The Argument of Mathematics

391 Pages·2013·4.76 MB·English

by  Andrew Aberdein

Logic, Epistemology, and the Unity of Science 30
Andrew Aberdein
Ian J Dove Editors
.
The Argument of
Mathematics
The Argument of Mathematics
LOGIC,EPISTEMOLOGY,ANDTHEUNITYOFSCIENCE
VOLUME30
Editors
ShahidRahman,UniversityofLilleIII,France
JohnSymons,UniversityofKansas,U.S.A.
ManagingEditor
AliAbasnezhad,UniversityofLilleIII,France
EditorialBoard
JeanPaulvanBendegem,FreeUniversityofBrussels,Belgium
JohanvanBenthem,UniversityofAmsterdam,theNetherlands
JacquesDubucs,UniversityofParisI-Sorbonne,France
AnneFagot-Largeault,CollègedeFrance,France
BasvanFraassen,PrincetonUniversity,U.S.A.
DovGabbay,King’sCollegeLondon,U.K.
JaakkoHintikka,BostonUniversity,U.S.A.
KarelLambert,UniversityofCalifornia,Irvine,U.S.A.
GrahamPriest,UniversityofMelbourne,Australia
GabrielSandu,UniversityofHelsinki,Finland
GöranSundholm,UniversiteitLeiden,TheNetherlands
HeinrichWansing,Ruhr-UniversityBochum,Germany
TimothyWilliamson,OxfordUniversity,U.K.
Logic,Epistemology, andtheUnityofScience aimstoreconsiderthequestionoftheunityofscience
in light of recent developments in logic. At present, nosingle logical, semantical ormethodological
frameworkdominatesthephilosophyofscience.However,theeditorsofthisseriesbelievethatformal
techniques like, forexample, independence friendly logic, dialogical logics,multimodal logics, game
theoreticsemanticsandlinearlogics,havethepotentialtocastnewlightonbasicissuesinthediscussion
oftheunityofscience.
Thisseriesprovidesavenuewherephilosophersandlogicianscanapplyspecifictechnicalinsightsto
fundamentalphilosophicalproblems.Whiletheseriesisopentoawidevarietyofperspectives,including
thestudyandanalysisofargumentationandthecriticaldiscussionoftherelationshipbetweenlogicand
thephilosophyofscience,theaimistoprovideanintegratedpictureofthescientificenterpriseinallits
diversity.
Forfurthervolumes:
http://www.springer.com/series/6936
Andrew Aberdein (cid:129) Ian J. Dove
Editors
The Argument
of Mathematics
123
Editors
AndrewAberdein IanJ.Dove
DepartmentofHumanities DepartmentofPhilosophy
andCommunication UniversityofNevada,LasVegas
FloridaInstituteofTechnology LasVegas,NV,USA
Melbourne,FL,USA
ISBN978-94-007-6533-7 ISBN978-94-007-6534-4(eBook)
DOI10.1007/978-94-007-6534-4
SpringerDordrechtHeidelbergNewYorkLondon
LibraryofCongressControlNumber:2013941250
©SpringerScience+BusinessMediaDordrecht2013
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Preface
Foundationalquestionsandthestatusofproofhavelongbeencentraltophilosophy
of mathematics, although mathematicians do a lot more than just prove results.
However,mostmathematicalpracticemaystillbeunderstoodintermsofargument.
Thus philosophy of mathematics needs an account of argument. Argumentation
theory seems like a good place to look. The intersection of the two, although
largelyunexplored,hasthepotentialtobehugelyfruitful.Thisbookbringstogether
important pioneering work from a variety of sources with newly commissioned
articlestoprovideanoverviewofthetopic.
Our first editorial venture on this theme was a special issue of the journal
Foundationsof Science published in 2009. We are grateful to Diederik Aerts, the
editor,forhissupport.Twoofthearticlesfromthatcollectionarerepublishedhere,
andtheauthorsoftwoothershavecontributednewchapters,howeverweshouldlike
tothankallofthecontributors:JodyAzzouni;EdwinColeman;BartVanKerkhove
andJeanPaulVanBendegem;DavidSherry;ZenonKulpa;MatthewInglisandJuan
PabloMejía-Ramos;AlisonPease,AlanSmaill,SimonColtonandJohnLee.Weare
gratefultooureditorsatSpringer,LucyFleet,TiesNijssenandChristiLue.
Edinburgh,Scotland AndrewAberdein
LasVegas,Nevada IanJ.Dove
December2012
v

Contents
1 Introduction................................................................. 1
AndrewAberdeinandIanJ.Dove
PartI WhatAreMathematicalArguments?
2 Non-deductiveLogicinMathematics:TheProbability
ofConjectures............................................................... 11
JamesFranklin
3 Arguments,Proofs,andDialogues........................................ 31
ErikC.W.Krabbe
4 ArgumentationinMathematics........................................... 47
JesúsAlcoleaBanegas
5 ArguingAroundMathematicalProofs................................... 61
MichelDufour
PartII ArgumentationasaMethodology
forStudyingMathematicalPractice
6 AnArgumentativeApproachtoIdealElementsinMathematics..... 79
PaolaCantù
7 HowPersuadedAreYou?ATypologyofResponses ................... 101
MatthewInglisandJuanPabloMejía-Ramos
8 RevealingStructuresofArgumentationsinClassroom
ProvingProcesses........................................................... 119
ChristineKnippingandDavidReid
9 CheckingProofs ............................................................ 147
JesseAlamaandReinhardKahle
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viii Contents
PartIII MathematicsasaTestbedforArgumentationTheory
10 DividingbyZero—andOtherMathematicalFallacies................. 173
LawrenceH.Powers
11 StrategicManeuveringinMathematicalProofs ........................ 181
ErikC.W.Krabbe
12 AnalogicalArgumentsinMathematics .................................. 199
PaulBartha
13 WhatPhilosophyofMathematicalPracticeCanTeach
ArgumentationTheoryAboutDiagramsandPictures................. 239
BrendanLarvor
PartIV An Argumentational Turn in the Philosophy
ofMathematics
14 MathematicsastheArtofAbstraction................................... 257
RichardL.Epstein
15 TowardsaTheoryofMathematicalArgument ......................... 291
IanJ.Dove
16 BridgingtheGapBetweenArgumentationTheory
andthePhilosophyofMathematics ...................................... 309
AlisonPease,AlanSmaill,SimonColton,andJohnLee
17 MathematicalArgumentsandDistributedKnowledge ................ 339
PatrickAllo,JeanPaulVanBendegem,
andBartVanKerkhove
18 TheParallelStructureofMathematicalReasoning .................... 361
AndrewAberdein
Index............................................................................... 381
Contributors
Andrew Aberdein Department of Humanities and Communication, Florida
InstituteofTechnology,Melbourne,FL,USA
JesseAlama FacultyofScienceandTechnology,CenterforArtificialIntelligence,
UniversidadeNovadeLisboa,Caparica,Portugal
JesúsAlcoleaBanegas DepartamentdeLògicaiFilosofiadelaCiència,Facultat
deFilosofiaiCC.del’Educació,UniversitatdeValència,València,Spain
Patrick Allo Center for Logic and Philosophy of Science, Vrije Universiteit
Brussel,Brussels,Belgium
Paul Bartha Department of Philosophy, University of British Columbia,
Vancouver,Canada
Paola Cantù CNRS & CEPERC, Université Aix-Marseille, Aix-en-Provence
Cedex1,France
SimonColton DepartmentofComputing,ImperialCollege,London,UK
Ian J. Dove Department of Philosophy, University of Nevada, Las Vegas,
LasVegas,NV,USA
Michel Dufour Departmentof Communication,SorbonneNouvelle,Paris Cedex
05,France
RichardL.Epstein AdvancedReasoningForum,Socorro,NM,USA
James Franklin School of Mathematics and Statistics, University of New South
Wales,Sydney,Australia
Matthew Inglis Mathematics Education Centre, Loughborough University,
Loughborough,UK
Reinhard Kahle CENTRIA and DM, FCT, Universidade Nova de Lisboa,
Caparica,Portugal
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