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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 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) 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 vii 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 ix