falsity。Thisisalsowhyitusedtobesaidthatwemustassume
somethingthatisfalse，asgeometersassumethelinewhichisnota
footlongtobeafootlong。Butthiscannotbeso。Forneitherdo
geometersassumeanythingfalse(fortheenunciationisextraneous
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，theypositedtherelativeterm(i。e。theunequal)，whichis
neitherthecontrarynorthecontradictoryofthese，andisonekind
ofbeingas’what’andqualityalsoare。
Theyshouldhaveaskedthisquestionalso，howrelativeterms
aremanyandnotone。Butasitis，theyinquirehowtherearemany
unitsbesidesthefirst1，butdonotgoontoinquirehowthereare
manyunequalsbesidestheunequal。Yettheyusethemandspeakof
greatandsmall，manyandfew(fromwhichproceednumbers)，longand
short(fromwhichproceedstheline)，broadandnarrow(fromwhich
proceedstheplane)，deepandshallow(fromwhichproceedsolids)；and
theyspeakofyetmorekindsofrelativeterm。Whatisthereason，
then，whythereisapluralityofthese？
Itisnecessary，then，aswesay，topresupposeforeachthing
thatwhichisitpotentially；andtheholderoftheseviewsfurther
declaredwhatthatiswhichispotentiallya’this’andasubstance
butisnotinitselfbeing-viz。thatitistherelative(asifhe
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
seestheinherentobjectionstotheIdeas(sothatitisnotfor
thisreasonthathepositsnumbers)，butwhopositsmathematical
number，whymustwebelievehisstatementthatsuchnumberexists，and
ofwhatuseissuchnumbertootherthings？Neitherdoeshewhosays
itexistsmaintainthatitisthecauseofanything(herathersaysit
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。For(i)extremesarenotsubstances，butratherallthesethings
arelimits。Forevenwalking，andmovementingeneral，hasalimit，so
thatontheirtheorythiswillbea’this’andasubstance。Butthat
isabsurd。Notbutwhat(ii)eveniftheyaresubstances，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。If(i)itconsistsofthegreatandsmall，itwillbethesame
astheother-ideal-number(hemakesspatialmagnitudesoutofsome
othersmallandgreat)。Andif(ii)henamessomeotherelement，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
