Introduction
Manyofthescientifictreatisesoftodayareformulatedina half-mysticallanguage,asthoughtoimpressthereaderwiththe uncomfortablefeelingthatheisinthepermanentpresenceofa superman.Thepresentbookisconceivedinahumblespiritandis writtenforhumblepeople.
CorneliusLanczos(1970,vii–viii)
Ourbestphysicaltheoriesareformulatedinabstractmathematicalterms.This createsdifficultiesfortheinterpretiveprojectoffiguringoutwhatthesetheories aresayingabouttheworld.Simplyput,itisnotobviouswhatamathematicalformalismsaysaboutthephysicalworld.Ofcourse,anymathematicalformulationwe devisewillbebasedonpiecesofexperimentalevidenceandmanifestlyobservable featuresoftheworld;butthosethingswillnotpindownthefullnatureoftheworld accordingtoatheory.Thereisnoconclusiveruleoralgorithmthattakesusfrom amathematicalformalismtothenatureofthephysicalworld,andyetwedoseem abletodrawreasonableconclusionsabouttheworldonthebasisofthesetheories.
Howdowedothis?Howdowefigureoutthenatureoftheworldfromamathematicallyformulatedphysicaltheory?Andwhatifatheorycanbeformulated mathematicallyindifferentways,asistypicallythecase—whatdoweinferabout theworldthen?
Thisbookisaboutthisinterpretiveproject.Itisabouttherelationshipbetween themathematicalstructuresinwhichourphysicaltheoriesareformulated,and thenatureofthephysicalworld(s)thesetheoriesdescribe.Iwillbesuggestingthat thereisacertainnotionofstructurethatisfamiliar(ifofteninexplicit)inphysics andmathematics,andthatpayingattentiontostructureinthissense,bothinthe mathematicalformalismandinthephysicalworld,isimportanttofiguringout whatphysics,especiallyfundamentalphysics,issayingabouttheworld.
Therehasbeenlotsofdiscussioninphilosophyrecently,inphilosophyof scienceandphilosophyofphysicsinparticular,centeringonvariousnotionsof structure.Letmesayalittleaboutthisbywayofcontrastwithmyownideas.
Thenotionthatismostfamiliarinphilosophyofsciencecomesfromtheliteratureonstructuralrealism,stemmingfromthediscussionofJohnWorrall(1989) (whofindssimilarideasinPoincaré).Worrallusesaparticularconceptionof structuretorespondtotheso-calledpessimisticmeta-inductionagainstscientific
realism.Thisisanargumentclaimingtoshowthatwehavenoreasontobelieve thatourscientifictheoriesaregettingatthetruthabouttheworld,astherealist maintains.Asamatterofhistoricalfact,manyscientifictheoriesthathadbeen successfulturnedouttobefalse.Inductivereasoningfromthesepastfailuresthen suggeststhatweshouldnotbelieveinthetruthofourcurrenttheorieseither:itis likelythatthesetheories,too,willeventuallybeshowntobefalse.Sincethepast islitteredwithfalsifiedyetsuccessfulscientifictheories,itseemsasthoughwe havenoreasontobelievethatourcurrenttheories,successfulastheyare,areeven gettinganyclosertothetruth.
Theviewthathascometobecalled“structuralrealism”aimstoacknowledge thisfactaboutpastscientifictheories,whileatthesametimeallowingtherealist toacceptthe“nomiracles”argumentfor scientificrealism—thatthesuccessofour scientifictheorieswouldbeamiracleiftheyweren’ttrue,oratleastapproximately true.Thestructuralrealistsaysthateventhoughmanypastscientifictheorieshave beenabandonedasfalse,thereisa structure tothesetheoriesthatremainsinplace throughouttheprocessoftheorychange,andaboutwhichwecanberealists.As WorrallsaysoftheshiftfromFresnel’stoMaxwell’stheoryoflight,forexample,the formerseeinglightasmechanicalvibrationsthroughasolidelasticmedium,the latteraswavesinanelectromagneticfield:“Therewascontinuityoraccumulation intheshift,butthecontinuityisoneof form or structure,notofcontent”(1989, 117;originalitalics).Fresnelwaswrongaboutthenatureoflight;yethistheory wasempiricallysuccessfulbecauseithituponthecorrectstructure,inthesenseof thecorrectformofequationsgoverninglight’stransmission.
Accordingtothestructuralrealist,ourscientifictheoriesaregettingatthetruth aboutthestructureoftheworld,ifnotitsintrinsicnature.Structureisthekernelof truththatsuccessfulscientifictheorieshaveincommon,andaboutwhichwecan safelyberealistseveninthefaceofthepessimisticmeta-induction.Oneversionof theview,knownasepistemicstructuralrealism,saysthatscientifictheoriesgive usknowledgeaboutthestructureoftheworldratherthanitsintrinsicnature. Anotherversion,onticstructuralrealism,goesfurthertosaythatallthereis torealityisstructure;orthatstructureisprimaryorfundamental,andobjects aresecondaryornonfundamental.Thisviewurges“ashiftinone’sontology, awayfromobjects,astraditionallyconceived,andtowardsstructures,typically conceivedofintermsofrelations,”sothat“inasmuchasobjectsexistatall,they derivetheirpropertiesandindividualityfromtherelationalnetworkinwhichthey areembedded”(RicklesandFrench,2006,25;4).Thesuggestionisthatwemust abandonastandardobject-orientedmetaphysics,asinJamesLadymanandDon Ross’sbook EveryThingMustGo (2009),thetitleofwhichconveysthegist.1
1 SeeLadyman(2016)andMcKenzie(2017)andreferencesthereinfordiscussionofalltheseideas andvariationsonthem.
Otherusesofstructureincludeaviewinphilosophyofphysicscalledspacetime structuralrealism,whichholdsthatspacetimeinparticularisnothingbutacertain kindofrelationalstructure.Thisviewclaimstobeanalternativetothetraditional relational-substantivaldichotomy,allowingustocircumventvariousdifficulties thatarisefromtakingspacetimepointstobeadditionalelementsintheontology: wearetobelieveintheexistenceofarelationalspacetimestructurewithouthaving tocommittotheexistenceorfundamentalityofspacetimepointsasobjects.Other conceptionsofstructureinrecentphilosophyofphysicsarebasedexplicitlyon mathematicalnotionsfrommodeltheoryorcategorytheory(asinHalvorson, 2019),whicharethenputtouseinaddressingvarioustopicsinphilosophyof science,suchasthenatureoftheoreticalequivalence.TedSider(2011)arguesthata particularnotionofstructure,distinctfromtheoneprevalentinrecentphilosophy ofscienceandphysics,isfruitfulthroughoutvariousareasofphilosophy,helping toclarifywhatisatissueinmanytraditionalphilosophicalpuzzlesanddebates. Itmayseemlikethelastthingweneedisanothertreatiseonthiswell-worn notion.Ihopetoconvinceyouotherwise.Itistruethattheterm“structure”has beenbandiedabout,butitisusedinverydifferentwaysbydifferentpeopleand inthecontextsofdifferentphilosophicaldebates.Iintenditinaparticularway thatisimplicitinvariousaspectsofourtheorizingaboutphysics,andisdistinct fromtheothernotionsontap.Notcompletelydistinct,mindyou:thereisareason forthecommonusage.Butitisdifferentenough.Itissomethingthat,taken seriously,yieldsprogressonthequestionsposedinthesecondparagraph.Thatthis notionisdistinctwillbecomeclearthroughthefollowingchapters.Onesignificant differenceisthatmyconceptionofstructuredoesnothaveanythingparticularly todowiththeexistenceorrelativepriorityofobjectsorintrinsicpropertiesas opposedtorelationsorrelationalstructures.Norismychiefconcerntotryto salvagerealisminthefaceofthepessimisticmeta-induction.(Idonotinanycase findthatargumentverycompelling.Itseemstomethatwecanacceptthehistory ofabandonedtheorieswhilestillhavinggoodreasontobelievethatourcurrent theories,withalltheadditionalevidencewehaveforthem,aregettingclosertothe truth,thoughIwon’targuethepointhere.2)Iwillbeadvocatingarealismabout structure,butonethatisquitedifferentfromwhatcurrentlygoesbythename.My overallaimisthustofurtherthis“structuralturn,”whileatthesametimedistancingmyapproachfromothernotionsandusesofstructureinrecentliterature.
Fortherestofthisintroductorychapter,Iwanttosketchmygeneraloutlookand somemotivatingthemes,tohelpsituatethediscussionwithinthephilosophical
2 ValiaAllorihasgenerouslysuggestedthatmyviewshouldcontaintheingredientstorespondto thatargument.Thatmaybe,butIamnotsure.Althoughmyaccountmayindirectlybolsterthecase forscientificrealism,Iamnotsurethatitdeflectsthepessimisticmeta-inductioninparticularinaway thatpeoplebotheredbytheargumentwillbehappywith.(Forinstance,Idonotarguethattheremust besomecommonstructurethatispreservedthroughtheorychange.)Iwillleaveittotheexpertsto saywhethermyideashaveanythingnoveltoofferbywayofdefusingthepessimisticmeta-induction.
landscape.Someofthesethemeswillbearguedfor,otherswillformbackground assumptions.Thesethemeswillbecontroversialtovaryingextents.Onething Ihopetoeventuallyshow,however,isthatmanyoftheseareimplicitinour physicaltheorizing,sothattheyshouldnotbeascontroversialastheymight initiallyseem.
Oneoverarchingthemeisthatweshouldtakethemathematicalstructuresof ourbestphysicaltheories seriously intellingusaboutthenatureofthephysical world.Ihavediscoveredmanypeopletobalkatthisthought,butitseemstome tofollowreasonablystraightforwardlyfromageneralcommitmenttoscientific realism,totheviewthat(roughly)ourbestscientifictheoriestellusabout,orare gettingincreasinglyclosetotellingusabout,thetruenatureoftheworld;thatthese theoriesarenotmerelypredictivedevicesorinstruments,andthisiswhytheyare assuccessfulastheyare.Thisisaviewthatstandsinmarkedcontrastto,forone, NielsBohr’sfamousstatementthat,“Itiswrongtothinkthatthetaskofphysicsis tofindouthownature is.Physicsconcernswhatwecan say aboutnature”(quoted inPetersen,1963,12).Iwon’tattempttoarguefor,letalonedefine,scientific realismhere:thiswillinsteadbeabackgroundassumption.However,notice hownaturallytheideaoftakingthemathematicalstructuresofourbestphysical theoriesseriouslygoeswiththerealistthoughtthatthesetheoriesaregettingatthe truthabouttheworld.Sincethesetheoriesareformulatedinmathematicalterms, itisnatural(fortherealist)tothinkthatthismathematicssomehowrepresents physicalreality,sothatwecanlearnaboutthatrealityfromthemathematical formalism.Inthisregard,keepinmindthatthemathematicalformulationofa theoryisnottheoreticallyinert,inthewaythatthetypeofinkinwhichwewrite downatheoryis,butisboundupwiththetheory’spredictivepower.
Thisisnottosaythatweshouldnaivelyreadoffeverythingaboutthephysical worlddirectlyfromthemathematics;thatwemusttakeeveryaspectofatheory’s formalismtorepresentgenuinefeaturesofthephysicalworld;orthatanymathematicalstructureusedtostateatheorymustrepresenttheworldbymeansofa simple,directcorrespondenceorisomorphism.Acceptingthebasicthoughtthat weshouldtakethemathematicalstructuresofourbestphysicaltheoriesseriously doesnotentailsuchacrudetypeofrealismorrecklessmethodofinterpretation. (Thinkingitdoessomaybethereasonpeoplebalkattheidea.)Anyviewthat takesthemathematicsseriouslyinwaysIwillsuggestisgoingtohavetopay attentiontothedifferencesamongvariousmathematicalfeaturesofaformulation; todistinguishbetweenthemathematicalfeaturesthatrepresentgenuinephysical featuresandthosethatdonot;andtodistinguishthemathematicalfeaturesthat directlyrepresentthephysicalworldfromthosethatdosolessdirectly.Itmay seemanimpossibletasktotrytodistinguishamongvarioustypesofmathematical featuresintheseways.ButthisissomethingthatIdo,forbetterorworse,take uphere.
Nordoestakingthemathematicsseriouslymeaneitherreifyingthemathematicalobjectsusedinatheory’sformulation,orsomehowtakingthephysicalworld itselftobeamathematicalobject.Wecanbecarefultodistinguishbetweenthe mathematicalstructuresusedtoformulateourtheoriesandthephysicalstructure oftheworld,whileatthesametimetakingtheformerasaguidetothelatter. AsTimMaudlinputsit,toattribute“amathematicalstructuretophysicalitems” istosaythatthoseitems“havesomephysicalfeaturesthatmakethemamenable toprecisemathematicaldescriptioninsomerespects”(2015,3).Itisnottosay thatthephysicalitemsthemselvesaremathematicalones,orthattherelevant mathematicalstructureisbeingreifiedintoaphysicalthing.Nor,again,isitto saythateverymathematicalfeatureofaformalismmustbedirectlypossessedby thephysicalthingsbeingrepresented.
Takingthemathematicalstructuresofourbestphysicaltheoriesseriouslysimplyamountstothefollowingepistemicidea:thatthesemathematicalstructurestell usaboutthephysicalworld;theyprovideevidenceaboutthenatureofthephysical world,sothatwecanlearnabouttheworldfromthesemathematicalstructures.I willbearguingthatfamiliarexamplesinwhichwetakethistobethecaseshowthat wedoimplicitly,andreasonably,adheretothisideainourphysicaltheorizing.
Maudlinsuggestssomethingalongtheselinesastheexplanationforthe “unreasonableeffectivenessofmathematics,”inEugeneWigner’sphrase,in describingthephysicalworld.Maudlinattributestheeffectivenesstothefactthat thestructureofthemathematics“directlyreflect[s]thestructureofthephysical world”(2015,4).Asheputsitelsewhere:“Thereisalongstandingpuzzleaboutwhy mathematicsshouldprovidesuchapowerfullanguagefordescribingthephysical world.Themostsatisfyingpossibleanswertosuchaquestionis: Becausethe physicalworldliterallyhasamathematicalstructure”(2014a,52;originalitalics). Myownviewdiffersincertainrespects—Ithinkthatamathematicalformalism cansuccessfullyrepresenttheworldevenwhilenotliterallydescribingit,in Maudlin’ssense,onereasonbeingthataformalismcansuccessfullyrepresentthe worldratherindirectly,inwayswewillsee—butthethoughtbehindtakingthe mathematicsseriouslyisinthesamespirit.Thethoughtissimplythattheremust be some kindofcorrespondencebetweenthemathematicsinwhichweformulate ourtheoriesandthenatureofthephysicalworld,acorrespondencethathelps explain,ontheonehand,theeffectivenessofthemathematicsindescribingthe world,andontheother,thesuccessofourinferencesabouttheworldonthebasis ofthatmathematics.
Ifwearegoingtotakethemathematicalstructuresofourphysicaltheories seriouslyintellingusaboutthenatureoftheworld,thenwewillhavetobe especiallycarefulwheneverthereseemtobedifferentmathematicalstructureswe canusetoformulateatheory.Anditisgenerallythoughtthattherearealternative mathematicalformulationsavailableforanygiventheory,formulationsthatare
regardedasmerenotationalvariants—differentwaysofstatingoneandthesame physicaltheory.
Takingthemathematicsseriouslymayrecommendadifferentattitude.The differentmathematicalformulationsmaybemorelikedistinctphysicaltheories, whichsaydifferentthingsaboutthenatureofthephysicalworld.Henceatheme Iwillarguefor:casesofmerenotationalvariantsinphysicsarehardertocome bythanpeopleusuallythink.Beforethatmakesyoutoouncomfortable,letme hastentoaddthatcasesofgenuineunderdeterminationoftheorybyevidence arelikewisehardertocomebythanpeopleusuallythink,forweoftenhavegood reasontochooseoneformulationoveranother.Iwillargueinparticularthat themathematicalstructureneededtoformulatethedynamicallaws,whatIcall atheory’sdynamicalstructure,3isimportantbothtointerpretingagiventheory, andtochoosingamongdifferenttheoriesorformulations(allthewhileallowing thatthisisnotthesolethingtotakeintoaccountwhenitcomestotheorychoice andinterpretation).
Inarguingforthis,Iwillmakeuseofaparticularwayofcomparingdifferent structureswithrespecttotheirrelativestrengthsoramounts,ameansofcomparisonthatgoesnaturallywiththenotionofstructureIhaveinmind.Using thismeansofcomparison,wewillseeatleastoneexampleofpairsofphysical theoriesthatarestandardlyclaimedtobeequivalent,yetwhosemathematical formulationsutilizedifferentamountsofstructure.Takingtheories’mathematical structuresseriouslythensuggeststhatthesearenotwhollyequivalenttheories, andthatwefurthermorehavereasontochooseoneovertheother.Althoughthis conclusionwillbecontentious,Iarguethatitfollowsfromsomegeneralprinciples wefamiliarlyrelyoninourphysicaltheorizing.Letmeaddaswellthatthisisonly tosaythatasimilarityinmathematicalstructureisanecessaryconditiononthe equivalenceofphysicaltheories.Iwillthroughoutthebookbepointingtoplaces wherewemustrelyonmorethantheories’mathematicalstructuresinorderto drawreasonableconclusionsaboutthephysics.
Anotherthemeabouttheorychoiceandinterpretation.Considerthefollowing passagefromDavidWallaceandChristopherTimpson:
[I]nourview,thereisnoguidetotheontologyofamathematicallyformulated theorybeyondthemathematicalstructureofthattheory....Butwhentryingto learnontologicallessonsfromthetheory,onedoeswelltopreferarepresentation whichmakesmanifestthestructurethatthetheoryascribestotheworld.
(WallaceandTimpson,2010,702)
3 Iusethistermalternatelytorefertothemathematicalstructurepresupposedbythe(mathematical) formulationofthedynamicallaws,andtothephysicalstructureintheworldpresupposedbythose laws.
WallaceandTimpsonusethisideatoargueagainstacertainviewonthemetaphysicsofquantummechanics,knownaswavefunctionrealism,whichholdsthat themathematicalwavefunctionusedinthequantumformalismrepresentsareal physicalfield.Againstsuchaview,theyclaim,“thereisafarmoreperspicuousway tounderstandthetheory”(2010,701).
IwillbedefendingsomethingsimilartoWallaceandTimpson’scriterionof perspicuousness,thoughItaketheideaabitfarther.(Atthesametime,wewillsee awayinwhichItakeitlessfar.)Iwillbesuggestingthatweadheretoacriterionof directnessinchoosingamathematicalformulationofaphysicaltheory:weshould, otherthingsbeingequal,preferaformulationthatmostdirectlycorrespondsto thenatureofthephysicalworld,forthisbringswithitalevelof“metaphysical perspicuousness”thatispreferable.ThereasonIsaythisgoesfartheristhateven thoughWallaceandTimpsonsuggestthatperspicuousnessisimportant,theydo notsaythattheperspicuousformulationispreferableforthereasonatypicalrealist wantstohear.WallaceandTimpsondenythattheperspicuousformulationis preferablebecauseitmostaccuratelyrepresentstheworld—oratleast,theyaimto remainneutralonthis.Theyareagnosticaboutwhether,whentherearecompeting mathematicalformulationsthatappeartodepictdifferentphysicalrealities,oneof themmustbethecorrectormostaccuraterepresentation,orwhetherthedifferent mathematicalrepresentationsmayinsteadbeequallycorrect,evenifoneismost perspicuous(whilenotbeingclosesttothetruth).
Iamgoingtosuggestthestrongerposition:themoreperspicuousformulation, theonethatispreferableforthatreason,moredirectlygetsatthetruenature ofphysicalreality.(Thatsaid,figuringoutwhichformulationismostdirectis acomplicatedandsubtlebusiness,forreasonsthatwon’tbefullyarticulated untiltheendofthebook.)Infact,IconfesstofindingtheWallace–Timpson positionsomewhatpuzzling.Whatdotheymeanbyaperspicuousrepresentation, ifnotaparticularlyclear-eyedrepresentationofthetruenatureofthephysical world?Againsttheiragnosticism,Ithinkthemoreperspicuousrepresentation providesthemoreaccuratedescriptionofphysicalreality,andthisiswhyitismore perspicuous.Itisforthatreasonpreferable(totherealist),otherthingsbeingequal. (AlthoughWallaceandTimpsonclaimtoberealists,theirviewraisesthespecter ofantirealism,inwaysIwilldiscuss.)
Onewayforaformulationtobemoredirectisbybeingstatedintermsof thingsthatarethemselvesmoredirectlyaboutthephysicalworld.Asaresult,by thelightsofthedirectnesscriterion,theoreticalformulationsgivenintermsof referenceframesorcoordinatesystems,althoughcommoninphysicsbooksand usefulformanypurposes,arelesspreferable,otherthingsbeingequal,because theyarelessdirect.Referenceframesandcoordinatesystemsaredeviceswebring tobearforthepurposesofdescribingphysicalsystems,notinherentinphysical systemsthemselves,andtheydonotdirectlycharacterizetheirnatures.Theyare indirect,ifuseful,descriptivetools.
Whyisadirectformulationpreferable?Isaymoreaboutthislateron,but onethoughtisthefollowing.ConsiderAristotle’sphysics,accordingtowhich certainelementstendtomovetowardthecenterofthesphericaluniverse,and otherelementstendtomoveawayfromit.Asaresultofthesetendencies,certain coordinatesystems—namely,thosewithanoriginlocatedatthecenterofthe universe—willbeparticularlynaturalorusefulfordescribingsystems’behavior. Ifsomeonewerethentoask why thoseareespeciallygoodcoordinatesystemsto use,itwouldbeunsatisfyingtorefrainfromgivingananswer.Weshouldbeableto giveananswerintermsofthenatureofthephysicalrealitythetheorydescribes— inthiscase,bypointingtothefactthatthereisadynamicallypreferredspatial location,whichiswell-reflectedbyanycoordinatesystemthathasitsoriginlocated there.Theexplanationshouldnotbottomoutatabrutereferencetocoordinate systems,letaloneparticularlywell-suitedcoordinatesystems,butatthenature ofthephysicalrealitythat makes thosecoordinatesystemsnaturaltouse.Direct formulationsareforanalogousreasonsmoreexplanatory:weunderstandmore readilywhatitisabouttheworldthatmakestheformulationasgoodandusefulas itis.(CompareFrankArntzeniusandCianDorronstandardmathematicaldefinitionsofdifferentiablemanifoldsintermsofcoordinatecharts.Suchdefinitionsare “spectacularlyunsatisfyingfromafoundationalpointofview”(2012,232),since theymentionthingslike“admissiblecoordinatefunctions”withoutsayingwhatit isabouttheintrinsicstructureofthespacethatmakesthosecoordinatefunctions admissible.)
Anotherthoughtinthebackgroundisthatindirectformulationsleavetoomuch roomtoappealtoanythingonelikesinformulatingthelaws,resultinginatype oftheorythattherealist,atleast,won’tbehappywith—onethatseemslikea “cheapinstrumentalistrip-off,”inthememorablephraseofJohnEarman(1989, 127).Considerthedebatebetweensubstantivalists,whosaythatspacetimeexists, andrelationalists,whodenythis.Oneimmediateconcernfortherelationalist isthatthelawsofstandardphysicaltheoriesseemtobestatedintermsof,or atleasttopresupposethingsabout,spacetimeanditsstructures.Inreply,the relationalistcansaythatweshouldnottakethesereferencestospacetimeso seriously:althoughwemakeuseof“spacetime”informulatingthelaws,there reallyisnosuchthing.Accordingtosucharelationalist,thelawsareformulated indirectly,intermsofsomethingthatisnotdirectlyaboutthephysicalworld,but isnonethelessmentionedforpurposesoftheorizingaboutsystems’behavior.This feelslikecheating.Itfeelsliketherelationalistissaying:“thingsbehaveasifthere werespacetime;wearejustifiedinreferringtospacetimeinourtheory;but[psssst!] therereallyisnosuchthing”—aninstrumentalistrip-off.Adirectformulation leaveslittleroomforcheating,andisforthatreasonpreferable.
Moregenerally,directnessbringswithitalevelofperspicuousnessthataids theinterpretiveprojectoffiguringoutwhatphysicsissayingabouttheworld.It canalsoyieldtheoreticalprogress:classicalelectromagnetismisacaseinpoint.
Maxwellformulatedtheequationsthatgobyhisnameintermsofthepotentials, andtheresultwasabitofamess.OliverHeavisidewastheonewhocameupwith theimprovedandstreamlinedversionwearefamiliarwithtoday.Asithappens, Heaviside’sformulationisalsomoredirect:theequationsareformulatedinterms oftheelectricandmagneticfields,whichthetheorytakestobegenuinephysical entitiesintheworld,ratherthanthepotentials,mathematicaldevicesthatdon’t directlycorrespondtophysicalthings.Heavisidewasledtohisformulationby thinkingcarefullyaboutwhatthetheoryissayingaboutthenatureofphysical reality,andtryingtodeviseamathematicalformulationthatdirectlymirrorsthat reality.
Oneunexpectedconsequenceofprizingbothdirectnessandperspicuousnessis thateventhoughweshouldgenerallypreferformulationsofphysicaltheoriesthat donotdependoncoordinatesorotherauxiliarydescriptivedevices—inaccord withcurrentthinkinginfoundationaldiscussions—atthesametimeweneedn’t eschewcoordinate-basedreasoninginphysicsaltogether—againstmuchcurrent thinking.Althoughthecurrentfashioninfoundationsofphysicsistoavoidall mentionofcoordinates,wewillseethattherearewaysofreasoningaboutphysics bymeansofcoordinatesthatareusefulandlegitimate,evenperspicuous.Another theme:theroleofcoordinatesystemsinphysicsismoresubtleandcomplicated thanusuallyacknowledged.Onedistinctiontobemadeinthiscontextisbetween formulationsorclaimsthatarecoordinate-orframe-dependent,inthesensethat theyvarywiththeparticularchoiceofcoordinatesystemorreferenceframe,and formulationsorclaimsthatarewhatwemightcallcoordinate-orframe-based, meaningsimplythattheymentionorrefertocoordinatesystemsorreference frames.Althoughtheformertypeofclaimcanbemisleadingastothetruenature ofphysicalreality—as,say,(frame-dependent)claimsabouttimeelapseinspecial relativitydonotgetattheunderlyingnatureoftheworldaccordingtothetheory— thelattercanbeauseful,evenstraightforward,guidetothenatureofphysical reality.OneexampleIwillemphasizeishowwecommonlytakethemathematical formofthelawsindifferentkindsofcoordinatesystemsorreferenceframesto indicatetheunderlyingnatureoftheworld,similarlytohowtheformofthemetric indifferenttypesofcoordinatesystemsindicatesthestructureoftheEuclidean plane.Thiskindoffeaturemakesreferencetocoordinatesystemsorreference frames,butitdoesnottherebyfailtoindicatetheunderlyingnatureof(physical orgeometrical)reality.Onthecontrary.
ItmaybeevidentbynowthatIendorsenotonlyastandardorold-fashioned typeofrealism,butalsothe“idealofpristineinterpretation,”inthephraseofLaura Ruetsche(2011,Ch.1).Ruetscheexplainsthisideal,andherreasonsforrejecting it,indetailthroughoutherbook,butthebasicideaiswhatwemightthinkofas thestandardrealistviewoftheoryinterpretation:thatourbestphysicaltheories tellusaboutwhattheworldislike;thatthereisone—andonlyone—waythe worldis,accordingtoatheory,sothatweshould“interpretatheoryinthesame
waynomatterwhatconditionsbefallit”(2011,343).Thepristineidealincludes thefurtherthoughtthatatheorytellsusaboutwhatotherphysicallypossible worldsarelike,inwaysdictatedbythelawsplussomegeneralphilosophical principles:somethingelseIendorse.ThereisoneplacethatIpartwiththepristine idealasRuetscheconceivesofit,though.Althoughtheinterpretationofatheory stemsfromthelawsinaccordwithsomegeneralprinciples,Idonotthinkthat thoseprinciplesarethemselveswholly“antecedent”orheld“comewhatmay,”as Ruetsche(2011,4)saysofthepristineideal.(Nordotheseprinciplessufficeto pickoutaninterpretation:someadditionalphysicalpositswillberequired,inways Iwilldiscuss.)Rather,thesortsofprinciplesIwilldefendareepistemicprinciples thatarebothinformedandjustifiedbyexamplesofsuccessfultheorizingthatrely onthem;andtheseprinciplesholdonlyceterisparibus.Wenonethelessdowellto abidebythem.
Againstthepristineideal,Ruetscheseestheoryinterpretationasamorepiecemeal,pluralistic,andpragmaticaffair.Shearguesthatthepristinealternativeis untenableoncewelookcarefullyatquantumfieldtheoryandthethermodynamic limitofquantumstatisticalmechanics.Iwon’tbeaddressingthosephysical theorieshere,butwillassumetheviabilityofapristineinterpretationforthe theoriesIdodiscuss:abackgroundassumption.Isimplyfindittoohardto reconcilemyrealisttendencieswiththepossibilityof“multipleinterpretations inthestandardsense”(Ruetsche,2011,10).Similarconsiderationsleadmeto assumeafundamentalismanduniversalismaboutthephysicallaws,againstNancy Cartwright’s(1999)viewthattheworldis“dappled,”bringingwithitapatchwork oflawsthataccuratelydescribethingsinonlyapiecemealway:anotheridea toodifficulttoreconcilewithathoroughgoingscientificrealism.PerhapsIwill eventuallybeforcedtosomesuchviewonthebasisofthephysicsRuetsche discusses;untilthen,Iaimtoholdontoamorecomprehensiverealismandsee whereitgetsme.(Infact,Iwouldputthingsmoreoptimistically,thoughIwon’t explorethisindetailhere:theapproachItakeandthelessonsIdrawforthe theoriesIdiscuss,includingtheviabilityofrealism,shouldcarryovertoanytheory wetaketobeagenuinecandidatefundamentaltheoryforourworld—solong asitisabonafidephysicaltheory,withabaseofempiricalevidenceandcertain coreprinciplesandpositsconcerningthephysicalontology.Thereasontofocus onthetheoriesIdoisthattheircoreprinciplesandmathematicalstructuresare comparativelywell-understoodand-delineated,makingiteasiertoextractthe sortsoflessonsIaimtoextract.)
Thatsaid,thereisroomforacertainstripeofantirealisttoagreewithmuchof whatIsayinthisbook.FornotallantirealistsareinstrumentalistsorBohr-type antirealists.Andtheantirealistwhodoesnotgosofarasthoseotherviewscan agreethatthemathematicalstructuresofourbestphysicaltheoriestellusabout thenatureofthephysicalworld,andcanevenendorsetheinterpretiveprojectof tryingtofigureoutwhattheyaretellingusabouttheworld,butsimplydenythatwe
shouldbelievewhattheyaretellingus—anattitudethatRuetscheherselfadopts. BasvanFraassen’sconstructiveempiricism(1980)isanotherviewthatwouldfall intothiscamp(atleastwhenitcomestotheories’claimsabouttheunobservable aspectsoftheworld).Indeed,onemightturnthetablestoquestionwhetheritis possibletomaintainthekindofrealismIadvocatetowardthevariousphysical theoriesIdiscuss—primarilyclassicalmechanicsandspacetimetheories,both classicalandrelativistic,butalsoclassicalelectromagnetismandnon-relativistic quantummechanics.Thesetheoriesfamouslyclashinwaysthatspelltroublefor adoptingablanketrealismtowardthemall.
Myinstinctsnonethelesspointelsewhere.Wecanbefull-throatedrealistsabout thesetheoriesseverally,treatingeachasacandidatefundamentaltheoryinitsturn (anapproachIadoptinthisbook).Weshouldevenbeabletoberealistsabout themall(albeitinawaythatmayrequiresuchnotoriouslytrickynotionsasrelative fundamentalityandapproximatetruth).Onceagain,itisnotmyaimtoarguefor suchapositionhere.Iwillsimplyassumeathoroughgoingrealism,evenifthisis notabsolutelyrequiredbymyapproach,andevenifthereisworklefttobedonein makingsenseofit.However,Ican’tcomplainifwhatIsayturnsouttobeacceptable tocertainantirealists,andIwillpointtoplaceswhereitdoesseemasthoughsuch anantirealistcanagreewithmydiscussion.
Iwillalsobesuggestingthatwetakewhatwemightcallatheory’s“metaphysical aspects”seriously:anothertheme.ValiaAllori(2015a)arguesthatwecannot investigatetheinvariancesorsymmetriesofaphysicaltheoryindependentlyof itsmetaphysics.Iwillbedefendingamoregeneralversionofthisidea.Although Idoarguethatthemathematicalformalismofatheoryisaguidetothenature ofthephysicalworld,itisalsothecasethatwecannotgetatthatnaturewholly independentlyofatheory’smetaphysicalaspects,whichgobeyondtheformalism. Simplyput,noteverythingaboutthephysicalworldcanbereadstraightoffthe mathematics.Onereasonisthatsomeinitialphysicalpositswillinvariablyplaya roleaswell.TogiveanexampleIwillreturnto:whatistheworldlikeaccordingto Newtoniangravitation?Theanswerwilldependonwhetherwetakethetheory tobeaboutparticleswithgravitationalforcesactingonthematadistance, assuggestedbythetraditionalformulationofthetheory,orparticleswhose motionsareinsteadaffectedbythelocalspacetimestructure,assuggestedby the“geometrized”formulationofthetheory.⁴Someinitialphysicalassumptions (suchaswhethertocountenancegravitationalforces)mustbemadebeforewe canfullydiscernthenatureofthephysicalworldaccordingtothetheory,aswell aswhichmathematicalformulationtofocusoninthisinterpretiveprojectinthe
⁴ GeometrizedNewtoniangravity,alsocalledNewton–Cartantheory,wasfirstdevelopedby ÉlieCartanandKurtFriedrichsinthe1920s.Itisformulatedintermsofthestandardmathematicalformalismofgeneralrelativity,resultinginatheorythatappearsto“geometrizeaway gravity”inamannersimilartogeneralrelativity.PresentationsareinFriedman(1983,Sec.3.4); Malament(2012,Ch.4).
firstplace.Themathematicsinwhichatheoryiscouchedisaguidetothenatureof physicalreality,inotherwords,butthisisnottosaythatthatnatureiscompletely determinedbythemathematics.Metaphysics—orwhat,forreasonstocome,may bebettercalledsimplyphysics—matterstoo.Wewillseethatthemathematicaland metaphysicalaspectsofaphysicaltheoryareintertwinedinvariouswaysthatare notcompletelystraightforward.
Moregenerally,thisbookwillinvolvesomemetaphysicsofphysics.Thisgoes againstwhatJuhaSaatsicallsthe“well-motivatedanti-metaphysicaltrendin theepistemologyofscientificrealism”(2019,146),especiallypopularamong structuralrealists.Saatsiisinfavorofthattrendandagainstany“deepmetaphysics”ofphysicsthatgoesbeyondthebasiccommitmentsoftherealist,which inhisviewincludeonlyanadherencetothetruthofourtheoriesinsofaras thisisneededforpredictingandexplainingthephenomena.Anythingbeyond thesebasiccommitmentsdevolvesintospeculativemetaphysicsthattherealist neednot—shouldnot—committo(somethinghesaysisparticularlyevidentin philosophicaldiscussionsofquantummechanics).
Althoughtherewillofcoursebeepistemologicalissuesthatariseforanyrealism daringtodelveintothemetaphysicsofphysics,tomymindthissortofthing doesnotgobeyondtherealist’scommitments,butispartandparcelofabasic realism,nottobeabandonedinthefaceofthedifficultepistemologicalquestions thatresult.Inparticular,itispartandparcelofarealismappliedtocandidate fundamentalphysicaltheories,whichareformulatedinabstractmathematical terms,andtherebyrequiresomemetaphysics—oragain,justplainphysics—in ordertopaintthepictureoftheworldtheydescribe.Inmyview,Saatsi’sown “minimalrealistattitude”simplyfallsshortofrealism:itisanuntenablestopping point,refusingtodigdeeperintothenatureoftherealityresponsibleforthe empiricallyconfirmedpredictionsandexplanationswegetfromscience.Indeed, thekindofrealismthatSaatsiendorseswouldwindupeschewingtoomuchof thescienceheclaimstowanttopreserve,suchasscientificexplanationsofthe phenomena,asIdiscussattheendofthebook.Forthesereasons,Idonotseemy approachas“metaphysicalhubris”(inSaatsi’sphrase),butabasicpartofscience asordinarilyunderstood.
Thisbringsmebacktoathemementionedearlier.Givenmyemphasisonboth themetaphysicalandmathematicalaspectsofphysicaltheories,Iwillfrequently seeanon-equivalencebetweentheoriesorformulationswhereothersseeequivalence.Forexample,Iarguethattwoformulationsofclassicalmechanicsthat areordinarilytakentobeequivalent(theLagrangianandNewtonianformulations)differnotonlyinmathematicalstructure,butalsoinvariousmetaphysical respects.Bothkindsofdifferencesaresignificantenoughtowarrantregarding theseasdistinctphysicaltheories,withdifferingaccountsofwhataclassical
mechanicalworldisreallylike.OrtaketheHeisenbergandSchrödingertheoriesor“pictures”ofnon-relativisticquantummechanics,generallyseenasmere notationalvariants.Althoughthesearemathematicallyequivalentinacertain sense,thereareplausiblysignificantmetaphysicaldifferencesbetweenthem.Ona naturalunderstandingoftheHeisenbergpicture,thereisonestateoftheworldthat isunchangingwithtime,whereasonanaturalunderstandingoftheSchrödinger picture,physicalstatesthemselvesevolveintime.Takingtheories’metaphysical aspectsseriouslythensuggeststhatthesearedistinctphysicaltheories,with differentaccountsofthephysicalworld(subtletiestobeelaboratedonlater).
Iwillbeemphasizingthesesortsofmetaphysicalaswellasmathematicaldifferencesbetweentheories,whichwillleadmetoseemorecasesofinequivalent theoriesthanmanypeoplewillbehappywith.Thatsaid,Icanstilltalkofthe variousrespectsinwhichtheoriesare,orarenot,equivalenttooneanother,andin thatwayretainwhatisofvaluebehindstandardclaimsofequivalenceinphysics.
Allofthesethemesareinterrelated,andinmyviewtheyareallboundupwitha commitmenttoscientificrealism.Again,Iwillnotargueforrealismhere.Norwill Iofferanaccountofscientifictheories,theoreticalequivalence,lawsofnature,scientificexplanation,orfundamentality—eventhoughallofthesehavearoletoplay inwhatfollows.WhatIsaycanhaveramificationsforthesethings,inwaysIwill discuss.YetItakeitthatmydiscussioncanproceedwithouthavingtogiveexplicit accountsoftheseothernotions,eachofwhichcouldtakeupabookonitsown.
Afinalthemehasmoretodowithphilosophicaltemperamentthananything else.Alotofrecentphilosophyofphysicshasbeenmarkedbyastronglyformal turn.Thisbookgoesagainstthattrend.AlthoughIwillbefocusingonthe mathematicalstructuresofphysicaltheories,thediscussionherecontainsmuch lessthanistypicalofrecentjournalarticlesinthewayofmathematicalformulasor proofsoftheorems.Moregenerally,Ieschewmanyofthemathematicalmethods adoptedbyphilosophersofphysicsthesedays,eventhoughIdodiscussandapply ideasthatcomedirectlyfrommathematics.Isimplytakeadifferentapproach, onethataimstominimizeexplicituseofmathematicsandtechnicalityasmuch aspossible.(Asmuchaspossible:occasionallythingswillunavoidablygetmore technical;whenthatisthecase,Iaimtomakethediscussionaccessibletothe uninitiated.)Itrytogettothebottomofthingsinasnon-technical,simpleand straightforward,amanneraspossible.InthisIaminfluencedbymydissertation advisors,DavidAlbert,BarryLoewer,andTimMaudlin,whodemonstratehow muchgoodphilosophycanbedoneintheabsenceofexplicitmathematics. Theirworkisofcoursedeeplygroundedinthemathematicsofphysics,butthe formalismtendstoremaininthebackground,broughttotheforeonanas-needed basis.Icannothopetoaccomplishafractionofwhattheydo,butIgreatlyadmire theirmodelofdoingphilosophyofphysics,andtrytoemulateit.
Toputitalittlemorestrongly:althoughIwillbeemphasizingtheimportance ofthemathematicalstructuresusedtoformulateourbestphysicaltheories,Ialso thinkthatonecanmisssignificantaspectsofwhatphysicsissayingbyfocusing too closelyonthemathematicalformalism.HansHalvorsonbemoansthe“decline instandardsofrigor”inrecentphilosophyofscience(recentdiscussionsof theoreticalequivalencebeingacaseinpoint),notingthatmanyatechnicalterm
hasmadeitswayintophilosophicaldiscussionbuthasthenlosttouchwithits technicalmoorings.Theresultisalmostalwaysthatphilosophersaddtothestock ofconfusionratherthanreducingit.Howunfortunateitisthatphilosophyof sciencehasfallenintothisstate,giventherolewecouldplayasprophetsofclarity andlogicalrigor.(Halvorson,2019,10)
Halvorsonsuggeststhatbringingtherigorandclarityoflogicandmathematics backintophilosophicaldiscussion,eventothepointofreframingvariousphilosophicalquestionsinexplicitlyformalormathematicalterms,willservetoclarify manyoutstandingissuesinphilosophyofscience.
Iagreethatformalmethodscanclarifypointsatissueandthatthemathematical turnhasproducedusefulresults.ButIalsothinkthatalaserfocusonthe mathematicalformalismcanleadonetomisstheforestforthetrees,foritcan leadonetomissimportantaspectsofthephysics.Inparticular,itcanleadone tomissthe“pictureoftheworld”presentedbyaphysicaltheory,inaphrase IuseinChapter7.AsRichardFeynmansays,withinadiscussionemphasizing theusefulnessandimportanceofmathematicstophysics,
Physicsisnotmathematics,andmathematicsisnotphysics.Onehelpstheother. Butinphysicsyouhavetohaveanunderstandingoftheconnectionofwords withtherealworld.Itisnecessaryattheendtotranslatewhatyouhavefigured outintoEnglish,intotheworld,intotheblocksofcopperandglassthatyouare goingtodotheexperimentswith.Onlyinthatwaycanyoufindoutwhetherthe consequencesaretrue.(Feynman,1965,55–6)
Whendoingphilosophyofphysics,theultimateaimofwhichistounderstand whatphysicsissayingabouttheworld,weshouldbecarefulnottooverplaythe usefulnessofmathematicalmethodsandreasoning,asusefulastheycanofcourse be.Forcertainthingsinphilosophyofphysics,includingthekindsofthings Iexaminehere,formalizationisnotappropriate—eventhoughIreadilyagree thattherearecasesinwhichonegainsrealinsightbyformalizingaquestionin philosophyofphysics.
Notetoothatalackofformalizationdoesnotnecessarilymeanalackofrigor, asIaimtoillustrateinthisbook,althoughrecentliteraturegivestheimpressionof assumingotherwise(asintheabovepassagefromHalvorson).Thatsaid,again,
thereissuchathingasfalseprecisionorrigororover-formalization,when thephenomenoninquestiondoesnotpossessthecorrespondinglevelofdetail orprecision.Considertheepistemicprinciplesthatguideourinferencesabout structure,whichIwilldefendandberelyingonthroughout.Theseepistemic rulesarenotcompletelyprecise,norwilltheyalwaysyieldconclusiveresults. Wenonethelesstaketheinferencestheyyieldtobegenerallysuccessful,anditis reasonabletorelyonthem.Suchistheway,Isubmit,withanyofourusualcriteria ofscientifictheorychoiceandinterpretation:scientifictheorizingisunavoidably messyinthisway.Andalthoughthephilosophicalidealofclaritymightseem todemandadecluttering,thisisnotalwaysfeasibleorevendesirable.Iaimto showthatthereisanyway“rigorenough”inthediscussion,inaphrasefromClark Glymour(1977,236)(quotedatmorelengthinChapter3).
Thefirstpartofthebook(Chapters2and3)discussestheideaofstructure Ihaveinmind;howtocomparedifferentkindsandamountsofstructure;and somegeneralprinciplesgoverningourinferencesaboutstructure,withexamples drawnfromphysics,mathematics,andphilosophyofphysics.Iarguethatthisidea ofstructureisfamiliar,ifoftenimplicit,inmuchofourtheorizing,andthatthe epistemicprinciplesgoverningourreasoningaboutitarefamiliarandgenerally successful.
Thesecondpartofthebookappliestheseideastoclassicalmechanics (Chapter4)andspacetimephysics(Chapter5).InChapter4,Idiscussthe LagrangianandNewtonianformulationsofclassicalmechanics.Iarguethat thesetwoformulationsdifferinstructure,inparticulartheirdynamicalstructure, thestructurerequiredbytheirrespectivedynamicallaws,sothat,accordingto thegenerallyacceptedprinciplesfromChapter3,weshouldconcludethatthey arenotfullyequivalent:theysaydifferentthingsaboutthefundamentalnature ofaclassicalmechanicalworld,contrarytotheusualviewthattheyaremere notationalvariants.InChapter5,Isuggestthatreformulatingthetraditional debatebetweenrelationalistsandsubstantivalistsaboutspacetimeintermsofa notionofspatiotemporalstructureservestoreorientthedebatesoastorender itdirectlyrelevanttocurrentphysics,againstrecentclaimsthatthisdisputeis non-substantive,outmoded,orwhollydivorcedfromphysics.
Thefinalpartofthebook(Chapters6and7)addressesconcernsthatarise fromtakingthisnotionofstructureseriously,includingtheworrythatItake themathematicalstructuresofourphysicaltheories too seriously,andrelated questionshavingtodowiththeequivalenceofphysicaltheories.ItisinChapter7 thatthemetaphysicalaspectsofatheorywillcometotheforeasbeingequally importanttoitsmathematicalstructureforunderstandingwhatthetheoryis sayingabouttheworld.Itishere,too,thattheentanglementbetweentheories’ metaphysicalandmathematicalaspectswillmorefullyemerge.
Iwon’tbediscussingmorecutting-edgephysics,suchasquantumfieldtheoryor variousprogramsinquantumgravity.NorwillIevenmuchdiscussnon-relativistic
