Mathesis metaphysica quadam · Mathesis metaphysica quadam Leibniz, entre Mathématiques et...
Transcript of Mathesis metaphysica quadam · Mathesis metaphysica quadam Leibniz, entre Mathématiques et...
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([email protected])orDavidRabouin([email protected]).