New Foundations for Physical Geometry: The Theory of Linear Structures PDF
Preview New Foundations for Physical Geometry: The Theory of Linear Structures
New Foundations for Physical Geometry New Foundations for Physical Geometry The Theory of Linear Structures Tim Maudlin 1 3 GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries #TimMaudlin2014 Themoralrightsoftheauthorhavebeenasserted FirstEditionpublishedin2014 Impression:1 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicence,orundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressControlNumber:2014931223 ISBN 978–0–19–870130–9 Asprintedandboundby CPIGroup(UK)Ltd,Croydon,CR04YY LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork. To V Volim te Icriticizebycreation,notbyfindingfault. Cicero Philosophyiswritteninthisgrandbook,theuniverse,whichstandscontinually opentoourgaze.Butthebookcannotbeunderstoodunlessonefirstlearnsto comprehend the language and read the letters in which it is composed. It is writteninthelanguageofmathematics,anditscharactersaretriangles,circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth. Galileo Contents Acknowledgments x Introduction 1 MetaphoricalandGeometricalSpaces 6 ALightDanceontheDustoftheAges 9 TheProliferationofNumbers 12 DescartesandCoordinateGeometry 14 JohnWallisandtheNumberLine 16 DedekindandtheConstructionofIrrationalNumbers 20 OverviewandTerminologicalConventions 25 1. TopologyandItsShortcomings 28 StandardTopology 31 ClosedSets,Neighborhoods,BoundaryPoints,andConnectedSpaces 33 TheHausdorffProperty 36 WhyDiscreteSpacesMatter 45 TheRelationalNatureofOpenSets 47 TheBillofIndictment(SoFar) 49 2. LinearStructures,Neighborhoods,OpenSets 54 MethodologicalMorals 54 TheEssenceoftheLine 57 The(First)TheoryofLinearStructures 59 Proto-LinearStructures 69 DiscreteSpaces,MrBush’sWildLine,theWovenPlane,andtheAffinePlane 74 ATaxonomyofLinearStructures 79 NeighborhoodsinaLinearStructure 81 OpenSets 85 Finite-PointSpaces 86 ReturntoIntuition 89 DirectedLinearStructures 92 LinearStructuresandDirectedLinearStructures 96 Neighborhoods,OpenSets,andTopologiesAgain 97 Finite-PointSpacesandGeometricalInterpretability 99 AGeometricallyUninterpretableTopologicalSpace 103 Segment-SplicedLinearStructures 104 LookingAhead 107 Exercises 107 Appendix:NeighborhoodsandLinearStructures 108 viii CONTENTS 3. ClosedSets,OpenSets(Again),ConnectedSpaces 113 ClosedSets:PreliminaryObservations 113 OpenandClosedIntervals 114 IP-closedandIP-openSets 115 IP-openSetsandOpenSets,IP-closedSetsandClosedSets 117 Zeno’sCombs 120 ClosedSets,OpenSets,andComplements 123 Interiors,BoundaryPoints,andBoundaries 127 FormalPropertiesofBoundaryPoints 136 ConnectedSpaces 140 ChainsandConnectedness 143 DirectednessandConnectedness 148 Exercises 150 4. SeparationProperties,Convergence,andExtensions 152 SeparationProperties 152 ConvergenceandUnpleasantness 155 SequencesandConvergence 160 Extensions 163 TheTopologist’sSineCurve 165 PhysicalInterlude:Thomson’sLamp 168 Exercises 172 5. PropertiesofFunctions 174 Continuity:anOverview 174 TheIntuitiveExplicationofContinuityandItsShortcomings 175 TheStandardDefinitionandItsShortcomings 178 WhattheStandardDefinitionof“Continuity”Defines 183 TheEssenceofContinuity 186 ContinuityataPointandinaDirection 190 AnHistoricalInterlude 192 RemarksontheArchitectureofDefinitions;LinealFunctions 194 LinesandContinuityinStandardTopology 199 Exercises 201 6. SubspacesandSubstructures;StraightnessandDifferentiability 203 TheGeometricalStructureofaSubspace:Desiderata 203 SubspacesinStandardTopology 205 SubspacesintheTheoryofLinearStructures 206 Substructures 211 OneWayForward 218 Euclid’sPostulatesandtheNatureofStraightness 220 ConvexAffineSpaces 227 Example:SomeConicalSpaces 233 Tangents 235 UpperandLowerTangents,Differentiability 244 Summation 253 Exercises 254 CONTENTS ix 7. MetricalStructure 256 ApproachestoMetricalStructure 256 RatiosBetweenWhat? 258 TheAdditivePropertiesofStraightLines 260 CongruenceandComparability 262 EudoxanandAnthyphaireticRatios 274 TheCompass 280 MetricLinearStructuresandMetricFunctions 285 OpenLines,CurvedLines,andRectification 287 ContinuityoftheMetric 291 Exercises 294 Appendix:ARemarkaboutMinimalRegularMetricSpaces 294 8. ProductSpacesandFiberBundles 297 NewSpacesfromOld 297 ConstructingProductLinearStructures 300 ExamplesofProductLinearStructures 303 NeighborhoodsandOpenSetsinProductLinearStructures 307 FiberBundles 309 Sections 313 AdditionalStructure 315 Exercises 318 9. BeyondContinua 320 HowCanContinuaandNon-ContinuaApproximateEachOther? 320 ContinuousFunctions 321 Homotopy 334 Compactness 339 SummaryofMathematicalResultsandSomeOpenQuestions 345 Exercises 346 AxiomsandDefinitions 347 Bibliography 358 Index 361
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