Mathesis metaphysica quadam

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2 Mathesismetaphysicaquadam Leibniz,entreMathématiquesetPhilosophie/Leibniz,betweenMathematicsandPhilosophy Colloqueinternational/InternationalConference M.Detlefsen(ANR chaired excellence«idealsofproof») &D.Rabouin(REHSEIS,laboratoireSPHERE,UMR7219) 8 10mars2010 TheIdealsofProof(IP)Project(ANR)andREHSEIS(UMR7219,SPHERE)arepleasedto announceaworkshoponinterrelationshipsbetweenmathematicsandphilosophyinthe thoughtofleibniz.theworkshopwilltakeplacemonday,march8throughwednesday, March10,2010.AllmeetingsthefirsttwodayswillbeinthesalleDusanneoftheENS (45rued'Ulm,Paris).Onthethirdday,themeetingswillbeintheKleeroom,whichis room454aofthecondorcetbuilding,onthegrandsmoulinscampusoftheuofparis Diderot.Thetalksanddiscussionsarefreeandopentothepublic.Allinterestedpersons arewarmlyinvitedtoattend. Monday,March8th.SalleDusanne,ENS SessionI:9h00 10h50 HerbertBreger(Leibniz Archiv,Hannover):"ThesubstructureofLeibniz'smetaphysics" Abstract Leibnizbelievedthathisphilosophywasnearlymathematicalorcouldbetransformedinto mathematicalcertainty.weknowthathetherebyalludedtohisprojectofacharacteristica universalis.althoughwearenotconvincedofleibniz'sclaim,wecanrecognizea substructureofleibniz'smetaphysicswhichismathematicalorisbuiltonnotionsof philosophyofmathematics. SessionII:11h00 12h50 MichelSerfati(IREM,UniversitéParisVII):"Mathematics,metaphysicsandsymbolismin Leibniz:theprincipleofcontinuity" Abstract OnthebasisoftheepistemologicalanalysisofseveralLeibnizianmathematicalexamplesIwillfirst attempttoshowhowleibniz s principleofcontinuity (whichis,infact,ameta principle)belongsto theconceptualframeworkofwhathecalls symbolicthought,atleastinsofarasitsmathematical implementations are concerned. I will then show how the ambiguity of the mathematical and metaphysical status of the principle engendered controversies between Leibniz and some of his correspondents.inlightoftheseseveralexampleswewillalsoappreciateleibniz sstandvis à visa centralquestion,underlyingthisstudy,namelythearchitectoniccharacterinmathematicalthought

3 of the use of signs, and especially this crucial point, namely the capacity of the mathematician to remain at the symbolic level, to continue to work at that level, retarding as much as possible the appeal to meanings; this is a primordial prescription for Leibniz deriving from what he calls the autarchic character of the sign. Finally, it will be shown how this seventeenth century principle remains to these days fully operational in research and teaching. Let us repeat, emphasizing Leibniz s glory: these epistemological procedures, this continuity schema seen as interiorizedmethodologicalguide,thisspecificconceptionofcontinuityasaregulativeprinciple all theseconceptsthataresofamiliartopresentdaymathematiciansthatonecanhardlyimaginethat they were not so before, were in fact ignored before him, especially by Descartes. Nowadays, therefore, 300 years after Leibniz, the principle of continuity belongs to an interiorized set of methodological rules. Like the principle of homogeneity (considered as normative) or that of generalization extension (considered as a standard procedure of construction of algebraic or topologicalobjects),theyconstitutepartofthedailymathematicalpractice. Lunch12h50 14h20 SessionIII:14h30 16h20 VincenzoDeRisi(HumboldtFellow,TechnischeUniversität,Berlin):"Leibniz'sstudiesonthe ParallelPostulate" Abstract Thedevelopmentofnon Euclideangeometriesisoftenregardedasanexemplartopictoconsiderthe relationsbetweenphilosophyandmathematics.althoughthephilosophicalimportofthe foundationalstudiesontheparallelpostulatewasclearlyrecognizedonlyinthesecondhalfofthe 19thcentury,thereisnodoubtthatanumberofmathematiciansandphilosopherswerewellaware ofthemetaphysicalandepistemologicalissuesraisedbythetheoryofparallelismalreadyinthe17th and18thcenturies.iwouldliketoshowthemostimportantcontributionsofleibniztothe geometricalfoundationsofatheoryofparallelsandtheirphilosophicalconsequencesforafull fledgedmonadology,framinghisconsiderationsinthemoregeneralconceptualdevelopmentwhich eventuallygavebirthtothemodernconceptofspace. Tuesday,March9,2010 SessionIV:9h00 10h50 PhilipBeeley(LinacreCollege,Oxford):"Indeliberationibusadvitampertinentibus.Method andcertaintyinleibniz smathematicalpractice" Abstract Alreadyinhisearliestphilosophicalwritings,Leibnizwasconcernedtoaccountforthesuccessinthe application of the mathematical sciences to our understanding of nature. In this context he developedasophisticatedconceptofnegligibleerrorwhichwouldlaterstandhimingoodsteadin

4 hismathematicalwritings,particularlythosewhicharenowseentohaveplayedadecisiveroleinthe emergence of modern analysis. The paper examines the historical and methodological context of Leibniz serrorconceptandshowshowitservestoexemplifytheprofoundwaysinwhichleibniz s philosophical deliberations on the application of mathematics informed his mathematical practice. SessionV:11h00 12h50 RichardArthur(DepartmentofPhilosophy,McMasterUniversity): Leibniz sactualinfinite inrelationtohisanalysisofmatter Abstract In this paper I examine some aspects of the relationship between Leibniz's thinking on the infinite and his analysis of matter. I begin by examining Leibniz's quadrature of the hyperbola, which demonstrates the connection between his work on infinite series, his upholding of the part whole axiom, and his consequent fictionalist understanding of the infinite and infinitely small. On that conception,tosaythatthereareactuallyinfinitelymanyprimes,forexample,istosaythatthereare morethancanbeassignedanyfinitenumbern.thereisnosuchthingasthecollectionofallprimes, noranumberofthemthatisgreaterthanalln,althoughonecancalculatewithsuchanumberasa fiction under certain well defined conditions. I then show how this conception fits with Leibniz's conceptionoftheinfinitedivisionofmatter,relatingittohisprincipleofcontinuity,andcontrasting hispositionontheinfiniteandthecompositionofmatterwiththoseofgeorgcantor. Lunch12h50 14h20 SessionVI:14h30 16h20 SamuelLevey(DepartmentofPhilosophy,DartmouthCollege):"Leibniz'sanalysisof Galileo'sparadox" Abstract In "Two New Sciences" Galileo shows that the natural numbers can be mapped into the square numbers,yieldingtheseeminglyparadoxicalresultthatthenaturalsarebothgreaterthanandequal to the squares. Galileo concludes that in the infinite the terms 'greater', 'less', and 'equal' do not apply.leibnizrejectsthisonthegroundthatitwouldviolatetheaxiom,fromeuclid,thatthewhole isgreaterthanthepart,andarguesinsteadthattheparadoxshowstheideaofaninfinitewholeto becontradictory.russellandothershaverejectedleibniz'sanalysisasrestingonacrucialambiguity ineuclid'saxiom,andgalileo'sparadoxistypicallyresolvedbyabandoningakeypremise.leibnizhas some defenders who point out that Russell's criticisms proceed from a distinctly post Cantorian standpoint. In this paper I argue that Leibniz's analysis involves a more subtle error, and even granting Galileo's premises as well as Euclid's Axiom it does not follow that the idea of an infinite wholeiscontradictory.thisfollowsonlyona"strong"definitionof'infinite',whereasleibniz'sown (now standard) definition allows infinite wholes consistently with Galileo's paradox. The details involvedcastlightwellintotherecessesofleibniz'sphilosophyofmathematics.

5 Wednesday,March10 SessionVII:9h00 10h50 EmilyGrosholz(DepartmentofPhilosophy,PennsylvaniaStateUniversity):"The RepresentationofTimeinGalileo,NewtonandLeibniz" Abstract I revisit the representation of time in the writings of Galileo, Newton and Leibniz, to see whether theytreattimeasconcrete,choosingrepresentationsthatallowustoreferwithreliableprecision,or as abstract, choosing representations that allow us to analyze successfully, locating appropriate conditionsofintelligibility.iinterprettheconflictbetweenleibnizandnewtonasduetodifferences intheirchoiceofrepresentation,andarguethattheconflicthasnotbeenresolved,butremainswith us. SessionVIII:11h00 12h50 Eberhard Knobloch (Institut für Philosophie, Wissenschaftstheorie, Wissenschafts und Technikgteschichte, Technische Universität Berlin): "Analyticité, équipollence et la théorie descourbeschezleibniz" Abstract Avantetaprèsl'inventiondesoncalculdifférentiel,Leibnizs'appuyaitsurlesnotionsd'analyticitéet d'équipollence de lignes et de figures. Ces deux notions jouent un rôle essentiel dans les mathématiquesleibniziennes.lesdeuxpartiesdelagéométrieontbesoindedeuxdifférentstypes d'analysetandisquel'existencedel'analyseentraînelagéométricitédesobjets.plustard,leibniza donnéuneautrejustificationpourlagéométricitéd'unecourbe.qu'est cequeveutdire'analytique' ou 'équipollent'? La conférence essaiera de clarifier cette question. En plus,, elle va expliquer la classificationleibniziennedescourbes,enparticuliercelledela'quadraturearithmétiqueducercle etc.', leur géométricité et va démontrer quelques théorèmes généraux sur les courbes analytiquessimples. Pourplusd informations/forfurtherinformation,pleasecontacteitheroftheworkshop organizers,michaeldetlefsen(mdetlef1@nd.edu)ordavidrabouin(rabouin@ens.fr).

It s a Small World After All Calculus without s and s

It s a Small World After All Calculus without s and s Dan Sloughter Department of Mathematics Furman University November 18, 2004 Smallworld p1/39 L Hôpital s axiom Guillaume François Antoine Marquis de