Abstractionism: Essays in Philosophy of Mathematics

367 Pages·2017·1.391 MB·English

by  Ebert, Philip A., Rossberg, Marcus (eds.)

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ABSTRACTIONISM
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Abstractionism
Essays in Philosophy of Mathematics
Editedby
PHILIP A. EBERT
and
MARCUS ROSSBERG
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3
Great Clarendon Street, Oxford, Ox2 6dp,
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First Edition published in 2016
Impression: 1
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Contents
I Introduction
1 IntroductiontoAbstractionism 3
PhilipA.EbertandMarcusRossberg
II Semantics and Ontology of Abstraction
2 CaesarandCircularity 37
WilliamStirton
3 TheExistence(andNon-existence)ofAbstractObjects 50
RichardG.Heck,Jr.
4 HaleandWrightontheMetaontologyofNeo-Fregeanism 79
MattiEklund
5 Neo-FregeanMeta-Ontology:
JustDon’tAskTooManyQuestions 94
FraserMacBride
6 TheNumberofPlanets,aNumber-ReferringTerm? 113
FriederikeMoltmann
III Epistemology of Abstraction
7 AFrameworkforImplicitDefinitionsandtheAPriori 133
PhilipA.Ebert
8 AbstractionandEpistemicEntitlement:
OntheEpistemologicalStatusofHume’sPrinciple 161
CrispinWright
9 Hume’sPrincipleandEntitlement:
OntheEpistemologyoftheNeo-FregeanProgram 186
NikolajJangLeeLindingPedersen
10 Neo-FregeanismReconsidered 203
AgustínRayo
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vi Contents
IV Mathematics of Abstraction
11 Conservativeness,Cardinality,andBadCompany 223
RoyT.Cook
12 ImpredicativityintheNeo-FregeanProgram 247
ØysteinLinnebo
13 AbstractionGrounded:ANoteonAbstractionandTruth 269
HannesLeitgeb
14 IneffabilitywithintheLimitsofAbstractionAlone 283
StewartShapiroandGabrielUzquiano
V Application Constraint
15 OnFrege’sApplicationsConstraint 311
PaulMcCallion
16 ApplicationsofComplexNumbersandQuaternions: Historical
Remarks,withaNoteonCliffordAlgebra 323
PeterSimons
17 DefinitionsofNumbersandTheirApplications 332
BobHale
Index 349
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Part I
Introduction
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1
Introduction to Abstractionism
Philip A. Ebert and Marcus Rossberg
1.1 WHATISABSTRACTIONISM?
Abstractionism in the philosophy of mathematics has its origins in Gottlob
Frege’slogicism—apositionFregedevelopedinthelatenineteenthandearly
twentiethcentury.Frege’smainaimwastoreducearithmeticandanalysisto
logic in order to provide a secure foundation for mathematical knowledge.
Asiswellknown,Frege’sdevelopmentoflogicismfailed.TheinfamousBasic
LawV—oneofthesixbasiclawsoflogicFregeproposedinhismagnumopus
GrundgesetzederArithmetik—issubjecttoRussell’sParadox.Thestrikingfea-
tureofFrege’sBasicLawVisthatittakestheformofanabstractionprinciple.
Thegeneralformofanabstractionprinciplecanbesymbolizedlikethis:1
x(cid:11)=x(cid:12) $ (cid:11)(cid:24)(cid:12)
where‘x’isaterm-formingoperatorapplicabletoexpressionofthetypeof(cid:11)
and(cid:12),and(cid:24)isanequivalencerelationonentitiesdenotedbyexpressionsof
that type. Accordingly, abstraction principles are biconditionals that feature
an equivalence relation on the right-hand side and an identity statement on
the left-hand side. The abstracta denoted by the terms featuring in the iden-
tity statement on the left are taken to be introduced, in some sense, by the
abstraction principle, giving the equivalence on the right-hand side concep-
tualpriorityoverthem.Moreonthisbelow.
Frege’sill-fatedBasicLawV,involvesco-extentionality(offunctions)asthe
relevantequivalencerelationontheright-handside,introducing,whatFrege
(cid:21)
termedvalue-ranges,"φ("),ontheleft:2
1Hereandbelow,wewillomitprefixeduniversalquantifiersinabstractionprinciples.We
aretherebyineffectneglectingthedistinctionbetweenschematic andaxiomatic (oruniversal)
formulationsofabstractionprinciples.Inthecontextoffullimpredicativesecond-orderlogic,
theseformulationsareequivalent,butinsystemswithweakersecond-ordercomprehension(see
page21below),thesecomeapart:theschematicformulationsentailtheaxiomaticones,butnot
viceversa;seee.g.Heck(1996,§1),Fine(2002,36–38),orLinnebo(2004,158).
2Inwords:Thevalue-rangeoffunctionf isidenticaltothevalue-rangeoffunctiongifand
onlyiff andghavethesamevalueforanyargument.Thevalue-rangeofafunctionisroughly
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