Table Of ContentBoston Studies in the Philosophy and History of Science 339
Gianluigi Oliveri
Claudio Ternullo
Stefano Boscolo Editors
Objects,
Structures,
and Logics
FilMat Studies in the Philosophy
of Mathematics
Boston Studies in the Philosophy and History
of Science
Volume 339
SeriesEditors
AlisaBokulich,BostonUniversity
JürgenRenn,MaxPlanckInstitutefortheHistoryofScience
ManagingEditor
LindyDivarci,MaxPlanckInstitutefortheHistoryofScience
FoundingEditor
RobertS.Cohen
EditorialBoardMembers
TheodoreArabatzis,UniversityofAthens
HeatherE.Douglas,UniversityofWaterloo
KostasGavroglu,UniversityofAthens
ThomasF.Glick,BostonUniversity
HubertGoenner,UniversityofGoettingen
JohnHeilbron,UniversityofCalifornia,Berkeley
DianaKormos-Buchwald,CaliforniaInstituteofTechnology
ChristophLehner,MaxPlanckInstitutefortheHistoryofScience
PeterMcLaughlin,UniversitätHeidelberg
AgustíNieto-Galan,UniversitatAutònomadeBarcelona
NuccioOrdine,UniversitádellaCalabria
AnaSimões,UniversidadedeLisboa
JohnJ.Stachel,BostonUniversity
BaichunZhang,ChineseAcademyofScience
TheseriesBostonStudiesinthePhilosophyandHistoryofSciencewasconceived
in the broadest framework of interdisciplinary and international concerns. Natural
scientists, mathematicians, social scientists and philosophers have contributed to
the series, as have historians and sociologists of science, linguists, psychologists,
physicians,andliterarycritics.
Theserieshasbeenabletoincludeworksbyauthorsfrommanyothercountries
aroundtheworld.
The editors believe that the history and philosophy of science should itself
be scientific, self-consciously critical, humane as well as rational, sceptical and
undogmaticwhilealsoreceptivetodiscussionoffirstprinciples.Oneoftheaimsof
Boston Studies, therefore, is to develop collaboration among scientists, historians
andphilosophers.
BostonStudiesinthePhilosophyandHistoryofSciencelooksintoandreflects
on interactions between epistemological and historical dimensions in an effort to
understandthescientificenterprisefromeveryviewpoint.
Moreinformationaboutthisseriesathttps://link.springer.com/bookseries/5710
Gianluigi Oliveri (cid:129) Claudio Ternullo
Stefano Boscolo
Editors
Objects, Structures,
and Logics
FilMat Studies in the Philosophy
of Mathematics
Editors
GianluigiOliveri ClaudioTernullo
AccademiaNazionaledelleScienze,Lettere DepartamentdeMatema`tiquesiInforma`tica
edArtidiPalermo UniversitatdeBarcelona
Universita`diPalermo Barcelona,Spain
Palermo,Italy
StefanoBoscolo
DipartimentodiFilosofiaeBeniCulturali
Universita`diVeneziaCa’Foscari
Trevio,Italy
ISSN0068-0346 ISSN2214-7942 (electronic)
BostonStudiesinthePhilosophyandHistoryofScience
ISBN978-3-030-84705-0 ISBN978-3-030-84706-7 (eBook)
https://doi.org/10.1007/978-3-030-84706-7
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland
AG2022
Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether
thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse
ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and
transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar
ordissimilarmethodologynowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Contents
1 Introduction................................................................. 1
GianluigiOliveri,ClaudioTernullo,andStefanoBoscolo
PartI MathematicalObjects
2 Aristotle’sProblem......................................................... 17
LucaZanetti
3 Hofweber’sNominalistNaturalism....................................... 31
EricSnyder,RichardSamuels,andStewartShapiro
4 ExploringMathematicalObjectsfromCustom-Tailored
MathematicalUniverses ................................................... 63
IngoBlechschmidt
5 RescuingImplicitDefinitionfromAbstractionism ..................... 97
DanielWaxman
PartII StructuresandStructuralisms
6 StructuralRelativityandInformalRigour.............................. 133
NeilBarton
7 Ontological Dependence and Grounding for a Weak
MathematicalStructuralism .............................................. 175
SilviaBianchi
8 TheStructuralistMathematicalStyle:BourbakiasaCaseStudy.... 199
Jean-PierreMarquis
9 Grothendieck Toposes as Unifying ‘Bridges’: A
MathematicalMorphogenesis............................................. 233
OliviaCaramello
v
vi Contents
PartIII LogicsandProofs
10 GameofGrounds........................................................... 259
DavideCattaandAntonioPiccolominid’Aragona
11 PredicativityandConstructiveMathematics............................ 287
LauraCrosilla
12 TruthandthePhilosophyofMathematics............................... 311
AndreaCantini
13 OnLakatos’sDecompositionoftheNotionofProof.................... 331
EnricoMoriconi
14 ACategoricalReadingoftheNumericalExistenceProperty
inConstructiveFoundations .............................................. 349
SamueleMaschio
About the Editors
ClaudioTernullo (PhD Liverpool, 2012) is currently Beatriu de Pinós (Marie-
Skłodowska Curie Actions COFUND) Postdoctoral Fellow at the University of
Barcelona.Previously,hehasheldpositionsaspost-docattheKurtGödelResearch
Center for Mathematical Logic at the University of Vienna and at the University
of Tartu. His research interests lie in logic and the philosophy of mathematics, in
particular in the philosophy of set theory. His work focuses on the set-theoretic
multiverse, new axioms (and their justification), and mathematical platonism (in
particular, Gödel’s Platonism). He has also done work on issues in ancient and
medievalphilosophy.
StefanoBoscolo (PhD Palermo, 2016) is an IT Solution Architect at Volkswagen
Group.Hiscurrentareasofexpertiseareadvancedanalytics,cloudcomputing,and
machine learning. After receiving his PhD in Logic and Philosophy of Science,
he worked at Ca’ Foscari University of Venice on truth pluralism. Throughout his
academiccareer,heworkedonthephilosophyofmathematics,inparticularonthe
debate between platonism and anti-platonism. His current research interests range
fromnatural-languageprocessingtodeepneuralnetworks.
GianluigiOliveriobtainedalaurea(BA)inphilosophyfromtheUniversityofBari
(Italy).Afterthat,hereceivedaDPhilinphilosophyfromtheUniversityofOxford
(GB)andaPhDincognitivesciencefromtheUniversityofMessina(Italy).
His research interests range from the philosophy of mathematics, artificial intelli-
gence,andthephilosophyofsciencetometaphysicsandthehistoryofphilosophy.
He has held teaching and research posts at the universities of Reading, Leeds,
Keele, Oxford, and Palermo, and has been a visiting scholar at the Sydney Centre
for the Foundations of Science, University of Sydney, Australia. He is, currently,
Associate Professor of Logic and the Philosophy of Science at the University of
vii
viii AbouttheEditors
Palermo (Italy); corresponding member of the Accademia Nazionale di Scienze,
Lettere ed Arti di Palermo (National Academy of Sciences, Letters and Arts of
Palermo); and chairman of the Centro Interdipartimentale per le Tecnologie della
Conoscenza (Interdepartmental Centre for the Technologies of Knowledge) at the
UniversityofPalermo.
Contributors
N.BartonFachbereichPhilosophie,UniversityofKonstanz,Konstanz,Germany
S.BianchiUniversitySchoolforAdvancedStudies(IUSS),Pavia,Italy
I.BlechschmidtInstitutfürMathematik,UniversitätAugsburg,Augsburg,
Germany
A.CantiniDILEF,UniversitàdiFirenze,Firenze,Italy
O.Caramello Dipartimento di Scienza e Alta Tecnologia, Università degli Studi
dell’Insubria,Como,Italy
InstitutdesHautesÉtudesScientifiques,Bures-sur-Yvette,France
D.CattaLIRMM–MontpellierUniversity,Montpellier,France
L.Crosilla Department of Philosophy, IFIKK, University of Oslo, Blindern,
Norway
A.P.d’Aragona Centre Gilles Gaston Granger, Aix-Marseille Univ, CNRS,
Aix-en-Provence,France
J.-P.MarquisDépartementdePhilosophie,UniversitédeMontréal,Montréal,QC,
Canada
S.MaschioDipartimentodiMatematica“TullioLevi-Civita”,Padova,Italy
E.MoriconiDepartmentofCivilisationsandFormsofKnowledge,viaP.Paoli15,
Pisa,Italia
R.SamuelsOhioStateUniversity,Columbus,OH,USA
S.ShapiroOhioStateUniversity,Columbus,OH,USA
E.SnyderLMU,Munich,Germany
ix
