Mathematics and Physical Reality

Mathematics and Physical Reality

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MathematicswithoutNumbers:TowardsaModal-StructuralInterpretationGeoffreyHellman

Printpublicationdate:1993PrintISBN-13:9780198240341PublishedtoOxfordScholarshipOnline:November2003DOI:10.1093/0198240341.001.0001

MathematicsandPhysicalRealityGeoffreyHellman(ContributorWebpage)

DOI:10.1093/0198240341.003.0004

AbstractandKeywords

Strategiesforextendingthemodalstructuralapproachtoapplicationsofmathematicsareexplored.Thebasicideaistoentertainwhatrelationshipswouldholdbetweenanonmathematicalsystemofinterestandnoninterferingadditionalhypotheticalobjectsthatmaybeneededtocarryrelevantmathematicalstructure.Usually,structuresforrealanalysissuffice,althoughstillmoreabstractstructuresmayariseinmodernphysics.Certaininterestingconnectionsbetweenthemodalstructuralapproachandaminimalscientificrealismareexposed(thelatteraidingtheformer),andcomparisonsaredrawnwithHartryField'snominalizationprogram.

Keywords:analytic,appliedmathematics,modernphysics,scientificrealism,synthetic

0.IntroductionAsisuniversallyrecognized,mathematicsthroughoutitshistoryhasbeenintimately


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boundupwithourinteractionswiththematerialworld,fromthemostmundanepracticalenterprisesofcountingandmeasuringtoourmostsophisticatedtheoreticaleffortstocomprehenditsworkingsastheunfoldingofphysicallaws.Fromahistoricalperspective,itwouldbenoexaggerationtosaythatphysicalapplicationhassustainedmathematicsasitsverylifeblood.

Thisperspectiveisreflectedinphilosophyofmathematics.Surelyoneofthestrongestreasonsifperhapsnottheonlyreasonfortakingmathematicaltruthseriouslystemsfromtheapparentlyindispensablerolemathematicaltheoriesplayintheveryformulationofscientificdescriptionsofthematerialworldaroundus.Assoonasweundertaketoconveytheinformationthat,say,therearemorespidersthanapes,weseemtobecommittedtonumbers,orclassesandfunctions.Describingthebehaviourofthestarsandgalaxiesapparentlyinvolvesusinagooddealoftheapparatusofdifferentialgeometry.Andtoprobetheatomicstructureofmatterandtheunderpinningsofchemistryandbiology,weseemtobeinvolvedinthetheoryofHilbertspaceandageneralizedformofmeasuretheory.Whateverthedetailsofthisentanglementbetweenphysicsandmathematics,surelyapurelyformalistapproachtomathematicswouldseemfarmoreplausibleweretherenoentanglement.Ifwestrainourimaginationsandsuppose(perimpossible!)thatmathematicaltheoriesandstructureshadnomaterialapplicationsthattheycouldsomehowbeisolatedfromtheempiricalscienceswhatobjectionwouldwehavetotreatingmathematicsasapurelyformalgame?Forsuchamathematics,thequestionoftruthmightnotevenseemtoarise.

Reflectionssuchastheseleadustoposethreeinterrelatedfundamentalquestionsconcerningmathematicsinitsapplications.Thefirstisthis:grantedthattheroleofmathematicsinordinaryandscientificapplicationsprovidessomegroundsfortakingmathematical(p.95) truthseriously(thatis,fortakingarealistasopposedtoaninstrumentaliststancetowardatleastsomemathematicaltheories),arethesetheexclusivegrounds,orarethereothers;and,ifthereareothers,whataretheyandhowpowerfularethey?Thesecondquestionreallyacompositeofquestionsaskshowmuchmathematicsisreallyindispensableforhowmuchscience?And,third,wemustask,justwhatdoessuchindispensabilitydemonstratewithregardtomathematicalobjectivityandmathematicalobjects?

Havingnowperhapspiquedthereader'sinterestinthegeneralsubject,Imustofferadisappointingapologyinadvance:noneofthesequestionswillbeanswereddefinitively.Atbest,partialandtentativeanswerstothesecondandthirdquestionswillemerge.Astothefirst,weshallbelefthanging.

Forthetaskthatdemandsimmediateattentionisthatofsketchinghowthemodalstructuralistframeworkalreadydevelopedforpuremathematicscanbeextendedtoappliedmathematics.How,inthefirstplace,arewetorepresentordinaryandscientificappliedmathematicalstatements?Whatarethemainassumptionsthatliebehindsucharepresentation?Havingsketchedthebasicideasandbroachedsomeofthemainproblems(in1),wemaythenturnourattentionbacktothebroaderquestionsconcerningindispensability.Asweshallsee(in2),thereisastrongcasethatmodern


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physicaltheoriesespeciallyGeneralRelativityandQuantumMechanicsrequire(thepossibilityof)mathematicalstructuressorichthateventhechancesofamodalnominalisminanyreasonablesensearedim.Thiscase,asweshallpresentit,dependsonaratherbroadinterpretationofappliedinthephraseappliedmathematics:questionsoftheoreticalphysicsofafoundationalcharacterareincluded.Butweseenorationalbasisforexcludingthem(e.g.bydrawingasharplinebetweenordinaryempiricalapplicationsandtheoreticalorevenmetatheoreticalapplications).And,itwouldbeironicindeediffoundationsofmathematicstookthestancethatfoundationsofphysicsneednotberespected!

Infact,weshallfurtherseethatrecentworkontheimplicationsofhighersettheoryraisesthetantalizingprospectthatstrongerandstrongerabstractmathematicalprinciplesmayhaveconsequencesofphysicalsignificance,undecidableinweakertheories,suggestingthatitwouldbefutiletoseekanyaprioriglobalframeworkforappliedmathematics.(Thiswillbebroughtoutin3.)

Aswehavealreadyseen,themodalstructuraltreatmentofpure(p.96) mathematicsinvokescounterfactualsofastrictkind:allrelevantconditionscanbestatedintheantecedents(asthecategoricityproofsshow,whereapplicable).Thus,thenotoriousproblemsconcerningwhatrelevantbackgroundconditionsaretobeunderstoodasheldfixedininterpretingordinarycounterfactuals(associatedwiththeproblemofcotenability,cf.above,Chapter2,1)didnotariseinthecontextofpuremathematics.Whenweturntoappliedmathematics,however,thereisasenseinwhichtheproblemreturnstohauntus,asweshallsoonsee.

Inthefinalsections(4and5),variousapproachestothisproblemwillbeexplored.Howonereactstoit,infact,seemstodependonone'srealistcommitmentsconcerningnonmathematicalreality.Ifone'srealismissufficientlystrong,theproblemsseemstoevaporate.Butifthemsapproachseekstomaintainametaphysicalneutralityonsuchquestions,thornyproblemsariseintheveryformulationoftheappliedmathematicalcounterfactuals.Effortstoovercomethemraisesomeinterestingpointsofcomparisonwithrecentsyntheticapproachestophysicaltheories(motivatedbynominalistconcernsandaimedatchallengingtheallegedscientificindispensabilityofmathematicsentirely).1Asweshallsee,someofthetechnicalportionsofsuchwork(e.g.Fieldstylerepresentationtheorems)canbeofrelevancetoamsprogramme,butsuchtheoremsgobeyondwhatisrequiredincrucialways.And,fromourownperspective,thephenomenonofnonconservativenessofmathematicallyrichtheories(highlightedin3)tendstoundermineanysweepingchallengetotheindispensabilityofabstractmathematicaltheories.

Ofnecessity,wehaveconcentratedhereonproblemsofformulationinvolvedinamstreatmentofappliedmathematics,andonsomeofthetechnicalandphilosophicalquestionsimmediatelysurroundingtheseproblems.However,thebroaderquestionsofjustificationoftheineliminablepostulatesofmsmathematicsespeciallythemodalexistenceaxiomsmustnotbeforgotten.Inthisconnection,appliedmathematicscanprovideacrucialepistemologicallink,muchasithasbeenthoughttoprovideunder


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familiarplatonisttreatments.ThepointhereistoadaptQuineanindispensabilityargumentstothemodalframework:ratherthancommitmenttocertainabstractobjectsreceivingjustificationviatheirroleinscientific(p.97) practice,itistheclaimsofpossibilityofcertaintypesofstructuresthataresojustified.Moreover,totheextentthatindispensabilityargumentscanbeadaptedtothemodalapproach,theirusualplatonistthrustisactuallyundermined:afixedrealmofabstractobjectsisnotreallyshowntobeindispensable;ratheritistheweakerclaimsofpossibilitythatoccupysuchaposition.

1.TheLeadingIdeasMuchasinthecaseofpuremathematics,wemayattempttorepresentordinaryappliedmathematicalstatementsasellipticalformodalconditionalsofaspecifiedform.Suchconditionalsspelloutwhatwouldobtainwerethereanysuitablyrich(pure)mathematicalstructureinadditiontotheactualnonmathematicalobjectsorsystemstowhichweareapplyingmathematicalconceptsandtheories.Herethemodalityofthecounterfactualisalogicomathematicalone,justasinthetreatmentofpuremathematics.Althoughwemaybeapplyingmathematicstophysicalobjects,wearenotautomaticallyconstrainedtoholdphysicalornaturallawsfixedincontemplatingapurelymathematicalstructureinadditionforthepurposesofcarryingappliedmathematicalinformation.(Thus,forexample,wearefreetoentertainthepossibilityofadditionalobjectsevenphysicalobjectsofagiventype,toserveascomponentsofamathematicalstructure.Suchobjectscouldbeconceivedasoccupyingacertainregionofspacetimebutasnotsubjecttocertaindynamicallawsnormallystateduniversallyforobjectsofthattype.)Justwhatmustbeheldfixedisamattertowhichweshallreturnbelow.

Buthow,itmaybeasked,cananadditionalstructureforpuremathematics(suchasansequenceoracompleteorderedseparablecontinuum)bebroughttobearonmaterialobjects?Imaginingsuchastructurewhetherthoughttooccupyspacetimeornotdoesnogoodunlesswecanalsospeakofrelationsbetweenthematerialsysteminquestionanditemsofthemathematicalstructure.Thus,torepresentsimplecounting,forexample,itdoeslittlegoodtoentertainthepossibilityofansequenceinadditiontotheactualobjectstobecountedunlesswecanalsospeakofcorrespondencesbetweenthoseobjectsandtheitemsservingasnumbersofthehypotheticalsequence.

Onesolutiontothisproblemistomoveimmediatelytomodelsof(p.98) settheory,thatis,toentertainhypotheticallymodelsofasuitablesettheoryinwhichactualobjectsaretakenasurelements.Thenwehavetheoperationofsetformationappliedtothoseactualobjects,andtheusualapparatusofmappingsandnumbersystems(settheoreticallyconstrued)isavailable.Thismightbecalledtheglobalapproach,since,ifthesettheorychosenissufficientlyrich,itcanbeinvokedtohandlevirtuallyanypresentorforeseeableinstanceofappliedmathematics.

Whilethereisagooddealtobesaidinfavourofsuchanapproach(especiallyconcerningitsintuitiveness,itspower,anditssimplicity),thereisalsoindependentinterestinpursuingapiecemealapproach,inwhichwelimitthehypotheticallyentertained(pure)mathematicalstructurestoalevelthatisactuallyneededforthepurposesoftheapplicationinquestion.Inpart,suchanapproachismotivatedbyanindependentinterest


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inthesecondleadingquestionposedabove(howmuchmathematicsforhowmuchscience),whichthepiecemealapproachisforcedtoface.Therearealsolegitimateconcernsovertheconsistencyofpowerfulsettheories,andovertheirabstractness.Dowereallyneedtoiteratethepowersetoperationbeyondanythingthatwecouldbesaidtoexperience,beyondsaythelevelofspacetimeregions?Ifso,howfarbeyondsuchalevelneedwego?2

Ifwepursuethepiecemealapproach,howarewetobringahypotheticallyentertainedmathematicalstructuretobearonthematerialworld?Themoststraightforwardandgeneralwayissimplytocontinueemployingsecondorderformulationsaswehaveinthetreatmentofpuremathematics.Thisallowsustospeakofclassesofwhatevernonmathematicalobjectswerecognizeandofrelationsbetweensuchobjectsandthoseofahypotheticallyentertainedpuremathematicalstructure.Insuchaframework,therepresentationofagreatdealofappliedmathematicsisthenquitestraightforward.

Toillustrate,letusconsiderasimplestatementofnumericalcomparison,say,Therearemorespidersthanapes(andadefinitefinitenumberofeach).(Theparentheticalclauseisaddedsothatsomeapparatusofnaturalnumbersisrequired.)Usingthesecondorder(p.99) formalismofChapters1and2,withourlanguageexpandedtoincludetherelevantnonmathematicalpredicates(inthiscase,justspider(S(x))andape(A(x)),wecanrepresentthestatementby,

Xf[(X,f)nm(isa11correspondencebetweentheclassofallxsuchthatS(x)andthefpredecessorsofninX&isa11correspondencebetweentheclassofallxsuchthatA(x)andthefpredecessorsofminX&m


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wemuststipulatefromtheoutsetthattheonlypossibilitiesweentertaininemployingthearesuchastoleavetheactualworldentirelyintact.

Ofcourse,inmostapplicationsofmathematics,onlyaportionoftheactualworldisinquestion,andinsuchcasesitwouldsufficetopermitabroaderinterpretationofthe,allowingworldswhich(p.100) differfromtheactualeveninmaterialrespectssolongassuchdifferencesoccuronlyoutsidetheregionofapplication.Insuchcases,thereisnoreasoninprinciplewhytheatomiccomponentsofhypotheticallyentertainedpuremathematicalstructurescouldnotthemselvesbetakentobematerialobjectsofthesamesortasoccuractually.Moreover,insuchcases,roomcanbeallowed(literally)forfurther(mutuallydiscrete)concreteobjectstoservetheroleoforderedktuplesofthehypotheticalmathematicalobjectstogetherwithwhateveractualobjectsarerecognized.AndthenthesecondordervariablescanbeinterpretednominalisticallyinthemannerofChapter1,6.Still,itwillbenecessarytostipulatethatwhateverextramaterialobjectsareentertainedforsuchpurposesarecausallyisolatedfromtheregionofactualitytowhichthemathematicsinquestionisbeingapplied.

Obviouslytherearelimitstosuchanapproach,sinceappliedmathematicsmustalsomakeroomforcosmology,indeedforanyscienceinwhichthelargescalestructureofspacetimeisatissue.Insuchcases,itmaybenecessarytoentertainobjectsascomponentsofpuremathematicalstructureswhicharenotthemselvesinspacetime.Theoptionshereareboundupwithotherissuesconcerningtherealityofspacetime,and,atthispoint,wewishonlytoalertthereadertothequestion.Thetopicwillariseagainbelow,atwhichpointweshallhavemoretosayonit.

Alreadyitshouldbeclearthateventhemostelementaryappliedmathematicsonthemodalapproachisintimatelyboundupwithconditionsstipulatingthatatleastpartof(perhapsthewholeof)theactualworldbeheldfixed,whenreasoningabouthypotheticallyentertainedmathematicalobjects.Sofar,wehavestatedsuchconditionsinrathergeneral,globalterms,bringinginexplicitreferencetotheactualworldortheactualconditionorstateof(somepartof)theactualworld.Astermssuchascausallyisolatedsuggest,conditionsoffixityornoninterference,thusphrased,appeartoembodysomeratherstrongassumptionsofphysicalrealism,especiallytheassumptionthatitevenmakessensetorefertotheactualworld,ortheconditionofthissystemofphysicalobjects,apartfromanyrelativizationtoalanguageortheoryorconceptualframework,etc.Thisraisesoneofthemostinterestingquestionsthataninquiryintoappliedmodalmathematicsuncovers:Isthisapparentdependenceofthecogencyofappliedmodalmathematicsonnontrivialassumptionsofphysicalrealismagenuinedependence,(p.101) orcanitinprinciplebeeliminated?And,ifeliminationprovestobeimpossible,justwhatconclusionshouldbedrawnastothenecessarycommitmentsofthemodalstructuralapproach?Weshallreturntothesequestionsbelow(4),afterhavingfirstdealtwiththerelativelymoretractableissuesconcerningthestrengthofsuitablemathematicalframeworks.

Beforeproceeding,letusnoteafurtherassumptionimplicitinappliedmodalmathematics.Justasinthecaseofpuremathematics,theremustalsobeaxiomsofmodalexistenceof


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thepossibilityofstructuresfulfillingtheconditionsoftheantecedentsofthecounterfactuals.Withoutsuchaxioms,ofcourse,allcounterfactualswiththeantecedentinquestioncouldbevacuouslytrue,andthetranslationpatternwouldbreakdown.Here,andinwhatfollows,itshouldbeunderstoodthattheappropriatemodalexistencepostulatemustaccompanyafullyexplicitformalizationoftheappliedmodaltheory.(Itshouldbynowbeclearhowtowriteoutsuchpostulates,andwewillnotstoptodoso.)Duetotheincreasinguncertaintiesofsuchpostulatesaswemovefurtherfromtherealmofexperience,thereisanaturalmotivationonthemodalapproach,asontheplatonistforseekingtocarryoutasmuchappliedmathematicsaspossiblewithinaminimalmathematicalframework.Someaspectsofthiswillbeconsideredinthefollowingsections.

Returningtoillustrations,letusconsiderscalarmagnitudessuchasmass(say,nonrelativistic,forsimplicity).Onstandardplatonisttreatments,suchaquantityisrepresentedasarealvaluedfunctiondefinedonadomainofobjects,eitherparticles,orspacetimepointsorregions,togetherwithagivenoperationallyspecifiedunit.Supposetheworst,thateachspacetimelocationistobeassignedarealmass(ordensity).Onthemodalstructuralapproach,itsufficestoentertainasingleseparableorderedcontinuum(asdefinedinChapter1,5),whoseelementsserveasrealnumbers.Thesenowcanservethedoubleroleofrepresentingspacetimepoints(viaapairingfunctionwhichallowsustospeakoforderedquadruplesofrealsintheusualway)andofrepresentingthevaluesofscalarquantities.Aquantitysuchasmassisthenasecondorderobjectandcanevenbetakenasasubsetofreals,eachsuchcodinganargumentandcorrespondingvalueviathefixedpairingfunction.Thus,torepresentastatement,ordinarilywrittenas

(p.102) wheremisafunctionconstantintroducedasabbreviatingmass,wecanwrite(followingthenotationofChapter1,5):

(3.2)wheretheclauseinquotesmustbespelledoutasfollows:

(i)Ftakeson(withinlimitsofexperimentalaccuracy)allactuallymeasuredvaluesexperimentallydeterminedasvaluesofmass;(ii)Fagrees(withinexperimentallimits)withalltheoreticallypredictedvaluesofmassunderrealworldconditions(thetheorybeingNewtonianmechanics).

Ifouroriginalsingularstatementisunderstoodasalowlevelempiricaloneonlylooselytiedtoatheory,asexpressedinclause(ii),theseconditionsareprobablyadequate.However,ifthestatementisunderstoodaspartofanapplicationofawholetheoryasitwouldbeinanysophisticatedapplicationofmathematics,thenafurtherconditionmustbeaddedtotheeffectthatthemassrepresenting(mathematical)objectsatisfiesthelawsofthetheory,orispartofamodelofthoselaws.(Whetheranintrinsicsecondorder

m(x) = r,

Xf[( ) F(FrepresentsmassRA2)X (


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statementofthisoveranRA2structurecanbegiven,orwhetherascenttoricherstructuresisrequired,willdependonthedetailedformulationofthetheoryinquestion.)Givensuchacondition,theutilityofnormalmathematicalapplicationsinpermittinginferencesastofurtherbehaviourofthesysteminquestionwillbeaccountedfor,muchasitisonfamiliarplatonist(modeltheoretic)treatments.

Intuitively,(3.2)canberead,Werethereanyseparableorderedcontinuum(noninterferingwiththeactualmaterialworld),therewouldbeamassrepresentingfunctionassigningthevaluer(ofthecontinuum)topointx,wheremassrepresentingisspelledoutassuggested.Nowitshouldbenotedthatthereferenceintheseclausestomeasuredvaluesofmassandpredictedvaluesofmassmustbeinterpretedintermsofoperationalproceduresandsymboliccalculations.Sincetheseconditionsenterintothehypotheticalconditionalsdesignedtoreplaceapparentreferencetomathematicalobjects,clearlyvaluescannotbetakenasreferringtomathematicalobjects.(Hencethehyphensinvaluesofmass.)Rather,measuredvaluesshouldbeunderstoodintermsofconcretepointerreadings,(p.103) generallyassociatedwithcertainsymbolsforrealnumbers.(Morerealistically,theywouldbeassociatedwithsymbolsforrationalnumbers;and,insomeinstances,somesequenceofsuccessivelymoreandmoreaccuratemeasurementsmaybespecifiedforgeneratingarealnumber.)Calculationsmaybeinvolvedaswell(astheytypicallyareinanysophisticatedmeasurementprocedure),andthesegeneratenumbersymbolsaswell(e.g.decimalorbinaryrepresentations,etc.).Thesethentakeonadefinitemeaningwhenahypotheticalstructureforthereals)isentertainedinaconditionalsuchas(3.2).(Togetherwithanyseparableorderedcontinuumthereisanassociatedcorrespondencebetweennotationsusedinpracticeandthepointsofthecontinuum.Thisisinducedbyarepresentationofrationalswithinthecontinuum(cf.Chapter1,5).)

Noristhisappealtooperationalproceduresandcorrespondencesbetweennotationsandvariouswaysofidentifyingrealnumbersandsoforthapeculiarityofamodalstructuraltreatment.Standardplatonisttreatmentsofappliedmathematicsimplicitlymustinvokequitesimilarmachinery,andmustrecognizearelativityofreferencetowaysoftakingnaturalnumbers,rationalnumbers,realnumbers,etc.Forthemostcommonformofplatonism,allsuchobjectsaresettheoreticconstructions,and,ofcourse,aninfinitevarietyofthesecanservethepurposesofmathematicalpracticeequallywell.Withinsettheory,ordinaryreferencetoarealnumber,say,isrelativetoaconstrualofthenaturalnumbersassets,toapairingfunction,toconstructionsofnegativeintegersandrationals,andtoconstructionsofreals(e.g.viaDedekindcuts,orCauchysequences,etc.).Themodalstructuralistmerelydoesallthisrelativityonebetterindispensingwithanyactualmathematicalobjectsatallintermsofwhichreferencetomathematicalobjectsisunderstood.

Inanyparticularcase,whetherahypotheticallyentertainedmathematicalobjectrepresentsaphysicalmagnitudeistosomeextentavaguematter,duetotheneedtotakeintoaccounttheapproximatenatureofmeasurementproceduresinmostscientificapplications.Thisis,ofcourse,reflectedinthereferencestolimitsofexperimental


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accuracyintheaboveclauses.Thismeansthat,ingeneral,therewillbemultiple,extensionallydivergentmathematicalobjects(functions)thatqualifyequallywellasrepresentationsofamagnitude.Whether,forinstance,torepresentthepathofanobjectthroughspaceasacontinuousfunctionoftimeoras,say,apiecewisecontinuousone,orevenahighlydiscontinuousone,orwhetherevento(p.104) representtimeasacontinuuminthefirstplace,aremattersunderdeterminedbyanydirectexperimentalprocedures.Thus,choicesmustbemadeonothergrounds,andprobablythemostdecisivegroundsinmanycasesareconsiderationsofsimplicityandconvenienceconsiderationssuchasthatawelldevelopedmathematicaltheoryofcontinuousfunctionsexistsenablingustoperformvitalcalculations,andthat,onsuchpracticalgrounds,weseekatheorycouchedinmathematicaltermsthatwecanhandle.(AsChihara[1973]bringsout,itisjustthisslackthatraisestheprospectsofconstructivistsubstitutesforclassicalappliedmathematics.)This,inturn,raisesthornyquestionsconcerningtheconventionalityofourscientifictheories,questionsthatcertainlycannotberesolvedhere.Inmyownview,itseemsobviousthatanysophisticatedapplicationofmathematicstothematerialworldinvolvesasignificantdegreeofidealization,whichimpliesasignificantdegreeofconventionalchoicebasedonpragmaticconsiderations.Atthesametime,thisbynomeansunderminestheobjectivityofourscientifictheories,providedthatthatobjectivityisproperlyunderstood.While,ingeneral,wecannotsaythattherearesuchandsuchmagnitudesinnaturerepresentedpreciselywithinauniquemathematizedtheory,westillmaybeabletosaythatnatureissuchastopermitrepresentationwithinarangeofmathematicalmodels,andthatthisrangeincludessuchandsuchmathematicallyprecisedescription.Perhapsthisisalltheobjectivityweeverrequire.Inanycase,withoutpursuingthisfurtherhere,sufficeittosaythatanythoroughaccountofappliedmathematicsmustatsomestagecometogripswiththesequestions.Theyarebynomeanspeculiartoastructuralisttreatment.

Withtheseessentialsoftheapproachinmind,letusnowturntothequestionofhowrich,mathematically,ourhypotheticalstructuresneedtobeinordertosupportapplicationsofourbestmodernphysicaltheories.

2.CarryingtheMathematicsofModernPhysics:RA2asaFrameworkAshasalreadybeenindicated,theRA2frameworkisknowntobeaverypowerfultheorywithregardtotherequirementsofapplied(p.105) mathematics.4Moreover,aswillbebroughtoutbelow,thereisasenseinwhichitdefinesalimitofnominalism,alimittothemathematicalrichnessofwhatcanbeconceivedofasconcretestructures.Thus,therearespecialreasonsforfocusingontherepresentingpowersofRA2.Canitreallydojusticetomodernphysicaltheories,especiallyGeneralRelativityandQuantumMechanics?Afullscaletreatmentofthiswouldtakeustoofarafield;butabriefglimpsewillsufficetocallattentiontosomeofthefascinatingissuesthatariseinthisarea.

InthecaseofGeneralRelativity,mattersarecomplicatedbythefactthatthetheoryisstandardlypresentedintwoverydifferentways.Ontheonehand,thereistheextrinsicpresentationfamiliartophysicistsinwhicheverythingiscarriedoutexplicitlyintermsofcoordinatesystemsandtransformationsamongthem;ontheother


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handthereistheintrinsicorinvariantpresentationintermsofabstractgeometricobjectsmakingnoreferencetocoordinatesystems.5Now,itistheintrinsicpresentationthatismathematicallymoreelegantand,moreover,canbearguedtoprovideaclearerideaofthecontentofthetheoryandofsuchmattersashowitcompareswithotherspacetimetheories(e.g.Newtoniangravitation).However,ordinaryphysicalapplicationsmakeuseofcoordinatesystems,sothatarepresentationoftheextrinsicpresentationmayberegardedasadequateformostpurposes.

Solongasweremainwiththeextrinsicpresentation,thereislittledoubtthatthesystemRA2ispowerfulenoughtoexpressandderivewhatisnormallyrequired.(Thatis,say,standardtextscouldbesystematicallytranslatedintoRA2andallresultsderived.)Geometricobjectsvectors,tensors,etc.aretreatedintermsoftheircomponentsinacoordinatesystemandtherulesfortransformingthemtootheradmissiblecoordinatesystems.Coordinatesystemscanbeviewedas11mapsfromregionsofspacetimeto4;andthecomponentsofageometricobjectaregivenbysuitablycontinuousrealvaluedfunctions.(Forexample,thecomponentsofatangent(p.106) vectorfieldTtocurve(=(u)beingasmoothmapfromanintervaloftospacetime),relativetocoordinatesxiconsistinfourfunctions, ,i=0,1,2,3.)Anysuchfunctioncangenerallybecodedbyasinglereal,beingdeterminedbyitsbehaviouronacountablesubdomain.6(Thinkofthesimplestcaseofacontinuousfunctionfromto:arealcancodeitsbehaviouratrationalarguments.Evencountablymanydiscontinuitiescanbeallowed.Inthecaseof,say,tensorfieldson(aregionof)spacetime,acountablesubdomainof4,asdescribedinagivencoordinatesystem,sufficestodeterminethefield.Solongasweremainwithinasinglecoordinatesystem,itmakesnodifferencewhetherweregardthefieldasdefinedon(partof)spacetimeoron(partof)4itself,viathecoordinatefunctions.Inaninvariant,coordinatefreepresentation,however,thedistinctionbecomessignificant,andinsomecasesleadstogreaterabstractness,sincewethenmustineffectkeeptrackofallcoordinatesystemsatonce.)Thetransformationsdetermining,say,ageneraltensoroperateonfinitelymanycomponentfunctions,hence,bycoding,onfinitelymanyreals,andyieldrealscodingtransformedfunctionsasvalues,clearlywithinRA2.And,asjustsuggested,a(suitablycontinuous)tensorfieldcanbedescribed,relativetoacoordinatesystem,byafunctionwhichgivesthecomponentfunctionsasvaluesonacountabledensesubdomainof4.Hence,atensorfieldcanbecodedasasinglerealnumber!Andalltheusualoperationsonsuchfields,includingcovariantdifferentiation,canbeintroducedasfunctionsfromrealstoreals,withinRA2(which,recall,includesthefullsecondordercomprehensionscheme).ThismuchshouldatleastmakeplausibletheclaimthatallthemathematicsactuallyrequiredinanyordinaryphysicalapplicationofGeneralRelativitycanbecarriedoutwithouttranscendingthirdordernumbertheory(equivalentlyRA2).(And,bymakingsufficientrelianceonapproximatingfunctions,agreatdealcanprobablybecarriedoutinapredicativesubsystemofanalysis,i.e.ofPA2.)7

Forordinaryapplications,thestorycouldendhere.However,notallapplicationsneedbeordinary.Thisforcesustoraiseadifficult(p.107) question:istheintrinsicformulationreallydispensableinfavouroftheextrinsicforallscientificpurposes?Evenif

d( )xidu


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alltheusualsortsofapplicationsinvolvingspecificcalculationscanbecarriedoutinRA2orinsomeweakersystem,thisdoesnotsettlethematter,fortherearequestionsoftheoreticalimportancethatgobeyondsuchapplications,butwhichamathematicalframeworkoughttobecapableofrepresentingifitistodojusticetothescientificenterprise.Asacaseinpoint,relevantinthepresentcontext,considerthewholeissueofrelativityprinciplesandrequirementsofgeneralcovariance,thoughtbymany(includingEinstein)todistinguishGeneralRelativityfromflatspacetimetheories.Remainingatthelevelofcoordinatebasedformulations,onecanreadilybemisledintothinkingthatGeneralRelativitydiffersfromflatspacetimetheoriesinsatisfyingsuchaprinciple,sinceoneconsiderstransformationsamonggeneralcurvilinearcoordinatesratherthanaprivilegedclassofinertialsystems.However,ifoneconsidersintrinsicformulations,itbecomesevidentthatthisismistaken,andthatthedemandofgeneralcovariancethatdynamicallawsretaintheirformunderarbitrarytransformationsamongcoordinatesystemsreallycomestonothingmorethanthedemandthatthoselawsbegivenanintrinsiccoordinateindependentformulation,somethingthatispossibleforflataswellascurvedspacetimetheories.8Infact,ifonelooksatspacetimetheoriesmodeltheoretically,oneseesthatextrinsicformulationsintermsofequationsinvolvingcoordinatespickoutawelldefinedclassofmodelsonlyrelativetoachoiceofcoordinates.Inadifferentsystemofcoordinates,thesamedifferentialequations(e.g.onelookinglikeageodesicequation)willpickoutadifferentclassofgeometricobjects(e.g.tangentvectorfields),henceadifferentclassofmodels.9Intrinsicformulationsautomaticallyovercomesuchproblems.

Now,ifweunderstandappliedmathematicsbroadlyenoughto(p.108) includethetheoreticalinsightthatsuchcomparisonsyield,whatevermathematicsisrequiredintheabstractintrinsicformulationcannotreadilybedismissedasdispensable.Granted,thisisanunusuallybroadinterpretationofappliedmathematics.Butthetheoreticalunderstandinginquestionisattheheartofthesciences,and,ifadispensabilityargument(totheeffectthatmathematicalstructuresricherthanXarenotneededtodonaturalscience)istocarryphilosophicalforce,suchconsiderationsmustbetakenintoaccount.

Thus,wecannotavoidconsideringtheproblemofrepresentingthemathematicsoftheintrinsicpresentation.Whilethemattercannotbedefinitivelysettledhere,thesituationseemstobethis:theabstracttheoryofmanifoldstranscendstheRA2framework,butessentiallyonlyattheearlieststages,namelyintheabstractcharacterizationofmanifoldsthemselves.Oncegivenamanifold,itappearsthat,infact,withsufficientrelianceoncodingdevices,thesystemRA2iscapableofrepresentingtherestofthemathematicalsuperstructureofabstractdifferentialgeometryemployedintheintrinsicpresentationofGeneralRelativity.Moreover,agreatmanyparticularmanifoldsactuallyencounteredinspacetimephysicscanbeintroducedexplicitlyinRA2,makinguseofsecondorderlogic.

Theintrinsicformulationbeginswiththeideathat,atleastlocally,spacetimehasthestructureofan4dimensionalsmooth(C)manifold.AnndimensionalCmanifoldconsistsinanarbitrarynonemptysetMtogetherwithamaximalsystemofcharts11


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mapsfromsubsetsUofMtoopensetsofnsuitablyinterrelatedsoastoinducethelocalsmoothnessstructureofEuclideannspaceonM.10Moreprecisely,annchartisapair(U,f)whereUMandfisa11mapfromUontoanopensetofn;annsubatlasonMisafamilyofnchartssuchthat(1)theycoverM,i.e.theunionofthedomainsofthechartsisM;(2)foranytwodistinctpointsp,qofM(p.109) therearecharts(U1,f),(U2,g)withpinU1,qinU2andU1U2=(Hausdorffproperty);and(3)anytwocharts(U1,f),(U2,g)ofthefamilyarecompatible,meaningthatfg1andgf1aresmooth(C),wheneverdefinedonopendomains.Finally,annatlasisobtainedfromannsubatlasbyaddingallnchartscompatiblewithallthoseofthensubatlas.MtogetherwithannatlasonMisanndimensionalCmanifold.

Notethat,whilewecouldperfectlywelltakeMtobe,i.e.toconsistofpointslabelledasrealnumbersforthepurposesofacodinginalogicalformalism(whilestillabstractingfromanyundefinedmetricortopologicalstructure),andwhilewecanalwayscodethecompositemapsfg1,etc.asreals(beingdeterminedbytheirbehaviouronacountabledensesubsetofn),wehavenomeansofcodingthefofthechartsasreals.ChartsaresecondorderRA2objects.Thismakesmaximalfamiliesofcharts(atlases)essentiallythirdorderoverthereals,whichiswhythegeneralnotionofmanifoldtranscendsRA2.

Moreover,whenwedevelopcalculusonmanifolds,itappearsthatwearebeyondRA2onceandforall,for,onceweleavecoordinatesbehind,theverynotionofavectorbecomesapparentlytooabstract:Avectorisstandardlytakenasaderivativeoperator(derivation),i.e.asamapfromallrealvaluedsmoothfunctionsaboutamanifoldpointmtothereals,meetingtherequirementsoflinearity((f+g)=(f)+(g),(af)=ay(f))andtheLeibnizianproperty((fg)=(f)g(m)+f(m)(g)).Assuchvectorsarethirdorderobjectsoverthereals,beyondRA2,andthentensorsandmoregeneralderivativeoperatorsareatleastasabstract.However,withoutappealingtocoordinates,wecanequivalentlytakevectorstobetangentvectorstosmoothcurves(fromaconnectedintervalIoftothemanifolddomainM).Givenasmoothcurveands0Iwith(s0)=p,atangentvector|ptoatpcanbeintroducedvia

forallsmoothrealvaluedfunctionsfonaneighbourhoodofp.(N.B.ThenotionsofsmoothnessformapsfromtoMorMouttoareintroducedviathesmoothnessofcompositemapsfromtonorfromntogivenbythechartfunctionsandtheirinverses.)Now,fisasmoothfunctionfromto,andthuscanbecodedbyareal.Moreover,thederivativeoperatorontherightiscontinuousso(p.110) it,too,canbecodedbyareal.Thus,abstractvectorsarebroughtdowntotheleveloffirstorderRA2objects.(TheproofthateveryoriginalabstractvectorcanbetakenasatangentvectortoasmoothcurveinMisstandard,but,unfortunatelyfortheRA2reductionist,itcannotbestatedinRA2.)

Butnow,tangentspaces,dualspaces,tensors,tensorfields,covariantderivativeoperatorsinsum,allthefurtherapparatusneededtocarryouttheintrinsicformulation

(f) = (f ) ,pdds

s0


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ofGeneralRelativitycomewithinthescopeofRA2.ThetangentspaceVmatapointmofthemanifoldisthesetofalltangentvectorsatm,andisavectorspace(ofdimensionn)overthereals.Itcanbetakenasasetofrealsviathecodingofvectorsjustindicated.ThedualspaceVm*oflinearrealvaluedmapsonVmisthenalsoidentifiableasasetofreals:eachinVm*isdetermined,bylinearity,byitsactiononfinitelymanybasisvectorsinVm(whichcanbechosenarbitrarilyfromanycoordinatesystemaboutm);hencecanbecodedbyareal.Next,ageneraltensorasamultilinearmapTfromfinitecrossproductsoftheformVmVm...VmVm*Vm*...Vm*intoRisdeterminedbyitsactiononbasisvectorsinthecomponentspacesmakingupthedomainofT;henceTitselfcanbecodedasasinglereal.Thus,smoothvectorandtensorfieldsmapsassigningvectorsortensors,respectively,topointsofopensubsetsofMcomewithinthepurviewofRA2,assetsofreals.(HerewemakeuseofreallabelsofthepointsofM.Thenafieldcanbecodedasasetoforderedpairsofreals,henceasasetofreals.)

Withonemorestepwehaveessentiallyallthatisneeded:wemusthaveawayoftalkingabout(quantifyingover)covariantderivativeoperators,orconnectionsonM.SuchaconnectionDisintroducedasanoperatorassigningtoCfieldsXandY,withadomainA,aCfieldDxYwithdomainA,obeyingfourconditions(ensuringappropriatelinearityandLeibnizianbehaviour).Primafacie,suchoperatorsarethirdorderoverthereals,andbeyondRA2byonelevel.However,theconditionsonDimplythatitisuniquelydeterminedinanyopendomainbyitsactionDeiejonafixedfinitebasefielde1...enofindependentCvectors.11Sincetheei(asfields)are(p.111) codedassetsofreals,thismeansthat,onthedomainofthesefields,Dcanberepresentedasasetofrealscodingathreeplacerelation(thatis,eiejDeiej).Now,ifoneassumes(asoneusuallydoesinthecontextofGeneralRelativity)thatthemanifoldMisseparablei.e.thatitcanbecoveredbycountablymanychartdomainsthencountablymanysuchthreeplacerelations,codableasasinglesuch,sufficetorepresentDthroughoutthemanifoldM.Inthismanner,evenquantificationoverconnectionsisbroughtwithinthescopeofRA2.

Still,remarkableasallthismaybe,wearenotabletostatemuchlessproveinRA2fundamentalgeneraltheoremsonmanifolds,suchasthetheoremthatthereexistsauniqueRiemannianconnectionona(semi)Riemannianmanifold,orthetheoremthatametrictensorgabonamanifolddeterminesauniquegeneralderivativeoperatorcompatiblewiththismetric.ForrecallthatthegeneralnotionofmanifoldisnotavailableinRA2.Thebestwecandoisintroduceparticularmanifoldse.g.themanifoldn,orthenspheremanifold,etc.byexplicitlyaxiomatizingasystemofcharts.(Forinstance,insecondorderlogic,wecanwritedownanexplicitdefinitionofthepredicateisachartofthenmanifoldatlas:wesimplyspecifytheidentityfunctionsonopensetsofn(asapointsetwiththeusualtopology),whichgivesaCsubatlas;thenwespecifythatanychartcompatiblewithallthoseofthesubatlasarechartsofthenmanifoldatlas.)Again,forordinaryapplications,suchproceduresareprobablyadequate.Butwithoutfundamentaltheoremsondifferentiablemanifoldsofthesortmentioned,wecanhardlyclaimtodojusticetotheintrinsicviewpoint.


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(p.112) Ourdiscussion,thusfar,haspresupposed,forthesakeofargument,thatvariouscodingdevicesarelegitimateinreducingmathematicalmachinerytolowlevelsofabstractionoverthenaturalnumbers.However,itshouldberecognizedthatdelicateproblemsofjustificationariseinconnectionwithsuchdevices.Thesepertaintothestatusofthemathematicalknowledgeemployedintheintroductionofacodinginthefirstplace,e.g.whenonesaysthatanabstractoperatorisuniquelydeterminedbyitsactiononfinitelymanyorcountablymanyargumentsofacertainsort,andthereforecanberepresentedby,say,arealnumberorasetofreals,etc.Inwhatsenseofcouldcouldamathematicalphysicistcarryoutallrelevantderivationsandcalculationswithinthecodingframework,withouteversteppingoutsideinordertoberemindedofwhatonearthisreallygoingon?Sinceournonreductionistconclusionscanbebasedonthemoredecisivecasethat,evenwithcoding,relevantmathematicsgoesbeyondeventhefullpowerofRA2,wehavepreferredtodevelopthatcasewithoutexaminingthedelicateepistemicproblemsraisedbyappealstocoding.But,webelieve,thoseproblemsaregenuineanddeservefurtherinvestigation.

WhenwecometoQuantumMechanics,thesituationisatleastasproblematicasinthecaseofGeneralRelativity.Inmanyordinaryapplications,agreatdealofthemathematicscancertainlybecarriedoutwithintheframeworkofRA2,butonceweconsidermoretheoreticalandfoundationalmatters,weseemtorequiremoreabstractstructures.Instandardcases,quantumstatescanberepresentedassquareintegrablecomplexvaluedfunctionsonanunderlyingrealspace,andmoreoveracountablecollectionofcontinuousfunctionsservesasabasisintheHilbertspaceofsuchfunctions.12Thus,arbitraryquantumstatesinsuchaspacecanberepresentedbyacountablesequenceofbasisfunctions,eachcodableasareal,hencebyareal.Linearoperatorsonsuchfunctionsarethen,primafacie,atthelevelofsecondorderRA2variables,asarethe(closed)subspacesoftheHilbertspace(identifiablewiththeprojectionoperators).Ifwenowconsiderprobabilitymeasurescountablydisjointlyadditive[01]valuedfunctionalsonthesubspacesoftheHilbertspacerepresentingthesystem(where,heredisjointnessofsubspacesmeanstheyareorthogonal)wehave,primafacie,climbedpast(p.113)RA2.However,givenseparabilityoftheHilbertspace,13eachsubspaceScanitselfbeidentifiedwithacountablecollectionfi=CofbasisvectorssuchthatCspansSandCisdenseinS.And,giventhateachofthefiiscodableasareal,soiseachsubspace.Andthen,anyprobabilitymeasurecanberepresentedasafunctionfromrealstoreals,i.e.atthesecondorderinRA2.

Infact,duetoanimportanttheoremofGleason[1957],allmeasuresonthesubspacesofaHilbertspaceofdimension3aregivenbythequantummechanicalalgorithm,thatis,theyareinducedbythepureandmixedquantumstatestogetherwiththeusualrulesforcalculatingprobabilities.Hence,againundertheaboveassumptions(separabilityoftheHilbertspaceandrealcodabilityofbasisfunctions),themeasuresthemselvescanbetakenasreals(i.e.codingthedensitymatriceswhichrepresentthepureandmixedstatesontheusualpresentationofthetheory).Thus,reasoninginvolvingprobabilitymeasuresinthevastpreponderanceofordinaryapplicationsofquantummechanicscanbecarriedoutinRA2,andprobablyevenintheoriesconsiderablyweakerthanRA2.


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However,inordertoarriveatthisconclusion,weassumedthattheHilbertspacerepresentingthephysicalsystembeseparable.And,furthermore,totakeadvantageofGleason'stheorem(permittingtherepresentationofmeasuresasrealsratherthansetsofreals),weimplicitlymadeuseofenoughmathematicstoproveGleason'stheorem.Infact,theproofofGleason'stheoremservesasaniceillustrationofthepotentialphysicalsignificanceofmathematicsthattranscendsRA2.

AsatheoremaboutprobabilitymeasuresonthesubspacesofseparableHilbertspaces(ofdimension3),Gleason'stheoremcansurelybeprovedinRA2.AllthehardworkintheprooftakesplaceinEuclideanthreedimensionalspace;moreover,everyinfinitedimensionalseparableHilbertspaceisisomorphicto2,thespaceofinfinitesequences(x1,x2...,xk,...)ofcomplexnumberssuchthat(p.114) k=1|xk|2isfinite,withtheinnerproductof(ak)and(bk)givenbyk=1ak*bk.(ThisfactcanbeprovedinRA2itself,makinguseofanintrinsicsecondorderstatementofwhatitistobeaninfinitedimensionalseparableHilbertspace,onexactlytheplanofthesecondorderlogicaldescriptionofmathematicalstructureswehavebeenusingthroughout.Secondorderlogicsufficeshere,takingvectorsinaHilbertspacetobefirstorderobjects.)Andeverythingweneedtosayaboutmeasuresonthesubspacesof2canbesaidwithinRA2alongthelinesalreadyindicated.

However,whathappensiftheassumptionofseparabilityisdropped?Theaboverepresentationviacodingthenclearlybreaksdown.Yet,thereisageneralizationofGleason'stheorem,provedinanabstractsetting,whichappliestononseparableaswellasseparableHilbertspaces.14Thistheoremprovestheexistenceofauniquefunctionw(p,b)definedontheatomspandpropositionsbofanarbitraryquantumpropositionsystem(representedbythelattice&Sscr;(H)ofclosedsubspacesofanarbitraryHilbertspaceofdimension3),wherew(p,b)satisfies:

(i)0w(p,b)1,(ii)p0,thereisanfisuchthat

wherethenormisdefinedbythescalarproductonHvia

Forfurtherdetailsonthemathematicalformalismofquantummechanics,seee.g.Jauch[1968].

(14)SeePiron[1976],pp.7381.Apreludetothisisanelaboraterepresentationtheoremtotheeffectthatanarbitraryquantumpropositionsystemcanberepresentedbyafamilyoflatticesofclosedlinearvarieties(subspaces)overabstractHilbertspaces,whichneednotbeseparable.Whatwouldserveasaminimalframeworkforprovingthisrepresentation?Sinceitsphysicalsignificanceisunclear,wehavepreferredtoconcentrateonGleason'stheorem,someofwhosephysicalcontentiseasiertospecify.Incidentally,thissourceprovidesaquitereadableproofoftheextremalcaseofGleason'stheorem.

(15)AnoncontextualhiddenvariablestheorycanbeunderstoodasdemandingdispersionfreemeasuresonthesubspacesofHilbertspace,i.e.measuresrepresentingstateswhichassignprobabilityeither0or1toeverystatementoftheformquantummagnitudeA(pertainingtothesystemrepresentedbythegivenHilbertspace)hasavalueinBorelsetS.Suchahiddenvariablesprogrammeiscallednoncontextualbecauseittreatsthequantummagnitudes(representedbylinearHermitianoperatorsontheHilbertspace)astheystand,withoutrelativizingthemtoexperimentalcontext(ortomaximalcompatiblesetsofoperators,etc.).ItisanimmediatecorollarytoGleason'stheoremthat,iftheHilbertspacehasdimension3,therecanbenodispersionfreemeasures(asquantumstatesalwaysexhibitdispersionforsomemagnitudes,astheyrespecttheHeisenberguncertaintyrelations).Forfurtherinformationonthetopicofhiddenvariablesandnohiddenvariablesproofs,seee.g.Belinfante[1973],Clauserand

f < ,fi

g = (g,g). 2


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Shimony[1978],andreferencescitedtherein.

(16)Thereisstillthepossibilitythatphysicaltheoriescanbenominalizedwithinapurelyrelational(synthetic)spacetimeframeworkemployingthemethodsofField[1980].Weshallhavemoretosayaboutthisbelow(4).ButnoteherethatFieldworkedoutasingleexampleNewtoniangravitationand,aswenotedabove,inthiscase,spacetimepointsandregionssufficeforrepresentingthemathematicsofthetheory.Infact,asourremarksaboveimply,itisnotevennecessarytointroduceapurelyrelationalspacetimeversionofthetheory;onecanreinterpretthemathematicsdirectlyviacoding.AndthenonebypassesthesortofrepresentationtheoremthatFieldhighlighted.(Theuseofsuchatheoremconfrontstechnicaldifficulties,tobereviewedbelow,4.)Moreover,onerespectsthemathematicsandneednotadoptaninstrumentaliststance.ButmoreimportantinthepresentcontextthereseemtobeseriousobstaclesinthewayofField'sprogrammeasastrategyfornominalizingmodernphysicaltheories,especiallyquantummechanics,inwhichthedomainsofthemodelsarealreadyhighlyabstract,andwhichdonotlendthemselvesreadilytoaspacetimereformulation.(Inthisconnection,seeMalament[1982].)Whethertheprogrammecanbemadetoworkevenfornonflatspacetimetheories,e.g.GeneralRelativity,is,Ibelieve,anopenquestion.

(17)AswassuggestedinPutnam[1967].

(18)Forasurveyofrecentresults,seeHarringtonetal.[1985].

(19)AtheoryT2isconservativeoverT1iffeverysentenceSinthelanguageofT1thatisprovableinT2isalsoprovableinT1.If,inaddition,everytheoremofT1isatheoremofT2,T2iscalledaconservativeextensionofT1.

(20)SeeGdel[1947].

(21)Forasurveyofresults,seeNerodeandHarrington,inHarringtonetal.[1985],pp.110.Anexampleofanintuitivemathematicalstatementthatrequiresuncountableiterationsofthepowersetoperationtoprove(specifically,thatisprovableinZFCbutnotinZC)isthestatement,EverysymmetricBorelsubsetoftheunitsquarecontainsorisdisjointfromthegraphofaBorelfunction.

AseriesofresultsisbasedonanalysingCantor'sdiagonalargumentthattheunitintervalIisuncountable.ThatargumentproducesaBoreldiagonalizationfunctionF:IIsuchthatnoF(y)isacoordinateofy.ButCantor'sFdependsontheorderinwhichthecoordinatesofyaregiven.IfFisrequiredtobeinvariantinthesensethatF(y)=F(y)wheneveryandyhavethesamecoordinates,thenthestatementIfF:IIisinvariant,thensomeF(x)isacoordinateofx,isprovableinZCbutnotinZFCwiththepowersetaxiomdeleted.VariousmodificationsofthisidealeadtoexamplesofsimilarstatementsthatareprovableinZFC+alargecardinalaxiom,butnotinZFC.AndthereareevenexamplesofstatementsofthissortwhichareprovableinZFC+thereisameasurablecardinalbutnotinZFC+thereisaRamseycardinal!Thus,evenattheleveloflarge,largecardinals(incompatiblewiththeAxiomofConstructibility,as


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Ramseycardinalsalreadyare),itisnecessarytogoevenfurthertoprovecertainnaturalmathematicalstatements.

(22)NB.TheindependenceproofsofCohen[1966]canbeadaptedtosecondorderaxiomatizationsofsettheory.SeeChuaqui[1972].

(23)Itwouldbebettertosayrealnonmathematicalsituation,sincepresumablytheproblemshereariseindependentlyofcommitmenttoamaterialistontology.Dualists,phenomenalists,etal.,mayseektoapplymathematicsinvariousways,andtheytoowouldneedtoinvokeanassumptionofnoninterferenceorfixityofthenonmathematicalworldrecognized.Hereandbelow,thisshouldbeunderstood,thoughforbrevityweshallsometimesspeakofmaterialconditions,etc.

(24)SomethinglikesuchastipulationliesbehindHorgan's[1987]reasonablesolutiontoarelatedproblemraisedbyHaleandResnik[1987]concerningHorgan'suseofcounterfactuals[1984]tonominalizeappliedmathematics.Theobjectionwasthatordinarypatternsofexplanationofmathematicsfreestatements,O(e.g.observationstatements),wouldbeupsetunderacounterfactualinterpretation,becauseinthemodalizedtheoryonewoulddeduce,notO,butsomethingoftheform(AO)(whereAisanappropriateantecedentinvolvingmathematicalaxioms).Horgan'ssolutionis,insuchcases,toaddasanaxiom,

onthegroundsthat,ifOweretoholdundertheconditionsenvisionedinA,thenitholdsinfact.ThisisreasonablebecauseinentertainingA,weentertainnoalterationofnonmathematicalmattersoffact.

(25)SeeespeciallyrecentwritingsofGoodman[1978]andPutnam[1981].

(26)IreferheretoworkofField[1980]andBurgess[1984]towhichwewillreturnbelow.

(27)Asthetermindependentisquiteambiguous,thissortofstatement(oritsnegation,favouredbyvariousantirealisms)mustbearticulatedwithsomecare.Infact,itisnontrivialtofindnotionsofmindindependencethatcanbeusedeventodifferentiaterealistandinstrumentalistpositionsfromoneanother.Foranattemptinthisdirectionandadiscussionofsomeoftheproblems,seeHellman[1983].

(28)ThismaybeconsideredapartialresponsetoGoodman[1978]whichemphasizestheemptinessofreferencetoauniqueworldunderlyingourownconstructions.Itisnotbyanymeanssuggestedherethattheroleofsuchreferenceinappliedmodalmathematicsisitsonlyorprincipaluse.

(29)Goodman[1972]and[1978].

(30)TheabovesketchofarealistpositionhasinterestingaffinitieswithPutnam'sOn

((A O)) O,


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Properties(in[1970]),reflectinganearlierrealistperspective.

(31)Ifthereisafiniteupperboundonthenumberofplacesrequiredinthepredicatesofasyntheticdeterminationbase,thenasyntheticapparatusoforderedtuplescanbedispensedwithinfavouroffinitelymanyk+1arytrueofpredicates,oftheform

inwhichtheorderofplacesisbuiltintothepredicate.Thepresumptionwouldbethatfinitelymanysuchdesignationrelationsarelearnableonebyone.

(32)DeterminationprinciplesofroughlythisgeneralformwereintroducedinHellmanandThompson[1975]asameansofexplicatingakindofphysicalismnotcommittedtostrongclaimsofreducibility(definabilityofdeterminedbydeterminingvocabulary).Relatedprinciples,knownassupervenienceprinciples,havebeendevelopedandappliedinavarietyofcontexts.SeeespeciallyKim[1984].Forasurvey,seeTeller[1983].SeealsoPost[1987]andHellman[1985]andreferencestherein.Thepresentsortofdeterminationprinciplediffersfromtheusualinthat,here,oneisexpandinguponawellworkedouttheoryforthehigherlevel,ratherthanforthelower.

Notethattherestrictiontomathematicallystandard(full)modelsobviatesanautomaticcollapsetoexplicitdefinability(ofdeterminedbydeterminingvocabulary)viatheBethdefinabilitytheorem.(Cf.e.g.Shoenfield[1967]foraprecisestatementandproofofthetheorem;fordiscussion,seeHellmanandThompson[1975].)

Notefurtherthat(vertical)determinationclaimsarecompatiblewith(horizontal)indeterminismintemporalevolution:theymerelyimplythatwhatevertemporalbranchingispossibleinthehigher(determined)levelmustalreadybereflectedincorrespondingbranchingatthelower(determining)level.

Finally,notethatitisnotrequiredthatthesyntheticvocabularyherebeobservationalinanyrestrictivesense.Itmay,initsownright,countashighlytheoretical;and,ifthedeviceoftrueofsemanticpredicatesisallowed,theoreticalvocabularyoftheoriginalappliedmathematicaltheorymaybeadaptedtothesyntheticlevelbyuseofcountablymanyinstancesinvolvingrationalvalues,assuggestedabove(cf.theexamplethatfollowsbelow).

(33)Cf.Quine[1969].

(34)ThisontologicalreductiopatternisthebackbonestructureofField[1980]:therepresentationtheoremisatheoremofmodeltheory,accessibletotheplatonist,butnottothenominalist(atleast,notwithoutfurtherargument,whichwasnotprovidedinField[1980]).Itshouldbepointedout,however,thatsuchreductioreasoningisproblematicinacertainsense:howcanitconvinceonewhostartsoutwiththeassumptionthatplatonistreferencetomodels,etc.,isunintelligible?Foroneinthatposition,thereductioargumentitselfshouldbeunintelligible.Buthow,then,couldanominalistscientistbecomeconvinced(byFieldtheoreticreasoning)thatthestandardplatonistreasoning(which,of

istrueof , , , ,Pk x1 x2 xk


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course,thenominalistscientistemployseveryday)isjustashortcut?Onepossibleanswerhereinvolvesdistinguishingbetweenanonconstructivenominalistpositionwhichholdsplatonistontologicalcommitmentstobefalse,butnotunintelligibleandastricterconstructivenominalistposition,whichdoesholdthosecommitmentstobeunintelligible.ThenperhapsweshouldsaythatonlythelatterisbarredfromfollowingFieldtheoreticreasoning.

(35)Craig'sobservationthatanyrecursivelyenumerabletheoryhasarecursiveaxiomatizationisinCraig[1953].Foradiscussionandcritiqueofitsuseinphilosophyofscience,seePutnam[1965].ItshouldbenotedthatField[1980],p.47,explicitlyeschewsaCraigianreplacementofplatonisticappliedmathematics(i.e.arecursiveaxiomatizationofitsnominalisticconsequences)asevenacandidateforanominalization,ongroundsofitsobviousunattractivenessasatheory.

(36)AsinKrantzetal.[1971].

(37)Fordetailsonrepresentationtheorems,seeField[1980],Ch.7;seealsoMalament[1982].InoursketchbelowwefollowthenotationofMalament'sreview(whichconcentratedontheelegantexampleoftheKleinGordonfield).

(38)TheprooftheoreticconservativenessclaimisnaturallyreadastherelevantclaimthatField[1980]soughttoestablish.However,Field[1985]emphasizesthatitisthesemanticconservativenessofmathematicalphysicsoversyntheticphysicsthatshouldbetheaimoftheprogramme.(Hetherewrites,WhatIshouldhavesaidisthatmathematicsisusefulbecauseitisofteneasiertoseethatanominalisticclaimfollowsfromanominalistictheoryplusmathematicsthantoseethatitfollowsfromthenominalistictheoryalone(p.241).)AsIwillhaveoccasiontoremarkbelow,Idonotthinkthatthisaimreallycomestogripswiththeissueofindispensabilityofhighermathematicsindispensabilityinitsprincipalroleofprovingtheorems.Ironically,this(sofarasweknow,indispensable)roleappearstobeconcededbyField[1985]inhisveryargumentthatsemanticconservativenessclaimscanbeputtousedespitethelackofacompleteproofprocedure(seeField[1985],p.252).

(39)SeeespeciallyShapiro[1983b]andBurgess[1984].Field[1980]acknowledgesthedifficultyatpp.104ff.,creditingMoschovakisandBurgesswiththeobservationthatGdelsentencesariseascounterexamplestotheconservativenessofToverTsyn.Hereweemphasizethepointthatincreasinglypowerfulmathematicaltheoriesmaywellberelevantforprovingphysicallymeaningfulsyntheticassertions.Thefactthat,asaxiomatizedtheories,thesemorepowerfulsystemshavetheirownundecidables,is,fromthepresentperspective,ifanything,areasonforthinkingthatwemayalwayshavetoconsiderstrongerandstrongertheories.

(40)Otherobjectionstoappealstosemanticconservativenesshavebeenraised,inparticularthatthenominalistshouldnotbeabletounderstandsemanticconsequenceintherelevant(full,secondorder)sense,sinceitinvolvesquantificationoverabstractmodels(cf.Malament[1982]).Primafacie,thisseemstelling.However,bymeansof


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codingdevices,agooddealofabstractmodeltheory(fortheoriessuchasPAandRA)canbecarriedoutnominalistically(cf.Ch.1,6),i.e.inmonadicsecondordersystems,especiallywheninterpretedoverspacetime.

Anotherpointintheresponsetotheunintelligibilityobjectionwouldbethatappealstosemanticconservativenessarepartoftheoverallreductioargument,whichmakesfreeuseofplatonistconstructions,asdoestherepresentationtheoremitself.Ourremarksabove,n.34,wouldthenapplyhereaswell.

(41)Fordetails,seeBurgess[1984].

(42)Cf.Shapiro[1983b].Where(r,p)abbreviatesthesyntheticstatementthatrisasumofequallyspaced,linearlyorderedpointswithinitialpointp(andhencecanservetomodelthenaturalnumbers),andwhereConTsyn(r,p)abbreviatesasyntheticstatementoftheconsistencyofTsyn,inTextitcannotbeprovedthat

(whereConTsyn(,0)abbreviatestheordinaryabstractmathematicalstatementofTsyn'sconsistency.For,sinceTextcanproveConTsyn(,0),if(*)wereTextprovable,(r,p)ConTsyn(r,p)wouldbealso.Butbyconservativeness,thislatterwouldthenbeprovableinTsynalso,contrarytoGdel's(second)incompletenesstheorem.

(43)Cf.Malament[1982].

(44)SeeEarmanandNorton[1987].

(45)SeeBohm[1957].

Accessbroughttoyouby: HINARIElSalvador

(r,p) [ (r,p) (,0)]Con Tsyn Con Tsyn

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