MathesismetaphysicaquadamLeibniz,entreMathématiquesetPhilosophie/Leibniz,betweenMathematicsandPhilosophy

Colloqueinternational/InternationalConferenceM.Detlefsen(ANR‐chaired’excellence«IdealsofProof»)&D.Rabouin(REHSEIS,laboratoireSPHERE,UMR7219)

8‐10mars2010

TheIdealsofProof(IP)Project(ANR)andREHSEIS(UMR7219,SPHERE)arepleasedtoannounceaworkshoponinterrelationshipsbetweenmathematicsandphilosophyinthe

thoughtofLeibniz.TheworkshopwilltakeplaceMonday,March8throughWednesday,March10,2010.AllmeetingsthefirsttwodayswillbeinthesalleDusanneoftheENS

(45rued'Ulm,Paris).Onthethirdday,themeetingswillbeintheKleeroom,whichisroom454AoftheCondorcetbuilding,ontheGrandsMoulinscampusoftheUofParis‐

Diderot.Thetalksanddiscussionsarefreeandopentothepublic.Allinterestedpersonsarewarmlyinvitedtoattend.

Monday,March8th.SalleDusanne,ENS

SessionI:9h00‐10h50HerbertBreger(Leibniz‐Archiv,Hannover):"ThesubstructureofLeibniz'smetaphysics"AbstractLeibnizbelievedthathisphilosophywasnearlymathematicalorcouldbetransformedintomathematicalcertainty.Weknowthathetherebyalludedtohisprojectofacharacteristicauniversalis.AlthoughwearenotconvincedofLeibniz'sclaim,wecanrecognizeasubstructureofLeibniz'smetaphysicswhichismathematicalorisbuiltonnotionsofphilosophyofmathematics.SessionII:11h00‐‐12h50MichelSerfati(IREM,UniversitéParisVII):"Mathematics,metaphysicsandsymbolisminLeibniz:theprincipleofcontinuity"

AbstractOnthebasisof theepistemologicalanalysisofseveralLeibnizianmathematicalexamples Iwill firstattempttoshowhowLeibniz’s“principleofcontinuity”(whichis,infact,ameta‐principle)belongsto

theconceptual frameworkofwhathecalls“symbolicthought”,at least insofaras itsmathematicalimplementations are concerned. I will then show how the ambiguity of the mathematical andmetaphysical status of the principle engendered controversies between Leibniz and some of his

correspondents.InlightoftheseseveralexampleswewillalsoappreciateLeibniz’sstandvis‐à‐visacentralquestion,underlyingthisstudy,namelythearchitectoniccharacterinmathematicalthought

of the use of signs, and especially this crucial point, namely the capacity of themathematician to

remain at the symbolic level, to continue towork at that level, retarding asmuch as possible theappeal tomeanings ; this is aprimordial prescription for Leibniz –deriving fromwhathe 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 asinteriorizedmethodologicalguide,thisspecificconceptionofcontinuityasaregulativeprinciple–all

theseconceptsthataresofamiliartopresentdaymathematiciansthatonecanhardlyimaginethatthey 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 ofgeneralization‐extension (considered as a standard procedure of construction of algebraic ortopologicalobjects),theyconstitutepartofthedailymathematicalpractice.

Lunch12h50‐‐14h20SessionIII:14h30‐‐16h20VincenzoDeRisi(HumboldtFellow,TechnischeUniversität,Berlin):"Leibniz'sstudiesontheParallelPostulate"AbstractThedevelopmentofnon‐Euclideangeometriesisoftenregardedasanexemplartopictoconsidertherelationsbetweenphilosophyandmathematics.Althoughthephilosophicalimportofthe

foundationalstudiesontheParallelPostulatewasclearlyrecognizedonlyinthesecondhalfofthe19thcentury,thereisnodoubtthatanumberofmathematiciansandphilosopherswerewellawareofthemetaphysicalandepistemologicalissuesraisedbythetheoryofparallelismalreadyinthe17th

and18thcenturies.IwouldliketoshowthemostimportantcontributionsofLeibniztothegeometricalfoundationsofatheoryofparallelsandtheirphilosophicalconsequencesforafull‐fledgedmonadology,framinghisconsiderationsinthemoregeneralconceptualdevelopmentwhich

eventuallygavebirthtothemodernconceptofspace.

Tuesday,March9,2010

SessionIV:9h00‐10h50PhilipBeeley(LinacreCollege,Oxford):"Indeliberationibusadvitampertinentibus.MethodandCertaintyinLeibniz’sMathematicalPractice"AbstractAlreadyinhisearliestphilosophicalwritings,Leibnizwasconcernedtoaccountforthesuccessinthe

application of the mathematical sciences to our understanding of nature. In this context hedevelopedasophisticatedconceptofnegligibleerrorwhichwouldlaterstandhimingoodsteadin

hismathematicalwritings,particularlythosewhicharenowseentohaveplayedadecisiveroleinthe

emergence of modern analysis. The paper examines the historical andmethodological context ofLeibniz’serrorconceptandshowshow itservestoexemplify theprofoundways inwhichLeibniz’sphilosophical deliberationson the applicationofmathematics informedhismathematical practice.

SessionV:11h00‐‐12h50RichardArthur(DepartmentofPhilosophy,McMasterUniversity):“Leibniz’sActualInfiniteinRelationtohisAnalysisofMatter”AbstractIn thispaper Iexamine someaspectsof the relationshipbetweenLeibniz's thinkingon the infinite

and his analysis of matter. I begin by examining Leibniz's quadrature of the hyperbola, whichdemonstrates the connectionbetweenhisworkon infinite series, his upholdingof thepart‐wholeaxiom, and his consequent fictionalist understanding of the infinite and infinitely small. On that

conception,tosaythatthereareactuallyinfinitelymanyprimes,forexample,istosaythattherearemorethancanbeassignedanyfinitenumberN.Thereisnosuchthingasthecollectionofallprimes,noranumberofthemthatisgreaterthanallN,althoughonecancalculatewithsuchanumberasa

fiction under certain well‐defined conditions. I then show how this conception fits with Leibniz'sconceptionoftheinfinitedivisionofmatter,relatingittohisprincipleofcontinuity,andcontrastinghispositionontheinfiniteandthecompositionofmatterwiththoseofGeorgCantor.

Lunch12h50‐‐14h20SessionVI:14h30‐‐16h20SamuelLevey(DepartmentofPhilosophy,DartmouthCollege):"Leibniz'sanalysisofGalileo'sparadox"

AbstractIn "Two New Sciences" Galileo shows that the natural numbers can be mapped into the square

numbers,yieldingtheseeminglyparadoxicalresultthatthenaturalsarebothgreaterthanandequalto the squares. Galileo concludes that in the infinite the terms 'greater', 'less', and 'equal' do notapply.Leibnizrejectsthisonthegroundthatitwouldviolatetheaxiom,fromEuclid,thatthewhole

isgreaterthanthepart,andarguesinsteadthattheparadoxshowstheideaofaninfinitewholetobecontradictory.RussellandothershaverejectedLeibniz'sanalysisasrestingonacrucialambiguity

inEuclid'sAxiom,andGalileo'sparadoxistypicallyresolvedbyabandoningakeypremise.Leibnizhassome defenders who point out that Russell's criticisms proceed from a distinctly post‐Cantorianstandpoint. In this paper I argue that Leibniz's analysis involves a more subtle error, and even

grantingGalileo'spremises aswell as Euclid'sAxiom it doesnot follow that the ideaof an infinitewholeiscontradictory.Thisfollowsonlyona"strong"definitionof 'infinite',whereasLeibniz'sown(now standard) definition allows infinite wholes consistently with Galileo's paradox. The details

involvedcastlightwellintotherecessesofLeibniz'sphilosophyofmathematics.

Wednesday,March10

SessionVII:9h00‐‐10h50EmilyGrosholz(DepartmentofPhilosophy,PennsylvaniaStateUniversity):"TheRepresentationofTimeinGalileo,NewtonandLeibniz"

AbstractI revisit the representationof time in thewritings ofGalileo,Newton and Leibniz, to seewhethertheytreattimeasconcrete,choosingrepresentationsthatallowustoreferwithreliableprecision,oras abstract, choosing representations that allow us to analyze successfully, locating appropriate

conditionsofintelligibility.IinterprettheconflictbetweenLeibnizandNewtonasduetodifferencesintheirchoiceofrepresentation,andarguethattheconflicthasnotbeenresolved,butremainswithus.SessionVIII:11h00‐‐12h50Eberhard Knobloch (Institut für Philosophie, Wissenschaftstheorie, Wissenschafts‐ undTechnikgteschichte, TechnischeUniversität Berlin): "Analyticité, équipollence et la théoriedescourbeschezLeibniz"

AbstractAvantetaprèsl'inventiondesoncalculdifférentiel,Leibnizs'appuyaitsurlesnotionsd'analyticitéetd'équipollence de lignes et de figures. Ces deux notions jouent un rôle essentiel dans lesmathématiques leibniziennes.Lesdeuxpartiesde lagéométrieontbesoindedeuxdifférentstypes

d'analysetandisquel'existencedel'analyseentraînelagéométricitédesobjets.Plustard,Leibnizadonnéuneautrejustificationpourlagéométricitéd'unecourbe.Qu'est‐cequeveutdire'analytique'ou 'équipollent'? La conférence essaiera de clarifier cette question. En plus,, elle va expliquer la

classificationleibniziennedescourbes,enparticuliercellede la 'Quadraturearithmétiqueducercleetc.', leur géométricité et va démontrer quelques théorèmes généraux sur les courbesanalytiquessimples.

Pourplusd’informations/Forfurtherinformation,pleasecontacteitheroftheworkshop

organizers,MichaelDetlefsen(mdetlef1@nd.edu)orDavidRabouin(rabouin@ens.fr).