The Metaphysics and Mathematics of Arbitrary Objects

250 Pages·2019·1.008 MB·English

by  Leon Horsten

TheMetaphysicsandMathematicsofArbitraryObjects
BuildingontheseminalworkofKitFineinthe1980s,LeonHorsten
heredevelopsanewtheoryofarbitraryentities.Heconnectsthisthe-
ory to issues and debates in metaphysics, logic, and contemporary
philosophyofmathematics,investigatingtherelationbetweenspecific
and arbitrary objects and between specific and arbitrary systems of
objects.Hisbookshowshowthisinnovativetheoryishighlyapplica-
bletoproblemsinthephilosophyofarithmetic,andexploresinpar-
ticularhowarbitraryobjectscanengagewiththenineteenth-century
conceptofvariablemathematicalquantities,howtheyarerelevantfor
debates around mathematical structuralism, and how they can help
our understanding of the concept of random variables in statistics.
This worked-through theory will open up new avenues within phi-
losophyofmathematics,bringingintheworkofotherphilosophers,
suchasSaulKripke,andprovidingnewinsightsintothedevelopment
ofthefoundationsofmathematicsfromtheeighteenthcenturytothe
presentday.
leon horsten is Professor of Philosophy at the University of
Bristol.HispublicationsincludeTheTarskianTurn:Deflationismand
AxiomaticTruth(2011)andGo¨del’sDisjunction:TheScopeandLimits
ofMathematicalKnowledge(co-editedwithPhilipWelch,2016).

The Metaphysics and Mathematics
of Arbitrary Objects
leon horsten
UniversityofBristol
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Informationonthistitle:www.cambridge.org/9781107039414
DOI:10.1017/9781139600293
©LeonHorsten2019
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Contents
ListofFigures [pagex]
Preface [xi]
Acknowledgements [xiv]
SymbolsandAbbreviations [xvi]
1 Introduction [1]
1.1 ArbitraryObjectsandthePhilosophyofMathematics [1]
1.2 ScratchingtheSurface [3]
1.3 StructureandMethod [4]
1.4 IntendedAudience [8]
1.5 OnNotationandTechnicalMatters [10]
2 MetaphysicsofMathematics [13]
2.1 TheMethodologyofPhilosophyofMathematics [13]
2.2 Quietism [16]
2.3 NaiveMetaphysicsandFoundationalMetaphysics [24]
2.4 NaiveMetaphysicsOnly [26]
2.5 MetaphysicsofMathematics [32]
2.6 AnObjectionfromPseudo-science [34]
2.7 MathematicalModels [36]
3 ArbitraryObjects [38]
3.1 TheMotivatingIdea [38]
3.2 EarlyRussellonVariables [40]
3.3 ArbitraryObjectsandTheirRelations [44]
3.4 IdentityandComprehension [45]
3.5 StateSpaces [48]
3.6 BeingandBeinginaState [49]
3.7 Hairdressers [53]
3.8 BabySteps [54]
3.9 Properties,Quantities [58]
4 MathematicalObjectsasArbitraryObjects [61]
4.1 ArbitraryNaturalNumbers [61]
4.2 MoreFregeanWorries [66]
4.3 Quasi-concreteObjects [68]
4.4 KnowledgeofArbitraryStrings [69]
4.5 KnowledgebyAcquaintanceandKnowledgebyDescription [74]
4.6 TheoryandApplications [75] vii
viii Contents
5 StructureinMathematics [78]
5.1 FormandStructure [78]
5.2 TheQuestionofRealism [80]
5.3 PlatonismandReductionism [83]
5.4 EliminativeStructuralism [85]
5.5 Non-eliminativeStructuralism [94]
5.6 StructureandComputation [98]
5.7 ThePeculiarCaseofSetTheory [102]
6 MathematicalStructures [105]
6.1 FromArbitraryObjectsoverArbitrarySystemstoGeneric
Systems [105]
6.2 FairDice [106]
6.3 TheGenericω-Sequences [108]
6.4 TheGenericNaturalNumberStructure [111]
6.5 OtherArbitrarySystems [113]
6.6 TheComputableGenericω-Sequence [114]
6.7 TheArbitraryCountableGraph [115]
6.8 IndividualStructures [117]
6.9 TheGenericHierarchy [120]
7 KitFine [123]
7.1 ConceptandObject [124]
7.2 VariableObjectsandDependence [127]
7.3 Identity,Systems,andComprehension [129]
7.4 CantorianAbstraction [134]
7.5 TheNaturalNumberStructure [136]
7.6 Predicativism? [139]
7.7 UniversalsversusVariableNumbers [141]
8 GenericSystemsandMathematicalStructuralism [143]
8.1 GenericStructuralism? [143]
8.2 ThemesfromEliminativeandNon-EliminativeStructuralism [146]
8.3 TheoriesandStructures [153]
8.4 ComparisonwithFine [155]
8.5 QuantificationandReference [159]
9 ReasoningaboutGenericω-Sequences [162]
9.1 IndividualConceptsandCarnapianModalLogic [162]
9.2 GenericTruth [165]
9.3 DefinabilityandIndiscernibility [172]
9.4 DeterminationandIndependence [177]
9.5 ComputationalComplexity [182]
9.6 AxiomatisingSecond-OrderArithmetic [184]
9.7 Well-Ordering [186]
Contents ix
10 ProbabilityandRandomVariables [189]
10.1 RandomVariables [189]
10.2 AConnectionwithIndividualConcepts? [191]
10.3 Uniformity [193]
10.4 Non-ArchimedeanProbabilityFunctions [195]
10.5 SetProbabilities [200]
11 DirectionsforFutureResearch [208]
11.1 SuccessorFailure? [208]
11.2 TowardsaGeneralTheory [210]
11.3 AspectsandApplications [211]
Bibliography [215]
Index [224]
Figures
3.1 Anarbitraryhairdresser [page54]
3.2 Aspecifichairdresser [54]
3.3 Totalspaceofarbitraryteachersandheadteachers [56]
3.4 Partialspaceofarbitraryteachersandheadteachers [57]
4.1 Arbitrarynaturalnumbers [63]
5.1 Aclassificationofmathematicaltheories [83]
6.1 Genericω-sequences [109]
6.2 Thediagonalω-sequenceN [111]
6.3 ThegraphGa−b,c [117]
6.4 ThesystemsS ,S ,S [118]
1 2 3
7.1 Dependence:ordertype2 [135]
7.2 Dependence:thenaturalnumberstructure [137]
7.3 Dependencealternative:ordertype2 [138]
9.1 D(Gc) [179]
10.1 Permutationπ [203]
x