Leibniz and the Structure of Sciences: Modern Perspectives on the History of Logic, Mathematics, Epistemology PDF
Preview Leibniz and the Structure of Sciences: Modern Perspectives on the History of Logic, Mathematics, Epistemology
Boston Studies in the Philosophy and History of Science 337 Vincenzo De Risi Editor Leibniz and the Structure of Sciences Modern Perspectives on the History of Logic, Mathematics, Epistemology Boston Studies in the Philosophy and History of Science Volume 337 SeriesEditors AlisaBokulich,BostonUniversity,Boston,MA,USA JürgenRenn,MaxPlanckInstitutefortheHistoryofScience,Berlin,Germany MichelaMassimi,UniversityofEdinburgh,Edinburgh,UK ManagingEditor Lindy Divarci, Max Planck Institute for the History of Science, Berlin, Berlin, Germany EditorialBoardMembers TheodoreArabatzis,UniversityofAthens,Athens,Greece HeatherE.Douglas,UniversityofWaterloo,Ontario,Canada JeanGayon,UniversitéParis1,Paris,France ThomasF.Glick,BostonUniversity,Boston,USA HubertGoenner,UniversityofGoettingen,Göttingen,Niedersachsen,Germany JohnHeilbron,UniversityofCalifornia,Berkeley,UK DianaKormos-Buchwald,CaliforniaInstituteofTechnology,Pasadena,CA,USA ChristophLehner,MaxPlanckInstitutefortheHistoryofScience,Berlin,Germany PeterMcLaughlin,UniversitätHeidelberg,Heidelberg,Baden-Württemberg, Germany AgustıNieto-Galan,UniversitatAutònomadeBarcelona,Bellaterra(Cerdanyola delV.),Spain NuccioOrdine,UniversitádellaCalabria,RENDE,Cosenza,Italy AnaSimões,UniversidadedeLisboa,Lisboa,Portugal JohnJ.Stachel,BostonUniversity,Brookline,MA,USA BaichunZhang,ChineseAcademyofScience,Beijing,China SylvanS.Schweber,HarvardUniversity,Cambridge,USA TheseriesBostonStudiesinthePhilosophyandHistoryofSciencewasconceived in the broadest framework of interdisciplinary and international concerns. Natural scientists, mathematicians, social scientists and philosophers have contributed to the series, as have historians and sociologists of science, linguists, psychologists, physicians,andliterarycritics. Theserieshasbeenabletoincludeworksbyauthorsfrommanyothercountries aroundtheworld. The editors believe that the history and philosophy of science should itself be scientific, self-consciously critical, humane as well as rational, sceptical and undogmaticwhilealsoreceptivetodiscussionoffirstprinciples.Oneoftheaimsof Boston Studies, therefore, is to develop collaboration among scientists, historians andphilosophers. BostonStudiesinthePhilosophyandHistoryofSciencelooksintoandreflects on interactions between epistemological and historical dimensions in an effort to understandthescientificenterprisefromeveryviewpoint. Moreinformationaboutthisseriesathttp://www.springer.com/series/5710 Vincenzo De Risi Editor Leibniz and the Structure of Sciences Modern Perspectives on the History of Logic, Mathematics, Epistemology 123 Editor VincenzoDeRisi LabratoireSPHère-CNRS Paris,France MaxPlanckInstitutefortheHistory ofScience Berlin,Germany ISSN0068-0346 ISSN2214-7942 (electronic) BostonStudiesinthePhilosophyandHistoryofScience ISBN978-3-030-25571-8 ISBN978-3-030-25572-5 (eBook) https://doi.org/10.1007/978-3-030-25572-5 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Foreword The present volume brings together eight essays on Gottfried Wilhelm Leibniz’s work on logic, mathematics, and epistemology. The essays have been arranged in broadly systematic order. The book begins with two studies on various aspects of Leibniz’s logic and its connection with other exact sciences (the contribution by Marko Malink and Anubav Vasudevan on the notion of logical coincidence and the one by Massimo Mugnai on Leibniz’s mereology). It continues with an essay on infinity and cardinality (by Richard Arthur) which succeeds in bridging the gap between pure logic and mathematics and with further contributions on the foundations of geometry (by Vincenzo De Risi on continuity in elementary geometry and David Rabouin and Valérie Debuiche on the possibility of non- EuclideangeometryinLeibniz’sthought).Thegeometricalperspectiveiscombined withconsiderationsonthecalculusinanessayonLeibniz’streatmentofcurves(by DavideCrippa),andaforayismadeintotherealmofmathematicalphysicswitha paperonLeibniz’scalculusofvariations(byJürgenJost).Thefinalessay(byNabeel Hamid)drawsonwhattheseprecedinginvestigationsinthehistoryofmathematics haveestablishedtooffersomenewviewsonLeibniz’sepistemologyofthephysical sciences. The book can serve to open up an interesting line of approach to many aspects of the work that Leibniz carried out in the exact sciences. The history of epistemology is transversally addressed in all the contributions, with a close connection being maintained throughout to Leibniz’s actual scientific practice as a working mathematician. Several essays deal with the history of science as such and enter, in considerable depth, into Leibniz’s technical achievements in logic, mathematics,ormathematicalphysics. The title of the book, Leibniz and the Structure of Sciences, stresses the “structural” approach which was characteristic of Leibniz’s own epistemological and scientific research and which may well represent the main step forward that Leibniz took as compared to several of his contemporaries. The structure of Leibnizianscience,however,isalsoillustratedbytheinternalconnectionamongthe essays published in this volume and their complex relations of grounding: so that logicseemstogroundmereology,mereologygroundsthetheoryofcardinalityand v vi Foreword mathematicsasawhole,thelattergroundsgeometryandthenanalysis,andanalysis forms the foundation of mathematical physics. In the course of the systematic exposition of the structures of these disciplines, however, the reader will come to realize that the connections between them are more complicated still and that, for instance,Leibniz’sepistemologyofgeometryhadrepercussionsonhisconceptions of whole and part and these latter on his conception of logic itself. Leibniz’s scientific practices (as opposed to his epistemological views) further enrich the picture, and the result is a complex relation of mutual grounding among various disciplines–arelationthatLeibnizhimselfwouldprobablyhavecalledapeculiar formofharmony–whichweourselvescanlookuponasaremarkablestructureof thefoundationsofsciences. Thesubtitleofthebook—ModernPerspectivesontheHistoryofLogic,Mathe- matics,Epistemology—underlinesthefactthatseveralessaysinthecollectionoffer someinsightfulcomparisonwithmoderntreatmentsofthesubject.Thus,theessay byMalinkandVasudevandealswithLeibniz’slogicalsystemfromtheperspective ofmodernlogicandBooleanalgebras;MugnaireadsLeibniz’sdefinitionsofwhole and part through the spectacles of contemporary mereology; Arthur compares Leibniz’s and Cantor’s views on the infinite; De Risi discusses Leibniz’s notion of continuity in relation to Dedekind completeness; Rabouin and Debuiche deal with Leibniz and non-Euclidean geometry; and Jost looks at Leibniz’s calculus of variationswiththetoolsoftoday’smathematics.ThesemodernreadingsofLeibniz do not, indeed, form the center of these essays, which are not written in order to draw such comparisons; but the authors nonetheless discovered that, in some cases, a modern perspective was apt to shed light on the actual force, structure, orlimitsofLeibniz’sendeavors.Thoughawareofthefactthatahistoriographical approachmakinguseofmoderncognitivetoolstointerpretearlymodernscientific results may easily resultinblatant anachronism, we think,and hope, that wehave avoided such a pitfall and have provided, on the contrary, some tools with which to better appreciate the actual historical dimension of Leibniz’s researches. In the past,indeed,Leibnizhasoftenbeenanachronisticallyaddressedastheunknowing fatherofmodernlogic,moderngeometry,topology,nonstandardanalysis,relativity theory,andsoforth.Thatistosay,noreallyattentivecomparisonbetweenhisviews andthesefurtherdevelopmentshasreallybeenundertaken—butratherananalogy or similarity has been uncritically assumed to exist. By confronting Leibniz with Cantor or Dedekind, the present volume wants to rectify such anachronisms, and to point up both the similarity and the distance between Leibniz’s project and the modern results. We hope that, in this way, the exact extent of Leibniz’s scientific innovationswillbebettergrasped.Atthesametime,thepresentvolumemayoffer materialsforreflectionnotjusttoLeibnizspecialistsbutalsotohistoriansofscience and epistemology, and philosophers of science, working on modern authors and problems. Several of the essays in this volume emerged from the renewed scholarly work around Leibniz that was fostered by the celebrations of Leibniz’s tercentenary in 2016. In particular, the editor of this volume, appointed Leibniz Chair at the UniversityofLeipzigintheWinterSemester2016–2017,contributedtoorganizing, Foreword vii incollaborationwiththeMaxPlanckInstituteforMathematicsintheSciencesand the Max Planck Institute for the History of Science, a Summer School on Leibniz that was held in Leipzig and Hannover on July 7–16, 2016, and a conference on “Leibniz and the Sciences” that took place in Leipzig on November 14–16, 2016. We thank the Leibniz Program of the University of Leipzig and the Max Planck Institutes for their generous support and their contributions to the advancement of Leibnizstudies. We also thank Andrew Arana, François Duchesneau, Daniel Garber, Eberhard Knobloch, Jeffrey McDonough, John Mumma, and Justin Smith for having con- tributedtoimprovingtheessaysinthepresentvolume. Contents 1 LeibnizontheLogicofConceptualContainmentandCoincidence ... 1 MarkoMalinkandAnubavVasudevan 2 Leibniz’sMereologyintheEssaysonLogicalCalculus of1686–1690.................................................................. 47 MassimoMugnai 3 LeibnizinCantor’sParadise:ADialogueontheActualInfinite....... 71 RichardT.W.Arthur 4 LeibnizontheContinuityofSpace ........................................ 111 VincenzoDe Risi 5 OnthePluralityofSpacesinLeibniz...................................... 171 ValérieDebuicheandDavidRabouin 6 One String Attached: Geometrical Exactness in Leibniz’s ParisianManuscripts........................................................ 203 DavideCrippa 7 LeibnizandtheCalculusofVariations.................................... 253 JürgenJost 8 TeleologyandRealisminLeibniz’sPhilosophyofScience.............. 271 NabeelHamid ix Contributors RichardT.W.ArthurMcMasterUniversity,Hamilton,ON,Canada DavideCrippaInstituteofphilosophy,CentreforScience,Technology,andSociety Studies,CzechAcademyofSciences,Prague,CzechRepublic ValérieDebuiche Aix Marseille Univ, CNRS, Centre Gilles-Gaston Granger, Aix-en-Provence,France NabeelHamidConcordiaUniversity,Montréal,Canada JürgenJost Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany MarkoMalinkNewYorkUniversity,NewYork,NY,USA MassimoMugnaiScuolaNormaleSuperiore,Pisa,Italy DavidRabouin Université de Paris, Laboratoire SPHERE, UMR 7219, CNRS, Paris,France VincenzoDeRisiLaboratoireSPHère,Paris,France MaxPlanckInstitutefortheHistoryofScience,Berlin,Germany AnubavVasudevanUniversityofChicago,Chicago,IL,USA xi
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