Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts PDF
Preview Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts
Synthese Library 407 Studies in Epistemology, Logic, Methodology, and Philosophy of Science Stefania Centrone Deborah Kant Deniz Sarikaya Editors Reflections on the Foundations of Mathematics Univalent Foundations, Set Theory and General Thoughts Synthese Library Studies in Epistemology, Logic, Methodology, and Philosophy of Science Volume 407 Editor-in-Chief OtávioBueno,DepartmentofPhilosophy,UniversityofMiami,USA Editors BeritBrogaard,UniversityofMiami,USA AnjanChakravarthy,UniversityofNotreDame,USA StevenFrench,UniversityofLeeds,UK CatarinaDutilhNovaes,VUAmsterdam,TheNetherlands TheaimofSyntheseLibraryistoprovideaforumforthebestcurrentworkinthe methodology and philosophy of science and in epistemology. A wide variety of different approaches have traditionally been represented in the Library, and every effortismadetomaintainthisvariety,notforitsownsake,butbecausewebelieve thattherearemanyfruitfulandilluminatingapproachestothephilosophyofscience andrelateddisciplines. Specialattentionispaidtomethodologicalstudieswhichillustratetheinterplay of empirical and philosophical viewpoints and to contributions to the formal (logical,set-theoretical,mathematical,information-theoretical,decision-theoretical, etc.) methodology of empirical sciences. Likewise, the applications of logical methodstoepistemologyaswellasphilosophicallyandmethodologicallyrelevant studiesinlogicarestronglyencouraged.Theemphasisonlogicwillbetemperedby interestinthepsychological,historical,andsociologicalaspectsofscience. BesidesmonographsSyntheseLibrarypublishesthematicallyunifiedanthologies andeditedvolumeswithawell-definedtopicalfocusinsidetheaimandscopeofthe book series. The contributions in the volumes are expected to be focused and structurally organized in accordance with the central theme(s), and should be tied togetherbyanextensiveeditorialintroductionorsetofintroductionsifthevolume isdividedintoparts.Anextensivebibliographyandindexaremandatory. Moreinformationaboutthisseriesathttp://www.springer.com/series/6607 Stefania Centrone (cid:129) Deborah Kant (cid:129) Deniz Sarikaya Editors Reflections on the Foundations of Mathematics Univalent Foundations, Set Theory and General Thoughts 123 Editors StefaniaCentrone DeborahKant TechnicalUniversityofBerlin UniversityofKonstanz Berlin,Germany Konstanz,Germany DenizSarikaya UniversityofHamburg Hamburg,Germany SyntheseLibrary ISBN978-3-030-15654-1 ISBN978-3-030-15655-8 (eBook) https://doi.org/10.1007/978-3-030-15655-8 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Introduction StefaniaCentrone,DeborahKant,andDenizSarikaya The present volume originates from the conference Foundations of Mathematics: UnivalentFoundationsandSetTheory(FOMUS),whichwasheldattheCenterfor InterdisciplinaryResearchofBielefeldUniversityfromthe18thtothe23rdofJuly 2016.Withinthisframeworkapproximately80graduatestudents,juniorresearchers and leading experts gathered to investigate and discuss suitable foundations for mathematics and their qualifying criteria, with an emphasis on homotopy type theory(HoTT)andunivalentfoundations(UF)aswellassettheory.Thisinterdis- ciplinaryworkshop,conceivedofasahybridbetweensummerschoolandresearch conference,wasaimedatstudentsandresearchersfromthefieldsofmathematics, computerscienceandphilosophy. Acollectedvolumerepresents,itgoeswithoutsaying,anexcellentopportunity to pursuing and deepening the lively discussions of a conference. This volume, however,isnotaconferenceproceedingsinthenarrowsensesinceitcontainsalso contributions from authors who were not present at FOMUS. Specifically, 6 from the19contributionshavebeendevelopedfrompresentationsattheconferenceand only9from24authorswerepresentatFOMUS. As to the conference, the concomitant consideration of different foundational theoriesformathematicsisanambitiousgoal.Thisvolumeintegratesbothunivalent foundations and set theory and aims to bring some novelty in the discussion on the foundations of mathematics. Indeed, a comparative study of foundational S.Centrone TechnicalUniversityofBerlin,Berlin,Germany e-mail:[email protected] D.Kant UniversityofKonstanz,Konstanz,Germany e-mail:[email protected] D.Sarikaya UniversityofHamburg,Hamburg,Germany e-mail:[email protected] v vi S.Centroneetal. frameworks with an eye to the current needs of mathematical practice is, even to thisday,adesideratum. The FOMUS conference was organized with the generous support of the AssociationforSymbolicLogic(ASL),theGermanMathematicalSociety(DMV), the Berlin Mathematical School (BMS), the Center of Interdisciplinary Research (ZiF), the Deutsche Vereinigung für Mathematische Logik und für Grundlagen- forschung der Exakten Wissenschaften (DVMLG), the German Academic Schol- arship Foundation (Stipendiaten machen Programm), the Fachbereich Grundlagen derInformatikoftheGermanInformaticsSociety(GI)andtheGermanSocietyfor AnalyticPhilosophy(GAP). TheeditorsreceivedfundingfromtheClaussen-Simon-Stiftung,Studienstiftung des deutschen Volkes, Heinrich-Böll-Stiftung, Hamburger Stiftung zur Förderung von Wissenschaft und Kultur, the Schotstek Network and the German Research Foundation (Temporary Positions for Principal Investigators, Heisenberg Pro- gramme). Very special thanks go to the above-mentioned associations for the support of theconferenceaswellasoftheeditors.Withoutthissupportthevolumeasitstands wouldnothavebeenpossible.Theopinionsinthevolumedonotnecessarilymatch withthoseoftheagencies. The editors warmly thank Lukas Kühne and Balthasar Grabmayr for their help in organizing the conference. The encouragement at a decisive moment and the friendly advice from Synthese Library’s editor-in-chief, Otávio Bueno and from Springer’s project coordinator, Palani Murugesan, were truly invaluable. A very specialthanksgoestotheauthorsofthecontributionsandtoallanonymousreferees whoreviewedeachsinglecontribution. The Topic Set theory is widely assumed to serve as a suitable framework for foundational issuesinmathematics.However,anincreasingnumberofresearchersarecurrently investigating Univalent Foundations as an alternative framework for foundational issues. This relatively young approach is based on HoTT. It links Martin-Löf’s intuitionistic type theory and homotopy theory fromtopology. Such developments showthenecessityofanoveldiscussiononthefoundationofmathematics,orsowe believe.Thevolumepursuestwocomplementarygoals: 1. To provide a systematic framework for an interdisciplinary discussion among philosophers,computerscientistsandmathematicians 2. Toencouragesystematicthoughtoncriteriaforasuitablefoundation General criteria for foundations of mathematics can be drawn from the single contributions.Somecandidatesthereofare: (cid:129) Naturalnesswithregardtomathematicalpractice Introduction vii (cid:129) Applicabilityinmathematicalpractice (cid:129) Expressivepower (cid:129) Possibilityofextendingthetheorybyjustifiednewaxioms (cid:129) Possibilityofimplementingthetheoryintoformalproofsystems (cid:129) Interpretabilityofnon-classical(e.g.constructive)approaches (cid:129) Plausibilityoftheontologicalimplications Asfarassettheoryisconcerned,theresearchliteratureisrichinperspectivesand argumentations on foundational criteria. Roughly, set theory is seen as a theory with much expressive power, in which almost every other mathematical theory canbeinterpretedandwhichalsomayverywellserveasontologicalfoundational frameworkformathematics.However,itisnotapplicableinallmathematicalareas andisnoteasilyimplementedinformalproofsystems. AsfarasHoTTisconcerned,discussiononspecificfoundationalcriteriahasnot yetbeenproperlycarriedout.ManyscholarsdefendthethesisthatHoTTisavery well applicable theory, easily implemented in formal proof systems (such as Coq andAgda)thatcanserveasagoodfoundationalframeworkbydwellingratheron intensionalthanonextensionalfeaturesofmathematicalobjects,atavariancewith settheory. Notleast,thequestionaboutalternativestothestandardset-theoreticfoundation of mathematics seems to be relevant also in view of the last developments of formal mathematics that appears to develop more and more in the direction of a mathematicalpracticefocusingontheuseofautomatedtheoremprovers. HistoricalBackground Foundationaldisputesarenotnewinthehistoryofmathematics.TheGrundlagen- streit at the beginning of the twentieth century is but one ultimate example of a controversy, sometimes acrimonious, between various schools of thought opposed toeachother. Among the questions that mathematics poses to philosophical reflection on mathematics, those concerning the nature of mathematical knowledge and the ontological status of mathematical objects are central, when it comes to the foundationsofmathematics.Ismathematicsasciencewithanowncontentorisita language,orrather,alanguage-schemathatadmitsdifferentinterpretations?Does mathematical activity describe objects that are there, or does it constitute them? Howdoweexplainourknowledgeofmathematicalobjects? Ifwebelievethatmathematicsisasciencewithanowncontent,wehavetosay whichobjectsmathematicsistalkingabout.Iftheseobjectsaremind-independent, howdowehaveaccesstothem?Ifweconstitutethem,howdoweexplainthefact thatthesamemind-dependentobjectsaregraspedbydifferentsubjects? Philosophyofmathematicsgenerallydistinguishesthreedifferentviewsastothe foundationsofmathematics:Logicism,FormalismandIntuitionism. viii S.Centroneetal. Logicism. The descriptive view at the turn of the century was represented by the Logicism of Gottlob Frege (1848–1925). Even the word “descriptive” hints at the fact that we are confronted with a version of the traditional standpoint of platonism. Numbers and numerical relations are abstract logical objects. Number systemsarewell-determinedmathematicalrealities.Thetaskoftheknowingsubject is to discover and to describe such realities that subsist independently of him via truepropositionsaboutsuchobjects.Thelatter,organizedsystematically,makeup the theory of that mathematical reality. Platonism is most often associated with an eminently non-epistemic conception of truth: the truth value of a proposition isindependentofitsbeingknown. The programme of logicism was to ground finite arithmetic on logic: the basic concepts of arithmetic (natural number, successor, order relation, etc.) had to be defined in purely logical terms, and arithmetical true propositions had to be ideographicallyderivedfromlogicalprinciples. The arithmetization of analysis initiated by Karl Weierstrass (1815–1897) had concluded with the simultaneous publication in 1872 of the foundations of the system of real numbers by Richard Dedekind (1831–1916)1 and Georg Cantor (1845–1918)2.Since,beforethen,itwaswellknownhowtodefinerationalnumbers intermsofintegersandthelatterintermsofnaturalnumbers,thelastquestiontobe answeredwashowtoleadbacknaturalnumberstologic.Toanswersuchquestion, Frege formulated in his Grundgesetze der Arithmetik (1901-1903) a system of principlesfromwhichtheaxiomsoffinitearithmeticshouldhavebeenderived.Just asthesecondvolumeoftheGrundgesetzewasgettingintoprint,Fregereceiveda letterfromBertrandRussell(1872–1970),whocalledhisattentiononanantinomy arising by an indiscriminate use of the principle of unlimited comprehension, that is,the assumption that each concept has an extension. Frege could only recognize themistakeinapostscript. Russell’s Antinomy. Russell’s antinomy, we recall it, says that class of all classes that are not elements of themselves is and is not element of itself. Indeed, if it is, it is not, since it is the class of all classes that do not have themselves as elements.Ifitisnot,itis,forthesamereason.Theantinomyturnsouttoberelevant associatedwithBasicLawVofFrege’sGrundgesetzethatappliestofunctionsand their course of values as well as to concepts (which for Frege are a special kind of functions) and their extensions. The gap is caused by the fact that Frege takes courses of values as well as extensions of concepts to be objects and takes Basic Law V as an identity criterium for such objects. Basic Law V stipulates that the extensionsoftwoconcepts(moregenerallythecoursesofvaluesoftwofunctions) are the same, if and only if the concepts apply exactly to the same objects (or the twofunctionshavethesameinput-outputbehaviour). 1HeretoseeDedekind1872. 2HeretoseeCantor1872. Introduction ix BasicLawVforconceptsrunsasfollows: Ext(P)=Ext(Q)↔∀x(P(x)↔Q(x)) whileitsmoregeneralversionforfunctionsrunsasfollows: ε(cid:4)f(ε)=ε(cid:4)g(ε)↔∀x(f(x)=g(x)). Russell’santinomycanbepresentedinFrege’ssystemthus: Let“R(x)”standforthepredicate“xisRussellian”: (∗) R(x)↔∃Y(x =Ext(Y)∧¬Y(x)). Thatis,xisRussellianiffxistheextensionofaconceptthatdoesnotapplytox. Let “r” be short for “Ext(R)”, the extension of the concept R. Then the contradiction R(r)↔¬R(r). iseasilyderivedasfollows. (1) ¬R(r)→R(r).Assume¬R(r);then,accordingtothedefinitionofr: r =Ext(R)and¬R(r). Byexistentialquantifierintroductionitfollowsthereof: (∗∗)∃Y. r =Ext(Y)∧¬Y(r), andso,by(*),wegetR(r). (2) R(r)→ ¬R(r).AssumeR(r);thenby(*) ∃Y(r =Ext(Y)∧¬Y(r)). Letthen(byekthesis)Y besuchthatr=Ext(Y)and¬Y(r). Sor =Ext(R)andr =Ext(Y),henceExt(Y)=Ext(R).AtthispointBasicLaw Vcomesintoplay,yielding ∀z(Y(z)↔R(z)). Thereforefrom¬Y(r)weget¬R(r). Thus, in conclusion, r is at once Russellian and not-Russellian, and Frege’s systemis,atleastwithoutamendments,inconsistent.
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