Acknowledgments
Overthecourseofwritingthisbook,Ihavegiventalksonsomeofthe contentsatvariousplaces,andIamgratefultohavereceivedfeedbackon thoseoccasions.Thebookwasalsoreadindraftforminareadinggroupat theUniversityofWesternOntario.Iexpressmythanksfortheirfeedback tomembersofthatgroup,andtothegraduatestudentsinacoursefor whichitwasusedasatext.Amongthosewhohaveprovidedvaluable commentsondraftsofthebookareDimitriosAthanasiou,MichaelCuffaro,ThomasdeSaegher,LucasDunlap,SonaGhosh,MarieGueguen,Bill Harper,MarcHolman,HaiyuJiang,MollyKao,AdamKoberinski,Joshua Luczak,MackenzieMarcotte,TimMaudlin,VishnyaMaudlin,JohnNorton, FilipposPapagiannopoulos,StathisPsillos,BrianSkyrms,ChrisSmeenk, andMartinZelko.ParticularthanksareduetoDavidWallaceandMarshall Abramsfortheirdetailedcomments.IthankZiliDongforassistancewith copy-editingandtheindex.
Partofthewritingofthisbookwasdonewhiletheauthorwasavisiting scholaratthePittsburghCenterforthePhilosophyofScience.Ithankthe Centerfortheirhospitality,andforfosteringalivelyintellectualatmosphere. Researchforthebookwassponsored,inpart,byGrahamandGaleWright, whogenerouslyfundtheGrahamandGaleWrightDistinguishedScholar AwardattheUniversityofWesternOntario.
Preface
Thisisabookabouttheuseofprobabilityinscience,withanemphasis onitsuseinphysics,and,particularly,instatisticalmechanics.Itsimpetus stemsfromapuzzle.Theapplicationofprobabilisticconceptsinphysicsis ubiquitous(moresothanmightatfirstappear,asIargueinthefirstchapter). Thisraisesthequestionofhowtointerprettheprobabilitiesbeingused. Ithaslongbeenrecognized(andisamplydocumentedbyIanHackingin hisbook TheEmergenceofProbability)thattheword“probability”hasbeen usedintwodistinctsenses.Oneoftheseisanepistemicsense,havingto dowithdegreesofbeliefofagents,suchasourselves,withlessthantotal knowledgeoftheworld.Followingstandardusageinphilosophy,Iwill callepistemicprobabilities credences.Inanothersense,calledbyHacking the aleatory sense,probabilitiesarefeaturesofcertainsortsofphysicalsetups:dicethrows,roulettewheels,andthelike.Weroutinelyraisequestions aboutwhetheracointossisfair,orwhetheraroulettewheelisbiased,and treatthesequestionsasquestionsabouttheobjects,questionsthatcanbe addressedbyexperimentation.Again,followingstandardusage,Iwillcall theseprobabilities chances,or objectivechances. Itwouldbeamistaketothinkofthesetwosensesinwhichtheword “probability”hasbeenusedasrivalinterpretationsofprobability,oneof whichmustbeacceptedtotheexclusionoftheother.Theyaresimplydistinct (albeitintertwined)concepts,andeachhasalegitimateuse.Indeed,much confusioncouldhavebeenpreventediftheword“probability”hadnever beenusedambiguously,and“chance”and“credence”usedinstead. However,thereisapuzzlehere.Formuchofthemodernperiod,itwas takenforgrantedthatthefundamentallawsofphysicsaredeterministic. Eventoday,whenitisacceptedthattheworldisquantum,notclassical,it isfarfromnoncontroversialthatthelessontobedrawnfromtheempirical successofquantumtheoryisindeterminismattheleveloffundamental physics.Onthemoststraightforwardconceptionofwhatanobjectivechance wouldbe,suchchancesareincompatiblewithdeterminism.Itisthisthatled writersonprobabilitytheorysuchasBernoulli(1713),Laplace(1814),and others,todeclarethatprobabilityiswhollyepistemic.Andyetthesesame
writersfall,onoccasion,intotakingitasamatterofphysicalfactwhethera givengameofchanceisfair,andthusthereisatensionintheirwritings.
Oneofthemainthesesofthisbookisthatthefamiliardichotomyof chance and credence isinadequate,andthatthereisaneedforaconceptof probabilitythatmakessenseinadeterministiccontextandwhichcombines epistemicandphysicalconsiderations.Thesehybridprobabilities,whichare neitherwhollyepistemicnorwhollyphysical,Icallepistemicchances.Iargue thattheyfulfilltherolerequiredofprobabilitiesingamesofchanceandin statisticalmechanics.
Inordertodoso,someground-clearingisrequired,asIneedtoexplain whytherearen’tready-madeserviceablenotionsthatfilltheroleforobjective chanceinadeterministiccontext.Oneofthese,connectedwithclassical probabilitytheory,istheideathatdefiningtheprobabilityofaneventisa simplematter:theprobabilityofanevent A isjusttheratioofthenumber ofpossibilitiescompatiblewiththeevent’soccurrencetothenumberof allpossibilities.Theotheristheideathatprobabilitystatementscanbe cashedoutasimplicitreferencestorelativefrequenciesinsomeactual orhypotheticalsequenceofevents.Neitheroftheseattemptsatdefining probabilitiesachievetheirpurpose.Theseapproacheshavebeenextensively criticized,butIdonotpresumethatallofmyreadersarefamiliarwiththese criticisms,and,indeed,expectthatsomereaderswillbegintheirreadingof thebookasproponentsofoneortheotheroftheseviews.Forthatreason, IexplainthereasonswhyIthinkthatneitheroftheseapproachesprovide anadequateconceptionofprobabilityinChapter3,andwhymyreaders should,too.
Oneofmymotivationsforwritingthebookisadissatisfactionwiththe wayprobabilitiesaretreatedinmuchofthephilosophicalliterature.Too muchoftheliteratureinthephilosophyofprobabilityislackinginhistorical perspective,and,asaconsequence,debatesthatwerecarriedoutinthe nineteenthcenturyarerecapitulatedinthetwenty-first.Wherehistorical referenceismade,itisoftenincorrect.Acommonnarrativehasitthat Laplace(1814)naïvelythoughtthatprobabilitiescouldbestraightforwardly definedonthebasisofaPrincipleofIndifference,andthatBertrand(1889) shatteredsomesortoforthodoxywithhis“paradox.”Thetruthisthatthe examplesinvokedatthebeginningofBertrand’s CalculdesProbabilités were meanttoillustrateaby-thenfamiliarpointmadealreadybyLaplace,and thatBertrandwas,infact,echoingLaplace.Ihopetorestoresomehistorical perspectivetocontemporarydiscussions.
Theliteratureonthefoundationsofstatisticalmechanicshasbeenheavily influencedbytheEhrenfests’encyclopediaarticle(1912).Thatarticlehas considerablemerits,buthistoricalaccuracyisnotoneofthem.TheEhrenfestssoughttosecuretheplaceoftheirlatefriendandmentor,Ludwig Boltzmann,asthechiefarchitectofstatisticalmechanics;andBoltzmann’s predecessors,inparticularJamesClerkMaxwell,andBoltzmann’ssuccessor, J.WillardGibbs,whobuiltonandextendedBoltzmann’swork,receive shortshrift.Thenarrativeinthemindsofmanyworkinginthephilosophy ofstatisticalmechanicsisthat,atthetimeBoltzmannpublishedhiscelebrated H-theorem(1872),thoseworkinginthefieldwenowcallstatistical mechanicspresumedthatamonotonicrelaxationtowardsequilibriumcould bederivedfromthelawsofmechanicsalone,andthatthereversibility objectionstemmingfromLoschmidt(1876)cameasaboltfromtheblue. Infact,theargumentfromthereversibilityofunderlyingmechanicallaws totheconclusionthatirreversibilityatthemacroscopiclevelhadtobea matterofprobabilityhadbeenfamiliartotheBritishphysicists,Maxwell, Kelvin,Tait,andRayleighforalmostadecadebeforeBoltzmanncameto thisrealization.AnothereffectoftheEhrenfests’influenceisthenotionthat therearetwoincompatiblerivalapproachestostatisticalmechanics:BoltzmannianandGibbsian.WhatisnowusuallycountedasBoltzmannian(or neo-Boltzmannian)statisticalmechanicsconsistsofextractingonestrand fromthemanytobefoundinBoltzmann’swork.Gibbssawhimself,notas offeringarivalapproachtoBoltzmann’s,butasbuildinguponanotherstrand ofBoltzmann’swork.Oneoftheseminaltextbooksofstatisticalmechanics, thatofR.C.Tolman(1938),weavestogetherconsiderationsdrawnfrom BoltzmannandGibbs,asdomanyofitssuccessors.Inmypresentation,the readerisintroducedtobothaGibbsianandneo-Boltzmannianapproach;my ownviewisthatthesupposedtensionsbetweenthemhavebeenexaggerated.
MychiefmotivationforwritingthebookisthefactthatIfrequentlyfind, indiscussionswithmyownstudentsandwithstudentsthatIencounterat conferencesandworkshops,thatagreatdealofmaterialthatIregardas prerequisitefordoingserious,professional-levelworkonthefoundations ofprobabilityandonthefoundationsofthermodynamicsandstatistical mechanics,isnotmaterialthatonecanpresumeisfamiliartothosewishing toembarkonsuchwork,andisnotreadilyavailable.ThisisabookthatI wishhadexisted,whenIwasagraduatestudentandthenajuniorresearcher, beginningtodelveintosuchmatters.
Thebookismeanttobemoreorlessself-contained,andcanserveas anintroductiontotheissuesdiscussedforthosenotalreadyfamiliarwith
them.Itcanbeused(asIhavedonewithanearlierdraft)asatextina graduatecourseonphilosophicalissuesinprobability,forstudentswithout backgroundinprobabilitytheory.Forthisreason,Ihaveincluded,asan Appendix,abriefintroductiontoprobabilitytheory,andincludealsochaptersintroducingthebasicsofthermodynamicsandstatisticalmechanics. ThesecontainmaterialthatIthinkeveryoneinterestedinthefoundationsof thesesubjectsshouldknow,andwhichisnoteasytogleanfromtheexisting philosophicalliteratureonthermodynamicsandstatisticalmechanics.
Iinvitereaderstothinkofthisbookasachoose-your-own-adventure book.Forthosewhowishtoreadthroughtheworkwithaminimumof mathematics,Ihaverelegatedmosttechnicaldetailstoappendices.But Ihopethatsomereaderswilltakethesepagesasaninvitationtoembark onseriousworkinphilosophyofprobability.Thetrainingofaphilosopher, whenitcomestorelevantbackgroundknowledge,isoftenunsystematic,and weoftenhavetopickupwhatweneedalongtheway.Forgraduatestudents andotherswithoutmuchbackgroundinprobability,Ihaveincludedintroductorymaterial,andIhopethatamongmyreaderswillbethosewhoare inclinedtoworkthroughit.
Noteonsources. Inthisbook,Iquotemanyworksthatwerepublished priortothetwentiethcentury.Thedatesandlocationoffirstpublication areimportantforhistoricalreasons,andforthisreasonIhaveprovided referencestotheoriginalpublications.Formanyofthefiguresquoted, suchasMaxwell,Kelvin,andBoltzmann,therearevolumesofcollected papers,whichmakestheworksmorereadilyaccessible.Forconvenience, Ihaveprovidedreferencetosuchcollections,inadditiontotheoriginal publications.Foranyquotationscontainedinthisbookfromworkswritten inlanguagesotherthanEnglish,Ihave,whereverpossible,lookedatthe originalpublication,andprovidecitationoftheoriginalforthosewhowant tolookatit,aswellasthetranslationfromwhichIamquoting.Incases whereanadequatetranslationwasnotreadilyavailable,thetranslationis myown,andIhaveprovidedtheoriginaltextinafootnote.Iamgratefulto MarieGueguenforassistancewiththeFrenchtranslations.
ThePuzzleofPredictability
1.1Introduction
Itisafamiliarfactthatsomethingscanbepredictedwithareasonabledegree ofconfidence;others,lessso.Thedailyrisingandsettingofthesun,the waxingandwaningofthemoon,andthemotionsofthemajorplanetsare familiarexamplesofthepredictable.Inotherdomains,suchasthestock market,themotionofaleafasitfallsfromatreeonablusteryday,orthe vagariesoftheweather,predictionishardertocomeby.
ThequestionIwanttoaskis:Howispredictionpossible,inthosedomains inwhichitis?
Avenerableanswer,andperhapsthefirstthatmightsuggestitselftosome readers,is:becausetherearedeterministiclawsofphysicsthat,togetherwith initialconditions,uniquelydeterminethefuturebehaviourofthings.This mightormightnotbetrue.Butit’sirrelevanttothequestionofwhatmakes predictionpossible.
Eveniftheunderlyinglawsofphysicsaredeterministic,weareneverin possessionofanythingevenremotelylikethesortofspecificationofthestate ofasystemthat,togetherwiththoselaws,sufficestodeterminewhatwewill observeinthefuture.Wenever,infact,takeintoaccountanythingmorethan atinyfractionofthephysicaldegreesoffreedomofanymacroscopicsystem. Instead,weengageinradicaldimensionalreductionofourproblems;we seekrelationsbetweenasmallnumberofmacroscopicvariables,discarding almosteverythingthereistobeknownaboutthesystem.
Itiswellknownthattherearesystemsthatexhibitextremelysensitive dependenceoninitialconditions,andtherehasbeenextensiveliteratureon theproblemsthatthisposesforprediction.Preciselong-termprediction, forsuchsystems,requiresabsurdlyprecisespecificationofinitialconditions. Butthesituationwefindourselvesinismuchworse;weknowvirtuallynothingabouttheprecisestateofmacroscopicsystemstowhichwesuccessfully applyourpredictivetechniques.
Togetasenseofthemagnitudeoftheproblem,considerwhatmightseem tobetheeasiestcase,thatofplanetarymotion.Astrophysicistscompute tablesofpredictedplanetarypositionswithanimpressivedegreeofprecision,millenniainadvance.Thecalculationisanintricateone,becauseit’s agravitationalmany-bodyproblem,whichistackledbyderivingsuccessive approximationstothemotion.Itiscomplicatedbythefactthat,atthelevelof precisionreachedbymodernastronomy,Newton’slawofgravitationisnot sufficient,andthetheoryofgeneralrelativitymustbetakenintoaccount. Butnevermindallthat;supposeyouwanttogetafirstapproximationtothe motionsofasingleplanet,say,Jupiter.Thegravitationalinfluenceofthesun onJupiterisbyfarthestrongestfactoraffectingitsmotion,andso,agood firstapproximationcanbeobtainedasatwo-bodyproblem.
Supposethen,thatyouareposedthefollowingproblem.Youaregiven therelevantlawsofphysics,themasses,currentpositions,andvelocitiesof theSunandJupiter,thevaluesofanyothermacroscopicallyascertainable quantitiesthatyoumightregardasrelevant,andpermissiontoignore everythingintheuniverseotherthanJupiterandtheSun,becauseyouare onlybeingaskedtoprovideafirstapproximationofthefuturebehaviour ofthesystem.Youareforbiddentoinvokeanythingelse.Theonlyassumptionthatyoumaymakeaboutthedetailedmicrostateofthesystemis thatitiscompatiblewiththevaluesofthemacrovariablesthatyouhave beengiven.
Question:Onthisbasis,whatcanyousayaboutthefuturecourseofthe system?
Answer:Virtuallynothing.
Ifyouwereposedaquestionlikethis,youwouldprobablytreatitasagravitationaltwo-bodyproblem.Thisisexactlysolvable,andanapproximation toinitialconditionsyieldsanapproximationtolaterconditions.
But,ifyouaretreatingitasatwo-bodyproblem,youareassumingthat theSunandJupiterwillremain,duringtheintervaloverwhichyouare interested,roughlysphericalbodiessmallinsizecomparedtothedistance betweenthem,andyouwilltreattheproblemasoneoffindingthemotionof thecentersofmassofthesetwospheres.Youareassumingthattheywillnot flyintopieces.Butthisis,bytherulesstipulated,anillicitassumption.The macrostateofthesystemiscompatiblewithawidevarietyofstatesonthe scaleof,say,individualmolecules,and,withinthescopeofthissetofstates aresomethatleadtoawidevarietyofsubsequentbehaviour.Theproblemis thatitisnotatwo-bodyproblem;it’satleastan N-bodyproblem,where
N isthetotalnumberofmoleculesthatcollectivelymakeupJupiterand thesun.
Whatweactuallyknowaboutanymacroscopicsystemdoesn’teven remotelycomeclosetobeingenoughtotightlyconstrainthefuture behaviourofthesystem.ThisiswhatIcallthe puzzleofpredictability.
Thekeytosolvingthepuzzleliesinthephenomenonof statisticalregularities.Ithaslongbeenrecognizedthataggregatesofeventsthatare,individually,unpredictable,cangiverisetoreliableregularities.Thus,forexample, thetotalnumberoftrafficaccidentsinagivenyearinPittsburghmight bemoreorlessconstant,fromyeartoyear,thoughexactlywhenandwhere thenextaccidentwilloccurisnotpredictable.Statisticalregularitiesarise whenweaggregatealargenumberofvariablesthatareeffectivelyrandom andeffectivelyindependent.Counterintuitiveasitmightseem,effective prediction,evenincasesofapparentdeterminism, always involvesthissort ofstatisticalregularity.Thisiswhytheissueofwhethertheunderlyinglaws aredeterministicisirrelevanttothequestionofhowitisthatwecanmake effectivepredictions.Evenifthelawsaredeterministic,whatisrequired fromthemis effectivelyrandom behaviourofmostofthevariablesthatare potentiallyrelevanttoapredictiontask.
Anupshotofthisisthatconsiderationsthatstemfromprobabilitytheory arerequiredforall—yes, all—ofourapplicationsofphysicaltheoryto prediction,andnotonlyincaseswheretheirpresenceisapparentbecause probabilitiesareexplicitlyinvoked.
Now,alittlemoredetailonthesepoints.
1.2Determinismandthepovertyofourknowledge
InapaperpresentedtotheAcadémieRoyaledesSciencesdeParisinFebruary 1773,Laplacewrote,
ThepresentstateofthesystemofNatureisevidentlyaconsequenceofwhat itwasattheprecedingmoment,andifweconceiveofanintelligencethat, foragiveninstant,embracedalltherelationsofthebeingsofthisUniverse, itwillbeabletodetermineforanygiventimeinthepastorthefuturethe respectivepositions,movements,andgenerallythepropertiesofallthose beings.(Laplace1776:§XXV,in OeuvresComplètes,vol.8,p.144)1
1 “L’étatprésentdusystemdelaNatureestévidemmentunesuitedequ’ilétaitaumoment précédent,et,sinousconcevonsuneintelligencequi,pouruninstantdonné,embrassetousles rapportsdesêtresdecetUniverse,ellepourradéterminerpouruntempsquelconqueprisdans
Thiswas,ofcourse,echoed40yearslaterinanoft-quotedpassagefromhis PhilosophicalEssayonProbabilities (Laplace1814:4;1902:4).2
SomehaveattributedtoLaplaceavisionofthefutureprogressofscience inwhichwehumansapproacheverclosertotheconditionofsuchabeing. NothingcouldbefurtherfromLaplace’sintention.Heisexplainingto hisaudiencewhy,inspiteofthepresumeddeterminismofthelawsof nature(takenbyLaplacetobean apriori truth,basedontheprincipleof sufficientreason),thereisneedforanysuchsubjectasprobabilitytheory. ThePerfectPredictorisintroducedasa contrast totheworkingsofthe humanmind.Thoughourpredictivesuccessesinastronomyfurnisha“feebleideaofthisintelligence,”thehumanmind“willalwaysremaininfinitely removedfromit”(Laplace1902:4),andthatiswhyweneedprobability theory.
Thisistruefortworeasons.Thefirstisthattheamountthatweactually knowisalwaysaminusculefractionofthecompletestateoftheworld. Acompletespecificationofthestatewouldrequirespecificationofthestate ofeveryatom,everyelementaryparticle.Thesecondisthat,evenif(per impossibile)wecouldbegiventhiscompletestate,wecouldnotprocessthe information,couldnotdotherequisitecomputation.Tomakeanyheadway, wewouldhavetoselectafewsalientvariables,andapplyaprobabilisticor statisticaltreatmenttotherest.
Ourvastignoranceis,ofcourse,onlyimportantifwhatwedon’tknow aboutispotentiallyrelevanttowhatwewanttopredict.Onepotential solutiontothepuzzleofpredictabilitythatmightsuggestitselfisthat,in ordertomakepredictionsaboutcertainmacrovariables,allweneedasinput isthevaluesofthosemacrovariables.
Takethecaseofplanetaryprediction.Wedon’tknowthedetailsofthe internalarrangementsoftheplanets,but,itmightbeargued,wedon’thaveto. AllweneedtoknowaboutJupiteristhatacertainmassisconcentrated,inan approximatelysphericallysymmetricfashion,withinasphericalregionthat lepasséoudansl’avenirlapositionrespective,lesmouvements,etgénéralementlesaffectations detouscesêtres.”
2 Laplaceisnot,inthispassage,formulatinganovelthesisorsayinganythingthatwouldhave beensurprisingorshockingtohisreaders;rather,heisremindinghisreadersofsomethingthat hepresumestheytakeforgranted.Indeed,markedlysimilarpassagescanbefoundinearlier writingsbyBoscovich,Condorcet,andd’Holbach;seeKožnjak(2015)fordiscussion.
issmallcomparedtointerplanetarydistances,andthatsomethingsimilar canbesaidabouttheSunandotherplanets.
Thisistrue;ifweknowthatthisholdsatthepresenttime,andcanbe expectedtoremaintruefortheforeseeablefuture,thenwehaveallweneed totreatthesolarsystemasan n-bodygravitationalproblem,where n isa relativelysmallnumberofastronomicalbodies.
However,asalreadymentioned,whatweknowaboutthecurrentmacrostateofJupiterisconsistentwithitsflyingapartinshortorder.Toseethis, imaginetwohalf-Jupitersizedblobsofgascolliding,andsettlingdowninto somethinglikethecurrentstateofJupiter.Therelevantphysicallawsare invariantundervelocityreversal,andso,ifthisisapastthatcouldleadto theobservedmacrostateofJupiter,itisalsoapossiblefutureoftheobserved macrostate.
ItmightseemthatsomefundamentallawofphysicsforbidsJupiterfrom spontaneouslydividingintotwohalf-planetsthatflyapartfromeachother. Butwhat?Conservationofenergy?Lettheresultinghalf-planetsbecooler thanJupiterisnow,sothatthedecreaseininternalenergyoffsetstheincrease inkineticenergy.
Thiswouldinvolvenoviolationoftheconservationofenergy,andneed notviolateanyotherconservationlaw,butitwouldviolatetheSecondLaw ofThermodynamics—atleast,asoriginallyconceived(seeChapter6).Thisis akeytosolvingthepuzzleofpredictability.Understandingthebasisforthe lawsofthermodynamicsisrequisiteforunderstandingpredictability,evenin domainswherethermodynamicsisnotexplicitlyevoked.Itwasoneofthe greatinsightsofnineteenth-centuryphysicstorealizethattheSecondLaw ofThermodynamicsrestsonstatisticalregularities.
Statisticalregularitiesarenotthesortofthingthatcanbesubjectsof certainknowledge.Unlikeregularitiesunderwrittenbydeterministiclaws, theyarenotexceptionless,andtheyareatbestthesortofthingonecan regardasverylikelytoobtain.Thepioneersofthesciencethatcameto becalledstatisticalmechanicscametotheconclusionthatphenomenathat wouldcountasviolatingtheSecondLaw,asoriginallyconceived,shouldbe regarded,notasphysicallyimpossible,butonlyashighlyimprobable.Thus, whatweareseekingisnotacriterionforrulingoutofconsiderationthesorts ofmicrostatesthatwouldleadtoJupiter-splittingbehaviour.Whatweare seekingismoresubtle:somewayofregardingitasunlikelythatamicrostate ofthatsortwillberealized.Andthismeansmakingsenseofprobabilistic assertions,eveninthecontextofdeterministicphysics.
1.3Statisticalregularitiesandthelawoflargenumbers
Today,statisticsisabranchofmathematics,closelyconnectedwith probabilitytheory.Itwasnotalwaysso.Theoriginalmeaningoftheword hadtodowithcollectionandpresentationinasystematicwayofdemographicdata.Here’showthesubjectwascharacterizedintheinauguralissue ofthe JournaloftheStatisticalSocietyofLondon,publishedinMayof1838.
ItiswithinthelastfewyearsonlythattheScienceofStatisticshasbeenatall activelypursuedinthiscountry;anditmaynot,evennow,beunnecessary toexplaintogeneralreadersitsobjects,andtodefineitsprovince.Theword StatisticsisofGermanorigin,andisderivedfromthewordStaat,signifying thesameasourEnglishword State,orabodyofmenexistinginasocial union.Statistics,therefore,maybesaid,inthewordsoftheProspectusof thisSociety,tobetheascertainingandbringingtogetherofthose“facts whicharecalculatedtoillustratetheconditionandprospectsofsociety” andtheobjectofStatisticalScienceistoconsidertheresultswhichthey produce,withtheviewtodeterminethoseprinciplesuponwhichthewellbeingofsocietydepends.
TheScienceofStatisticsdiffersfromPoliticalEconomy,because, althoughithasthesameendinview,itdoesnotdiscusscauses,norreason uponprobableeffects;itseeksonlytocollect,arrange,andcompare,that classoffactswhichalonecanformthebasisofcorrectconclusionswith respecttosocialandpoliticalgovernment.
(StatisticalSocietyofLondon1838:1)
Nearly50yearslater,inthevolumecelebratingthejubileeofthesociety,its president,RawsonW.Rawson,characterizedthescienceas
thesciencewhichtreatsofthestructureof“humansociety,” i.e.,ofsociety inallitsconstituents,howeverminute,andinallitsrelations,however complex;embracingalikethehighestphenomenaofeducation,crime,and commerce,andtheso-called“statistics”ofpin-makingandLondondust bins.(Rawson1885:8)
Bythattimethemathematicalmethodsthatwenowthinkofasthe provinceofthescienceofstatisticshadbeguntomaketheirwayintothe profession,thoughtheywerenotyetdominant.WeseethisintheJubilee volume,whichcontainsapaperbyF.Y.Edgeworththatintroduceshis
beyondchanceandcredence7
audiencetosuchtopicsasthenormaldistributionandsignificancetests,3 totheperplexityofsomemembersoftheaudience(Galton1885:266).
Systematiccollectionofstatisticsburgeonedinthefirsthalfofthe nineteenthcentury,givingriseto(inHacking’swords)an“avalancheof printednumbers”(Hacking1990:3)Withthisavalancheofnumberscame anincreasedawarenessofstatisticalregularities,thatis,stablefrequencies, inlargepopulations,oftheoccurrenceofindividuallyunpredictableevents. Theseextended,notonlytosuchthingsasbirthsanddeaths,butevento matterssuchastheoccurrencesofviolentcrimes.AsQueteletdramatically putit,⁴
Itisabudgetthatwepaywithfrighteningregularity,thatwhichwepay totheprisons,laborcamps,andscaffolds…everyyearthenumbershave confirmedmypredictions,tothepointthat,Icouldhavesaidwithmore exactitude:itisatributethatmanacquitswithgreaterregularitythanthat whichheowestonatureortothetreasuryoftheState,thathepaystocrime!
(Quetelet1835:9)
InEngland,oneconduitofthissortofthinkingwasThomasHenryBuckle, describedbyTheodorePorteras“possiblythemostenthusiasticandbeyond doubtthemostinfluentialpopularizerofQuetelet’sideasonstatistical regularity”(Porter1986:65).
Alsointhenineteenthcenturycameanincreasedawarenessthat,paradoxicalasitmightseem,wecanseetheseregularitiesasarising,notinspite ofchance,but becauseof chance.ThebasicideaisfoundalreadyinJacob Bernoulli’s Arsconjectandi (1713),andhastodowithwhatPoisson(1835; 1837)namedthe lawoflargenumbers.
Thingsofanynaturearesubjecttoauniversallawthatcanbecalledthelaw oflargenumbers.Itconsistsinthefactthat,ifweobserveveryconsiderable numbersofeventsofthesamenature,dependingonconstantcausesand causesthatvaryirregularly,sometimesinonedirectionandsometimesin theother,withouttheirvariationbeingprogressiveinanygivendirection,
3 Yes,thoseexistedbeforeFisherwasborn!
⁴ “Ilestunbudgetqu’onpaieavecunerégularitéeffrayante,c’estceluidesprisons,desbagnes etdeséchafauds;…et,chaqueannée,lesnombressontvenusconfirmermesprévisions,àtel point,quej’auraispudire,peut-êtreavecplusd’exactitude:Ilestuntributquel’hommeacquitte avecplusderégularitéqueceluiqu’ildoitàlanatureouautrésordel’État,c’estceluiqu’ilpaie aucrime!”
wefindratiosbetweenthesenumbersthatareverynearlyconstant.For eachkindofthings,theseratioshaveaspecialvaluefromwhichtheywill deviatelessandless,astheseriesofobservedeventsincreasesfurther,and whichtheywouldreachexactlyifitwerepossibletoextendtheseriesto infinity.(Poisson1837:7)⁵
Weobservestablerelativefrequenciesofeventsinacertainclass,not becausethereisaProvidencewatchingoverthesequence,ensuringthat thingsbalanceout,butbecausethereisn’t.Itistheresultofaggregationof independentchanceeventsthatleadstothesestablefrequencies.
Thebasicideacanbeillustratedbyacointoss.Supposeacoinistossed n times,andthattheprobabilityofheadsonanygiventossis p (forafaircoin, p=1/2),andsupposethatthesetossesareprobabilisticallyindependentof eachother;theprobabilityof Heads onanygiventossis p,regardlessofthe resultsoftheothertosses.⁶Nowconsidertherelativefrequencyof Heads, thatis,thefraction,outofallthetosses,ofthosethatland Heads.This won’t,typically,beexactlyequalto p.Butforalargenumberoftosses,it willprobablybecloseto p,andtheprobabilitythattherelativefrequency ismorethanagivendistancefrom p getssmallerasthenumber n oftosses increases.SeeAppendixforfurtherdiscussion.
1.4Theimportationofstatisticsintophysics
Thisistheintellectualbackgroundagainstwhichseriousworkonthekinetic theoryofgasesbegan,inthemid-nineteenthcentury.Statisticalregularities wereafamiliarphenomenoninthesocialsciences,andtherewasrecognition thattheexplanationfortheseregularitieswastobebasedonprobabilistic considerations.
⁵ “Leschosesdetoutenaturesontsoumisesàuneloiuniversellequ’onpeutappelerlalois desgrandsnombres.Elleconsisteenceque,sil’onobservedesnombrestrèsconsidérables d’événementsd’unemêmenature,dépendantsdecausesconstantesetdecausesquivarient irrégulièrement,tantôtdansunsens,tantôtdansl’autre,c’est-à-diresansqueleurvariationsoit progressivedansaucunsensdéterminé,ontrouvera,entrecesnombres,desrapportsàtrèspeu prèsconstants.Pourchaquenaturedechoses,cesrapportsaurontunevaleurspécialedontils s’écarterontdemoinsenmoins,àmesurequelasériedesévénementsobservésaugmentera davantage,etqu’ilsatteindraientrigoureusements’ilétaitpossibledeprolongercettesérieà l’infini.”
ThisdiffersfromthecorrespondingpassageinPoisson(1835)onlybytheinsertionofthe phrase“decausesconstantes”inthesecondsentence.
⁶ AsequenceofeventsofthissortisreferredtoasequenceofBernoullitrials.
Thekinetictheoryofgasespositsthatagasconsistsofalargenumber ofmoleculesmovinginahaphazardfashion.Thistheory,pioneeredby Clausius(thoughtherewereimportantprecursors,includingDaniel Bernoulli,Herapath,Waterston,Joule,andKrönig),wasdeveloped,atthe handsofMaxwell,Boltzmann,andGibbs,intothesciencethatwe,following Gibbs’coinage,nowcallstatisticalmechanics,andwhosescopehasbeen extendedwellbeyondtreatmentofgases(seeBrush1976forsomeofthe earlyhistory).
Poissonhimself,whogavethelawoflargenumbersitsname,cites moleculartheoryasanapplication.Inabodycomposedofdiscrete moleculesseparatedbyemptyspace,theintermoleculardistancesmay varywidely.Nonetheless,forsufficientlylargevolumes,equalvolumeswill containroughlyequalnumbersofmolecules(Poisson1835:481;1837:10).
Krönig(1856)alsoarguesfororderfromaggregateddisorder,inaccordance withthelawoflargenumbers.
Incomparisonwiththegas-atomseventhesmoothestwallistoberegarded asveryuneven,andthepathofeachgas-atommustthereforebeonethatis soirregularastoescapecalculation.Accordingtothelawsoftheprobability calculusonemayassume,however,inplaceofthiscompleteirregularity,a completeregularity.(Krönig1856:316)⁷
ItwasJamesClerkMaxwell,however,whomadetheuseofstatisticalmethodsacentralpartofkinetictheory.Recognizingthattheeffectofcollisions betweenmoleculeswouldrenderthedetailsoftheirmotionseffectively unpredictable,headoptedthestrategyofasking,notforthedetailedstate ofthegas,butforastatisticalsummaryofitspropertiescomparabletothe tablescompiledbythestatisticians.
Icarefullyabstainfromaskingthemoleculeswhichenterwheretheylast startedfrom.Ionlycountthemandregistertheirmeanvelocities,avoiding allpersonalenquirieswhichwouldonlygetmeintotrouble.
(ReportonapaperbyOsborneReynoldsontheflow ofrarifiedgases,March28,1879,inHarman2002:776, andGarberetal.1995:422)
⁷ “DenGasatomengegenüberistauchdieebensteWandalssehrhöckerigzubetrachten, unddieBahnjedesGasatomsmussdeshalbeinesounregelmässigeseyn,dasssiesichder Berechnungentzicht.NachdenGesetzenderWahrscheinlichkeitsrechnungwirdmanjedoch stattdieservollkommenenUnregelmässigkeiteinevollkommeneRegelmässigkeitannehmen dürfen.”
