第43章

Further，whyshouldtherealwaysbebecoming，andwhatisthe causeofbecoming？-thisnoonetellsus。Andthosewhosupposetwo principlesmustsupposeanother，asuperiorprinciple，andsomust thosewhobelieveintheForms；forwhydidthingscometo participate，orwhydotheyparticipate，intheForms？Andallother thinkersareconfrontedbythenecessaryconsequencethatthereis somethingcontrarytoWisdom，i。e。tothehighestknowledge；butwe arenot。Forthereisnothingcontrarytothatwhichisprimary；for allcontrarieshavematter，andthingsthathavematterexistonly potentially；andtheignorancewhichiscontrarytoanyknowledge leadstoanobjectcontrarytotheobjectoftheknowledge；butwhat isprimaryhasnocontrary。 Again，ifbesidessensiblethingsnoothersexist，therewillbe nofirstprinciple，noorder，nobecoming，noheavenlybodies，but eachprinciplewillhaveaprinciplebeforeit，asintheaccounts ofthetheologiansandallthenaturalphilosophers。Butifthe Formsorthenumbersaretoexist，theywillbecausesofnothing； orifnotthat，atleastnotofmovement。Further，howisextension， i。e。acontinuum，tobeproducedoutofunextendedparts？Fornumber willnot，eitherasmoverorasform，produceacontinuum。Butagain therecannotbeanycontrarythatisalsoessentiallyaproductive ormovingprinciple；foritwouldbepossibleforitnottobe。Or atleastitsactionwouldbeposteriortoitspotency。Theworld， then，wouldnotbeeternal。Butitis；oneofthesepremisses，then， mustbedenied。Andwehavesaidhowthismustbedone。Further，in virtueofwhatthenumbers，orthesoulandthebody，oringeneral theformandthething，areone-ofthisnoonetellsusanything； norcananyonetell，unlesshesays，aswedo，thatthemovermakes themone。Andthosewhosaymathematicalnumberisfirstandgoon togenerateonekindofsubstanceafteranotherandgivedifferent principlesforeach，makethesubstanceoftheuniverseamere seriesofepisodes(foronesubstancehasnoinfluenceonanotherby itsexistenceornonexistence)，andtheygiveusmanygoverning principles；buttheworldrefusestobegovernedbadly。 ’Theruleofmanyisnotgood；onerulerlettherebe。’ WEhavestatedwhatisthesubstanceofsensiblethings，dealing inthetreatiseonphysicswithmatter，andlaterwiththesubstance whichhasactualexistence。Nowsinceourinquiryiswhetherthere isorisnotbesidesthesensiblesubstancesanywhichisimmovable andeternal，and，ifthereis，whatitis，wemustfirstconsiderwhat issaidbyothers，sothat，ifthereisanythingwhichtheysay wrongly，wemaynotbeliabletothesameobjections，while，if thereisanyopinioncommontothemandus，weshallhavenoprivate grievanceagainstourselvesonthataccount；foronemustbecontent tostatesomepointsbetterthanone’spredecessors，andothersno worse。 Twoopinionsareheldonthissubject；itissaidthattheobjects ofmathematics-i。e。numbersandlinesandthelike-aresubstances，and againthattheIdeasaresubstances。And(1)sincesomerecognize theseastwodifferentclasses-theIdeasandthemathematicalnumbers， and(2)somerecognizebothashavingonenature，while(3)some otherssaythatthemathematicalsubstancesaretheonlysubstances， wemustconsiderfirsttheobjectsofmathematics，notqualifyingthem byanyothercharacteristic-notasking，forinstance，whethertheyare infactIdeasornot，orwhethertheyaretheprinciplesand substancesofexistingthingsornot，butonlywhetherasobjectsof mathematicstheyexistornot，andiftheyexist，howtheyexist。Then afterthiswemustseparatelyconsidertheIdeasthemselvesina generalway，andonlyasfarastheacceptedmodeoftreatment demands；formostofthepointshavebeenrepeatedlymadeevenby thediscussionsoutsideourschool，and，further，thegreaterpart ofouraccountmustfinishbythrowinglightonthatinquiry，viz。 whenweexaminewhetherthesubstancesandtheprinciplesof existingthingsarenumbersandIdeas；forafterthediscussionofthe Ideasthisremansasathirdinquiry。 Iftheobjectsofmathematicsexist，theymustexisteitherin sensibleobjects，assomesay，orseparatefromsensibleobjects (andthisalsoissaidbysome)；oriftheyexistinneitherof theseways，eithertheydonotexist，ortheyexistonlyinsome specialsense。Sothatthesubjectofourdiscussionwillbenot whethertheyexistbuthowtheyexist。 Thatitisimpossibleformathematicalobjectstoexistin sensiblethings，andatthesametimethatthedoctrineinquestionis anartificialone，hasbeensaidalreadyinourdiscussionof difficultieswehavepointedoutthatitisimpossiblefortwo solidstobeinthesameplace，andalsothataccordingtothesame argumenttheotherpowersandcharacteristicsalsoshouldexistin sensiblethingsandnoneofthemseparately。Thiswehavesaid already。But，further，itisobviousthatonthistheoryitis impossibleforanybodywhatevertobedivided；foritwouldhaveto bedividedataplane，andtheplaneataline，andthelineata point，sothatifthepointcannotbedivided，neithercantheline， andifthelinecannot，neithercantheplanenorthesolid。What difference，then，doesitmakewhethersensiblethingsaresuch indivisibleentities，or，withoutbeingsothemselves，have indivisibleentitiesinthem？Theresultwillbethesame；ifthe sensibleentitiesaredividedtheotherswillbedividedtoo，or elsenoteventhesensibleentitiescanbedivided。 But，again，itisnotpossiblethatsuchentitiesshouldexist separately。Forifbesidesthesensiblesolidstherearetobeother solidswhichareseparatefromthemandpriortothesensible solids，itisplainthatbesidestheplanesalsotheremustbeother andseparateplanesandpointsandlines；forconsistencyrequires this。Butiftheseexist，againbesidestheplanesandlinesand pointsofthemathematicalsolidtheremustbeotherswhichare separate。(Forincompositesarepriortocompounds；andifthere are，priortothesensiblebodies，bodieswhicharenotsensible，by thesameargumenttheplaneswhichexistbythemselvesmustbeprior tothosewhichareinthemotionlesssolids。Thereforethesewillbe planesandlinesotherthanthosethatexistalongwiththe mathematicalsolidstowhichthesethinkersassignseparateexistence； forthelatterexistalongwiththemathematicalsolids，whilethe othersarepriortothemathematicalsolids。)Again，therefore， therewillbe，belongingtotheseplanes，lines，andpriortothem therewillhavetobe，bythesameargument，otherlinesandpoints； andpriortothesepointsinthepriorlinestherewillhavetobe otherpoints，thoughtherewillbenootherspriortothese。Now(1) theaccumulationbecomesabsurd；forwefindourselveswithonesetof solidsapartfromthesensiblesolids；threesetsofplanesapartfrom thesensibleplanes-thosewhichexistapartfromthesensible planes，andthoseinthemathematicalsolids，andthosewhichexist apartfromthoseinthemathematicalsolids；foursetsoflines，and fivesetsofpoints。Withwhichofthese，then，willthe mathematicalsciencesdeal？Certainlynotwiththeplanesandlines andpointsinthemotionlesssolid；forsciencealwaysdealswithwhat isprior。And(thesameaccountwillapplyalsotonumbers；for therewillbeadifferentsetofunitsapartfromeachsetof points，andalsoapartfromeachsetofrealities，fromtheobjectsof senseandagainfromthoseofthought；sothattherewillbevarious classesofmathematicalnumbers。 Again，howisitpossibletosolvethequestionswhichwehave alreadyenumeratedinourdiscussionofdifficulties？Forthe objectsofastronomywillexistapartfromsensiblethingsjustasthe objectsofgeometrywill；buthowisitpossiblethataheavenandits parts-oranythingelsewhichhasmovement-shouldexistapart？ Similarlyalsotheobjectsofopticsandofharmonicswillexist apart；fortherewillbebothvoiceandsightbesidesthesensible orindividualvoicesandsights。Thereforeitisplainthatthe othersensesaswell，andtheotherobjectsofsense，willexist apart；forwhyshouldonesetofthemdosoandanothernot？Andif thisisso，therewillalsobeanimalsexistingapart，sincethere willbesenses。 Again，therearecertainmathematicaltheoremsthatareuniversal， extendingbeyondthesesubstances。Herethenweshallhaveanother intermediatesubstanceseparatebothfromtheIdeasandfromthe intermediates，-asubstancewhichisneithernumbernorpointsnor spatialmagnitudenortime。Andifthisisimpossible，plainlyitis alsoimpossiblethattheformerentitiesshouldexistseparatefrom sensiblethings。 And，ingeneral，conclusioncontraryaliketothetruthandtothe usualviewsfollow，ifoneistosupposetheobjectsofmathematicsto existthusasseparateentities。Forbecausetheyexistthustheymust bepriortosensiblespatialmagnitudes，butintruththeymustbe posterior；fortheincompletespatialmagnitudeisintheorderof generationprior，butintheorderofsubstanceposterior，asthe lifelessistotheliving。 Again，byvirtueofwhat，andwhen，willmathematicalmagnitudes beone？Forthingsinourperceptibleworldareoneinvirtueofsoul， orofapartofsoul，orofsomethingelsethatisreasonable enough；whenthesearenotpresent，thethingisaplurality，and splitsupintoparts。Butinthecaseofthesubjectsof mathematics，whicharedivisibleandarequantities，whatisthecause oftheirbeingoneandholdingtogether？ Again，themodesofgenerationoftheobjectsofmathematics showthatweareright。Forthedimensionfirstgeneratedislength， thencomesbreadth，lastlydepth，andtheprocessiscomplete。If， then，thatwhichisposteriorintheorderofgenerationispriorin theorderofsubstantiality，thesolidwillbepriortotheplane andtheline。Andinthiswayalsoitisbothmorecompleteandmore whole，becauseitcanbecomeanimate。How，ontheotherhand，could alineoraplanebeanimate？Thesuppositionpassesthepowerof oursenses。 Again，thesolidisasortofsubstance；foritalreadyhasina sensecompleteness。Buthowcanlinesbesubstances？Neitherasaform orshape，asthesoulperhapsis，norasmatter，likethesolid；for wehavenoexperienceofanythingthatcanbeputtogetheroutof linesorplanesorpoints，whileifthesehadbeenasortof materialsubstance，weshouldhaveobservedthingswhichcouldbe puttogetheroutofthem。 Grant，then，thattheyarepriorindefinition。Stillnotall thingsthatarepriorindefinitionarealsopriorin substantiality。Forthosethingsarepriorinsubstantialitywhich whenseparatedfromotherthingssurpasstheminthepowerof independentexistence，butthingsarepriorindefinitiontothose whosedefinitionsarecompoundedoutoftheirdefinitions；andthese twopropertiesarenotcoextensive。Forifattributesdonotexist apartfromthesubstances(e。g。a’mobile’orapale’)，paleis priortothepalemanindefinition，butnotinsubstantiality。Forit cannotexistseparately，butisalwaysalongwiththeconcrete thing；andbytheconcretethingImeanthepaleman。Thereforeit isplainthatneitheristheresultofabstractionpriornorthat whichisproducedbyaddingdeterminantsposterior；foritisby addingadeterminanttopalethatwespeakofthepaleman。 Ithas，then，beensufficientlypointedoutthattheobjectsof mathematicsarenotsubstancesinahigherdegreethanbodiesare，and thattheyarenotpriortosensiblesinbeing，butonlyindefinition， andthattheycannotexistsomewhereapart。Butsinceitwasnot possibleforthemtoexistinsensibleseither，itisplainthat theyeitherdonotexistatallorexistinaspecialsenseand thereforedonot’exist’withoutqualification。For’exist’hasmany senses。