Introduction
StewartShapiroandGeoffreyHellman
Theidea(or,perhapsbetter,theneed)forthisvolumebecamecleartouswhen wewereworkingonourmonograph, Varietiesofcontinua:fromregionstopoints andback.1Wedeveopedaninterestinvariouscontemporaryaccountsofcontinuity:theprevailingDedekind–Cantoraccount,smoothinfinitesimalanalysis (orsyntheticdifferentialgeometry),andintuitionisicanalysis.Eachofthese theoriessanctionssomelong-standingpropertiesthathavebeenattributedtothe continuous,attheexpenseofotherpropertiessoattributed.Theintuitionistic theoriesviolatetheintermediatevaluetheorem,whiletheDedekind–Cantorone givesupthethesisthatcontinuaareunifiedwholes,andcannotbedividedcleanly. Thesloganisthatcontinuaareviscous,orsticky.
WeweresurprisedtolearnthatmanyphilosophersandevensomemathematicianstaketheDedekind–Cantorconceptionofcontinuitytobenotonlythe rightone,buttheonlyone.Someweresurprisedtolearnthatthereareanyother notions.Theymayhaveheardsomethingofthehistory,butmanytakeitfor grantedthatwenowhavetheonecorrectaccountofcontinuity.Theoncelongstanding“intuitions”thatsupporttheotheraccountsarenottobetakenseriously (andperhapsnevershouldhavebeen).
Ourviewisthatthereisnosingle,monolithicpropertyofcontinuity.Itis moreofaclusterconceptthatcanbesharpened,anddevelopedrigorously,in mutuallyincompatibleways.Earlyon,wewereledtotheAristotelianideathat continuaarenotcomposedofpoints,orindivisibleparts.Intermsofcontemporarymetaphysics,thethemeisthatcontinuaaregunky:everypartofacontinous substancehasaproperpart.This,ofcourse,isnotsanctionedinthecontemporary Dedekind–Cantoraccountnor,arguably,intheintuitionisticaccountseither (dependingonwhatcountsasa“point”inthosecontexts).
Thebulkofourprojectin Varietiesofcontinua wastodevelopvariousgunky, orpoint-free,accountsofcontinuity:one-dimensionalandtwo-(andthree-…) dimensional,Euclideanandnon-Euclidean,withactualinfinityandwithoutactual infinity(anotherthemederivedfromAristotleandmaintainedfortwomillennia).
1 Oxford,OxfordUniversityPress,2018.
StewartShapiroandGeoffreyHellman, Introduction In: TheHistoryofContinua:Philosophical andMathematicalPerspectives.Editedby:StewartShapiroandGeoffreyHellman,Oxford UniversityPress(2021).©StewartShapiroandGeoffreyHellman. DOI:10.1093/OSO/9780198809647.003.0001
Wewentontocomparetheseaccountswiththeircontemporarycounterpartson variousmathematical,logical,andmetaphysicalgrounds.
Althoughitisobviousthatmathematicalandphilosophicalthoughtabout thecontinuousdevelopedconsiderablyovertheages,wecouldnotfindany comprehensivetreatmentofthishistory.2Sothetimeisrightforsuchaproject. Thetwoofushaveadeepinterestinthehistoryofphilosophyandmathematics, butwearenotscholarsintheseareas.Soaneditedvolumeseemedtheright coursetofollow.PeterMomtchiloff,fromOxfordUniversityPress,enthusiastically receivedourproposal,andwasverygenerousconcerningcontentandlength.The presentvolumeistheresult.
Ourideathatavolumeonthistopicismosttimely,andmostneeded,musthave beencorrect,judgingfromthestellargroupofscholarswewereabletorecruit. Theyareamongthetopresearchersineachperiodthatiscovered.Atonepoint, earlyon,someonesuggestedthatwekeepthisprojectquietforatime,sothatwe wouldnotbe“scooped”bysomeoneelse.Ourreplywasthatwewould,ofcourse, welcomemoreworkinthisarea,butwecouldnotseeanyonecomingupwitha lineupanywherenearasgoodastheonewehavehere.
Thepapershereallspeakforthemselves,andthereadercandeterminethe contentofeachfromitstitleanditsplaceinthevolume.Sowewillrestcontent herewithaverybriefremarkoneachpaper.
WebegininancientGreece.BarbaraSattler’scontributionconcernsthemetaphysicalandnaturalphilosophythatunderliestheancientdiscussions,arguing thatthemathematicsisalesscentralconcern.Themainfocusis,ofcourse, Aristotle,andhisprecursors,notablyParmenidesandZeno.Acentralthemeofthe paperisAristotle’sresponsetoZeno’sparadoxes.3OrnaHararicoverstheperiod inantiquityafterAristotle,focusingprimarilyonAlexanderofAphrodisias.The closelookatoneofAristotle’ssuccessorshelpsilluminatebothaccounts.Edith DudleySyllaturnstothemedievalperiod.Hermainfocusisa(relatively)recently discoveredmanuscript,byThomasBradwardine,anditsrelationtomedievalviews before,during,andafterBradwardine’stime.Manyoftheissuesunderdebate todaywereprevalentthen.
Nextistheso-calledearlymodernperiod,whenmathematiciansdevelopedthe calculusand,withthat,theriseofinfinitesimaltechniques.Ineffect,continuous magnitudesaretreatedasinfinitesumsofindivisibleelements,eachofwhichis infinitelysmall.Thephilosophicalissuesdominatedthinkingduringthatperiod. WearedelightedtohavetwocontributionsbySamuelLevey.Thefirstison GalileoGalilei,whoseentrytothethemeofthecontinuumistheanalysisof
2 Thereis Infinityandcontinuityinancientandmedievalthought, editedbyN.Kretzmann(Ithaca, CornellUniversityPress,1983).
3 WemightaddthatBarbara,andhercolleagueSarahBroadie,weremosthelpfultousinourwork on Varietiesofcontinua, aidingusinunderstandingthemainAristotelianthemes.Weeagerlyawait Barbara’sforthcomingbookonAristotleoncontinuity.
continuousuniformandacceleratedmotion,acommonconcernofmathematicians,scientists,andphilosophersduringthismostfascinatingera.Levey’ssecond contributionisonGottfriedWilhelmLeibniz,whowasprofoundlyinfluencedby Galileo.Leibnizfamouslydubbedthecontinuuma“labyrinth”.Thereasonforthis is,inlargepart,that“thediscussionofcontinuityandoftheindivisiblesthatappear tobeitselements”requires“considerationoftheinfinite”.Inbetweenthesetwo papers,thereisonebyDouglasJesseph,whofocusesattentiononBonaventura Cavalieri,whosementorwasGalileo.Cavaliericontributedgreatlytotheso-called “methodofindivisibles”thatformedthebasisfortheinfinitesimaltechniques developedbyLeibniz,Newton,andothers.
DanielSutherland’sarticleturnsthereader’sattentiontoImmanuelKant,and theroleandplaceofcontinuityandintuitionineighteenth-centuryanalysis.It focusesonissuesraisedbycontinuityfortherepresentationoftheinfinitelysmall and,inparticular,onthestatusofgeometricalandkinematicrepresentations.
PaulRusnockcoversBernardBolzano,afascinatingfigureoftheearlynineteenthcentury.Bolzanowasoneofthefirstimportantmathematiciansand philosopherstoinsistthatcontinua are composedofpoints,andhemadeaheroic attempttocometogripswiththeunderlyingissuesconcerningtheinfinite.
Nextuparethetwofiguresmostresponsibleforthecontemporaryhegemony. AkihiroKanamoricoversGeorgCantor.ThisarticleprovidestherichmathematicalandhistoricalbasisforCantor’sinitialworkonlimitsandcontinuityand ascentfromearlyconceptualizationstonewones,frominteractiveresearchtosolo advance.Cantorproceededtomoreandmorespecificresults,justashedeveloped moreandmoresettheory.
EmmylouHaffnerandDirkSchlimmcoverRichardDedekind,providinga detailedviewofbothfoundationalandmathematicalaspects.Dedekind,ofcourse, characterizedthepropertyofcontinuityfortherealnumbersintermsofwhat arenowcalled“Dedekindcuts”ontherationalnumbers—thuspresupposingthat continuaarecomposedofpoints,orpoint-likeelements.HaffnerandSchlimmgo ontoconsidersomeofDedekind’smoremathematicaltreatmentsofcontinuity, notablythedefinitionoftheRiemannsurfaceinhisjointworkwithHeinrichWeber(1882).TheyshowhowDedekind’sapproachesbecameincreasinglyabstract, whileatthesametimeretainingacommonmethodology.
WehavetwooutstandingcontributionsbyCharlesMcCarty.Thefirstuses alucidanalysisofthemathematicianPaulduBois-Reymondtoarguefora constructiveaccountofcontinuity,inoppositiontothecontemporaryDedekind–Cantordominance.McCarty’ssecondpaper,anicecompaniontothefirst,treats HermannWeyland,moreimportantly,L.E.J.Brouwer.
FranciscoVargasandMatthewE.MoorecoverCharlesSandersPeirce,who oncedubbedthenotionofcontinuity“themaster-keywhich…unlocksthearcana ofphilosophy”.Roughly,Peirce’saccounthasitthatacontinuoussubstancehasa lotmorepointsthanaregionofDedekind–Cantorspace—ineffectthereisno
setofallsuchparts.VargasandMoorecoverthedefiningfeaturesofPeirce’s mathematicaltheoryofcontinuity,givingamodelforthattheoryinZermelo–Fraenkelsettheory.TheygoontosummarizePeirce’sownattemptstoputhis conceptionintoarigorousform.
AlfredNorthWhiteheadisknownforpresentingapoint-free,orgunky,account ofcontinuity,andheshowedhowtorecoverpointsasakindof“extensiveabstraction”,alimitofsetsorsequencesofregions.⁴AchilleVarzipresentsWhitehead’s variousattemptsalongtheselines.
EachofthefinalfourpapersinthevolumepresentsamoreorlesscontemporarytakeoncontinuitythatisoutsidetheDedekind–Cantorframework.Peter Koellnergivesusanaccountbasedonpredicativity—therejectionofimpredicative definitions—derivedfromtheworkofHenriPoincaré,BertrandRussell,Hermann Weyl,and,especially,SolomonFeferman.Sofarasweknow,allotherviewstake acontinuoussubstance,likespaceorspace-time,tobegivenasawhole,inits entirety.IncontrasttotheDedekind–Cantorviews,manytheorists(Aristotle, mostofthemedievalwriters,Leibniz,Kant,…)insistthatthepartsofacontinuous substanceconstituteapotentialinfinity:ourabilitytocarveoutanddescribe partsofcontinuaisonlypotential,inthesensethatthereisnocompletedtotality ofallsuchparts.Adistinctivefeatureofthepredicativistviewisthatittakesa continuumtobeitselfpotentialwhile—atleastaspresentedrigorouslybythe leadingproof-theoristSolomonFeferman—usingclassicallogic(butnotmodal logic).ThisseemstobetheresultofacceptingsomeaspectsoftheDedekind–Cantorpicture.Roughly,(1)wethinkofaline,say,asacollectionofpoints;(2) wethinkofthepointsasrealnumbers(asinanumber-line);and(3)wethink ofarealnumberasasetofnaturalnumbers.Butwetheninsistthatallsuchsets mustbedefinedinapredicativemanner.Sothecontinuum(orthiscontinuum)is producedinstages:aswedefinesomesetsofnaturalnumbers,wearethenableto definemoresuchsets,andthereisnostagewhereallsuchsetshavebeen,orcan be,defined.
GiangiacomoGerlapresentsasurveyofcontemporaryaccounts(including Varietiesofcontinua)thatdonottakecontinuatobecomposedofpoints.Henotes thatacentralissue,ineachcase,istorecoverthenotionofbeingapoint,typically viasomesortofabstractioninthemouldofWhitehead’sextensiveabstraction.
JohnBellcoverscontemporaryaccountsofcontinuitythatinvokeintuitionistic logic.Mostbecomeinconsistentifageneralizedversionofexcludedmiddleis imposed.Acentralconcernistheextenttowhicheachaccountsanctionsa longstandingintuitionthatcontinuaarewholes,andcannotbedividedcleanly.As notedabove,this“indecomposibility”islostontheDedekind–Cantoraccounts.
⁴ Similartechniquesareusedinour Varietiesofcontinua.
Indeed,suchaccountsviewacontinuumasasetofpoints.Withclassicallogic, eachsubsethasacomplement.
PhilipEhrlichprovidesarichpresentationoftheoriesthat,unlikethe Dedekind–Cantoraccounts,accepttheexistenceofinfinitesimals.Suchaccounts thusviolatetheArchimedeanprinciple,adoptedinbothAristotleandEuclid, butthepayoffisconsiderable.Ehrlichshowshowsuchaccountsderivefrom forerunnersfromthelatenineteenthcenturyandtheearlydecadesofthetwentieth century.
DivisibilityorIndivisibility
TheNotionofContinuityfromthePresocratics toAristotle
BarbaraM.Sattler
1.Introduction
Whilemathematicalpracticeinancienttimesprovidedsomeinspirationforthe debateaboutcontinuityinearlyGreekthinkinguptothetimeofAristotle, mathematicsisnotwherethemaindebate—asfarasithasbeenhandeddown tous—happens.Rather,thediscussionaboutcontinuityisadebatewithinmetaphysicsandnaturalphilosophy.Wewillseethatthemainthinkerstocontribute tothedevelopmentofanunderstandingofcontinuityareParmenides,Zeno, andAristotle.Andwhileamodernunderstandingofcontinuitymayseemtobe essentiallyanti-Aristotelian,1Aristotlewillprovetobethethinkerwhoprepared manyofthecrucialfeaturesofamodernaccountofcontinuity.2
ThemainpointofcontroversyaboutcontinuityinearlyGreektimesis divisibility,asthischapteraimstoshow.Allpartiestothisdisputeagreethatmagnitudes whicharecontinuous(suneches)arehomogeneousandwithoutanygaps.3They disagree,however,onwhichinferencestodrawfromthisforthepossibilityof divisibility—whetheritimpliesdivisibilityorindivisibility.
Thefirstphilosophicallyinterestingandsystematicusageofthenotionof continuitywefindinParmenidespoem.Heunderstandsbeingcontinuousas beingcompletelyhomogeneousandwithoutanydifferences.Parmenidesinfers fromthelackofanydifferencesthatwhatiscontinuousisalsoindivisible,since whatiscompletelyhomogeneousdoesnotprovideany(sufficient)reasonforitto bedividedanywhere;thusitisnotdivisible.
1 ForanotableexceptionofamodernconceptionofthecontinuumthatisinspiredbyAristotle,and thuseitherassumesnopointsasbasicconstituents,orevennoactualinfinity;seeLinnebo,Hellman, andShapiro2016andHellmanandShapiro2018.
2 Evenifnotoriouslyherejectstheassumptionofanactualinfinity,seeespecially Physics bookIII.
3 Gaplessnessishereusedinanintuitivesenseasbeingwithoutanyholes,interruptions,orsudden changes;notinthemodernmathematicalsenseintermsofcompletenessaccordingtowhichrational numbersdonotformacontinuum,whilerealnumbersdo.
BarbaraM.Sattler, DivisibilityorIndivisibility:TheNotionofContinuityfromthePresocraticsto Aristotle In: TheHistoryofContinua:PhilosophicalandMathematicalPerspectives. Editedby:StewartShapiroandGeoffreyHellman,OxfordUniversityPress(2021). ©StewartShapiroandGeoffreyHellman.DOI:10.1093/OSO/9780198809647.003.0002
ZenowillbeshowntostrengthenParmenides’understandingofcontinuity, bydemonstratingthatwewouldgetintoinconsistenciesifweweretoassume divisibility:giventhattherearenointernaldifferencesthatcouldgiverisetoany division,ifweassumedwhatiscontinuoustobedivisibleatanyparticularpoint, thenitseemsitcouldbedividedeverywhere.Butifitcanbedividedeverywhere, thepartswewouldthusderivecannotbethoughtofconsistentlyaccordingto Zeno,aswewillseebelow.
Bycontrast,Aristotleembracestheideaofcontinuaasbeingdivisibleanywhere, whichhetakesupfromtheactivityofthemathematiciansofhistime.While geometersofthistimepresupposethemagnitudestheydealwithtobedivisible anywhere,interestinglytheydonotdiscusscontinuityinthemathematicaltexts handeddowntous.Wefind,however,afulldiscussionofthisnotionasappropriatedfornaturalphilosophyinAristotle’s Physics,anddivisibility adinfinitum isacrucialfeature.AristotlereactstothedivisibilityproblemraisedbyZeno’s paradoxeswithacomplexoflogicaltools:heshowsthattheseproblemscanbe avoidedwiththehelpofanewunderstandingofthepart–wholerelationship,a two-foldunderstandingoflimits,anewunderstandingofthenotionofinfinity, andacarefuldistinctionbetweenactualdivisionandpotentialdivisibility.
2.Parmenides’AccountofContinuity
TheGreektermforbeingcontinuous, suneches,literallymeans‘holdingtogether’. Wewillseethatthisholdingtogethercanbeunderstoodinratherdifferentways— thingsaretemporallyuninterrupted,spatiallyconnected,orontologicallyholding together.InGreekliteraturebeforeParmenides,thewordsunechesrefersmainlyto uninterruptedactivity⁴andassuchimpliesacertaintemporalextension(twodays ortenyears)duringwhichthisactivitytakesplace.WithParmenides,however, beingcontinuousnolongerreferstoanactivity,butrathertoanontological feature:beingcontinuousisacharacteristicofwhattrulyorultimatelyis(toeon, Being),whichistheonlythingthatcanbethoughtconsistently.⁵Whattrulyishas nothingtodowithanykindofactivity;indeed,Parmenidesexplicitlyclaimsthat itisunmovedorunmovable.⁶
ForParmenides,beingcontinuousimpliescompletehomogeneityand,ultimately,indivisibility.Threepassagesfromfragment8ofhispoemsetoutParmenides’notionofcontinuityinparticular.
⁴ See,forexample, Odyssey IX,lines74–75(“therefortwonightsandtwodayscontinuouslywelay, eatingourheartsforwearinessandsorrow”),orHesiod, Theogony 635–636(“they,withbitterwrath, werefightingcontinuallywithoneanotheratthattimefortenfullyears”).Formoredetails,cf.Sattler 2019,onwhichtheParmenidespartofthepaperdraws.
⁵ ForanaccountofhowtounderstandwhattrulyisaccordingtoParmenides,seeSattler2011.
⁶ Theverbaladjective akinêton canindicateeitherapossibility(unmovable)orapassiveresulting state(unmoved).
2.1BeingContinuousExcludesAnyTemporalDifferences
Infragment8,lines5–6,Parmenidesclaimsthat“neitherwasit[whattrulyis]nor willitbe,sinceitisnowalltogether,one,continuous”.Being“nowalltogether,one, continuous”isnamedasthereasonwhytemporaldifferencesthatarecapturedas “was”and“willbe”cannotbesaidofwhattrulyis.“Whatwasandwillbe”seemto bethethingswedealwithinoureverydayworld(whichforParmenidescannotbe objectsofknowledgebutarewhatweordinarymortalsrefertoinouropinions). Thesethingsarespreadouttemporally:theyweretherein(somepartof)thepastor willbetherein(somepartof)thefuture.Bycontrast,whattrulyis,isnotsubject tothesetemporaldifferences,because,accordingtoParmenides,itisaltogether now,one,continuous.Itis now—thishasbeenunderstoodeitherasindicating atemporality,beingbeyondtime;⁷orasindicatingsomepresentthatwecannever addressaspastorfuture.⁸Inbothcases,‘now’cannotbetemporallyextendedifit istobestrictlydistinguishedfromwasandwillbe;otherwisetherewillbeatime whenitwouldberighttosayofitthatitwasorthatitwillbe.⁹Soaccordingto thefirstpassage,Parmenidesdeniesthatwhattrulyisisextendedintimeinthe wayeverydayperceptiblethingsare;itis continuous inthesenseofnotallowing foranytemporaldifferences,likewasandwillbe.Being“nowalltogether”, “one”,and“continuous”thuspreventswhattrulyisfrombeingstretchedout intime.
2.2BeingContinuousImpliesBeingHomogeneous,FullofBeing, andNoMoreorLess
Inlines22–25Parmenidesmakesitclearthatbeingcontinuousexcludesnotonly temporaldifferences,butalsootherkindsofdifferences:
(1) Anditisnotdivisiblesinceitisallhomogeneous.1⁰
(2) Norisitmoreanywhere(oratanypoint),whichwouldpreventitfrom beingonecontinuous,norless,butitisasawholefullofbeing.
⁷ SeeOwen1966;similarlyMourelatos2008,pp.105–107.
⁸ SeeCoxon2009,p.196,whounderstandsitas“totalcoexistenceinthepresent”.Anditneeds tobeapresentthathasneithercomeintobeingnorwillpassaway,sinceParmenidesarguesagainst generationforwhattrulyis.
⁹ Somescholarshavereadthe‘now’asindicatingeternaltemporalduration,forexample,Tarán 1965,p.179,Gallop1984,p.15,andPalmer2009;cf.alsoSchofield1970.Foracritiqueofthisreading, seeSattler2019.
1⁰ Oudediaireton couldeithermean‘notdivisible’or‘notdivided’.SinceParmenidesseemsto deducenecessaryfeaturesofwhattrulyisinfragment8,‘notdivisible’seemstobethemoreappropriate translation.
(3) Throughthatitisallcontinuous,forBeingisincontactwithBeing(fr.8, lines22–25).
ThefirststepinthisargumentclaimsParmenides’Beingtobeallhomogeneous; thisimpliesbeingindivisible.Thesecondsteprulesoutthatitismoreorless—a conditionthatwouldpreventitfrombeingcontinuous.Instead,itisasawhole fullofbeing,whichseemstomeanequallyfull,neithermorenorless.Thelastpart ofthethirdstep,“BeingisincontactwithBeing”,pointsoutthatallofBeingis connected,andso,presumably,thereisnothinginbetweenanywherethatisnot Being,whichwouldunderminethehomogeneityofwhattrulyis.Letusclarify thesenseinwhichthecontinuityreferredtohereshouldbeunderstoodandhow itrelatestobeinghomogeneous.
Thefirstpartofthethirdstep,“throughthatitisallcontinuous”,readsasa conclusion—whatprecedesthusshouldexplainwhywhattrulyisisallcontinuous.11Thefeatureswearegivenin(1)and(2)thatshouldguaranteecontinuityare thatitisnotdivisible,itisallhomogeneous,itisnotmoreanywherenorless,and itisasawholefullofbeing.Whatisimportantforushereisthatbeingcontinuous includesallthesefeatures.Accordingly,beinghomogeneous(homoion)isaweaker notionthanbeingcontinuous,sincebeingcontinuousmeansbeinghomogeneous plusfulfillingsomefurthercriteria.TheGreekword homoion basicallymeans‘of thesamekind’.12Thus,being homoion hereseemsnaturallyunderstoodasbeing homogeneousandonewithrespecttokindorgenus—whattrulyisisnotdivisible intodifferentkindsorgenera.Thiswouldstillleaveopenthepossibilityofother, internaldifferences,likequantitativeorqualitativedifferences,whichareatleast inpartexcludedwiththefollowinglinesthatthereisno‘moreorless’thatwould preventBeingfrombeingcontinuous.
Beingcontinuoushasbeenunderstoodinverydifferentsensesinthispassage— inatemporal,spatial,ontological,andlogicalsense.13Forthetimebeing,I willsimplysuggestthatbeingcontinuousnotonlyimpliesindivisibilityinkind andgenus,butalsoexcludessomeotherkindofdifferences(betheytemporal,spatial,ontological,orlogical);thenextpassagewillbringmoreclarityon thisquestion.
11 Thefollowingclause,“BeingisincontactwithBeing”,iseitherasummaryorreformulationof (1)–(2),or,asissometimesthecasewithParmenides,anadditionalreasonthatisonlyprovidedafter theconclusion.
12 Seealso,forexample,Mourelatos2008,pp.113–114,131andTarán1965.
13 Forexample,Owen1960,pp.96–97understandsittemporally;Schofield1970,pp.132–134takes itinaspatialsense;Tarán1965,pp.106–108understandsitontologically;andCoxon2009,pp.325–326suggestsalogicalsense.Sincethetemporaldifferencesweencounteredinfragment8,lines5–6 weredifferencesoftenseandthuscompletelydifferentfromdifferencesof‘morenorless’,itwouldbe strangeifthesetemporaldifferenceswerenowtakenupby‘moreorless’.
2.3ConditionsThatWouldPreventContinuity
Inlines43–49Parmenidesspellsoutconditionsunderwhichwhattrulyiswould nolongerbecontinuous:whatwouldpreventitfrombeingcontinuouscouldbe eithernon-BeingoranunequallydistributedBeing.Theformerwouldleadtoit beinglargerorsmallerhereorthere,thelattertomoreorlessBeing.Butsincethere isnonon-Being1⁴andnotmoreofBeinghereandlessthere,1⁵thecontinuityof Beingasawholeisgranted.Theexplicitaimoftheselinesistodemonstratethe completenessofwhattrulyis;butpartofitscompletenessconsistsinitsbeing continuous.1⁶
While“largerandsmaller”suggestquantitativedifference—itiswhatwewould callquantitiesthatarelargerorsmaller—“moreorless”couldalsocoverqualitative differences(forexample,moreorlesshotness,blueness,etc.).Accordingly,itseems plausiblethatParmenideswantstoruleoutwhatwewouldcallquantitativeas wellasqualitativedifferenceshere—notanyspecificquantitativeandqualitative differences,butratherquantitativeandqualitativedifferencesingeneral.1⁷
Summingupthediscussionofallthreepassages,wecansaythatbeingcontinuousforParmenidesrequiressomethingtobehomogeneousandtoexcludeany kindsofdifferenceswhichcanbefurtherspecifiedasfollows:temporaldifferences, aswellaswhatwemaytermqualitativeandquantitativedifferences.Thenecessary resultofcontinuitythusunderstoodis indivisibility.ForParmenidesseemsto assumethatadivisionispossibleonlywherethereisinhomogeneityandthus differences—ifsomethingisdivisiblethenitmustbedivisiblebyvirtueofa differencewithinitselfsuchthatonepartofitcanbeseparatedfromthepartfrom whichitdiffers.ButsinceforParmenideswhatiscontinuousishomogeneousin everyrespectandexcludesdifferences,itisnecessarilyindivisible.1⁸
3.Zeno’sParadoxes:NegativeConsequencesof InfiniteDivisibility
Parmenidesarguesforwhatiscontinuoustobeindivisible.Zenostrengthens Parmenides’understandingofcontinuitybyshowingthenegativeconsequences theassumptionofdivisibilitywouldhave:itwouldundermineanystrongnotion ofunity,andthepartsofsuchadivisioncouldnotbeconsistentlythought.
1⁴ Asshowninfragments2,6,and7. 1⁵ Asdemonstratedinlines22–25.
1⁶ Whiletheword suneches isnotmentioned,thediscussionsofarshowsthatthispassagesystematizestheconditionsthatwouldpreventsomethingfrombeingcontinuous.
1⁷ Ofcourse,itisnotobviousthatParmenideswouldhavedistinguishedbetweenqualityand quantityinthewayfamiliartousatleastsinceAristotle;butheclearlyexcludestwodistinctkinds ofdifferencesthatwemaycaptureasquantitativeandqualitative.
1⁸ SeeSattler2019forsupportofthisclaim.
Letusstartwithabrieflookatthechallengedivisibilityseemstoposeforan understandingofunityaccordingtoZenoinfragments1and2:1⁹
Fragment1:
AndThemistiussaysthatZeno’sargumenttriestoprovethatwhatis,isone,from itsbeingcontinuousandindivisible.‘For’runstheargument,‘ifitweredivided, itwouldnotbeoneinthestrictsensebecauseoftheinfinitedivisibilityofbodies.’ Fragment2:2⁰
For,heargues,ifitweredivisible,thensupposetheprocessofdichotomytohave takenplace:theneithertherewillbeleftcertainultimatemagnitudes,which areminimaandindivisible,butinfiniteinnumber,andsothewholewillbe madeupofminimabutofaninfinitenumberofthem;orelseitwillvanish andwillbedividedawayintonothing,andsobemadeupofpartsthatare nothing.Bothofwhichconclusionsareabsurd.Itcannotthereforebedivided, butremainsone.Further,sinceitiseverywherehomogeneous,ifitisdivisibleit willbedivisibleeverywherealike,andnotdivisibleatonepointandindivisibleat another.Supposeitthereforeiseverywheredivided.21 Thenitisclearagainthat nothingremainsanditvanishes,andsothat,ifitismadeupofparts,itismade upofpartsthatarenothing.Forsolongasanyparthavingmagnitudeisleft,the processofdivisionisnotcomplete.Andso,heargues,itisobviousfromthese considerationsthatwhatisisindivisible,withoutparts,andone.
Theseparadoxesclaimthatifweassumesomeonethingtobedivisibleandthus tohaveparts,thisonewillnotbeoneinastrictsenseanylonger.Itrestsonthe backgroundassumptionthatifitweredivisible,itwouldhavepartsandthusit wouldnotonlybeone(whole),butalsoatthesametimemany(parts).Thisis, however,impossible,sincebeingoneandmanyaremutuallyexclusivenotionsfor theEleatics,asZenomakesclearinfragment8.22Henceassumingdivisibilityleads tothecontradictoryresultthatoneisalsomany.Onlyifouroneiscontinuousand thusindivisiblewillwebedealingwithwhatisreallyone.
Furthermore,theassumptionthattheoneisdivisibleanddividedeverywhere alsoshowsthatwecannotconceivethesepartsinanyconsistentway(ashemakes clearinthepluralityparadoxinfragment2justquoted).Avariantofthisproblem canalsobeseenwithoneofZeno’sprobablymorefamousparadoxes,hisfirst
1⁹ IamusingH.D.P.Lee’s1967editionofZeno’sfragments,sinceitismoreencompassingthan Diels/Krantz;Iwillalsousehistranslations.Fragment1inLeeisfoundinSimplicius, Physics 139.19–22;fragment2inSimplicius, Physics 139.27ff.ThediscussionofZenodrawsfromSattler 2020b,ch.3.
2⁰ Porphyryattributesfragment2toParmenides,butAlexanderandSimpliciusthinkitmorelikely byZeno;cf.alsoLee1967,p.12.
21 ThissentencedemonstratesZeno’smovefromdivisibilitytobeingdivided—acentralpointin Zeno’sparadoxesthatwillbeattackedbyAristotle.
22 Leefragment8=DK29A21.
paradoxofmotion.Accordingtothisparadox,ifsomethingmovesoveracertain distance,forexamplearunnerwantstocoveracertainfinitetrackABinafinite timeFG,shefirsthastocoverhalfofthisdistanceAC(Figure1.1).Butbefore therunnercancoverthedistanceAC,shemusthavecoveredalreadyhalfofthis distance,AD,andbeforethat,halfofthishalf,AE,etc.Soshewillhavetopassan infinitenumberofspatialsegmentsbeforereachingtheend.23
Thisseeminglysimpleparadoxraisesseveralproblems,buttheonlyonethat concernsushereisthatitseemstoshowthatbycoveringa finitedistance,arunner hastopassaninfinitenumberofspatialsegments,which,accordingtoZeno,cannot bedone.
Zeno’sparadoxesstrengthenParmenides’understandingofcontinuity,by demonstratingthatevenifweweretoleaveParmenides’positiontotheside, divisibilitystillseemstogetusintoinconsistenciesforthefollowingreason:given thattherearenointernaldifferencesthatwouldaccountforanydivision,ifwe assumewhatiscontinuoustobedivisibleatanyparticularpointthenitseemsto bedivisibleanywhere—aswejustsaw,thefiniteracetrackseemstobedivisible ad infinitum.Butifitisdivisibleanywhere,itseemsitcanbedividedeverywhere.2⁴ Thepartswewouldthusderiveare,however,problematicaccordingtoZeno(as wesawinfragment2).Foreitherweassumethattheprocessofdivisioncangoon withoutanyrestrictions,inwhichcasethepartseither(1)havetobedivideduntil thereisnothingleft;thenwewillhavetoassumethatthesepartsofnilextension makeupanextendedwhole,whichZenoclaimstobeabsurd.Or(2)theseparts havesomeextension,butthisonlymeansthatthedivisionisnotcompleteyet, sincethesepartswouldinprinciplebefurtherdivisible,andthepartsarethus undetermined.
Alternatively,theprocessofdivisioncannotgoonforever,but,atacertainpoint, reachessomethinglikeindivisibleminima.Then(3)theonefinitewholewillbe madeupofinfinitelymanyextendedminima,whichaccordingtoZenoisnot thinkable.Wemaybepuzzledwhyheassumesinfinitelymanyminimainthiscase. Fragment11suggestsananswerbypointingoutthatifthereisaplurality,ithasto
23 InAristotle’s Physics 263a4–11thisparadoxispresentedintwoforms,whichinaccordancewith thesecondaryliteraturecanbecalled‘progressive’and‘regressive’.Theregressivevariantisjustthe givenstateofaffairs,whiletheprogressiveformassumesthataftertherunnerhascoveredthefirsthalf, shethenagainhastocoverthefirsthalfoftheremainingdistanceandthenagainthefirsthalfofthestill remainingdistance,etc.However,inlogicalterms,bothversionsareequivalent.Foramoredetailed discussionofthisparadox,seeSattler2020b,ch.3.
2⁴ Themovefrombeingdivisibleanywheretobeingdividedeverywherewillbequestionedby Aristotle.
Figure1.1 Zeno’sparadox.
bebothfinite(thenumbertheyare)andinfinite(forinorderfortheretobetwo separatethings,therealwaysneedstobesomeotherthirdthinginbetween,and anotherfourththingbetweenthesecondandthird,etc.).
WeknowthatinfactallthreeavenuesconsideredbyZenotobeabsurdhave beenpursuedbythetraditiontofollow,thelastone,inmodifiedform,bythe atomists,thesecondonebyAristotle,andthefirstonebymodernmathematics.2⁵
ForZeno,however,assumingdivisibilityofwhatiscontinuous2⁶leadstotwo majorproblems:suchacontinuouswholewouldonlybeoneinaweaksense,as itwouldbemanypartsatthesametime,andallpossibilitiesofconceivingofsuch partsleadtoabsurdities.Accordingly,whatiscontinuoushastobeindivisiblefor theEleatics.2⁷
4.AMathematicalConceptionofContinuity
Whileourcontemporaryunderstandingofcontinuitydevelopedwithinthefield ofmathematics,sofarwehaveseenthatfortheearlyGreeksthenotionwas discussedwithinthearenaofmetaphysicsandnaturalphilosophy.2⁸Surprisingly, wedonotfindadiscussionorexplicitdefinitionofcontinuityassuchinthe mathematicaltextshandeddowntousfromthetimebeforeorjustafterAristotle. AlsoinEuclid,ourbestmathematicalsourceclosetothetimeofAristotle,the term suneches isnotdefinedandnotveryoftenused.2⁹Nevertheless,wefind clearindicationsthatcrucialfeaturesofthemostprominentunderstandingof continuitylateroncanbefoundwiththemathematicians,asthefollowingpoints suggest.Thereareacoupleofpassagesinthe Elements inwhichEuclidusesthe term suneches asbeingsuccessiveinthewaycontinuouslinesare.3⁰Inthese passages, suneches seemstobeunderstoodastwo-place—onestraightlineiscontinuouswithanotherstraightline.Butthereisalsoaone-placeunderstandingof continuaasbeinginfinitelydivisible,whichispresupposedgenerallyingeometricalconstructions:geometershavetounderstandtheirgeometricalobjects—lines,
2⁵ Grünbaum1968,forexample,adoptsthisfirstroute,thattheextendedwholewouldhavetobe madeupofunextendedparts,andpointsoutthatthereisawaytoallowforsuchpart–wholerelations inmathematics,wherealine,andthussomethingextended,isunderstoodtoconsistofextensionless parts.Sinceextensionissimplyafeatureofthesetmakinguptheline,notofanyoftheindividual membersofthisset,thereis,accordingtoGrünbaum,noparadox.AndGrünbaumsimplyassumes thatthesameholdsinthephysicalrealm.Itisnotclear,however,thatwhatholdstrueofmathematical thingscanalsobesaidofphysicalthings—thatanextendedphysicalthingcan consist ofunextended physicalpartsandthatthisisnotjustamathematical description.
2⁶ WesawthatforZenothismeansassumingunrestricteddivisibility,sincethereisnomorereason toassumeadivisionhereratherthanthere.
2⁷ Thereareacoupleofphilosopherswhodealwithsomenotionof suneches betweenZenoand Aristotle,suchasPhilolaos,Gorgias,andAnaxagoras,butIdonothavespacetodiscussthemhere.
2⁸ Thefollowingsection,aswellassections5and6,drawonSattler2020b,ch.7.
2⁹ Andmostofthetimethetermreferstoacontinuedproportion—toacontinuousratioasin Elements bookVIII,proposition8.
3⁰ As,forexample,inbookI,postulate2,orbookXI,proposition1.
surfaces,andsolids—asmagnitudesthatarealwaysfurtherdivisibleandthus divisiblewithoutend.31Thisbecomesclear,forexample,fromthediscussionof theanonymoustreatise OnIndivisibleLines,wherethereplytothepostulateof indivisiblelinesfrequentlyreliesontheassumptionthatmathematicianstreat theirgeometricalobjectsasbeingalwaysfurtherdivisible.32Whilenotexplicitly discussed,thisideaofmagnitudesbeingdivisiblewithoutendseemstohavebeen takensomuchforgrantedbymathematiciansthattheydonothavetopayit anyspecialattention.Itisclearthatwhentheyassumecrossinglinesandsimilar constructions,thereisnoreflectionofatomisticworries,suchasthatalinecrossing anotherlinewouldneedtogobetweentwoatoms;ratherinfinitedivisibilityjust seemstobepresupposed.Finally,alsoAristotleclaimsthemathematiciansto understandmagnitudesinthesenseofbeinginfinitelydivisibleinhisdiscussion oftheinfinitein Physics bookIII.33
Weseethatthemathematicalunderstandingofmagnitudesiscruciallydifferent fromtheEleaticnotion.WhilethemathematiciansseemtosharetheEleatic assumptionthatbeingcontinuousimpliesbeinghomogeneous—forthemathematiciansthismeansthateachpossiblepartofacontinuousmagnitudeistreated alike—theinfinitedivisibilityofthemathematicalcontinuumistheveryopposite oftheindivisibilityParmenidesassumed.3⁴Themathematicalunderstandingof continuityinthesenseofinfinitedivisibilityispresupposedbyanygeometrical operationthatinvolvesthemathematicalbisectionofaline,surface,orbody.3⁵
ForhisphysicsAristotlecantakeupthisimplicitunderstandingofmagnitudes fromthemathematicians,3⁶eveniftheydonotprominentlycaptureitwiththe term suneches.Aristotle’saccountofcontinuitythusseemstocombineEleatic terminologywithareflectiononGreekmathematics.3⁷
31 Fromacontemporarypointofview,wemayaskwhetherthisunderstandingofgeometrical magnitudesasbeingalwaysfurtherdivisiblewouldmapontotherealnumbers,astheDedekindCantorcontinuumdoes,oronlyontotherationalnumbers.Sodoestheinfinitedivisibilitysimply meanthatwecouldinprinciplemakeinfinitelymanycuts(whichwouldbethecasewiththerational numbers),ordoesitmeanthatwecandivideitexactlywhereverwewant,evenat,letussay, �� (for examplewiththehelpofacirclethatwerollalongthelinewewanttocut)?
32 See DeLineisInsecabilibus 969b20ff.
33 203b17–18.Andin Physics 200b18–20Aristotlereferstoexistingaccountsofbeing suneches as beinginfinitelydivisiblethatarelikelytobemathematicalaccounts,cf.Sattler2020b,ch.7.
3⁴ Zenoattackstheassumptionofinfinitedivisibilityinhisparadoxes,butwedonothavethetextual evidencetosaywhetherthemathematiciansrespondedtothisattack,orsimplyignoredZeno,or perhapsdidnotfeelattackedbecauseZenoshowedinfinitedivisibilitytoleadintoparadoxesforour basicconceptionsconcerningthephysicalrealm(likemotionandapluralityofphysicalthings),not explicitlyforthemathematicalrealm.
3⁵ See,forexample, DeCaelo 303a2andalso DeLineisInsecabilibus 969b29–70a5.
3⁶ Cf.alsoWaschkies1977.
3⁷ Karasmanis2009,p.250claimsthatAristotleunderstoodthecontinuuminaphysicalway andthusdidnotapproachcontinuity(whichheunderstandsasZenonianinfinitedivisibility)and magnitudefromthephenomenonofincommensurability.Karasmanisclaimsthata“Zenonianinfinite divisibilitynevercutsalineintoincommensurablesegments.However,wedonothaveanyevidence thatAristotlehadunderstoodthisproblem.”IthinkKarasmanisisrightinclaimingthatAristotle didnotapproachcontinuityandmagnitudefromtheproblemofincommensurability.Butthisisnot
5.Aristotle’sAccountofContinuity
WhileAristotlecantakeovertheunderstandingofcontinuaasbeinginfinitely divisiblefromamathematicalcontext,hedevelopsitinsuchawayastobeable todealwithproblemsspecifictomotionandtoaccountforthespecificformof unityrequiredforphysicalbutnotmathematicalcontinua.Forexample,inthe physicalrealmtherecanbeadifferencebetweenbeingcontinuousandbeing contiguous,ashepointsoutin Physics V,3,whilethisdifferenceisnottobefound inmathematics.Formathematicians,twolinesinthesameplanethattouchare one,3⁸aslongastheydonotsimplyintersectorformanangle;theyarecontinuous. Inaphysicalcontext,however,twothingsthatarecontinuousinthemselvesand nexttoeachotherdonotbecomeonethingsimplybytouching,atleastnotif theyaresolidthings.Accordingly,in Physics V,3Aristotlegivesustwocriteria forbeingcontinuous.Continuousarethosethings(a)whoselimitstouch—this iswhatmakesforcontinuousthingsastheyareinmathematics.Additionally, however,inthephysicalrealm,(b)theselimitshavetobeone,forotherwisewe wouldbedealingonlywithneighbouringthings.3⁹Inaphysicalcontext,wecan distinguishmerelytouchinglimitsfromthosethathavebecomeone—onlywhere thelimitsareonedowehaveanobjectthatmovesasawhole.
Weshouldalsobearinmindthatinspiteofsomemathematicalbackground,the paradigmcontinuaforAristotlearecertainphysicalmagnitudes,suchasphysical bodies,time,distance,andmotion.Liquidcontinua,likewater,andthingslike mouldingdoughmayhaveinpartdifferentfeatures—forexample,thedistinction betweencontiguityandcontinuityworkswithbodies,butdoesnotseemtowork withwater.⁴⁰Aristotle’saccountofcontinuityismeanttofitthephysicalrealm⁴1
becausehedidnotunderstandthisproblem,butratherbecausea‘Zenonianinfinitedivisibility’can butneednotcutalineintoincommensurablesegments,sothisisaproblemnotessentialforgivinga basicaccountofcontinuity.AsAristotlehimselfpointsoutinhis PriorAnalytics 65b16–20(apassage Karasmanisexplicitlyrefersto),infinitedivisibilityandincommensurabilityaretwodifferentproblems. WhatthisshowsforourimmediateinvestigationisthatAristotledoesnotnecessarilytakeoverallthe problemsthatcanbefoundinthevicinityofamathematicalnotion.
3⁸ Asaretwosurfacesorbodies.
3⁹ Accordingly,Aristotlesometimestalksaboutlimitsintheplural(soin Physics 231a22)and sometimesaboutlimitinthesingular(227a12).Wemaybeworriedabouthisuseoftheplural,since thelimitsoflinesarepointsandAristotleexplicitlyclaimsthatpointscannottouch(andiflimitstouch, wouldtheynothavetohavelimitsthemselves?).Aristotlemaybetalkingloosely,whenhetalksabout limitsintheplural(ashedoeswhenmentioningpointstouchingin227a29),andactuallythinkabout alimitat whichthingstouch;orhemaythinkthatwithphysicalthingsweareneverreallydealingwith points.
⁴⁰ Furthermore,oncewephysicallyactualizeapartofAristotle’sparadigmaticcontinua,wecannot simplygetthewholeback(wewillneedglue,etc.);thisdoes,however,notholdforwater.
⁴1 Thedistinctionbetweenphysicalandtheoreticaldivisibilitywillbecomehighlyimportantinthe continuumdiscussioninthethirteenthandfourteenthcenturieswithThomasAquinas’s‘Minimatheorie’inhiscommentaryonAristotle’s Physics andDunsScotus’sreactiontothatinhis OpusOxoniense ButitisnotsomethingAristotleisconcernedwithashisunderstandingofcontinuaisclearlytailoredto thephysicalandnotthemathematicalrealm(eventhoughderivedfromareflectiononmathematics).
