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第49章

  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