What，then，isreallyimpliedinthedoctrinethatallknowledgerestsuponexperience？OneofMill’sintellectualancestorslaysdownthefundamentalprinciple。Itisabsurd，saysHume，39*totrytodemonstratematteroffactbyaprioriarguments。’Nothingisdemonstrableunlessthecontraryimpliesacontradiction。Nothingthatisdistinctlyconceivableimpliesacontradiction。Whateverweconceiveasexistentwecanalsoconceiveasnon-existent。Thereisnobeing，therefore，whoanon-existenceimpliesacontradiction。’’Matteroffact，’then，mustbeprovedbyexperience；but，givena’fact’wemaydeducenecessaryconsequences。Allnecessitymaybehypothetical；thereisan’if’toevery’must，’butrememberingthe’if’the’must’
willbeharmless。Itcannevertakeusbeyondexperience。Theexistenceofspaceitselfcannotbecallednecessary；butspaceoncegiven，allgeometrymay’necessarily’follow，andimplyrelationsrunningthroughthewholefabricofscientificknowledge。Millagreesthata’hypothetical’necessityofthiskindbelongstogeometry；andadds，thatinanysciencewhatever，wemight，bymakinghypotheses，arriveatanequalnecessity。40*Butthen，hegoesontourge，thehypothesesofgeometryarenot’absolutetruths，’but’generalisationsfromobservation，’or’inductionsfromtheevidenceofoursenses，’41*which，therefore，arenotnecessarilytrue。Thisledtohiskeenestcontroversies，and，inmyopinion，tohisleastsuccessfulanswers。HeespeciallyclaimscreditinhisAutobiographyforhavingattackedthe’stronghold’oftheintuitionistsbyupsettingbeliefintheaprioricertaintyofmathematicalaphorisms。Infact，hisopponentsconstantlyappealedtothecaseofmathematics，andMillassumesthattheycanbemetonlybyreducingsuchtruthstothecaseofpurelyempiricaltruths。Hearguesboldlythatthe’characterofnecessityascribedtothetruthsofmathematics’is’anillusion。’42*Geometryandarithmeticarebothfoundeduponexperienceorobservation。Hegoesindeedstillfurtherattimes。
Atoneplaceheevenholdsthattheprincipleofcontradictionitselfissimply，oneofourfirstandmostfamiliargeneralisationsfromexperience。’Weknow，’bythesimplestobservationofourownminds，’thatbeliefanddisbeliefexcludeeachother，andthatwhenlightispresentdarknessisabsent。43*Millthoughthimselfbound，wesee，torefertoexperiencenotonlyourknowledgeoffacts，buteventhecapacities，whicharesaidbyanotherschooltobetheconditionsofperceivingandthusacquiringexperience。IfhehadstudiedKant，hemighthavereachedabetterversionofhisownview。Asitwas，hewasledtoacceptingparadoxeswhichhewasnotreallyconcernedtomaintain。Hehadtochoosebetweenatheoryof’intuitions’——sounderstoodastoentitleustoassertmatteroffactindependentlyofexperience——andatheorywhichseemstomakeeventheprimaryintellectualoperationsmerestatementsofempiricalfact。Sincenecessarystatementsaboutmattersoffactmustbeimpossible，hearguesthatwecannotevendrawnecessaryinferencesfromobservedfact。Notcontentwithsayingthatallnecessityishypothetical，hearguesthatallnecessity，eventhelogicalnecessityofcontradiction，isafigment。Ifhedoesnotcarryoutatheorywhichwouldseemtomakeallreasoningunsatisfactory，hemaintains，atleast，thatthehypothesesorassumptionsinvolvedingeometry，andeveninarithmetic，aregeneralisedfromexperience，and’seldom，ifever，exactlytrue。’Iftheassumptionsareinaccurateoruncertain，thewholesuperstructureofsciencemustalsobeuncertain。
Thenatureofhisargumentfollowsfromhispreviouspositions。Hetreatsspaceandnumberassomehowqualitiesofthe’things，’orasattributeswhichweobservewithoutinanysensesupplyingthem。Hisargumentupongeometrybeginsbyassertingthattherearenosuch’realthings’aspointsorlinesorcircles。Nay，theyarenotevenpossible，sofaraswecansee，consistentlywiththeactualconstitutionoftheuniverse。Itis’customary’toanswerthatsuchlinesonlyexistinourminds，andhavethereforenothingtodowithoutwardexperience。44*
This，however，isincorrectpsychologically，becauseourideasarecopiesoftherealities。Alinewithoutbreadthis’inconceivable，’andthereforedoesnotexisteveninthemind。
Hencewemustsupposethatgeometrydealseitherwith’non-entities’orwith’naturalobjects。’45*Arithmeticfareslittlebetter。Whenwesaythattwoandonemakethree，weassertthatthesamepebblesmay，’byanalterationofplaceandarrangement’——thatis，bybeingformedintooneparcelortwo——bemadetoproduceeithersetofsensations。46*Eachofthenumbers，2，3，4，etc。，hesayselsewhere，’denotesphysicalphenomenaandconnotesaphysicalpropertyofthosephenomena。’47*Arithmeticowesitspositiontothe’fortunateapplicability’toitofthe’inductivetruth’thatthesumsofequals’areequal。’48*Itisobvioustoremarkthatthisisonlytrueofcertainapplicationsofarithmetic。Whenwespeakofthenumbersofapopulation，weimply，asMilladmits，noequalityexceptthateachpersonisaunit。49*Wemayspeakwithequalproprietyofanumberofsyllogismsorofmetaphors，inwhichwehavenothingtodowith’equality’or’physicalproperties’atall。Further，asheobserves，50*itisthepeculiarityofthecasethatcountingonethingistocountallthings。WhenIseethatfourpebblesaretwopairsofpebbles，I
seethesametruthforallcases，including，forexample，syllogisms。Milladmits，accordingly，that’inquestionsofpurenumber’——thoughonlyinsuchquestions——theassumptionsare’exactlytrue，’andapparentlyholdsthatwemaydeduceexactlytrueconclusions。Thatoughttohavebeenenoughforhim。Hehadreallynosufficientreasonfordeprivingusofourarithmeticalfaith。Hecanhimselfpointoutitsharmlessness。Ashetrulysays，’fromlawsofspareandnumberalonenothingcanbededucedbutlawsofspareandnumber。’51*Werannevergetoutsideoftheworldofexperienceandobservationbyapplyingthem。Ifwecount，wedonotsaythattheremustbefourthings，butthatwherevertherearefourthingstherearealsotwopairsofthings。Theunlucky’pebble’argumentillustratesoneconfusion。
’Twoandtwoarefour’ischangedinto’twoandtwomakefour。’
Thestatementofaconstantrelationismadeintoastatementofanevent。Twopebblesaddedtotwomightproduceafifth，buttheoriginaltwopairswouldstillbefour。Thespace-problemsuggestsgreaterdifficulties。Space，heargues，musteitherbeapropertyofthingsoranideainourminds，andthereforea’non-entity。’Ifweconsiderit，however，tobeaformofperception，thedisjunctionceasestobevalid。Thespace-perceptionsmarktheborder-linebetween’object’and’subject，’andwecannotplaceitsproductineithersphereexclusively。Thespace-relationsare’subjective，’becausetheyimplyperceptionbythemind，butobjectivebecausetheyimplytheactionofthemindasmind，anddonotvaryfromonepersonor’subject’toanother。Tosaywhethertheywereobjectiveorsubjectiveabsolutelyweshouldhavetogetoutsideofourmindsaltogether——whichisanimpossiblefeat。Therefore，again，itisnotreallytothepurposetoallegethatsucha’thing’asastraightlineoraperfectcircleneverexists。Whetherwesaythatacurvedeviatesfromorconformstoperfectcircularity，weequallyadmittheexistenceofaperfectcircle。Wemaybeunabletomarkitwithfingerormicrometer，butitisthere。Ifnotwolinesareexactlyequal，thatmustbebecauseonehasmoresparethantheother。Mill’sargumentseemstoinvolvetheconfusionbetweenthestatementthatthingsdifferinspaceandthestatement，whichwouldbesurelynonsense，thatthespareitselfdiffers。Itistotransferthedifferencefromthethingsmeasuredtothemeasureitself。Itisjustthepeculiarityofspacethatitcanonlybemeasuredbyspace；andthattosayonespaceisgreaterthananother，issimplytosay，’thereismorespace。’Asinthecaseofnumber，heisreallymakinganillegitimatetransferfromonespheretoanother。Astraightlineisasymmetricaldivisionofspace，whichmustbetakentoexist，thoughwecannotmakeaperfectlystraightline。Ourinabilitydoesnottendtoprovethatthe’space’itselfisvariable。Inapplyingameasurewenecessarilyassumeitsconstancy；anditisdifficulteventounderstandwhat’variability’means，unlessitisvariabilityinreferencetosomeassumedstandard。If，asMillseemstothink，spaceisapropertyofthings，varyinglikeotherproperties，wehavetoask，Inwhat，then，doesitvary？Allotherpropertiesvaryinrespectoftheirspace-relations；but，ifspaceitselfbevariable，weseemtobereducedtohopelessincoherence。
Thus，toascribenecessitytogeometryaswellastoarithmeticisnottoascribe’necessity’topropositionstouseHume’slanguageagainabout’mattersoffact。’The’necessity’
isimpliedinapeculiaritywhichMillhimselfputsveryforcibly，52*andwhichseemstobeallthatiswanted。Anarithmeticalformulaofthesimplestormostcomplexkindisanassertionthattwowaysofconsideringafactareidentical。WhenIsaythattwoandtwomakefour，orlaydownsomealgebraicalformula，suchasTaylor’stheorem，Iamassertingthepreciseequivalenceoftwoprocesses。Idonotevensaythattwoandtwomustmakefour，butthat，iftheymakefour，theycannotalsoorevermakefive。Thenumberisthesameinwhateverorderwecount，solongaswecountalltheunits，andcountthemcorrectly。SomuchisimpliedinMill’sobservationthatcountingonesetofthingsiscountingallthings。Theconcretecircumstancesmakenodifference。Thesameistrueofgeometry。
Thecomplexfiguremaybealsoregardedasacombinationofsimplerfigures。Itremainspreciselythesame，thoughweperceivethatbesidesbeingonefigureitisalsoacombinationoffigures。Thisrunsthroughallmathematicaltruths，and，I
think，indicatesMill’sprecisedifficulty。Hesaysquitetrulythattoknowtheexistenceofafactyoumustalwayshavesomethinggivenbyobservationorexperience。Themostcomplexmathematicalformulaemaystillberegardedasequatingdifferentstatementsofthesameexperience。Thedifferenceisonlythattheexperienceisevolvedintomorecomplexforms，notbyanychangeinthedatasupplied，butbyanintellectualoperationwhichconsistsessentiallyinorganisingthedatainvariousways。Thereasonerdoesnotforaninstantdesertfact；heonlyperceivesthatitmaybecontemplatedindifferentways，andthatverydifferentstatementsasserttheverysamefactorfacts。Ourexperiencemaybeincreased，eitherbytheentranceofnewobjectsintoourfieldofobservation，orbythedifferentmethodsofcontemplation。Themathematiciandealswithpropositionswhichremainequallytrueifwesupposenochangewhatevertotakeplaceintheworld，or，asMillputsit，’ifalltheobjectsoftheuniversewereunchangeablyfixed。’53*Histheories，inshort，constructamaponwhichhecanafterwardslaydownthechangeswhichinvolvetime。Thefillingupofthemapdependsentirelyuponobservationandexperience；buttomakethemapitselfamerebundleofaccidentalcoexistencesistodestroytheconditionsofexperience。Themapisourownfacultyofperception。
’Thereissomethingwhichseemstorequireexplanation，’saysMill，54*’inthefactthatanimmensemultitudeofmathematicaltruths……canbeelicitedfromsosmallanumberofelementarylaws。’ItispuzzlingwhenyouidentifyNewtonwithDiamondonthegroundthattheybothseethesame’fact。’Butitisnomorepuzzlingthananythingelse，asindeedMillproceedstoshow，whenweobservethemethodbywhichinarithmetic，forexample，anindefinitenumberofrelationsisimpliedbythesimpleprocessofcounting。Thefactisthesameforallobservers，insofarastheyhavethesamedata；buttoperceivethedataalreadyimpliesthegermofthoughtfromwhichallthedemonstrativesciencesareevolved。Theknowledgecanbetransformedandcomplicatedtoanindefinitedegreebysimplyidentifyingdifferentwaysofcombiningthedata。Mill，inhisanxietytoadheretofactsandexperience，failstorecogniseadequatelytheprocessbywhichsimpleobservationisevolvedintocountlessmodifications。Thedifficultyappearsinitsextremeforminthecurioussuggestionthateventheprincipleofcontradictionisaproductofexperience。Millissoresolvedtoleavenothingforthemindtodo，thathesupposesaprimitivemindwhichisnotevenabletodistinguish’isnot’from’is。’Itishardtounderstandhowsucha’mind，’ifitwerea’mind，’