VarietiesofContinua
GeoffreyHellmanand StewartShapiro
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Contents
Preface vii
.TheOldOrthodoxy(Aristotle)vstheNewOrthodoxy (Dedekind–Cantor)
.TheClassicalContinuumwithoutPoints
.AristotelianandPredicativeContinua WithØysteinLinnebo
.RealNumbersonanAristotelianContinuum
.Regions-basedTwo-dimensionalContinua:TheEuclideanCase .Non-EuclideanExtensions
References
Preface
Thisworkbeganseveralyearsagowithaseriesofdiscussionsoncompetingconceptionsofcontinuainmathematics,informedbylatetwentieth-centurydevelopments,suchasRobinsoniannon-standardanalysisandthemoreradical,non-classical smoothinfinitesimalanalysis,thatrehabilitatedearliernotionsofinfinitesimals.At firstourplanswererathermodest—todoacomparativestudyofleadingapproaches tothecontinuous,assessingthetrade-offsinvolved,inamoresystematicandcomprehensivewaythanwhatwasalreadyavailableintheliterature.Inparticular,we werestruckbythehistoricaltransitionfromthelong-standing,Aristoteliannonpunctiformconceptiontomodernpunctiformanalysisbasedonthemethodof limitsduetoBolzano,Cauchy,Weierstrass,Cantor,Dedekind,etal.,whichrapidly becamethedominantmainstreamapproach.Alongwiththecontrastbetweennonpunctiform,orregions-based,conceptionsandthemodernviewofcontinuaas entirelycomposedofpoints,thereisalsothecontrastbetweenpotentialinfinity— againgoingbacktoAristotle—andtherecognitionofactualinfinitiesinmainstream modernmathematics.Herewewerestruckbythecircumstancethatrestrictionsto intuitionisticlogic,alsoassociatedwithpotentialinfinity,havebeenmotivatedin radicallydifferentways,ontheonehandbydemandsofconstructivity(byBrouwer, Bishop,etal.),and,ontheother,byrestrictionsto“smoothmanifolds”(asinsmooth infinitesimalanalysis(SIA),especiallyasdescribedbyJohnBell),althoughthislatter wayofmotivatingtherestrictionhasbeenchallengedbyoneofus.Also,wewere motivatedtoassessintermediate,semi-constructivesystemsofpredicativeanalysis (goingbacktoRussell,Poincaré,Weyl,andperfectedinworkofFefermanetal.), basedonclassicallogic,recognizingactualinfinitybutonlyofpredicativelyjustifiable countabletotalities.
Giventhisvarietyofsystemsandmotivatingideas,itseemedcleartousthat “continuity”isacluster-concept,realizableinmultiple,oftencompetingways,andthat nosinglesystemcoulddojusticetoallthedesiderataemergingfrommathematical andscientificpractice,pastandpresent.
Itthenoccurredtousthatitoughttobepossibletoavoidtheshifttointuitionistic logic,alongwiththeapparatusofnil-potentinfinitesimals,whilestillbasinganalysis andgeometryontheideaofregionsor“chunks”ofagivenspaceormanifold, eschewingpoints,atleastaspartsofregions.Couldnotoneremain“classical”inone’s prooftheoryandtreatmentoftheinfinitewhilestillrealizingathoroughgoingnonpunctiformconceptionofcontinua?Wewere,ofcourse,alreadyfamiliarwithideas ofWhiteheadandothersonrecovering“points”asdefinedbysuitablynestedregions ofaspace.Wewere,however,surprisedathowlittlehadbeendoneinworkingout
suchverynaturalideasrigorously,especiallybydeployingtheresourcesofatomless mereology,whichseemswell-suitedforsuchaproject.Inparticular,Tarskihadbegun suchaprojectinearlyworkwithLe´sniewski,supplementingmereologywiththesole primitive“sphere”inordertorecoverclassical,three-dimensionalgeometry.Buton closerexaminationwerealizedthatTarski’singeniousdefinitionswerenotsupplied withautonomousaxioms,beyondthoseofmereology,onthemathematicalprimitive, “sphere”,frameddirectlyintermsofregions,sothattheprojectremainedinan incompletestate.
Sowesetourselvesthetaskofcompletingsuchwork,butbeginningwiththesimple caseofacontinuumofonedimensionandthentryingtoextendsuchatreatmentto higher-dimensionalspaces.Thisinturninvolvedusinseekingtorecover,onthesame basisofregions,metricalstructuresofEuclideanandthennon-Euclideangeometries, ultimatelyresultinginthecorepositivedevelopmentsofthepresentwork.
Itwascleartousfromtheoutsetthatthereisastronghistoricaldimensiontoour project,rootedasitisinideasgoingbacktoAristotle.Indeed,wecalloursystemfor aone-dimensionalregions-based(“gunky”)continuum“semi-Aristotelian”,sinceit adheresstrictlytoAristotle’sdesideratumofnotbeingconstitutedofpointswhileat thesametimemakingfreeuseofactualinfinities,bothgeometricandset-theoretic (equivalentlydescribedvialogicofplurals).Thisinturnnaturallyledustoaskhowwe mightformulatemorefullyAristoteliantheoriesofcontinua,enforcingrestrictionto potentialinfinitiesonly.AsØysteinLinnebohadalreadyworkedoutformalismbased onmodallogicforexpressingpotentialinfinity,wepursuedathree-waycollaboration withhimforachapterdevotedtothistopic.
Atthisstage,then,wehaddevelopedsystemsofanalysisandgeometrythatare “semi-Aristotelian”,non-punctiformbut“classical”withrespecttoactualinfinities, andwehad“moreAristotelian”systemsthatwerebothnon-punctiformandconfined tothepotentiallyinfinite.Thatleftthefurtherpossibilityofpredicativesystems,lying betweenthemoreAristotelianandoursemi-Aristotelianones,basedonregions,using classicallogic(withoutmodality),butadmittingonlycountablyinfinitetotalities (correspondingtowell-foundedsettheorywithoutthepower–setsaxiom).Thedevelopmentofsuchpredicativesystemsthenroundsoutourformalworkpresentedhere.
Wethenturnedourattentiontodiscussingsomeoftheinterestingphilosophical implicationsoftheformaldevelopments,includingthesystematiccomparisonsof thedifferentapproachesthatmotivatedourprojectinthefirstplace,butalsoimplicationsforvariousdiscussionsofcontemporaryanalyticmetaphysicspertainingto applicationsofgeometryandanalysistomaterialobjectsofthenaturalworld,such ashowtounderstandormodel“contact”ofmaterialobjects,whethertoadmitpointlikeobjects,andsoforth.Asexplainedbelow,manyofthesediscussionserrinsimply assumingthatthegeometrybeingappliedpertainstopunctiformspaces,whereas variousconundraevaporateinthecontextof“gunky”spaces.
Finally,toroundoutthisvolume,wereturntoouroriginalthemeofcomparisons andtrade-offsamongthediverseconceptionsofcontinua,nowincludingtheregionsbasedsystemsconstructedhere.
Weincurredmanydebtsalongtheway.Intheinitialdevelopmentoftheonedimensional,semi-Aristoteliansystem,wewerehelpedbyRoyCook,aformerstudentofoneofusandacurrentcolleagueoftheother,andbygraduatestudents atourhomeinstitutions,notablyShayLoganatMinnesotaandPatrickReeder atOhioState.Eithertogetherorseparately,wegavepresentationsofearlyversionsofsomeofthismaterialatavarietyofconferences,workshops,andcolloquia,andwereenlightenedbycommentsfromtheaudiences.Thelistincludesthe UniversityofMinnesota,theOhioStateUniversity,severalmeetingsoftheMidwest PhilosophyofMathematicsWorkshop,heldannuallyatNotreDame(andoncein Illinois),theFoundationsInterestGroupattheMinnesotaCenterforPhilosophyof Science,Logica,thePhilosophyofScienceAssociation,theUniversityofSheffield,the UniversityofOslo,theUniversityofCaliforniaatDavis,theHebrewUniversity ofJerusalem,theUniversityofConnecticut,theUniversityofSouthernCalifornia, andtheUniversityofStAndrews.ThankstoanonymousreviewersfromOxford UniversityPress, PhilosophiaMathematica, LogicandLogicalPhilosophy, Iyyun,and the ReviewofSymbolicLogic.ThanksalsotoFrankArntzenius,JeremyAvigad,John Bell,EinarDuengerBohn,SarahBroadie,JohnBurgess,BenCaplan,TimCarlson, AaronCotnoir,PhilipEhrlich,ThomasForster,JamesFranklin,HarveyFriedman, GiangiacomoGerla,CharlesGeyer,RafałGruszczy ´ nski,JeremyHeis,EliHirsch, ChristopherHookway,JesseHouchens,ØysteinLinnebo,CharlesDavidMcCarty, MarcoPanza,AndrzejPietruszczak,CarlPosy,AgustínRayo,BarbaraSattler,Wilfried Sieg,AllanSilverman,ReedSolomon,GabrielUzquiano,AchilleVarzi,andKeren Wilson.Sincereapologiestoallofthosewhoshouldbementionedhere,butarenot. WeowealargedebtofgratitudetoHaidarAl-Dhalimyforamostcarefulproofreading ofearlierdrafts,developingtheLaTeXcodeforourfigures,andforhelpwiththe finalproofreadingandtheindex.Finally,thankstoPeterMomtchiloff,ofOxford UniversityPress,andhisstaff,forencouragingthisprojectandforguidingitthrough thevariousstagesofpublication.
TheOldOrthodoxy(Aristotle) vstheNewOrthodoxy (Dedekind–Cantor)
.CommonGround
Insomeways,Aristotlecameremarkablyclosetothecontemporary,“classical”,or “orthodox”accountofcontinuity,especiallyincomparisonwithhismainopponents onthesematters,theatomists(seeWhite[127]andMiller[87]).1 First,forAristotle, acontinuumis infinitelydivisible,inthesensethatithasnosmallestparts.Inthecase ofonedimension,theideaisthatanylinesegmentcanbebisected(ordividedinto anyfinitenumberofparts).Theresultsofabisectionaretwosegments,eachofwhich canbebisected,andsoonindefinitely.Theatomists,oratleastthoseradicalatomists whoappliedtheirviewtospaceitselfandtimeitself,insistthattherearesmallest regions,thosethatcannotbefurtherdivided.Zeno’sparadoxoftheStadiumseems tobeaimedatthosetheorists.ForDedekind–Cantor,ofcourse,acontinuumhasno smallest extended parts,althoughonecanthinkofapointasanunextendedpartofa lineorlinesegment.
Inatleastoneplace,Aristotleseemstosaythatinfinitedivisibilityis sufficient for continuity:
... nocontinuousthingisdivisibleintothingswithoutparts.Norcantherebeanythingofany otherkindbetween;foritwouldbeeitherindivisibleordivisible,andifitisdivisible,divisible eitherintoindivisiblesorintodivisiblesthatarealwaysdivisible,inwhichcaseitiscontinuous. (Physics VI,231b11–15)
Second,Aristotle’scontinuumisArchimedean.Thishastwoformulations,the secondasortofinversetothefirst:
(i)Let a and b beanytwomagnitudes,ofthesamedimension.Thenthereisa naturalnumber m,suchthattheresultofadding a toitself m-timesislarger
1 ItisironicthattheDedekind–Cantoraccountiscalled“classical”since,inthepresentcontext,we contrast thataccountwithAristotle’s.
oldorthodoxy(aristotle)vsnew(dedekind–cantor)
than b.AsAristotleputsit(Physics,8,10):“Bycontinuallyaddingtoafinite quantity,Ishallexceedanydefinitequantity”.
InEuclid’s Elements (BookV,Definition4),thisprincipleappearsasa definition:“Magnitudesaresaidtohavearatiotooneanotherwhichare capable,whenmultiplied,ofexceedingoneanother”.Theassumptionisthat anytwomagnitudesofthesamedimension“havearatio”toeachother.
(ii)Let a and b beanytwomagnitudes.Supposeweremoveatleasthalfof a, andthenremoveatleasthalfoftheremainder,etc.Continuinginthisway, wewilleventuallyproduceamagnitudesmallerthan b.AsEuclid(Book10, Proposition1)putsit:2
Twounequalmagnitudesbeingsetout,iffromthegreatertherebesubtracteda magnitudegreaterthanitshalf,andfromthatwhichisleftamagnitudegreaterthan itshalf,andifthisprocessisrepeatedcontinually,therewillbeleftsomemagnitudeless thanthelessermagnitudesetout.
Theradicalatomistspresumablyrejectthesecondprinciple,althoughsomeofthem mayacceptthefirstversion.
.Points
Forpresentpurposes,ofcourse,thedifferencesbetweentheAristotelianandthe contemporary,Dedekind–Cantoraccountofcontinuityaremoresalient.Almostall ofthesedifferencesaretied,inonewayoranother,toAristotle’srejectionofthe actualinfinite,althoughweareloathtospeculateontheconceptualandexplanatory connectionsbetweenthevariousfeatures.Webeginwiththematterofpoints.
Toputthingsabitanachronistically,Aristotlearguesthattime,space,andmotion allhavethesamestructure.Thereasonseemstobethatmotionistobeunderstoodas movementinspacethroughtime.Itfollowsthat,forAristotle,ifanyoneoftime,space, ormotionispunctiform—composedentirelyofpoints,orpoint-likethings—thenso aretheothers.Andifanyoneoftime,space,ormotioniscomposedofextended atoms,thensoaretheothers,etc.Itseemsthateveryoneintheancient,medieval,and earlymoderndebatesacceptthatmuch.
Aswejustsaw,Aristotlerejectstheatomistichypothesisthatthereareextended atomsinspaceortime.Forpresentpurposes,thecentralfeatureoftheAristotelian account—andoneprimarydifferencewiththereceivedDedekind–Cantorone—is thatcontinuaare not composedofpoints.Inotherwords,forAristotle,continuaare notpunctiform.Sticking,here,toonedimension,thepartsofalinesegmentareother linesegments(orsumsthereof).
2 EuclidprovesthisfromthefirstArchimedeanprinciple(Book5,Definition4),thuspresupposingthat thegivenmagnitudes“havearatio”toeachother.Healsoinvokessomepropertiesofunequalmagnitudes. Themethodofexhaustion,tracedtoEudoxus,reliesonthisprinciple.
MichaelJ.White[127],p.29attributesthefollowing“principleofnonsupervenienceofcontinuity”toAristotle:
N-SC.Eachpartitionofacontinuousmagnitudeintoproperpartsyieldspartseachofwhichis pairwisecontinuouswithatleastoneotherpart.
WewillsoongettowhatAristotlemeansfortwothingstobe“continuouswith eachother”.Fornow,thecrucialfeatureisthatiftwothingsarecontinuous,then nothingofthesamekindcancomebetweenthem.ForAristotle,asforEuclid,asin contemporaryEuclideangeometry,betweenanytwopoints,thereisalinesegment.As Aristotleputsit,“thatwhichisintermediatebetweenpointsisalwaysaline,andthat whichisintermediatebetweenmomentsisaperiodoftime”(Physics,Book6,231a1). Sonotwopointsorinstantscanbecontinuous(orcontiguous)witheachother.Thus, theprincipleN-SCentailsthatalinesegment,oranintervaloftime,doesnothave points,orinstants,asparts.3
Torepeat,andtolabortheobvious,thisisonekeyplacewherethecontemporary Dedekind–CantorconceptionofthecontinuumpartscompanywithAristotle.The now“orthodox”accountdoesconstruealinesegmentas(nothingmorethan)aset ofpoints,eachofzerolength.Andatimeintervalisasetofinstants,eachofzero duration.TheDedekind–Cantortheoryissometimescalleda“point–set”account. Tobesure,Aristotledoesrecognizepoints,insomesense.Indeed,pointsplaya crucialroleinhisviewsoncontinuityandindecomposability.ForAristotle,pointsare thelimits,orendpoints,oflinesegments.4 Thepointsinteriortoalinesegmentexist onlypotentially,notactually.Aninteriorpointrepresentsaplacewhereasegment couldbe divided.Sotobisectalinesegmentistoindicateaplacewhereitcould bebroken.Butunlessthesegment is dividedthere,thepointdoesnotexist—itis notactual.
This,wetakeit,isthemainfeatureofAristotle’sresponsetoZeno’sparadox ofDivision.Supposethatsomeone,Ms.Walker,wishestocoveradistanceof,say, 100meters.Shemustfirstwalkthefirst50meters.CallthatWalk1.Thenshemust walkthenext25meters.Walk2.Thenthenext12.5meters.Walk3.Andonitgoes. Soinordertoreachherdestination,Ms.Walkermustcompleteaninfinitenumberof Walks.Zenoconcludesthatshecannotdoso,and,thus,thatmotionisimpossible. Aristotle,ofcourse,followscommonsense,anddisagreeswiththeconclusion. Motionisindeedpossible(presumablybecauseitisactual).Aristotle’sviewsonpoints takeonmuchoftheburdenofexplainingwhatgoeswrongwithZeno’sargument.
3 White’sprincipleofnon-superveniencepresupposesthatcontinuousmagnitudes(suchaslinesegments) can bepartitionedintoproperparts.Thisrunsagainstalong-standingthemethatacontinuous magnitude cannot bedivided,cleanly,intoproperparts.Asweshallsoonsee,Aristotlehimselfheldaversionofthis“indecomposability”.Accordingtocontemporaryintuitionism,continuaareindecomposable inaratherstrongsense.SeeChapter8.
4 Also,forAristotle,linesarejusttheedgesofplanefigures,andplanefiguresarejustthefacesof three-dimensionalphysicalobjects.
oldorthodoxy(aristotle)vsnew(dedekind–cantor)
ConsidertheendpointofWalk1,apoint50metersfromthestartofthejourney.For Aristotle,thatrepresentsaplacewhereMs.Walker canpause onherjourney,either temporarilyorpermanently.Thereareinfinitelymanyplaceswhereshecouldstop. Butunlessshe does stopthere(orsomeplaceelse),theresimplyisnoseparateWalk1 toconsider.Thereisjustasingleevent,herwalkoftheentire100meters.Thatwalk isnotthesumofWalk1,Walk2, ...
Wedonotclaimthatthisresolveseveryversionoftheparadox.Aristotledoes notconsideraso-called“staccato”scenariolikethefollowing:Ms.Walkertakes 30minutestocompleteWalk1,andthenpausesfor15minutes.Thenshetakes 7.5minutesforWalk2,andpausesfor3.75minutes.Thenshetakes1.875minutes forWalk3andpausesforhalfofthat,etc.Itstilltakes60minutestocoverthefull 100metersand,byAristotle’sownlights,Ms.Walker does havetocompleteaninfinite numberofwalksfirst.5
Tosummarize,itseemsthat,forAristotle,linesegmentshaveactualendpoints— butsuchendpointsaredecidedlynot part ofthelinesegment.Theyarejustits boundaries.Eachendpointismetaphysicallytiedtothesegmentitbounds,and cannotbeconsideredinisolationfromthesegment,inthesamesenseasthesmile oftheCheshirecatcannotbeconsideredapartfromthecat.Theinteriorpointsona linesegmentexistonlypotentially.Thereisapotentialinfinityofsuchinteriorpoints, butnotanactualinfinityofthem(seeLear[70]).
In Physics 8.8(263bg3–6),wefind:
Sothatifsomeoneaskswhetheryoucantraverseaninfinityeitherintimeorinlength,we mustsaythatinawayyoucan,andinawayyoucannot.Foryoucannottraverseaninfinityof actuallyexisting[divisions]butyoucanofpotentiallyexistingones. (quotedinSorabji[116],213)
Wewillturntomattersofinfinitysoon.
.Succession,Contiguity,andContinuity
Inthe Categories,Aristotledistinguishesdiscretefromcontinuousentities:
Discretearenumberandlanguage;continuousarelines,surfaces,bodies,andalso,besides these,timeandplace.Forthepartsofanumberhavenocommonboundaryatwhichtheyjoin together Norcouldyoueverinthecaseofnumberfindacommonboundaryofitsparts,but theyarealwaysseparate.Hencenumberisoneofthediscretequantities.Similarly,languagealso isoneofthediscretequantities Foritspartsdonotjointogetheratanycommonboundary. Forthereisnocommonboundaryatwhichthesyllablesjointogether,buteachisseparatein itself.Aline,ontheotherhand,isacontinuousquantity.Foritispossibletofindacommon boundaryatwhichitspartsjointogether,apoint.(4b22)
5 Itisunclearwhetherthis“staccato”scenariowouldbeatallrelevanttothedebateatthetime.Zenowas arguingthatmotion(asaninstanceofchange)isimpossible.Nooneintheancientdebatewouldregard the“staccato”walkaspossible.ThankstoSarahBroadiehere.See[127],53–72and[70].
oldorthodoxy(aristotle)vsnew(dedekind–cantor)
Number,here,means“positivenaturalnumber”.ForAristotle,naturalnumbersare always numbersof somethings,treatedasdiscrete.And“line”iswhatwewouldcalla linesegment.ForAristotle,whatmakeslinesegments“continuous”(witheachother) isthatwhentheyjointogether,thereisacommonboundary,namelyapoint.
InBook5ofthe Physics,Aristotleintroducesacommon-sensenotion:
Athingisinsuccessionwhenitisafterthebeginninginpositionorinform andwhen furtherthereisnothingofthesamekindasitselfbetweenitandthattowhichitisinsuccession, e.g.alineorlines,ifitisaline,aunitorunitsifitisaunit,ahouseifitisahouse.(226b34)
Tofollowoneofhisexamples,thinkofthehouses(afterthefirst)inarowon agivenstreet.Theyareinsuccessionbecausetherearenohousesbetweenany consecutivetwo.
Aristotledefinestwothingstobe contiguous,or incontact,iftheyareinsuccession, insuchawaythatnothingofthesamekind cangobetween:“Thingsaresaidto beincontactwhentheirextremitiesaretogether”(226b21).Thinkofapairof adjacentbooksonatightlypackedshelf.Thoseare(orappeartobe)insuccession andcontiguous(i.e.incontact).
Hegoesontodefineanothernotion:
Thecontinuousisjustwhatiscontiguous,butIsaythatathingiscontinuouswhenthe extremitiesofeachatwhichtheyareincontactbecomeoneandthesameandare(asthe nameimplies)containedineachother.Continuityisimpossibleiftheseextremitiesaretwo. Thisdefinitionmakesitplainthatcontinuitybelongstothingsthatnaturally,invirtueoftheir mutualcontactformaunity.(227a6)
Sotheaforementionedbooksontheshelfarenotcontinuous,sincenomatterhow closetoeachothertheyget,eachretainsitsownboundaries.Whencontiguous,they remain twobooks,eachwithitsownboundary.Incontrast,whencontinuousthings, suchaslinesegments,arebroughtintocontact,thecommonboundarybetweenthem isabsorbed,andtheybecomeasingleline,a“unity”.Foramorephysicalexample, thinkoftwobodiesof(liquid)water,orconsiderwhathappenswhenoneleavestwo loavesofwetdoughtorisebeforebeingbakedintobread.Iftheloavesareplacedtoo closetogether,whentheycomeintocontactwitheachother,theywillabsorbeach other,andmergeintoasingleloaf—withnodiscernibleboundarybetweenthem.
ItiscuriousthatAristotledefinescontinuityasa relation betweenthings:they arecontinuous(ornot) witheachother.Yethealsospeaksofcontinuous things, suggestingthatcontinuityisa(monadic)property.Perhapstheideaisthatasubstance iscontinuousifwheneverneighboringpartsareconsidered,theyarecontinuous witheachother.Orperhapscontinuousthingsonlyhavepartspotentially.Think,for example,ofalimbofatree.Tobreakitupintopartswouldbetodestroyit,oratleast todestroyitsunity.
Inthislastpassage,Aristotlesoundsathemethatplaysaroleinthesubsequent developmentofcontinuity,upto,butnotincluding,theadventoftheDedekind–Cantoraccount.Acontinuousthing,forAristotle,formsa unity.Thereissomething
oldorthodoxy(aristotle)vsnew(dedekind–cantor)
thatbindsittogether,andmakesitOnething.Inaslogan,continuoussubstancesare viscous.6 Thecontemporary,Dedekind–Cantorcontinuaarenotlikethis.Aline,for example,isjustasetofpoints.Andasetcanbebrokencleanlyandarbitrarily(atleast assumingclassicallogic)intoitsvarioussubsets.
InBook6of Physics,Aristotlesummarizesthedefinitions,andsoundsathemethat motivatesthepresentproject:
Nowiftheterms“continuous”,“incontact”,and“insuccession”areunderstoodasdefined above—thingsbeing“continuous”iftheirextremitiesareone,“incontact”iftheirextremities aretogether”,and“insuccession”ifthereisnothingoftheirownkindintermediatebetween them—nothingthatiscontinuouscanbecomposedofindivisibles:e.g.alinecannotbe composedofpoints,thelinebeingcontinuousandthepointindivisible.(231a1)
... sinceindivisibleshavenoparts,theymustbeincontactwithoneanotheraswholewith whole.Andiftheyareincontactwithoneanotheraswholewithwhole,theywillnotbe continuous:forthatwhichiscontinuoushasdistinctparts ... aswesaw,nocontinuousthing isdivisibleintothingswithoutparts ... Moreover,itisplainthateverythingcontinuousis divisibleintodivisiblesthatareinfinitelydivisible.(231b)
In Metaphysics 3.5(1002a28–b11),Aristotleconnectsthiswithhisviewsonpoints: Forassoonasbodiescomeintocontactoraredivided,theboundariessimultaneouslybecome oneiftheytouchandtwoiftheyaredivided.Hence,whenthebodieshavebeenputtogether, oneboundarydoesnotexist,buthasceasedtoexist,andwhentheyhavebeendivided,the boundariesexistwhichdidnotexistbefore(forthepoint,beingindivisible,wasnotdivided intotwo).(quotedin[116],11)
Itistheseviewsoncontinuityandboundaries—pointsinparticular—thatunderlie theAristotelianthesisofindecomposability.ForAristotle,thereasononecannot breakacontinuousobject,suchasalinesegment,cleanlyintopiecesisthatdoing so creates,ormakesactual,theboundariesoftheresultingsegments.Somethingnew comesintobeing(orintoactuality).
Othertheoriesofcontinuity,suchasintuitionisticanalysisandsmoothinfinitesimalanalysisshowamuchstrongersenseofindecomposability(see,forexample, Dummett[39]andBell[15],andChapter8forabriefsummary).Inthosetheories, itisnotthecasethatanintervalcanbebrokenupintopieces,cleanly,withnothing leftout.Anyattempttobreakupasegment willleavesomethingout.Themetaphoris thatwhenyoucutalinesegment,somethingwillsticktotheknife.
Wenowturntowhatisprobablythemostfar-reachingdifferencebetweenAristoteliancontinuaandthecontemporaryDedekind–Cantoraccount.Beginningwith Aristotle,almosteverymajorphilosopherandmathematicianbeforethenineteenth
6 ThankstoMarcoPanza,SarahBroadie,KerenWilson,CarlPosy,andBarbaraSattlerhere.
oldorthodoxy(aristotle)vsnew(dedekind–cantor)
centuryrejectedthenotionoftheactualinfinite.Theyallarguedthattheonlysensible notionisthatofpotentialinfinity—atleastforscientificor,later,non-theological purposes.In Physics 3.6(206a27–9),Aristotlewrote,“Forgenerallytheinfiniteisas follows:thereisalwaysanotherandanothertobetaken.Andthethingtakenwill alwaysbefinite,butalwaysdifferent.”AsRichardSorabji[116](322–3)putsit,for Aristotle,“infinityisanextendedfinitude”(seealso[70],[71],[81]).
Aristotle,alongwithancient,medieval,andearlymodernmathematicians,recognizedtheexistenceofcertain procedures thatcanbeiteratedindefinitely,withoutlimit. Examplesarethebisectionandtheextensionoflinesegments.Ancientmathematiciansmadebrilliantuseofsuchprocedures.Forexample,themethodofexhaustion, akindofforerunnertointegration,wasemployedtocalculatetheareasofcurved figuresintermsofrectilinearones.
Whatwasrejectedarewhatwouldbetheendresultsofapplyingtheseprocedures infinitelyoften:self-standingpoints,infinitelylongregions,andinfinite(or,inmodernterms,transfinite)sets.This,ofcourse,isconnectedtoAristotle’sviewsonpoints. If,contrarytoAristotle(andcontrarytotheradicalatomists),wearetothinkofa linesegmentasacollectionof(actuallyexisting)points,thenthatcollectionisindeed infinite—actuallyinfinite.Aristotle,ofcourse,rejectstheveryexistenceofactually infinitecollectionsandthethesisthatlinesarecomposedofpoints.
In Ongenerationandcorruption,hewrote:
For,sincenopointiscontiguoustoanotherpoint,magnitudesaredivisiblethroughand throughinonesense,andyetnotinanother.When itisadmittedthatamagnitudeis divisiblethroughandthrough,itisthoughtthatthereisapointnotonlyanywhere,butalso everywhereinit:henceitfollowsthatthemagnitudemustbedividedawayintonothing.For thereisapointeverywherewithinit,sothatitconsistseitherofcontactsorofpoints.Butitis onlyinonesensethatthemagnitudeisdivisiblethroughandthrough,viz.insofarasthereis onepointanywherewithininandallitspointsareeverywherewithinitifyoutakethemsingly.
(317a3–8)
JonathanLear[70]arguesthatitisnottheexistenceofiteratedproceduresthat makesforAristotelianpotentialinfinity.Thematterconcernsthe structure ofgeometricmagnitudes:
itiseasytobemisledintothinkingthat,forAristotle,alengthissaidtobepotentially infinitebecausetherecouldbeaprocessofdivisionthatcontinuedwithoutend.Thenitis naturaltobeconfusedastowhysuchaprocesswouldnotalsoshowthelinetobeactually infinitebydivision. ... [I]twouldbemoreaccuratetosaythat,forAristotle,itisbecausethe lengthispotentiallyinfinitethattherecouldbesuchaprocess.Moreaccurate,butstillnot true ... Strictlyspeakingtherecouldnotbesuchaprocess,butthereasonwhytherecouldnot beisindependentofthestructureofthemagnitude:howeverearnestadividerImaybe,Iam alsomortal. ... evenatthatsadmomentwhentheprocessofdivisiondoesterminate,therewill remaindivisionswhichcouldhavebeenmade.Thelengthispotentiallyinfinitenotbecauseof theexistenceofanyprocess,butbecauseofthestructureofthemagnitude.(p.193)
oldorthodoxy(aristotle)vsnew(dedekind–cantor)
Learhighlightstheabovethemethat,forAristotle,alinesegmentispotentiallyinfinite becausethereareinfinitelymanyplaceswhereit canbe divided.So,nomatterhow manytimesonedividesaline,therewillstillbesomeofthelineleft.Learconcludes thatAristotle’sthesisis“thatthestructureofthemagnitudeissuchthatanydivision willhavetobeonlyapartialrealizationofitsinfinitedivisibility:therewillhavetobe possibledivisionsthatremainunactualized”(p.194).7
ForAristotle,linesthemselvesarealsoonlyoffinitelength.Thatis,Aristotelian linesarewhatarenowcalledlinesegments.Asnoted,eachsuchlinehastwo endpoints.Euclid,too,seemstoallowonlylinesegments,atleastforthemostpart. Thefirstpostulateis“Todrawastraightlinefromanypointtoanypoint”.This,of course,onlyproducessegments.Thesecondpostulateis“Toproduceafinitestraight linecontinuouslyinastraightline”.Theidea,itseems,isthatastraightlinecanbe extendedasfarasonewishes,butthereislittleornoneedtocountenanceinfinitely longlines.Thelanguageofpotentialextensionisinvokedexplicitlyinthe(in)famous fifthpostulate:“That,ifastraightlinefallingontwostraightlinesmaketheinterior angleonthesamesidelessthantworightangles,thetwostraightlines, ifproduced indefinitely,meetonthatsideonwhichtheanglesarelessthanthetworightangles” (emphasisadded,ofcourse).
Givenhisoverallphilosophyofmathematics,Aristotlemayhavehadaproblem witheventhe potential infinityoflinesegments,andthuswithEuclid’ssecond andfifthpostulates.Asnotedabove,Aristotleheldthatmathematicalentitiesare abstractedfromphysicalentities.Numbers,forexample,arenumbersofobjects,such asnumbersofcows,construedasindivisible.Linesaretheedgesofphysicalobjects, suchascubes.Aristotlealsoheldthattheentireuniverseisfiniteinextent.Itisa sphere.Sotherecannotbeanylinelongerthanthediameterofthatsphere(Physics, Book3,7:207b27).SeeKnorr[68],121–2,andHintikka[62].Wemakenoattempt torecapturethisaspectofAristotle’sviews.
Tosummarize,forAristotle,asforEuclid,linesarenotactuallyinfinite(i.e.not infinitelylong).ButforEuclid,atleast,linesarepotentiallyinfinite,inthesensethat anylinecanbeextended.
Asnoted,whenitcomestotheinfinite—oratleastinfinitecollections—viewslike Aristotle’swerestandardthroughoutthemedievalandearlymodernperiod,through mostofthenineteenthcentury.Thegreatestmathematicalmindsinsistedthatonly thepotentiallyinfinitemakessense.Leibniz,forexample,wrote:
Itcould wellbearguedthat,sinceamonganytentermsthereisalastnumber,which isalsothegreatestofthosenumbers,itfollowsthatamongallnumbersthereisalast number,whichisalsothegreatestofallnumbers.ButIthinkthatsuchanumberimpliesa
7 ItisgenerallyheldthatAristotle’sownviewsonmodalityarevexed.Wemakenoattempttosort thatout.
oldorthodoxy(aristotle)vsnew(dedekind–cantor)
contradiction ... Whenitissaidthatthereareinfinitelymanyterms,itisnotbeingsaidthat thereissomespecificnumberofthem,butthattherearemorethananyspecificnumber. (lettertoBernoulli,[72],III566,translatedin[76],76–7,87)8
... weconclude ... thatthereisnoinfinitemultitude,fromwhichitwillfollowthatthereis notaninfinityofthings,either.Or[rather]itmustbesaidthataninfinityofthingsisnotone whole,orthatthereisnoaggregateofthem.([73],6.3,503,translatedin[76],86)
YetM.Descartesandhisfollowers,inmakingtheworldouttobeindefinitesothatwecannot conceiveofanyendtoit,havesaidthatmatterhasnolimits.Theyhavesomereasonfor replacingtheterm“infinite”by“indefinite”,forthereisneveraninfinitewholeintheworld, thoughtherearealwayswholesgreaterthanothersadinfinitum.AsIhaveshownelsewhere, theuniversecannotbeconsideredtobeawhole.([74],151)
AndGauss[47]:
Iprotestagainsttheuseofinfinitemagnitudeassomethingcompleted,whichisneverpermissibleinmathematics.
ForGaussandLeibniz,asforAristotle,asforahostofothers,theinfinitejust is thelimitlessnessofcertainprocesses;noactualinfinitiesexist.Theonlyintelligible notionofinfinityisthatofpotentialinfinity—thetranscendenceofany(finite)limit.9
Foratleastthecasesofinteresthere—regions,naturalnumbers,andthelike— GeorgCantorarguedfortheexactoppositeofthis,claimingthatthepotentially infiniteisdubious,unlessitissomehowbackedbyanactualinfinity: Icannotascribeanybeingtotheindefinite,thevariable,theimproperinfiniteinwhatever formtheyappear,becausetheyarenothingbuteitherrelationalconceptsormerelysubjective representationsorintuitions(imaginationes),butneveradequateideas.([27],205,n.3)
... everypotentialinfinite,ifitistobeapplicableinarigorousmathematicalway,presupposes anactualinfinite.([28],410–11)
WethinkitsafetosaythatthisCantorianorientationisnowdominantinthe relevantintellectualcommunities,especiallyconcerningthemathematicaldomains mentionedabove,withvariousconstructivistsasnotableexceptions(staytuned).
.Plan
Asnotedabove,thenoworthodoxDedekind–Cantoraccountofcontinuitymakes twomajordeparturesfromtheAristoteliantradition.Ittakescontinuatobecomposedofpoints,anditmakes(heavy)useofactualinfinity,àlaCantor.Oneofthe
8 The“contradiction”mentionedheremightbethe so-called“Galileoparadox”,thatwithinfinite collections,apropersubsetcanbeequinumerouswithaset.This,ofcourse,isnowastandard feature ofinfinitesets;afeatureandnotabug.
9 Tobesure,Leibniz,Gauss,andahostofotherswerenotcompletetelyconsistentonthis.Theywere, afterall,pioneersintheemergenceofmodernmathematics,nottomentiontheiruseofinfinitesimals. AreaderremindsusthatwhileLeibnizrejectedinfinite number,hewas,attimes,achampionofactually infinitetotalities.Theexegeticalissuesare(well)beyondthescopeofthiswork.
oldorthodoxy(aristotle)vsnew(dedekind–cantor)
mainconclusionsofthisstudyisthat,ofthesetwo,theemergenceofinfinityisfar moresignificant.Thematterofwhethercontinuaarecomposedofpointsis,atmost, aconvenience.
Chapter2developsa“semi-Aristotelian”accountofaone-dimensionalcontinuum. UnlikeAristotle,itmakessignificantuseofactualinfinity,inlinewithcurrent practice,ofcourse.LikeAristotle,however,thisaccountdoesnotrecognizepoints, atleastnotaspartsofregionsinthespace.Theformalbackgroundisclassical mereologytogetherwitheither(aweak)settheory,higher-orderlogic(understood with“standardsemantics”),oralogicofpluralquantification.Thefirst-ordervariables rangeoverregionsofthespace,andthereisanaxiomthatentailsthateveryregion hasaproperpart.This,ofcourse,isinconsistentwiththeexistenceofpoints(atleast withpointsaspartsofregions,astheyareinordinary,Dedekind–Cantorspaces).
Incontrasttointuitionisticanalysis,smoothinfinitesimalanalysis,andErrett Bishop’s[19]constructivism,weshowthatitispossibletopartitionour“gunkyline” intomutuallyexclusiveandexhaustivediscreteparts(e.g.tocleanlybisectintervals). Thisdemonstratestheindependenceof“viscosity”and“indecomposability”froma nonpunctiformconceptionofthecontinuum.
ItissurprisingthatsuchsimpleaxiomsasoursimplytheArchimedeanproperty andanintervalanalogueofDedekindcompleteness(least-upper-boundprinciple). Thisiswithoutan“extremalaxiom”(totheeffectthat“thesearetheonlywaysofgeneratingregions”),aprinciplethat,ineffect,justimposescompleteness.Wealsoestablish anisomorphismwiththeDedekind–Cantorstructureoftherealline R asacomplete, separable,orderedfield,andshowhowtodefine“points”asasortofWhiteheadian “extensiveabstraction”.A“point”,soconstrued,isanequivalenceclassofcertainsets ofregions.The“points”thusdefinedareisomorphictothestandardrealline.
Wealsopresentsomesimpletopologicalmodelsofoursystem,establishingconsistencyrelativetoclassicalanalysis.Indeed,ourgunkytheoryandthemorestandard Dedekind–Cantoronearemutuallyinterpretableand,indeed,definitionallyequivalent.So,atleastforaone-dimensionalcontinuum,theexistenceofpoints,aspartsof regions,isnotnecessary.
Tobesure,oursisnotthefirstattemptatarigorouspoint-free(orpointless), regions-basedaccountofthecontinuous,althoughwethinkithassomeinteresting features.Weprovidecomparisonswithsomeearlieraccounts(althoughother comparisonsarepostponeduntilafterwedevelophigher-dimensionalversionsof thetheory).
Chapter3,coauthoredwithØysteinLinnebo,presentsamoreAristoteliantheory, onethateschewsboththeexistenceofpoints(aspartsofregions)andtheuse ofactualinfinity.Infinityentersthesemi-Aristoteliantheory,ofChapter2,with ourmereologicalprincipleofunrestrictedfusion.Indevelopingthattheory,we sometimestakethefusionofanexplicitlyinfinitesetofregions.Thisoccurs,for example,intheproofoftheArchimedeanpropertyandintherecoveryofthestandard Dedekind–Cantorrealline.Inotherplaces,wefusesomeregionswithoutbothering
tocheckhowmanyregionsaresofused.WetakethisplaywithinfinityinChapter2 tobe“actual”,sincethe(actual)existenceofa single region—apartofthespace—is establishedwitheachfusion.
ThestraightforwardAristotelian“fix”istoreplacetheunrestrictedfusionprinciple withoneallowingthefusionofanytwo(andthusanyfinitenumberof)regions.The resultingtheory,however,isextremelyweak,failingtoprovetheexistenceofmeets, differences,bisections,andbiextensions,andalsofailingtoproveanArchimedean property.Sothoseareaddedasadditionalaxioms.Itisperhapsinterestingthatadding anotablyunAristotelianprincipleofunrestrictedfusionallowsonetoproveseveral theoremsthatare,infact,legitimatefortheAristotelian.
BuildingonsomepreviousworkbyLinnebo,wethenshowhowtotakethe “potential”natureoftheusualoperationsseriously,byusingamodallanguage (thusdown–playingthefusionprinciple),andweshowthatmodalandnon-modal approachesareequivalent.Wethinkthatthemodalapproachbettercapturesthe matterofpotentiality(butwemakenoattempttoexplicate,letaloneconformto,the oftenvexedaccountofmodalityproposedbyAristotlehimself).
Thechapterconcludeswithapredicativeaccountofthecontinuum.Intuitively, thattheoryliesbetweenthesemi-AristotelianaccountofChapter2andthe moreAristotelianaccounthere.Thepredicativistacceptstheexistenceofsome infinitecollections(orpluralities)ofregions,thosethatcanbedefinedina predicativelyacceptableway.Theusualrangeofissuesconcerningpredicativeanalysis re-emergehere.
Chapter4isanattempttorecover“points”and(somethinglike)realnumbersin themoreAristotelianframeworkpresentedinChapter3,alongsimilarlinestothe analogousdevelopmentinthe“semi-Aristotelian”Chapter2.Gettinganaloguesofthe rationalnumbersisstraightforwardenough,but,wethink,thatexhaustsAristotelian resources.
Asnotedabove,Aristotleand,justabouteveryoneelseuntilthenineteenthcentury, rejectedtheexistenceoftheactualinfinite.Ancient,medieval,andearlymodern mathematiciansworkedwithspecificproceduresthatcanbeiteratedindefinitely often,thusintroducingpotentialinfinities.Whattheydidnothave,sofaraswe know,isa theoryof potentiallyinfinitesetsorsequences.Toputthematterabit anachronistically,theancient,medieval,andearlymodernmathematiciansdidnot countenanceboundvariablesthatrangeoverpotentiallyinfiniteentities.
Thathadtowaitfortheadventofintuitionisticandconstructivemathematics,in thetwentiethcentury.Theplanforthischapteristofollowtheirleadanddevelop a“superstructure”of“points”and“realnumbers”onthegunky,fullyAristotelian framework,understoodintermsofpotentiallyinfiniteprocesses.Itturnsoutthat, unliketheforegoingsemi-Aristotelianaccount,thepunctiformsuperstructurehere isverydifferentfromtheunderlying,gunkyspace—wedonotfindanythinglike mutualinterpretabilitybetweenthegunkyandmorepunctiformtheories.Inmostof thesystems,wereproducetheconstructiveresultthatallfunctionsonrealnumbers
oldorthodoxy(aristotle)vsnew(dedekind–cantor)
arecontinuous.Itfollowsthatthepunctiformsuperstructurehasastrongformof indecomposability:itisnotpossibletodividean“interval”cleanlyintotwoparts. Thisisincontrastwiththeunderlying,Aristoteliangunkyframeworkwhichboastsa principleofbisection.
Again,inthesemi-Aristotelianframework,thepunctiformsuperstructureisisomorphictotheDedekind–Cantorrealnumbers,andtheDedekind–Cantortheory ismutuallyinterpretableand,indeed,definitionallyequivalenttothegunkysemiAristotelianone.Wetakethisdifferencebetweenthesemi-Aristotelianandthemore Aristotelianframeworkstobesignificant.This,too,highlightsthesignificanceofthe actualinfinite,especiallywhenconstruedimpredicatively.
Thenextorderofbusinessistopresenthigher-dimensional,regions-basedtheories (fromthesemi-Aristotelianperspective).Theone-dimensionalcaseissimplified somewhatbythefeaturethatthereareonlytwo“directions”fortheline,whichwe label“left”and“right”.Atwo-dimensionalspaceismuchricherhaving,intuitively, infinitelymany“directions”.
Chapter5developsaEuclidean,two-dimensional,regions-basedtheory.Aswith thesemi-AristotelianaccountinChapter2,thegoalhereistorecoverthenow orthodoxDedekind–Cantorcontinuumonapoint-freebasis.
WefirstderivetheArchimedeanpropertyforaclassofreadilypostulatedorientationsofcertainspecialregions,whatwecall“generalizedquadrilaterals”(intended asparallelograms),bywhichwecovertheentirespace.Thenwegeneralizethisto arbitraryorientations,andthenestablishanisomorphismbetweenthespaceandthe usualpoint-based R × R.Asintheone-dimensionalcase,thisisdoneonthebasisof axiomswhichcontainnoexplicit“extremalclause”,andwehavenoaxiomofinduction otherthanordinarynumerical(mathematical)induction.Afterexplicitlydefining “point”and“line”,asextensiveabstractions,wederivethecharacteristicParallel’s Postulate(Playfairaxiom)fromourregions-basedaxioms,andpointthewaytoward derivingkeyEuclideanmetricalproperties.
Theextensionfromtwotothreeandhigherdimensionsisfairlystraightforward. Wesketchhowitcanbeaccomplished.
Chapter6adaptstheforegoingresultstopresenttwonon-Euclideantheories,both inlinewiththe(semi-)Aristotelianthemeofrejectingpoints,aspartsofregions (butworkingwithactualinfinity).Thefirsttheoryisatwo-dimensional hyperbolic space,thatis,onethathasanegativeconstantcurvature.Thisspacearises,inthe standardpoint-basedtreatments,fromadoptingtheEuclideanaxiomsfortheplane except,ofcourse,theParallel’sPostulate,whichisreplacedbyanaxiomstating thatforany“line” l and“point” p,thereismorethanone“line”(usuallyinfinitely manysuch“lines”) l through p thatnevermeets l.Furthermore,theanglesumsof polygonsarelessthantheirEuclideancounterparts,withthedifference(“defect”) shrinkingasthepolygonsconsideredgetsmaller,approaching,butneverreaching, theEuclideanvalues.UnliketheEuclideancase,the“shrinking”polygonscannot besimilar.
Oursecondtheorycapturesaspaceofconstant positive curvature,atwodimensional spherical geometry.Thetaskhereistoformulateaxiomsonregions whichallowustoprovethat(i)therearenoinfinitesimalregionsand(ii)thatthere are no parallelstoanygiven“line”throughany“point”notonthegiven“line”.
Inbothcases,ofcourse,wehavetodefine“point”and“line”,since,inlinewith thethemeofthebook,regionsdonotcontainpointsorlines,asparts.Ouraxioms, anddefinitions,implytheotherEuclideanaxioms,except,ofcourse,thestatement thatanytwopointsdetermineauniqueline.Inthesphericalcase,theexceptionsare “antipodalpoints”.
Chapter7turnstomattersofmetaphysics.Someanalyticmetaphysicianshave occupiedthemselveswiththenature(orthepossiblenature)ofspaceandtime(or space–time),andwiththerelationshipbetweenphysicalobjectsandtheregionsof spaceorspace–timetheyoccupy.Someoftheissuesconcerntheboundariesof objectsandthenotionofcontact.Much,butnotall,ofthemetaphysicalliteraturejust assumes,withoutcomment,thattheunderlyingspaceorspace–timeispunctiform, followingthenoworthodoxDedekind–Cantorpicture.Thatis,manywritersassume thatspaceorspace–timeiscomposedofpoints—itisstructuredlike R3 or R4 Physicalobjectsthusoccupyregionsthatarenothingbutsetsofpoints.Manyofthe authorsgettangledupoverwhattosayaboutpoint-sizedregions,orevenpoint-sized objects.Ourfirstgoalinthischapteristogiveasomewhatbiasedoverviewofaportion ofthisliterature,arguingthatmanyoftheissuesaremucheasiertonegotiateifwe assumearegions-basedspaceorspace–time.
Somemetaphysiciansclaimthatphysicalobjectscanfullyoccupyaregion—asetof points—onlyiftheregionisregularopen(orregularclosed).Thosearethe“regions” inthemaintopologicalmodelsofourowntheories.Sothosemetaphysicianswould haveexactlythesamepictureiftheybeganwitharegions-basedaccountofspace,or space–time,likeours.
Wethenturntosomeapparentlimitationsofoursemi-Aristotelianaccountsof spaceorspace–time.Forexample,thenaturalanalogueofLebesguemeasureis notcountablyadditive(althoughitisfinitelyadditive),andthereseemstobeno straightforwardwaytoaccountforcontinuousvariationinourframeworks—other thanbyjustintroducing“points”via“extensiveabstraction”,andworkingwiththose (asintheDedekind–Cantorpicture).Alsodiscussedisthetopicofdifferentialgeometry,ofcentralimportanceinmodernspace–timephysics.Ratherthanattempting adirect,regions-basedtheorymathematicallyequivalenttostandarddifferential geometry(say,semi-Riemannianfour-dimensional),wepointoutthatourreduction ofpunctiformrealanalysisalreadyiscapableofincorporatingsuchatheory,despite itssweepingcomplexity.Concludingthissection,wediscusstheextenttowhichour responsesinthefaceofthesechallengesrepresentshortcomingsoftheregions-based, gunkyapproaches.
Wethenbroachthequestionofadjudicatingwhetherspaceorspace–timereally ispunctiform.Thetightconnectionbetweenourregions-based,gunkytheoriesand
oldorthodoxy(aristotle)vsnew(dedekind–cantor)
themorestandardDedekind–Cantorpunctiformtheoriesindicatesthatspaceor space–timecanbedescribed,completelyandadequatelyeitherway.Ifonetheoryis adequate,thensoistheothertoexactlythesameextent.Thissuggeststhat,given thecontemporarynotionoftheactuallyinfinite,a“dispute”betweenanadvocateof pointsandonewhorejectspointsissomethinglikeaCarnapian,externalquestion. Wesupportthiswithatreatmentofverbaldisputes.
Ourstudyconcludeswithabriefsketchofdifferentaccountsofcontinuity:punctiform,intuitionistic,smooth,predicative,etc.,andindicatestheextenttowhich eachcapturesseverallong-standingfeaturesthathavebeenattributedtocontinuous entities.Noonetheorycapturesthemalland,indeed,noonetheorycancapture themall.Theleadingintuitivepropertiesofthecontinuousexhibittensionswithone another,and,wesubmit,pertaintoaconceptthatismoreupforsharpeninginvarious incompatibleways,ratherthanonethatwouldyieldtomoretraditional,univocal philosophicalanalysis.
