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
seriesofepisodesforonesubstancehasnoinfluenceonanotherby
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。And1sincesomerecognize
theseastwodifferentclasses-theIdeasandthemathematicalnumbers，
and2somerecognizebothashavingonenature，while3some
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。Now1
theaccumulationbecomesabsurd；forwefindourselveswithonesetof
solidsapartfromthesensiblesolids；threesetsofplanesapartfrom
thesensibleplanes-thosewhichexistapartfromthesensible
planes，andthoseinthemathematicalsolids，andthosewhichexist
apartfromthoseinthemathematicalsolids；foursetsoflines，and
fivesetsofpoints。Withwhichofthese，then，willthe
mathematicalsciencesdeal？Certainlynotwiththeplanesandlines
andpointsinthemotionlesssolid；forsciencealwaysdealswithwhat
isprior。Andthesameaccountwillapplyalsotonumbers；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
apartfromthesubstancese。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。