Leibniz on Time, Space, and Relativity

423 Pages·2021·2.113 MB·English

by  Richard T. W. Arthur

Leibniz on Time, Space, and Relativity
Richard T. W. Arthur
https://doi.org/10.1093/oso/9780192849076.001.0001
Published: 2021 Online ISBN: 9780191944345 Print ISBN: 9780192849076
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https://doi.org/10.1093/oso/9780192849076.002.0003 Page iv
Published: December 2021
Subject: Philosophy of Science, History of Western Philosophy
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Preface
IfirstdiscoveredhowinterestingaphilosopherLeibnizwaswhileworkingonmy
PhDdissertation on time andthe foundations of physics atWestern inLondon,
Ontario in the late 1970s. Reading his discussion of space and time in his
controversy with Samuel Clarke, I found I was finally getting some insight into
how to interpret the ‘t’ that physicists manipulated in their equations. On my
returntoWesternafter ayear inCalabar,Nigeria, myteachingappointments in
the Department of Applied Mathematics allowed me leeway to follow up this
interestinearnest.IauditedRobertButts’graduateseminaronLeibniz,andgavea
talkonLeibniz’stheoryoftimetothePhilosophyDepartment.
Initially I had assumed, perhaps naively, that when Leibniz wrote about “the
phenomena” in physics he simply meant “what are observed,” and that his
accountofphenomenalbodiesasresultingfrommorebasicentities,hismonads
orsimplesubstances,wasanalogoustothesituationinmodernphysics.Theidea
that the extendedness of bodies is not fundamental, but derives from more
primitive entities having the nature of force, did not seem a far cry from the
situation as described by quantum physics. Similarly, his notion of the states of
substances following one another in a continuous series as a result of their
“appetition”(ortendencytowardssubsequentstates)seemedtohaveananalogue
in quantum theory, where the Hamiltonian operator acts on a given quantum
stateofanisolatedsystemtogeneratelaterstatesofthesamesystem.
The more I learned of Leibniz’s metaphysics, however, the more perplexing
I found it. As is well known, Leibniz insisted that substances do not strictly
speaking interact with one another. He equated their states with perceptions,
where perception is taken in the broad sense of a monad’s representation of the
universe(moreorlessconfusedly)fromitsparticularpointofview,makingtheir
appetition more like a generalization of desire, and rendering monads decidedly
mind-like. Yet, Leibniz held, physical bodies are infinite aggregates of monads,
andanychangeoccurringinsuchcompositespresupposeschangeinthequalities
ofthesimple.Howcouldthisbe?Bodiescouldnotbecomposedfromminds,nor
physicalchangesfrompsychicones(asLeibnizhimselfstressed).¹
¹ Fortheimpossibilityofsuchcompositions,wehavetheauthorityofLeibnizhimself.Ashewrote
toJohannBernoulli(September30,1698):“Youwereafraidthatmatterwouldbecomposedofnon-
quanta.Irespondthatitisnomorecomposedofsoulsthanofpoints”;andtoMichelangeloFardellain
1690:“itshouldnotbesaidthatindivisiblesubstanceentersintothecompositionofabodyasapart,
butratherasanessentialinternalrequisite”(AVI4,1669/AG103),and“asoulisnotapartofmatter,
butabodyinwhichthereisasoulissuchapart”(AG105).
viii 
Puzzlementaboutthewaybodiesandtheirchangesresultfrommonadsis,of
course, par for the course. But at least as perplexing for me was the general
consensus that Leibniz excluded relations from his fundamental ontology.
FollowingRussell,itiswidelybelievedthat,appearancestothecontrary,Leibniz
denied relations at the deepest level of his metaphysics. Since it is incontestable
thatheregardedspaceandtimeasrelational,thiswould(itisthought)accountfor
hisregardingthemasideal.Monads,onthisinterpretation,couldhavenolocation
in space and time, and would exist timelessly, like Kant’s noumena, only in the
intelligible realm. But such an interpretation, it seemed to me, was directly
contradicted by Leibniz himself in many places. In 1703 he assured his corres-
pondent De Volder, for example, that there is a place for all changes of both
spiritual and material things both “in the order of coexistents, that is, in space,”
and“intheorderofsuccessives,thatis,intime”(LDV266/267).Eventhoughthey
arenotthemselvesextended,simplesubstancescannotexistwithoutabody,“and
tothatextenttheydonotlacksituationororderwithrespecttoothercoexisting
thingsintheuniverse”(LDV266–269).Itwasonthisfoundation—namely,onthe
mutual situations of coexisting substances through their extended bodies—that
Leibnizbuilthistheoryofspace,asheexplained(alltoobriefly)toClarke.Here,it
istrue,one may argue that since monads areonlysituated throughtheirbodies,
andbodiesarephenomena,thentheserelationsareonlyamongthephenomena
andnotamongmonadsthemselves.Itisdifferentwithtime,however,sincethere
(as I have long argued) Leibniz bases temporal relations directly on relations
among monadic states. This calls into question the idea that the ideality of
relations precludes theexistence of monadsin time,or that temporal succession
appliesonlytothestatesofphenomena.Butifmonadicstatesareorderedintime,
and each state expresses the situations of the bodies of coexisting monads,
providing the basis for their spatial ordering, this suggests that space and time
arenotmerementalconstructions,butalsohavesomebasisinreality.Howthis
couldbeso,andinwhatsense,hasmotivatedthelineofresearchIhavepursued
thathasculminatedinthisbook.
ItbeganasthreechaptersofaprojectedvolumeonLeibniz’sLabyrinthofthe
Continuum,whichIhadoriginallytitledAriadneanThreads.Theideawastohave
each chapter corresponding to one of the topics Leibniz himself had included
under the rubric of a book project he had conceived in 1676 ‘de Compositione
continui,tempore,loco,motu,atomis,indivisibilietinfinito’(AVI3,77/DSR90)—
that is, on the composition of the continuum, time, place, motion, atoms, the
indivisible,andtheinfinite.Thatproject,however,becametoobigandunwieldy,
so I separated off what was pertinent to the theory of substance as a solution to
the labyrinth of the continuum, and published that in 2018 as Monads,
Composition, and Force, postponing the treatment of time, space, and the more
mathematical topics for another volume. Now that remainder has undergone a
furtherfission,asIrecognizethatatreatmentoftime,space,andmotion—allof
 ix
them relational—would form a coherent monograph all by itself, saving treat-
ments of the mathematics of the infinite and the infinitely small for further
projects. How the arguments of the present work relate to and depend on those
ofthepreviousoneIdescribeindetailintheintroductionbelow.
In addition to that introduction, this book is comprised by three substantial
chapters, each of seven sections, with its own introduction and conclusion, and
supplemented by four appendices and a glossary of the technical terms Leibniz
used, particularly in relation to the infinite. I have chosen to write a conclusion
specifictoeachchapterratherthanwritingageneralconclusion,andIincludein
eachsomeobservationsonhowLeibniz’sviewsrelatetomodernthinkingonthe
samesubject.
Chapter1isbuiltaroundmyfirstpublicationonLeibniz(‘Leibniz’sTheoryof
time’, 1985), which I had extensively reworked for intended inclusion in
AriadneanThreadsin2008–9.Itissupplementedbymaterialfrommytreatment
of the causal theory of temporal precedence (2016) in response to criticisms, as
well as from a forthcoming paper on vague states and discontinuous change to
appearinaforthcomingFestschriftforMassimoMugnai(thanksareduehereto
PeterMomtchiloffforgrantingmepermissiontousemuchofthematerialin§1.5
for that paper). This chapter also includes substantial new material on time and
contingency,andonreductionandthenatureofLeibniz’snominalismabouttime.
I present a formal exposition of the theory in Appendix 1: the relational core in
two versions, compossibility, temporal counterparts, and Leibniz’s complex and
innovative views on change and the continuity of time. On this last topic in
particular,IbelieveIhavebrokennewgroundhere.
Chapter2 builds upon my ‘Leibniz’s Theory of Space’ (2013b)—which itself
drew on ideas from my(1987) and (1994b)—although it mainly consistsin new
material.IexpanduponthegenesisofLeibniz’sviewsonspace,presentasuccinct
accountofthemainfeaturesofanalysissitusasamathematicaltreatmentofspace,
andtwosectionsonhowthisrelatestohismetaphysicsofspace.Leibniz’sanalysis
situs remained an unfinished project, and our understanding of it will almost
certainlyundergochangesandimprovementsasthecollectionandeditingofhis
manuscriptsonitproceeds.ButifIhavesucceededingivingsomesemblanceof
anaccountofitcompatiblewithmyreadingofLeibniz’smetaphysics,illuminated
by the contrast with De Risi’s phenomenalistic interpretation, I will be well
satisfied.
Chapter3 is a substantial reworking of a paper I finished in the summer of
2019, ‘Causes and the Relativity of Motion in Leibniz’. That paper drew on my
(1994a),andincorporatedelementsofotherpapersIpublishedwhileworkingon
it,(2013c),(2015a),and(2015b),butitturnedouttobetoolongforpublicationin
a journal. It is now about twice as long as it was in 2019, since it incorporates a
newsectiononCopernicanismandinstrumentalism,andasubstantialtreatment
of the whole question of whether Leibniz’s space could accommodate motion
x 
throughspaceandtime,andwhatkindof‘spacetime’isimplicitinthis.Idoubtif
this is the last word on Leibniz’s views on the relativity of motion, but I believe
Ihaveatleastmadeitseemfarmorecoherentthanitisgenerallyportrayedtobe.
Since the status of relations in Leibniz’s thought is both crucial to the inter-
pretationIgiveinthechapters,andyettooinvolvedforinclusioninthemaintext,
Ipresentanessaytreatingthisquestioninthesecondappendix.Inthethird,Igive
translations of extracts from Leibniz’s writing on analysis situs over the years,
since there is very little available in English translation. In the fourth, I give
translations of three drafts Leibniz wrote in Rome in 1689 on the question of
therelativityofmotion,CopernicanismandtheCensure.Finally,intheglossary
IexplainsomeofthetechnicaltermsLeibnizused,particularlyinconnectionwith
theinfinite.
It is a pleasure to acknowledge the generous feedback I have received from
colleaguesondraftsofthiswork.PreeminentamongthesehasbeenVincenzoDe
Risi, with whom I have been discussing and corresponding about Leibniz’s
metaphysics of space (and learning from him) ever since I was an examiner for
his PhD thesis at the Scuola Normale Superiore in Pisa in 2005. In response to
materialIhadaskedhimtolookover(thepenultimateversionsofchapter2and
section3.5),hesentmeanexquisite13-pageessay,whichwashugelyhelpfulfor
meinclarifyingmyownviewsaswellashis;healsoprovidedemendationsforthe
glossary.OsvaldoOttavianialsoreadthroughthewholemanuscriptandprovided
me with extremely valuable responses, sources, links, and corrections. Many
thanks, too, to the OUP readers, for their feedback on the draft manuscript
IsubmittedinSeptember2020,andsuggestionsforitsimprovement.Thathelped
metoclarifymythoughtandmyexpositionsofseveralpoints,andalsoprompted
metoprovidetheintroductorychapterandglossaryoftechnicalterms.Iamalso
very grateful to David Rabouin, Lucia Oliveri, Laurynas Adomaitis, Jeffrey
Elawani,andAngelaAxworthyfortheirsubstantialcriticalresponsestosamples
Isentthem;toFilippoCostantiniforwelcomeadviceandcommentary,especially
onthemereologyinAppendix1;toPaulLodge,JeffreyMcDonough,andMattia
Brancato for their suggestions and comments on some of the material; and to
Massimo Mugnai, Samuel Levey, Ed Slowik, Nico Bertoloni Meli, Pauline
Phemister, Daniel Garber, Tzuchien Tho, Jan Cover, Stefano Di Bella, Ohad
Nachtomy, Ursula Goldenbaum, Don Rutherford, Doug Jesseph, Jean-Pascal
Anfray, Enrico Pasini, Stephen Puryear, Martha Bolton, Mic Detlefsen, Marco
Panza, Laurence Bouquiaux, Arnaud Pelletier, and Gianfranco Mormino for
fruitfulexchangesofviewsovertheyearsonvariousaspectsofwhatisdiscussed
here.Thanks,also,toDavidRabouinfordrawingmyattentiontotextsonanalysis
situs recently prepared from manuscript sources by his team in the ANR
MATHESIS project in collaboration with the Leibniz Research Centre in
Hanover (Leibniz-Archiv),andtohim,Siegmund Probst,VincenzoDeRisi,and
MichaelKempeforpermissiontopublishtranslationsofthreeofthemhere(they
 xi
are to appear under the Creative Commons licence CC-by-NC 4.0.). I am
particularly indebted to Siegmund Probst for his prompt and invaluable expert
helpindatingthesemanuscripts.
Thanks, finally, to Peter Momtchiloff of Oxford University Press for his
unstinting help in seeing this project through to completion, and to family and
friendsfortheirforbearanceandsupportinthecreativeprocess.