Mathematical Anti-Realism and Modal Nothingism

86 Pages·2022·1.203 MB·English

by  Mark Balaguer

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This Element defends mathematical anti-realism against l
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mathematical anti-realism, defends that view against a number
of well-known objections, and raises a less widely discussed
objection to anti-realism – an objection based on the fact that The Philosophy of
(a) mathematical anti-realists need to commit to the truth of Mathematics
certain kinds of modal claims, and (b) it’s not clear that the
truth of these modal claims is compatible with mathematical
anti-realism. The second part considers various strategies that
anti-realists might pursue in trying to solve this modal-truth
problem with their view, argues that there’s only one viable view
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that anti-realists can endorse in order to solve the modal-truth a
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problem, and argues that the view in question – which is here e
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ElementsinthePhilosophyofMathematics
editedby
PenelopeRush
UniversityofTasmania
StewartShapiro
TheOhioStateUniversity
MATHEMATICAL
ANTI-REALISM AND
MODAL NOTHINGISM
Mark Balaguer
California State University
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781009346016
DOI:10.1017/9781009346030
©MarkBalaguer2022
Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisions
ofrelevantcollectivelicensingagreements,noreproductionofanypartmaytake
placewithoutthewrittenpermissionofCambridgeUniversityPress&Assessment.
Firstpublished2022
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
ISBN978-1-009-34601-6Paperback
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Mathematical Anti-Realism and Modal Nothingism
ElementsinthePhilosophyofMathematics
DOI:10.1017/9781009346030
Firstpublishedonline:December2022
MarkBalaguer
CaliforniaStateUniversity
Authorforcorrespondence:MarkBalaguer,[email protected]
Abstract:ThisElementdefendsmathematicalanti-realismagainstan
underappreciatedproblemwiththatview–aproblemhavingtodo
withmodaltruthmaking.Thefirstpartdevelopsmathematicalanti-
realism,defendsthatviewagainstanumberofwell-knownobjections,
andraisesalesswidelydiscussedobjectiontoanti-realism–an
objectionbasedonthefactthat(a)mathematicalanti-realistsneedto
committothetruthofcertainkindsofmodalclaims,and(b)it’snot
clearthatthetruthofthesemodalclaimsiscompatiblewith
mathematicalanti-realism.Thesecondpartconsidersvarious
strategiesthatanti-realistsmightpursueintryingtosolvethismodal-
truthproblemwiththeirview,arguesthatthere’sonlyoneviableview
thatanti-realistscanendorseinordertosolvethemodal-truth
problem,andarguesthattheviewinquestion–whichisherecalled
modalnothingism–istrue.
ThisElementalsohasavideoabstract:www.cambridge.org/Philosophy
ofMathematics_Balaguer_abstract
sserP Keywords:mathematicalanatib-rsetraalicstmo,bmjeocdtsality,errortheory,truthmaking,
y
tisrev ©MarkBalaguer2022
in
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e ISBNs9781009346016(PB),9781009346030(OC)
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Contents
1 Introduction 1
MathematicalAnti-Realism 3
2 WhatIsMathematicalAnti-Realism? 3
3 MathematicalErrorTheoryDefended 14
4 ParaphraseNominalismandDeflationary-Truth
NominalismRevisited 29
ModalNothingism 33
5 ModalSemanticsandModalTruthmaking 33
6 PossibleWorlds 39
7 ModalPrimitivism 47
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ytisre 8 ModalNothingismtotheRescue 57
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MathematicalAnti-RealismandModalNothingism 1
1Introduction
Mathematicalanti-realismis(somewhatroughly)theviewthatourmathematical
theoriesdon’tprovidetruedescriptionsofmathematicalobjects(i.e.,thingslike
numbersandsets)becausetherearenosuchthingsasmathematicalobjects.The
reasonthere are nomathematicalobjects,onthisview, isthat(a) ifthere were
mathematical objects, they would be abstract objects – that is, nonphysical,
nonmental,nonspatiotemporalobjects–and(b)therearenoabstractobjects.
MycentralaiminthisElementistoarticulatewhatIthinkisanunderappre-
ciatedproblemformathematicalanti-realism–aproblemhavingtodowiththe
truthmaking of modal claims – and to develop and defend what I think is the
onlyplausiblesolutiontothatproblem.
In the first part of this Element, I’ll provide a more thorough and precise
articulation of mathematical anti-realism, I’ll explain how I think that view
shouldbedeveloped,andI’llrespondtoanumberofobjectionsthatyoumight
raise against that view. But at the end of the first part, another objection will
emerge,anobjectionbasedonthefactthat(a)mathematicalanti-realistsneedto
committothetruthofcertainkindsofmodalclaims(i.e.,possibilityclaimsand/
ornecessityclaimsand/orcounterfactuals),and(b)it’snotclearthatthetruthof
thesemodalclaimsiscompatiblewithmathematicalanti-realism.Inthesecond
part of this Element, I’ll run through the various strategies that anti-realists
mightpursueintryingtosolvethismodal-truthproblem–thatis,theproblemof
explaining how modal claims of the relevant kind could be true, given that
mathematical anti-realism is also true – and I’ll argue that there’s only one
viableviewavailabletoanti-realistshere.I’llcallthisviewmodalnothingism,
sserP andI’llargueattheendofthesecondpartthatmodalnothingismistrueandthat
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tisre mathematicalanti-realistscanusethisviewtoblockthemodal-truthobjection
vin totheirview.
U
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irbm threethings:I’lldefinemathematical realismandanti-realism;I’lldistinguish
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C y threedifferentversionsofanti-realism,namely,paraphrasenominalism,defla-
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en tionary-truthnominalism,andmathematicalerrortheory;andI’llargueagainst
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o d paraphrasenominalismanddeflationary-truthnominalism.
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hsilb InSection3,I’llrespondtofourobjectionstoerrortheory.I’llspendmostof
uP 0 my time responding to the worry that error theorists can’t account for the
3
06 factualness and objectivity of mathematics. After that, I’ll respond to three
4
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90 more objections to error theory – a Moorean objection, a Lewisian objection,
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7 and a Quine-Putnam indispensability objection. During the course of the dis-
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2 ThePhilosophyofMathematics
Inparticular,I’llarguethattheyneedtocommittothetruthofeithercounter-
factualslike
[CF]Iftherehadactuallyexistedaplenitudinousrealmofabstractobjects,
thenitwouldhavebeenthecasethat3wasprime,
ornecessitarianconditionalslike
[N]Necessarily,ifthereexistsaplenitudinousrealmofabstractobjects,then
3isprime.
Finally,attheendofSection3,I’llpointoutthatthiscreatesaproblemforerror
theory.Forgiventhaterrortheoristsdenythatthereareabstractobjects,it’snot
clearthattheyhaveaccesstoanyplausibleaccountofhowmodalsentenceslike
[CF]and[N]couldbetrue.
InSection4,I’llarguethat,likeerrortheorists,paraphrasenominalistsand
deflationary-truth nominalists need to commit to the truth of modal sentences
like[CF]or[N],andso–again,likeerrortheorists–theyencountertheworry
thattheydon’thaveaccesstoanyplausibleaccountofhowsentenceslike[CF]
and[N]couldbetrue,giventhattherearenosuchthingsasabstractobjects.
The secondpartofthisElementconsistsofSections5–8.InSection5,I’ll
do two things. First, I’ll introduce the two most prominent views of the
semantics of ordinary modal sentences, namely, the possible-worlds analysis
(whichsays,roughly,thatordinarymodalsentencesareclaimsaboutpossible
worlds) and modal primitivism (which says, roughly, that ordinary modal
sentences involve primitive modal operators). Second, I’ll introduce
a principle that, prima facie, seems extremely plausible – namely, that for
sserP any true sentence S, there’s something about reality that makes S true – and
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tisre I’ll point out that (a) given the plausibility of this principle, mathematical
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irbm in abstract objects, it’s not clear that they have access to any tenable view of
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y what makes these modal sentences true.
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niln In Section 6, I’ll argue that, if they want to, mathematical anti-realists can
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silb endorseamodalerrortheory;butI’llalsoarguethatiftheydothis,theninorder
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64 discourse, they’ll have to introduce novel primitive modal operators (i.e.,
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MathematicalAnti-RealismandModalNothingism 3
truthmaking problem that’s essentially equivalent to the modal truthmaking
problemthattheyfaceiftheyendorsemodalprimitivism.
InSection7,I’llturntothequestionofwhethermathematicalanti-realistscan
solvethemodaltruthmakingproblemiftheyendorsemodalprimitivism,andI’ll
arguethattheydon’tseemtohaveanyviableoptionshere.Morespecifically,I’ll
argue that anti-realists don’t have access to any plausible view of what makes
ordinarymodalsentencestrue.I’lldothisbyrunningthroughanumberofoptions–
including conventionalist views, essentialist views, and potentiality views – and
arguingthatnoneoftheavailableviewsgivesanti-realistswhattheyneed.
Finally,inSection8,I’llarguethatthere’sawayoutofthisproblemforanti-
realists.I’lldothisbyarguingagainsttheassumptionthateverytruesentenceis
madetruebyreality.Morespecifically,I’llarguethatwhileourmodalsentences
are true – indeed, substantively and objectively true – there’s nothing about
reality that makes them true. I’ll call this view modal nothingism, and in
Section 8, I’ll argue that it’s true and that it gives mathematical anti-realists
asolutiontothemodaltruthmakingproblemwiththeirview.
MATHEMATICALANTI-REALISM
2WhatIsMathematicalAnti-Realism?
2.1OpeningRemarks
In order to understand mathematical anti-realism, we first need to understand
mathematical realism – because the former can be defined as simply the
sse negationofthelatter.Thus,I’llbegin,inSection2.2,bydiscussingmathemat-
rP ical realism (in particular, I’ll define realism and explain why one might be
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30 2.2.1MathematicalRealismDefined
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00 The best way to understand mathematical realism is to begin by thinking about
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4 ThePhilosophyofMathematics
beatrueclaimaboutacertainobject,namely,thenumber3;‘3isprime’seems
analogousinthiswayto,forinstance,‘Marsisround.’Justasthelattertellsusthat
acertainobject(Mars)hasacertainproperty(roundness),sotheformertellsusthat
theobject3hasthepropertyofbeingprime.And–again,primafacie–itseems
that our mathematical theories give us a bunch of true claims about a bunch of
objects; for example, Peano Arithmetic tells us that every natural number has
asuccessor,andthatthereareinfinitelymanyprimes,andsoon.
Mathematicalrealismistheviewthattheseappearances(abouttheorieslike
Peano Arithmetic and sentences like ‘3 is prime’) are indeed true. More
precisely, realism is the view that (a) there exist mathematical objects (e.g.,
numbersandsets),and(b)ourmathematicaltheories(whichincludeordinary
sentenceslike‘3isprime’)providetruedescriptionsoftheseobjects.
2.2.2TheArgumentforMathematicalRealism
Considerthefollowingargument:
[1] Mathematicalsentenceslike‘3isprime’shouldbereadatfacevalueand,
hence,asmakingstraightforwardclaimsaboutthenatureofcertainspecific
objects;forexample,‘3isprime’shouldbereadasbeingoftheform‘Fa’
and, hence, as making a straightforward claim about the nature of the
number3.But
[2] Ifsentenceslike‘3isprime’shouldbereadatfacevalue,andifmoreover
they’retrue,thentheremustactuallyexistobjectsofthekindsthatthey’re
about;forexample,if‘3isprime’makesastraightforwardclaimaboutthe
sse natureofthenumber3,andifthissentenceisliterallytrue,thentheremust
rP y actuallyexistsuchathingasthenumber3.Moreover,
tisre [3] Mathematicalsentenceslike‘3isprime’aretrue.Therefore,from[1]–[3],
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dirb [4] There actually exist objects that mathematical sentences like ‘3 is prime’
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a areabout,andthesesentencesprovidetruedescriptionsofthoseobjects;for
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yb e example, the number 3 actually exists, and ‘3 is prime’ provides a true
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silb Thisargumentis valid, and [4] isessentially equivalent to mathematical real-
uP 0 ism,andsoif[1]–[3]arealltrue,thenmathematical realismisalsotrue.But,
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06 primafacie,itseemsthat[1]–[3]aretrue(andI’llarticulatesomeargumentsfor
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0 these three premises in Sections 2.4, 3.2.1, and 3.4), and so it seems that
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