But，further，allotherthingscannotcomefromtheFormsinany
oftheusualsensesof’from’。Andtosaythattheyarepatternsand
theotherthingsshareinthemistouseemptywordsandpoetical
metaphors。Forwhatisitthatworks，lookingtotheIdeas？Andany
thingcanbothbeandcomeintobeingwithoutbeingcopiedfrom
somethingelse，sothat，whetherSocratesexistsornot，amanlike
Socratesmightcometobe。Andevidentlythismightbesoevenif
Socrateswereeternal。Andtherewillbeseveralpatternsofthe
samething，andthereforeseveralForms；e。g。’animal’and
’two-footed’，andalso’man-himself’，willbeFormsofman。Again，the
Formsarepatternsnotonlyofsensiblethings，butofForms
themselvesalso；i。e。thegenusisthepatternofthevarious
forms-of-a-genus；thereforethesamethingwillbepatternandcopy。
Again，itwouldseemimpossiblethatsubstanceandthatwhose
substanceitisshouldexistapart；how，therefore，couldtheIdeas，
beingthesubstancesofthings，existapart？
InthePhaedothecaseisstatedinthisway-thattheFormsare
causesbothofbeingandofbecoming。YetthoughtheFormsexist，
stillthingsdonotcomeintobeing，unlessthereissomethingto
originatemovement；andmanyotherthingscomeintobeing(e。g。a
houseoraring)ofwhichtheysaytherearenoForms。Clearly
thereforeeventhethingsofwhichtheysaythereareIdeascanboth
beandcomeintobeingowingtosuchcausesasproducethethingsjust
mentioned，andnotowingtotheForms。ButregardingtheIdeasitis
possible，bothinthiswayandbymoreabstractandaccurate
arguments，tocollectmanyobjectionslikethosewehaveconsidered。
Sincewehavediscussedthesepoints，itiswelltoconsideragain
theresultsregardingnumberswhichconfrontthosewhosaythat
numbersareseparablesubstancesandfirstcausesofthings。Ifnumber
isanentityanditssubstanceisnothingotherthanjustnumber，as
somesay，itfollowsthateither(1)thereisafirstinitanda
second，eachbeingdifferentinspecies，-andeither(a)thisistrue
oftheunitswithoutexception，andanyunitisinassociablewith
anyunit，or(b)theyareallwithoutexceptionsuccessive，andanyof
themareassociablewithany，astheysayisthecasewith
mathematicalnumber；forinmathematicalnumbernooneunitisin
anywaydifferentfromanother。Or(c)someunitsmustbeassociable
andsomenot；e。g。supposethat2isfirstafter1，andthencomes3
andthentherestofthenumberseries，andtheunitsineachnumber
areassociable，e。g。thoseinthefirst2areassociablewithone
another，andthoseinthefirst3withoneanother，andsowiththe
othernumbers；buttheunitsinthe’2-itself’areinassociablewith
thoseinthe’3-itself’；andsimilarlyinthecaseoftheother
successivenumbers。Andsowhilemathematicalnumberiscounted
thus-after1，2(whichconsistsofanother1besidestheformer1)，
and3whichconsistsofanother1besidesthesetwo)，andtheother
numberssimilarly，idealnumberiscountedthus-after1，adistinct
2whichdoesnotincludethefirst1，anda3whichdoesnotinclude
the2andtherestofthenumberseriessimilarly。Or(2)onekind
ofnumbermustbelikethefirstthatwasnamed，onelikethatwhich
themathematiciansspeakof，andthatwhichwehavenamedlastmustbe
athirdkind。
Again，thesekindsofnumbersmusteitherbeseparablefrom
things，ornotseparablebutinobjectsofperception(nothowever
inthewaywhichwefirstconsidered，inthesensethatobjectsof
perceptionconsistsofnumberswhicharepresentinthem)-eitherone
kindandnotanother，orallofthem。
Theseareofnecessitytheonlywaysinwhichthenumberscan
exist。Andofthosewhosaythatthe1isthebeginningand
substanceandelementofallthings，andthatnumberisformedfrom
the1andsomethingelse，almosteveryonehasdescribednumberinone
oftheseways；onlynoonehassaidalltheunitsareinassociable。
Andthishashappenedreasonablyenough；fortherecanbenoway
besidesthosementioned。Somesaybothkindsofnumberexist，that
whichhasabeforeandafterbeingidenticalwiththeIdeas，and
mathematicalnumberbeingdifferentfromtheIdeasandfromsensible
things，andbothbeingseparablefromsensiblethings；andothers
saymathematicalnumberaloneexists，asthefirstofrealities，
separatefromsensiblethings。AndthePythagoreans，also，believe
inonekindofnumber-themathematical；onlytheysayitisnot
separatebutsensiblesubstancesareformedoutofit。Forthey
constructthewholeuniverseoutofnumbers-onlynotnumbers
consistingofabstractunits；theysupposetheunitstohavespatial
magnitude。Buthowthefirst1wasconstructedsoastohave
magnitude，theyseemunabletosay。
Anotherthinkersaysthefirstkindofnumber，thatofthe
Forms，aloneexists，andsomesaymathematicalnumberisidentical
withthis。
Thecaseoflines，planes，andsolidsissimilar。Forsomethink
thatthosewhicharetheobjectsofmathematicsaredifferentfrom
thosewhichcomeaftertheIdeas；andofthosewhoexpress
themselvesotherwisesomespeakoftheobjectsofmathematicsandina
mathematicalway-viz。thosewhodonotmaketheIdeasnumbersnor
saythatIdeasexist；andothersspeakoftheobjectsof
mathematics，butnotmathematically；fortheysaythatneitheris
everyspatialmagnitudedivisibleintomagnitudes，nordoanytwo
unitstakenatrandommake2。Allwhosaythe1isanelementand
principleofthingssupposenumberstoconsistofabstractunits，
exceptthePythagoreans；buttheysupposethenumberstohave
magnitude，ashasbeensaidbefore。Itisclearfromthisstatement，
then，inhowmanywaysnumbersmaybedescribed，andthatalltheways
havebeenmentioned；andalltheseviewsareimpossible，butsome
perhapsmorethanothers。
First，then，letusinquireiftheunitsareassociableor
inassociable，andifinassociable，inwhichofthetwowayswe
distinguished。Foritispossiblethatanyunityisinassociable
withany，anditispossiblethatthoseinthe’itself’are
inassociablewiththoseinthe’itself’，and，generally，thatthosein
eachidealnumberareinassociablewiththoseinotherideal
numbers。Now(1)allunitsareassociableandwithoutdifference，we
getmathematicalnumber-onlyonekindofnumber，andtheIdeas
cannotbethenumbers。Forwhatsortofnumberwillman-himselfor
animal-itselforanyotherFormbe？ThereisoneIdeaofeachthing
e。g。oneofman-himselfandanotheroneofanimal-itself；butthe
similarandundifferentiatednumbersareinfinitelymany，sothat
anyparticular3isnomoreman-himselfthananyother3。Butifthe
Ideasarenotnumbers，neithercantheyexistatall。Forfromwhat
principleswilltheIdeascome？Itisnumberthatcomesfromthe1and
theindefinitedyad，andtheprinciplesorelementsaresaidtobe
principlesandelementsofnumber，andtheIdeascannotberankedas
eitherpriororposteriortothenumbers。
But(2)iftheunitsareinassociable，andinassociableinthe
sensethatanyisinassociablewithanyother，numberofthissort
cannotbemathematicalnumber；formathematicalnumberconsistsof
undifferentiatedunits，andthetruthsprovedofitsuitthis
character。Norcanitbeidealnumber。For2willnotproceed
immediatelyfrom1andtheindefinitedyad，andbefollowedbythe
successivenumbers，astheysay’2，3，4’fortheunitsintheidealare
generatedatthesametime，whether，asthefirstholderofthetheory
said，fromunequals(comingintobeingwhenthesewereequalized)or
insomeotherway-since，ifoneunitistobepriortotheother，it
willbeprioralsoto2thecomposedofthese；forwhenthereisone
thingpriorandanotherposterior，theresultantofthesewillbe
priortooneandposteriortotheother。
Again，sincethe1-itselfis
first，andthenthereisaparticular1whichisfirstamongthe
othersandnextafterthe1-itself，andagainathirdwhichisnext
afterthesecondandnextbutoneafterthefirst1，-sotheunitsmust
bepriortothenumbersafterwhichtheyarenamedwhenwecountthem；
e。g。therewillbeathirdunitin2before3exists，andafourthand
afifthin3beforethenumbers4and5exist-Nownoneofthese
thinkershassaidtheunitsareinassociableinthisway，but
accordingtotheirprinciplesitisreasonablethattheyshouldbe
soeveninthisway，thoughintruthitisimpossible。Foritis
reasonableboththattheunitsshouldhavepriorityandposteriority
ifthereisafirstunitorfirst1，andalsothatthe2’sshouldif
thereisafirst2；forafterthefirstitisreasonableandnecessary
thatthereshouldbeasecond，andifasecond，athird，andsowith
theotherssuccessively。(Andtosayboththingsatthesametime，
thataunitisfirstandanotherunitissecondaftertheideal1，and
thata2isfirstafterit，isimpossible。)Buttheymakeafirstunit
or1，butnotalsoasecondandathird，andafirst2，butnotalsoa
secondandathird。Clearly，also，itisnotpossible，ifallthe
unitsareinassociable，thatthereshouldbea2-itselfanda
3-itself；andsowiththeothernumbers。Forwhethertheunitsare
undifferentiatedordifferenteachfromeach，numbermustbecounted
byaddition，e。g。2byaddinganother1totheone，3byadding
another1tothetwo，andsimilarly。Thisbeingso，numberscannot
begeneratedastheygeneratethem，fromthe2andthe1；for2
becomespartof3and3of4andthesamehappensinthecaseofthe
succeedingnumbers，buttheysay4camefromthefirst2andthe
indefinitewhichmakesittwo2’sotherthanthe2-itself；ifnot，the
2-itselfwillbeapartof4andoneother2willbeadded。And
similarly2willconsistofthe1-itselfandanother1；butifthisis
so，theotherelementcannotbeanindefinite2；foritgenerates
oneunit，not，astheindefinite2does，adefinite2。
Again，besidesthe3-itselfandthe2-itselfhowcantherebe
other3’sand2’s？Andhowdotheyconsistofpriorandposterior
units？Allthisisabsurdandfictitious，andtherecannotbea
first2andthena3-itself。Yettheremust，ifthe1andthe
indefinitedyadaretobetheelements。Butiftheresultsare
impossible，itisalsoimpossiblethatthesearethegenerating
principles。
Iftheunits，then，aredifferentiated，eachfromeach，these
resultsandotherssimilartothesefollowofnecessity。But(3)if
thoseindifferentnumbersaredifferentiated，butthoseinthesame
numberarealoneundifferentiatedfromoneanother，evensothe
difficultiesthatfollowarenoless。E。g。inthe10-itselftheir
aretenunits，andthe10iscomposedbothofthemandoftwo5’s。But
sincethe10-itselfisnotanychancenumbernorcomposedofany
chance5’s——or，forthatmatter，units——theunitsinthis10must
differ。Foriftheydonotdiffer，neitherwillthe5’sofwhichthe
10consistsdiffer；butsincethesediffer，theunitsalsowill
differ。Butiftheydiffer，willtherebenoother5’sinthe10but
onlythesetwo，orwilltherebeothers？Iftherearenot，thisis
paradoxical；andifthereare，whatsortof10willconsistofthem？
Forthereisnootherinthe10butthe10itself。Butitis
actuallynecessaryontheirviewthatthe4shouldnotconsistof
anychance2’s；fortheindefiniteastheysay，receivedthe
definite2andmadetwo2’s；foritsnaturewastodoublewhatit
received。
Again，astothe2beinganentityapartfromitstwounits，and
the3anentityapartfromitsthreeunits，howisthispossible？
Eitherbyone’ssharingintheother，as’paleman’isdifferent
from’pale’and’man’(foritsharesinthese)，orwhenoneisa
differentiaoftheother，as’man’isdifferentfrom’animal’and
’two-footed’。
Again，somethingsareonebycontact，somebyintermixture，
somebyposition；noneofwhichcanbelongtotheunitsofwhichthe2
orthe3consists；butastwomenarenotaunityapartfromboth，
somustitbewiththeunits。Andtheirbeingindivisiblewillmakeno
differencetothem；forpointstooareindivisible，butyetapair
ofthemisnothingapartfromthetwo。
