falsity。Thisisalsowhyitusedtobesaidthatwemustassume
somethingthatisfalse,asgeometersassumethelinewhichisnota
footlongtobeafootlong。Butthiscannotbeso。Forneitherdo
geometersassumeanythingfalsefortheenunciationisextraneous
totheinference,norisitnon-beinginthissensethatthethings
thatarearegeneratedfromorresolvedinto。Butsince’non-being’
takeninitsvariouscaseshasasmanysensesastherearecategories,
andbesidesthisthefalseissaidnottobe,andsoisthepotential,
itisfromthisthatgenerationproceeds,manfromthatwhichisnot
manbutpotentiallyman,andwhitefromthatwhichisnotwhitebut
potentiallywhite,andthiswhetheritissomeonethingthatis
generatedormany。
Thequestionevidentlyis,howbeing,inthesenseof’the
substances’,ismany;forthethingsthataregeneratedarenumbers
andlinesandbodies。Nowitisstrangetoinquirehowbeinginthe
senseofthe’what’ismany,andnothoweitherqualitiesor
quantitiesaremany。Forsurelytheindefinitedyador’thegreat
andthesmall’isnotareasonwhythereshouldbetwokindsof
whiteormanycoloursorflavoursorshapes;forthenthesealsowould
benumbersandunits。Butiftheyhadattackedtheseothercategories,
theywouldhaveseenthecauseofthepluralityinsubstancesalso;
forthesamethingorsomethinganalogousisthecause。This
aberrationisthereasonalsowhyinseekingtheoppositeofbeingand
theone,fromwhichwithbeingandtheonethethingsthatare
proceed,theypositedtherelativetermi。e。theunequal,whichis
neitherthecontrarynorthecontradictoryofthese,andisonekind
ofbeingas’what’andqualityalsoare。
Theyshouldhaveaskedthisquestionalso,howrelativeterms
aremanyandnotone。Butasitis,theyinquirehowtherearemany
unitsbesidesthefirst1,butdonotgoontoinquirehowthereare
manyunequalsbesidestheunequal。Yettheyusethemandspeakof
greatandsmall,manyandfewfromwhichproceednumbers,longand
shortfromwhichproceedstheline,broadandnarrowfromwhich
proceedstheplane,deepandshallowfromwhichproceedsolids;and
theyspeakofyetmorekindsofrelativeterm。Whatisthereason,
then,whythereisapluralityofthese?
Itisnecessary,then,aswesay,topresupposeforeachthing
thatwhichisitpotentially;andtheholderoftheseviewsfurther
declaredwhatthatiswhichispotentiallya’this’andasubstance
butisnotinitselfbeing-viz。thatitistherelativeasifhe
hadsaid’thequalitative’,whichisneitherpotentiallytheoneor
being,northenegationoftheonenorofbeing,butoneamongbeings。
Anditwasmuchmorenecessary,aswesaid,ifhewasinquiringhow
beingsaremany,nottoinquireaboutthoseinthesamecategory-how
therearemanysubstancesormanyqualities-buthowbeingsasa
wholearemany;forsomearesubstances,somemodifications,some
relations。Inthecategoriesotherthansubstancethereisyetanother
probleminvolvedintheexistenceofplurality。Sincetheyarenot
separablefromsubstances,qualitiesandquantitiesaremanyjust
becausetheirsubstratumbecomesandismany;yetthereoughttobe
amatterforeachcategory;onlyitcannotbeseparablefrom
substances。Butinthecaseof’thises’,itispossibletoexplainhow
the’this’ismanythings,unlessathingistobetreatedasbotha
’this’andageneralcharacter。Thedifficultyarisingfromthe
factsaboutsubstancesisratherthis,howthereareactuallymany
substancesandnotone。
Butfurther,ifthe’this’andthequantitativearenotthe
same,wearenottoldhowandwhythethingsthatarearemany,but
howquantitiesaremany。Forall’number’meansaquantity,andso
doesthe’unit’,unlessitmeansameasureorthequantitatively
indivisible。If,then,thequantitativeandthe’what’are
different,wearenottoldwhenceorhowthe’what’ismany;butif
anyonesaystheyarethesame,hehastofacemanyinconsistencies。
Onemightfixone’sattentionalsoonthequestion,regarding
thenumbers,whatjustifiesthebeliefthattheyexist。Tothe
believerinIdeastheyprovidesomesortofcauseforexistingthings,
sinceeachnumberisanIdea,andtheIdeaistootherthings
somehoworotherthecauseoftheirbeing;forletthissuppositionbe
grantedthem。Butasforhimwhodoesnotholdthisviewbecausehe
seestheinherentobjectionstotheIdeassothatitisnotfor
thisreasonthathepositsnumbers,butwhopositsmathematical
number,whymustwebelievehisstatementthatsuchnumberexists,and
ofwhatuseissuchnumbertootherthings?Neitherdoeshewhosays
itexistsmaintainthatitisthecauseofanythingherathersaysit
isathingexistingbyitself,norisitobservedtobethecause
ofanything;forthetheoremsofarithmeticianswillallbefoundtrue
evenofsensiblethings,aswassaidbefore。
Asforthose,then,whosupposetheIdeastoexistandtobe
numbers,bytheirassumptioninvirtueofthemethodofsettingout
eachtermapartfromitsinstances-oftheunityofeachgeneralterm
theytryatleasttoexplainsomehowwhynumbermustexist。Since
theirreasons,however,areneitherconclusivenorinthemselves
possible,onemustnot,forthesereasonsatleast,assertthe
existenceofnumber。Again,thePythagoreans,becausetheysawmany
attributesofnumbersbelongingtesensiblebodies,supposedreal
thingstobenumbers-notseparablenumbers,however,butnumbersof
whichrealthingsconsist。Butwhy?Becausetheattributesof
numbersarepresentinamusicalscaleandintheheavensandin
manyotherthings。Those,however,whosaythatmathematicalnumber
aloneexistscannotaccordingtotheirhypothesessayanythingofthis
sort,butitusedtobeurgedthatthesesensiblethingscouldnot
bethesubjectofthesciences。Butwemaintainthattheyare,aswe
saidbefore。Anditisevidentthattheobjectsofmathematicsdo
notexistapart;foriftheyexistedaparttheirattributeswould
nothavebeenpresentinbodies。NowthePythagoreansinthispoint
areopentonoobjection;butinthattheyconstructnaturalbodies
outofnumbers,thingsthathavelightnessandweightoutofthings
thathavenotweightorlightness,theyseemtospeakofanother
heavenandotherbodies,notofthesensible。Butthosewhomake
numberseparableassumethatitbothexistsandisseparablebecause
theaxiomswouldnotbetrueofsensiblethings,whilethe
statementsofmathematicsaretrueand’greetthesoul’;andsimilarly
withthespatialmagnitudesofmathematics。Itisevident,then,
boththattherivaltheorywillsaythecontraryofthis,andthatthe
difficultyweraisedjustnow,whyifnumbersareinnowaypresentin
sensiblethingstheirattributesarepresentinsensiblethings,has
tobesolvedbythosewhoholdtheseviews。
Therearesomewho,becausethepointisthelimitandextreme
oftheline,thelineoftheplane,andtheplaneofthesolid,
thinktheremustberealthingsofthissort。Wemusttherefore
examinethisargumenttoo,andseewhetheritisnotremarkably
weak。Foriextremesarenotsubstances,butratherallthesethings
arelimits。Forevenwalking,andmovementingeneral,hasalimit,so
thatontheirtheorythiswillbea’this’andasubstance。Butthat
isabsurd。Notbutwhatiieveniftheyaresubstances,theywill
allbethesubstancesofthesensiblethingsinthisworld;forit
istothesethattheargumentapplied。Whythenshouldtheybecapable
ofexistingapart?
Again,ifwearenottooeasilysatisfied,wemay,regardingall
numberandtheobjectsofmathematics,pressthisdifficulty,that
theycontributenothingtooneanother,thepriortotheposterior;
forifnumberdidnotexist,nonethelessspatialmagnitudeswould
existforthosewhomaintaintheexistenceoftheobjectsof
mathematicsonly,andifspatialmagnitudesdidnotexist,souland
sensiblebodieswouldexist。Buttheobservedfactsshowthatnature
isnotaseriesofepisodes,likeabadtragedy。Asforthe
believersintheIdeas,thisdifficultymissesthem;forthey
constructspatialmagnitudesoutofmatterandnumber,linesoutof
thenumberplanesdoubtlessoutofsolidsoutofortheyuseother
numbers,whichmakesnodifference。Butwillthesemagnitudesbe
Ideas,orwhatistheirmannerofexistence,andwhatdothey
contributetothings?Thesecontributenothing,astheobjectsof
mathematicscontributenothing。Butnotevenisanytheoremtrueof
them,unlesswewanttochangetheobjectsofmathematicsandinvent
doctrinesofourown。Butitisnothardtoassumeanyrandom
hypothesesandspinoutalongstringofconclusions。These
thinkers,then,arewronginthisway,inwantingtounitetheobjects
ofmathematicswiththeIdeas。Andthosewhofirstpositedtwokinds
ofnumber,thatoftheFormsandthatwhichismathematical,neither
havesaidnorcansayhowmathematicalnumberistoexistandof
whatitistoconsist。Fortheyplaceitbetweenidealandsensible
number。Ifiitconsistsofthegreatandsmall,itwillbethesame
astheother-ideal-numberhemakesspatialmagnitudesoutofsome
othersmallandgreat。Andifiihenamessomeotherelement,he
willbemakinghiselementsrathermany。Andiftheprincipleof
eachofthetwokindsofnumberisa1,unitywillbesomethingcommon
tothese,andwemustinquirehowtheoneisthesemanythings,
whileatthesametimenumber,accordingtohim,cannotbegenerated
exceptfromoneandanindefinitedyad。
Allthisisabsurd,andconflictsbothwithitselfandwiththe
probabilities,andweseemtoseeinitSimonides’longrigmarole’for
thelongrigmarolecomesintoplay,likethoseofslaves,whenmen
havenothingsoundtosay。Andtheveryelements-thegreatandthe
small-seemtocryoutagainsttheviolencethatisdonetothem;for
theycannotinanywaygeneratenumbersotherthanthosegotfrom1by
doubling。
Itisstrangealsotoattributegenerationtothingsthatare
eternal,orratherthisisoneofthethingsthatareimpossible。
ThereneedbenodoubtwhetherthePythagoreansattributegeneration
tothemornot;fortheysayplainlythatwhentheonehadbeen
constructed,whetheroutofplanesorofsurfaceorofseedorof
elementswhichtheycannotexpress,immediatelythenearestpartof
theunlimitedbegantobeconstrainedandlimitedbythelimit。But
sincetheyareconstructingaworldandwishtospeakthelanguage
ofnaturalscience,itisfairtomakesomeexaminationoftheir
physicaltheorics,buttoletthemofffromthepresentinquiry;for
weareinvestigatingtheprinciplesatworkinunchangeablethings,so
thatitisnumbersofthiskindwhosegenesiswemuststudy。
Thesethinkerssaythereisnogenerationoftheoddnumber,which
evidentlyimpliesthatthereisgenerationoftheeven;andsome
presenttheevenasproducedfirstfromunequals-thegreatandthe
small-whentheseareequalized。Theinequality,then,mustbelongto
thembeforetheyareequalized。Iftheyhadalwaysbeenequalized,
theywouldnothavebeenunequalbefore;forthereisnothingbefore
thatwhichisalways。Thereforeevidentlytheyarenotgivingtheir
accountofthegenerationofnumbersmerelytoassistcontemplationof
theirnature。
Adifficulty,andareproachtoanyonewhofindsitno
difficulty,arecontainedinthequestionhowtheelementsandthe
principlesarerelatedtothegoodandthebeautiful;thedifficulty
isthis,whetheranyoftheelementsissuchathingaswemeanbythe
gooditselfandthebest,orthisisnotso,butthesearelaterin
originthantheelements。Thetheologiansseemtoagreewithsome
thinkersofthepresentday,whoanswerthequestioninthe
negative,andsaythatboththegoodandthebeautifulappearinthe
natureofthingsonlywhenthatnaturehasmadesomeprogress。This
theydotoavoidarealobjectionwhichconfrontsthosewhosay,as
somedo,thattheoneisafirstprinciple。Theobjectionarisesnot
fromtheirascribinggoodnesstothefirstprincipleasan