterityofapersonconsistsofallwhocanbereachedbygoingroundthetablefromrighttoleft.Thisincludeseverybodyatthetable,includingthepersonhimself,since\nTheDefnirionofOrder37twelvestepsbringusbacktoourstarting-point.Thusinsucht'ProPeracase,thoughtherelationR-ancestor"isconnected,andthoughRitselfisanaliorelative,wedonotgetaseriesttttbecauseproperR-ancestorisnotanaliorelative.ItisforthisreasonthatwecannotsaythatonePersoncomesbeforeanotherwithrespecttotherelation"rightof"ortoitsancestralderivative.Theabovewasaninstanceinwhichtheancestralrelationwasconnectedbutnotcontainedindiversity.Aninstancewhereitiscontainedinfiversitybutnotconnectedisderivedfromthet(ordinarysenseofthewordancestor."IfrisaProPerancestorofy,xandycannotbethesameperson;butitisnottruethatofanytwopersonsonemustbeanancestoroftheother.Thequestionofthecircumstancesunderwhichseriescanbegeneratedbyancestralrelationsderivedfromrelationsofcon-secutivenessisoftenimportant.Someofthemostimportantcasesarethefollowing:LetRbeamany-onerelation,andletusconfineourattentiontotheposterityofsometerm#.Whenttttsoconfined,therelationproperR-ancestormustbeconnected;thereforeallthatremainstoensureitsbeingserialisthatitshallbecontainedindiversity.Thisisageneralisationoftheinstanceofthefinner-table.AnothergeneralisationconsistsintakingRtobeaone-onerelation,andincludiogtheancestryofraswellastheposterity.Hereagain,theoneconditionrequiredtosecurethegenerationofaseriesisthattherelation"ProPerR-ancestor"shallbecontainedindiversity.Thegenerationoforderbymeansofrelationsofconsecutive-ness,thoughimportantinitsownsphere,islessgeneralthanthemethodwhichusesatransitiverelationtodefinetheorder.Itoftenhappensinaseriesthatthereareaninfinitenumberofinter-mediatetermsbetweenanytwothatmaybeselected,howeverneartogetherthesemaybe.Take,forinstance,fractionsinorderofmagnitude.Betweenanytwofractionsthereareothers-forexample,thearithmeticmeanofthetwo.Consequentlythereisnosuchthingasapairofconsecutivefractions.Ifwedepended\n38fntroductiont0MatltenaticalPltilosopltyuponconsecutivenessfordefiningorder,weshouldnotbeabletodefinetheorderofmagnitudeamongfractions.Butinfacttherelationsofgreaterandlessarnongfractionsdonotdemandgenerationfromrelationsofconsecutiveness,andtherelationsofgreaterandlessamongfractionshavethethreecharacteristicswhichweneedfordefiningserialrelations.Inallsuchcasestheordermustbedefinedbymeansofatransitioerelation,sinceonlysucharelationisabletoleapoveraninfinitenumberofintermediateterms.Themethodofconsecutiveness,likethatofcountingfordiscoveringthenumberofacollection,isappro-priatetothefinite;itmayevenbeextendedtocertaininfiniteseries,namely,thoseinwhich,thoughthetotalnumberoftermsisinfinite,thenumberoftermsbetweenanytwoisalwaysfinite;butitmustnotberegardedasgeneral.Notonlyso,butcaremustbetakentoeradicatefromtheimaginationallhabitsofthoughtresultingfromsupposingitgeneral.Ifthisisnotdone,seriesinwhichtherearenoconsecutivetermswillremaindifficultandpuzzling.Andsuchseriesareofvitalimportancefortheunderstandingofcontinuity,space,time,andmotion.Therearemanywaysinwhichseriesmaybegenerated,butalldependuponthefindingorconstructionofanasymmetricaltransitiveconnectedrelation.Someofthesewayshavecon-siderableimportance.Wemaytakeasillustrativethegenera-tionofseriesbymeansofathree-termrelationwhichwemaycall"between."Thismethodisveryusefulingeometry,andmayserveasanintroductiontorelationshavingmorethantwoterms;itisbestintroducedinconnectionwithelementarygeometry.Givenanythreepointson4straightlineinordinaryspace,theremustbeoneofthemwhichisbetweentheothertwo.Thiswillnotbethecasewiththepointsonacircleoranyotherclosedcurve,because,givenanythreepointsonacircle,wecantravelfrgmanyonetoanyotherwithoutpassingthroughthethird.fnfact,thenotionttbetweenttischaracteristicofopenseries-orseriesinthestrictsense-asopposedtowhatmaybecalled\nTheDefnitionofOrder39ttt'cyclicseries,where,aswithpeopleatthedinner-table,asufficientjourneybringsusbacktoourstarting-point.Thist'betweennotionof"maybechosenasthefundamentalnotionofordinarygeometty;butforthepresentwewillonlyconsideritsapplicationtoasinglestraightlineandtotheorderingofthepointsonastraightline.lTakinganytwopointsa,b,theline(aD)consistsofthreeparts(besidesaand&themselves):(r)Pointsbetweenaandb.(z)Pointsrsuchthataisbetweenxandb.(3)Pointsysuchthatbisbetweenyanda.Thustheline(ob)canbedefinedintermsoftherelationttbetween.tt63Inorderthatthisrelationbetween"mayarrangethepointsofthelineinanoiderfromlefttoright,weneedcertainassump-tions,namely,thefollowing:-(r)Ifanythingisbetweenaandb,aandbarenotidentical.(z)AnythingbetweenaandDisalsobetweenbanda.ft)Anythingbetweenaand&isnotidenticalwitha(nor,consequently,withbrinvirtueof(z)).(+)IfrisbetweenAandb,anythingbetweenaandxisalsobetweenaandb.G)If*isbetweerraandb,andbisbetweenrandy,then&isbetweenaandy.(6)IfxandyarcbetweenaandD,theneitherrandyareidentical,orNisbetweenaandltor*isbetrveenyandb.(Z)If&isbetweenaand*andalsobetweenaandy,theneitherxandyarcidentical,ortcisbetweenbandltoryisbetweenbandx.Thesesevenpropertiesareobviouslyverifi.edinthecaseofpointsonastraightlineinordinaryspace.Aoythree-termrelationwhichverifiesthemgivesrisetoseries,asmaybeseenfromthefollowingdefinitions.Forthesakeofdefiniteness,letusassumeICl.Riadstad,iMatematiaa,iv.pp.55ft'.;Pri'nciptresofMathemat'ics,p.3e4($375).\n+ofnroductiontoMathematicalPltiloso2ltythataistotheleftof.b.Thenthepointsoftheline(ab)arcQ)thosebetweenwhichandb,alies-thesewewillcalltotheleftof.a;(r)oitself;(3)thosebetweenaandb;(+)bitself;(5)thosebetweenwhichandalies&-thesewewillcalltotherightof.b.Wemaynowdefinegenerallythatoftwopointsx,y,ontheline(ab),weshallsaythatffis"totheleftof"yinanyofthefollowingcases'-(r)Whenrandyarcbothtotheleftofa,andyisbetweenxanda;(z)Whenristotheleftofa,andyisaorDorbetweenaandDortotherightof.b;(3)Whenxisa,andyisbetweenaandborisDoristotherightof.b;(4)Whenxandyarcbothbetweenaandb,andyisbetweenxandb;(5)When*isbetweenaandb,andyis&ortotherightolb;(6)Whenxisbandyistotherightof.b;f)Whenrandyarcbothtotherightof.bandrisbetweenbandy.Itwillbefoundthat,fromthesevenpropertieswhichwehave3'betweenr"assignedtotherelationitcanbededucedthattherelation"totheleftofr"asabovedefined,isaserialrelationaswedefinedthatterm.Itisimportanttonoticethatnothinginthedefinitionsortheargumentdependsuponourmeaningttby"betweentheactualrelationofthatnamewhichoccursinempiricalspace:anythree-termrelationhavingtheabovesevenpurelyformalpropertieswillservethepurposeoftheargumentequallywell.Cyclicorder,suchasthatofthepointsonacircle,cannotbettgeneratedbymeansofthree-termrelationsofbetween.t'Weneedarelationoffourterms,whichmaybecalled"separationofcouples."Thepointmaybeillustratedbyconsideringajourneyroundtheworld.OnemaygofromEnglandtoNewZealandbywayofSuezorbywayofSanFrancisco;wecannot\nTlreDcfnitionofOrder+rsaydefinitelythateitherofthesetwoplacesis"between"EnglandandNewZealand.Butifamanchoosesthatroutetogoroundtheworld,whicheverwayroundhegoes,histimesinEnglandandNewZealandareseParatedfromeachotherbyhistimesinSuezandSanFrancisco,andconversely.Generalising,ifwetakeanyfourpointsonacircle,wecanseparatethemintotwocouples,sayaandbandrand1lrsuchthat,inordertogetfromatobonemustpassthrougheither#or!,andinordertogetfromtctoyonemustpassthrougheitheraorb.Undertheset'circumstanceswesaythatthecouple(o,b)are"seParatedbythecouple(r,y).Outofthisrelationacyclicordercanbegen-erated,inawayresemblingthatinwhichwegeneratedanoPent'betweenr"butsomewhatmorecomplicated.lorderfromThepurposeofthelatterhalfofthischapterhasbeentosuggestthesubjectwhichonemaycall"generationofserialrelations."Whensuchrelationshavebeendefined,thegenerationofthemfromotherrelationspossessingonlysomeoftheProPertiesrequiredforseriesbecomesveryimportant,especiallyinthephilosophyofgeometryandphysics.Butwecannot,withinthelimitsofthepresentvolume,domorethanmakethereaderawarethatsuchasubjectexists.tCf.PyinaiplesofMathematias,p.zo5($r94),andreferencestheregiven.\nCHAPTERVKINDSOFRELATIONSAcnsarpartofthephilosophyofmathematicsisconcernedwithrelations,andmanydifierentkindsofrelationshavedifierenrkindsofuses.Itoftenhappensthatapropertywhichbelongstoallrelationsisonlyimportantasregardsrelationsofcertainsorts;inthesecasesthereaderwillnotseethebearingofthepropositionassertingsuchapropertyunlesshehasinmindthesortsofrelationsforwhichitisuseful.Forreasonsofthisdescription,aswellasfromtheintrinsicinterestofthesubject,itiswelltohaveinourmindsaroughlistofthemoremathematicallyserviceablevarietiesofrelations.wedealtintheprecedingchapterwithasupremelyimportantclass,namely,serialrelations.Eachofthethreepropertieswhichwecombinedindefiningseries-namely,asymmetry,transitivencss,andconncxity-hasitsownimportance.Wewillbeginbysayingsomethingoneachofthesethree.Asymmetry,i.e.rhepropertyofbeingincompatiblewiththeconverse,isacharacteristicoftheverygreatestinterestandimportance.Inordertodevelopitsfunctions,wewillconsid.ervariousexamples.Therelationltusbandisasymmetrical,andsoistherelationwtfe;i.e.if,aishusbandofb,bcannotbehusbandofa,andsimilarlyinthecaseofwife.ontheotherhand,therelation"spousettissymmetrical:if.aisspouseof.b,then6isspouseofa.Supposenowwearegiventherelationspouse,and,wewishtoderivetherelationhusband.Husbandisthesameasrnalespouseorspouseofafemale;thustherelationbusband.can4"\nKindsofRelations+3bederivedfromspoaseeitherbylimitingthedomaintomalesorbylimitingtheconversetofemales.Weseefromthisinstancethat,whenasymmetricalrelationisgiven,itissometimespossible,withoutthehelpofanyfurtherrelation,toseParateitintotwoasymmetricalrelations.Butthecaseswherethisispossiblearerareandexceptional:theyarecaseswheretherearetwomutuallyexclusiveclasses,sayaandB,suchthatwhenevertherelationholdsbetweentwoterms,oneofthetermsisamemberofaandtheotherisamemberofF-"t,inthecaseofspouse,onetermoftherelationbelongstotheclassofmalesandonetotheclassoffemales.Insuchacase,therelationwithitsdomainconfinedtoowillbeasymmetrical,andsowilltherelationwithitsdomainconfinedtoB.Butsuchcasesarenotofthesortthatoccurwhenwearedealingwithseriesofmorethantwoterms;forinaseries,allterms,exceptthefirstandlast(iftheseexist),belongbothtothedomainandtotheconversedomainofthegeneratingrelation,sothatarelationlikehusband,wherethedomainandconversedomaindonotoverlap,isexcluded.ThequestionhowtoclnstructrelationshavingsomeusefulpropertybymeansofoperationsuPonrelationswhichonlyhaverudimentsofthepropertyisoneofconsiderableimportance.Transitivenessandconnexityareeasilyconstructedinmanycaseswheretheoriginallygivenrelationdoegnotpossessthem:forexample,ifRisanyrelationwhatever,theancestralrelationderivedfromRbygeneralisedinductionistransitive;andifRisamany-onerelation,theancestralrelationwillbeconnectedifconfinedtotheposterityofagiventerm.Butasymmetryisamuchmoredifficultpropertytosecurebyconstruction.Themethodbywhichwederivedhusbandfromspouseis,aswehaveseen,nOtavailableinthemostimportantcases,suchasgreater,before,totherightof,whercdomainandconversedomainoverlap.Inallthesecases,wecanofcourseobtainasymmetricalrelationbyaddingtogetherthegivenrelationanditsconverse,butwecannotpassbackfromthissymmetricalrelationtotheoriginalasymmetricalrelationexcePtbythehelpofsomeasymmetrical\n++[ntroauctiontoMathematicalPltilosophyrelation.Take,forexample,therelationgreater:therelationgreatcrorless-,i,e.unequal-issymmetrical,butthereisnothinginthisrelationtoshowthatitisthesumoftwoasymmetricalrelations.Takesucharelationas"difieringinshape."Thisisnotthesumofanasymmetricalrelationanditsconverse,sinceshapesdonotformasingleseries;butthereisnothingtoshowthatitdifiersfrom"differinginmagnitude"ifwedidnotalreadyknowthatmagnitudeshaverelationsofgreaterandless.ThisillustratesthefundamentalcharacterofasymmetryasaProPertyofrelations.Fromthepointofviewoftheclassificationofrelations,beingasymmetricalisamuchmoreimportantcharacteristicthanimplyingdiversity.fuy*metricalrelationsimplydiversity,ttbuttheconverseisnotthecase.Lfnequalrttforexample,impliesfiversity,butissymmetrical.Broadlyspeaking,w€maysaythat,ifwewishedasfaraspossibletodispensewithrelationalpropositionsandreplacethembysuchasascribedpredicatestosubjects,wecouldsucceedinthissolongasweconfinedourselvestosyrnnetricalrelations:thosethatdonotimplydiversity,iftheyaretransitive,mayberegardedasassert-ingacommonpredicate,whilethosethatdoimplydiversitymayberegardedasassertingincompatiblepredicates.Forexample,considertherelationofsimilaritybetweenclasses,bymeansofwhichwedefinednumbers.Thisrelationissym-metricalandtransitiveanddoesnotimplydiversity.Itwouldbepossible,thoughlesssimplethantheprocedureweadopted,toregardthenumberofacollectionasapredicateofthecollec-tion:thentwosimilarclasseswillbetwothathavethesamenumericalpredicate,whiletwothatarenotsimilarwillbetwothathavedifierentnumericalpredicates.Suchamethodofreplacingrelationsbypredicatesisformallypossible(thoughoftenveryinconvenient)solongastherelationsconcernedaresymmetrical;butitisformallyimpossiblewhentherelationsareasymmetrical,becausebothsamenessanddifferenceofpredi-catesaresymmetrical.Asymmetricalrelationsarerwemay\nKindsofRelations+ssay,themostcharacteristicallyrelationalofrelations,andthemostimportanttothephilosopherwhowishestostudytheultimatelogicalnatureofrelations.Anotherclassofrelationsthatisofthegreatestuseistheclassofone-manyrelations,i.e,relationswhichatmostonetermcanhavetoagiventerm.Sucharefather,mother,husband(exceptinTibet),squareof,sineof,andsoon.Butparent,squareroot,andsoon,arenotone-many.Itispossible,formally,toreplaceallrelationsbyone-manyrelationsbymeansofadevice.Take(t^y)therelationlessamongtheinductivenumbers.Givenanynumberagreaterthant,therewillnotbeonlyonenumberhavingtherelationlesston,butwecanformthewholeclassofnumbersthatarelessthanz.Thisis,oneclass,anditsrelationtoaisnotsharedbyanyotherclass.Wemaycalltheclassofnumbersthatarelessthannthe"properancestry"ofn,inthesenseinwhichwespokeofancestryandposterityinconnectionwithmathematicalinduction.Then"properancestry"isaone-manyrelation(one-manywillalwaysbeusedsoastoincludeone-one),sinceeachnumberdeterminesasingleclassofnumbersasconstitutingitsproperancestry.Thustherelationlessthancanbereplacedbybeingamemberoftheproperancestryof.Inthiswayaone-manyrelationinwhichtheoneisaclass,togetherwithmembershipofthisclass,canalwaysformallyreplacearelationwhichisnotone-many.Peano,whoforsomereasonalwaysinstinctivelyconceivesofarelationasone-many,dealsinthiswaywiththosethatarenaturallynotso.Reductiontoone-manyrelationsbythismethod,however,thoughpossibleasamatterofform,doesnotrepresentatechnicalsimplification,andthereiseveryreasontothinkthatitdoesnotrepresentaphilosophicalanalysis,ifonlybecauseclassesm,ustberegardedas"logicalfictions."Weshallthere-forecontinuetoregardone-manyrelationsasaspecialkindofrelations.One-manyrelationsareinvolvedinallphrasesoftheform"theso-and-soofsuch-and-such.""TheKingofEngland,"\n46fntroductiont0MatlrernaticalPhilosoplzyt'the"thewifeofSocratesr"fatherofJohnStuartMill,"andsoon,alldescribesomepersonbymeansofaone-manyrelationtoagiventerm.Apersoncannothavemorethanonefather,therefore"thefatherofJohnStuartMill"describedsomeoneperson,evenifwedidnotknowwhom.Thereismuchtosayonthesubjectofdescriptions,butforthepresenritisrelationsthatweareconcernedwith,anddescriptionsareonlyrelevantasexemplifyingtheusesofone-manyrelations.Itshouldbeobservedthatallmathematicalfunctionsresultfromone-manyrelations:thelogarithmofx,thecosineofr,etc.,are,likethefatherofx,termsdescribedbymeansofaone-manyrelation(logarithm,cosine,etc.)toagiventerm(r).Thenotionof.functionneednotbeconfinedtonumbers,ortotheusestowhichmathematicianshaveaccustomedus;itcanbeextendedtoallcasesofone-manyrelations,and"thefatherofx"isjustaslegitimatelyafunctionofwhichristheargumentasis"thelogarithmof*."Functionsinthissensearedescriptiaefunctions.Asweshallseelater,therearefunctionsofastillmoregeneralandmorefundamentalsort,namely,propositionalfunctions;butforthepresentweshallconfineourattentiontodescriptivefunctions,i.e.t'thetermhavingtherelationRtotrcr"or,forshort,tttheRoffrr"whereRisanyone-manyrelation.Itwillbeobservedthatif"theRofr"istodescribeadefiniteterm,#mustbeatermtowhichsomethinghastherelationR,andtheremustnotbemorethanonetermhavingtherelationttRto,c,sincethert'correctlyused,mustimplyuniqueness.Thuswemayspeakof"thefatherofx"ifrisanyhumanbeingexceptAdamandEve;butwecannotspeakoft,thefatherof.x"ifxisatableorachairoranythingelsethatdoesnothaveafather.WeshallsaythattheRofr"exists"whenthereisjustoneterm,andnomore,havingtherelationRto#.ThusifRisaone-manyrelation,theRofrexistswhenever,cbelongstotheconversedomainofR,andnototherwise.Regarding"theRofr"asafunctioninthemathematical\nKindsofRclations+7ttttsense,wesaythatxistheargumentofthefunction,andifyisthetermwhichhastherelationRtox,i.e.ifyistheRofx,thenyisthe"value"ofthefunctionfortheargumentff.IfRisaone-manyrelation,therangeofpossibleargumentstothefunctionistheconversedomainofR,andtherangeofvaluesisthedomain.Thustherangeofpossibleargumentstothefunction"thefatherof*"isallwhohavefathers,i.t.thecon-versedomainoftherelationfatber,whiletherangeofpossiblevaluesforthefunctionisallfathers,i,e.thedomainoftherelation.Manyofthemostimportantnotionsinthelogicofrelationsaredescriptivefunctions,forexampleicont)erse,domain,con-sersedornain,f.eld,Otherexampleswilloccurasweproceed.Amongone-manyrelations,one-onerclationsareaspeciallyimportantclass.Wehavealreadyhadoccasiontospeakofone-onerelationsinconnectionwiththedefinitionofnumber,butitisnecessarytobefamiliarwiththem,andnotmerelytoknowtheirformaldefinition.Theirformaldefinitionmaybederivedfromthatofone-manyrelations:theymaybedefinedasone-manyrelationswhicharealsotheconversesofone-manyrelations,i.e.asrelationswhicharebothone-manyandmany-one.One-manyrelationsmaybedefinedasrelationssuchthat,ifrhastherelationinquestiontoy,thereisnoothertermr'whichalsohastherelationtoy.Or,again,theymaybedefinedasfollows:Giventwotermsrandr',thetermstowhichrhasthegivenrelationandthosetowhichr'hasithavenomemberincommon.Ot,again,theymaybedefinedasrelationssuchthattherelativeproductofoneofthemanditsconverseimpliesidentity,wherethe"relativeproduct"oftworelationsRandSisthatrelationwhichholdsbetweenxandzwhenthereisanintermediatetermy,suchthatrhastherelationRtoyandyhastherelationStoz.Thus,forexample,ifRistherelationoffathertoson,therelativeproductofRanditsconversewillbetherelationwhichholdsbetweenrandamanzwhenthereisapersony,suchthatristhefatherofyandyisthesonofz.Itisobviousthat*andzmustbe\n+8IntroducttontoMatlzernaticalPhilosopltythesameperson.If,ontheotherhand,wetaketherelationofparentandchild,whichisnotone-manbwecannolongerarguethatrifrisaparentofyandyisachildofz,xandzmustbethesameperson,becauseonemaybethefatherofyandtheotherthemother.Thisillustratesthatitischaracteristicofone-manyrelationswhentherelativeproductofarelationanditsconverseimpliesidentity.Inthecaseofone-onerelationsthishappens,andalsotherelativeproductoftheconverseandtherelationimpliesidentity.GivenarelationR,itisconvenient,ifrhastherelationRtoy,tothinkofyasbeingreachedfromt'ttxbyarL"R-steporanR-vector."Inthesamecase*willbereachedfromybya"backwardR-step."Thuswemaystatethecharacteristicofone-manyrelationswithwhichwehavebeendealingbysayingthatanR-stepfollowedbyaback-wardR-stepmustbringusbacktoourstarting-point.Withotherrelations,thisisbynomeansthecase;forexample,ifRistherelationofchildtoparent,therelativeproductofRanditsconverseistherelation"selforbrotherorsisterr"andifRistherelationofgrandchildtograndparent,therelativeproductofRanditsconverseis"selforbrotherorsisterorfirstcousin."Itwillbeobservedthattherelativeproductoftworelationsisnotingeneralcommutative,i.e.therelativeproductofRandSisnotingeneralthesamerelationastherelativeproductofSandR,^0.g.therelativeproductofparentandbrotherisuncle,buttherelativeproductofbrotherandparentisparent.One-onerelationsgiveacorrelationoftwoclasses,termforterm,sothateachtermineitherclasshasitscorrelateintheother.Suchcorrelationsaresimplesttograspwhenthetwoclasseshavenomembersincommon,liketheclassofhusbandsandtheclassofwives;forinthatcaseweknowatoncewhetheratermistobeconsideredasonefromwhichthecorrelatingrelationRgoes,orasonelowhichitgoes.Itisconvenienttousethewordreferentforthetermfromwhichtherelationgoes,andthetermrelatumforthetermtuwhichitgoes.Thusifrandyarchusbandandwife,then,withrespecttotherelation\nKinasofRelations+g(thusbandr"*isreferentandyrelatum,butwithrespecttotherelatron"wifer"yisreferentandrrelatum.Wesaythatattrelationanditsconversehaveoppositesenses"Ithusthettsense"ofarelationthatgoesfromtctoyistheoppositeofthatofthecorrespondingrelationfromytotc.Thefactthatarelationhasa"sense"isfundamental,andispartofthereaso4whyordercanbegeneratedbysuitablerelations.Itwillbeobservedthattheclassofallpossiblereferentstoagivenrelationisitsdomain,andtheclassofallpossiblerelataisitsconversedomain.Butitveryoftenhappensthatthedomainandconversedomainofaone-onerelationoverlap.Take,forexample,thefirsttenintegers(excludingo),andaddrtoeachIthusinsteadofthefirsttenintegerswenowhavetheintegers2,3,4,5r617r8rg,ro,rr.Thesearethesameasthosewehadbefore,exceptthatrhasbeencutoffatthebeginningandrrhasbeenjoinedonattheend.Therearestilltenintegers:theyarecorrelatedwiththeprevioustenbytherelationof.ntonlr,whichisaone-onerelation.Or,again,insteadofaddingrtoeachofouroriginaltenintegers,wecouldhavedoubledeachofthem,thusobtainingtheintegers2,+r618,ro,t2,t+,t6,t8,zo.Herewestillhavefiveofourprevioussetofintegers,namelR22126t8,to.Thecorrelatingrelationinthiscaseistherelationofanumbertoitsdouble,whichisagainaone-onerelation.Orwemighthavereplacedeachnumberbyitssquare,thusobtainingthesett,*r9,16,25r36,*9r6+,8r,roo.Onthisoccasiononlythreeofouroriginalsetareleft,namely,t,1.r9.Suchprocessesofcorrelationmaybevariedendlessly.Themostinterestingcaseoftheabovekindisthecasewhereourone-onerelationhasaconversedomainwhichispart,but\nSofntroductiontoMathematicalPltihsoplt1notthewhole,ofthedomain.If,insteadofconfiningthedomaintothefirsttenintegers,wehadconsideredthewholeoftheinductivenumbers,theaboveinstanceswouldhaveillustratedthiscase.Wemayplacethenumbersconcernedintworows'puttingthecorrelatedirectlyunderthenumberwhosecorrelateitis.Thuswhenthecorrelatoristherelationofnton{r,wehavethetworows:lr213r*r5r"'tt"'2r314r516r"'n{I"Whenthecorrelatoristherelationofanumbertoitsdouble,wehavethetworows:lr2r3r4,5t'''n'',2141618rror..2n,..Whenthecorrelatoristherelationofanumbertoitasquare,therowsare:lr213,4,5,nl,419,t6,25,n2Inallthesecases,allinductivenumbersoccurinthetoProw,andonlysomeinthebottomrow.Casesoft{rissort,wheretheconversedomainisa"properpart"ofthedomain(i.e.apartnotthewhole),willoccuPyusagainwhenwecometodealwithinfinity.Forthepresent,wewishonlytonotethattheyexistanddemandconsideration.Anotherclassofcorrelationswhichareoftenimportantis((theclasscalledpermutationsrttwherethedomainandconversedomainareidentical.Consider,forexample,theaixpossiblearrangementsofthreeletters:a,b,carcrbbrcrabrarccrarbcrbra\nKindsofRelationsjrEachofthesecanbeobtainedfromanyoneoftheothersbymeanJof.acorrelation.Take,forexample,thefirstandlast,(",b,c)and(c,b,a).Hereaiscorrelatedwithc,bvmthitself,andcwitha.Itisobviousthatthecombinationoftwopermu-tationsisagainapermutation,r'.a.thepermutationsofagivenclassformwhatiscalledat'group.ttThesevariouskindsofcorrelationshaveimportanceinvariousconnections,someforonepurpose,someforanother.Thegeneralnotionofone-onecorrelationshasboundlessimportanceinthephilosophyofmathematics,aswehavepartlyseenalre^dy,butshallseemuchmorefullyasweproceed,oneofitsuseswilloccupyusinournextchapter.\nCHAPTERVISIMILARITYOFRELATIONSWnsawinChapterII.thattwoclasseshavethesamenumberoftermswhentheyate"similarrt'i.e,whenthereisaone-onerelationwhosedomainistheoneclassandwhoseconversedomainistheother.Insuchacasewesaythatthereisa(tttone-onecorrelationbetweenthetwoclasses.Inthepresentchapterwehavetodefinearelationbetweenrelations,whichwillplaythesamepartforthemthatsimilarityofclassesplaysforclasses.Wewillcallthisrelation"similarityofrelationsr"or"likeness"whenitseemsdesirabletouseadifierentwordfromthatwhichweuseforclasses.HowislikenesstobedefinedIWeshallemploystillthenotionofcorrelation:weshallassumethatthedomainoftheonerelationcanbecorrelatedwiththedomainoftheother,andtheconversedomainwiththeconversedomain;butthatisnotenoughforthesortofresem-blancewhichwedesiretohavebetweenourtworelations.Whatwedesireisthat,whenevereitherrelationholdsbetweentwoterms,theotherrelationshallholdbetweenthecorrelatesofthesetwoterms.TheeasiestexampleofthesortofthingwedesireisamaP.Whenoneplaceisnorthofanother,theplaceonthemapcorresPondingtotheoneisabovetheplaceonthemapcorrespondingtotheother;whenoneplaceiswestofanother,theplaceonthemaPcorresPondi.stotheoneistotheleftoftheplaceonthemaPcorresPondingtotheotherIandsoon.ThestructureofthemaPcorresPondawiththatof52\nSimilaritlofRelationsj3thecountryofwhichitisamap.Thespace-relationsinthemaphave"likeness"tothespace-relationsinthecountrymapped.Itisthiskindofconnectionbetweenrelationsthatwewishtodefine.We*"y,inthefirstplace,profitablyintroduceacertainrestriction.Wewillconfineourselves,indefininglikeness,tosuchrelationsashavettfieldsr"i.e.tosuchaspermitoftheformationofasingleclassoutofthedomainandtheconversedomain.Thisisnotalwaysthecase.Take,forexample,therelation"domainr"i.e.therelationwhichthedomainofarelationhastotherelation.Thisrelationhasallclassesforitsdomain,sinceeveryclassisthedomainofsomerelation;andithasallrelationsforitsconversedomain,sinceeveryrelationhasadomain.Butclassesandrelationscannotbeaddedto-gethertoformanewsingleclass,becausetheyareofdifierentlogical"types."Wedonotneedtoenteruponthedifficultdoctrineoftypes,butitiswelltoknowwhenweareabstainingfromenteringuponit.Wemaysay,withoutenteringuponthegroundsfortheassertion,thatarelationonlyhasa"field"ttwhenitiswhatwecallhomogeneousr"f.e.whenitsdomainandconversedomainareofthesamelogicaltype;andasarough-and-readyindicationofwhatwemeanby^"typ"r"wemaysaythatindividuals,classesofindividuals,relationsbetweenindividuals,relationsbetweenclasses,relationsofclassestoindividuals,andsoon,aredifferenttypes.Nowthenotionoflikenessisnotveryusefulasappliedtorelationsthatarenothomogeneous;weshall,therefore,indefininglikeness,simplifyourproblembyspeakingofthe"field"ofoneoftherelationsconcerned.Thissomewhatlimitsthegeneralityofourdefinition,butthelimitationisnotofanypracticalimpor-tance.Andhavingbeenstated,itneednolongerberemembered.WemaydefinetworelationsPandQas"similarr"orastthavinglikenessr"whenthereisaone-onerelationSwhosedomainisthefieldofPandwhoseconversedomainisthefieldofQ.andwhichissuchthat,ifonetermhastherelationP\n,0MatltematicalPltihsopltyS+fnroductiontoanother,thecorrelateoftheonehastherelationQtothecorrelateoftheother,andoiceversa.Afigurewillmakethisclearer.LetxandYbetwovtermshavingtherelationP.Thentherearetobetwoterms^t,u,suchthatrhastherela-SJDtionStoz,yhastherelationStow,andzhastherelationQtow.Ifthishappenswith;toeverypairoftermssuchasrandy,andiftheconversehappenswitheverypairoftelmssuchaszandw,itisclearthatforeveryinstanceinwhichtherelationPholdsthereisacorrespondinginstanceinwhichtherelationQholds,andoiceoersa;andthisiswhatwedesiretosecurebyourdefinition.Wecaneliminatesomered.undanciesintheabovesketchofadefinition,byobservingthat,whentheaboveconditionsarerealised,therelationPisthesameastherelativeproductofSandQandtheconverseofS,i.e.theP-stepfrom,ctoymaybereplacedbythesuccessionoftheS-stepfromrtoz,theQstepfromt'tow,andthebackwardS-stepfromwtoy.Thuswemaysetupthefollowingdefinitions:-ttttArelationSissaidtobeacorrelator"oranordinalcorrelator"oftworelationsPandQitSisone-one,hasthefieldofQforitsconversedomain,andissuchthatPigtherelativeproductofSandQandtheconverseofS.t'similarr"TworelationsPandQaresaidtobeortohavettlikenessr"whenthereisatleastonecorrelatorofPandQ.Thesedefinitionswillbefoundtoyieldwhatweabovedecidedtobenecessary.Itwillbefoundthat,whentworelationsaresimilar,theyshareallpropertieswhichdonotdependuPontheactualtermsintheirfields.Forinstance,ifoneimpliesdiversity,sodoestheother;ifoneistransitive,soistheother;ifoneiscon-nected,soistheother.Henceifoneisserial,soistheother.Again,ifoneieone-manyorone-one,theotherisone-many\nSimilarityofRelationsSsorone-one;andsoon,throughallthegeneralpropertiesofrelations.Evenstatementsinvolvingtheactualtermsofthefieldofarelation,thoughtheymaynotbetrueastheystandwhenappliedtoasimilarrelation,willalwaysbecapableoftranslationintostatementsthatareanalogous.Weareledbysuchconsiderationstoaproblemwhichhas,inmathematicalphilosophrr2nimportancebynomeansadequatelyrecognisedhitherto.Ourproblemmaybestatedasfollows.-Givensomestatementinalanguageofwhichweknowthegrammarandthesyntax,butnotthevocabulary,whatarethepossiblemeaningsofsuchastatement,andwhatarethemean-ingsoftheunknownwordsthatwouldmakeittrue?Thereasonthatthisquestionisimportantisthatitrepresents,muchmorenearlythanmightbesupposed,thestateofourknowledgeofnature.Weknowthatcertainscientificpro-positions-which,inthemostadvancedsciences,areexpressedinmathematicalsymbols-aremoreorlesstrueoftheworld,butweareverymuchatseaastotheinterpretationtobeputuponthetermswhichoccurinthesepropositions.Weknowmuchmore(touse,foramoment,anold-fashionedpairofterms)abouttheiformofnaturethanaboutthematter.Accordingly,whatwereallyknowwhenweenunciatealawofnatureisonlythatthereisprobablysotneinterpretationofourtermswhichwillmakethelawapproximatelytrue.Thusgreatimportanceattachestothequestion:Whatarethepossiblemeaningsofalawexpressedintermsofwhichwedonotknowthesubstantivemeaning,butonlythegrammarandsyntaxiAndthisquestionistheonesuggestedabove.Forthepresentwewillignorethegeneralquestion,whichwilloccupyusagainatalaterstage;t}resubjectoflikenessitselfmustfirstbefurtherinvestigated.Owingtothefactthat,whentworelationsaresimilar,theirpropertiesarethesameexceptwhentheydependuponthefieldsbeingcomposedofjustthetermsofwhichtheyarecom-posed,itisdesirabletohaveanomenclaturewhichcollects\n56fnroductiont0MathematicalPhihsophytogetheralltherelationsthataresimilartoagivenrelation.Justaswecalledthesetofthoseclassesthataresimilartoat'numberofthatclass,sowemaycallthesetlin.r,classthe"ofthoserelationsthataresimilartoagivenrelationthett"Uwithnumber"ofthatrelation.Butinordertoavoidconfusionthenumbersappropriatetoclasses,wewillspeak,inthiscase,ofa"relation-number."Thuswehavethefollowingdefinitions:-The"relation-number"ofagivenrelationistheclassofallthoserelationsthataresimilartothegivenrelation.,,Relation-numberst'arethesetofallthoseclassesofrelationscomestothatarerelation-numbersofvariousrelationsIor,whatthesamething,arelationnumberisaclassofrelationsconsistingofallthoserelationsthataresimilartoonememberoftheclass.Whenitisnecessarytospeakofthenumbersofclassesinawaywhichmakesitimpossibletoconfusethemwithrelation-numbers,weshallcallthem"cardinalnumbers."Thuscardinalnumbersarethenumbersappropriatetoclasses.Theseincludetheordinaryintegersofdailylife,andalsocertaininfinitenumbers,ofwhichweshallspeaklater.Whenwespeakofttt'numberswithoutqualification,wearetobeunderstoodasmeaningcardinalnumbers.Thedefinitionofacardinalnumber,itwillberemembered,isasfollows:-The"cardinalnumber"ofagivenclassisthesetofallthoseclassesthataresimilartothegivenclSss'Themostobviousapplicationofrelation-numbersistosedes'Twoseriesmayberegardedasequallylongwhentheyhavethesamerelation-number.Twof'niteserieswillhavethesamerelation-numberwhentheirfieldshavethesamecardinalnumberofterms,andonlythen-i.a.aseriesof(say)I5termswillhavethesamerelation-numberasanyotherseriesoffifteenasaseriesterms,butwillnothavethesamerelation-numberofI4or16terms,nor,ofcourserthesamerelation-numberasarelationwhichisnotserial.Thus,inthequitespecialcaseoffiniteseries,thereisparallelismbetweencardinalandrelation-numbers.Therelation-numbersapplicableto$eriesmaybe\nSinilariryofRelartons57calledttserialnumbers"(whatarecommonlycalled,,ordinalnumberst'areasub-classofthese);thusafiniteserialnumberisdeterminatewhenweknowthecardinalnumberoftermsinthefieldof.aserieshavingtheserialnumberinquestion.Ifzisafinitecardinalnumber,therelation-numberofaserieswhichhasntermsiscalledthe'6ordinal"numbera.(Therearealsoinfiniteordinalnumbers,butofthemweshallspeakinalaterchapter.)Whenthecardinalnumberoftermsinthefieldofaseriesisinfinite,therelation-numberoftheseriesisnotdeterminedmerelybythecardinalnumber,indeedaninfinitenumberofrelation-numbersexistforoneinfinitecardinalnumber,asweshallseewhenwecometoconsiderinfiniteseries.whenaseriesisinfinite,whatwemaycallits"lengthr,,i,e.itsrelation-number,mayvarywithoutchangeinthecardinalnumber;butwhenaseriesisfinite,thiscannothappen.wecandefineadditionandmultiplicationforrelation-numbersaswellasforcardinalnumbers,andawholearithmeticofrelation-numberscanbedeveloped.Themannerinwhichthisistobedoneiseasilyseenbyconsideringthecaseofseries.suppose,forexample,thatwewishtodefinethesumoftwonon-overlappingseriesinsuchawaythattherelation-numberofthesumshallbecapableofbeingdefinedasthesumoftherelation-numbersofthetwoseries.Inthefirstplace,itisclearthatthereisanorderinvolvedasbetweenthetwoseries:oneofthemmustbeplacedbeforetheother.Thusifpandearethegeneratingrelationsofthetwoseries,intheserieswhichistheirsumwithPputbeforeQ,everymemberofthefieldofPwillprecedeeverymemberofthefieldofQ.Thustheserialrelationwhichistobedefinedasthesumofpandeisnot"PorQ"simply,but"PorQortherelationofanymemberofthefieldofPtoanymemberofthefieldofe.',AssumingthatPandQdonotoverlap,thisrelationisserial,but,,pore"isnotserial,beingnotconnected,sinceitdoesnotholdbetweenamemberofthefieldofPandamemberofthefieldofe.ThusthesumofPandQ,asabovedefined,iswhatweneedinorder\n58IntrodactiontaMathematicalPhilostphytodefinethesumoftworelation-numbers.Similarmodifica-tionsareneededforproductsandpowers.Theresultingarith-meticdoesnotobeythecommutativelaw:thesumorproductinoftworelation-numbersgenerallydependsuPontheorderwhichtheyaretaken.Butitobeystheassociativelaw,oneformofthedistributivelaw,andtwooftheformallawsforpowers,notonlyasappliedtoserialnumbers,butasappliedtorelation-numbersgenerally.Relation-arithmetic,infact,thoughrecent,isathoroughlyrespectablebranchofmathematics.Itmustnotbesupposed,merelybecauseseriesaffordthemostobviousapplicationoftheideaoflikeness,thattherearenootherapplicationsthatareimportant.Wehavealreadymentionedmaps,andwemightextendourthoughtsfromthisillustrationtogeometrygenerally.Ifthesystemofrelationsbywhichageometryisappliedtoacertainsetoftermscanbebroughtfullyintorelationsoflikenesswithasystemapplyingtoanothersetofterms,thenthegeometryofthetwosetsisindistinguishablefromthemathematicalpointofview,i'a'allthepropositionsarethesame,exceptforthefactthattheyareappliedinonecasetoonesetoftermsandintheothertoanother.beW"-"yillustratethisbytherelationsofthesortthatmaycalled"betweenr"whichweconsideredinChapterIV.Wetheresawthat,providedathree-termrelationhascertainformallogicalproperties,itwillgiverisetoeeries,andmaybecalledttabetween-relation.ttGivenanytwopointS,wecanusethebetween-relationtodefinethestraightlinedeterminedbytJrosetwopointsIitconsistsofaandDtogetherwithallpointsr,suchthatthebetween-relationholdsbetweenthethreepointsa,b,xinsomeorderorothef.IthasbeenshownbyO.VeblenthatwemayregardourwholesPaceasthefieldofathree-termbetween-relation,anddefineourgeometrybytheProPertiesweassigntoourbetween-relation.lNowlikenessisjustaseasilyrThiedoesnotaPPlytoellipticsP:rce,butonlytospacesinwhichthestraightlineisanopenseries.MoilernMathematics,editedbyJ.W.A.Young,pp.3-5r(monographbyo.veblenon"TheFoundationsofGeonetry").\nSimilariryofRelationsSgdefinablebetweenthree-termrelationsaEbetweentwo-termrelations.IfBandB'aretwobetween-relations,sothat"*B(y,z)"means"*isbetweenyandzwithrespecttoBr"weshallcallSacorrelatorofBandB'ifithasthefieldofB'foritsconversedomain,andissuchthattherelationBholdsbetweenthreetermswhenB'holdsbetweentheirS-correlates,andonlythen.AndweshallsaythatBislikeB'whenthereisatleastooecorrelatorofBwithB'.Thereadercaneasilyconvincehimselfthat,ifBislikeB'inthissense,therecanbenodifferencebetweenthegeometrygeneratedbyBandthatgeneratedbyB'.Itfollowsfromthisthatthemathematicianneednotconcernhimselfwiththeparticularbeingorintrinsicnatureofhispoints,lines,andplanes,evenwhenheisspeculatingasanapplied,mathematician.Wemaysaythatthereisempiricalevidenceoftheapproximatetruthofsuchpartsofgeometryasarenotmattersofdefinition.Butthereisnoempiricalevidenceastot'pointwhata"istobe.Ithastobesomethingthatasnearlyaspossiblesatisfiesouraxioms,butitdoesnothavetobettverysmall"or"withoutparts.ttWhetherornotitisthosethingsisamatterofindifference,solongasitsatisfiestheaxioms.Ifwecan,outofempiricalmaterial,constmctalogicalstructure,nomatterhowcomplicated,whichwillsatisfyourgeometricalarioms,thatstructuremaylegitimatelybecalleda"point."Wemustnotsaythattlereisnothingelsethatcouldlegitimatelyt'3'Thisbecalleda"point;wemustonlysayiobjectwehaveconstructedissufficientforthegeometer;itmaybeoneofmanyobjects,anyofwhichwouldbesufficient,butthatisnoconcernofours,sincethisobjectisenoughtovindicatetheempiricaltruthofgeometry,insofarasgeometryisnotamatterofdefinition."Thisisonlyanillustrationofthegeneralprinciplethatwhatmattersinmathematics,andtoaverygreatertentinphysicalscience,isnottheintrinsicnatureofourterms,butthelogicalnatureoftheirinterrelations.Wemayaay,oftwosimilarrelations,thattheyhavetherame\n6ofntroductiontoMath.ematicalPhilosophlt'structure."Formathematicalpurposes(thoughnotforthoseofpurephilosophy)theonlythingofimportanceaboutarelationisthecasesinwhichitholds,notitsintrinsicnature.Justasaclassmaybedefinedbyvariousdifferentbutco-extensiveconcepte-e.g,,,mant,and..featherlessbipedrt'-rotworelationswhichareconceptuallydifierentmayholdinthesamesetofinstances.An"instance"inwhicharelationholdsistobeconceivedasacoupleofterms,withanorder,sothatoneofthetermscomesfirstandtheothersecond;thecoupleistob.,ofcourse,suchthatitsfirsttermhastherelationinquestiontoitssecond.t'Take(r"y)therelationfather":wecandefinewhatwemaycallthe"extension"ofthisrelationastheclassofallorderedcouples(*,y)whicharesuchthat*isthefatherofy.Fromthemathematicalpointofview,theonlythingofimportanceabouttherelation"father"isthatitdefinesthissetoforderedcouples.Speakinggenerally'wesay:Thettextension"ofarelationistheclassofthoseorderedcouples(*,y)whicharesuchthat*hastherelationinquestiontoy.WecannowgoastepfurtherintheProcessofabstraction,ttandconsiderwhatwemeanbystructure.ttGivenanyrelation,wecan,ifitisasufficientlysimpleone,constructamaPofit.Forthesakeofdefiniteness,letustakearelationofwhichtheextensionisthefollowingcouplesi4b,ac,ad,bc,ce,dc,de,wherea,b,c,d,earefivetermsrnomatterwhat'Wemaymakea.-!"ff';"::'*:':HITJJJ:f:Til::,H::|\-|intheaccomPanyingfigure.Whatis"t|\-|,.rr."ledbythemapiswhatwecallthe'l'\'ttt-7structure"oftherelation.;'\Itisclearthatthe"structure"ofthe\{telationdoesnotdependupontheparticular;termsthatmakeuPthefieldoftherelation.Thefieldmaybechangedwithoutchangingthestructure,andthestructulemaybechangedwithoutchangingthefield-for\nSinilarityofRelations6rexample,ifweweretoaddthecoupleaeintheaboveillustrationweshouldalterthestructurebutnotthefield.Tworelations((havethesamestructurerttweshallsay,whenthesamemapwilldoforboth-or,whatcomestothesamething,wheneithercanbeamapfortheother(sinceeveryrelationcanbeitsownmap).Andthat,asamoment'sreflectionshows,istheverysamethingaswhatwehavecalled"likeness.ttThatistosayrtworelationshavethesamestructurewhentheyhavelikeness,i.a.whentheyhavethesamerelation-number.Thuswhatwe('relation-numbert'definedastheistheverysamethingasisobscurelyintendedbytheword66structuls"-awordwhich,importantasitis,isnever(sofaraswektrow)definedinprecisetermsbythosewhouseit.Therehasbeenagreatdealofspeculationintraditionalphilosophywhichmighthavebeenavoidediftheimportanceofstructure,andthefifficultyofgettingbehindit,hadbeenrealised,.Forexample,itisoftensaidthatspaceandtimearesubjective,buttheyhaveobjectivecounterparts;orthatphenomenaaresubjective,butarecausedbythingsinthemselves,whichmusthavedifferencesinterJecorrespondingwiththedifierencesinthephenomenatowhichtheygiverise.Wheresuchhypothesesaremade,itisgenerallysupposedthatwecanknowverylittleabouttheobjectivecounterparts.Inactualfact,however,ifthehypothesesasstatedwerecorrect,theobjectivecounterpartswouldformaworldhavingthesamestructureasthephenomenalworld,andallowingustoinferfromphenomenathetruthofallpropositionst}atcanbestatedinabstracttermsandareknowntobetrueofphenomena.Ifthephenomenalworldhasthreedimensions,somusttheworldbehindphenomenaIifthepheno-menalworldisEuclidean,somusttheotherbe;andsoon.Inshort,everypropositionhavingacommunicablesignificancemustbetrueofbothworldsorofneither:theonlydifierencemustlieinjustthatessenceofindividualitywhichalwayseludeswordsandbaffiesdescription,butwhich,forthatveryreason,isirrelevanttoscience.Nowtheonlypurposethatphilosophers\n6zfnroductionr0MathematicalPhilasophlhaveinviewincondemningphenomenaisinordertopersuadefromthemselvesandothersthattherealworldisverydifierenttheworldofappearance.wecanallsympathisewiththeirwishtoprovesuchaverydesirableproposition,butwecannotcon-gr"iolatethemontheirsuccess.Itistruethatmanyofthemdonotasseltobjectivecounterpartstophenomena,andtleseescapefromtheaboveargument.Thosewhodoassertcounter-partsare,asarule,veryreticentonthesubject,probablybecausetheyfeelinstinctivelythat,ifpursued,itwillbringabouttoomuchof.arapprocltemenlbetweentherealandthephenomenalworld.Iftheyweretopursuethetopic,theycouldhardlyavoidtheconclusionswhichwehavebeensuggesting.Insuchways,aswellasinmanyothers,thenotionofstructureorrelation-numbericimportant.\nCHAPTERVIIRATIONAL,REAL,ANDCOMPLEXXUMBERSwnhavenowseenhowtodefinecardinalnumbers,andalsorelation-numbers,ofwhichwhatarecommonlycalledordinalnumbersareaparticularspecies.Itwiltbefoundthateachofthesekindsofnumbermaybeinfinitejustaswellasfinite.Butneitheriscapable,asitstands,ofthemorefamiliarexten-sionsoftheideaofnumber,namely,theextensionetonegative,fractional,irrational,andcomplexnumbers.rndrepresentchapterweshallbrieflysupplylogicaldefinitionsofthesevariousextensions.oneofthemistakesthathavedelayedthediscoveryofcorrectdefinitionsinthisregionisthecommonideathateachextensionofnumberincludedtheprevioussortsasspecialcases.Itwasthoughtthat,indealingwithpositiveandnegativeintegers,thepositiveintegersmightbeidentifiedwiththeoriginalsignlessintegers.Againitwasthoughtthatafractionwhosedenominatorisrmaybeidentifiedwiththenaturalnumberwhichisitsnumerator.Andtheirrationalnumbers,euchasthesquarerootof2,weresupposedtofindtheirplaceamongrationalfrac-tions,asbeinggreaterthansomeofthemandlessthantheothers,sothatrationalandirrationalnumberscouldbetakentogetherasoneclass,called"realnumbers.,,Andwhentheideaofnumberwasfurtherextendedsoastoincludettcomplex',numbers,i.c,numbersinvolvingthesquarerootof-r,itwasthoughtthatrealnumberscouldberegardedasthoseamongcomplexnumberginwhichtheimaginaq/part(i.t.thepart63\n6+fntoductiont0MathematicalPhilosaplty-r)waszero.Allwhichwasamultipleofthesquarerootofaswethesesuppositionswereerroneous,andmustbediscarded,shallfind,ifcorrectdefinitionsaretobegiven'Letusbeginwithpositivcandnegativeintegers.Itisobvious-Imustbothbeonamoment'sconsiderationthatfrandTherelations,andinfactmustbeeachother'sconverses'obviousandsufficientdefinitionisthat*ristherelationofifmrr+rtofr,and-ristherelationofnton+\.Generally,isanyinductivenumber,*mwtllbetherelationof.n{nt'ton(foranyn),and-mwillbetherelationofnton+rn.Accord-itgtothisdefinition,*misarelationwhichisone-onesolongasnisacardinalnumber(finiteorinfinite)andmisaninductivecardinalnumber.Butlmisundernocircumstancescapableofbeingidentifiedwithza,whichisnotarelation,butaclassofclasses.Indeed,*miseverybitasdistinctftommas-tLis.Fractionsaremoreinterestingthanpositiveornegativeintegers'WeneedfractionsformanyPurPoses,butperhapsmostobviouslyforpurposesofmeasurement'MyfriendandcollaboratorDrA.N.Whiteheadhasdevelopedatheoryoffractionsspeciallywhichissetforthadaptedfortheirapplicationtomeasurement,isneededistodefinein.FrinripiaMathciatica.LButifallthatproperties,thisobjectshavingtherequiredpurelymathematicalporpor"."oL"achievedbysimplermethod,whichweshall"nlnasbeingthatir"r.adopr.Weshalldefinethefractionnumbersr,ywhenrelation*tti.ttholdsbetweentwoinductiveccn:yfl;.ThisdefinitionenablesustoProvethatrnfnisaone-onerelation,providedneithermorniszeto.Andof'coursenfmistheconverserelationtont'ln.Fromtheabovedefinitionitisclearthatthefractionnlristhatrelationbetweentwointegers#andywhichconsistsinthefactthatx:m!.Thisrelation,liketherelation*m,isbynomeanscapableofbeingidentifiedwiththeinductivecardinalofclassesareobjectsnumberi,b"r^osearelationandaclassrVol.iii.*3ooff.,especia[y3o3.\nRational,Real,andComplexNumbers65ofutterlydifferentkinds.lItwillbeseenthatofnisalwaysthesamerelation,whateverinductivenumbernmaybe;itis,inshort,therelationofotoanyotherinductivecardinal.Wemaycallthisthezeroofrationalnumbers;itisnot,ofcourse,identicalwiththecardinalnumbero.Conversely,therelationmfoisalwaysthesame,whateverinductivenumberrnmaybe.Thereisnotanyinductivecardinaltocorrespondtomfo,Wemaycallit"theinfinityofrationals."Itisaninstanceofthesortofinfinitethatistraditionalinmathematics,andthatisrepresentedby"*."ThisisatotallydifferentsortfromthetrueCantorianinfinite,whichweshallconsiderinournextchapter.Thein-finityofrationalsdoesnotdemand,foritsdefinitionoruse,anyinfiniteclassesorinfiniteintegers.Itisnot,inactualfact,averyimportantnotion,andwecoulddispensewithitaltogetheriftherewereanyobjectindoingso.TheCantorianinfinite,ontheotherhand,isofthegreatestandmostfundamentalimpor-tance;theunderstandingofitopensthewaytowholenewrealmsofmathematicsandphilosophy.Itwillbeobservedthatzeroandinfinity,aloneamongratios,arenotone-one.Zercisone-many,andinfinityismany-one.Thereisnotanydifficultyindefininggreaterandlessamongratios(orfractions).Giventworatiosmfnandpfq,weshallsaythatmfnislessthanplqifzegislessthanpn.Thereisnodifficultyinprovingthattherelation"lessthan,"sodefined,isserial,sothattheratiosformaseriesinorderofmagnitude.Inthisseries,zeroisthesmallesttermandinfinityisthelargest.Ifweomitzeroandinfinityfromourseries,thereisnolongeranysmallestorlargestratio;itisobviousthatifmfnisanymtiootherthanzeroandinfinity,mfznissmallerandzmfnislarger,thoughneitheriszeroorinfinit/,sothatmfnisneitherthesmallestIOfcourseinpracticeweshallcontinuetospeakofafractionas(say)greaterorlessthanr,meaninggreaterorlessthantheratior/r.Solongasitisunderstoodthattheratior/randthecardinalnumberraredifferent,itisnotnecessarytobealwayspedanticinemphasisingthedifference.\n66fntroductiontoMathematicalPltilosoplt1northelargestratio,andtherefore(whenzeroandinfinityareomitted)thereisnosmallestorlargest,sincemfnwaschosenarbitrarily.Inlikemannerwecanprovethathowevernearlyequaltwofractionsmaybe,therearealwaysotherfractionsbetweenthem.For,letmfnandplqb,twofractions,ofwhichplqi"thegreater.Thenitiseasytosee(ortoprove)that@*il1(n*{)willbegreaterthanmfnandlessthanp/9.Thustheseriesofratiosisoneinwhichnotwotermsareconsecutive,buttherearealwaysothertermsbetweenanytwo.Sincethereareothertermsbetweentheseothers,andsoonadinf'nitum,itisobviousthatthereareaninfinitenumberofratiosbetweenanytwo,howevernearlyequalthesetwomaybe.rAserieshavingthepropertythattherearealwaysothertermsbetweenttanytwo,sothatnotwoareconsecutive,iscalledcompact."Thustheratiosinorderofmagnitudeforma"compact"series.Suchserieshavemanyimportantproperties,anditisimportanttoobservethatratiosaffordaninstanceofacomPactseriesgeneratedpurelylogically,withoutanyappealtospaceortimeoranyotherempiricaldatum.Positiveandnegativeratioscanbedefinedinawayanalogoustothatinwhichwedefinedpositiveandnegativeintegers.Havingfirstdefinedthesumoftworatiosmlnandplqas(mq*pn)fng,wedefine+plqastherelationof.mfnlplqto*ln,wherernfnisanyratio;and-plqisofcoursetheconverseof*plq.Thisisnottheonlypossiblewayofdefiningpositiveandnegativeratios,butitisawaywhich,forourpurpose,hasthemeritofbeinganobviousadaptationofthewayweadoptedinthecaseofintegers.Wecomenowtoamoreinterestingextensionoftheideaofttttnumber,i,e.theextensiontowhatarecalledrealnumbers,whicharethekindthatembraceirrationals.InChapterI.wet'incommensurableshadoccasiontomention"andtheirdis-rSirictlyspeaking,thisstatement,aswellasthosefollowingtotheendoftheparagraph,involveswhatiscalledthe"axiomofinfinity,"whicbwillbediscussedinalaterchapter.\nRational,Real,andComplexNumbers67coverybyPythagoras.Itwasthroughthem,i.e.throughgeometry,thatirrationalnumberswerefirstthoughtof.Asquareofwhichthesideisoneinchlongwillhaveadiagonalofwhichthelengthisthesquarerootof.zinches.But,astheancientsdiscovered,thereisnofractionofwhichthesquareisz.ThispropositionisprovedinthetenthbookofEuclid,whichisoneofthosebooksthatschoolboyssupposedtobefortunatelylostinthedayswhenEuclidwasstillusedasatext-book.Theproofisextraordinarilysimple.Ifpossible,letmfnbethesquarerootofi,,sothatmzfn,-z,i.e.m2-2n2.Thusmzisanevennumber,andthereforernm:ustbeanevennumber,becausethesquareofanoddnumberisodd.Nowif.rniseven,m2mustdivideby+,forifm.:2?tthenrn2:4?2.Thusweshallhave*?2:2n2,wherepishalfof.m.Hence2p2:n2,andthereforenlpwillalsobethesquarerootofz.Butthenwecanrepeattheargument:ifn:2{tplgwillalsobethesquarerootoIz,andsoon,throughanunendingseriesofnumbersthatareeachhalfofitspredecessor.Butthisisimpossible;ifwedivideanumbetbyz,andthenhalvethehalf,andsoon,wemustreachanoddnumberafterafinitenumberofsteps.Orwemayputtheargumentevenmoresimplybyassumingthatthemfnwestartwithisinitslowestterms;inthatcase,mandncannotbothbeeven;yetwehaveseenthat,if.mzfnz:zttheymustbe.Thustherecannotbeanyfractionnt.fnwhosesquareisz.Thusnofractionwillexpressexactlythelengthofthediagonalofasquarewhosesideisoneinchlong.Thisseemslikeachallengethrownoutbynaturetoarithmetic.Howeverthearithmeticianmayboast(asPythagorasdid)aboutthepowerofnumbers,natureseemsabletobaffiehimbyexhibitinglengthswhichnonumberscanestimateintermsoftheunit.Buttheproblemdidnotremaininthisgeometricalform.Assoonasalgebrawasinvented,thesameproblemaroseasregardsthesolutionofequations,thoughhereittookonawiderform,sinceitalsoinvolvedcomplexnumbers.Itisclearthatfractionscanbefoundwhichapproachnearer\n68Introductiont0Math.enaticalPltihsoplt1andnearertohavingtheirsquareequaltoz.Wecanformanascendingseriesoffractionsallofwhichhavetheirsquareslessthanz,butdifieringfromzintheirlatermembersbylessthananyassignedamount.Thatistosay,supposeIassignsomesmallamountinadvance,sayone-billionth,itwillbefoundthatallthetermsofourseriesafteracertainone,saythetenth,havesquaresthatdifferfromzbylessthanthisamount.AndifIhadassignedastillsmalleramount,itmighthavebeennecessarytogofurtheralongtheseries,butweshouldhavereachedsoonerorlateratermintheseries,saythetwentieth,afterwhichalltermswouldhavehadsquaresdifferingfromzbylessthanthisstillsmalleramount.Ifwesettoworktoextractthesquarerootof.zbytheusualarithmeticalrulerw€shallobtainanunendingdecimalwhich,takentoso-and-somanyplaces,exactlyfulfilstheaboveconditions.Wecanequallywellformadescendingseriesoffractionswhosesquaresareallgreaterthanz,butgreaterbycontinuallysmalleramountsaswecometolatertermsoftheseries,anddifiering,soonerorlater,bylessthananyassignedamount.Inthiswayweseemtobedrawingacordonroundthesquaierootof.z,anditmayseemdifficulttobelievethatitcanpermanentlyescaPeus.Nevertheless,itisnotbythismethodthatweshallactuallyreachthesquarerootof,z.Ifwedivideallratiosintotwoclasses,accordingastheirsquaresarelessthan2ornot,wefindthat,amongthosewhosesquaresarenotlessthanz,allhavetheirsquaresgreaterthanz.Thereisnomaximumtotheratioswhosesquareislessthanz,andnominimumtothosewhosesquareisgreaterthanz,Thereisnolowerlimitshortofzerotothedifferencebetweenthenumberswhosesquareisalittlelessthanzandthenumberswhosesquareisalittlegreaterthanz,Wecan,inshort,divideallratiosintotwoclassessuchthatallthetermsinoneclassarelessthanallintheother,thereisnomaximumtotheoneclass,andthereisnominimumtotheother.Betweenthese-fhustwoclasses,wher{zoughttobe,thereisnothing.our"\nRational,Real,andConplexNumbers69cordon,thoughwehavedrawnitastightaspossible,hasbeendrawninthewrongplace,andhasnotcaught\/;.Theabovemethodofdividingallthetermsofaseriesintotwoclasses,ofwhichtheonewhollyprecedestheother,wasbroughtintoprominencebyDedekindrlandisthereforecalleda"Dedekindcut."Withrespecttowhathappensatthepointofsection,therearefourpossibilities:(t)theremaybeamaximumtothelowersectionandaminimumtotheuppersection,(z)theremaybeamaximumtotheoneandnominimumtotheother,(3)theremaybenomaximumtotheone,butaminimumtotheother,(4)theremaybeneitheramaximumtotheonenoraminimumtotheother.Ofthesefourcases,thefirstisillustratedbyanyseriesinwhichthereareconsecutiveterms:intheseriesofintegerrs,forinstance,alowersectionmustendwithsomenumberaandtheuppersectionmustthenbeginwithn+r.Thesecondcasewillbeillustratedintheseriesofratiosifwetakeasourlowersectionallratiosuptoandincludingt,andinouruppersectionallratiosgreaterthanr.Thethirdcaseisillustratedifwetakeforourlowersectionallratioslessthant,andforouruppersectionallratiosfromrupward(includingritselfl.Thefourthcase,aswehaveseen,isillustratedifweputinourlowersectionallratioswhosesquareislessthanz,andinouruppersectionallratioswhosesquareisgreaterthanz.Wemayneglectthefirstofourfourcases,sinceitonlyarisesinserieswherethereareconsecutiveterms.Inthesecondofourfourcases,wesaythatthemaximumofthelowersectionisthelouterlimitoftheuppersection,orofanysetoftermschosenoutoftheuppersectioninsuchawaythatnotermoftheuppersectionisbeforeallofthem.Inthethirdofourfourcases,wesaythattheminimumoftheuppersectionistheupperlimitofthelowersection,orofanysetoftermschosenoutofthelowersectioninsuchawaythatnotermofthelowersectionisafterallofthem.Inthefourthcase,wesaythatIStetigheitunil,irrationaleZahlen,zndedition,Brunswick,1892.\n70fntoductiontoMatlrematicalPltilosopltyt'thereisa"gap:neithertheuppersectionnorthelowerhasalimitoralastterm.Inthiscase,wemayalsosaythatwehavean"irrationalsectionrt'sincesectionsoftheseriesofratioshave"g"ps"whentheycorresPondtoirrationals.Whatdelayedthetruetheoryofirrationalswasamistakenbeliefthattheremustbe"limits"ofseriesofratios.Thenotionof"limit"isoftheutmostimportance,andbeforeproceedingfurtheritwillbewelltodefineit.t'Atermrissaidtobean"upperlimitofaclassowithrespecttoarelationPif(r)ahasnomaximuminP,(z)everymemberofawhichbelongstothefieldofPprecedes#'(3)everymemberofthefieldofPwhichprecedesrprecedessomememberttofo.(Byprecedes"wemean"hastherelationPto'")'-Thispresupposesthefollowingdefinitionofa"maximum"ttt'AtermrissaidtobeamaximumofaclassowithrespecttoarelationPifrisamemberofoandofthefieldofPanddoesnothavetherelationPtoanyothermemberofo.Thesedefinitionsdonotdemandthatthetermstowhichtheyareappliedshouldbequantitative.Forexample,givenaseriesofmomentsoftimearrangedbyearlierandlater,their"maximum"(ifany)willbethelastofthemoments;butift'theyarearrangedbylaterandearlier,their"maximum(ifany)willbethefirstofthemoments.The"minimum"ofaclasswithrespecttoPisitsmaximumt'lowerwithrespecttotheconverseofP;andthelimit"withrespecttoPistheupperlimitwithrespecttotheconverseofP.Thenotionsoflimitandmaximumdonotessentiallydemandthattherelationinrespecttowhichtheyaredefinedshouldbeserial,buttheyhavefewimportantapplicationsexcepttocaseswhentherelationisserialorquasi-serial.Anotionwhichisoftenimportantisthenotion"upperlimitormaximuor"towhichwemaygivethename"upperboundary."Thusthettupperboundary"ofasetoftermschosenoutofaseriesistheirlastmemberiftheyhaveone,but,ifnot,itisthefirsttermafterallofthem,ifthereissuchaterm.Ifthereisneither\nRational,Real,andComplexNurnbers7|amaximumnoralimit,thereisnoupperboundary.The"lowerboundary"isthelowerlimitorminimum.RevertingtothefourkindsofDedekindsection,weseethatinthecaseofthefirstthreekindseachsectionhasaboundary(upperorlowerasthecasemaybe),whileinthefourthkindneitherhasaboundary.Itisalsoclearthat,wheneverthelowersectionhasanupperboundary,theuppersectionhasalowerboundary.Inthesecondandthirdcases,thetwoboundariesareidentical;inthefirst,theyareconsecutivetermsoftheseries.Aseriesiscalled"Dedekindian"wheneverysectionhasaboundary,upperorlowerasthecasemaybe.WehaveseenthattheseriesofratiosinorderofmagnitudeisnotDedekindian.Fromthehabitofbeinginfluencedbyspatialimagination,peoplehavesupposedthatseriesmusthavelimitsincaseswhereitseemsoddiftheydonot.Thus,perceivingthattherewasnorationallimittotheratioswhosesquareislessthanz,theyallowedthemselvesto"postulate"anirrationallimit,whichwastofilltheDedekindgr'c.Dedekind,intheabove-mentionedwork,setuptheaxiomthatthegapmustalwaysbefilled,i.a.thateverysectionmusthaveaboundary.Itisforthisreasonthatserieswherehisaxiomisverifiedarecalled"Dedekindian."Butthereareaninfinitenumberofseriesforwhichitisnotverified.Themethodof"postulating"'whatwewanthasmanyadvan-tages;theyarethesameastheadvantagesoftheftoverhonesttoil.Letusleavethemtoothersandproceedwithourhonesttoil.ItisclearthatanirrationalDedekindcutinsomeway'orePre-sents"aD.irrational.Inordertomakeuseofthis,whichtobeginwithisnomorethanavaguefeeling,wemustfindsomewayof.elicitingfromitaprecisedefinition;andinsrdertodothis,wemustdisabuseourmindsofthenotionthatanirrationalmustbethelimitoLasetofratios.Justasratioswhosede-nominatorisrarenotidenticalwithintegers,soth,,serational\n72fntroductiont0MatltenaticalPltilosopltynumberswhichcanbegreaterorlessthanirrationa.ls,orcanhaveirrationalsastheirlimits,mustnotbeidentifiedwithratios.Wehavetodefineanewkindofnumberscalled"realnumbersr"ofwhichsomewillberationalandsomeirrational.Thosethatarerational"correspond"toratios,inthesamekindofwayinwhichtherationftcorcespondstotheintegerz;buttheyarenotthesameasratios.Inordertodecidewhattheyaretobe,letusobservethatanirrationalisrepresentedbyanirrationalcut,andacutisrepresentedbyitslowersection.Letusconfineourselvestocutsinwhichthelowersectionhasnomaximum;inthiscasewewillcallthelowersectiona"segment."Thenthosesegmentsthatcorrespondtoratiosarethosethatconsistofallratioslessthantheratiotheycorrespondto,whichistheirboundary;whilethosetJratrepresentirrationalsarethosethathavenoboundary.Segments,boththosethathaveboundariesandthosethatdonot,aresuchthat,ofanytwopertainingtooneseries,onemustbepartoftheotherIhencetheycanallbearrangedinaseriesbytherelationofwholeandpart.AseriesinwhichthereareDedekindBaps,i.e.inwhichtherearesegmentsthathavenoboundary,willgiverisetomoresegmentsthanithasterms,sinceeachtermwilldefineasegmenthavingthattermforboundary,andthenthesegmentswithoutboundarieswillbeextra.Wearenowinapositiontodefinearealnumberandanirrationalnumber.ttttArealnumberisasegmentoftheseriesofratiosinorderofmagnitude.ttt'Anirrationalnumberisasegmentoftheseriesofratioswhichhasnoboundary.ttA"rationalrealnumberisasegmentoftheseriesofratioswhichhasaboundary.Thusarationalrealnumberconsistsofallratioslessthanacertainratio,anditistherationalrealnumbercorrespondingtothatratio.Therealnumberl,forinstance,istheclassofproperfractions.\nRational,Real,andConplexNumbers73Inthecasesinwhichwenaturallysupposedthatanirrationalmustbethelimitofasetofratios,thetruthisthatitisthelimitofthecorrespondingsetofrationalrealnumbersintheseriesofsegmentsorderedbywholeandpart.Forexample,t/iittheupperlimitofallthosesegmentsoftheseriesofratiosthatcorrespondtoratioswhosesquareislessthanz.Moresimplystill,\/;isthesegmentconsistingofallthoseratioswhosesquareislessthanz.ItiseasytoprovethattheeeriesofsegmentsofanyseriesisDedekinfian.For,givenanysetofsegments,theirboundarywillbetheirlogicalsum,a.a.theclassofallthosetermsthatbelongtoatleastonesegmentoftheset.lTheabovedefinitionofrealnumbersisanexampleof"con-t(struction"asagainstpostulationr"ofwhichwehadanotherexampleinthedefinitionofcardinalnumbers.Thegreatadvantageofthismethodisthatitrequiresnonewassumptions,butenablesustoproceeddeductivelyfromtheoriginalapparatusoflogic.Thereisnodifficultyindefiningadditionandmultiplicationforrealnumbersasabovedefined.Giventworealnumbersp,andv,eachbeingaclassofratios,takeanymemberofpandanymemberofyandaddthemtogetheraccordingtotherulefortheadditionofratios.Formtheclassofallsuchsumsobtainablebyvaryingtheselectedmembersofp,andv.Thisgivesanewclassofratios,anditiseasytoprovethatthisnewclassisasegmentoftheseriesofratios.Wedefineitasthesumofp,andv.Wemaystatethedefinitionmoreshortlyasfollows:-Thearitbmeticalsumoftutorealnumbersistheclassofthearithmeticalsumsofamemberoftheoneandamemberoftheotherchoseninallpossibleways.1ForafuiiertreatmentofthesubjectofsegmentsandDedekindianrelations,seePrinci,piaMathemati,aa,vol.ii.x2ro-2r4.Forafullertreatmentofrealnumbers,seeibid.,vol.iii.*3roff.,andPrinciplesofMathematics,chaps.xxxiii.andxxxiv.\n7+[ntroductiontaMatltematicalPhilosoplt1,Wecandefinethearithmeticalproductoftworealnumbersinexactlythesameway,bymultiplyingamemberoftheonebyamemberoftheotherinallpossibleways.Theclassofratiosthusgeneratedisdefinedastheproductofthetworealnumbers.(Inallsuchdefinitions,theseriesofratiosistobedefinedasexcludingoandinfinity.)Thereisnodifficultyinextendingourdefinitionstopositiveandnegativerealnumbersandtheiradditionandmultiplication.ftremainstogivethedefinitionofcomplexnumbers.Complexnumbers,thoughcapableofageometricalinterpreta-tion,arenotdemandedbygeometryinthesameimperativewayttinwhichirrationalsaredemanded.Acomplex"numbermeansanumberinvolvingthesquarerootofanegativenumber,whetherintegral,fractional,orreal,Sincethesquareofanegativenumberispositive,anumberwhosesquareistobenegativehastobeanewsortofnumber.Usingtheletterrforthesquarerootof-r,anynumberinvolvingthesquarerootofanegativenumbercanbeexpressedintheformx*!i,whererandyarcreal.Thepartyeiscalledthe"imaginary"partofthisnumber,('realrbeingthe"real"part.(Thereasonforthephrasenumberg"isthattheyarecontrastedwithsuchasare"ima-ginary.")Complexnumbershavebeenforalongtimehabituallyusedbymathematicians,inspiteoftheabsenceofanyprecisedefinition.Ithasbeensimplyassumedthattheywouldobeytheusualarithmeticalrules,andonthisassumptiontheiremploy-menthasbeenfoundprofitable.Theyarerequiredlessforgeometrythanforalgebraandanalysis.Wedesire,forexample,tobeabletosaythateveryquadraticequationhastworoots,andeverycubicequationhasthree,andsoon.Butifweareconfinedtorealnumbers,suchanequationas12*r-ohasnoroots,andsuchanequationasf-t:ohasonlyone.Everygeneralisationofnumberhasfirstpresenteditselfasneededforsomesimpleproblem:negativenumberswereneededinorderthatsubtractionmightbealwayspossible,sinceotherwisea-bwouldbemeaninglessifawerelessthanb;ftaction$wereneeded\nRational,Real,andComplexNumbers75inorderthatdivisionmightbealwayspossible;andcomplexnumbersareneededinorderthatextractionofrootsandsolu-tionofequationsmaybealwayspossible.Butextensionsofnumberarenotcreatedbythemereneedforthem:theyarecreatedbythedefinition,anditistothedefinitionofcomplexnumbersthatwemustnowturnourattention.Acomplexnumbermayberegardedanddefinedassimplyanorderedcoupleofrealnumbers.Here,aselsewherermanydefinitionsarepossible.Allthatisnecessaryisthatthedefini-tionsadoptedshallleadtocertainproperties.Inthecaseofcomplexnumbers,iftheyaredefinedasorderedcouplesofrealnumbers,wesecureatoncesomeoftheProPertiesrequired,namely,thattworealnumbersarerequiredtodetermineacom-plexnumber,andthatamongthesewecandistinguishafirstandasecond,andthattwocomplexnumbersareonlyidenticalwhenthefirstrealnumberinvolvedintheoneisequaltothefirstinvolvedintheother,andthesecondtothesecond.Whatisneededfurthercanbesecuredbydefiningtherulesofadditionandmultiplication.Wearetohave(x*yi)*(x'*y'i):(tcIx')*0*y')i(xtyi)(x'+y'i)-(xx'-!y')+(xy'*x'y)i.Thusweshalldefinethat,giventwoorderedcouplesofrealnumbers,(*,y)and(x',y'),theirsumistobethecouple(rf*',y*!'),andtheirproductistobethecouple(**'-yy',xy'{x'y).Bythesedefinitionsweshallsecurethatourorderedcouplesshallhavethepropertieswedesire.Forexample,taketheproductofthetwocouples(o,y)and(o,y').Thiswill'bytheaboverule,bethecouple(-yy',o).Thusthesquareofthecouple(o,r)willbethecoupl.(-r,o).Nowthosecouplesinwhichthesecondtermisoarethosewhich,accordingtotheusualnomenclature,havetheirimaginarypartzero;inthenotationx*!i,theyarex{oi,whichitisnaturaltowritesimplyr.Justasitisnatural(buterroneous)toidentifyratioswhosede-nominatorisunitywithintegers,soitisnatural(buterroneous)\n76IntrodactiontoMathematicalPhilosopltytoidentifycomplexnumberswhoseimaginarypartiszerowithrealnumbers.Althoughthisisanerrorintheory,itisacon-('venienceinpractice;"x{oi"*^ybereplacedsimplyby,c"ttt'ando*ti"by"yir"providedwerememberthatthe"*isnotreallyarealnumber,butaspecialcaseofacomplexnumber.c(Andwhenyisr,"yi"mayofcoursebereplacedbyi."Thusthecouple(o,r)isrepresentedbyi,andthecouple(-r,o)isrepresentedby-r.Nowourrulesofmultiplicationmakethesquareof(o,r)equalto(-r,o),i.e.thesquareofiis-r.Thisiswhatwedesiredtosecure.ThusourdefinitionsserveallnecessaryPurPoses.Itiseasytogiveageometricalinterpretationofcomplexnumbersinthegeometryoftheplane.Thissubjectwasagree-ablyexpoundedbyW.K.CliffordinhisCommonSenseoftbeExactSciences,abookofgreatmerit,butwrittenbeforetheimportanceofpurelylogicaldefinitionshadbeenrealised.Complexnumbersofahigherorder,thoughmuchlessusefulandimportantthanthosewhatwehavebeendefining,havecertainusesthatarenotwithoutimportanceingeometry,asmaybeseen,forexample,inDrWhitehead'sUniversalAlgebra.Thedefinitionofcomplexnumbersoforderaisobtainedbyanobviousextensionofthedefinitionwehavegiven.Wedefineacomplexnumberofordernasaone-manyrelationwhosedomainconsistsofcertainrealnumbersandwhoseconversedomainconsistsoftheintegersfromtton.tThisiswhatwouldordi-narilybeindicatedbythenotation(*r,fr22fis'...*n),wherethesuffixesdenotecorrelationwiththeintegersusedassuffixes,andthecorrelationisone-ma\!tnotnecessarilyone-one,becauser,andx,maybeequalwhenrand.rarenotequal.Theabovedefinition,withasuitableruleofmultiplication,willserveallpurposesforwhichcomplexnumbersofhigherordersareneeded.Wehavenowcompletedourreviewofthoseextensionsofnumberwhichdonotinvolveinfinity.Theapplicationofnumbertoinfinitecollectionsmustbeournexttopic.tCt.PrinciplesofMathematics,g36o,p.3Zg.\nCHAPTBRVIIIINFINITECARDINALNUMBERSTHrdefinitionofcardinalnumberswhichwegaveinChapterII.wasappliedinChapterIII.tofinitenumbers,i.e.totheordinarynaturalnumbers.Tothesewegavethenamettinductivenumbersr"becausewefoundthattheyaretobedefinedasnumberswhichobeymathematicalinductionstartingfromo.Butwehavenotyetconsideredcollectionswhichdonothaveaninductivenumberofterms,norhaveweinquiredwhethersuchcollectionscanbesaidtohaveanumberatall.Thisisanancientproblem,whichhasbeensolvedinourownday,chieflybyGeorgCantor.InthepresentchapterweshallattempttoexplainthetheoryoftransfiniteorinfinitecardinalnumbersasitresultsfromacombinationofhisdiscoverieswiththoseofFregeonthelogicaltheoryofnumbers.ftcannotbesaidtobecertainthatthereareinfactanyinfinitecollectionsintheworld.Theassumptionthatthereareiswhatwecallthe"axiomofinfinity."Althoughvariouswayssuggestthemselvesbywhichwemighthopetoprovethisaxiom,thereisreasontofearthattheyareallfallacious,andthatthereisnoconclusivelogicalreasonforbelievingittobetrue.Atthesametime,thereiscertainlynologicalreasonagainstinfinitecollections,andwearethereforejustified,inlogic,ininvestigatingthehypo-thesisthattherearesuchcollections.Thepracticalformofthishypothesis,forourpresentpurposes,istheassumptionthat,ifnisanyinductivenumber,zisnotequalton+r.Varioussubtletiesariseinidentifyingthisformofourassumptionwith77\n78fntroductiontoMath'ematicalPltilosoplr.1theformthatassertstheexistenceofinfinitecollections;butwewillleavetheseoutofaccountuntil,inalaterchapterrwecometoconsidertheaxiomofinfinityonitsownaccount.Forthepresentweshallmerelyassumethat,ifnisaninductivenumber,aisnotequalton{t.ThisisinvolvedinPeano'sassumptionthatnotwoinductivenumbershavethesamesuc-cessor1for,if.n-n*r,thenn-randnhavethesamesuccessor,r,ramelyn.ThusweareassumingnothingthatwasnotinvolvedinPeano'sprimitivepropositions.Letusnowconsiderthecollectionoftheinductivenumbersthemselves.Thisisaperfectlywell-definedclass.Inthefirstplace,acardinalnumberisasetofclasseswhichareallsimilartoeachotherandarenotsimilartoanythingexcepteachother.ttWethendefineastheinductivenumbers"thoseamongcardinalswhichbelongtothePosterityofowithresPecttotherelationof.nton*t,i,e.thosewhichPossesseveryProPertypossessedbyoandbythesuccessorsofpossessors,meaningbyttthesuccessor"ofnthenumbern+r.Thustheclassofgeneral"inductivenumbers"isperfectlydefinite.Byourdefinitionofcardinalnumbers,thenumberoftermsintheclass(6ofinductivenumbersistobedefinedasallthoseclassesthataresimilartotheclassofinductivenumbers"-i.a.thissetofclasseszsthenumberoftheinductivenumbersaccordingtoourdefinitions.Nowitiseasytoseethatthisnumberisnotoneoftheinductivenumbers.Ifzisanyinductivenumber,thenumbero{numbersfromoton(bothincluded)isn{r;thereforethetotalnumberofinductivenumbersisgreaterthann,nomatterwhichoftheinductivenumbersnmaybe.Ifwearrangetheinductivenumbersinaseriesinorderofmagnitude,thisserieshasnolastterm;butilnisaninductivenumber,everyserieswhosefieldhasntermshasalastterm,asitiseasytoProve.Suchdifferencesmightbemultipliedadlib.Thusthenumberofinductivenumbersisanewnumber,difierentfromallofthem,notPossess-ingallinductiveproperties.Itmayhappenthatohasacertain\nInfininCardinalNumbers79property,andthatifnhasitsohasafr,andyetthatthisnewnumberdoesnothaveit.Thedifficultiesthatsolongdelayedthetheoryofinfinitenumberswerelargelyduetothefactt}atsome,atleast,oftheinductivepropertieswerewronglyjudgedtobesuchasmustbelongtoallnumbers;indeeditwasthoughtthattheycouldnotbedeniedwithoutcontradiction.Thefirststepinunderstandinginfinitenumbersconsistsinrealisingthemistakennessofthisview.Themostnoteworthyandastonishingdifierencebetweenaninductivenumberandthisnewnumberisthatthisnewnumberisunchangedbyaddingrorsubtractingrordoublingorhalvingoranyofanumberofotheroperationswhichwethinkofasnecessarilymakinganumberlargerorsmaller.ThefactofbeingnotalteredbytheadditionofrisusedbyCantorforthedefini-tionofwhathecalls"transfinite"cardinalnumbers;butforvariousreasons,someofwhichwillappearasweproceed,itisbettertodefineaninfinitecardinalnumberasonewhichdoesnotpossessallinductiveproperties,i.e.simplyasonewhichisnotaninductivenumber.Nevertheless,thepropertyofbeingunchangedbytheadditionofrisaveryimportantone,andwemustdwellonitforatime.Tosaythataclasshasanumberwhichisnotalteredbytheadditionofristhesamethingastosaythat,ifwetakeaterm#whichdoesnotbelongtotheclass,wecanfindaone-onerelationwhosedomainistheclassandwhoseconversedomainisobtainedbyadding#totheclass.Forinthatcase,theclassissimilartothesumofitselfandthetermx,i.e.toaclasshavingoneextraterm;sothatithasthesamenumberasaclasswithoneextraterm,sothatifzisthisnumber,n:n{t.Inthiscase,weshallalsohaven:n-rti.e.therewillbeone-onerelationswhosedomainsconsistofthewholeclassandwhoseconversedomainsconsistofjustonetermshortofthewholeclass.Itcanbeshownthatthecasesinwhichthishappensarethesameastheapparentlymoregeneralcasesinwhichsorrrepart(shortofthewhole)canbeputintoone-onerelationwitb.thewhole.Whenthiscanbedone,\n8oIntroductiontoMathernaticalPlt'ilosopltythecorrelatorbywhichitisdonemaybesaidto"reflect"thewholeclassintoapartofitself;forthisreason,suchclasseswillbecalled"reflexive."Thus:t'reflexiveA"classisonewhichissimilartoaProPerPartofitself.(A"properpart"isapartshortofthewhole.)A"reflexive"cardinalnumberisthecardinalnumberofareflexiveclass.Wehavenowtoconsiderthispropertyofreflexiveness.Oneofthemoststrikinginstancesofa"reflexion"isRoyce'sillustrationofthemap:heimaginesitdecidedtomakeamaPofEnglanduponapartofthesurfaceofEngland.Amap,ifitisaccurate,hasaperfectone-onecorresPondencewithitsoriginal;thusourmaP,whichispart,isinone-onerelationwiththewhole,andmustcontainthesamenumberofpointsasthewhole,whichmustthereforebeareflexivenurnber.Royceisinterestedinthefactthatthemap,ifitiscorrect,mustcontainamapofthem"p,whichmustinturncontainamaPofthemaPofthernaprandsoonadinf,nitum.Thispointisinteresting,butneednotoccupyusatthismoment.Infact,weshalldowelltopassfromPicturesqueillustrationstosuchasaremorecompletelydefinite,andforthisPurPosewecannotdobetterthanconsiderthenumber-seriesitself.Therelationofnton*r,confinedtoinductivenumbers,isone-one,hasthewholeoftheinductivenumbersforitsdomain,andallexceptoforitsconversedomain.Thusthewholeclassofinductivenumbersissimilartowhatthesameclassbecomest'whenweomito.Consequentlyitisareflexive"classaccordingt'reflexivetothedefinition,andthenumberofitstermsisa"number.Again,therelationofntozn,confinedtoinductivenumbers,isone-one,hasthewholeoftheinductivenumbersforitsdomain,andtheeveninductivenumbersaloneforitsconversedomain.Hencethetotalnumberofinductivenumbersisthesameasthenumberofeveninductivenumbers.ThispropertywasusedbyLeibniz(andmanyothers)asaProofthatinfinitenumbersareimpossible;itwasthoughtself-contradictorythat\nInfninCardinalNumbers8l"thepartshouldbeequaltothewhole."Butthisisoneofthosephrasesthatdependfortheirplausibilityuponanunperceivedvagueness:thewordt'equalt'hasmanymeanings,butifitistakentomeanwhatwehavecalled"similarrt'thereisnocontra-diction,sinceaninfinitecollectioncanperfectlywellhavepartssimilartoitself.Thosewhoregardthisasimpossiblehave,unconsciouslyasarule,attributedtonumbersingeneralpro-pertieswhichcanonlybeprovedbymathematicalinduction,andwhichonlytheirfamiliaritymakesusregard,mistakenly,astruebeyondtheregionofthefinite.ttWheneverwecanreflect"aclassintoapartofitself,thesamerelationwillnecessarilyreflectthatpartintoasmallerpart,andsoonadinf.nitum.Forexample,wecanreflect,aswehavejustseen,alltheinductivenumbersintotheevennumbers;wecan,bythesamerelation(thatofntozn)reflecttheevennumbersintothemultiplesof+,theseintothemultiplesof8,andsoon.ThisisanabstractanaloguetoRoyce'sproblemofthemap.Theevennumbersatea"*"p"ofalltheinductivenumbers;themultiplesof.4areamapofthemap;themultiplesof8areamapofthemapofthemap;andsoon.Ifwehadappliedthesameprocesstotherelationofnton*r,our"map"wouldhaveconsistedofalltheinductivenumbersexcepto;themapofthemapwouldhaveconsistedofallfrom2onward,themapofthemapofthemapofallfrom3onward;andsoon.Thechiefuseofsuchillustrationsisinordertobecomefamiliarwiththeideaofreflexiveclasses,sothatapparentlyparadoxicalarithmeticalpropositionscanbereadilytranslatedintothelanguageofreflexionsandclasses,inwhichtheairofparadoxismuchless.Itwillbeusefultogiveadefinitionofthenumberwhichisthatoftheinductivecardinals.Forthispurposewewillfirstdefinethekindofseriesexemplifiedbytheinductivecardinalsinorderofmagnitude.Thekindofserieswhichiscalleda"progression"hasalreadybeenconsideredinChapterI.Itisaserieswhichcanbegeneratedbyarelationofconsecutiveness:\n8zInnoductiontoMatltematicalPlt'ilosopltyeverymemberoftheseriesistohaveaSuccessor,butthereistobejustonewhichhasnopredecessor,andeverymemberoftheseriesistobeintheposterityofthistermwithrespecttotherelation"immediatepredecessor.ttThesecharacteristicsImaybesummedupinthefollowingdefinition:-ttt'Aprogessionisaone-onerelationsuchthatthereisjustonetermbelongingtothedomainbutnottotheconversedomain,andthedomainisidenticalwiththeposterityofthisoneterm.Itiseasytoseethataprogression,sodefined,satisfiesPeano'sfiveaxioms.Thetermbelongingtothedomainbutnottothe'6conversedomainwillbewhathecallso";thetermtowhicht'atermhastheone-onerelationwillbethe"successoroftheterm;andthedomainoftheone-onerelationwillbewhathecalls"number."Takinghisfiveaxiomsinturn,wehavethefollo*irgtranslatit'o(t)isanumber"becomes:"Thememberofthedomainwhichisnotamemberoftheconversedomainisamemberofthedomain."Thisisequivalenttotheexistenceofsuchamember,whichisgiveninourdefinition.Wewillcallthisttmemberthefirstterm.tt(z)ttThesuccessorofanynumberisanumberttbecomes:"Thetermtowhichagivenmemberofthedomainhastherela-tioninquestionisagainamemberofthedomain."Thisisprovedasfollows:Bythedefinition'everymemberofthedomainisamemberoftheposterityofthefirsttermIhencethesuccessorofamemberofthedomainmustbeamemberoftheposterityofthefirstterm(becausetheposterityofatermalwayscontainsitsownsuccessors,bIthegeneraldefinitionofposterity),andthereforeamemberofthedomain,becausebythedefinitiontheposterityofthefirsttermisthesameasthedomain.(3)"Notwonumbershavethesamesuccessor."Thisisonlytosaythattherelationisone-many,whichitisbydefinition(beingone-one).tCt.PrincifiaMathematica',vol.ii.*rz3.\nInfniteCardinalNumbers83(+)"oisnotthesuccessorofanynumberttbecomes:ttThefirsttermisnotamemberoftheconversedomainr"whichisagainanimmediateresultofthedefinition.$)Thisismathematicalinduction,andbecomes:"Everymemberofthedomainbelongstotheposterityofthefirsttermr"whichwaspartofourdefinition.ThusprogressionsaswehavedefinedthemhavethefiveformalpropgrtiesfromwhichPeanodeducesarithmetic.,Itisttt'easytoshowthattwoprogessionsaresimilarinthesensedefinedforsimilarityofrelationsinChapterVI.Wecan,ofcourse,derivearelationwhichisserialfromtheone-onerelationbywhichwedefineaprogression:themethodusedisthatexplainedinChapterIV.,andtherelationisthatofatermtoamemberofitsproperposteritywithrespecttotheoriginalone-onerelation.Twotransitiveasymmetricalrelationswhichgeneratepro-gressionsaresimilar,forthesamereasonsforwhichthecor-respondingone-onerelationsaresimilar.Theclassofallsuchtttransitivegeneratorsofprogressionsisaserialnumber"inthesenseofChapterVI.;itisinfactthesmallestofinfiniteserialnumbers,thenumbertowhichCantorhasgiventhenamettt,bywhichhehasmadeitfamous.Butweareconcerned,forthemoment,withcardinalnumbers.Sincetwoprogressionsaresimilarrelations,itfollowsthattheirdomains(ortheirfields,whicharethesameastheirdomains)aresimilarclasses.Thedomainsofprogressionsformacardinalnumber,sinceeveryclasswhichissimilartothedomainofaprogressioniseasilyshowntobeitselfthedomainofaprogression.Thiscardinalnumberisthesmallestoftheinfinitecardinalnumbers;itistheonetowhichCantorhasappropriatedtheHebrewAlephwiththesuffixo,todistinguishitfromlargerinfinitecardinals,whichhaveothersuffixes.Thusthenameofthesmallestofinfinitecardinalsisr*0.TosaythataclasshasnotermsisthesamethingastosaythatitisamemberofNo,andthisisthesamethingastosay\n8+[ntroductiontoMatltertaticalPltilosoplrythatthemembersoftheclasscanbearrangedinaprogression.Itisobviousthatanyprogressionremainsaprogressionifweomitafinitenumberoftermsfromit,oreveryothertelm,orallexcepteverytenthtermoreveryhundredthterm.Thesemethodsofthinningoutaprogressiondonotmakeitceasetobeaprogression,andthereforedonotdiminishthenumberofitsterms,whichremainsN6.Infact,anyselectionfromapro-gressionisaprogressionifithasnolastterm,howeversparselyitmaybedistributed.Take(t"y)inductivenumbersoftheformnn,orrln*,SuchnumbersgrowveryrareinthehigherPartsofthenumberseries,andyettherearejustasmanyofthemasthereareinductivenumbersaltogether,namely,No.Conversely,wecanaddtermstotheinductivenumberswithoutincreasingtheirnumber.Take,forexample,ratios.Onemightbeinclinedtothinkthattheremustbemanymoreratiosthanintegers,sinceratioswhosedenominatorisIcorrespondtotheintegers,andseemtobeonlyaninfinitesimalProPortionofratios.Butinactualfactthenumberofratios(orfractions)isexactlythesameasthenumberofinductivenumbers,namely,No.Thisiseasilyseenbyarrangingratiosinaseriesonthefollowingplan:Ifthesumofnumeratoranddenominatorinoneislessthanintheother,puttheonebeforetheother;ifthesumisequalinthetwo,putfirsttheonewiththesmallernumerator.Thisgivesustheseriesr,rf2,2,rl1,,3,t/+,zl3,ilz,4,rl5,...Thisseriesisaprogression,andallratiosoccurinitsoonerorIater.Hencewecanarrangeallratiosinaprogression,andtheirnumberisthereforeNo.Itisnotthecase,however,thataIIinfrnitecollectionshaveN0terms.Thenumberofrealnumbers,forexample,isgreaterthan*oIitis,infact,2No,anditisnothardtoProvethatznisgreaterthannevenwhenaisinfinite.The.easiestwayofprovingthisistoprove,first,thatifaclasshasnmembers,itcontainsz"sub-classes-inotherwords,thatthereareznways\nInfninCarainalNumbers85ofselectingsomeofitsmembers(includingtheextremecaseswhereweselectallornone);andsecondly,thatthenumberofsub-classescontainedinaclassisalwaysgreaterthanthenumberofmembersoftheclass.Ofthesetwopropositions,thefirstisfamiliarinthecaseoffinitenumbers,andisnothardtoextendtoinfinitenumbers.Theproofofthesecondissosimpleandsoinstructivethatweshallgiveit:Inthefirstplace,itisclearthatthenumberofsub-classesofagivenclass(sayo)isatleastasgreatasthenumberofmembers,sinceeachmemberconstitutesasub-class,andwethushaveacorrelationofallthememberswithsomeofthesub-classes.Henceitfollowsthat,ifthenumberofsub-classesisnotequa.ltothenumberofmembers,itmustbegreater.Nowitiseasytoprovethatthenumberisnotequal,byshowingthat,givenanyone-onerelationwhosedomainisthemembersandwhoseconversedomainiscontainedamongthesetofsub-classes,theremustbeatleastonesub-classnotbelongingtotheconversedomain.Theproofisasfollows:1Whenaone-onecorrelationRisestablishedbetweenallthemembersofoandsomeofthesub-classes,itmayhappenthatagivenmemberriscorrelatedwithasub-classofwhichitisamemberIor,again,itmayhappenthatxiscorrelatedwithasub-classofwhichitisnotamember.Letusformthewholeclass,Ft^y,ofthosemembersrwhicharecorrelatedwithsub-classesofwhichtheyarenotmembers.Thisisasub-classofo,anditisnotcorrelatedwithanymemberofo.For,takingfirstthemembersofF,eachofthemis(bythedefinitionof0correlatedwithsomesub-classofwhichitisnotamember,andisthereforenotcorrelatedwithp.TakingnextthetermswhicharenotmembersofP,eachofthem(bythedefinitionofp)iscorrelatedwithsomesub-classofwhichitisamember,andthereforeagainisnotcorrelatedwithB.ThusnomemberofoiscorrelatedwithB.SinceRwasanyone-onecorrelationofallmembers1ThisproofistakenfromCantor,withsomesimplifications:seeJahresberichtilerd,eutsahenMathemati,her-Vereini6ung,i.(r892),p.7I.\n86IntroductiontoMatltematicalPltilosoplt1withsomesub-classes,itfollowsthatthereisnocorrelationofallmemberswithallsub-classes.Itdoesnotmattertotheproofifphasnomembers:allthathappensinthatcaseisthatthesub-classwhichisshowntobeomittedisthenull-class.Henceinanycasethenumberofsub-classesisnotequaltothenumberofmembers,andtherefore,bywhatwassaidearlier,itisgreater.Combiningthiswiththepropositionthat,if.nisthenumberofmembers,znisthenumberofsub-classesrwehavethetheoremthatznisalwaysgreaterthanfL,eYen.whenaisinfinite.Itfollowsfromthispropositionthatthereisnomaximumtotheinfinitecardinalnumbers.Howevergreataninfinitenumbernmaybe,znwillbestillgreater.Thearithmeticofinfinitenumbersissomewhatsurprisinguntilonebecomesaccustomedtoit.Wehave,forexample,No*r:N0,Nsfa:No,wherenisanyinductivenumber,No8:No'(Thisfollowsfromthecaseoftheratios,for,sincearatioisdeterminedbyapairofinductivenumbers,itiseasytoseethatthenumberofratiosisthesquareofthenumberofinductivenumbers,i,e.itisnos;butwesawthatitisalsono.)Nsn:No,wherenisanyinductivenumber'(ThisfollowsfromNo2:Nobyinduction;forifNo':No,thenNor+1-Ns2:N..)But2No)No'fnfact,asweshallseelater,2Noisaveryimportantnumber,namely,thenumberoftermsinaserieswhichhas"continuity"inthesenseinwhichthiswordisusedbyCantor.Assumingspaceandtimetobecontinuousinthissense(aswecommonlyd;inanalyticalgeometryandkinematics),thiswillbethenumberofpointsinspaceorofinstantsintime;itwillalsobethenumberofpointsinanyftniteportionofspace,whether\nInfniaCardinalNumbers87Iine,area,orvolume.AfterNo,2Noisthemostimportantandinterestingofinfinitecardinalnumbers.Althoughadditionandmultiplicationarealwayspossiblewithinfinitecardinals,subtractionanddivisionnolongergivedefiniteresults,andcannotthereforebeemployedastheyareemployedinelementaryarithmetic.Takesubtractiontobeginwith:solongasthenumbersubtractedisfinite,allgoeswell;iftheothernumberisreflexive,itremainsunchanged.ThusNe-z:No,ifzisfinite;sofar,subtractiongivesaperfectlydefiniteresult.ButitisotherwisewhenwesubtractN0fromitself;wemaythengetanyresult,fromouptoN0.Thisiseasilyseenbyexamples.Fromtheinductivenumbers,takeawaythefollowingcollectionsofxoterms.-(t)Alltheinductivenumbers-remainder,zero.(z)Alltheinductivenumbersfromfl.onwards-remainder,thenumbersfromoton-1,numberingntermsinall.(g)AUtheoddnumbers-remainder,alltheevennumbers,numberingNoterms.AllthesearedifferentwaysofsubtractingnofromNo,andallgivedifferentresults.Asregardsdivision,verysimilarresultsfollowfromthefactthatnoisunchangedwhenmultipliedbyzor3oranyfinitenumberzorbyxo.ItfollowsthatnodividedbyN0mayhaveanyvaluefromruptoNo.Fromtheambiguityofsubtractionanddivisionitresultsthatnegativenumbersandratioscannotbeextendedtoinfinitenumbers.Addition,multiplication,andexponentiationproceedquitesatisfactorily,buttheinverseoperations-subtraction,division,andextractionofroots-areambiguous,andthenotionsthatdependuponthemfailwheninfinitenumbersareconcerned.Thecharacteristicbywhichwedefinedfinitudewasmathe-maticalinduction,i.e.wedefinedanumberasfinitewhenitobeysmathematicalinductionstartingfromo,andaclassasfinitewhenitsnumberisfinite.Thisdefinitionyieldsthesorrofresultthatadefinitionoughttoyield,namely,thatthefinite\n88fnroductiontoMatlternaticalPltihsopltynumbersarethosethatoccurintheordinarynumber-seriese,!,2,3,Butinthepresentchapter,theinfinitenum-berswehavediscussedhavenotmerelybeennon-inductive:theyhavealsobeenref.exiuc.Cantorusedreflexivenessasthedcf.nitionoftheinfinite,andbelievesthatitisequivalenttonon-inductiveness;thatistosay,hebelievesthateveryclassandeverycardinaliseitherinductiveorreflexive,Thismaybetrue,andmayverypossiblybecapableofproof;buttheproofshithertoofieredbyCantorandothers(includingthePresentauthorinformerdays)arefallacious,forreasonswhichwillbettexplainedwhenwecometoconsiderthemultiplicativeaxiom.t'Atpresent,itisnotknownwhetherthereareclassesandcardinalswhichareneitherreflexivenorinductive.If.nweresuchacardinal,weshouldnothaven:fl*r,butzwouldnotbeone66ofthenaturalnumbersr"andwouldbelackinginsomeoftheinductiveproperties.Allknowninfiniteclassesandcardinalsarereflexive;butforthepresentitiswelltopreserveanoPenmindastowhetherthereareinstances,hithertounknown,ofclassesandcardinalswhichareneitherreflexivenorinductive.Meanwhile,weadoptthefollowingdefinitions:-Af,niteclassorcardinalisonewhichisinductive.Aninj.niteclassorcardinalisonewhichisnotinductive.Allref.exiveclassesandcardinalsareinfinite;butitisnotknownatpresentwhetherallinfiniteclassesandcardinalsarereflexive.WeshallreturntothissubjectinChapterXII.\nCHAPTERIXINFINITESERIESANDORDINALSAN"infiniteseries"*^ybedefinedasaseriesofwhichthefieldisaninfiniteclass.Wehavealreadyhadoccasiontoconsideronekindofinfiniteseries,namely,progressions.Inthischapterweshallconsiderthesubjectmoregenerally.Themostnoteworthycharacteristicofaninfiniteseriesisthatitsserialnumbercanbealteredbymerelyre-afiangingitsterms.Inthisrespectthereisacertainoppositenessbetweencardinalandserialnumbers.Itispossibletokeepthecardinalnumberofareflexiveclassunchangedinspiteofaddingtermstoit;ontheotherhand,itispossibletochangetheserialnumberofaserieswithoutaddingortakingawayanyterms,bymerere-arrangement.Atthesametime,inthecaseofanyinfiniteseriesitisalsopossible,aswithcardinals,toaddtermswithoutalteringtheserialnumber:everythingdependsuponthewayinwhichtheyareadded.Inordertomakemattersclear,itwillbebesttobeginwithexamples.Letusfirstconsidervariousfifferentkindsofserieswhichcanbemadeoutoftheinductivenumbersarrangedonvariousplans.Westartwiththeseriesrr21314,n,which,aswehavealreadyseen,representsthesmallestofin-finiteserialnumbers,thesortthatCantorcallser.Letusproceedtothinoutthisseriesbyrepeatedlyperformingthe89\n9ofnnoductiontoMath.ematicalPhilosoplt1operationofremovingtotheendthefirstevennumberthatoccurs.Wethusobtaininsuccessionthevariousseries:lr3r4r5r"'lL\'2,tr3r5r6,.n*r1...2,4,lr3r5r7,"n*Zr':!2,4,6,andsoon.Ifweimaginethisprocesscarriedonaslongaspossible,wefinallyreachtheseriesr,3,5,7,..,un+r,.2,+,618,.,2n,inwhichwehavefirstalltheoddnumbersandthenalltheevennumbers.Theserialnumbersofthesevariousseriesarealft,a{2,ttat*3,...zcD.Eachofthesenumbersis"greaterthananyofitspredecessors,inthefollowingsense:-Oneserialnumberissaidtobe"greater"thananotherifanyserieshavingthefirstnumbercontainsaParthavingthesecondnumber,butnoserieshavingthesecondnumbercontains^parthavingthefirstnumber.Ifwecomparethetwoserieslt2r3r4r''n,lr314r5,'n*1,j'weseethatthefirstissimilartothepartofthesecondwhichomitsthelastterm,namely,thenumberz,butthesecondisnotsimilartoanypartofthefirst.(Thisisobvious,butiseasilydemonstrated.)Thusthesecondserieshasagreaterserialnumberthanthefirst,accordingtothedefinition-i.s,@+risgreaterthanar.Butifweaddatermatthebeginningofaprogressioninsteadoftheend,westillhaveaProgression.Thusrfar,-o.r.Thusr*arisnotequaltoar+r.Thisischaracteristicofrelation-arithmeticgenerally;ifp,andvaretworelation-numbers,thegeneralruleisthatp*visnotequaltorfp.Thecaseoffiniteordinals,inwhichthereisequality,isquiteexceptional.Theserieswefinallyreachedjustnowconsistedoffirstalltheoddnumbersandthenalltheevennumbers,anditsserial\nInfniteSeriesandOrdinalsgrnumberiszu.Thisnumberisgreaterthancoora*n,whereaisfinite.Itistobeobserveo',that,inaccordancewiththegeneraldefinitionoforder,eachofthesearrangementsofintegersistoberegardedasresultingfromsomedefiniterelation.E.g.theonewhichmerelyremovesztotheendwillbedefinedbythefollo*irgrelationi"tcandyarefiniteintegers,andeitheryiszandrisnot2,otneitheriszandrislessthany."Theonewhichputsfirstalltheoddnumbersandthenalltheevenoneswillbedefinedby:"xandyarcfiniteintegers,andeitherrisoddandyisevenorrislessthanyandbothareoddorbothareeven."Weshallnottrouble,asarule,togivetheseformulainfuture;butthefactthattheycouldbegivenisessential.Thenumberwhichwehavecalledza),namely,thenumberofaseriesconsistingoftwoprogressions,issometimescalled@.2.Multiplication,likeaddition,dependsupontheorderofthefactors:aprogressionofcouplesgivesaseriessuchasfit,tu#2,tz,fru,tr,'''tc17!n,''',whichisitselfaprogression;butacoupleofprogressionsgivesaserieswhichistwiceaslongasaprogression.Itisthereforenecessarytodistinguishbetweenz@anda.2.Usageisvariable;weshallusezarforacoupleofprogressionsandar.zforapro-gressionofcouples,andthisdecisionofcoursegovernsourgeneralinterpretationof"a.B"whenoandBarerelation-numbersi"o..p"otillhavetostandforasuitablyconstructedsumoforelationseachhavingBterms.Wecanproceedindefinitelywiththeprocessofthinningouttheinductivenumbers.Forexample,wecanplacefirsttheoddnumbers,thentheirdoubles,thenthedoublesofthese,andsoon.Wethusobtaintheseriesr,3,5r7,...i2,6,tO,t4,...i4,tzrzor2gr..;8,24,40,56,,..,ofwhichthenumberiscrrz,sinceitisaprogressionofprogressions.Aryoneoftheprogressionsinthisnewseriescanofcoursebe\ng2fntoductiontoMathematicalPhilosoplrythinnedoutaswethinnedoutouroriginalprogression.Wecanproceedtocr3,@4,.".(D-,anlisoonIhoweverfarwehavegonetwecanalwaysgofurther.TheseriesofalltheordinalsthatcanbeobtainedinthiswaY,i.e.allthatcanbeobtainedbythinningoutaprogression,isitselflongerthananyseriesthatcanbeobtainedbyre-arrangingthetermsofaprogression.(Thisisnotdifficulttoprove.)Thecardinalnumberoftheclassofsuchordinalscanbeshowntobegreaterthant*oIitisthenumberwhichCantorcallsNl.Theordinalnumberoftheseriesofallordinalsthatcanbemadeoutofan!{0,takeninorderofmagnitude,iscalledc.r1.Thusaserieswhoseordinalnumberiso\hasafieldwhosecardinalnumberisnr.Wecanproceedfromar1andNltoargandNsbyaProcessexactlyanalogoustothatbywhichweadvancedfromarandNstoar1andttr.AndthereisnothingtoPreventusfromadvancingindefinitelyinthiswaytonewcardinalsandnewordinals.ItisnotknownwhetherzNoisequaltoanyofthecardinalsintheseriesofAlephs.Itisnotevenknownwhetheritiscomparablewiththeminmagnitude;foraughtweknow,itmaybeneitherequaltonorgreaternorlessthananyoneoftheAlephs-Thisquestionisconnectedwiththemultiplicativeaxiom,ofwhichweshalltreatlater.Alltheserieswehavebeenconsideringsofarinthischapterhavebeenwhatiscalled"well-ordered."Awell-orderedseriesisonewhichhasabeginning,andhasconsecutiveterms'andhasatermnextaf.tetanyselectionofitsterms,providedthereareanytermsaftertheselection.Thisexcludes,ontheonehand,compactseries,inwhichtherearetermsbetweenanytwo,andontheotherhandserieswhichhavenobegindog,orinwhichtherearesubordinatePartshavingnobeginoiog.Theseriesofnegativeintegersinorderofmagnitude,having-1,isnotwell-ordered;butnobeginning,butendingwith-I'itiswell-ordered,takeninthereverseorder,beginningwithbeinginfactaProgression.Thedefinitionis:\nInfniteSeriesandOrdinals93ttA"well-orderedseriesisoneinwhicheverysub-class(except,ofcourse,thenull-class)hasafirstterm.An"ordinal"numbermeanstherelation-numberofawell-orderedseries.Itisthusaspeciesofserialnumber.Amongwell-orderedseries,ageneralisedformofmathematicalinductionapplies.Apropertymaybesaidtobe"transfinitelyhereditary"if,whenitbelongstoacertainselectionofthetermsinaseries,itbelongstotheirimmediatesuccessorpro-videdtheyhaveone.Inawell-orderedseries,atransfinitelyhereditarypropertybelongingtothefirsttermoftheseriesbelongstothewholeseries.Thismakesitpossibletoprovemanypropositionsconcerningwell-orderedserieswhicharenottrueofallseries.Itiseasytoarrangetheinductivenumbersinserieswhicharenotwell-ordered,andeventoarrangethemincompactseries.Forexample,wecanadoptthefollowingplan:considertHedecimalsfrom'r(inclusive)tor(exclusive),arrangedinorderofmagnitude.TheseformacompactseriesIbetweenarrytwotherearealwaysaninfinitenumberofothers.Nowomitthedotatthebeginningofeach,andwehaveacompactseriesconsistingofallfiniteintegersexceptsuchasdividebyro.Ifwewishtoincludethosethatdividebyto,thereisnodifficulty;insteadofstartingwith'r,wewillincludealldecimalslessthanr,butwhenweremovethedot,wewilltransfertotherightanyo'sthatoccuratthebeginningofourdecimal.Omittingthese,andreturningtotheonesthathavenoo'satthebeginning,wecanstatetheruleforthearrangementofourintegersasfollows:Oftwointegersthatdonotbeginwiththesamedigit,theonethatbeginswiththesmallerdigitcomesfirst.Oftwothatdobeginwiththesamedigit,butdifierattheseconddigit,theonewiththesmallerseconddigitcomesfirst,butfirstofalltheonewithnoseconddigit;andsoon.Generally,iftwointegersagreeasregardsthefirstndigits,butnotasregardsthe(afr)tn,thatonecomesfirstwhichhaseitherno(n*t)'hdigitorasmalleronethantheother.Thisruleofarrangement,\ng+[ntroductiontoMatlternaticalPltilosoplt1asthereadercaneasilyconvincehimself,givesrisetoacompactseriescontainingalltheintegersnotdivisiblebyro;and,aswesaw,thereisnodifficultyaboutincludingthosethataredivisiblebyIo.ItfollowsfromthisexamplethatitispossibletoconstructcompactserieshavingN0terms.fnfact,wehavealreadyseenthatthereareN0ratios,andratiosinorderofmagnitudeformacomPactseries;thuswehavehereanotherexample.Weshallresumethistopicinthenextchapter.Oftheusualformallawsofaddition,multiplication,andex-ponentiation,allareobeyedbytransfinitecardinals,butonlysomeareobeyedbytransfiniteordinals,andthosethatareobeyedbythemareobeyedbyallrelation-numbers.Bythe"usualformallaws"wemeanthefollowing:-I.Thecommutativelaw:o1_B:BlaandoxB-pxo.II.Theassociativelaw:("*Ofy:a+(F+y)and("x0xy:ax(pxy).III.Thedistributivelaw:a(F*y):o,F*ay.Whenthecommutativelawdoesnothold,theaboveformofthedistributivelawmustbedistinguishedfrom(F*y)"-Fotyo.Asweshallseeimmediately,oneformmaybetrueandtheotherfalse.IV.Thelawsofexponentiation:aF.ar:aF*t,a't.Fr_-(o|)r,(a)t-64t.Alltheselawsholdforcardinals,whetherfiniteorinfinite,andforf,niteordinals.Butwhenwecometoinfiniteordinals,orindeedtorelation-numbersingeneral,someholdandsomedonot.Thecommutativelawdoesnothold;theassociativelawdoeshold;thedistributivelaw(adoptingtheconvention\nInfniteSeriesandOrdinalsgswehaveadoptedaboveasregardstheorderofthefactorsinaproduct)holdsintheform(F+y1o:Fo*yo,butnotintheforma(B+y)-oF*o"y;theexponentiallawsaF.q"r-qs+yand(oTr:afistillhold,butnotthelaway.Fr-("pr,whichisobviouslyconnectedwiththecommutativelawformultiplication.Thedefinitionsofmultiplicationandexponentiationthatareassumedintheabovepropositionsaresomewhatcomplicated.ThereaderwhowishestoknowwhattheyareandhowtheabovelawsareprovedmustconsultthesecondvolumeofPrincipiaMatltematica,*172-176.OrdinaltransfinitearithmeticwasdevelopedbyCantoraranearlierstagethancardinaltransfinitearithmetic,becauseithasvarioustechnicalmathematicaluseswhichledhimtoit.Butfromthepointofviewofthephilosophyofmarhematicsitislessimportantandlessfundamentalthanthetheoryoftransfinitecardinals.Cardinalsareessentiallysimplerthanordinals,anditisacurioushistoricalaccidentthattjreyfirstappearedasanabstractionfromthelatter,andonlygraduallycametobestudiedontheirownaccount.ThisdoesnotapplytoFrege'swork,inwhichcardinals,finiteandtransfinite,weretreatedincompleteindependenceofordinals;butitwasCantor'sworkthatmadetheworldawareofthesubject,whileFrege'sremainedalmostunknown,probablyinthemainonaccountofthedifficultyofhissymbolism.Andmathematicians,likeotherpeople,havemoredifficultyinunderstandingandusingnotionswhicharecomparatively"simple"inthelogicalsensethaninmanipulatingmorecomplexnotionswhichare\ns6fnroductiont0MathenaticalPltihsoplt1moreakintotheirordinarypractice.Forthesereasons,itwasonlygraduallythatthetrueimportanceofcardinalsinmathe-maticalphilosophywasrecognised.Theimportanceofordinals,thoughbynomeanssmall,isdistinctlylessthanthatofcardinals,andisverylargelymergedinthatofthemoregeneralconcePtionofrelation-numbers.\nCHAPTERXLIMITSANDCONTINUITYTnrconceptionofa"limit"isoneofwhichtheimportanceinmathematicshasbeenfoundcontinuallygreaterthanhadbeenthought.Thewholeofthedifferentialandintegralcalculus,indeedpracticallyeverythinginhighermathematics,dependsuponlimits.Formerly,itwassupposedthatinfinitesimalswereinvolvedinthefoundationsofthesesubjects,butWeierstrassshowedthatthisisanerror:whereverinfinitesimalswerethoughttooccur,whatreallyoccursisasetoffinitequantitieshavingzerofortheirlowerlimit.Itusedtobethoughtthat"limit"wasanessentiallyquantitativenotion,namely,thenotionofaquantitytowhichothersapproachednearerandnearer,sothatamongthoseotherstherewouldbesomedifferingbylessthananyassignedquantity.Butinfactthenotionof"Iimit"isapurelyordinalnotion,notinvolvingquantityatall(exceptbyaccidentwhentheseriesconcernedhappenstobequantitative).Agivenpointonalinemaybethelimitofasetofpointsontheline,withoutitsbeingnecessarytobringinco-ordinatesormeasure-mentoranythingquantitative.ThecardinalnumberHoisthelimit(intheorderofmagnitude)ofthecardinalnumbersr,2,3,.,,n2...,althoughthenumericaldifferencebetweeoN6andafinitecardinalisconstantandinfinite:fromaquantitativepointofview,finitenumbersgetnonearertoNoastheygrowlarger.WhatmakesNothelimitofthefinitenumbersisthefactthat,intheseries,itcomesimmediatelyafterthem,whichisanordinalfact,notaquantitativefact.97\n98fnroductiontoMatlternaticalPltihsoplt1Therearevariousformsofthenotionof"limitr"ofin-creasingcomplexity.Thesimplestandmostfundamentalform,fromwhichtherestarederived,hasbeenalreadydefined,butwewillhererepeatthedefinitionswhichleadtoit,inageneralforminwhichtheydonotdemandthattherelationconcernedshallbeserial.Thedefinitionsareasfollows:-The"minima"ofaclassowithrespecttoarelationParethosemembersofoandthefieldofP(ifany)towhichnomemberofohastherelationP.The"maxima"withrespecttoParetheminimawithrespecttotheconverseofP.The"sequents"ofaclassowithrespecttoarelationParetttttheminimaofthe"successors"ofo,andthesuccessorsofd.arethosemembersofthefieldofPtowhicheverymemberofthecommonpartofoandthefieldofPhastherelationP.t'precedentsttThewithrespecttoParethesequentswithrespecttotheconverseofP.The"upperlimits"ofowithrespecttoParethesequentsprovidedohasnomaximum;butifohasamaximum,ithasnoupperlimits.The"lowerlimits"withrespecttoParetheupperlimitswithrespecttotheconverseofP.WheneverPhasconnexity,aclasscanhaveatmostonemaximum,oneminimum,onesequent,etc.Thus,inthecasest'weareconcernedwithinpractice,wecanspeakofthelimit"(ifany).WhenPisaserialrelation,wecangreatlysimplifytheabovedefinitionofalimit.Wecan,inthatcase,definefirstthet'boundary"ofaclassa,i.e.itslimitsormaximum,andthenproceedtodistinguishthecasewheretheboundaryisthelimitfromthecasewhereitisamaximum.Forthispurposeitist'segment.t'besttousethenotionofWewillspeakofthe"segmentofPdefinedby^classo"asallthosetermsthathavetherelationPtosomeoneormoreofthemembersofo.Thiswillbeasegmentinthesensedefined\nLimitsandContinuity99inChapterVII.;indeed,everysegmentinthesensetheredefinedisthesegmentdefinedbysomeclasso.IfPisserial,thesegmentdefinedbyoconsistsofallthetermsthatprecedesometermorotherofo.Ifohasamaximum,thesegmentwillbeallthepredecessorsofthemaximum.ButifcLhasnomaximum,everymemberofoprecedessomeothermemberofo,andthewholeofoisthereforeincludedinthesegmentdefinedbyo,Take,forexample,theclassconsistingofthefractionst,t,*,+8'"''ri.e.of.allfractionsoftheformr-fordifierentfinitevalues2nof.n.Thisseriesoffractionshasnomaximum,anditisclearthatthesegmentwhichitdefines(inthewholeseriesoffractionsinorderofmagnitude)istheclassofallproperfractions.Or,again,considertheprimenumbers,consideredasaselectionfromthecardinals(finiteandinfinite)inorderofmagnitude.Inthiscasethesegmentdefinedconsistsofallfiniteintegers.AssumingthatPisserial,the"boundary"ofaclassowillbetheterm#(ifitexists)whosepredecessorsarethesegmentdefinedbyo.A"maximum"of.aisaboundarywhichisamemberofo.An"upperlimit"ofoisaboundarywhichisnotamemberofcaIfaclasshasnoboundarl,ithasneithermaximumnorlimit.Thisisthecaseofan"irrational"Dedekindcut,orofwhatiscalledu"gp."((Thustheupperlimit"oIasetofterrnsowithrespecttoaseriesPisthattermx(if.itexists)whichcomesafteralltheo's,butissuchthateveryearliertermcomesbeforesomeoftheots.Wemaydefineallthe"upperlimiting-points"ofasetoftermsFallthosethataretheupperlimitsofsetsofterms"rchosenoutofB.Weshall,ofcourse,havetodistinguishupperlimiting-pointsfromlowerlimiting-points.Ifweconsider,forexample,theseriesofordinalnumbers:lr2r3t...(t)ral*Ir..,z(t)rza{t,.\nrooIntrodactiont0MatlzematicalPhihsoplrytheupperlimiting-pointsofthefieldofthisseriesarethosethathavenoimmediatepredecessors,i.e.lr@t2cDrJtDr.,.@2ra2*@,..2cr2,,..aa,.."Iheupperlimiting-pointsofthefieldofthisnewserieswillbel,e)2,2e)2,.,(!)97-"*a'Ontheotherhand,theseriesofordinals-andindeedeverywell-orderedseries-hasnolowerlimiting-points,becausetherearenotermsexceptthelastthathavenoimmediatesuccessors.Butifweconsidersuchaseriesastheseriesofratios,everymemberofthisseriesisbothanupperandalowerlimiting-pointforsuitablychosensets.Ifweconsidertheseriesofrealnumbers,andselectoutofittherationalrealnumbers,thisset(therationals)willhavealltherealnumbersasupperandlowerlimiting-points.Thelimiting-pointsofasetarecalledits"firstderivativer"andthelimiting-pointsofthefirstderivativearecalledthesecondderivative,andsoon.Withregardtolimits,w€maydistinguishvariousgradesofttttwhatmaybecalledcontinuity"inaseries.Thewordcon-tinuity"hadbeenusedforalongtime,buthadremainedwithoutanyprecisedefinitionuntilthetimeofDedekindandCantor.Eachofthesetwomengaveaprecisesignificancetotheterm,butCantor'sdefinitionisnarrowerthanDedekind's:aserieswhichhasCantoriancontinuitymusthaveDedekindiancon-trnuity,buttheconversedoesnothold.Thefirstdefinitionthatwouldnaturallyoccurtoamanseekingaprecisemeaningforthecontinuityofserieswouldbetodefineitasconsistinginwhatwehavecalled"compactnessr"i.e.inthefactthatbetweenanytwotermsoftheseriesthereareothers.Butthiswouldbeaninadequatedefinition,becauseofthet'gapsexistenceof"inseriessuchastheseriesofratios.WesawinChapterVII.thatthereareinnumerablewaysinwhichtheseriesofratioscanbedividedintotwoparts,ofwhichonewhollyprecedestheotler,andofwhichthefirsthasnolastterm,\nLimitsandContinuitlIOIwhilethesecondhasnofirstterm.Suchastateofaffairsseemscontrarytothevaguefeelingwehaveastowhatshouldcharacter-ttisecontinuityr"and,whatismore,itshowsthattheseriesofratiosisnotthesortofseriesthatisneededformanymathematicalpurposes.Takegeometry,forexample:wewishtobeabletosaythatwhentwostraightlinescrosseachothertheyhaveapointincommon,butiftheseriesofpointsonalineweresimilartotheseriesofratios,thetwolinesmightcrossina"gap"andhavenopointincommon.Thisisacrudeexample,butmanyothersmightbegiventoshowthatcompactnessisinadequateasamathematicaldefinitionofcontinuity.Itwastheneedsofgeometty,asmuchasanything,thatledtothedefinitionof"Dedekindian"continuity.Itwillbere-memberedthatwedefinedaseriesasDedekindianwheneverysub-classofthefieldhasaboundary.(Itissufficienttoassumethatthereisalwaysanupperboundarnorthatthereisalwaysalowerboundary.Ifoneoftheseisassumed,theothercanbededuced.)Thatistosay,aseriesisDedekindianwhentherearenogaps.Theabsenceofgapsmayariseeitherthroughtermshavingsuccessors,orthroughtheexistenceoflimitsintheabsenceofmaxima.Thusafiniteseriesorawell-orderedseriesisDedekindian,andsoistheseriesofrealnumberg.TheformersortofDedekindianseriesisexcludedbyassumingthatourseriesiscompact;inthatcaseourseriesmusthaveaProPertywhichmaltformanypurposes,befittinglycalledcontinuity.Thusweareledtothedefinition:ttAserieshasDedekindiancontinuity"whenitisDedekindianandcompact.Butthisdefinitionisstilltoowideformanypurposes.Suppose,forexample,thatwedesiretobeabletoassignsuchpropertiestogeometricalspaceasshallmakeitcertainthateverypointcanbespecifiedbymeansofco-ordinatesrvhicharerealnumbers:thisisnotinsuredbyDedekindiancontinuityalone.Wewanttobesurethateverypointwhichcannotbespecifiedbyrutionalco-ordinatescanbespecifiedasthelimitof.aprogressionofpoints\nrozIntroductiontoMatltematicalPltilosoplt1whoseco-ordinatesarerational,andthisisafurtherpropertywhichourdefinitiondoesnotenableustodeduce.Wearethusledtoacloserinvestigationofserieswithrespecttolimits.ThisinvestigationwasmadebyCantorandformedthebasisofhisdefinitionofcontinuity,although,initssimplestform,thisdefinitionsomewhatconcealstheconsiderationswhichhavegivenrisetoit.Weshall,therefore,firsttravelthroughsomeofCantor'sconceptionsinthissubjectbeforegivinghisdefinitionofcontinuity.t'perfectttCantordefinesaseriesaswhenallitspointsarelimiting-pointsandallitslimiting-pointsbelongtoit.Butthisdefinitiondoesnotexpressquiteaccuratelywhathemeans.Thereisnocorrectionrequiredsofarasconcernsthepropertythatallitspointsaretobelimiting-points;thisisapropertybelongingtocompactseries,andtonoothersifallpointsaretobeupperlimiting-oralllowerlimiting-points.Butifitisonlyassumedthattheyarelimiting-pointsone.^y,withoutspecify-ingwhich,therewillbeotherseriesthatwillhavethepropertyinquestion-forexample,theseriesofdecimalsinwhichadecimalendinginarecurring9isdistinguishedfromthecorrespondingterminatingdecimalandplacedimmediatelybeforeit.Suchaseriesisverynearlycompact,buthasexceptionaltermswhichareconsecutive,andofwhichthefirsthasnoimmediateprede-cessor,whilethesecondhasnoimmediatesuccessor.Apartfromsuchseries,theseriesinwhicheverypointisalimiting-pointarecompactseries;andthisholdswithoutqualificationifitisspecifiedthateverypointistobeanupperlimiting-point(orthateverypointistobealowerlimiting-point).AlthoughCantordoesnotexplicitlyconsiderthematter,wemustdistinguishdifferentkindsoflimiting-pointsaccordingtothenatureofthesmallestsub-seriesbywhichtheycanbedefined.Cantorassumesthattheyaretobedefinedbyprogressionszorbyregressions(whicharetheconversesofprogressions).Wheneverymemberofourseriesisthelimitofaprogressionorregres-sion,Cantorcallsourseries"condensedinitself"(insicbdicht).\nLimitsandContinuitlr03Wecomenowtothesecondpropertybywhichperfectionwastobedefined,namely,thepropertywhichCantorcallsthatofbeing"closed"(abgescblossen).This,aswesaw,wasfirstdefinedasconsistinginthefactthatallthelimiting-pointsofaseriesbelongtoit.Butthisonlyhasanyeffectivesignificanceifourseriesisgivenascontainedinsomeotherlargerseries(asisthecase,e.g.,:vrrrthaselectionofrealnumbers),andlimiting-pointsaretakeninrelationtothelargerseries.Otherwise,ifaseriesisconsideredsimplyonitsownaccount,itcannotfailtocontainitslimiting-points.WhatCantortneansisnotexactlywhathesays;indeed,onotheroccasionshesayssomethingratherdifierent,whichzswhathemeans.Whathereallymeansisthateverysubordinateserieswhichisofthesortthatmightbeex-pectedtohavealimitdoeshave.alimitwithinthegivenseriesIi.e.everysubordinateserieswhichhasnomaximumhasalimit,i.e.everysubordinateserieshasaboundary.ButCantordoesnotstatethisforet)erysubordinateseries,butonlyforprogres-sionsandregressions.(Itisnotclearhowfarherecognisesthatthisisalimitation.)Thus,finally,wefindthatthedefinitionwewantisthefollowing:-Aseriesissaidtobe"closed"(abgeschlossen)wheneveryPro-gressionorregressioncontainedintheserieshasalimitintheseries.Wethenhavethefurtherdefinition:-Aseriesis"perfect"whenitiscondensedinitselfandclosed,f.a.wheneverytermisthelimitofaprogressionorregression,andeveryprogressionorregressioncontainedinthesedeshasalimitintheseries.Inseekingadefinitionofcontinuity,whatCantorhasinmindisthesearchforadefinitionwhichshallaPPlytotheseriesofrealnumbersandtoanyseriessimilartothat,buttonoothers.ForthispurposewehavetoaddafurtherProPerty.Amongtherealnumberssomearerational,someareirrational;althoughthenumberofirrationalsisgreaterthanthenumberofrationals,yettherearerationalsbetweenanytworealnumbers,however\nro+fntoductiontoMatltematicalPlrilosoplrylittlethetwomaydiffer.Thenumberofrationals,aswesa%isllo.Thisgivesafurtherpropertywhichsufficestocharacterisecontinuitycompletely,namely,thepropertyofcontainingaclassofN0membersinsuchawaythatsomeofthisclassoccurbetweenanytwotermsofourseries,howeverneartogether.Thisproperty,addedtoperfection,sufficestodefineaclassofserieswhichareallsimilarandareinfactaserialnumber.ThisclassCantordefinesasthatofcontinuousseries.Wemayslightlysimplifyhisdefinition.Tobeginwith,wesay:A"medianclass"ofaseriesisasub-classofthefieldsuchthatmembersofitaretobefoundbetweenanytwotermsoftheseries.Thustherationalsareamedianclassintheseriesofrealnumbers.Itisobviousthattherecannotbemedianclassesexceptincompactseries.WethenfindthatCantor'sdefinitionisequivalenttotJlefollowing:-((Aseriesiscontinuous"when(r)itisDedekindian,(z)itcontainsamedianclasshavingNnterms.Toavoidconfusion,weshallspeakofthiskindag'6Cantoriancontinuity."ItwillbeseenthatitimpliesDedekindiancon-tinuity,buttheconverseisnotthecase.AllserieshavingCantoriancontinuityaresimilar,butnotallserieshavingDedekindiancontinuity.Thenotionsoflinitandcontinuitywhichwehavebeendefiningmustnotbeconfoundedwiththenotionsofthelimitofafunctionforapproachestoagivenargument,orthecontinuityofafunctionintheneighbourhoodofagivenargument.Thesearedifierentnotions,veryimportant,butderivativefromtheaboveandmorecomplicated.Thecontinuityofmotion(ifmotioniscontinuous)isaninstanceofthecontinuityofafunction;ontheotherhand,thecontinuityofspaceandtime(iftheyarecontinuous)isaninstanceofthecontinuityofseries,or(tospeakmorecautiously)of.akindofcontinuitywhichcan,bysufficientmarhemarical\nLimitsandContinuityr05manipulation,bereducedtothecontinuityofseries'Inviewofthefundamentalimportanceofmotioninappliedmathe-matics,aswellasforotherreasons,itwillbewelltodealbrieflywiththenotionsoflimitsandcontinuityasappliedtofunctions;butthissubjectwillbebestreservedforaseParatechapter.Thedefinitionsofcontinuitywhichwehavebeenconsidering,namely,thoseofDedekindandCantor,donotcorresPondverycloselytothevagueideawhichisassociatedwiththewordinthemindofthemaninthestreetorthephilosopher.TheyconceivecontinuityratherasabsenceofseParatenessrthesortofgeneralobliterationofdistinctionswhichcharacterisesathickfog.Afoggivesanimpressionofvastnesswithoutdefinitemultiplicityordivision.Itisthissortofthingthatameta-physicianmeansby"continuityr"declaringit,verytruly,tobecharacteristicofhismentallifeandofthatofchildrenandanimals.Thegeneralideavaguelyindicatedbytheword"continuity"whensoemployed,orbytheword"flux,"isonewhichiscertainlyquitedifierentfromthatwhichwehavebeendefining.Take,forexample,theseriesofrealnumbers.Eachiswhatitis,quitedefinitelyanduncompromisingly;itdoesnotPassoverbyimperceptibledegreesintoanother;itisahard,seParateunit,anditsdistancefromeveryotherunitisfinite,thoughinitcanbemadelessthananygivenfiniteamountassignedadvance.Thequestionoftherelationbetweenthekindofex-continuityexistingamongtherealnumbersandthekindhibited,e.g.bywhatweseeatagiventime,isadifficultandintricateone.Itisnottobemaintainedthatthetwokindsaresimplyidentical,butitmay,Ithink,beverywellmain-tainedthatthemathematicalconcePtionwhichwehavebeenconsideringinthischaptergivestheabstractlogicalschemetowhichitmustbepossibletobringempiricalmaterialbysuitablemanipulation,ifthatmaterialistobecalled"continuous"inanypreciselydefinablesense.Itwouldbequiteimpossible\nr06IntroductiontoMathematicalPhilosoplrytojustifythisthesiswithinthelimitsofthepresentvolume.ThereaderwhoisinterestedmayreadanattempttojustifyitasregardstimeinparticularbythepresentauthorintheMonistforr9r4-5raswellasinpartsof.ourKnowledgeoiftlteExternalWorld.Withtheseindications,wemustleavethisproblem,interestingasitis,inordertoreturntotopicsmorecloselyconnectedwithmathematics.\nCHAPTERXILIMITSANDCONTINUITYOFFUNCTIONSh'rthischapterweshallbeconcernedwiththedefinitionofthelimitofafunction(ifany)astheargumentapproachesagivent'con-value,andalsowiththedefinitionofwhatismeantby"tinuousfunction."Bothoftheseideasaresomewhattechnical,andwouldhardlydemandtreatmentinamereintroductiontomathematicalphilosophybutforthefactthat,especiallythroughtheso-calledinfinitesimalcalculus,wrongviewsuponourpresenttopicshavebecomesofirmlyembeddedinthemindsofprofessionalphilosophersthataprolongedandconsiderableeffortisrequiredfortheiruprooting.IthasbeenthoughteversincethetimeofLeibnizthatthedifferentialandintegralcalculusrequiredinfinitesimalquantities.Mathematicians(especiallyWeierstrass)provedthatthisisanerror;buterrorsincorporated,e.g.inwhatHegelhastosayaboutmathematics,diehard,andphilosophershavetendedtoignoretheworkofsuchmenasWeierstrass.Limitsandcontinuityoffunctions,inworksonorfinarymathematics,aredefinedintermsinvolvingnumber.Thisisnotessential,asDrWhiteheadhasshown.lWewill,however,beginwiththedefinitionsinthetext-books,andproceedafter-wardstoshowhowthesedefinitionscanbegeneralisedsoastoapplytoseriesingeneral,andnotonlytosuchasarenumericalornumericallymeasurable.Letusconsideranyordinarymathematicalfunction,fx,where1SeePrincipiaMathematice,vo'I.ii.xz3u+,34.L07\nr08fnroductiontoMathematicalPhilosoplryxandfxarebothrealnumbers,and_fxisone-valued-i.e.whenrisgiven,thereisonlyonevaluethatfxcanhave.Wecallrt'argumentr"theandifxthe"valuefortheargumentx.',Whenafunctioniswhatwecall"continuousr"theroughideaforwhichweareseekingaprecisedefinitionisthatsmalldifierencesinrshallcorrespondtosmalldifierencesinfx,andifwemakethedifferencesinrsmallenough,wecanmakethedifierencesinfxfallbelowanyassigned.amount.Wedonotwant,ifafunctionistobecontinuous,thatthereshallbesuddenjumps,sothat,forsomevalueofx,anychanee,howeversmall,willmakeachangeinfxwhichexceedssomeassignedfiniteamount.Theordinarysimplefunctionsofmathematicshavethisproperty:itbelongs,forexample,tox2,#,...logr,sinx,andsoon.Butitisnotatalldifficulttodefinediscontinuousfunctions.Take,asanon-mathematicalexampl.,"theplaceofbirthoftheyoungestpersonlivingattimet."Thisisafunctionoft;itsvalueisconstantfromthetimeofoneperson'sbirthtothetimeofthenextbirth,andthenthevaluechangessuddenlyfromonebirthplacetotheother.Ananalogousmathematicalexamplewouldbetttheintegernextbelow*rttwhererisarealnumber.Thisfunctionremainsconstantfromoneintegertothenext,andthengivesasuddenjo*p.Theactualfactisthat,thoughcontinuousfunctionsaremorefamiliar,theyaretheexceptions:thereareinfinitelymorediscontinuousfunctionsthancontinuousones.Manyfunctionsarediscontinuousforoneorseveralvaluesofthevariable,butcontinuousforallothervalues.Takeasanexamplesinr/r.Thefunctionsin0passesthroughallvaluesfrom-rtoreverytimethat0passesfrom-filztozrfz,orfromnfzto3nlz,orgenerallyfrom(zn-t)nlzto(znfr)rrfz,wherenisanyinteger.Nowifweconsiderr/rwhen*isverysmall,weseethatasrdiminishestfxgrowsfasterandfaster,sothatitpassesmoreandmorequicklythroughthecycleofvaluesfromonemultipleofnfztoanotherasrbecomessmallerandsmaller.Consequentlysinr/rpassesmoreandmorequicklyfrom-r\nLinitsandContinuityofFunctionsr09torandbackagain,astcgrowssmaller.Infact,ifwetakeanyintervalcontainingo,saytheintervalfrom-eto*ewhereeissomeverysmallnumber,sinr/rwillgothroughaninfinitenumberofoscillationsinthisinterval,andwecannotdiminishtheoscillationsbymakingtheintervalsmaller.Thusround.abouttheargumentothefunctionisdiscontinuous.Itiseasytomanufacturefunctionswhicharefiscontinuousinseveralplaces,orinttsplaces,oreverywhere.Exampleswillbefoundinanybookonthetheoryoffunctionsofatealvariable.Proceedingnowtoseekaprecisedefinitionofwhatismeantbysayingthatafunctioniscontinuousforagivenargument,whenargumentandvaluearebothrealnumbers,letusfirstttdefineaneighbourhood"ofanumberlcasallthenumbersfromtc-eto*fe,whereeissomenumberwhich,inimportantcases,willbeverysmall.Itisclearthatcontinuityatagivenpointhastodowithwhathappensinanyneighbourhoodofthatpoint,howeversmall.WhatwedesireistlriszIf.aistheargumentforwhichwewishourfunctiontobecontinuous,letusfirstdefineaneighbourhood(osay)containingthevaluefawhichthefunctionhasfortheargumentaIwedesirethat,ifwetakeasufficientlysmallneighbourhoodcontaininga,allvaluesforargumentsthroughoutthisneighbourhoodshallbecontainedintheneighbourhoodo,nomatterhowsmallwemayhavemadeo.Thatistosay,ifwedecreethatourfunctionisnottodifferftomfabymorethansomeverytinyamount,wecanalwaysfindastretchofrealnumbers,havingainthemiddleofit,suchthatthroughoutthisstretchfxwillnotdifierfromfabymorethanthepre-scribedtinyamount.Andthisistoremaintruewhatevertinyamountwemayselect.Henceweareledtothefollo*itgdefinition:-'6Thefunctionfir)issaidtobecontinuous"fortheargu-mentaif.,foteverypositivenumbero,differentfromo,butassmallasweplease,thereexistsapositivenumbere,difierentfromo,suchthat,forallvaluesofEwhicharenumerically\nIIOfntroductiontoMathenaticalPhilosoplrylesslthane,thedifierenc,-f(o*S)1@)isnumericallylessthano.Inthisdefinition,ofirstdefinesaneighbourhoodof.f(a),namely,theneighbourhoodfromf(a)-otof(a)*o.Thedefini-tionthenproceedstosaythatwecan(bymeansofe)defineaneighbourhood,namely,thatfroma-etoa{e,suchthat,forallargumentswithinthisneighbourhood,thevalueofthefunctionlieswithintheneighbourhoodfrom-f(n)-otof(a){o.Ifthiscanbedone,howeveromaybechosen,thefunctionisttcon-tttinuousfortheargumenta.Sofarwehavenotdefinedthet'limit"ofafunctionforagivenargument.Ifwehaddoneso,wecouldhavedefinedthecontinuityofafunctiondifferently:afunctioniscontinuousatapointwhereitsvalueisthesameasthelimitofitsvalueforapproacheseitherfromaboveorfrombelow.Butitisonlytheexceptionally"tame"functionthathasadefinitelimitastheargumentapproachesagivenpoint.Thegeneralruleisthatafunctionoscillates,andthat,givenanyneighbourhoodofagivenargument,howeversmall,awholestretchofvalueswilloccurforargumentswithinthisneighbourhood.Asthisisthegeneralrule,letusconsideritfirst.Letusconsiderwhatmayhappenastheargumentapproachessomevalueafrombelow.Thatistosay,wewishtoconsiderwhathappensforargumentscontainedintheintervalfromA-etoa,,whereeissomenumberwhich,inimportantcases,willbeverysmall.ThevaluesofthefunctionforargumentsfromA-etoa(aexcluded)willbeasetofrealnumberswhichwilldefineacertainsectionofthesetofrealnumbers,namely,thesectionconsistingofthosenumbersthatarenotgreaterthanallthevaluesforargumentsfroma-etoa..Givenanynumberinthissection,therearevaluesatleastasgreatasthisnumberforargumentsbetweena-eanda,i.e.IorargumentsthatfallverylittleshortrAnumberissaidtobe"numericallvless"thanewhenitliesbetween-eand*s.\nLirnixandCantinuityafFunctionsIIIof.a(if.eisverysmall).Letustakeallpossiblee'sandallpossiblecorrespondingsections.Thecommonpartofalltheset'ttsectionswewillcalltheultimatesectionastheargumentapproachesa.Tosaythatanumberzbelongstotheultimatesectionistosaythat,howeversmallwemaymakee,thereareargumentsbetweena,-eandaforwhichthevalueofthefunctionisnotlessthanz.Wemayapplyexactlythesameprocesstouppersections,i.e.tosectionsthatgofromsomepointuptothetop,insteadoffromthebottomuptosomepoint.Herewetakethosenumbersthatarenotlessthanallthevaluesforargumentsfroma-etoa1thisdefinesanuppersectionwhichwillvaryasevaries.Takingthecommonpartofallsuchsectionsforallpossiblee's,weobtainthe"ultimateuppersection."Tosaythatanumberzbelongstotheultimateuppersectionistosaythat,howeversmallwemakee,thereareargumentsbetweenA-eandaforwhichthevalueofthefunctionisnotgreaterthanz.Ifatermzbelongsbothtotheultimatesectionandtotheultimateuppersection,weshallsaythatitbelongstothe"ultimateoscillation."Wemayillustratethematterbycon-sideringoncemorethefunctionsintfxasxapproachesthevalueo.Weshallassume,inordertofitinwiththeabovedefinitions,thatthisvalueisapproachedfrombelow.Letusbeginwiththe"ultimatesection."Between-eando,whateveremaybe,thefunctionwillassumethevalueiforcertainarguments,butwillneverassumeanygreatervalue.Hencetheultimatesectionconsistsofallrealnumbers,positiveandnegative,uptoandincludingr;i.e.itconsistsofallnegativenumberstogetherwitho,togetherwiththepositivenumbersuptoandincludingr.Similarlythe"ultimateuppeisection"consistsofallpositivenumberstogetherwitho,togetherwiththenegativenumbersdowntoandincluding-r.('ultimateThustheoscillation"consistsofallrealnumbersfrom-rtor,bothincluded.\nrrzfnnoductiontoMathematicalPlrilosoplryWemaysaygenerallythatthe"ultimateoscillation"ofafunctionastheargumentapproachesafrombelowconsistsofallthosenumbers.rwhicharesuchthat,howevernearwecometoa,weshallstillfindvaluesasgreatasrandvaluesassmallas,c.Theultimateoecillationmaycontainnoterms,oroneterm,ormanyterms.Inthefirsttwocasesthe{unctionhasadefinitelimitforapproachesfrombelow.Iftheultimateoscillationhasoneterm,thisisfairlyobvious.Itisequallytrueifithasnone;foritisnotdifficulttoprovethat,iftheultimateoscilla-tionisnull,thebound"ryoftheultimatesectionisthesameasthatoftheultimateuppdrsection,andmaybedefinedasthelimitofthefunctionforapproachesfrombelow.Butiftheultimateoscillationhasmanyterms,thereisnodefinitelimittothefunctionforapproachesfrombelow.Inthiscasewecantakethelowerandupperboundariesoftheultimateoscillation(i.e.thelowerboundaryoftheultimateuppersectionandtheupperboundaryoftheultimatesection)asthelowerandupperlimitsofits"ultimate"valuesforapproachesfrombelow.Similarlyweobtainlowerandupperlimitsofthe"ultimate"valuesforapproachesfromabove.Thuswehave,inthegenerafcaserifourlimitstoafunctionforapproachestoagivenargument.Ihelimitforagivenargumentaonlyexistswhenallthesefourareequal,andisthentheircommonvalue.Ifitisalsothevaluefortheargumenta,thefunctioniscontinuousforthisargument.Thismaybetakenasdefiningcontinuity:itisequivalenttoourformerdefinition.Wecandefinethelimitofafunctionforagivenargument(ifitexists)withoutpassingthroughtheultimateoscillationandthefourlimitsofthegeneralcase.Thedefinitionproceeds,inthatcase,justastheearlierdefinitionofcontinuityproceeded.Letusdefinethelimitforapproachesfrombelow.Ifthereistobeadefinitelimitforapproachestoafrombelow,itisnecessaryandsufficientthat,givenanysmallnumbero,twovaluesforargumentssufficientlyneartoa(butbothlessthana)willdifier\nLimitsandContinuityofFunctionsrr3bylessthanoIi.e.if.eissufficientlysmall,andourargumentsbothliebetweenA-canda(aexcluded),thenthedifferencebetweenthevaluesfortheseargumentswillbelessthano.Thisistoholdforanyo,howeversmall;inthatcasethefunctionhas^limitforapproachesfrombelow.Similarlywedefinethecasewhenthereisalimitforapproachesfromabove.Thesetwolimits,evenwhenbothexist,neednotbeidentical;andiftheyareidentical,theystillneednotbeidenticalwiththeaaluefortheargumenta.Itisonlyinthislastcasethatwecallthefunctioncontinulusfortheargumentz.66Afunctioniscalledcontinuous"(withoutqualification)whenitiscontinuousforeveryargument.Anotherslightlydifferentmethodofreachingthedefinitionofcontinuityisthefollowing:-Letussaythatafunction"ultimatelyconvergesintoaclassa"ifthereissomerealnumbersuchthat,forthisargumentandallargumentsgreaterthanthis,thevalueofthefunctionisamemberoftheclasso.Similarlyweshallsaythatafunctiontt"convergesintooastheargumentapproachesnfrombelowifthereissomeargumentylessthanrsuchthatthroughouttheintervalfromy(included)tor(excluded)thefunctionhas'Wemaynowsaythatavalueswhicharemembersofc.functioniscontinuousfortheargumenta,forwhichithasthevalueifa,ifitsatisfiesfourconditions,namely:-(r)Givenanyrealnumberlessthanfa,thefunctioncon-vergesintothesuccessorsofthisnumberastheargumentapproachesafrombelow;(z)Givenanyrcalnumbergreaterthan.fa,thefunctioncon-vergesintothepredecessorsofthisnumberastheargumentapproachesafrombelow;(3)and(4)Similarconditionsforapproachestoafromabove.Theadvantagesofthisformofdefinitionisthatitanalysestheconditionsofcontinuityintofour,derivedfromconsideringargumentsandvaluesrespectivelygreaterorlessthantheargumentandvalueforwhichcontinuityistobedefined.\nrr+IntroductiontoMatlrentaticalPltilosoplr1Wemaynowgeneraliseourdefinitionssoastoapplytoserieswhicharenotnumericalorknowntobenumericallymeasurable.Thecaseofmotionisaconvenientonetobearinmind.ThereisastorybyH.G.Wellswhichwillillustrate,fromthecaseofmotion,thedifferencebetweenthelimitofafunctionforagivenargumentanditsvalueforthesameargument.Theheroofthestory,whopossessed,withouthisknowledge,thepowerofrealisinghiswishes,wasbeingattackedbypoliceman,buton"ejaculating"Goto-"hefoundthatthepolicemandisappeared.Ifflt)wasthepoliceman'spositionattimet,and/othemomentoftheejaculation,thelimitofthepoliceman'spositionsasIapproachedtolofrombelowwouldbeincontactwiththehero,whereasthevaluefortheargumentlowas-.Butsuchoccur-rencesaresupposedtoberareintherealworld,anditisassumed,thoughwithoutadequateevidence,thatallmotionsarecontinu-ous,i./.that,givenanybody,iff(t)isitspositionattimet,-f(t)((ttisacontinuousfunctionoft.Itisthemeaningofcontinuityinvolvedinsuchstatementswhichwenowwishtodefineassimplyaspossible.Thedefinitionsgivenforthecaseoffunctionswhereargumentandvaluearerealnumberscanreadilybeadaptedformoregeneraluse.LetPandQbetworelations,whichitiswelltoimagineserial,thoughitisnotnecessarytoourdefinitionsthattheyshouldbeso.LetRbeaone-manyrelationwhosedomainiscontainedinthefieldofP,whileitsconversedomainiscon-tainedinthefieldofQ.ThenRis(inageneralisedsense)afunction,whoseargumentsbelongtothefieldofQ,whileitsvaluesbelongtothefieldofP.Suppose,forexample,thatwearedealingwithaparticlemovingonaline:letQbethetime-series,Ptheseriesofpointsonourlinefromlefttoright,RtherelationofthepositionofourparticleonthelineattimeatothetimeA,sothat"theRofa"isitspositionattimea,Thisillustrationmaybeborneinmindthroughoutourdefinitions.WeshallsaythatthefunctionRiscontinuousfortheargument\nLimixandContinuitlofFunctionsII5ait,givenanyintervalcrontheP-seriescontainingthevalueofthefunctionfortheargumenta,thereisanintervalontheQ-seriescontaininganotasanend-pointandsuchthat,through-outthisinterval,thefunctionhasvalueswhicharemembersofo.(Wemeanbyan"intervalt'allthetermsbetweenanytwo1i.e.tl*andyatetwomembersoIthefieldofP,andrhastherelationPtoltweshallmeanbythe"P-intervalxtoy"alltermszsuchthatrhastherelationPtoxandzhastherela-tionPto),-together,whensostated,withxorythemselves.)Wecaneasilydefinethe"ultimatesection"andthe"ulti-t'ultimatemateoscillation.t'Todefinethesection"forapproachestotheargumentafrombelow,takeanyargltmentywhichprecedesa(i.e.hastherelationQtoa),takethevaluesofthefunctionforallargumentsuptoandincludingy,andformthesectionofPdefinedbythesevalues,i.e.thosemembersoftheP-serieswhichareearlierthanoridenticalwithsomeofthesevalues.Formallsuchsectionsforally'sthatprecedea,andtaketheircommonpart;thiswillbetheultimatesection.Theultimateuppersectionandtheultimateoscillationarethendefinedexactlyasinthepreviouscase.Theadaptationofthedefinitionofconvergenceandtheresultingalternativedefinitionofcontinuityofiersnodifficultyofanykind.WesaythatafunctionRis"ultimatelyQ-convergentintoa"iIthereisamemberyoftheconversedomainofRandthefieldofQsuchthatthevalueofthefunctionfortheargumentyandforanyargumenttowhichyhastherelationQisamemberttofa.WesaythatRQ-convergesintocrastheargumentt'approachesagivenargumentaifthereisatermyhavingtherelationQtoaandbelongingtotheconversedomainofRandsuchthatthevalueofthefunctionforanyargumentintheQ-intervalfromy(inclusive)toa(exclusive)belongstoo.Ofthefourconditionsthatafunctionmustfulfilinordertobecontinuousfortheargumenta,thefirstis,puttingbf.otthevaluefortheargumentd:\nTr5fntroductiontoMath'entaticalPhilosoplryGivenanytermhavingtherelationPtob,RQ-convergesintothesuccessorsofD(withrespecttoP)astheargumentapproachesafrombelow,ThesecondconditionisobtainedbyreplacingPbyitsconverse;thethirdandfourthareobtainedfromthefirstandsecondbyreplacingQbyitsconverse.Thereisthusnothing,inthenotionsofthelimitofafunctionorthecontinuityoLafunction,thatessentiallyinvolvesnumber,Bothcanbedefinedgenerally,andmanypropositionsaboutthemcanbeprovedforanytwoseries(onebeingtheargument-seriesandtheotherthevalue-series).Itwillbeseenthatthedefinitionsdonotinvolveinfinitesimals.Theyinvolveinfiniteclassesofintervals,growingsmallerwithoutanylimitshortofzero,buttheydonotinvolveanyintervalsthatarenotfinite.Thisisanalogoustothefactthatifalineaninchlongbehalved,thenhalvedagain,andsoonindefinitely,weneverreachinfini-tesimalsinthiswayiafterabisections,thelengthofourbitisIofaninch:andthisisfinitewhateverfinitenumbernmay2nbe.Theprocessofsuccessivebisectiondoesnotleadtodivisionswhoseordinalnumberisinfinite,sinceitisessentiallyaone-by-oneprocess.Thusinfinitesimalsarenottobereachedinthisway.Confusionsonsuchtopicshavehadmuchtodowiththedifficultieswhichhavebeenfoundinthediscussionofinfinityandcontinuity.\nCHAPTERXIISELECTIONSANDTHEMULTIPLICATIVEAXIOMInthischapterwehavetoconsideranaxiomwhichcanbeenunciated,butnotproved,intermsoflogic,andwhichiscon-venient,thoughnotindispensable,incertainportionsofmathe-matics.Itisconvenient,inthesensethatmanyinterestingpropositions,whichitseemsnaturaltosupposetrue,cannotbeprovedwithoutitshelp;butitisnotindispensable,becauseevenwithoutthosepropositionsthesubjectsinwhichth.yoccurstillexist,thoughinasomewhatmutilatedform.Beforeenunciatingthemultiplicativeaxiom,wemustfirstexplainthetheoryofselections,andthedefinitionofmulti-plicationwhenthenumberoffactorsmaybeinfinite.Indefiningthearithmeticaloperations,theonlycorrectpro-cedureistoconstructanactualclass(orrelation,inthecaseofrelation-numbers)havingtherequirednumberofterms.ThissometimesdemandsacertainamountofingenuitRbutitisessentialinordertoprovetheexistenceofthenumberdefined.Take,asthesimplestexample,thecaseofaddition.Supposewearegivenacardinalnumberp,,andaclassowhichhaspterms.Howshallwedefinep+piForthispurposewemusthavetwoclasseshavingt,terms,andt]r.ymustnotoverlap.Wecanconstructsuchclassesfromoinvariousways,ofwhichthefollowingisperhapsthesimplest:Formfirstalltheorderedcoupleswhosefirsttermisaclassconsistingofasinglememberofa,andwhosesecondtermisthenull-class;then,secondly,formalltheorderedcoupleswhosefirsttermis]17\nrr8IntroductiontoMatlrematicalPltilosopltythenull-classandwhosesecondtermisaclassconsistingofasinglememberofa.Thesetwoclassesofcouplmhavenomemberincommon,andthelogicalsumofthetwoclasseswillhavep*pterms.Exactlyanalogouslywecandefiney'*v,giventhatp,isthenumberofsomeclassoandyisthenumberofsomeclassp.Suchdefinitione,asarule,aremerelyaquestionofasuitabletechnicaldevice.Butinthecaseofmultiplication,wherethenumberoffactorsmaybeinfinite,importantProblemsariseoutofthedefinition.Multiplicationwhenthenumberoffactorsisfiniteofiersnodifficulty.Giventwoclassesoandp,ofwhichthefirsthasptermsandthesecondzterms,wecandefinep4vasthenumberoforderedcouplesthatcanbeformedbychoosingthefirsttermoutofcrandthesecondoutofp.Itwillbeseenthatthisde-finitiondoesnotrequirethataandpshouldnotoverl^P;itevenremainsadequatewhenoandBareidentical.Forexample,letobetheclasswhosemembergatetr,',fi2,frs.Thentheclasswhichisusedtodefinetheproductpr,Xpistheclassofcouples:(*r,*),(tcr,tc),(xr,xa)i(xz,,c),(xz,x),(rz,xt);(xt,tc),(tcs,x),(xa,xe).Thisdefinitionremainsapplicablewhen6r,orvotbothareinfinite,anditcanbeextendedstepbystePtothreeorfouroranyfinitenumberoffactors.Nodifficultyarisesasregardsthisdefinition,exceptthatitcannotbeextendedtoan,inf'niUnumberoffactors.TheproblemofmultiplicationwhenthenumberoffactorsmaybeinfinitearisesinthiswayISupposewehaveaclassrcconsistingofclasses;supposethenumberoftermsineachoftheseclassesisgiven.HowshallwedefinetheproductofalltheeenumbersIIfwecanframeourdefinitiongenerallynitwillbeapplicablewhetherKisfiniteorinfinite.Itistobeobservedthattheproblemistobeabletodealwiththecasewhenrcisinfinite,notwiththecasewhenitsmembersare.If\nSelectionsandtlteMultiplicatizteAxiomII9risnotinfinite,themethoddefinedaboveisjustasapplicablewhenitsmembersareinfiniteaswhentheyarefinite.Itisthecasewhenrcisinfinite,eventhoughitsmembersmaybefinite,thatwehavetofindawayofdealingwith.Thefollo*irgmethodofdefiningmultiplicationgenerallyisduetoDrWhitehead.ItisexplainedandtreatedarlengthinPrincipiaMathematicarvol.i.*8ofi.,andvol.ii.*rr+.Letussupposetobeginwiththatrcisaclassofclassesnotwoofwhichoverlap-saytheconstituenciesinacountrywherethereisnopluralvoting,eachconstituencybeingconsideredasaclassofvoters.Letusnowsettoworktochooseonetermoutofeachclasstobeitsrepresentative,asconstituenciesdowhentheyelectmembersofParliament,assumingthatbylaweachconstituencyhastoelectamanwhoisavoterinthatconstituency.Wethusarriveataclassofrepresentatives,whomakeupourParliament,onebeingselectedoutofeachcon-stituency.HowmanydifferentpossiblewaysofchoosingaParliamentarethere?Eachconstituencycanselectanyoneofitsvoters,andthereforeiftherearcpvotersinaconstituency,itcanmake;r,choices.Thechoicesofthedifierentconstituenciesareindependent;thusitisobviousthat,whenthetotalnumberofconstituenciesisfinite,thenumberofpossibleParliamentsisobtainedbymultiplyingtogetherthenumbersofvotersinthevariousconstituencies.Whenwedonotknowwhetherthenumberofconstituenciesisfiniteorinfinite,wemaytakethenumberofpossibleParliamentsasdcfiningtheproductofthenumbersoftheseparateconstituencies.Thisisthemethodbywhichinfiniteproductsaredefined.Wemustnowdropourillustration,andproceedtoexactstatements.Letrcbeaclassofclasses,andletusassumetobeginwiththatnotwomembersofrcoverlap,i.c.thatifoandBaretwodifierentmembersofr,thennomemberoftheoneisamemberoftheother.Weshallcallaclassa"selection"fromrcwhenitcon-sistsofjustonetermfromeachmemberof.rc;i,t.t"isa.,selec-t'tionfromrifeverymemberof;r,belongstosomemember\nr2a^fntroducriontoMatlrematicalPhilosoplt1ofr,andifobeanymemberofrc,y,andohaveexactlyonetermincommon.Theclassofall"selections"fromKweshallcallthe6'multiplicativeclass"ofrc.Thenumberoftermsinthemultiplicativeclassofr,i.e.thenumberofpossibleselectionsfromr,isdefinedastheproductofthenumbersofthemembersofr.Thisdefinitionisequallyapplicablewhetherrisfiniteorinfinite.Beforewecanbewhollysatisfiedwiththesedefinitions'wemustremovetherestrictionthatnotwomembersoftcaretooverlap.Forthispurposezinsteadofdefiningfirstaclasscalleda"selectionr"wewilldefinefirstarelationwhichwewillcalla"selector."ArelationRwillbecalleda"selector"fromrif,fromeverymemberofrc,itpicksoutonetermastherepresentativeofthatmember,i.e.if.,givenanymemberaofrc,thereisjustonetermxwhichisamemberofcuandhastherelationRtoo;andthisistobeallthatRdoes.Theformaldefinitionis:ttAselector"fromaclassofclassesrisaone-manyrelation,havingrcforitsconversedomain,andsuchthat,ifrhastherelationtoo,thenrisamemberofcl.IfRisaselectorfromr,andoisamemberofrc,andxisthetermwhichhastherelationRtocr,wecallrthe"rePresentative"ofoinrespectoftherelationR.A"selection"fromrcwillnowbedefinedasthedomainofaselector;andthemultiplicativeclass,asbefore,willbetheclassofselections.Butwhenthemembersofrcoverlap,theremaybemoreselectorsthanselections,sinceatermrwhichbelongstotwoclassesoandpmaybeselectedoncetorePresentoandoncetorePresentp,grvingrisetodifierentselectorsinthetwocases,buttothesameselection.Forpurposesofdefiningmultiplication,itistheselectorswerequireratherthantheselections'Thuswedefine:t'Theproductofthenumbersofthemembersofaclassofttclassesrcigthenumberofselectorsfromrc.WecandefineexPonentiationbyanadaptationpftheabovq\nSelectionsandtlreMultiplicativelxiomT2lplan.Wemight,ofcourse,definep"asthenumberofselectorsfromyclasses,eachofwhichhasp,terms.Butthereareobjectionstothisdefinition,derivedfromthefactthatthemultiplicativeaxiom(ofwhichweshallspeakshortly)isunneces-sarilyinvolvedifitisadopted.Weadoptinsteadthefollowingconstruction:-LetobeaclasshavingF,terms,andBaclasshavingyterms.Letybeamemberofp,andformtheclassofallorderedcouplesthathaveyfortheirsecondtermandamemberofofortheirfirstterm.Therewillbe4rsuchcouplesforagiven/2sinceanymemberofq,maybechosenforthefirstterm,andohaspr,members.Ifwenowformalltheclassesofthissortthatresultfromvaryingltweobtainaltogetheryclasses,sinceymaybeanymemberofp,andphasymembers.Theseyclassesareeachofthemaclassofcouples,namely,allthecouplesthatcanbeformedofavariablememberofoandafixedmemberofp.Wedefinep'asthenumberofselectorsfromtheclassconsistingofthesezclasses.Orwemayequallywelldefinep,"asthenumberofselections,for,sinceourclassesofcouplesaremutuallyexclusive,thenumberofselectorsisthesameasthenumberofselections.Aselectionfromourclassofclasseswillbeasetoforderedcouples,ofwhichtherewillbeexactlyonehavinganygivenmemberofBforitssecondterm,andthefirsttermmaybeanymemberoIq,.Thuspuisdefinedbytheselectorsfromacertainsetofvclasseseachhavingpterms,butthesetisonehavingacertainstructureandamoremanageablecompositionthanisthecaseingeneral.Therelevanceofthistothemultiplicativeaxiomwillappearshortly.Whatappliestoexponentiationappliesalsototheproductoftwocardinals.Wemightdefine"lrxr"asthesumofthenumbersofyclasseseachhavingFcterms,butweprefertodefineitasthenumberoforderedcouplestobeformedconsistingofamemberofofollowedbyamemberofflwhereohas;r,termsandBhasvterms.Thisdefinition,also,isdesignedtoevadethenecessityofassumingthemultiplicativeaxiom.\nr22IntroductiontoMatlrematicalPlrilosopltyWithourdefinitions,wecanprovetheusualformallawsofmultiplicationandexponentiation.Butthereisonethingwecannotprove:wecannotProvethataproductisonlyzerowhenoneofitsfactorsiszero.Wecanprovethiswhenthenumberoffactorsisfinite,butnotwhenitisinfinite.Inotherwords,wecannotprovethat,givenaclassofclassesnoneofwhichisnull,theremustbeselectorsfromthem;orthat,givenaclassofmutuallyexclusiveclasses,theremustbeatleastoneclassconsistingofonetermoutofeachofthegivenclasses.Thesethingscannotbeproved;andalthough,atfirstsight,theyseemobviouslytrue,yetreflectionbringsgraduallyincreasingdoubt,untilatlastwebecomecontenttoregistertheassumPtionanditsconsequences,asweregistertheaxiomofparallels,withoutassumingthatwecanknowwhetheritistrueorfalse.Thea$sumption,looselyworded,isthatselectorsandselectionsexistwhenweshouldexpectthem.Therearemanyequivalentwaysofstatingitprecisely.Wemaybeginwiththefollowing:-"Givenanyclassofmutuallyexclusiveclasses,ofwhichnoneisnull,thereisatleastoneclasswhichhasexactlyonetermincommonwitheachofthegivenclasses."1Thispropositionwewillcallthe"multiplicativeaxiom."Wewillfirstgivevariougequivalentformsoftheproposition,andthenconsidercertainwaysinwhichitstruthorfalsehoodisofinteresttomathematics.Themultiplicativeaxiomisequivalenttothepropositionthataproductisonlyzerowhenatleastoneofitsfactorsiszero;i.e.thatrifanynumberofcardinalnumbersbemultipliedtogether,theresultcannotbeounlessoneofthenumbersconcernediso.Themultiplicativeaxiomisequivalenttothepropositionthat,ifRbeanyrelation,andrcanyclasscontainedintheconversedomainofR,thenthereisatleastoneone-manyrelationimplyingRandhavingrcforitsconversedomain.Themultiplicativeaxiomisequivalenttotheassumptionthatifobeanyclass,andrallthesub-classesofowiththeexceptionrSeePrincipioMathematiaa,vol.i.I88.Alsovql.iii.c257-258,\nSelectionsandtheMultiplicative'lxiom123ofthenull-class,thentlereisatleastoneselectorfromrc.ThisistheforminwhichtheaxiomwasfirstbroughttothenoticeofthelearnedworldbyZermelo,inhis"Beweis,dassjedeMengewohlgeordnetwerdenkann.t'tZetmeloregardstheaxiomasanunquestionabletruth.Itmustbeconfessedthat,untilhemadeitexplicit,mathematicianshaduseditwithoutaqualm;butitwouldseemthattheyhaddonesounconsciously.AndthecreditduetoZermeloforhavingmadeitexplicitisentirelyindependentoftleguestionwhetlreritistrueorfalse.ThemultiplicativeaxiomhasbeenshownbyZermelo,intheabove-mentionedproof,tobeequivalenttothepropositionthateveryclasscanbewell-ordered,i.e.canbearrangedinaseriesinwhicheverysub-classhasafirstterm(except,ofcourse,thenull-class).Thefullproofofthispropositionisdifficult,butitisnotdifficulttoseethegeneralprincipleuponwhichitproceeds.Itusestheformwhichwecall"Zermelot$axiomr"i.e.itassumesthat,givenanyclasso,thereisatleastoneone-manyrelationRwhoseconversedomainconsistsofallexistentsub-classesofoandwhichissuchthat,ifrhastherelationRtof,thenrisamemberoff.Sucharelationpicksouta"representative"fromeachsub-class;ofcourse,itwilloftenhappenthattwosub-classeshavethesamerepresentative.WhatZermelodoes,ineffect,istocountoffthemembersofa,onebyone,bymeansofRandtransfiniteinduction.Weputfirsttherepresentativeofo;callitrr.Thentaketherepresentativeoftheclassconsistingofallofoexcepttcr;callitxr.Itmustbedifierentfromrr,becauseeveryrepresentativeisamemberofitsclass,and*,isshutoutfromthisclass.Proceedsimilarlytotakeawayxr,andlet*sbetherepresentativeofwhatisleft.Inthiswaywefirstobtainaprogression#r,rr,.xn,...,assumingthatoisnotfinite.Wethentakeawaythewholeprogression;letro,betherepresentativeofwhatisleftofo.Inthiswaywecangoonuntilnothingisleft.ThesuccessiverepresentativeswillformaIMathematischeAnnalen,vol.lix.pp.514-6.InthisformweshallspeakofitasZermelo'saxiom.\nrz+InroductiontoMatltematicalPltilosoplr1well-orderedseriescontainingallthemembersofo.(Ih.aboveis,ofcourse,onlyahintofthegenerallinesoftheproof.)Thispropositioniscalled"Zermelo'stheorem."Themultiplicativeaxiomisalsoequivalenttotheassumptionthatofanytwocardinalswhicharenotequal,onemustbethegreater.Iftheaxiomisfalse,therewillbecardinalsp,andvsuchthatp,isneitherlessthan,equalto,norgreaterthanv.Wehaveseenthattt,and2bl.possiblyformaninstanceofsuchapait.Manyotherformsoftheaxiommightbegiven,buttheabovearethemostimportantoftheformsknownatPresent.Astothetruthorfalsehoodoftheaxiominanyofitsforms,nothingisknownatpresent.Thepropositionsthatdependupontheaxiom,withoutbeingknowntobeequivalenttoit,arenumerousandimportant.Takefirsttheconnectionofadditionandmultiplication.Wenaturallythinkthatthesumofymutuallyexclusiveclasses,eachhavingpterms,musthavepxvterms.Whenvisfinite,thiscanbeproved.Butwhenzisinfinite,itcannotbeprovedwithoutthemultiplicativeaxiom,exceptwhere,owingtosomespecialcir-cumstance,theexistenceofcertainselectorscanbeproved.Thewaythemultiplicativeaxiomentersinisasfollows:Supposewehavetwosetsofymutuallyexclusiveclasses,eachhavingg,terms,andwewishtoprovethatthesumofonesethasasmanytermsasthesumoftheother.InordertoProvethis,wemustestablishaone-onerelation.Now,sincethereareineachcaseyclasses,thereissomeone-onerelationbetweenthetwosetsofclasses;butwhatwewantisaone-onerelationbetweentheirterms.Letusconsidersomeone-onerelationSbetweentheclasses.ThenifrcandIarethetwosetsofclasses,andcr,issomememberofr,therewillbeamemberpof)\whichwillbethecorrelateofowithrespecttoS.Nowaandpeachhavepterms'andarethereforesimilar.Thereare,accordingly,one-onecor-relationsofoandB.Thetroubleisthattherearesomany.Inordertoobtainaone-onecorrelationofthesumofrwiththesumofl,wehavetopickoutoneselectionfromatetofclasges\nSelectionsandtheMultiplicati,ueAxiomt2sofcorrelators,oneclassofthesetbeingalltheone-onecorrelatorsofowithB.IfrcandIareinfinite,wecannotingeneralknowthatsuchaselectionexists,unlesswecanknowthatthemulti-plicativeaxiomistrue.Hencewecannotestablishtheusualkindofconnectionbetweenadditionandmultiplication.Thisfacthasvariouscuriousconsequences.Tobeginwith,weknowthatN'2:N'XNq:Ns.Itiscommonlyinferredfromthisthatthesumof*oclasseseachhavingHomembersmustitselfhaveNomembers,butthisinferenceisfallacious,sincewedonotknowthatthenumberoftermsinsuchasumis*oXNo,norconsequentlythatitislso.Thishasabearinguponthetheoryoftransfiniteordinals.ItiseasytoprovethatanordinalwhichhasHopredecessorsmustbeoneofwhatCantorcallsthet6secondclass,'?i.r.suchthataserieshavingthisordinalnumberwillhaveN0termsinitsfield.Itisalsoeasytoseethat,ifwetakeanyprogressionofordinalsofthesecondclass,thepredecessorsoftheirlimitformatmostthesumofnoclasseseachhavingnoterms.Itisinferredthence-fallaciously,unlessthemulti-plicativeaxiomistrue-thatthepredecessorsofthelimitareN0innumber,andthereforethatthelimitisanumberofthet'secondclass."Thatistosay,itissupposedtobeprovedthatanypro-gressionofordinalsofthesecondclasshasalimitwhichisagainanordinalofthesecondclass.Thisproposition,withthecorol-larythata.,,(thesmallestordinalofthethirdclass)isnotthelimitofanyprogression,isinvolvedinmostoftherecognisedtheoryofordinalsofthesecondclass.Inviewofthewayinwhichthemultiplicativeaxiomisinvolved,thepropositionanditscorollarycannotberegardedasproved.Theymaybetrue,ortheymaynot.Allthatcanbe.saidatpresentisthatwedonotknow.Thusthegreaterpartofthetheoryofordinalsofthesecondclassmustberegardedasunproved.Anotherillustrationmayhelptomakethepointclearer.WeknowthatzXN0-N0.HencewemightsupposethatthesumofttopairsmusthaveNoterms.Butthis,thoughwecanprovethatitissometimesthecase,cannotbeprovedtohappenalways\nr26IntroductiontoMatlternaticalPltilosopltyunlessweassumethemultiplicativeaxiom.Thisisillustratedbythemillionairewhoboughtapairofsockswheneverheboughtapairofboots,andnevetatanyothertime,andwhohadsuchapassionforbuyingboththatatlasthehadHopairsofbootsandHopairsofsocks.Theproblemis:Howmanybootshadh.,andhowmanysocks?OnewouldnaturallysuPPosethathehadtwiceasmanybootsandtwiceasmanysocksashehadpairsofeach,andthatthereforehehadHoofeach,sincethatnumberisnotincreasedbydoubling.Butthisisaninstanceofthedifficulty,alreadynoted,ofconnectingthesumofvclasseseachhavingptermswithpxv.Sometimesthiscanbedone,sometimesitcannot.Inourcaseitcanbedonewiththeboots,butnotwiththesocks,excePtbysomeveryartificialdevice.Thereasonforthedifierenceisthis:Amongbootswecandis-tinguishrightandleft,andthereforewecanmakeaselectionofoneoutofeachpair,namely,wecanchoosealltherightbootsoralltheleftboots;butwithsocksnosuchprincipleofselectionSuggestsitself,andwecannotbesure,unlessweassumethemultiplicativeaxiom,thatthereisanyclassconsistingofonesockoutofeachpair.Hencetheproblem.Wemayputthematterinanotherway.ToprovethataclasshasHoterms,itisnecessaryandsufficienttofindsomewayofarrangingitstermsinaprogression.Thereisnodifficultyindoingthiswiththeboots.ThepairsaregivenasforminganN0,andthereforeasthefieldofaProgression.Withineachpair,taketheleftbootfirstandtherightsecond,keepingtheorderofthe,pairunchanged;inthiswayweobtainaProgressionofalltheboots.Butwiththesocksweshallhavetochoosearbi-trarily,witheachpair,whichtoPutfirst;andaninfinitenumberofarbitrarychoicesisanimpossibility.Unlesswecanfindarulef.orselecting,i.e,arelationwhichisaselector,wedonotknowthataselectioniseventheoreticallypossible.Ofcourse,inthecaseofobjectsinspace,likesocks,wealwayscanfindsomeprincipleofselection.Forexample,takethecentresofmassofthesocks:therewillbepointspinspacesuchthat,withany\nSelectionsandtheMultiplicatiztet{xiomr27pair,thecentresofmassofthetwosocksarenotbothatexactlythesamedistancefromp;thuswecanchoose,fromeachpair,thatsockwhichhasitscentreofmassnearertop.Butthereisnotheoreticalreasonwhyamethodofselectionsuchasthisshouldalwaysbepossible,andthecaseofthesocks,withalittlegoodwillonthepartofthereader,mayservetoshowhowaselectionmightbeimpossible.Itistobeobservedthat,if.itwereimpossibletoselectoneoutofeachpairofsocks,itwouldfollowtharthesockscouldnotbearrangedinaprogression,andthereforethattherewerenotNoofthem.Thiscaseillustratesthat,ifpisaninfinitenumber,onesetofp,pairsmaynotcontainthesamenumberoftermsasanothersetofp,pairsIfor,givennopairsofboots,therearecertainlyNoboots,butwecannotbesureofthisinthecaseofthesocksunlessweassumethemultiplicativeaxiomorfallbackuponsomefortuitousgeometricalmethodofselectionsuchastheabove.Anotherimportantprobleminvolvingthemultiplicativeaxiomistherelationofreflexivenesstonon-inductiveness.ItwillberememberedthatinChapterVIII.wepointedoutthatareflexivenumbermustbenon-inductive,butthattheconverse(sofarasisknownatpresent)canonlybeprovedifweassumethemultiplicativeaxiom.Thewayinwhichthiscomesaboutisasfollows:-Itiseasytoprovethatareflexiveclassisonewhichcontainssub-classeshavingNoterms.(Theclassmay,ofcourse,itselfhaveNoterms.)Thuswehavetoprove,ifwecan,that,givenanynon-inductiveclass,itispossibletochooseaprogressionoutofitsterms.Nowthereisnodifficultyinshowingtharanon-inductiveclassmustcontainmoretermsthananyinductiveclass,or,whatcomestothesamething,thatifoisanon-induc-tiveclassandzisanyinductivenumber,therearesub-classesofathathaveyterms.Thuswecanformsetsoffinitesub-classesofo:Firstoneclasshavingnoterms,thenclasseshavingrterm(asmanyastherearemembersofo),t}enclasseshaving\nrzgfnnoductiontoMatltematicalPltihsoplty2terms,andsoon.Wethusgetaprogressionofsetsofsub-classes,eachsetconsistingofallthosethathaveacertaingivenfinitenumberofterms.Sofarwehavenotusedthemultiplica-tiveaxiom,butwehaveonlyprovedthatthenumberofcollec-tionsofsub-classesofoisareflexivenumber,i.e.that,ifpisthenumberofmembersof.a,sothatz*isthenumberofsub-classesofaarrdzzpisthenumberofcollectionsofsub-classes,then,providedpisnotinductive,zlumustbereflexive.Butthisisalongway{romwhatwesetouttoProve.Inordertoadvancebeyondthispoint,wemustemploythemultiplicativeaxiom.Fromeachsetofsub-classesletuschooseoutone,omittingthesub-classconsistingofthenull-classalone.Thatistosay,weselectonesub-classcontainingoneterm,aL,say;onecontainingtwoterms,o.bsay;onecon-tainingthree,os,say;andsoon.(\Mecandothisifthemultipli-cativeaxiomisassumedIotherwise,wedonotknowwhetherwecanalwaysdoitornot.)Wehavenowaprogressionottebes,,..ofsub-classesofo,insteadofaProgressionofcollectionsofsub-classes;thusweareoneStePnearertoourgoal.Wenowknowthat,assumingthemultiplicativeaxiom,ifg,isanon-inductivenumber,2t'mustbeareflexivenumber.Thenextstepistonoticethat,althoughwecannotbesurethatnewmembersofq.comeinatanyonespecifiedstageintheprogression01,ez,@b...wecanbesurethatnewmembergkeeponcominginfromtimetotime.Letusillustrate.Theclasso1rwhichconsistsofoneterm,isanewbeginning;lettheonetermbe11.Theclassor,consistingoftwotermstmayormaynotcontainxr;ifitdoes,itintroducesonenewterm;andifitdoesnot,itmustintroducetwonewterms,sayxz,xg.Inthiscaseitispossiblethato,consistsoffryfr22!ts,andsointroducesnonewterms,butinthatcasecr4mustintroduceanewterm.Thefirstvclasseseb@ztQs,...o/contain,attheverymost,tlz*3*...{vterms,i.e,v(v4.r)/zterms;thusitwouldbepossible,iftherewerenorePetitionsinthefirstyclasses,togoonwithonlyrepetitionsfromthe(zfr)tn\nSelectionsandtheMultiplicati'uelxiomrzgclasstothev(v*r)lr'oclass.Butbythattimetheoldtermswouldnolongerbesufficientlynumeroustoformanextclasswiththerightnumberofmembers,7.e.v(v*t)lz+t,thereforenewtermsmustcomeinatthispointifnotsooner.Itfollowsthat,ifweomitfromourprogressionabclz>@s,..,allthoseclassesthatarecomposedentirelyofmembersthathaveoccurredinpreviousclasses,weshallstillhaveaprogression.LetournewprogressionbecalledFr,Fr,F....(Weshallhaveat:Frandar:82,becauseolanda2nlustintroducenewterms.Wemayormaynothaveas:Fs,but,speakinggenerally,pnwillbeo,,wherevissomenumbergreaterthanp;i.e.theB'saresorleofthea's.)NowtheseB'saresuchthatanyoneofthem,sayF*,containsmemberswhichhavenotoccurredinanyofthepreviousB's.Lety*bethePartofFnwhichconsistsofnewmembers.ThuswegetanewprogressiotTt,TbTs,..(Againy1willbeidenticalwithB,andwitho1;if.a2doesnotcontaintheonememberofar,weshallhavefz:Fz:o2,butifo2doescontainthisonemember,y2willconsistoftheothermemberofo2.)Thisnewprogressionofy'sconsistsofmutuallyexclusiveclasses.HenceaselectionfromthemwillbeaPro-gression;i.e.if.x,isthememberof!r,*,isamemberoIyr,x,isamemberofyu,andsoon;thenxL,xz,xs,...isaprogression,andisasub-classofa.Assumingthemultiplicativeaxiom,suchaselectioncanbemade.Thusbytwiceusingthisaxiomwecanprovethat,iftheaxiomistrue,everynon-inductivecardinalmustbereflexive.ThiscouldalsobededucedfromZermelo'stheorem,that,iftheaxiomistrue,everyclasscanbewellordered;forawell-orderedseriesmusthaveeitherafiniteorareflexivenumberoftermsinitsfield.Thereisoneadvantageintheabovedirectargurnent,asagainstdeductionfromZermelo'stheorem,thattheaboveargumentdoesnotdemandtheuniversaltruthofthemulti-plicativeaxiom,butonlyitstruthasappliedtoasetofNoclasses.ItmayhappenthattheaxiomholdsforHqclasses,thoughnotforlargernumbersofclasses.Forthisreasonitisbetter,when\nr30fntroductiontoMatlternaticalPlrilosoplt1itispossible,tocontentourselveswiththemorerestrictedassumption.Theassumptionmadeintheabovedirectargu-mentisthataproductof*ofactorsisneverzerounlessoneofthefactorsiszero.Wemaystatethisassumptionintheform:"Noisamultipliablenumberr"whereanumberyisdefinedasttmultipliable"whenaproduct.of.ufactorsisneverzerounlessoneofthefactorsiszero.Wecan?rot)ethataf,nitenumberisalwaysmultipliable,butwecannotprovethatanyinfinitenumberisso.Themultiplicativeaxiomisequivalenttotheassumptionthatallcardinalnumbersaremultipliable.Butinordertoidentifythereflexivewiththenon-inductive,ortodealwiththeproblemofthebootsandsocks,ortoshowthatanyprogressionofnumbersofthesecondclassisofthesecondclass,weonlyneedtheverymuchsmallerassumptionthatnoismultipliable.Itisnotimprobablethatthereismuchtobediscoveredinregardtothetopicsdiscussedinthepresentchapter.Casesmaybefoundwherepropositionswhichseemtoinvolvethemultiplicativeaxiomcanbeprovedwithoutit.Itisconceivablethatthemultiplicativeaxiominitsgeneralformmaybeshowntobefalse.Fromthispointofview,Zermelo'stheoremofiersthebesthope:thecontinuumorsomestillmoredenseseriesmightbeprovedtobeincapableofhavingitstermswellordered,whichwouldprovethemultiplicativeaxiomfalse,invirtueofzermelo'stheorem.Butsofar,nomethodofobtainingsuchresultshasbeendiscovered,andthesubjectremainswrappedinobscurity.\nCHAPTERXIIITHEAXIOMOFINFINITYANDLOGICALTYPESTsBaxiomofinfinityisanassumPtionwhichmaybeenunciatedasfollows:-"Ifnbeanyinductivecardinalnumber,thereisatleastoneclassofindividualshavingnterms."Ifthisistrue,itfollows,ofcourse,thattherearemanyclassesofindividualshavingnterms,andthatthetotalnumberofindividualsintheworldisnotaninductivenumber.For,bytheaxiom,thereisatleastoneclasshavingn+rterms'fromwhichitfollowsthattherearemanyclassesofntermsandthatnisnotthenumberofindividualsintheworld.Sincenisanyinductivenumber,itfollowsthatthenumberofindividualsintheworldmust(ifouraxiombetrue)exceedanyinductivenumber.Inviewofwhatwefoundintheprecedingchapter,aboutthepossibilityofcardinalswhichareneitherinductivenorreflexive,wecannotinferfromouraxiomthatthereareatleastt*oindividuals,unlessweassumethemultiplicativeaxiom.ButwedoknowthatthereareatleastNoclassesofclasses,sincetheinductivecardinalsareclassesofclasses,andformaprogressionifouraxiomistrue.Thewayinwhichtheneedforthisaxiomarisesmaybeexplainedasfollows:-OneofPeano'sassumptionsisthatnotwoinductivecardinalshavetheSamesuccessor,i.e.thatweshallnothavem+l:nltunlessffi-fl,if.marrdnareinductivecardinals.InChapterVIII.wehadoccasiontousewhatisvirtuallythesameastheaboveassumptionofPeano's,namely,that,ifaisaninductivecardinal,r3t\n|32fntrodactiontoMatltematicalPlrihsoplryzrisnotequalton{r.Itmightbethoughtthatthiscouldbeproved.Wecanprovethat,ifoisaninductiveclass,andnisthenumberofmembersofo,thenaisnotequalton{t.Thispropositioniseasilyprovedbyinduction,andmightbethoughttoimplytheother.Butinfactitdoesnot,sincetheremightbenosuchclassaso.Whatitdoesimplyisthis:Ifnisaninductivecardinalsuchthatthereisatleastoneclasshavingrzmembers,thenaisnotequalton+r.Theaxiomofinfinityassuresus(whethertrulyorfalsely)thatthereareclasseshavingamembers,andthusenablesustoassertthatnisnotequalton{t.Butwithoutthisaxiomweshouldbeleftwiththepossibilitythatnandnlrmightbothbethenull-class.Letusillustratethispossibilitybyanexample:Supposetherewereexactlynineindividualsintheworld.(Astowhatismeantbytheword"individualr"Imustaskthereadertobepatient.)Thentheinductivecardinalsfromoupto9wouldbesuchasweexpect,butro(definedasg*l)wouldbethenull-class.Itwillberememberedthatn+rmaybedefinedasfollowsin+risthecollectionofallthoseclasseswhichhaveaterrn*suchthat,whenristakenaway,thereremainsaclassofnterms.Nowapplyingthisdefinition,weseethat,inthecasesupposed,9*risaclassconsistingofnoclasses,f.a.itisthenull-class.ThesamewillbetrueoIg*2,orgenerallyofg*n,unlessniszero.Thusroandallsubsequentinductivecardinalsr,villallbeidentical,sincetheywillallbethenull-class.Insuchacasetheinductivecardinalswillnotformaprogression,norwillitbetruethatnotwohavethesamesuccessor,for9androwillbothbesucceededbythenull-class(robeingitselfthenull-class).Itisinordertopreventsucharithmeticalcatastrophesthatwerequiretheaxiomofinfinity.Asamatteroffact,solongaswearecontentwiththearith-meticoffiniteintegers,anddonotintroduceeitherinfiniteintegersorinfiniteclassesorseriesoffiniteintegersorratios,itispossibletoobtainalldesiredresultswithouttheaxiomofinfinity.Thatistosay,wecandealwiththeaddirion,multi-\nTlzeAxiomofInfinitlandLogicalTyesr33plication,andexponentiationoffiniteintegersandofratios,butwecannotdealwithinfiniteintegersorwithirrationals.Thusthetheoryofthetransfiniteandthetheoryofrealnumbersfailsus.Howthesevariousresultscomeaboutmustnowbeexplained.Assumingthatthenumberofindividualsintheworldisn,thenumberofclassesofindividualswillbezn.ThisisinvirtueofthegeneralpropositionmentionedinChapterVIII.thatthenumberofclassescontainedinaclasswhichhasnmembersiszn.Nowznisalwaysgreaterthann.Hencethenumberofclassesintheworldisgreaterthanthenumberofindividuals.If,now,wesupposethenumberofindividualstobe9taswedidjustno%thenumberofclasseswillbezs,i.e.5rz.Thusifwetakeournumbersasbeingappliedtothecountingofclassesinsteadoftothecountingofindividuals,ourarithmeticwillbenormaluntilwereach5rzithefirstnumbertobenullwillbe!rJ.Andifweadvancetoclassesofclassesweshalldostillbetter:thenumberofthemwillbe2512,^numberwhichissolargeastostaggerimagination,sinceithasaboutr53digits.Andifweadvancetoclassesofclassesofclasses,weshallobtainanumberrepresentedbyraisedtoaPowerwhichhasabout"rJ3digits;thenumberofdigitsinthisnumberwillbeaboutthreetimes10162.Inatimeofpapershortageitisundesirabletowriteoutthisnumber,andifwewantlargeroneswecanobtainthembytravellingfurtheralongthelogicalhierarchy.Inthiswayarryassignedinductivecardinalcanbemadetofinditsplaeeamongnumberswhicharenotnull,merelybytravellingalongthehierarchyforasufficientdistance.lAsregardsratios,wehaveaverysimilarstateofafiairs.Ifaratioy,lvistohavetheexpectedproperties,theremustbeenoughobjectsofwhateversortisbeingcountedtoinsurethatthenull-classdoesnotsuddenlyobtrudeitself.Butthiscanbeinsured,foranygivenratioy'fv,withouttheaxiomofrOnthissubjectseePilncipiaMathematica,vol.ii.xrzofr'.Onthecorrespondingproblemsasregardsratio,seeibid.,vol.iii.*3o3ff.\nr3+fnnoductiontoMatlrematicalPhilosoplt1infinity,bymerelytravellingupthehierarchyasufficientdistance.Ifwecannotsucceedbycountingindividuals,wecantrycountingclassesofindividuals;ifwestilldonotsucceed,wecantryclassesofclasses,andsoon.Ultimately,howeverfewindi-vidualstheremaybeintheworld,weshallreachastagewheretherearemanymorethanpobjects,whateverinductivenumberpmaybe.Eveniftherewerenoindividualsatall,thiswouldstillbetrue,fortherewouldthenbeoneclass,namely,thenull-class,zclassesofclasses(namely,thenull-classofclassesandtheclasswhoseonlymemberisthenull-classofindividuals),4classesofclassesofclasses,t6atthenextstage,65rfi6atthenextstage,andsoon.Thusnosuchassumptionastheaxiomofinfinityisrequiredinordertoreachanygivenratiooranygiveninductivecardinal.Itiswhenwewishtodealwiththewholeclassorseriesofinductivecardinalsorofratiosthattheaxiomisrequired.WeneedthewholeclassofinductivecardinalsinordertoestablishtheexistenceofNo,andthewholeseriesinordertoestablishtheexistenceofprogressions:fortheseresults,itisnecessarythatweshouldbeabletomakeasingleclassorseriesinwhichnoinductivecardinalisnull.Weneedthewholeseriesofratiosinorderofmagnitudeinordertodefinerealnumbersassegments:thisdefinitionwillnotgivethedesiredresultunlesstheseriesofratiosiscompact,whichitcannotbeifthetotalnumberofratios,atthestageconcerned,isfinite.Itwouldbenaturaltosuppose-asIsupposedmyselfinformerdays-that,bymeansofconstructionssuchaswe'havebeenconsidering,theaxiomofinfinitycouldbeproved.Itmaybesaid:Letusassumethatthenumberofindividualsisz,wherenmaybeowithoutspoilingourargument;thenifweformthecompletesetofindividuals,classes,classesofclasses,etc.,alltakentogether,thenumberoftermsinourwholesetwillben{zn{z\n,..adinf.,whichislr0.Thustakingallkindsofobjectstogether,andnot\nThe'lxiomofInfnitlandLogicalTlpesr35confiningourselvestoobjectsofanyonetyPe,weshallcertainlyobtainaninfiniteclass,andshallthereforenotneedtheaxiomofinfinity.Soitmightbesaid.Now,beforegoingintothisargument,thefirstthingtoobserveisthatthereisanairofhocus-pocusaboutit:somethingremindsoneoftheconjurerwhobringsthingsoutofthehat.Themanwhohaslenthishatisquitesuretherewasn'taliverabbitinitbefore,butheisatalosstosayhowtherabbitgotthere.Sothereader,ifhehasarobustsenseofreality,willfeelconvincedthatitisimpossibletomanufactureaninfinitecollectionoutofafinitecollectionofindividuals,thoughhemaybeunabletosaywheretheflawisintheaboveconstruction.Itwouldbeamistaketolaytoomuchstressonsuchfeelingsofhocus-Pocus;likeotheremotions,theymayeasilyleadusastray.Buttheyafiordapriruafaciegroundforscrutinisingverycloselyanyargumentwhicharousesthem.Andwhentheaboveargumentisscrutiniseditwill,inmyopinion,befoundtobefallacious,thoughthefallacyisasubtleoneandbynomeanseasytoavoidconsistently.Thefallacyinvolvedisthefallacywhichmaybecalled"con-fusionoftypes."Toexplainthesubjectof"types"fullywouldrequireawholevolume;moreover,itisthepurposeofthisbooktoavoidthosepartsofthesubjectswhicharestillobscureandcontroversial,isolating,fortheconvenienceofbeginners,thosepartswhichcanbeacceptedasembodyingmathematicallyascer-tainedtruths.Nowthetheoryoftypesemphaticallydoesnotbelongtothefinishedandcertainpartofoursubject:muchofthistheoryisstillinchoate,confused,andobscure.Buttheneedofsomedoctrineoftypesislessdoubtfulthanthepreciseformthedoctrineshouldtake;andinconnectionwiththeaxiomofinfinityitisparticulariyeasytoseethenecessityofsomesuchdoctrine.Thisnecessityresults,forexample,fromthe"conffadictionofthegreatestcardinal."WesawinChapterVIII.thatthenumberofclassescontainedinagivenclassisalwaysgreaterthanthe\nr36fntroductiontoMathematicalPhilosopltynumberofmembersoftheclass,andweinferredthatthereisnogreatestcardinalnumber.Butifwecould,aswesuggestedamomentago,addtogetherintooneclasstheindividuals,classesofindividuals,classesofclassesofindividuals,etc.,weshouldobtainaclassofwhichitsownsub-classeswouldbemembers.Theclassconsistingofallobjectsthatcanbecounted,ofwhateversort,must,iftherebesuchaclass,haveacardinalnumberwhichisthegreatestpossible.Sinceallitssub-classeswillbemembersofit,therecannotbemoreofthemthantherearemembers.Hencewearriveatacontradiction.WhenIfirstcameuponthiscontradiction,intheyearr9or,IattemptedtodiscoversomeflawinCantor'sproofthatthereisnogreatestcardinal,whichwegaveinChapterVIII.Apply-ingthisprooftothesupposedclassofallimaginableobjects,Iwasledtoanewandsimplercontradiction,namely,thefollowing:-Thecomprehensiveclassweareconsidering,whichistoembraceeverything,mustembraceitselfasoneofitsmembers.Inotherwords,ifthereissuchathingastteverythingr"thenttevery-thing"issomething,andisamemberoftheclass"everythirg."Butnormallyaclassisnotamemberofitself.Mankind,forexample,isnotaman.Formnowtheassemblageofallclasseswhicharenotmembersofthemselves.Thisisaclass:isitamemberofitselfornot?Ifitis,itisoneofthoseclassesthatarenotmembersofthemselves,i.e.itisnotamemberofitself.Ifitisnot,itisnotoneofthoseclassesthatarenotmembersofthemselves,i.e.itisamemberofitself.Thusofthetwohypo-theses-thatitis,andthatitisnot,amemberofitself-eachimpliesitscontradictory.Thisisacontradiction.Thereisnodifficultyinmanufacturingsimilarcontradictionsadlib.ThesolutionofsuchcontradictionsbythetheoryoftypesissetforthfullyinPrincipiaMathematica,randalso,morebriefly,inarticlesbythepresentauthorintheAmericanJournalrVol.i.,Introduction,chap.ii.,xrzand*20ivolii.,PrefatoryStatement.\nThez{xionofInfnitlandLogicalTypesr37ofMatltematics1andintheRevuedeMetapbysiqueetdeMorul.e.zForthepresentanoutlineofthesolutionmustsuffi.ce.Thefallacyconsistsintheformationofwhatwemaycallt'"impureclasses,i.t.classeswhicharenotPureasto"tyPr-"Asweshallseeinalaterchapter,classesarelogicalfictions,andastatementwhichappearstobeaboutaclasswillonlybesigni-ficantifitiscapableoftranslationintoaforminwhichnomentionismadeoftheclass.Thisplacesalimitationuponthewaysinwhichwhatarenominally,thoughnotreally,namesforclassescanoccursignificantly:asentenceorsetofsymbolsinwhichsuchpseudo-namesoccurinwrongwaysisnotfalse,butstrictlydevoidofmeaning.Thesuppositionthataclassis,orthatitisnot,amemberofitselfismeaninglessinjustthisway.Andmoregenerally,tosupposethatoneclassofindividualsisamember,orisnotamember,ofanotherclassofindividualswillbetosupposenonsense;andtoconstructsymbolicallyanyclasswhosernembersarenotallofthesamegradeinthelogicalhierarchyistousesymbolsinawaywhichmakesthemnolongersymboliseanything.Thusiftherearenindividualsintheworld,andznclassesofindividuals,wecannotformanewclass,consistingofbothindividualsandclassesandhavingn{znmembers.Inthiswaytheattempttoescapefromtheneedfortheaxiomofinfinitybreaksdown.IdonotPretendtohaveexplainedthedoctrineo{types,ordonemorethanindicate,inroughoutline,whythereisneedofsuchadoctrine.Ihaveaimedonlyatsayingjustsomuchaswasrequiredinordertoshowthatwecannotprot)etheexistenceofinfinitenumbersandclassesbysuchconjurer'smethodsaswehavebeenexamining.Thereremain,however,certainotherpossiblemethodswhichmustbeconsidered.VariousargumentsprofessingtoProvetheexistenceofinfiniteclassesaregiveninthePrinciplesofMathematics,Sffg(p.lSil.1"MathematicalLogrcasbasedontheTheoryofTypes,"vol.xxx.'r9o8,pp.zzz-262.2"Lesparadoxesdelalogique,"19o6,pp,62745o.\nr38[ntroductiontoMatlrematicalPltilosoplryInsofarastheseargumentsassumethat,if.nisaninductivecardinal,aisnotequalton{r,theyhavebeenalreadydealtwith.Thereisanargument,suggestedbyapassageinPlato'sParmenides,totheeffectthat,ifthereissuchanumberasr,thenrhasbeing;butrisnotidenticalwithbeing,andthereforerandbeingaretwo,andthereforethereissuchanumberasz,andztogetherwithrandbeinggivesaclassofthreeterms,andsoon.Thisargumentisfallacious,partlybecause"being"isnotatermhavinganydefinitemeaning,andstillmorebecause,ifadefinitemeaningwereinventedforit,itwouldbefoundthatnumbersdonothavebeing-theyare,infact,whatarecalledttlogicalfictionsrttasweshallseewhenwecometoconsiderthedefinitionofclasses.Theargumentthatthenumberofnumbersfromoton(bothinclusive)iszfrdependsupontheassumptionthatuptoandincludingntonumberisequaltoitssuccessor,which,aswehaveseen,willnotbealwaystrueiftheaxiomofinfinityisfalse.Itmustbeunderstoodthattheequationn:fl*r,whichmightbetrueforafinitenif.nexceededthetotalnumberofindividualsintheworld,isquitedifierentfromthesameequationasappliedtoareflexivenumber.Asappliedtoareflexivenumber,itmeansthat,givenaclassofaterms,thisclassis"similart,tothatobtainedbyaddinganotherterm.Butasappliedroanumberwhichis,toogreatfortheactualworld,itmerelymeansthatthereisnoclassofzindividuals,andnoclassoIn{rindi-viduals;itdoesnotmeanthat,ifwemountthehierarchyoftypessufficientlyfartosecuretheexistenceofaclassofaterms,weshallthenfindthisclass"similar"tooneof.nfrterms,forifaisinductivethiswillnotbethecase,quiteindependentlyofthetruthorfalsehoodoftheaxiomofinfinity.ThereisanargumentemployedbybothBolzano1andDede-kind2toprovetheexistenceofreflexiveclasses.Theargument,inbrief,isthis:AnobjectisnotidenticalwiththeideaoftherBolzano,Paradoxiend,esUncnillicken,t3.rDedekind,Wassinilund,wassotrlenilieZahlenINo.66.\nTheAxiomofInfnityandLogicalTypesr3gobject,butthereis(atleastintherealmofbeing)anideaofanyobject.Therelationofanobjecttotheideaofitisone-one,andideasareonlysomeamongobjects.Hencetherelation"ideaof"constitutesareflexionofthewholeclassofobjectsintoapartofitself,namely,intothatpartwhichconsistsofideas.Accordingly,theclassofobjectsandtheclassofideasarebothinfinite.Thisargumentisinteresting,notonlyonitsownaccount,butbecausethemistakesinit(orwhatIjudgetobemistakes)areofakindwhichitisinstructivetonote.Themainerrorconsistsinassumingthatthereisanideaofeveryobject.Itis,ofcourse,exceedinglydifficulttodecidewhatisttmeantbyan"ideaIbutletusassumethatweknow.Wearethentosupposethat,startingG"y)withSocrates,thereistheideaofSocrates,andsoonadinf.Nowitisplainthatthisisnotthecaseinthesensethatalltheseideashaveactualempiricalexistenceinpeople'sminds.Beyondthethirdorfourthstagetheybecomemythical.Iftheargumentistobeupheld,the"ideas"intendedmustbePlatonicideaslaidupinheaven,forcertainlytheyarenotohearth.Butthenitatoncebecomesdoubtfulwhethertherearesuchideas.Ifwearetoknowthatthereare,itmustbeonthebasisofsomelogicaltheory,provingthatitisnecessarytoathingthatthereshouldbeanideaofit.Wecertainlycannotobtainthisresultempirically,orapplyit,tt-1hgasDedekinddoes,to"meineGedankenweltworldofmythoughts.Ifwewereconcernedtoexaminefullytherelationofideaandobject,weshouldhavetoenteruponanumberofpsychologicalandlogicalinquiries,whicharenotrelevanttoourmainpurpose.Butafewfurtherpointsshouldbenoted.If"idea"istobeunderstoodlogically,itmaybeidenticalvnththeobject,oritmaystandforadesmiption(inthesensetobeexplainedinasubsequentchapter).Intheformercasetheargumentfails,becauseitwasessentialtotheproofofreflexivenessthatobjectandideashouldbedistinct.Inthesecondcasetheargumentalsofails,becausetherelationofobjectanddescriptionisnot\n|+oIntroductiontoMatltematicalPltilosaplryone-one:thereareinnumerablecorrectdescriptionsofanygivenobject.Socrates(t.5.)maybedescribedas"themasterofPlator"oras"thephilosopherwhodrankthehemlockr"oras"thehusbandofXantippe."If-totakeuptheremaininghypothesis-(6idea"istobeinterpretedpsychologically,itmustbemaintainedthatthereisnotanyonedefinitepsychologicalentitywhichcouldbecalledtheideaoftheobject:therearein-numerablebeliefsandattitudes,eachofwhichcouldbecalledanideaoftheobjectinthesenseinwhichwemightsay"myideaofSocratesisquitedifferentfromyours,"butthereisnotanycentralentity(exceptSocrateshimself)tobindtogethervariousttideasofSocratesrttandthusthereisnotanysuchone-onerela-tionofideaandobjectastheargumentsupposes.Nor,ofcourse,aswehavealreadynoted,isittruepsychologicallythatthereareideas(inhoweverextendedasense)ofmorethanatinyproportionofthethingsintheworld.Forallthesereasons,theaboveargumentinfavourofthelogicalexistenceofreflexiveclassesmustberejected.Itmightbethoughtthat,whatevermaybesaidof.logicalarguments,theernpiricalargumentsderivablefromspaceandtime,thediversityofcolours,etc.,arequitesuficienttoprovetheactualexistenceofaninfinitenumberofparticulars.Idonotbelievethis.Wehavenoreasonexceptprejudiceforbeliev-ingintheinfiniteextentofspaceandtime,atanyrateinthesenseinwhichspaceandtimearephysicalfacts,notmathematicalfictions.Wenaturallyregardspaceandtimeascontinuous,or,atleast,ascompact;butthisagainismainlyprejudice.Thet'quantat'theoryofinphysics,whethertrueorfalse,illustratesthefactthatphysicscanneveraffordproofofcontinuity,thoughitmightquitepossiblyafforddisproof.Thesensesarenotsufficientlyexacttodistinguishbetweencontinuousmotionandrapidfiscretesuccession,asanyonemaydiscoverinacinema.Aworldinwhichallmotionconsistedofaseriesofsmallfinitejerkswouldbeempiricallyindistinguishablefromoneinwhichmotionwascontinuous.Itwouldtakeuptoomuchspaceto\nTheAxiomofInfinityandLogicalTypesr+rdefendthesethesesadequately;forthepresentIammerelysuggestingthemforthereader'sconsideration.Iftheyarevalid,itfollowsthatthereisnoempiricalreasonforbelievingthenumberofparticularsintheworldtobeinfinite,andthattherenevercanbeIalsothatthereisatpresentnoempiricalreasontobelievethenumbertobefinite,thoughitistheoreticallyconceivablethatsomedaytheremightbeevidencepoindog,thoughnotconclusively,inthatdirection.Fromthefactthattheinfiniteisnotself-contrafictory,butisalsonotdemonstrablelogically,wemustconcludethatnothingcanbeknownaprioriastowhetherthenumberofthingsintheworldisfiniteorinfinite.Theconclusionis,therefore,toadoptaLeibnizianphraseology,thatsomeofthepossibleworldsarefinite,someinfinite,andwehavenomeansofknowingtowhichofthesetwokindsouractualworldbelongs.Theaxiomofinfinitywillbetrueinsomepossibleworldsandfalseinothers;whetheritistrueorfalseinthisworld,wecannottell.Throughoutthischapterthesynonyms"individual"and((particular"havebeenusedwithoutexplanation.Itwouldbeimpossibletoexplainthemadequatelywithoutalongerdisquisi-tiononthetheoryoftypesthanwouldbeappropriatetothepresentwork,butafewwordsbeforeweleavethistopicmaydosomethingtodiminishtheobscuritywhichwouldotherwiseenvelopthemeaningoIthesewords.Inanordinarystatementwecandistinguishaverb,expressinganattributeorrelation,fromthesubstantiveswhichexpressthesubjectoftheattributeorthetermsoftherelation."Casar'6lived"ascribesanattributetoCasar;BrutuskilledCasar"expressesarelationbetweenBrutusandCasar.Usingtheword6'subjectttinageneralisedsense,wemaycallbothBrutusandCasarsubjectsofthisproposition:thefactthatBrutusisgram-maticallysubjectandCasarobjectislogicallyirrelevant,sincettthesameoccurrencemaybeexpressedinthewordsCasarwaskilledbyBrutus,"whereCasaristhegrammaticalsubject.\n|+zfnroductiont0MatltematicalPlrilosopltyThusinthesimplersortofpropositionweshallhaveanattributettorrelationholdingoforbetweenone,twoormore"subjectsintheextendedsense.(Arelationmayhavemorethantwotermsie.g."AgivesBtoC"isarelationof.threeterms.)Nowitoftenhappensthat,onacloserscrutiny,theapparentsubjectsarefoundtobenotreallysubjects,buttobecapableofanalysis;theonlyresultofthis,however,isthatnewsubjectstaketheirplaces.Italsohappensthattheverbmaygrammaticallybemadesubjectie.g.wemaysay,"KillingisarelationwhichholdsbetweenBrutusandCasar."Butinsuchcasesthegrammarismisleading,andinastraightforwardstatement,followingtherulesthatshouldguidephilosophicalgrammar,BrutusandCasarwillappearasthesubjectsandkillingastheverb.Wearethusledtotheconceptionoftermswhich,whentheyoccurinpropositions,canonlyoccurassubjects,andneverinanyotherway.ThisispartoftheoldscholasticdefinitionoLsubstance;butpersistencethroughtime,whichbelongedtothatnotion,formsnopartofthenotionwithwhichwearecon-ttttcerned.Weshalldefinepropernamesasthosetermswhichcanonlyoccurassubiectsinpropositions(using"subject"intheextendedsensejustexplained).Weshallfurtherdefine66"individuals"orparticulars"astheobjectsthatcanbenamedbypropernames.(Itwouldbebettertodefinethemdirectly,ratherthanbymeansofthekindofsymbolsbywhichtheyaresymbolised;butinordertodothatweshouldhavetoplungedeeperintometaphysicsthanisdesirablehere.)Itit,ofcourse,possiblethatthereisanendlessregress:thatwhateverappearsasaparticularisreally,oncloserscrutiny,aclassorsomekindofcomplex.Ifthisbethecase,theaxiomofinfinitymustofcoursebetrue.Butifitbenotthecase,itmustbetheoreticallypossibleforanalysistoreachultimatesubjects,anditisthesethatgivethemeaningof"particulars"or"individuals."Itistothenumberofthesethattheaxiomofinfinityisassumedtoapply.Ifitistrueofthem,itistrue\nThe'lxiornofInfnitlandLogicalTypesr+3ofclassesofthem,andclassesofclassesofthem,andsoon;similarlyifitisfalseofthem,itisfalsethroughoutthishierarchy.Henceitisnaturaltoenunciatetheaxiomconcerningthemratherthanconcerninganyotherstageinthehierarchy.Butwhethertheaxiomistrueorfalse,thereseemsnoknownmethodofdiscovering.\nCHAPTERXIVINCOMPATIBILITYANDTHETHEORYOFDEDUCTIONWuhavenowexplored,somewhathastilyitistrue,thatPartofthephilosophyofmathematicswhichdoesnotdemandacriticalexaminationoftheideaofclass.Intheprecedingchapter,however,w€foundourselvesconfrontedbyproblemswhichmakesuchanexaminationimperative.Beforewecanundertakeit,wemustconsidercertainotherpartsofthephilos-ophyofmathematics,whichwehavehithertoignored.Inasynthetictreatment,thepartswhichweshallnowbeconcernedwithcomefirst:theyaremorefundamentalthananythingthatwehavediscussedhitherto.Threetopicswillconcernusbeforewereachthetheoryof,classes,namely:(r)thetheoryofdeduction,(z)propositionalfunctions,(l)descriptions.Ofthese,thethirdisnotlogicallypresupposedinthetheoryofclasses,butitisasimplerexampleofthekind,oftheorythatisneededindealingwithclasses.Itisthefirsttopic,thetheoryofdeduction,thatwillconcernusinthepresentchaPter.Mathematicsisadeductivescience:startingfromcertainpremisses,itarrives,byastrictProcessofdeduction,atthevarioustheoremswhichconstituteit.Itistruethat,inthePast,mathematicaldeductionswereoftengreatlylackinginrigour;itistruealsothatperfectrigourisascarcelyattainableideal.Nevertheless,insofarasrigourislackinginamathematicalproof,theproofisdefective;itisnodefencetourgethatcommonsenseshowstheresulttobecorrect,forifweweretorelyuPonthat,itwouldbebettertodispensewithargumentaltogether,r++\n[ncompatibiliryandtlteTlreoryofDedactionr+Sratherthanbringfallacytotherescueofcommonsense.Noappealtocommonsense,orttintuitionr"oranythingexceptstrictdeductivelogic,oughttobeneededinmathematicsafterthepremisseshavebeenlaiddown.Kant,havingobservedthatthegeometersofhisdaycouldnotprovetheirtheoremsbyunaidedargument,butrequiredanappealtothefigure,inventedatheoryofmathematicalreasoningaccordingtowhichtheinferenceisneverstrictlylogical,butalwaysrequiresthesupportofwhatiscalled"intuition."Thewholetrendofmodernmathematics,withitsincreasedpursuitofrigour,hasbeenagainstthisKantiantheory.ThethingsinthemathematicsofKant'sdaywhichcannotbeproved,cannotbeknown-forexample,theaxiomofparallels.Whatcanbeknown,inmathematicsandbymathe-maticalmethods,iswhatcanbededucedfrompurelogic.Whatelseistobelongtohumanknowledgemustbeascertainedother-wise-empirically,throughthesensesorthroughexperienceinsomeform,butnotapriori.ThepositivegroundsforthisthesisaretobefoundinPrincipiaMathematica,passirn1acontroversialdefenceofitisgiveninthePrinciplesofMatbe-matics.Wecannotheredomorethanreferthereadertothoseworks,sincethesubjectistoovastforhastytreatment.Mean-while,weshallassumethatallmathematicsisdeductive,andproceedtoinquireastowhatisinvolvedindeduction.Indeduction,wehaveoneormorepropositionscalledpre-tnisses,fromwhichweinferapropositioncalledtheconclusion.Forourpurposes,itwillbeconvenient,whenthereareoriginallyseveralpremisses,toamalgamatethemintoasingleproposition,soastobeabletospeakof.thepremissaswellasofthecon-clusion.Thuswemayregarddeductionasaprocessbywhichwepassfromknowledgeofacertainproposition,thepremiss,toknowledgeofacertainotherproposition,theconclusion.Butweshallnotregardsuchaprocessaslogicaldeductionunlessitiscorrect,i.e.unlessthereissucharelationbetweenpremissandconclusionthatwehavearighttobetrievetheconclusion\nr+6fntroductiontoMarhematicalPlriloso7lr1ifweknowthepremisstobetrue.Itisthisrelationthatischieflyofinterestinthelogicaltheoryofdeduction.Inordertobeablevalidlytoinferthetruthofaproposition,wemustknowthatsomeotherpropositionisuue,andthatthereisbetweenthetwoarelationofthesortcalled"implicationr"i.e.that(aswesay)thepremiss"implies"theconclusion.(Weshalldefinethisrelationshortly.)Orwemayknowthatacertainotherpropositionisfalse,andthatthereisarelationbetweenIthetwoofthesortcalled"disjunctionr"expressedby"porqr"sothattheknowledgethattheoneisfalseallowsustoinferthattheotheristrue.Again,whatwewishtoinfermaybethefalsehood.ofsomeproposition,notitstruth.Thismaybeinferredfromthetruthofanotherproposition,providedweknowthatthetwoare"incompatibler"i.e,thatifoneistrue,theotherisfalse.Itmayalsobeinferredfromthefalsehoodofanotherproposition,injustthesamecircumstancesinwhichthetruthoftheothermighthavebeeninferredfromthetruthoftheone;i.e.fromthefalsehoodofpwemayinferthefalsehoodofg,whengimpliesp.Allthesefourarecasesofinference.Whenourttmindsarefixeduponinference,itseemsnaturaltotakeimpli-cation"astheprimitivefundamentalrelation,sincethisistherelationwhichmustholdbetweenpandgifwearetobeabletoinferthetruthofgfromthetrutbofP.Butfortechnicalreasonsthisisnotthebestprimitiveideatochoose.Beforeproceedingtoprimitiveideasanddefinitions,letusconsiderfurtherthevariousfunctionsofpropositionssuggestedbytheabove-mentionedrelationsofpropositions.Thesimplestofsuchfunctionsisthenegative,"not1t."Thisisthatfunctionofpwhichistruewhen2isfalse,andfalsewhenpistrue.Itisconvenienttospeakofthetruthofapro-2position,oritsfalsehood,asits"truth-value";i.e.trutltisthe"truth-value"ofatrueproposition,andifalseboodofafalseone.Thusnot?hastheoppositetruth-valuetop.1Weshallusetheletterspropositions.f,g,t,s,ItodenotevariableaThistermisduetoFrege.\nfncompatibilityandtlreTlteoryofDeductionr+7Wemaytakenextdisiunctionr"?or{."Thisisafunctionwhosetruth-valueistruthwhenpistrueandalsowhengistrue,butisfalsehoodwhenbothpandqarefalse.Nextwemaytakeconiunctiofl,"pandq."Thishastruthforitstruth-valuewhenpandgarcbothtrue;otherwiseithasfalsehoodforitstruth-value.Takenextincompatibility,i.e."pandqarcnotbothtrue."Thisisthenegationofconjunction;itisalsothedisjunction((ofthenegationsofpandg,i.e.itisnot-pornot-q."ftstruth-valueistruthwhenpisfalseandlikewisewhengisfalse;itstruth-valueisfalsehoodwhenpandgarebothtrue.ccLasttakeimplication,i.e."pimpliesqr"otifp,theng."ThisistobeunderstoodinthewidestsensethatwillallowustoinferthetruthofgifweknowthetruthoIp.Thusweinter-pretitasmeaning:"Unlesspisfalse,gistruert'ort'eitherpisfalseorgistrue."(Thefactthat"implies"iscapableofothermeaningsdoesnotconcernus;thisisthemeaningwhich'cisconvenientforus.)Thatistosafrpimpliesg"istomean(cnot-porq,":itstruth-valueistobetruthifpisfalse,likewiseifgistrue,andistobefalsehoodif.pistrueand.gisfalse.Wehavethusfivefunctions:negation,disjunction,conjunction,incompatibility,andimplication.Wemighthaveaddedothers,ccforexample,jointfalsehood,not-pandnot-gr"buttheabovefivewillsuffice.Negationdiffersfromtheotherfourinbeingafunctionofoneproposition,whereastheothersarefunctionsof.two.Butallfiveagreeinthis,thattheirtruth-valuedependsonlyuponthatofthepropositionswhicharetheirarguments.Giventhetruthorfalsehoodofptorof.pandg(asthecasemaybe),wearegiventhetruthorfalsehoodofthenegation,disjunc-tion,conjunction,incompatibility,orimplication.Afunctionofpropositionswhichhasthispropertyiscalleda"truth-function."Thewholemeaningofatruth-functionisexhaustedbythestatementofthecircumstancesunderwhichitistrueorfalse."Not7r"forexample,issimplythatfunctionofpwhichistruewhenpisfalse,andfalsewhenpistrue:thereisnofurther\nr48[ntroductiontoMathematicalPltilosopltymeaningtobeassignedtoit.Thesameappliesto"porqandtherest.Itfollowsthattwotruth-functionswhichhavethesametruth-valueforallvaluesoftheargumentareindis-tinguishable.Forexample,"pandq"isthenegationofccttnot-pornot-gandsiceoersa.;thuseitherofthesemaybedej.nedasthenegationoftheother.Thereisnofurthermeaninginatruth-functionoverandabovetheconditionsunderwhichitistrueorfalse.Itisclearthattheabovefivetruth-functionsarenotallinde-pendent.Wecandefinesomeofthemintermsofothers.Thereisnogreatdifficultyinreducingthenumbertotwo;thetwochoseninPrincipiaMathematicaarenegationanddisjunction.Implicationisthendefinedas"not'porq";incompatibilityKasnot-pornot-g"Iconjunctionasthenegationofincompati-1thatwecanbecontentbility.ButithasbeenshownbyShefier2thatthisenableswithonepimitiveideaforallfive,andbyNicodustoreducetheprimitivepropositionsrequiredinthetheor;'ofdeductiontotwonon-formalprinciplesandoneformalone.Forthispurpose,wemaytakeasouroneindefinableeitherincompatibilityorjointfalsehood.Wewillchoosetheformer.Ourprimitiveidea,now,isacertaintruth-functioncalled'6incompatibilityr"whichwewilldenotebyplq.Negationcanbeatoncedefinedastheincompatibilityofaproposition(3withitself,i.e.not-p"isdefinedas"plpl'Disjunctionistheincompatibilityofnot-pandnot-ll,i.e-itis(plillllil-ImplicationistheincomPatibilityof-pandnot-{,i.e.pl(qld.Conjunctionisthenegationofincompatibility,i.e.itit(plill(plq).Thusallourfourotherfunctionsaredefinedintermsofincompatibility.Itisobviousthatthereisnolimittothemanufactureoftruth-functions,eitherbyintroducingmoreargumentsorbyrepeatingarguments.Whatweareconcernedwithistheconnectionofthissubjectwithinference.tTrans.Am.Math.Soc.,vol.xiv.pp.48r-488.8Ptoc.Camb.Phil.Soa.,vol.xix.,i.,Januaryr9\?.\nIncompatibilityandtheTlteoryofDeductionr+gIfweknowthatpisuueandthatpimpliesbwecanproceedtoassert{.Thereisalwaysunavoidablysornethingpsycho-logicalaboutinference:inferenceisamethodbywhichwearriveatnewknowledge,andwhatisnotpsychologicalaboutitistherelationwhichallowsustoinfercorrectly;buttheactualpassagefromtheassertionofptotheassertionofgisapsychologicalprocess,andrvemustnotseektorepresentitinpurelylogicalterms.Inmathematicalpractice,whenweinfer,wehavealwayssomeexpressioncontainingvariablepropositions,saypandq,whichisknown,invirtueofitsform,tobetrueforallvaluesoIpandIiwehavealsosomeotherexpression,partoftheformer,whichisalsoknowntobetrueforallvaluesolpandq;andinvirtueoftheprinciplesofinference,weareabletodropthispartofouroriginalexpression,andassertwhatisleft.Thissomewhatabstractaccountmaybemadeclearerby^fewexamples.LetusassumethatweknowthefiveformalprinciplesofdeductionenumeratedinPrincipiaMathematica.(M.Nicodhasreducedthesetoone,butasitisacomplicatedproposition,wewillbeginwiththefive.)Thesefivepropositionsareasfollows:-(r)"porp"impltesp-i.e.ifeitherpistrueorpistrue,thenpistrue.(z)gimplies"po,q"-i.e.thedisjunction"potg"istruewhenoneofitsalternativesistrue.(3)"potg"implies"gorp."Thiswouldnotberequiredifwehadatheoreticallymoreperfectnotation,sinceintheconceptionofdisjunctionthereisnoorderinvolved,sothat"potg"and"qorp"shouldbeidentical.Butsinceoursymbols,inanyconvenientform,inevitablyintroduceanorder,weneedsuitableassumptionsforshowingthattheorderisirrelevant.(+)Ifeitherpismueor"gorr"istrue,theneithergistrueor"porr"istrue.(Thetwistinthispropositionservestoincreaseitsdeductivepower.)\nrjofntroductiontoMath.ematicaPltilosopltyG)Ifgimpliesr,then"po,g"implies"Porr."TheseatetheformalprinciplesofdeductionemployedinPrincipiaMathematica.Aformalprincipleofdeductionhasadoubleuse,anditisinordertomakethisclearthatwehavecitedtheabovefivepropositions.Ithasauseasthepremissofaninference,andauseasestablishingthefactthatthepre-missimpliestheconclusion.Intheschemaofaninferencewehaveapropositionp,andaproPosition"pimplies(lr"fromwhichweinferg.Nowwhenweareconcernedwiththeprinci-plesofdeduction,ourapparatusofprimitivepropositionshastoyieldboththepandthe"pimpliesg"ofourinferences.Thatistosay,ourrulesofdeductionaretobeused,notonlyasrules,whichistheiruseforestablishitg"pimpliesgr"butalsoassubstantivepremisses,i.e.asthe2ofourschema.Suppose,forexample,wewishtoprovethatifpimpliesg,thenifgimpliesritfollowsthatpimpliesr.Wehaveherearelationofthreepropositionswhichstateimplications.Putpr:PimpliesI,?z:Iimpliesr,and?s:Pimpliesr.Thenwehavetoprovethatp,impliesthatp,impliespr.Nowtakethefifthofouraboveprinciples,substitutenot?forp,ccandrememberthattot-porg"isbydefinitionthesameas"pimpliesg."Thusourfifthprincipleyields:'2'2"Ifqimpliesr,thenimpliesg'impliesimpliesrr'"i.e."p,impliesthatfuimpliespu."CallthisProPo-sitionA.Butthefourthofourprinciples,whenwesubstitutenot-p,not-q,forpandg,andrememberthedefinitionofimplication,becomes:"Ifpimpliesthatgimpliesr,thengimpliesthatpimpliesr."WritingprinplaceofP,Prinplaceofq,andprinplaceofr,thisbecomes:"Ifprimpliesthatprimpliespr,thenplimpliesthatp,impliespr."CallthisB.\nfncompatibilityandtheTlteoryofDeductionr5rNowweprovedbymeansofourfifthprinciplethat"primptiesthatp1implies?sr"whichwaswhatwecalledA.Thuswehavehereaninstanceoftheschemaofinference,sinceArepresentsthepofourscheme,andBrepresentsthettpimplirtOt'r,Hencewearrive^tg,namely,'p,impliesthatp,impliesps,"whichwasthepropositiontobeproved.Inthisproof,theadaptationofourfifthprinciple,whichyieldsA,occursasasubstantivepremiss;whiletheadaptationofourfourthprinciple,whichyieldsB,isusedtogivetheformoftheinference.Theformalandmaterialemploymentsofpremissesinthetheoryofdeductionarecloselyintertwined,anditisnotveryimportanttokeepthemseparated,providedwerealisethattheyareintheorydistinct.Theearliestmethodofarrivingatnewresultsfromapremissisonewhichisillustratedintheabovededuction,butwhichitselfcanhardlybecalleddeduction.Theprimitivepropositions,whatevertheymaybe,aretoberegardedasassertedforallpossiblevaluesofthevariablepropositionsp,g,rwhichoccurinthem.Wemaythereforesubstitutefor(r"y)panyexpressionwhosevalueisalwaysaproposition,e.g.not-p,"simpliest,"andsoon.Bymeansofsuchsubstitutionswereallyobtainsetsofspecialcasesofouroriginalproposition,butfromaprac-ticalpointofviewweobtainwhatarevirtuallynewpropositions.Thelegitimacyofsubstitutionsofthiskindhastobeinsuredbymeansofanon-formalprincipleofinference.lWemaynowstatetheoneformalprincipleofinferencetowhichM.Nicodhasreducedthefivegivenabove.Forthispurposewewillfirstshowhowcertaintruth-functionscanbedefinedintermsofincompatibility.Wesawalreadythatplklg)means"pimpliesf."rNosuchprincipleisenunciatedinPrincipiaMathematicaorinM.Nicod'sarticlementionedabove.Butthiswouldseemtobeanomission,\nrSzfntroductiontoMatltematicalPltilosoplt1Wenowobservethatplklr)means"Pimphesbothgandr."Forthisexpressionmeans"pi,incompatiblewiththeincom-patibilityof.gandrr"i.e."pimpliesthatgandratenotincom-patibler"i.e."2impliesthatqandrareboth11ug"-for'aswesaw,theconjunctionofqandristhenegationoftheirincompatibility.ObservenextthatIl(tlt)means"Iimpliesitself."Thisisaparticularcaseo!?|(qld.LetuswriteIto,rthenegationof2;thusp/swillmeanthenegationofpfs,i.e.itwillmeantheconjunctionofpands.Itfollowsthat$lq)lpI'expressestheincompatibilityofslqwiththeconjunctionofpands;inotherwords,itstatesthatifpandrarebothtrue,slqisfalse,i.e.sandqarebothtrue;instillsimplerwords,itstatesthatpandsjointlyimplysandgjointly.Now,putP-p|(qlr),T:tl(tlt),_e-('ldlpl'.ThenM.Nicod'ssoleformalprincipleofdeductionispl.rle,inotherwords,PimpliesbothzrandQ.Heemploysinadditiononenon-formalprinciplebelongingtothetheoryoftypes(whichneednotconcernus),andonecorrespondingtotheprinciplethat,givenp,andgiventhatpimplies%wecanassertg.Thisprincipleis:thengistrue."From"IfplQlg)istrue,andpistrue,thisapparatusthewholetheoryofdeductionfollows,excePtinsofarasweareconcernedwithdeductionfromortotheexistenceortheuniversaltruthof"propositionalfunctionsr"whichweshallconsiderinthenextchapter.Thereis,ifIarlnotmistaken,acertainconfusioninthe\nIncompatibilityandtlteTlteoryofDeauctionrj3mindsofsomeauthorsastotherelation,betweenpropositions,invirtueofwhichaninferenceisvalid.Inorderthatitmaybesali.dtoinfergfromp,itisonlynecessarythatpshouldbe((trueandthatthepropositionnot-potq"shouldbetrue.Wheneverthisisthecase,itisclearthatgmustbetrue.ButinferencewillonlyinfacttakeplacewhenthepropositionKnotnotg"isknownotherwisethanthroughknowledgeofnot-porknowledgeof.q.Wheneverpisfalse,'cnot-porq,,istrue,butisuselessforinference,whichrequiresthatpshouldbetrue.Whenevergisalreadyknowntobetrue,Knot-porg"isofcoursealsoknowntobetrue,butisagainuselessforinference,sincegisalreadyknown,andthereforedoesnotneedtobeinferred.Infact,inferenceonlyariseswhenc'not-porq,',canbeknownwithoutourknowingalreadywhichofthetwoalternativesitisthatmakesthedisjunctiontrue.Now,thecircumstancesunderwhichthisoccursarethoseinwhichcertainrelationsofformexistbetweenpandq.Forexample,weknowthatifrimpliesthenegationofs,thensimpliesthenegationof.r.Between"rimpliesnot-J"andttsimpliesnot-rttthereisaformalrelationwhichenablesustoknowthatthefirstimpliesthesecond,withouthavingfirsttoknowthatthefirstisfalseortoknowthatthesecondistrue.Itisundersuchcircum-stancesthattherelationofimplicationispracticallyusefulfordrawinginferences.Butthisformalrelationisonlyrequiredinorderthatwemaybeabletohnowthateitherthepremissisfalseortheconclusionistrue.Itisthetruthof"r.ot+org"thatisrequiredforthevalidityof.theinferenceIwhatisreguiredfurtJrerisonlyrequiredforthepracticalfeasibilityoftheinference.ProfessorC.I.LewisIhasespeciallystufiedthenarrower,formalrelationwhichwemaycall"formaldeducibilityl'Heurgesthatthewiderrelation,thatexpressedby"not-porgr"shouldnotbettcalledimplication."Thatir,however,amatterofwords.1SeeMdnil,vol.xxi.,rgt2,pp.Szz-53r;andvol.xxiii.,rgr4,pp.240447.\nr5+fntoductiontoMathernaticalPhilosopltyProvidedouruseofwordsisconsistent,itmatterslittlehowwedefinethem.TheessentialpointofdifierencebetweenthetheorywhichIadvocateandthetheoryadvocatedbyProfessorLewisisthis:Hemaintainsthat,whenoneProPositiongis,,formallydeducible"fromanotherp,therelationwhichwe6(perceivebetweenthemisonewhichhecallsstrictimplicatiodr"whichisnottherelationexpressedby"not-porq,"butanarrowerrelation,holdingonlywhentherearecertainformalconnectionsbetweenpandq.Imaintainthat,whetherornottherebesucharelationashespeaksof,itisinanycaseonethatmathe-maticsdoesnotneed,andtlereforeonethat,ongeneralgroundsofeconomy,oughtnottobeadmittedintoouraPParatusoffundamentalnotions;that,whenevertherelationof"formaldeducibility"holdsbetweentwopropositions,itisthecasethatwecanseelfiateitherthefirstisfalseorthesecondtrue'andthatnothingbeyondthisfactisnecessarytobeadmittedintoourpremisses;andt}at,finally,thereasonsofdetailwhichProfessorLewisadducesagainsttheviewwhichIadvocatecanallbemetindetail,anddependfortheitplausibilityuPonacovertandunconsciousassumptionofthepointofviewwhichIreject.Iconclude,therefore,thatthereisnoneedtoadmitasafunda-mentalnotionanyformofimplicationnotexpressibleasatruth-function.\nCHAPTERXVPROPOSITIONALFUNCTIONSWnru,intheprecedingchapter,wewerediscussingpropositions,wefidnotattempttogiveadefinitionoftheword"proposition."Butalthoughthewordcannotbeformallydefined,itisnecessarytosaysomethingastoitsmeaning,inordertoavoidtheverycommonconfusionwith"propositionalfunctionsrt'whicharetobethetopicofthepresentchapter.Wemeanby^"proposition"primarilyaformofwordswhichexpresseswhatiseithertrueorfalse.Isay"primarilyr"becauseIdonotwishtoexcludeotherthanverbalsymbols,orevenmerethoughtsiftheyhaveasymboliccharacter.ButIthinktheword"proposition"shouldbelimitedtowhatfray,insomesense,becalledttsymbolsr"andfurthertosuchsymbolsasgiveexpressiontotruthandfalsehood.Thus"twoandtwottarefour"andtwoandtwoarefive"willbepropositions,ttttandsowillSocratesisaman"andSocratesienotaman.ttThestatement:"Whatevernumbersdandbmaybe,(alb)z:az{zab{bt"isaproposition;butthebareformula'3(aql1z:azlzab{bz"aloneisnot,sinceitassertsnothingdefiniteunlesswearefurthertold,orledtosuppose,thataandbarctohaveallpossiblevalues,oraretohavesuch-and-suchvalues.Theformeroftheseistacitlyassumed,asarule,intheenunciationofmathematicalformula,whichthusbecomepropositions;butifnosuchassumptionweremade,th.ywouldbe',proposi-tionalfunctions."A"propositionalfunctionrttinfact,isanexpressioncontaioingoneormoreundeterminedconstituents,155\nr56fntroductiontoMatltematicalPhilosopltysuchthat,whenvaluesareassignedtotheseconstituents,theexpressionbecomesaProPosition.Inotherwords,itisafunctionwhosevaluesareProPositions.Butthislatterdefinitionmustbeusedwithcaution.Adescriptivefunction,e.g."thehardestpropositioninA'smathematicaltreatise,"willnotbeaPro-positionalfunction,althoughitsvaluesareProPositions.Butinsuchacasethepropositionsareonlydescribed:inaproposi-tionalfunction,thevaluesmustactuallyenunciatepropositions.Examplesofpropositionalfunctionsareeasytogive:"xishuman"isaproPositionalfunction;solongasfrremainsundetermined,itisneithertruenorfalse,butwhenavalueisassignedtofritbecomesatrueorfalseproposition.Atymathematicalequationisapropositionalfunction.Solongasthevariableshavenodefinitevalue,theequationismerelyanexpressionawaitingdeterminationinordertobecomeatrueorf"iseproposition.Ifitisanequationcontainingonevariable,itbecomestruewhenthevariableismadeequaltoarootoftheequation,otherwiseitbecomesfalse;butifitisan"identity"itwillbetruewhenthevariableisanynumber.Theequationtoacurveinaplaneortoasurfaceinspaceisabelong-propositionalfunction,trueforvaluesoftheco-ordinatesiogtopointsonthecurveorsurface,falseforothervalues'Expressionsoftraditionallogicsuchas"allAisB"arepro-positionalfunctions:AandBhavetobedeterminedasdefinite.l"rr.rbeforesuchexpressionsbecometrueorfalse.Thenotionoft'cases"orttinstancesttdependsuPonPro-positional{unctions.Consider,forexample,thekindofprocesssuggestedbywhatiscalled"generalisationr"andletustakesomeveryprimitiveexample,saf,"lightningisfollowedbyttthunder.t'Wehaveanumberofinstances"ofthis,i.e.anumberofpropositionssuchas:"thisisafashoflightningandisfollowedbythunder."whataretheseoccurrences"instances"of?Theyareinstancesofthepropositionalfunction:"If*isaflashoflightning,xisfollowedbythunder'"Theprocessofgeneralisation(withwhosevaliditywearefortun-\nPropositionalFunctiontt57atelynotconcerned)consistsinpassingfromanumberofsuchinstancestotheuni,uersaltruthofthepropositionalfunction:"Ifxisaflashoflightning,*isfollowedbythunder."Itwillbefoundthat,inananalogousway,propositionalfunctionsarealwaysinvolvedwheneverwetalkofinstancesorcasesorexamples.Wedonotneedtoask,orattempttoanswer,thequestion:"What,Japropositionalfunctioni"Apropositionalfunctionstandingallalonemaybetakentobeamereschema,amereshell,anemptyreceptacleformeaning,notsomethingalreadysignificant.Weateconcernedwithpropositionalfunctions,broadlyspeaking,intwoways:first,asinvolvedinthenotionstttttttrneinallcasesandtrueinsomecases"Isecondly,asinvolvedinthetheoryofclassesandrelations.Thesecondofthesetopicswewillpostponetoalaterchapter;thefirstmustoccuPyusnow.Whenwesaythatsomethingisttalwaystrue"or"trueint'allcasesrt'itisclearthatthesomethirg"involvedcannotbeaproposition.Apropositionisjusttrueorfalse,andthereisanendofthematter.Therearenoinstancesorcasesoft')'((SocratesisamanorNapoleondiedatStHelena."Thesearepropositions,anditwouldbemeaninglesstospeakoftheirbeingtrue"inallcases."Thisphraseisonlyapplicabletopropositionalfunctions.Take,forexample,thesortofthingthatisoftensaidwhencausationisbeingdiscussed.(Wearenotconcernedwiththetruthorfalsehoodofwhatissaid,butonlywithitslogicalanalysis.)WearetoldthatAis,ineveryinstance,followedbyB.Nowifthereare"instances"ofA,AmustbesomegeneralconceptofwhichitissignificanttosayK"xrisAr""x,isL"r,isAr"andsoon,wherefrbJi,zstl,sateparticularswhicharenotidenticalonewithanother.Thisapplies,a.g.,toourpreviouscaseoflightning.Wesaythatlightning(A)isfollowedbythunder(B).Buttheseparateflashesareparticulars,notidentical,butsharingthecommonpropertyofbeinglightning.Theonlywayofexpressinga\nrj8fntoductiontoMatltematicalPltilosopltycommonpropertygenerallyistosaythatacommonProPertyofanumberofobjectsisapropositionalfunctionwhichbecomestruewhenanyoneoftheseobjectsistakenasthevalueofthevariable.Inthiscasealltheobjectsare"instances"ofthetruthofthepropositionalfunction-forapropositionalfunction,thoughitcannotitselfbetrueorfalse,istrueincertaininstancesttt't'andfalseincertainothers,unlessitisalwaystrueoralwaysfalse."When,toreturntoourexample,wesaythatAisineveryinstancefollowedbyB,wemeanthat,whateverxmaybe,ifrisanA,itisfollowedbyaB;thatis,weareassertingthatt'acertainpropositionalfunctionisalwaystrue-"((((Sentencesinvolvingsuchwordsas"allr"ever/rt'ar"ttthert'"some"requirepropositionalfunctionsfortheirinter-pretation.Thewayinwhichpropositionalfunctionsoccurcanbeexplainedbymeansoftwooftheabovewords,namely,tt"all"andsome.t'Thereare,inthelastanalysis,onlytwothingsthatcanbedonewithapropositionalfunction:oneistoassertthatitistrueinallcases,theothertoassertthatitistrueinatleastonecase,orinsomecases(asweshallsay,assumingthatthereistobenonecessaryimplicationofapluralityofcases).Alltheotherusesofpropositionalfunctionscanbereducedtothesetwo.t'inWhenwesaythatapropositionalfunctionistrueallcasesr"or"always"(asweshallalsosay,withoutanytemporalsugges-tion),wemeanthatallitsvaluesaretrue.If'"Sx"isthefunction,andaistherightsortofobjecttobeanargumentto"6*r"then$aistobetrue,howevera'mayhavebeenchosen.Forexample,"ifaishuman,aismortal"istruewhetheraishumanornot;infact,everypropositionofthisformistrue.'3Thusthepropositionalfunctionifxishuman,rismortal"ttisalwaystruer"or"trueinallcases."Or,again,thestate-ttmenttherearenounicorns"istheSameaSthestatement'*isnotaunicorntistrueinall"thepropositionalfunctioncases."TheassertionsintheprecedingchapteraboutPro-positions,e.g.K6?org'implies'q,ot?r'"arereallyassertions\nPropositionalFunctionsr59thatcertainpropositionalfunctionsaretrueinallcases.Wedonotasserttheaboveprinciple,forexample,asbeingtrueonlyofthisorthatparticularporg,butasbeingtrueofanyporqconcerningwhichitcanbemadesignificantly.Theconditionthatafunctionistobesignif,cantforagivenargumentisthesameastheconditionthatitshallhaveavalueforthatargument,eithertrueorfalse.Thestudyoftheconditionsofsignificancebelongstothedoctrineoftypes,whichweshallnotpursuebeyondthesketchgivenintheprecedingchapter.Notonlytheprinciplesofdeduction,butalltheprimitivepropositionsoflogic,consistofassertionsthatcertainproposi-tionalfunctionsarealwaystrue.Ifthiswerenotthecase,theywouldhavetomentionparticularthingsorconcepts-socrates,orredness,oreastandwest,orwhatnotr-andclearlyitisnottheprovinceoflogictomakeassertionswhicharetrueconcerningonesuchthingorconceptbutnotconcerninganother.Itispartofthedefinitionoflogic(butnorthewholeofitsdefinition)thatallitspropositionsarecompletelygeneral,i.e.theyallconsistoftheassertionthatsomepropositionalfunctioncon-tainingnoconstanttermsisalwaystrue.Weshallreturninourfinalchaptertothediscussionofpropositionalfunctionscontainingnoconstantterms.Forthepr$entwewillproceedtotheotherthingthatistobedonewithapropositionalfunction,namely,theassertionthatitisttsometimestruer"i.e.trueinatleastoneinstance.t'thereWhenwesayaremenrttthatmeansthatthepro-positionalfunction"Nisaman"issometimestrue.WhenwettsaysomemenareGreeksrttthatmeansthatthepropositionalfunction"xisamanandaGreek"issometimestrue.Whenwesay"cannibalsstillexistinAfrica,"thatmeansthatthepro-positionalfunction"tcisacannibalnowinAfrica"issometimestrue,i.e.istrueforsomevaluesofti.,Tosayttthereareatleastzindividualsintheworld"istosaythatthepropositionalfunction"oisaclassofindividualsandamemberofthecardinalnumbern"issometimegtrue,or,aswemaysay,istrueforcertain\nt6ofnroductiontoMathematicalPhilosoplt1valuesof.'.Thisformofexpressionismoreconvenientwhenitisnecessatytoindicatewhichisthevariableconstituentwhichwearetakingastheargumenttoourpropositionalfunction.Forexample,theabovepropositionalfunction,whichwemayc'shortentoaisaclassoInindividualsr"containstwovariables,s'andn.Theaxiomofinfinity,inthelanguageofpropositionalfunctions,is:"Thepropositionalfunction'if.nisaninductivenumber,itistrueforsomevaluesofothataisaclassofaindi-viduals'istrueforallpossiblevaluesofn."Herethereisasubordinatefunction,ttq.isaclassof.nindividuals,"whichissaidtobe,inrespectofo,sometime-rtrue;andtheassertionthatthishappensifaisaninductivenumberissaidtobe,inrespectofn,alwaystrue.Thestatementthatafunction$xisalwaystrueisthenegationofthestatementthatnot-|tcissometimestrue,andthestate-mentthat$xissometimestrueisthenegationofthestate-mentthatnot-{risalwaystrue.Thusthestatement,,allmenaremortalsttisthenegationofthestatementthatthettfunctionrisanimmortalmant'issometimestrue.Andthestatement6tthereareunicorns"igthenegationofthestate-mentthatthefunction"*isnotaunicorn"isalwaystrue.lWesaythat$xis"nevertrue"ort'alwaysfalse,,ifnot-{xisalwaystrue.Wecan,ifwechoose,takeoneofthepair,,alwaysr,t'sometimes"asaprimitiveidea,anddefinetheotherbymeansoftheoneandnegation.Thusifwechoose(6sometimes,,asourprimitiveidea,wecandefine.c(($xisalwaystrue'isto'itmeanisfalsethatnot-$xissometimestrue.z',2Butforreasonsconnectedwiththetheoryoftypesitseemsmorecorrecttotakeboth"always"andttsometimest'asprimitiveideas,anddefinebytheirmeansthenegationofpropositionsinwhichtheyoccur.Thatistor"y,assumingthatwehavealreadyrrhemethodofdeductionisgiveninPfinoipiaMathematica,vol.i.*9.IForlinguisticreasons,toavoidsuggestingeitherthepluralorthesingular,itisoftenconvenienttosay"gxisnotalwaysfalse"ratherthan"grsometimes"ot"gfrissometimestrue."\nPropositionall.'uncfionst6tdefined(oradoptedasaprimitiveidea)thenegationofpro'positionsofthetypetowhichrbelongs,wedefine:"The''negationof.'$xalways'isnot-d#sometimes;andthenega-t'tionof{rsometimes'isnot-$xalways.'"Inlikemannerwecanre-definedisjunctionandtheothertruth-functions,asappliedtopropositionscontainingapparentvariables,intermsofthedefinitionsandprimitiveideasforpropositionscontainingnoapparentvariables.Propositionscontainingnoapparentvariablesarecalled"elementarypropositions."Fromthesewecanmountupstepbystep,usingsuchmethodsashavejustbeenindicated,tothetheoryoftruth-functionsasappliedtopropositionscontainingone,two,thre€...variables,oranynumberupton,wherenisanyassignedfinitenumber.Theformswhicharetakenassimplestintraditionalformallogicarereallyfarfrombeingso,andallinvolvetheassertionofallvaluesorsomevaluesofacompoundpropositionalfunction.Take,tobeginwith,"allSisP."WewilltakeitthatSisdefinedby^propositionalfunction{r,andPby^propositionalfunctionry'r.8.g.,ifSisrnen,{rwillbe"rishuman")ifPismortals,r/rwillbe"thereisatimeatwhichrdies."Then"allSisP"means'$'f*implies:y',x'isalwaystrue."Itis'(tobeobservedthatallSisP"doesnotapplyonlytothosetermsthatactuallyareS's;itsayssomethingequallyabouttermswhicharenotS's.SupposewecomeacrossarltcofwhichwedonotknowwhetheritisanSornot;still,ourstatement"al|SisP"tellsussomethitgaboutr,namelyrthatifxisanS,thenxisaP.AndthisiseverybitastruewhenrisnotanSaswhenxisanS.Ifitwerenotequallytrueinbothcases,thereductioadabsurdunt.wouldnotbeavalidmethod;fortheessenceofthismethodconsistsinusingimplicationsincaseswhere(asitafterwardsturnsout)thehypothesisisfalse.Wemayt'putthematteranotherway.InordertounderstandallSisPrttitisnotnecessarytobeabletoenumeratewhattermsareS'sIprovidedweknowwhatismeantbybeinganSandwhatbybeingaP,wecanunderstandcompletelywhatisactuallyaffirmed\n16zIntroductiontoMathematiealPhilosophyby"allSisI?,"howeverlittlewemayknowofactualinstancesofeither.ThisshowsthatitisnotmerelytheactualtermsthatareS'sthatarerelevantinthestatement"allSisPrt'butallthetermsconcerningwhichthesuppositionthattheyareS'sissignificant,i.e.allthetermsthatareS's,togetherwithallthetermsthatarenotS's-i.a.thewholeoftheappropriatelogical"typ"."Whatappliestostatementsaboutallappliesalsotottstatementsaboutsome.Therearemenrtte.g,,meansthatt'"rishumanistrueforsomevaluesofx.Hereallvaluesofr(i.e.allvaluesforwhich"rishuman"issignificant,whethertrueorfalse)arerelevant,andnotonlythosethatinfactarehuman.(Thisbecomesobviousifweconsiderhowwecouldprovesuchastatementtobefalse.)Everyassertionaboutttt'"all"orsomethusinvolvesnotonlytheargumentsthatmakeacertainfunctiontrue,butallthatmakeitsignificant,i.e.allforwhichithasavalueatall,whethertrueorfalse.Wemaynowproceedwithourinterpretationofthetraditionalformsoftheold-fashionedformallogic.WeassumethatSisthosetermsxf.orwhich{ristrue,andPisthoseforwhichry'ristrue.(Asweshallseeinalaterchapter,allclassesarederivedinthiswayfrompropositionalfunctions.)Then:'"AllSisP"means"6*implies$x'isalwaystrue."tttt'tSomeSisPmeans"6*andtlxissometimestrue.t''"NoSisP"means"6*impliesrlot4ltx'isalwaystrue."tttt'tSomeSisnotP"meansSxandnot-{xissometimestrue.ttItwillbeobservedthatthepropositionalfunctionswhicharehereassertedforallorsomevaluesarenotSxandtlxthem-selves,buttruth-functionsof$xandr\xf.orthesarneargument,c.Theeasiestwaytoconceiveofthesortofthingthatisintendedistostartnotfromt'rancltlxingeneral,butfrom$aandt[a,whereaissomeconstant.Supposeweareconsider-ingall"menaremortal":wewillbeginwithttIfSocratesishuman,Socratesismortalrtt\nPropositionalFunctionsr53andthenwewillregard"Socrates"asreplacedby^variabler('t'whereverSocratesoccurs.Theobjecttobesecuredisthat,although*remainsavariable,withoutanydefinitevalue,yetttitistohavethesamevaluein"$x"asin"{*whenweare((assertingthatt'rimpliesr/r"isalwaystrue.Thisrequiresthatweshallstartwithafunctionwhosevaluesaresuchas"6oimplies{or"ratherthanwithtwoseparatefunctions{xand$x;forifwestartwithtwoseparatefunctionswecanneversecurethatthe*,whileremainingundetermined,shallhavethesamevalueinboth.Forbrevitywesay"S*alwaysimplies**"whenwemeanthat"6*implies**"isalwaystrue.Propositionsoftheform"6*alwaysimplies{*"arecalled"formalimplications";thisnameisgivenequallyifthereareseveralvariables.Theabovedefinitionsshowhowfarremovedfromthesimplestformsaresuchpropositionsas"allSisP,"withwhichtradi-tionallogicbegins.Itistypicalofthelackofanalysisinvolvedthattraditionallogictreats"allSisP"asapropositionofttthesameformas"tcisP"-r.g.,ittreatsallmenaremortal"ttasofthesameformasSocratesismortal.ttAswehavejustseen,thefirstisoftheform"#*alwaysimplies{*r"whilethesecondisoftheform"{*."Theemphatic$eparationofthesetwoforms,whichwasefiectedbyPeanoandFrege,wasaveryvitaladvanceinsymboliclogic.Itwillbeseenthat"allSisP"and"noSisP"donotreallydifferinform,exceptbythesubstitutionofnot-tpxfortltx,6'somet'someandthatthesameappliestoSisP"andSisnotP."Itshouldalsobeobservedthatthetraditionalrulesofconversionarefaulty,ifweadopttheview,whichistheonlytechnicallytolerableone,thatsuchpropositionsas"allSisP"(tttdonotinvolvetheexistenceofS's,i.e.donotrequirethatthereshouldbetermswhichareS's.Theabovedefinitionsleadtotheresultthat,if.$xisalwaysfalse,i.e.if.therearenoS's,then"allSisP"and"noSisP"willbothbetrue,what-\n,6+fnftoductiontoMathematicalphihsoph,leverPmaybe.For,accordingtothedefinitioninthelastchapter,t'r,means,,not-$xor6*implies{***r,whichisalwaystrueifnot-t'*isalwaystrue.Atthefirstmoment,thisresultmightleadthereadertodesiredifierentdefinitions,butalittlepracticalexperiencesoonshowsthatanydifierentdefinitionswouldbeinconvenientandwouldconcealtheimportantideas.Theproposition,,6*alwaysimplies{*,and.$xissometimestrue"isessentiallycomposite,anditwouldbeveryawkwardtogivethisasthedefinitionof',allSisp,"forthenweshouldhavenolanguageleftfor,,$xalvtaysimplies{*r"whichisneededahundredtimesforoncethattheotherisneeded.But,withourdefinitions,"a[sisp"doesnotimplyt'somesisPrt'sincethefirstallowsthenon-existenceofsandtheseconddoesnot;thusconversionperaccidensbecomeginvalid,andsomemoodsofthesyliogismatefallaciou\c.g.Darapti:"A1lMiss,allMisP,thereforesomesispr"whichfailsifthereisnoM.Thenotionofttexigtence"hasseveralforms,oneofwhichwilloccupyusinthenextchapter;butthefundamenralformisthatwhichisderivedimmediatelyfromthenotionof,,some-timestrue."wesaythatanargumenta"satisfiesttafunctionSxif.$aistrue;thisisthesamesenseinwhichtherootsofanequationaresaidtosatisfytheequation.Nowif$xissometimestrue,wemaysaytherearetc'sforwhichitistrue,otwemaysay"argumentssatisfying$xexist."Thisisthefundamentalmean-('existence."ingofthewordOthermeaningsareeitherderivedfromthis,orembodymereconfusionofthought.wemaycorrectlysay"menexistrt'meaningthatttrisamanttissome-timestrue.Butifwemakeapseudo-syllogism:"Menexisr,Socratesisaman,thereforeSocratesexistsrttwearetalkingnonsense,sincettSocratest'isnot,like.,menrttmerelyanun-determinedargumenrtoagivenpropositionalfunction.Thefallacyiscloselyanalogoustothatoftheargument:"Menarenumerous,Socratesisaman,thereforeSocratesisnumerous,ttInthiscaseitisobviousthattheconclusionisnonsensical,but\nPropositionaIFunctionsr65inthecaseofexistenceitisnotobvious,forreasonswhichwillappearmorefullyinthenextchapter.ForthePresentletusmerelynotethefactthat,thoughitiscorrecttosay"menexistr"itisincorrect,orrathermeaningless,toascribeexistencetoagivenparticularrwhohappenstobeaman.Generally,"termstt'3asatisfying,f*exist"means"{*issometimestrue;butexists"(whereaisatermsatisfying#x)isamerenoigeorshape,devoidofsignificance.Itwillbefoundthatbybearinginmindthissimplef.allacywecansolvemanyancientphilosophicalpuzzlesconcerningthemeaningofexistence.Anothersetofnotionsastowhichphilosophyhasalloweditselftofallintohopelessconfusionsthroughnotsufficientlyseparatingpropositionsandpropositionalfunctionsarethenotionsof"modality":necessary,possible,andimpossible.(Sometimescontingentorassertoricisusedinsteadof.possible.)Thetraditionalviewwasthat,amongtruePropositions,somewerenecessary,whileothersweremerelycontingentorassertoric;whileamongfalsepropositionssomewereimpossible,namely,thosewhosecontradictorieswerenecessary,whileothersmerelyhappenednottobetrue.Infact,however,therewasneveranyclearaccountofwhatwasaddedtotruthbytheconceptionofnecessity.Inthecaseofpropositionalfunctions,thethree-folddivisionisobvious.If."Sx"isanundeterminedvalueofacertainpropositionalfunction,itwillbenecessaryifthefunctionisalwaystrue,possibleifitissometimestrue,andimpossibleif'itisnevertrue.Thissortofsituationarisesinregardtoprob-ability,forexample.Supposeaballrisdrawnfromabagwhichcontainsanumberofballs:ifalltheballsarewhite,ttriswhite"isnecessary;ifsomearewhite,itispossible;ifnone,itisimpossible.Hereallthatishnownaboutristhatitsatisfiesacertainpropositionalfunction,namely,"ccwasaballinthebag."Thisisasituationwhichisgeneralinprob-abilityproblemsandnotuncommoninpracticalliie-e.g.whenapersoncallsofwhomweknownothingexceptthathebringsaletterofintroductionfromourfriendso-and-so.Inallsuch\nfi6InuoductiontoMathematicalPhilosoplrycases,asinregardtomodalityingeneral,thepropositionalfunctionisrelevant.Forclearthinking,inmanyverydiversedirections,thehabitofkeepingpropositionalfunctionssharplyseparatedfrompropositionsisoftheutmostimportance,andthefailuretodosointhepasthasbeenadisgracetophilosophy,\nCHAPTERXVIDESCRIPTIONSWrdealtintheprecedingchapterwiththewordsallandsotfteiinthischapterweshallconsiderthewordtbeinthesingular,andinthenextchapterweshallconsiderthewordtheintheplural.Itmaybethoughtexcessivetodevotetwochapterstooneword,buttothephilosophicalmathematicianitisawordofverygreatimportance:likeBrowning'sGrammarianwiththeenclitic8e,IwouldgivethedoctrineofthiswordifIttweredeadfromthewaistdown"andnotmerelyinaprison.63Wehavealreadyhadoccasiontomentiondescriptivettttfunctionsr"i.r.suchexpressionsasthefatherof.r"orthesinet'descriptions.n'oftc."Thesearetobedefinedbyfirstdefining,,Adescription"^^ybeoftwosorts,definiteandindefinite(orambiguous).Anindefinitedescriptionisaphraseofthettformaso-and-sor"andadefinitedescriptionisaphraseoftheform"theso-and-so"(iothesingular).Letusbeginwiththeformer.ttWhodidyoumeet?"ttImetaman.tt'3Thatisaveryindefinitedescription."Wearethereforenotdepartingfromusageinourterminology.Ourquestionis:WhatdoIreallyttassertwhenIaSsertImetaman"?Letusassume,forthemoment,thatmyassertionistrue,andthatinfactImetJones.ItisclearthatwhatIassertisnot"ImetJones."Imaysay.,Imetaman,butitwasnotJones";iothatcase,thoughIlie,Idonotcontadictmyself,asIshoulddoifwhenIsayImetar67\nr68fntoductiontoMatltenaticalPlrilosopltymanIreallymeanthatImetJones.ItisclearalsothatthepersontowhomIamspeakingcanunderstandwhatIsay,evenifheisaforeignerandhasneverheardofJones.Butwemaygofurther:notonlyJones,butnoactualman,entersintomystatement.Thisbecomesobviouswhenthestate-mentisfalse,sincethenthereisnomorereasonwhyJonesshouldbesupposedtoenterintothepropositionthanwhyany-oneelseshould.Indeedthestatementwouldremainsignificant,thoughitcouldnotpossiblybetrue,eveniftherewerenomanatall.ttImetaunicorn"ort'Imetasea-serpent,tisaperfectlysignificantassertion,ifweknowwhatitwouldbetobeaunicornorasea-serpent,i.e.whatisthedefinitionofthesefabulousmonsters.Thusitisonlywhatwemaycalltheconccptthatentersintotheproposition.Inthecaseof,'unicornr,'forexample,thereisonlytheconcept:thereisnotalso,soms,whereamongtheshades,somethingunrealwhichmaybecalled"aunicorn."Therefore,sinceitissignificant(thoughfalse)tosay"Imetaunicornr"itisclearthatthisproposition,rightlyanalysed,doesnotcontainaconstituentt'aunicornrrtthoughitdoescontaintheconceptttunicorn."Thequestionof"unrealityr"whichconfrontsusatthispoint,isaveryimportantone.Misledbygrammar,thegreatmajorityofthoselogicianswhohavedealtwit}tJrisquestionhavedealtwithitonmistakenlines.Theyhaveregardedgrammaticalformasasurerguideinanalysisthan,infact,itis.Andtheyhavenotknownwhatdifierencesingram-maticalformareimportant.',TmetJones,,and.,Imetaman"wouldcounttraditionallyaspropositionsofthesameform,butinactualfacttheyareofquitedifierentforms:thefirstnamesanactualperson,Jones;whilethesecondinvolvesapropositionalfunction,andbecomes,whenmadeexplicit:"ThetfunctionImetrand*ishumantissometimestrue.t'(Itwillberememberedthatweadoptedtheconventionofusing('sometimes"asnotimplyingmorethanonce.)Thisproposi-tionisobviouslynotoftheform"Imetxr"whichaccounts\nDescriptionsr69ttfortheexistenceoftheproposition"Imetaunicorninspiteofthefactthatthereisnosuchthingas"aunicorn."Forwantoftheapparatusofpropositionalfunctions'manylogicianshavebeendriventotheconclusionthatthereareunrealobjects.Itisargued,e,g.byMeinong,rthatwecanspeakaboutt'thegoldenmountainrtttttheroundsquarerttandsoon;wecanmaketruepropositionsofwhichthesearethesubjects;hencetheymusthavesomekindoflogicalbeing,sinceotherwisethepropositionsinwhichtheyoccurwouldbemeaningless.Insuchtheories,itseemstome,thereisafailureofthatfeelingforrealitywhichoughttobepreservedeveninthemostabstractstudies.Logic,Ishouldmaintain,mustnomoreadmitaunicornthanzoologycan;forlogicisconcernedwiththerealworldjustastrulyaszoology,thoughwithitsmoreabstractandgeneralfeatures.Tosaythatunicornshaveanexigtenceinheraldry,orinliterature,orinimagination,isamostpitifulandpaltryevasion.Whatexistsinheraldryisnotananimal,madeoffleshandblood,movingandbreathingofitsowninitiative.Whatexistsisapicture,oradescriptioninwords.Similarly,tomaintainthatHamlet,forexample,existsinhisownworld,namely,intheworldofShakespeare'simagination,justastrulyasG"y)Napoleonexistedintheordinaryworld,istosaysomethingdeliberatelyconfusing,drelseconfusedtoadegreewhichisscarcelycredible.Thereisonlyoneworld,the"realt'world:Shakespeare'simaginationispartofit,andthethoughtsthathehadinwritingHamletarereal.SoarethethoughtsthatwehaveinreadingthePlay.Butitisoftheveryessenceoffictionthatonlythethoughts,feelings,etc.,inShakespeareandhisreadersarereal,andthatthereisnot,inadditiontothem,anobjectiveHamlet.WhenyouhavetakenaccountofallthefeelingsrousedbyNapoleoninwritersandreadersofhistory,youhavenottouchedtheactualman;butinthecaseofHamletyouhavecometotheendofhim.IfnoonethoughtaboutHamlet,therewouldbenothing|(JnteysuchungenzwrGegenstandstheovieundPsychologie,rgo4.\nr70IntraductiontoMathematicalPhilosopltyleftofhim;ifnoonehadthoughtaboutNapoleon,hewouldhavesoonseentoitthatsomeonedid.Thesenseofrealityisvitalinlogic,andwhoeverjuggleswithitbypretendingthatHamlethasanotherkindofrealityisdoingadisservicetothought.Arobustsenseofrealityisverynecessaryinframingacorrectanalysisofpropositionsaboutunicorns,goldenmoun-tains,roundsquares,andothersuchpseudo-objects.Inobediencetothefeelingofrealitlweshallinsistthat,intheanalysisofpropositions,nothing"unreal"istobeadmitted.But,afterall,ifthere,.rnothingunreal,ho*,itmaybeasked,couldweadmitanythingunrealtThereplyisthat,indealing*ithpropositions,wearedealinginthefirstinstancewithsymbols,andifweattributesignificancetogroupsofsymbolswhichhavenosignificance,weshallfallintotheerrorofadmittingunrealities,intheonlysenseinwhichthisispossible,namely,asobjectsdescribed.Intheproposition('Imetaunicornrttthewholefourwordstogethermakeasigni-ficantproposition,andtheword"unicorn"byitselfissignificant,ttinjustthesamesenseasthewordman."Butthetwowordsttttaunicorndonotformasubordinategrouphavingameaningofitsown.Thusifwefalselyattributemeaningtothesetwottwords,wefindourselvessaddledwithaunicornr"andwiththeproblemhowtherecanbesuchathinginaworldwheretttherearenounicorns."Aunicornisanindefinitedescrip-tionwhichdescribesnothing.Itisnotanindefinitedescriptionwhichdescribessomethingunreal.Suchapropositionastt('risunreal"onlyhasmeaningwhen,c"isadescription,definiteorindefinite;inthatcasethepropositionwillbetrueif."x"isadescriptionwhichdescribesnothing.Butwhether(()'thedescriptiontcdescribessomethingordescribesnothing,itisinanycasenotaconstituentofthepropositioninwhichitoccurs;likettaunicornt'justnow,itisnotasubordinategrouphavingameaningofitsown.Allthisresultsfromthefactthat,'(tt3cttwhentc''isadescription,risunreal"otrdoesnotexistisnotnonsense,butisalwayssignificantandsometimestrue.\nDescriptionst7rWemaynowproceedtodefinegenerallythemeaningofpropositionswhichcontainambiguousdescriptions.Supposettwewishtomakesomestatementaboutaso-and-sorttwherett"so-and-sotsarethoseobjectsthathaveacertainproperty6,i.e,thoseobjectsrforwhichthepropositionalfunctionfristrue.(8.g,ifwetakettaman"asourinstanceof'oaso-and-sorttfrwillbe"rishuman.")Letusnowwishtoassertthepropertyassertthatttaso-and-sotthas*of"aso-and-sor"i.e.wewishtothatpropertywhichrhaswhen$ristrue.(8.g,inthecaseof"Imetamanr"r/rwillbe"Imetr.t')Nowtheproposition66thataso-and-so"hasthepropertytlisnotaProPositionoftttttheform"**."Ifitwere,aso-and-sowouldhavetobeidenticalwithrforasuitabletc;andalthough(inasense)thismaybetrueinsomecases,itiscertainlynottrueinsuchacaseas"aunicorn."Itisjustthisfact,thatthestatementthataso-and-sohasthepropertyry'isnotoftheformr/4whichmakesitpossiblefor"aso-and-so"tobe,inacertainclearlydefinablesense,"unreal."Thedefinitionisasfollows:-56Thestatementthatanobjecthavingthepropertyrfhastheprope*ytl"means:"Thejointassertionof.$ccand$xisnotalwaysfalse."Sofaraslogicgoes,thisisthesamepropositionasmightbeexpressedby"someS'sarc{'"";butrhetoricallythereisadifierence,becauseintheonecasethereisasuggestionofsingularity,andintheothercaseofplurality.This,however,isnottheimportantpoint.Theimportantpointisthat,whenrightlyanalysed,propositionsverballyabout"aso-and-go"arefoundtocontainnoconstituentrepresentedbythisphrase.Andthatiswhysuchpropositionscanbesignificantevenwhenthereisnosuchthingasaso-and-so.Thedefinitionof.existence,asappliedtoambiguousdescrip-tions,resultsfromwhatwassaidattheendofthepreceding'6ttttchapter.Wesaythatmenexist"oramanexistsifthe\nr72IntroductiontoMatltematicalPltilosopltypropositionalfunction"rishuman"issometimestrue;andttgenerallyaso-and-so"existsiI"xisso-and-so"issometimestrue.Wemayputthisinotherlanguage.ThepropositionttttttSocratesisamanisnodoubteguiaalenttoSocratesishumanr"butitisnottheverysameproposition.Theisofttt'Socratesishumanexpressestherelationofsubjectandpredicate;theasof."Socratesisaman"expressesidentity.Itisadisgracetothehumanracethatithaschosentoemploy(cthesamewordis"forthesetwoentirelydifierentideas-adisgracewhichasymboliclogicallanguageofcourseremedies.Theidentityin"Socratesisaman"isidentitybetweenant'ttobjectnamed(acceptingSocratesasaname,subjecttoqualificationsexplainedlater)andanobjectambiguouslydescribed.Anobjectambiguouslydescribedwill"exist"whenatleastonesuchpropositionistrue,i.a.whenthereisatleastonetruepropositionoftheformt'risaso-and-sorttwhere3cN"isaname.Itischaracteristicofambiguous(asopposedtodefinite)descriptionsthattheremaybeanynumberoftruepropositionsoftheaboveform-Socratesisaman,Platoisaman,etc.Thus"amanexistst'followsfromSocrates,orPlato,oranyoneelse.Withdefinitedescriptions,ontheotherhand,thecorrespondingformofproposition,namely,,'risthet'(eso-and-so(wheretc"isaname),canonlybetrueforonevalueoLtcatmost.Thisbringsustothesubjectofdefinitedescriptions,whicharetobedefinedinawayanalogoustothatemployedforambiguousdescriptions,butrathermorecomplicated.Wecomenowtothemainsubjectofthepresentchapter,namely,thedefinitionofthewordthe(inthesingular).Oneveryimportantpointaboutthedefinitionof,'aso-and-so"((appliesequallytotheso-and-so"Ithedefinitiontobesoughtisadefinitionofpropositionsinwhichthisphraseoccurs,notadefinitionofthephraseitselfinisolation.Inthecaseof"aso-and-sor"thisisfairlyobvious:noonecouldsupposethat"aman"wasadefiniteobject,whichcouldbedefinedbyitself.\nDescriptions173Socratesisaman,Platoisaman,Aristotleisaman,butwettttttcannotinferthat"amanmeansthesameasSocratesttmeansandalsothesameasPlato"meansandalsothesamettt'asAristotlemeans,sincethesethreenameshavedifferentmeanings.Nevertheless,whenwehaveenumeratedallthemenintheworld,thereisnothingleftofwhichwecansay,"Thisisaman,andnotonlyso,butitisthe'amanr'thequintes-sentialentitythatisjustanindefinitemanwithoutbeingany-bodyinparticular."Itisofcoursequiteclearthatwhateverthereisintheworldisdefinite:ifitisamanitisonedefinitemanandnotanyother.Thustherecannotbesuchanentityt'as"amantobefoundintheworld,asopposedtospecificman.Andaccordinglyitisnaturalthatwedonotdefine"aman"itself,butonlythepropositionsinwhichitoccurs.Inthecaseof"theso-and-so"thisisequallytrue,thoughatfirstsightlessobvious.Wemaydemonstratethatthismustbethecase,by^considerationofthedifferencebetweenanarreandadef,nitedescription.Taketheproposition,"Scottistheauthorof.Waveiley."Wehavehereaname,"Scottr"andattdescription,theauthorof.Waverleyr"whichareassertedtoapplytothesameperson.Thedistinctionbetweenanameandallothersymbolsmaybeexplainedasfollows:-Anameisasimplesymbolwhosemeaningissomethingthatcanonlyoccurassubject,i.e.somethingofthekindthat,inChapterXIII.,wedefinedasan"individual"ora"particular."Anda"simple"symbolisonewhichhasnopartsthatare((symbols.ThusScott"isasimplesymbol,because,thoughithasparts(namely,separateletters),thesepartsarenotsymbols.t'Ontheotherhand,theauthorof.Waverlty"isnotasimplesymbol,becausetheseparatewordsthatcomposethephrasearepartswhicharesymbols.If,asmaybethecase,whateversecn$tobean"individual"isreallycapableoffurtheranalysis,weshallhavetocontentourselveswithwhatmaybecalled66relativeindividualsr"whichwillbetermsthat,throughoutthecontextinquestion,areneveranalysedandneveroccur\n17+IntrudacrtontuMathematicalPhitosophlottrerwisethanaseubjects.Andinthatcaseweshallhavecorrespondinglytocontentourselveswith..relativenames."Fromthestandpointofourpresentproblem,namely,thedefini-tioqofdescriptions,thisproblem,whethertheseareabsolutenamesoronlyrelativenames,maybeignored,sinceitcon-cernsdifferentstagesinthehierarchyof..typesr"whereaswehavetocomparesuchcouplesasttScott,tand..theauthorofwaonlcy,"whichbothapplytothesameobject,anddonotraiset}eproblemoftypes.Wemant}erefore,fortlemomenr,treatnamesascapableofbeingabsolute;nothingthatweshallhavetosaywilldependuponthisassumption,butthewordingmaybealittleshortenedbyit.wehave,then,twot}ingstocompare:(r)aflatnc,whichisasimplesymbol,directlydesignatinganindividualwhichisitsmeaning,andhavingthismeaninginitsownrighr,in-dependentlyofthemeaningsofallotherwords;(z)adesniption,whichconsistsofseveralwords,whosemeaningsarealreadyfixed,andfromwhichresult!whateveristobetakenasthettttmeaningofthedescription.Apropositioncontainingadescriptionisnotidenticalwithwhatthatpropositionbecsmeswhenanameissubstituted.,evenifthenamenamesttesameobjectasthedescriptiondescribes.'3Scottistheauthorofwaveilcy"isobvioorlyadifferentpropositionfrom33ScottisScott',:thefirstisio.tinliteraryhistory,thesecondatrivialtruism.Andifwe"putanyoneotherthanScottinplaceof',theauthorof.Waveileyr,ourpropositionwouldbecomefalse,andwouldthereforecertainlynolongerbethesameproposition,But,itmaybesaid,ourpropositionisessentiallyofrhesameformas(say),.ScottissirWalterr"inwhichtwonamesaresaidtoapplytothesameperson.Thereplyisthat,if"scottissirwalter"reallymeansttthepersonnamedtScotttisthepersonnamedtSirWalterrrrtthenthenamesarebeingusedasdescriptionszi.c.ttreindividual,insteadofbeingnamed,isbeingdescribedasthepersonhavingthatname.Thisisawayinwhichnamesarefrequentlyused\nDcscriprtons175inpractice,andtherewill,asarule,benothinginthephraseologytoghowwhethertheyarebeingusedint*riswayorzJnames.Whenanameisuseddirectly,merelytoindicatewhatwearespeakingabout,itisnopartofthefactasserted,orofthefalsehoodifourassertionhappenstobefalse:itismerelypartofthesymbolismbywhichweexpressourthought.Whatwewanttoexpressissomethingwhichmight(forexample)betranslatedintoaforeignlanguage;itissomethingforwhichtheactualwordsareavehicle,butofwhichtheyarenoPart.Ontheotherhand,whenwemakeapropositionabout"thepersoncalled3Scottrttttheactualname"Scottttentersintowhatweareasserting,andnotmerelyintothelanguageusedinmakingtheassertion.Ourpropositionwillnowbeadifrerentoneifwe'Sirsubstitute"thepersoncalledWalter."'Butsolongas66ttweareusingnameszJnames,whetherwesayScottorwhetherttttwesaySirWalterisasirrelevanttowhatweareassertingaswhetherwespeakEnglishorFrench.Thussolongasnameeareused4Jnames,"ScottisSirWalter"isthesametrivialpropositionas"ScottisScott."Thiscompletestheproofthat3(Scottistheauthorof,Wavcrby"isnotthesamepropositionasresultsfromsubstitutinganamefor"ttreauthorofWaoerlcyr"nomatterwhatnamemaybesubstituted.Whenweu8eavariable,andspeakofapropositionalfunction,$rsay,theprocesgofapplyinggeneralstatementsaboutrtoparticularcaseswillconsistinsubstitutinganamefortheletter"tcr"assumingthatfisafunctionwhichhasindividualsforitst'alwaysarguments.Suppose,forexample,thatf*istrue"Iletitbe,say,the"lawofidentitn"16:1s.Thenwemaysub-stitutefor"r"anynamewechoose,andweshallobtainatnre3tproposition.AssumingforthemomentthatSocratesr"ttttt'Plator"andAristotlearenames(averyrashassumption),wecaninferfromthelawofidentitythatSocratesisSocrates,PlatoisPlato,andAristotleisAristotle.Butweshallcommitaf.allacyifweattempttoinfer,withoutfurtlerpremisses,thattheauthorofWaocrhyistheauthorof.Waoerlry.Thisresultg\nt76IntroductiontoMathernaticalPltilosopltyfromwhatwehavejustproved,that,ifwesubstituteanamefor"theauthorofWaverley"inaproposition,thepropositionweobtainisadifferentone.Thatistosay,applyingtheresult(g,:v"isnotthesametoourpresentcase:If"x"isaname,propositionas"theauthorof.Waveileyistheauthorof.Waverleyr"c()'nomatterwhatname,cmaybe.Thusfromthefactthatallpropositionsoftheform"x--tc"aretruewecannotinfer,withoutmoreado,thattheauthorofWaverleyistheauthorofWaacrley.Infact,propositionsoftheform"theso-and-soistheso-and-so"arenotalwaystrue:itisnecessarythattheso-and-soshouldexist(atermwhichwillbeexplainedshortly).ItisfalsethatthepresentKingofFranceisthepresentKingofFrance,orthattheroundsquareistheroundsquare.Whenwesubstituteadescriptionforaname,propositionalfunctionswhichare"alwaystrue"maybecomefalse,ifthedescriptiondescribesnothing"Thereisnomysteryinthisassoonaswerealise(whatwasprovedintheprecedingparagraph)thatwhenwesubstituteadescriptiontheresultisnotavalueofthepropositionalfunctioninquestion.Wearenowinapositiontodefinepropositionsinwhichadefinitedescriptionoccurs.Theonlythingthatdistinguishestttheso-and-so"from"aso-and-so"istheimplicationofuniqueness.Wecannotspeakof"tbeinhabitantofLondonr"becauseinhabitingLondonisanattributewhichisnotunique.ttWecannotspeakaboutthepresentKingofFrancer"becausethereisnone;butwecanspeakabout"thepresentKingofooEngland."Thuspropositionsabouttheso-and-so"alwaysimplythecorrespondingpropositionsabout"aso-and-sor"withtheaddendumthatthereisnotmorethanoneso-and-so.Suchapropositionas"ScottistheauthorofWaverley"couldnotbetrueifWaverleyhadneverbeenwritten,orifseveralpeoplehadwrittenit;andnomorecouldanyotherpropositionresultingfromapropositionalfunction?rbythesubstitutionof"theauthorof.fVaveilty"for"x."Wemaysaythat"thettt#anthorof.Waverlty"meansthevalueofrforwhichwtote\nDescriptions177t'theWaveiley'istrue."ThusthepropositionauthorofWaverleywasScotchrt'forexample,involves:(r)",cwroteW/averley"isnotalwaysfalse;(z)"if.xandywroteWaveiley,xandyareidentical"isalwaystrueI(3)"ifxwroteWaaerley,rwasScotch"isalwaystrue.Thesethreepropositions,translatedintoordinarylanguage,state:(r)atleastonePersonwroteWaoerley;(z)atmostonepersonwroteWaaeiley;(3)whoeverwrotell/averleywasScgtch.Allthesethreeareimpliedby"theauthorofWaverleywasScotch."Conversely,thethreetogether(butnotwoofthem)implythattheauthorofWaaerleywasScotch.Hencethethreetogethermaybetakenasdefiningwhatismeantbytheproposition"theauthorof.WaverleywasScotch."Wemaysomewhatsimplifythesethreepropositions.Thefirstandsecondtogetherareequivalentto:"Thereisaterm(#'rsuchthatwroteWaverleyistruewhenxiscandisfalsewhenrisnotc."Inotherwords,t'Thereisatermcsuchthat'*wroteWaverley'isalwaysequivalentto'*isc."'(Twopropositionsare"equivalent"whenbotharetrueorbothare'cfalse.)Wehavehere,tobeginwith,twofunctionsofx,tct'andttrwroteWaverleyiscr"andweformafunctionofcbyconsideringtheequivalenceofthesetwofunctionsofxforallvaluesof.x;wethenproceedtoassertthattheresultingfunctionofeist'sometimestruer"i.e.thatitistrueforatleastonevalueofr.(Itobviouslycannotbetrueformorethanonevalueofr.)Thesetwoconditionstogetheraredefinedasgivingthemeaningof"theauthorof.Waverlryexists.ttWemavnowdefine"thetermsatisfyingthefunction$xexists."Thisisthegeneralformofwhichtheaboveisapar-ticularcase."Theauthorof.Waveilty"is"thetermsatisfying6thefunctionrwroteWaverley."'And"theso-and-so"will\nr78fnroductiontoMathematicalPlrihsoplr1alwaysinvolvereferencetosomepropositionalfunction,namely,thatwhichdefinesthepropertythatmakesathingaso-and-so.Ourdefinitionisasfollows:-"Thetermsatisfyingthefunctionfxexists"means:6"Thereisatermrsuchthat$xisalwaysequivalentto#istr.'"Inordertodefine"theauthoroffVaaeileywasScotchr"wehavestilltotakeaccountofthethirdofourthreeproposi-tions,namely,"WhoeverwroteWaveileywasScotch.t'ThiswillbesatisfiedbymerelyaddingthattheeinquestionistobeScotch.Thus"theauthorof.WaveileywasScotch"is:('"Thereisatermrsuchthat(l)xwroteWaverleyisalways'equivalenttoxiscr'(z)cisScotch."Andgenerally:"thetermsatisfying$xsatisfiest*"isdefinedasmeaning:"Thereisatermesuchthat(r)$xisalwaysequivalentto'riscr'(2)tlcistrue."Thisisthedefinitionofpropositionsinwhichdescriptionsoccur.Itispossibletohavemuchknowledgeconcerningatermdescribed,i.t.toknowmanypropositionsconcerning"theso-and-sor"withoutactuallyknowingwhattheso-and-sois,i.e.withoutknowinganypropositionoftheform"ristheso-and*sor"3'whereN"isaname.Inadetectivestorypropositionsaboutt'themanwhodidthedeed"areaccumulated,inthehopethatultimatelytheywillsuffi.cetodemonstratethatitwasAwhodidthedeed.Wemayevengosofarastosaythat,inallsuchknowledgeascanbeexpressedinwords-withthett'3exceptionof"thisandthat"andafewotherwordsofwhichthemeaningvariesondifferentoccasions-nonames,inthestrictsense,occur,butwhatseemlikenamesarereallydescriptions.WemayinquiresignificantlywhetherHomert'Homerexisted,whichwecouldnotdoif"wereaname.Theproposition"theso-and-soexists"issignificant,whethertrueorfalse;butif.aistheso-and-so(where"a"isaname),thewords"a.exists"aremeaningless.Itisonlyofdescriptions\nDescriptions17g-definiteorindefinite-thatexistencecanbesignificantlyasserted;for,if"attisaname,itmustnamesomething:whatdoesnotnameanythingisnotaname,andtherefore,ifintendedtobeaname,isasymboldevoidofmeaning,whereasadescrip-tion,like"thepresentKingofFrancer"doesnotbecomein-capableofoccurringsignificantlymerelyonthegroundthatitdescribesnothing,thereasonbeingthatitisacomplarsymbol,ofwhichthemeaningisderivedfromthatofitsconstituentsymbols.Andso,whenweaskwhetherHomerexisted,weareusingtheword"Homer"asanabbreviateddescription:wemayreplaceitby(t"y)"theauthorofthelliadandtheOdyssey."Thesameconsiderationsapplytoalmostallusesofwhatlooklikepropernames.Whendescriptionsoccurinpropositions,itisnecessarytodistinguishwhatmaybecalled"primary"and"secondary"occurrences.Theabstractdistinctionisasfollows.Adescrip-t'primarytionhasa"occurrencewhenthepropositioninwhichitoccursresultsfromsubstitutingthedescriptionfor'(N"insomepropositionalfunctionfr;adescriptionhasattsecondary"occurrencewhentheresultofsubstitutingthedescriptionfor*in$xgivesonlypartof,thepropositioncon-cerned.Aninstancewillmakethisclearer.Consider"thepresentKingofFranceisbald."Here"thepresentKingofFrance"hasaprimaryoccurrence,andthepropositionisfalse.Everypropositioninwhichadescriptionwhich