New Foundations for Physical Geometry: The Theory of Linear Structures

374 Pages·2014·2.104 MB·English

by  Tim Maudlin

New Foundations for Physical Geometry

New Foundations for
Physical Geometry
The Theory of Linear Structures
Tim Maudlin
1
3
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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