Butthisconsequencealsowemustnotforget，thatitfollowsthat
therearepriorandposterior2andsimilarlywiththeother
numbers。Forletthe2’sinthe4besimultaneous；yettheseareprior
tothoseinthe8andasthe2generatedthem，theygeneratedthe
4’sinthe8-itself。Thereforeifthefirst2isanIdea，these2’s
alsowillbeIdeasofsomekind。Andthesameaccountappliestothe
units；fortheunitsinthefirst2generatethefourin4，sothat
alltheunitscometobeIdeasandanIdeawillbecomposedof
Ideas。Clearlythereforethosethingsalsoofwhichthesehappentobe
theIdeaswillbecomposite，e。g。onemightsaythatanimalsare
composedofanimals，ifthereareIdeasofthem。
Ingeneral，todifferentiatetheunitsinanywayisan
absurdityandafiction；andbyafictionImeanaforcedstatement
madetosuitahypothesis。Forneitherinquantitynorinqualitydo
weseeunitdifferingfromunit，andnumbermustbeeitherequalor
unequal-allnumberbutespeciallythatwhichconsistsofabstract
units-sothatifonenumberisneithergreaternorlessthan
another，itisequaltoit；butthingsthatareequalandinnowise
differentiatedwetaketobethesamewhenwearespeakingofnumbers。
Ifnot，noteventhe2inthe10-itselfwillbeundifferentiated，
thoughtheyareequal；forwhatreasonwillthemanwhoallegesthat
theyarenotdifferentiatedbeabletogive？
Again，ifeveryunit+anotherunitmakestwo，aunitfromthe
2-itselfandonefromthe3-itselfwillmakea2。Now(a)thiswill
consistofdifferentiatedunits；andwillitbepriortothe3or
posterior？Itratherseemsthatitmustbeprior；foroneoftheunits
issimultaneouswiththe3andtheotherissimultaneouswiththe2。
Andwe，forourpart，supposethatingeneral1and1，whetherthe
thingsareequalorunequal，is2，e。g。thegoodandthebad，oraman
andahorse；butthosewhoholdtheseviewssaythatnoteventwo
unitsare2。
Ifthenumberofthe3-itselfisnotgreaterthanthatofthe2，
thisissurprising；andifitisgreater，clearlythereisalsoa
numberinitequaltothe2，sothatthisisnotdifferentfromthe
2-itself。Butthisisnotpossible，ifthereisafirstandasecond
number。
NorwilltheIdeasbenumbers。Forinthisparticularpointthey
arerightwhoclaimthattheunitsmustbedifferent，ifthereare
tobeIdeas；ashasbeensaidbefore。FortheFormisunique；butif
theunitsarenotdifferent，the2’sandthe3’salsowillnotbe
different。Thisisalsothereasonwhytheymustsaythatwhenwe
countthus-’1，2’-wedonotproceedbyaddingtothegivennumber；
forifwedo，neitherwillthenumbersbegeneratedfromthe
indefinitedyad，norcananumberbeanIdea；forthenoneIdeawill
beinanother，andallFormswillbepartsofoneForm。Andsowith
aviewtotheirhypothesistheirstatementsareright，butasa
wholetheyarewrong；fortheirviewisverydestructive，sincethey
willadmitthatthisquestionitselfaffordssome
difficulty-whether，whenwecountandsay-1，2，3-wecountby
additionorbyseparateportions。Butwedoboth；andsoitis
absurdtoreasonbackfromthisproblemtosogreatadifferenceof
essence。
Firstofallitiswelltodeterminewhatisthedifferentiaof
anumber-andofaunit，ifithasadifferentia。Unitsmustdiffer
eitherinquantityorinquality；andneitheroftheseseemstobe
possible。Butnumberquanumberdiffersinquantity。Andifthe
unitsalsodiddifferinquantity，numberwoulddifferfromnumber，
thoughequalinnumberofunits。Again，arethefirstunitsgreateror
smaller，anddothelateronesincreaseordiminish？Alltheseare
irrationalsuppositions。Butneithercantheydifferinquality。For
noattributecanattachtothem；foreventonumbersqualityissaid
tobelongafterquantity。Again，qualitycouldnotcometothemeither
fromthe1orthedyad；fortheformerhasnoquality，andthe
lattergivesquantity；forthisentityiswhatmakesthingstobe
many。Ifthefactsarereallyotherwise，theyshouldstatethis
quiteatthebeginninganddetermineifpossible，regardingthe
differentiaoftheunit，whyitmustexist，and，failingthis，what
differentiatheymean。
Evidentlythen，iftheIdeasarenumbers，theunitscannotall
beassociable，norcantheybeinassociableineitherofthetwoways。
Butneitheristhewayinwhichsomeothersspeakaboutnumbers
correct。ThesearethosewhodonotthinkthereareIdeas，either
withoutqualificationorasidentifiedwithcertainnumbers，butthink
theobjectsofmathematicsexistandthenumbersarethefirstof
existingthings，andthe1-itselfisthestarting-pointofthem。Itis
paradoxicalthatthereshouldbea1whichisfirstof1’s，asthey
say，butnota2whichisfirstof2’s，nora3of3’s；forthesame
reasoningappliestoall。If，then，thefactswithregardtonumber
areso，andonesupposesmathematicalnumberalonetoexist，the1
isnotthestarting-point(forthissortof1mustdifferfrom
the-otherunits；andifthisisso，theremustalsobea2whichis
firstof2’s，andsimilarlywiththeothersuccessivenumbers)。Butif
the1isthestarting-point，thetruthaboutthenumbersmustrather
bewhatPlatousedtosay，andtheremustbeafirst2and3and
numbersmustnotbeassociablewithoneanother。Butifontheother
handonesupposesthis，manyimpossibleresults，aswehavesaid，
follow。Buteitherthisortheothermustbethecase，sothatif
neitheris，numbercannotexistseparately。
Itisevident，also，fromthisthatthethirdversionisthe
worst，-theviewidealandmathematicalnumberisthesame。Fortwo
mistakesmustthenmeetintheoneopinion。(1)Mathematicalnumber
cannotbeofthissort，buttheholderofthisviewhastospinitout
bymakingsuppositionspeculiartohimself。And(2)hemustalsoadmit
alltheconsequencesthatconfrontthosewhospeakofnumberinthe
senseof’Forms’。
ThePythagoreanversioninonewayaffordsfewerdifficultiesthan
thosebeforenamed，butinanotherwayhasotherspeculiarto
itself。Fornotthinkingofnumberascapableofexistingseparately
removesmanyoftheimpossibleconsequences；butthatbodiesshouldbe
composedofnumbers，andthatthisshouldbemathematicalnumber，is
impossible。Foritisnottruetospeakofindivisiblespatial
magnitudes；andhowevermuchtheremightbemagnitudesofthissort，
unitsatleasthavenotmagnitude；andhowcanamagnitudebecomposed
ofindivisibles？Butarithmeticalnumber，atleast，consistsofunits，
whilethesethinkersidentifynumberwithrealthings；atanyrate
theyapplytheirpropositionstobodiesasiftheyconsistedof
thosenumbers。
If，then，itisnecessary，ifnumberisaself-subsistentreal
thing，thatitshouldexistinoneofthesewayswhichhavebeen
mentioned，andifitcannotexistinanyofthese，evidentlynumber
hasnosuchnatureasthosewhomakeitseparablesetupforit。
Again，doeseachunitcomefromthegreatandthesmall，
equalized，oronefromthesmall，anotherfromthegreat？(a)Ifthe
latter，neitherdoeseachthingcontainalltheelements，norare
theunitswithoutdifference；forinonethereisthegreatandin
anotherthesmall，whichiscontraryinitsnaturetothegreat。
Again，howisitwiththeunitsinthe3-itself？Oneofthemisanodd
unit。Butperhapsitisforthisreasonthattheygive1-itselfthe
middleplaceinoddnumbers。(b)Butifeachofthetwounitsconsists
ofboththegreatandthesmall，equalized，howwillthe2whichis
asinglething，consistofthegreatandthesmall？Orhowwillit
differfromtheunit？Again，theunitispriortothe2；forwhenit
isdestroyedthe2isdestroyed。Itmust，then，betheIdeaofanIdea
sinceitispriortoanIdea，anditmusthavecomeintobeing
beforeit。Fromwhat，then？Notfromtheindefinitedyad，forits
functionwastodouble。
Again，numbermustbeeitherinfiniteorfinite；forthese
thinkersthinkofnumberascapableofexistingseparately，sothatit
isnotpossiblethatneitherofthosealternativesshouldbetrue。
Clearlyitcannotbeinfinite；forinfinitenumberisneitherodd
noreven，butthegenerationofnumbersisalwaysthegeneration
eitherofanoddorofanevennumber；inoneway，when1operates
onanevennumber，anoddnumberisproduced；inanotherway，when2
operates，thenumbersgotfrom1bydoublingareproduced；in
anotherway，whentheoddnumbersoperate，theotherevennumbers
areproduced。Again，ifeveryIdeaisanIdeaofsomething，andthe
numbersareIdeas，infinitenumberitselfwillbeanIdeaof
something，eitherofsomesensiblethingorofsomethingelse。Yet
thisisnotpossibleinviewoftheirthesisanymorethanitis
reasonableinitself，atleastiftheyarrangetheIdeasastheydo。
Butifnumberisfinite，howfardoesitgo？Withregardtothis
notonlythefactbutthereasonshouldbestated。Butifnumber
goesonlyupto10assomesay，firstlytheFormswillsoonrunshort；
e。g。if3isman-himself，whatnumberwillbethehorse-itself？The
seriesofthenumberswhicharetheseveralthings-themselvesgoes
upto10。Itmust，then，beoneofthenumberswithintheselimits；
foritisthesethataresubstancesandIdeas。Yettheywillrun
short；forthevariousformsofanimalwilloutnumberthem。Atthe
sametimeitisclearthatifinthiswaythe3isman-himself，the
other3’saresoalso(forthoseinidenticalnumbersaresimilar)，so
thattherewillbeaninfinitenumberofmen；ifeach3isanIdea，
eachofthenumberswillbeman-himself，andifnot，theywillat
leastbemen。Andifthesmallernumberispartofthegreater
(beingnumberofsuchasortthattheunitsinthesamenumberare
associable)，thenifthe4-itselfisanIdeaofsomething，e。g。of
’horse’orof’white’，manwillbeapartofhorse，ifmanisItis
paradoxicalalsothatthereshouldbeanIdeaof10butnotof11，nor
ofthesucceedingnumbers。Again，therebothareandcometobe
certainthingsofwhichtherearenoForms；why，then，aretherenot
Formsofthemalso？WeinferthattheFormsarenotcauses。Again，
itisparadoxical-ifthenumberseriesupto10ismoreofareal
thingandaFormthan10itself。Thereisnogenerationofthe
formerasonething，andthereisofthelatter。Buttheytryto
workontheassumptionthattheseriesofnumbersupto10isa
completeseries。Atleasttheygeneratethederivatives-e。g。thevoid，
proportion，theodd，andtheothersofthiskind-withinthedecade。
Forsomethings，e。g。movementandrest，goodandbad，theyassign
totheoriginativeprinciples，andtheotherstothenumbers。This
iswhytheyidentifytheoddwith1；foriftheoddimplied3how
would5beodd？Again，spatialmagnitudesandallsuchthingsare
explainedwithoutgoingbeyondadefinitenumber；e。g。thefirst，
theindivisible，line，thenthe2&c。；theseentitiesalsoextendonly
upto10。
Again，ifnumbercanexistseparately，onemightaskwhichis
prior-1，or3or2？Inasmuchasthenumberiscomposite，1isprior，
butinasmuchastheuniversalandtheformisprior，thenumberis
prior；foreachoftheunitsispartofthenumberasitsmatter，
andthenumberactsasform。Andinasensetherightangleisprior
totheacute，becauseitisdeterminateandinvirtueofits
definition；butinasensetheacuteisprior，becauseitisapart
andtherightangleisdividedintoacuteangles。Asmatter，then，the
acuteangleandtheelementandtheunitareprior，butinrespect
oftheformandofthesubstanceasexpressedinthedefinition，the
rightangle，andthewholeconsistingofthematterandtheform，
areprior；fortheconcretethingisnearertotheformandtowhatis
expressedinthedefinition，thoughingenerationitislater。How
thenis1thestarting-point？Becauseitisnotdivisiable，they
say；butboththeuniversal，andtheparticularortheelement，are
indivisible。Buttheyarestarting-pointsindifferentways，onein
definitionandtheotherintime。Inwhichway，then，is1the
