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。