Varieties of Continua: From Regions to Points and Back

219 Pages·2018·3.8 MB·English

by  Geoffrey Hellman

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Varieties of Continua
OUPCORRECTEDPROOF – FINAL,30/12/2017,SPi
OUPCORRECTEDPROOF – FINAL,30/12/2017,SPi
Varieties of Continua
From Regions to Points and Back
Geoffrey Hellman and
Stewart Shapiro
1
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3
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Contents
Preface vii
. TheOldOrthodoxy(Aristotle)vstheNewOrthodoxy
(Dedekind–Cantor) 
. TheClassicalContinuumwithoutPoints 
. AristotelianandPredicativeContinua 
WithØysteinLinnebo
. RealNumbersonanAristotelianContinuum 
. Regions-basedTwo-dimensionalContinua:TheEuclideanCase 
. Non-EuclideanExtensions 
. TheMatterofPoints 
. Scorecard 
References 
Index 
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Preface
This work began several years ago with a series of discussions on competing con-
ceptions of continua in mathematics, informed by late twentieth-century develop-
ments,suchasRobinsoniannon-standardanalysisandthemoreradical,non-classical
smoothinfinitesimalanalysis,thatrehabilitatedearliernotionsofinfinitesimals.At
firstourplanswererathermodest—todoacomparativestudyofleadingapproaches
tothecontinuous,assessingthetrade-offsinvolved,inamoresystematicandcom-
prehensive way than what was already available in the literature. In particular, we
were struck by the historical transition from the long-standing, Aristotelian non-
punctiform conception to modern punctiform analysis based on the method of
limits due to Bolzano, Cauchy, Weierstrass, Cantor, Dedekind, et al., which rapidly
becamethedominantmainstreamapproach.Alongwiththecontrastbetweennon-
punctiform, or regions-based, conceptions and the modern view of continua as
entirely composed of points, there is also the contrast between potential infinity—
againgoingbacktoAristotle—andtherecognitionofactualinfinitiesinmainstream
modernmathematics.Herewewerestruckbythecircumstancethatrestrictionsto
intuitionistic logic, also associated with potential infinity, have been motivated in
radicallydifferentways,ontheonehandbydemandsofconstructivity(byBrouwer,
Bishop,etal.),and,ontheother,byrestrictionsto“smoothmanifolds”(asinsmooth
infinitesimalanalysis(SIA),especiallyasdescribedbyJohnBell),althoughthislatter
way of motivating the restriction has been challenged by one of us. Also, we were
motivated to assess intermediate, semi-constructive systems of predicative analysis
(going back to Russell, Poincaré, Weyl, and perfected in work of Feferman et al.),
basedonclassicallogic,recognizingactualinfinitybutonlyofpredicativelyjustifiable
countabletotalities.
Given this variety of systems and motivating ideas, it seemed clear to us that
“continuity”isacluster-concept,realizableinmultiple,oftencompetingways,andthat
nosinglesystemcoulddojusticetoallthedesiderataemergingfrommathematical
andscientificpractice,pastandpresent.
Itthenoccurredtousthatitoughttobepossibletoavoidtheshifttointuitionistic
logic,alongwiththeapparatusofnil-potentinfinitesimals,whilestillbasinganalysis
and geometry on the idea of regions or “chunks” of a given space or manifold,
eschewingpoints,atleastaspartsofregions.Couldnotoneremain“classical”inone’s
prooftheoryandtreatmentoftheinfinitewhilestillrealizingathoroughgoingnon-
punctiformconceptionofcontinua?Wewere,ofcourse,alreadyfamiliarwithideas
ofWhiteheadandothersonrecovering“points”asdefinedbysuitablynestedregions
ofaspace.Wewere,however,surprisedathowlittlehadbeendoneinworkingout
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viii preface
suchverynaturalideasrigorously,especiallybydeployingtheresourcesofatomless
mereology,whichseemswell-suitedforsuchaproject.Inparticular,Tarskihadbegun
suchaprojectinearlyworkwithLe´sniewski,supplementingmereologywiththesole
primitive“sphere”inordertorecoverclassical,three-dimensionalgeometry.Buton
closerexaminationwerealizedthatTarski’singeniousdefinitionswerenotsupplied
withautonomousaxioms,beyondthoseofmereology,onthemathematicalprimitive,
“sphere”, framed directly in terms of regions, so that the project remained in an
incompletestate.
Sowesetourselvesthetaskofcompletingsuchwork,butbeginningwiththesimple
caseofacontinuumofonedimensionandthentryingtoextendsuchatreatmentto
higher-dimensionalspaces.Thisinturninvolvedusinseekingtorecover,onthesame
basisofregions,metricalstructuresofEuclideanandthennon-Euclideangeometries,
ultimatelyresultinginthecorepositivedevelopmentsofthepresentwork.
Itwascleartousfromtheoutsetthatthereisastronghistoricaldimensiontoour
project,rootedasitisinideasgoingbacktoAristotle.Indeed,wecalloursystemfor
a one-dimensional regions-based (“gunky”) continuum “semi-Aristotelian”, since it
adheresstrictlytoAristotle’sdesideratumofnotbeingconstitutedofpointswhileat
thesametimemakingfreeuseofactualinfinities,bothgeometricandset-theoretic
(equivalentlydescribedvialogicofplurals).Thisinturnnaturallyledustoaskhowwe
mightformulatemorefullyAristoteliantheoriesofcontinua,enforcingrestrictionto
potentialinfinitiesonly.AsØysteinLinnebohadalreadyworkedoutformalismbased
onmodallogicforexpressingpotentialinfinity,wepursuedathree-waycollaboration
withhimforachapterdevotedtothistopic.
At this stage, then, we had developed systems of analysis and geometry that are
“semi-Aristotelian”, non-punctiform but “classical” with respect to actual infinities,
andwehad“moreAristotelian”systemsthatwerebothnon-punctiformandconfined
tothepotentiallyinfinite.Thatleftthefurtherpossibilityofpredicativesystems,lying
betweenthemoreAristotelianandoursemi-Aristotelianones,basedonregions,using
classical logic (without modality), but admitting only countably infinite totalities
(correspondingtowell-foundedsettheorywithoutthepower–setsaxiom).Thedevel-
opmentofsuchpredicativesystemsthenroundsoutourformalworkpresentedhere.
Wethenturnedourattentiontodiscussingsomeoftheinterestingphilosophical
implications of the formal developments, including the systematic comparisons of
thedifferentapproachesthatmotivatedourprojectinthefirstplace,butalsoimpli-
cationsfor various discussionsof contemporary analyticmetaphysics pertaining to
applicationsofgeometryandanalysistomaterialobjectsofthenaturalworld,such
ashowtounderstandormodel“contact”ofmaterialobjects,whethertoadmitpoint-
likeobjects,andsoforth.Asexplainedbelow,manyofthesediscussionserrinsimply
assuming that the geometry being applied pertains to punctiform spaces, whereas
variousconundraevaporateinthecontextof“gunky”spaces.
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preface ix
Finally,toroundoutthisvolume,wereturntoouroriginalthemeofcomparisons
andtrade-offsamongthediverseconceptionsofcontinua,nowincludingtheregions-
basedsystemsconstructedhere.
We incurred many debts along the way. In the initial development of the one-
dimensional,semi-Aristoteliansystem,wewerehelpedbyRoyCook,aformerstu-
dent of one of us and a current colleague of the other, and by graduate students
at our home institutions, notably Shay Logan at Minnesota and Patrick Reeder
at Ohio State. Either together or separately, we gave presentations of early ver-
sions of some of this material at a variety of conferences, workshops, and collo-
quia, and were enlightened by comments from the audiences. The list includes the
UniversityofMinnesota,theOhioStateUniversity,severalmeetingsoftheMidwest
Philosophy of Mathematics Workshop, held annually at Notre Dame (and once in
Illinois),theFoundationsInterestGroupattheMinnesotaCenterforPhilosophyof
Science,Logica,thePhilosophyofScienceAssociation,theUniversityofSheffield,the
University of Oslo, the University of California at Davis, the Hebrew University
of Jerusalem, the University of Connecticut, the University of Southern California,
and the University of St Andrews. Thanks to anonymous reviewers from Oxford
UniversityPress,PhilosophiaMathematica,LogicandLogicalPhilosophy,Iyyun,and
theReviewofSymbolicLogic.ThanksalsotoFrankArntzenius,JeremyAvigad,John
Bell,EinarDuengerBohn,SarahBroadie,JohnBurgess,BenCaplan,TimCarlson,
Aaron Cotnoir, Philip Ehrlich, Thomas Forster, James Franklin, Harvey Friedman,
Giangiacomo Gerla, Charles Geyer, Rafał Gruszczyn´ski, Jeremy Heis, Eli Hirsch,
Christopher Hookway, Jesse Houchens, Øystein Linnebo, Charles David McCarty,
MarcoPanza,AndrzejPietruszczak,CarlPosy,AgustínRayo,BarbaraSattler,Wilfried
Sieg, Allan Silverman, Reed Solomon, Gabriel Uzquiano, Achille Varzi, and Keren
Wilson.Sincereapologiestoallofthosewhoshouldbementionedhere,butarenot.
WeowealargedebtofgratitudetoHaidarAl-Dhalimyforamostcarefulproofreading
of earlier drafts, developing the LaTeX code for our figures, and for help with the
final proofreading and the index. Finally, thanks to Peter Momtchiloff, of Oxford
UniversityPress,andhisstaff,forencouragingthisprojectandforguidingitthrough
thevariousstagesofpublication.