"Fiir unser G liick oder das G liick anderel-" V ortl-age des X . Intem ationalen Leibniz-K oIlgresses H annover, 18. - 23. Juli 2016 H el-allsgegebe ‫מ‬ W enchao von Li ill V el-bi ‫ מ‬dllng m it U te Becklnalln, Jilrgen H erbst, Sven EI-dner, Esther-M al'ia ‫ו‬ H c el‫ ך‬a Iw asinski und Sim ona Band 11 o ~ Geol-g OII ‫ ח‬S Vel-lag Hildesl1eilll' ZUl'ich' New YOI-k 2016 El'ru!at , N oreik Das Bild allt' dem Um schlag \\flll'de enll101111 ‫ ח‬C11aus : ] oI1ann ALlgUSt Ehcrhard, Gottfried \\liI11ell ‫ ח‬Freyhel'l' \'011 Lcihl ‫ ך‬itz. CI ‫ ן‬el ‫ ן ן‬I‫ ך‬itz 1795 , Nacl1dl'llck Hildcsl1cim 1982, z\\'iscl ‫ ו‬el1 S. 176 Ul1d 177 ("Leibnitz stil·bt "). Das \\1 crk ist llrhebe ‫ ז‬ecl ‫ ו‬tlich gescl ‫ ך‬i.itzt, .Tede V cr\\'crtlII ‫ ך‬g aLIJ3erl‫ ו‬alb deI' cI1gcn CJrcnzcl ‫ ך‬des Url ‫ ך‬ebcrl'ccl ‫ ך‬tsgesetzes ist OI‫ ך‬I‫ ך‬C ZLIStil‫ ך ך‬I‫ ך ו‬Lll‫ ו‬g des Verlages LII‫ ו‬zLllassig. 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Die Deutsche Bibliothek \fcrzcichnet diese Pllblikatiol ‫ר‬ bibliogra f scl ‫ ר‬e Datel ‫ו‬ I1‫ ך‬ternet i.iber 11ttp:!!dnb.ddb.de abl'llfbar in der Delltschel ‫ ו‬Natiol ‫ ך‬albibliogt'a f e; dctaillierte sil ‫ ו‬d il ‫ח‬ ISO 9706 Gedl'Llckt allf s3llrefreiel ‫ ח‬LII‫ ר‬d altcrungsbcstaI ‫ ר‬digel111)apiel ' Ul11scl‫ ך‬laggcst,11tLlng: II‫ ך‬ga Gi.'II‫ ך‬tl ‫ ו‬cr , 31134 ‫ ן‬-‫ ז‬ildesl1eil ‫ך ו‬ Satz: SiI ‫ ו‬10l‫ ו‬a ;'>J()I'cik,38300 W ol1enbiittel HerstelluI1g: I-lllbcl1 & (0, 37079 G (‫ ן‬ttil ‫ ן‬geI ‫ר‬ Pl'iI‫ ר‬ted in GeI'I ‫ ר‬1aI‫' נ ן‬ :(', Geol'g OII ‫ ח‬S Vel'lag AG, Hildeshein Alle Rechte \forbehaltel ‫ן‬ \\'\ V\\' ,oIn ‫ ו‬s.tic I.SBN t)7S-3-4S7-15429-9 ‫ן‬ 2016 Leibniz, the Young Kant, and Boscovich on the Relationality of Space Idan Shimonв (Tel Aviv) δeiЛniz’s main thesis regarding the nature of spaМe is that spaМe is relational.1 This means that spaМe is not an independent oЛjeМt or eбistent in itself, Лut rather a set of relations Лetаeen oЛjeМts eбisting at the same time. The realitв of spaМe, therefore, is derived from oЛjeМts and their relations. For δeiЛniz and his suММessors, this vieа of spaМe аas intimatelв МonneМted аith the understanding of the Мomposite nature of material oЛjeМts. The nature of the relation Лetаeen spaМe and matter аas МruМial to the МonМeptualization of Лoth spaМe and matter. In this paper, I disМuss δeiЛniz’s aММount of relational spaМe and eбamine its novel elaЛorations Лв tаo of his suММessors, namelв, the вoung Immanuel Kant and the Croat natural philosopher Roger BosМoviМh. Kant’s and BosМoviМh’s studies of δeiЛniz’s aММount lead them to original versions of the relational vieа of spaМe. Thus, δeiЛniz’s relational spaМe proved to Лe a philosophiМallв fruitful notion, as it вielded Лold and intriguing attempts to deМipher the nature of spaМe and аas a keв part in innovative sМientifiМ ideas. 1. Leibniz’s Account of Relational Space δeiЛniz’s main thesis regarding spaМe is that it is relational. He defines spaМe as a set of relations Лetаeen things аhiМh eбist at the same time. The folloаing is a Мlear eбposition of δeiЛniz’s definition of spaМe from his МorrespondenМe аith Samuel Clarkeμ “As for mв oаn opinion, I have said more than onМe, that I hold spaМe to Лe something merelв relative, as time isν that I hold it to Лe an order of МoeбistenМes, as In addition to the standard aЛЛreviations for Gerhardt’s (GP, Gε) and the AМademв’s (A) editions of δeiЛniz’s teбts, I use the folloаing aЛЛreviationsμ AG = G. W. δeiЛnizμ Philosophical Essays, transl. Лв Roger Arieа/Daniel GarЛer, Indianapolis 1989ν H = Id.μ Theodicy, transl. Лв E. ε. Huggard/ed. Лв Austin Farrer, δondon 19η1ν δ = Id.μ Philosophical Papers and Letters, transl. Лв δeroв E. δoemker, DordreМht 219θ9ν δA = Id./Antoine Arnauldμ The Leibniz-Arnauld Correspondence, transl. Лв Haвdn T. εason, εanМhester 19θ7ν δC = Id./Samuel Clarkeμ The Leibniz-Clarke Correspondence, ed. Лв Henrв G. Aleбander, εanМhester 19ηθν PP = G. W. δeiЛnizμ Philosophical Writings, ed. Лв George H. R. Parkinson, δondon 1973ν W = Kant’s Critique of Pure Reason. Background Source Materials, transl. and ed. Лв EriМ Watkins, CamЛridge 2009. 1 7ζ Idan Shimonв time is an order of suММessions. For spaМe denotes, in terms of possiЛilitв, an order of things аhiМh eбist at the same time, Мonsidered as eбisting together, аithout enquiring into their manner of eбisting. And аhen manв things are seen together, one perМeives that order of things among themselves.”2 δater in the МorrespondenМe, δeiЛniz provides an eluМidation of his definition Лв means of an analogв аith “genealogiМal spaМe” or familв МonneМtionsμ “Д...] ДspaМe] Мan onlв Лe an ideal thing, Мontaining a Мertain order, аherein the mind МonМeives the appliМation of relations. In like manner, as the mind Мan fanМв to itself an order made up of genealogiМal lines, аhose Лigness аould Мonsist onlв in the numЛer of generations, аherein everв person аould have his plaМeν and if to this one should add the fiМtion of a metempsвМhosis, and Лring in the same human souls again, the persons in those lines might Мhange plaМeμ he аho аas a father, or a grandfather, might ЛeМome a son, or a grandson, etМ. And вet those genealogiМal plaМes, lines, and spaМes, though theв should eбpress real truth, аould onlв Лe ideal things.”3 A familв tree and its genealogiМal lines depend on the eбistenМe of Мertain persons and their МonneМtions. These persons, as familв memЛers, Лear speМifiМ genealogiМal relations to one another and form a Мertain order among themselves. There is no “genealogiМal realm” over and aЛove the familв memЛers and the relations theв Лear to one another, and one need not assume suМh a realm in аhiМh these ordered relations somehoа inhere, for these relations hold independentlв of anв suМh presumed realm. The plaМe of a familв memЛer in the genealogiМal tree is determined Лв the relations she Лears to other “δeiЛniz’s 3rd letter”, § ζ, δC, pp. 2η–2θ. See alsoμ “Д…]ДspaМe] is that order, аhiМh renders Лodies МapaЛle of Лeing situated, and Лв аhiМh theв have a situation among themselves аhen theв eбist together” (“δeiЛniz’s ζth letter”, § ζ1, δC, p. ζ1)ν “Д…] spaМe is nothing Лut the order of eбistenМe of things possiЛle at the same time” (“δetter to de Volder, 30 June 170ζ”ν GP II, 2θ9 / δ, p. η3θ)ν “Д…] spaМe is the order of Мoeбisting phenomena” (“δetter to Des Bosses, 1θ June 1712”ν GP II, ζη0 / δ, p. θ0ζ)ν “Д…] just as in time аe МonМeive nothing Лut the verв order Дdispositio] or series of Мhanges that Мan take plaМe in time, so too, аe understand nothing in spaМe Лut the possiЛle order of Лodies” (“τn Bodв and ForМe, against the Cartesians”ν GP IV, 39ζ / AG, p. 2η1)ν “ДspaМe] is a relationshipμ an order, not onlв among eбistents, Лut also among possiЛles as though theв eбisted” (A VI, θ, 1ζ9ν G. W. δeiЛnizμ New Essays on Human Understanding, transl. Лв Peter Remnant/ Jonathan Bennett, CamЛridge 199θ, p. 1ζ9)ν “SpaМe is the order of Мoeбisting things, or the order of eбistenМe for things аhiМh are simultaneous” (“εetaphвsiМal Foundations of εathematiМs”ν Gε VII, 18 / δ, p. θθθ)ν “Д…] spaМe is no more real than time, that is, Д…] spaМe is nothing Лut the order of Мoeбistents, just as time is the order of things that have eбisted Лefore” (“Remarks on George Berkeleв’s Principles of Human Knowledge”, AG, p. 307). 3 “δeiЛniz’s ηth δetter”, § ζ7, δC, pp. 70–71. 2 δeiЛniz, the Вoung Kant, and BosМoviМh on the Relationalitв of SpaМe 7η familв memЛers, and not Лв her supposed aЛsolute plaМe in an independent genealogiМal sphere. In like manner, spaМe is nothing Лut a set of relations Лetаeen Мoeбistent things, and there is no independent spatial entitв over and aЛove these things and their relations. Again, the plaМe of a thing is determined Лв its spatial relations to other things, namelв hoа it is situated in relation to them and its distanМe from them, and not Лв some aЛsolute position in an independent spaМe. The plaМe of a Мar driving northЛound along a tаo kilometers one-аaв street Мan Лe determined Лв its relation to the street and the Лuildings along it. First, it is on the southern side of the street, near the Лuildings at the Лeginning of the street. After a Мouple of minutes, it is on the northern side of the street, near the Лuildings at the end of the street and tаo kilometers from the Лeginning of the street. A seМond Мar, starting noа the same route, is at the same place аhere the first аas a Мouple of minutes ago. To saв that it is “at the same plaМe” is not to saв that it fills the same part of an aЛsolute spaМe earlier filled Лв the first Мar, Лut simplв that it Лears the same relation to the street and the surrounding Лuildings as the first Мar did a Мouple of minutes ago.ζ τn δeiЛniz’s aММount, then, spaМe has no independent Лut merelв derivative realitв. It is not a self-suЛsistent “Мontainer” аhiМh makes the eбistenМe of oЛjeМts possiЛle. Rather, oЛjeМts are logiМallв prior to spaМeμ its realitв is derived from and dependent on oЛjeМts. AММordinglв, one Мannot literallв mean that “things are in spaМe”μ things Мannot aМtuallв Лe in spaМe, ЛeМause spaМe is nothing over and aЛove Мoeбistent oЛjeМts. Instead, this phrase has to Лe understood as indiМating that things Лear spatial relations to one another as eбplained aЛove.η The realitв of spaМe is further doаngraded in δeiЛniz’s aММount Лв his additional tаo theses, that is, the theses of idealitв and phenomenalitв of spaМe. The idealitв thesis folloаs from the definition of spaМe as relational and from δeiЛniz’s aММount of relations. δeiЛniz maintains that relations are essentiallв ideal, sinМe realitв ultimatelв Мonsists of suЛstanМes and their aММidents and relations Мan Лe neither one of these modes of realitв.θ Consequentlв, See δeiЛniz’s definition of plaМe in his “ηth δetter to Clarke”, § ζ7, δC, pp. θ9–70, and his Мonsequent definition of spaМeμ “Д…] that аhiМh Мomprehends all those plaМes, is Мalled spaМe”ν “Д…] spaМe results from plaМes taken together.” η For a luМid eбposition and МritiМism of δeiЛniz’s arguments for his relational definition of spaМe in the МorrespondenМe аith Clarke, see σiМk Huggett (ed.)μ Space from Zeno to Einstein. Classic Readings with a Contemporary Commentary, CamЛridge 1999, pp. 1θ0–1θ8. θ A relation Лetаeen tаo things is not itself a thing or a suЛstanМe. Furthermore, it Мannot Лe a propertв of a thing or an aММident, “for if so, аe should have an aММident in tаo suЛjeМts, аith ζ 7θ Idan Shimonв spaМe, Лeing nothing Лut a set of relations, is not a real Лut an ideal thing or a mental МonstruМt. The thesis that spaМe is phenomenal is an independent one and does not folloа from the former tаo theses regarding spaМe. AММording to this thesis, spaМe Лelongs to the phenomenal realm of phвsiМal things, and not to the fundamentallв real sphere of suЛstanМes or monads. That is, spaМe Мonsists in relations Лetаeen phвsiМal things, and not Лetаeen ultimatelв real suЛstanМes. This does not, hoаever, turn spaМe into a fiМtion. Spatial relations still eбpress truths, sinМe theв denote aМtual relationships Лetаeen phвsiМal oЛjeМts, аhiМh, although not ultimatelв real, are still grounded in true realities and, therefore, Мonsidered “аell founded phenomena”7. This aММount introduМes a metaphвsiМs of three levels of realitвμ at the ground level are the genuinelв real Мonstituents of all realitв, namelв, suЛstanМesν the seМond level Мonsists of phвsiМal oЛjeМts, аhiМh are aggregates or resultants of suЛstanМes and, therefore, qualifв as аell founded phenomenaν8 at the third level stand spaМe and time, аhiМh Мonsist in relations Лetаeen phвsiМal oЛjeМts and, therefore, maв qualifв as аell founded phenomena of seМond order.9 In a letter to Des Bosses, δeiЛniz eбplains the reason that led him to his unique metaphвsiМs and vieа of spaМeμ “I Мonsider the eбplanation of all phenomena solelв through the perМeptions of monads funМtioning in harmonв аith eaМh other, аith Мorporeal suЛstanМes rejeМted, to Лe useful for a fundamental investigation of things. In this аaв of eбplaining things, spaМe is the order of Мoeбisting phenomena, as time is the order of suММessive phenomena, and there is no spatial or aЛsolute nearness or one leg in one, and the other in the otherν аhiМh is Мontrarв to the notion of aММidents.” And “Лeing neither a suЛstanМe, nor an aММident, it must Лe a mere ideal thing”, “δeiЛniz’s ηth δetter”, § ζ7, δC, p. 71. 7 For a disМussion of δeiЛniz’s three main theses regarding spaМe, see σiМholas Jolleвμ Leibniz, δondon 200η, pp. 8ζ–89. 8 This idea appears at a multitude of plaМes. See for eбample “δetter to Arnauld, 9 τМtoЛer 1θ87”ν A II, 2, 2ζ9 / δA, p. 1η2. “First Truths”ν A VI, ζ, 1θζ8 / δ, p. 270. “δetter to de Volder, 30 June 170ζ”ν GP II, 2θ8–270 / δ, p. η3θ–η37. “δetter to de Volder, 170ζ–170η”ν GP II, 27η– 27θ / AG, pp. 181–182. “δetter to Des Bosses, 1η εarМh 171η”ν GP II ζ92 / δ, p. θ09. “δetter to σiМolas Remond, 10 Januarв 171ζ”ν GP III, θ0θ / δ, p. θηη. “δetter to σiМolas Remond, 11 FeЛruarв 171η”ν GP III, θ3θ / δ, p. θη9. “Remarks on George Berkeleв’s Principles of Human Knowledge”, AG, p. 307. “Against BarЛariМ PhвsiМs”ν GP VII, 3ζζ / AG, p. 319. 9 τn the three ontologiМal levels in δeiЛniz’s metaphвsiМs, see Jill VanМe Burokerμ Space and Incongruence. The Origin of Kant’s Idealism, DordreМht 1981, pp. 3η–37. It is sometimes argued that the asМription of аell founded phenomena to spaМe and time is less appropriate, sinМe theв are merelв mental МonstruМts, and that δeiЛniz aММordinglв in later teбts prefers to Мlassifв them as ideal rather than аell founded phenomena. τn this point, see RoЛert εerriheа Adamsμ Leibniz. Determinist, Theist, Idealist, σeа Вork 199ζ, pp. 2η3–2ηη. δeiЛniz, the Вoung Kant, and BosМoviМh on the Relationalitв of SpaМe 77 distanМe Лetаeen monads… In this МonМeption, also, there is involved no eбtension or Мomposition of the Мontinuum, and all diffiМulties aЛout points disappear. It is this that I tried to saв someаhere in mв Theodicy – that the diffiМulties in the Мomposition of a Мontinuum ought to аarn us that аe must think far differentlв of things.”10 δeiЛniz suggests here that it is the Мonsideration of the diffiМulties in the Мomposition of Мontinuous entities that led him to “think far differentlв of things” and to advanМe his original metaphвsiМs and aММount of spaМe. In the Theodicy, he Мlaims that this puzzle Мonstitutes one of the “tаo famous laЛвrinths аhere our reason verв often goes astraв” and that it arises due to “laМk of a true МonМeption of the nature of suЛstanМe and matter.”11 σoа the proЛlem of the Мomposition of Мontinuous things and the nature of suЛstanМe and matter essentiallв involves tаo Мentral issuesμ the Мomposite nature of phвsiМal oЛjeМts and the relation Лetаeen oЛjeМts and spaМe. τn the one hand, δeiЛniz argues that Мomposite oЛjeМts (“Лeings Лв aggregation”) draа their realitв from their parts. If theв аere Мomposed of parts аhiМh are also Мomposite and so on to infinitв, theв аould laМk a foundation for their realitв and аould not Лe genuinelв real. Therefore, to Лe real, theв must ultimatelв Лe Мomposed of true entities or suЛstanМes, the hallmark of аhiМh is unitв. “Д…] everв Лeing Лв aggregation presupposes Лeings endoаed аith real unitв, ЛeМause everв Лeing derives its realitв onlв from the realitв of those Лeings of аhiМh it is Мomposed, so that it аill not have anв realitв at all if eaМh Лeing of аhiМh it is Мomposed is itself a Лeing Лв aggregation, a Лeing for аhiМh аe must still seek further grounds for its realitв, grounds аhiМh Мan never Лe found in this аaв, if аe must alаaвs Мontinue to seek for them.”12 “δetter to des Bosses”, 1θ June 1712ν GP II, ζη0–ζη1 / δ, p. θ0ζ. Seeμ “There are tаo famous laЛвrinths аhere our reason verв often goes astraвμ one МonМerns the great question of the Free and the σeМessarв, aЛove all in the produМtion and the origin of Evilν the other Мonsists in the disМussion of Мontinuitв and of the indivisiЛles аhiМh appear to Лe the elements thereof, and аhere the Мonsideration of the infinite must enter in. The first perpleбes almost all the human raМe, the other eбerМises philosophers onlв. I shall have perМhanМe at another time an opportunitв to deМlare mвself on the seМond, and to point out that, for laМk of a true МonМeption of the nature of suЛstanМe and matter, people have taken up false positions leading to insurmountaЛle diffiМulties, diffiМulties аhiМh should properlв Лe applied to the overthroа of these verв positions” (“TheodiМв”, prefaМeν GP VI, 29 / H, p. η3–ηζ). Cf. “TheodiМв – Preliminarв Dissertation”, § 2ζν GP VI, θζ–θη / H, p. 88–89ν “τn Freedom”ν A VI, ζ, 1θηζ / δ, p. 2θζ. 12 “δetter to Arnauld, 30 April 1θ87”ν A II, 2 18ζ / AG, p. 8η. Cf.μ “The monad Д…] is nothing Лut a simple suЛstanМe аhiМh enters into Мompounds Д…] There must Лe simple suЛstanМes, sinМe there are Мompounds, for the Мompounded is Лut a МolleМtion or an aggregate of simples Д…] The monads are the true atoms of natureν in a аord, theв are elements of things. We need 10 11 78 Idan Shimonв Although this line of argument ЛeМame a strong rationalistiМ tenet,13 it maв also draа its ground from the nature of meМhaniМal eбplanation. To eбplain a phвsiМal oЛjeМt meМhaniМallв is to derive its properties from the properties of its parts. The properties of the parts depend, in turn, on the properties of the parts of the parts, and so on. σoа if this mode of eбplanation is to аork, it seems that a Мertain ground floor of fundamental parts аith primitive properties has to Лe postulated.1ζ τn the other hand, the spatial eбtension of oЛjeМts maв seem to implв that their division into parts proМeeds indefinitelв. SpaМe, as a Мontinuous magnitude, is infinitelв divisiЛle. A spatial region, hoаever small, Мan alаaвs Лe further dividedν its division never ends in indivisiЛle, simple parts of аhiМh it is Мomposed. And if this essential МharaМteristiМ of spaМe is somehoа Мarried over to oЛjeМts, then theв too Мannot Лe Мomposed of simple parts. This is Мlearlв the Мase in DesМartes’s aММount. DesМartes identifies spaМe аith matter and maintains that the nature of Лodв Мonsists in eбtension.1η Therefore, on his aММount, oЛjeМts, just like spaМe, are not made up of simple parts.1θ In addition, aЛsolutist vieаs of spaМe maв also Лe taken as implвing the same МonМlusion. τn fear no dissolution in them, and there is no МonМeivaЛle аaв in аhiМh a simple suЛstanМe Мan Лe destroвed naturallв” (“εonadologв” §§ 1–ζν GP VI, θ07 / δ, p. θζ3). See also “σeа Sвstem”, § 3ν GP IV, ζ78–ζ79 / δ, p. ζηζ. “σeа Sвstem”, § 11ν GP IV, ζ82 / δ, p. ζηθ. “PrinМiples of σature and GraМe”, § 1ν GP VI, η98 / δ, p. θ3θ. For a disМussion of the unitв and simpliМitв of suЛstanМes, see Jolleвμ Leibniz, pp. 37–ζ1. 13 Christian Wolff argues that simples are the ground of Мompositesμ “If there are Мomposite things, there must also Лe simple Лeings. For if no simple Лeings аere present, then all parts – theв Мan Лe taken to Лe as small as вou might ever like, even inМonМeivaЛlв small parts – аould have to Мonsist of other parts. But then, sinМe one Мould provide no reason аhere the Мomposite parts аould ultimatelв Мome from, just as little as one Мould Мomprehend аhere a Мomposite numЛer аould arise from if it Мontained no unities in itself, and вet nothing Мan Лe аithout a suffiМient ground Д…], one must ultimatelв admit simple things from аhiМh the Мomposites arise” (Christian Wolffμ “Rational Thoughts on God, the World and the Soul of Human Beings, Also All Things in General”, § 7θ, inμ W, p. 17). Aleбander Baumgarten also reasons that if there are Мomposites, there must Лe simplesμ “A Мomposite Мannot eбist, eбМept as the determinations of others Д…] σoа, apart from Мomposites, there are onlв simples Д…] Therefore, if Мomposites eбist, monads eбist” (Aleбander Baumgartenμ “εetaphвsiМs” § 2ζη, inμ W, p. 103). 1ζ This line of thought seems to Лe implied in IsaaМ σeаtonμ Opticks, σeа Вork 1979, pp. ζ00– ζ0ζ. 1η Seeμ “Д…] the same eбtension аhiМh Мonstitutes the nature of Лodв also Мonstitutes the nature of spaМe” (Renц DesМartesμ Principles of Philosophy II, § 11, transl. Лв Valentin R. εiller and Reese P. εiller, DordreМht 1983, p. ζ3). 1θ Seeμ “We also easilв understand that it is not possiЛle for anв atoms, or parts of matter аhiМh are Лв their oаn nature indivisiЛle, to eбist. The reason is that if there аere suМh things, theв аould neМessarilв have to Лe eбtended, no matter hoа tinв theв are imagined to Лe. We Мan, therefore, still МonМeive of eaМh of them Лeing divided into tаo or more smaller ones, and thus δeiЛniz, the Вoung Kant, and BosМoviМh on the Relationalitв of SpaМe 79 these vieаs, spaМe, as a reМeptaМle in аhiМh Лodies eбtend, is a Мondition for the eбistenМe and struМture of Лodies. Therefore, Лodies are essentiallв eбtended entities. Put differentlв, spatial eбtension is a primarв qualitв of Лodies and this implies their infinite divisiЛilitв.17 σoа δeiЛniz aММepts Лoth seeminglв МonfliМting vieаs. He Лoth agrees that phвsiМal oЛjeМts are, in some sense, infinitelв divisiЛle and insists that simple suЛstanМes essentiallв Мonstitute Мomposite oЛjeМts. His аaв out of this laЛвrinth rests on his original МonМeption of spaМe and his threefold ontologвμ Лв doаngrading the ontologiМal status of spaМe, δeiЛniz is aЛle to disМard the idea that eбtension in spaМe is a primarв propertв of oЛjeМts,18 and to propose an entirelв different understanding of the Мonstitution of oЛjeМts. Thus, on the phenomenal level, phвsiМal oЛjeМts are indeed infinitelв divisiЛle eбtended masses in spaМe. But on a deeper metaphвsiМal level, theв are ultimatelв Мonstituted Лв “metaphвsiМal points” or simple suЛstanМes. Construed in a аaв analogous to spirits, these metaphвsiМal points are aМtive simple entities, and their Мounterparts in the phвsiМal realm are Мorporeal suЛstanМes essentiallв made up of phвsiМal forМes.19 аe knoа that theв are divisiЛle. For it is impossiЛle to ДМlearlв and distinМtlв] МonМeive of dividing anвthing аithout knoаing, from that verв faМt, that it is divisiЛle” (iЛid., § 20, pp. ζ8– ζ9). 17 It is important of Мourse to distinguish Лetаeen different meanings of divisiЛilitв in this Мonteбtμ see the distinМtion Лetаeen formal and actual divisiЛilitв in note 21 Лeloа. 18 See “DisМourse on εetaphвsiМs”, § 12ν A II, ζ, 1ηζη / δ, 309–310. “δetter to Arnauld, 30 April 1θ87”ν A II, 2, 187–188 / AG, pp. 8θ–87. “δetter to Arnauld, 9 τМtoЛer 1θ87”ν A II, 2, 2ζ9–2η0 / δA, pp. 1η2–1η3. “A SpeМimen of DisМoveries”ν A VI, ζ, 1θ22 / PP, pp. 81. “τn Bodв and ForМe, against the Cartesians”ν GP IV, 393–39ζ / AG, pp. 2η1. “SpeМimen DвnamiМum”ν Gε, VI, 2ζθ–2ζ7 / δ, pp. ζζζ–ζζη. “δetter to De Volder, 2ζ εarМh 1θ99”ν GP II, 1θ9– 170 / δ, pp. η1θ. “δetter to De Volder, 23 June 1θ99”ν GP II, pp. 182–183 / δ, p. η19. τn δeiЛniz on the idea of eбtension, see Charlie D. Broadμ Leibniz. An Introduction, δondon 197η, pp. ηζ–ηην Glenn A. Hartzμ Leibniz’s Final System. Monads, Matter and Animals, δondon 2007, pp. θ3–θη. 19 In “σeа Sвstem” (§ 3, GP IV, ζ78–ζ79 / δ, p. ζηζν § 11, GP IV, ζ82 / δ, p. ζηθ) δeiЛniz МonМludes Лв elimination that the ЛasiМ elements Мonstituting Мomposites must Лe metaphвsiМal points. These elements Мan Лe neither material atoms, sinМe theв are “Мontrarв to reason,” nor mathematiМal points, sinМe theв are limits rather than parts of eбtended oЛjeМts. HenМe, the ЛasiМ elements must Лe metaphвsiМal points. For an analвsis of δeiЛniz’s disМussions of divisiЛilitв and simple parts of matter, see Hartzμ Leibniz’s Final System, Мh. 2 and 3. Hartz shoаs that δeiЛniz is МonМerned аith tаo aspeМts of division of matter. From one point of vieа, matter is regarded as infinitelв divisible into ever smaller mass-parts. From another, matter is aМtuallв divided into simple suЛstanМes. This distinМtion, Hartz suggests, enaЛles δeiЛniz to finallв find his аaв out of the “laЛвrinth of the Мontinuum” (Hartzμ Leibniz’s Final System, p. 72). 80 Idan Shimonв 2. Kant’s Version of Relational and Real Space In his 17ηθ Physical Monadology, Kant uses a δeiЛnizian МonМeption of relational spaМe, together аith a dвnamiМal model of matter, to resolve the Мontroversв regarding the theorв of monads and the Мonstitution of matter Лetаeen δeiЛnizian-Wolffian metaphвsiМians and σeаtonian thinkers in the 17ζ0s and 17η0s Berlin AМademв.20 As δeonard Euler oЛserved, the keв point in the Мontroversв аas the МonМeption of spatial eбtension.21 He suММinМtlв summarized the δeiЛnizian line of thought as folloаs. SinМe it is “a Мompletelв estaЛlished truth that eбtension is divisiЛle to infinitв, and that it is impossiЛle to МonМeive parts so small as to Лe unsusМeptiЛle of further division,” philosophers аho rejeМt the infinite divisiЛilitв of Лodies “do not impugn this truth itself, Лut denв that it takes plaМe in eбisting Лodies” Лв doаngrading eбtension to the status of a merelв aЛstraМt, ideal propertв.22 The neаlв reestaЛlished Berlin AМademв аas torn Лetаeen σeаtonians and Wolffians. The former аere led Лв the president of the AМademв, Pierre δouis εaupertuis, and the AМademв’s preeminent mathematiМian and sМientist, δeonard Euler. Despite the faМt that Wolff deМlined King FrederiМk the Great’s invitation to Мo-lead the Berlin AМademв, Wolffians аere still dominant and serving at keв positions in the AМademв (e.g. Samuel Formeв, the seМretarв of the AМademв). The AМademв made monadologв the suЛjeМt for the prize Мontest of 17ζ7. SМholars аere asked to Мlearlв formulate the doМtrine of monads and then to deМide аhether it Мould Лe deМisivelв proved or refuted. Anti-monadists oЛjeМted that monads violated the estaЛlished laа of inertia. To posit that material elements are endoаed аith spontaneous aМtivitв and inner poаers МapaЛle of altering the states of the Лodies аhiМh theв Мonstitute is at odds аith the essential passivitв of oЛjeМts professed Лв this laа. τn their vieа, oЛserved Мhanges in the aМtual аorld must Лe eбМlusivelв aММounted for Лв means of eбternal forМes. Theв also argued that the simpliМitв of monads МonfliМts аith the infinite divisiЛilitв of the spaМe oММupied Лв material Лodies. It appeared ineбpliМaЛle to them that infinitelв divisiЛle Лodies Мould Лe Мonstituted Лв simple suЛstanМes or elements. An anti-monadist essaв аon the Мontest, Лut the deЛate аas not settled thereЛв. For detailed historiМal aММounts of the МonfliМt, see δeаis White BeМkμ Early German Philosophy. Kant and His Predecessors, CamЛridge 19θ9, pp. 31ζ–319ν Ronald Calingerμ “The σeаtonian-Wolffian Controversв. 17ζ0–17η9,” inμ Journal of the History of Ideas 30 (19θ9), pp. 319–330ν Irving Polonoffμ Force, Cosmos, Monads and other Themes of Kant’s Early Thought, Bonn 1973, pp. 77–89ν Alison δaваineμ Kant’s Early Metaphysics and the Origins of the Critical Philosophy, AtasМadero 1993, pp. 27–31. 21 See δeonard Eulerμ “δetters to a German PrinМess”, inμ W, p. 213μ “The Мontroversв Лetаeen modern philosophers and geometers Д…] turns on the divisiЛilitв of Лodв. This propertв is undouЛtedlв founded on eбtension, and it is onlв in so far as Лodies are eбtended that theв are divisiЛle and МapaЛle of Лeing reduМed to parts”. 22 IЛid., The deЛate spread Лeвond the аalls of the AМademв and attraМted the attention of nonprofessional intelleМtuals and laвmen as аell. Euler, one of the Мentral figures in the deЛate, vividlв depiМts the agitation it aroused in the German аorldμ “There аas a time аhen the dispute aЛout monads emploвed suМh general attention and аas МonduМted аith so muМh аarmth that it forМed its аaв into the Мompanв of everв desМription, that of the guardroom not eбМepted. 20 δeiЛniz, the Вoung Kant, and BosМoviМh on the Relationalitв of SpaМe 81 Although Kant advoМated a relational МonМeption of spaМe, he did not suЛmit to this line of thought. Kant sought for a theorв that аould aМknoаledge Лoth the infinite divisiЛilitв of spaМe and the Мlaim that phвsiМal oЛjeМts Мonsist of simple suЛstanМes, Лut аithout doаngrading spaМe to merelв ideal thing and аithout Мonstruing suЛstanМes as mental. He thus proposed a theorв of relational Лut real spaМe and of monads or simple suЛstanМes аhiМh are essentiallв physical. This theorв of phвsiМal monads has the advantage that it avoids the proЛlem of eбplaining hoа the phвsiМal and spatial properties of material oЛjeМts arise from non-eбtended, mental δeiЛnizian monads, of аhiМh Лodies are supposed to Лe Мomposed. The question is hoа it is possiЛle for simple phвsiМal suЛstanМes to Лe in spaМe or to fill a spaМe аithout thereЛв loosing their simpliМitв, given that spaМe is infinitelв divisiЛle. The ansаer turns on Kant’s insight that filling a spaМe does not neМessarilв entail Лeing Мomposed of independent parts. Something filling a spaМe must Лe Мomposite onlв on the additional assumption that the formal divisiЛilitв of spaМe entails the actual divisiЛilitв of things in spaМe and henМe their non-simpliМitв.23 To avoid this additional assumption and the МonМlusion that it implies, Kant advanМes a relational vieа of spaМe and a dвnamiМal model of matter. These tаo doМtrines alloа him to maintain Лoth that Лodies eбtend in spaМe and that theв Мonsist of simple parts. Regarding spaМe, Kant Мlaims that “sinМe spaМe is not a suЛstanМe Лut a Мertain appearanМe of the eбternal relation of suЛstanМes, it folloаs that the possiЛilitв of dividing the relation of one and the same suЛstanМe into tаo parts is not inМompatiЛle аith the simpliМitв Дor] unitв of the suЛstanМe”2ζ In There аas sМarМelв a ladв at Мourt аho did not take a deМided part in favor of monads or against them. In a аord, all Мonversation аas engrossed Лв monads – no other suЛjeМt Мould find admission” (Eulerμ “δetters”, inμ W, p. 218). 23 Bв “aМtual” divisiЛilitв I understand the logiМal possiЛilitв to separate and distanМe parts from one another. It is Мlear that spaМe is not divisiЛle in this meaningμ it is logiМallв impossiЛle to distanМe one region of spaМe from the regions adjaМent to it. In the Мase of spaМe, it is possiЛle merelв to identifв parts or regions and mark Лorders Лetаeen them, аhiМh means that spaМe is onlв “formallв” divisiЛle. And see the quotation from Kant in the folloаing footnote. 2ζ “PhвsiМal εonadologв” = Pε, 1, ζ80. Writings Лв Kant are Мited Лв seМtion numЛer (if appliМaЛle) and volume and page numЛer of Kants Gesammelte Schriften, ed. Лв the German AМademв of SМienМes, Berlin 1900–. Translations to English are from the Cambridge Edition of the Works of Immanuel Kant, ed. Лв Paul Guвer and Allen Wood, CamЛridge 1992–. See alsoμ “The division of spaМe Д…] is not the separation of things, of аhiМh one is set apart from another and has a self-suffiМient eбistenМe of its oаn. It rather displaвs a Мertain pluralitв or quantitв in an eбternal relation. SinМe this is the Мase, it is oЛvious that a pluralitв of suЛstantial parts does not folloа from the division of spaМe. SinМe it is this pluralitв alone аhiМh аould Лe 82 Idan Shimonв other аords, spaМe depends on eбternal relations Лetаeen phвsiМal suЛstanМes. It is not a suЛstanМe or an independent entitв аhiМh is logiМallв prior to phвsiМal suЛstanМes. HenМe, it is not a Мondition for their eбistenМe and, therefore, theв need not assume its struМture and admit infinite divisiЛilitв. Put differentlв, spatial properties are not primitive properties of phвsiМal oЛjeМts. Theв are not something аhiМh oЛjeМts possess simplв ЛeМause theв eбist. For eбample, a thing eбisting on its oаn аithout anв aМtual МonneМtions to other things has no spatial place. To have a plaМe, it must Лe a part of a аorld, namelв, a sвstem of things аhiМh Лear actual relations and interaМtions among one another.2η Having a spatial position thus presupposes eбternal relations. And these, in turn, are derived from the reМiproМal aМtions аhiМh Лodies eбerМise upon one another Лв means of their forМes. Similarlв, the spatial extension of phвsiМal oЛjeМts depends on the forМes аhiМh Мonstitute the verв essenМe of matter. τn Kant’s dвnamiМal model of matter, material oЛjeМts are Мomposed of phвsiМal monads, аhiМh are point-like elements eбerting repulsive and attraМtive forМes. Repulsion is stronger near the point-like element, аhile attraМtion is stronger at greater distanМes. The set of equiliЛrium points Лetаeen repulsion and attraМtion (i.e. the points аhere the monad neither repels nor attraМts other monads) defines the limit of the spatial eбtension of a phвsiМal monad. It is Мlear, then, that though it is possiЛle to disМern different spatial parts in the sphere of eбtension of the monad, it is not possiЛle to separate or distanМe these parts from one another, sinМe theв all depend on and radiate from the same point-like sourМe of forМes. HenМe, although it eбtends in spaМe and fills a spaМe, the monad is nonetheless simple.2θ To sum up, Kant’s version of relational spaМe and his model of matter provide an alternative аaв out of the laЛвrinth of the Мontinuum. σote that for Kant, spaМe is not derived, as for δeiЛniz, from mere apparent eбternal relations Лetаeen phenomenal oЛjeМts Мonstituted Лв “аindoаless” monads, Лut rather from genuine reМiproМal aМtions of material elements upon one another in virtue of their phвsiМal forМes. Therefore, for Kant spaМe is relational and opposed to the suЛstantial simpliМitв of the monad, it is suffiМientlв Мlear that the divisiЛilitв of spaМe is not at all opposed to the simpliМitв of the monad” (Pε, 1, ζ80). 2η This definition of a аorld reМurs throughout Kant’s pre-МritiМal teбts. See for eбample “δiving ForМes”, § 8, 1, 22–23ν “σeа EluМidation”, 1, ζ1ζν “Inaugural Dissertation”, § 2, 2, 390. 2θ For a detailed aММount of Kant’s dвnamiМal model of matter in the “PhвsiМal εonadologв”, see Idan Shimonвμ The Antinomies and Kant’s Conception of Nature, Tel Aviv 2013, pp. 3η–ζ3. δeiЛniz, the Вoung Kant, and BosМoviМh on the Relationalitв of SpaМe 83 real.27 And this is enough to resolve the proЛlem of the Мomposition of phвsiМal oЛjeМts. 3. Boscovich on Derivative and Discrete Space Roger BosМoviМh also put forаard an aММount аhiМh regards spaМe as derivative. But he approaМhed the suЛjeМt from a different angle. His main task in his 17η8 Theory of Natural Philosophy аas to settle the proЛlem regarding Мollisions Лetаeen Лodies. This led him to a dвnamiМal theorв of material elements similar to that of Kant.28 BosМoviМh аas reluМtant to enter the puzzle МonМerning the nature of spaМe and time, sinМe he thought “that this is merelв a question of terminologв.”29 σevertheless, he made some Мontentious Мlaims on this issue and presented an intriguing vieа of derivative spaМe. τn BosМoviМh’s aММount, loМal and temporal properties are real modes of eбistenМe of material elementsμ “Anв point Дof matter] has a real mode of eбistenМe, through аhiМh it is аhere it isν and another, due to аhiМh it eбists at the time аhen it does eбist. These real modes of eбistenМe are to me real time and spaМe.”30 SinМe real time and spaМe are nothing Лut these modes of As δeаis White BeМk noted, for Kant “SpaМe must Лe ontologiМallв real, Лut it need not Лe ontologiМallв primitive”ν BeМkμ Philosophy, p. ζζ7). 28 BosМoviМh aММepted δeiЛniz’s МritiМism of the vieа that matter is Мomposed of perfeМtlв hard and undeformaЛle elements (for δeiЛniz’s МritiМism, see Idan Shimonвμ “What is (the) εatter – δoМke, δeiЛniz, and the Controversв that Could not Take PlaМe,” inμ HerЛert Breger/Jürgen HerЛst/Sven Erdner (eds.)μ Natur und Subjekt. IX. Internationaler Leibniz-Kongress, Hannover 2011, pp. 1070–1079, here p. 1078). If that аere the Мase, then аhile ЛounМing in Мollision, elements аould Мhange their motion in an instant. This is a violation of the prinМiple of Мontinuitв, аhiМh BosМoviМh regarded as a primarв prinМiple of nature. Furthermore, suМh a Мhange involves infinite aММeleration and therefore, in aММord аith σeаton’s forМe laа, infinite forМe. HenМe, BosМoviМh proposed a dвnamiМal vieа of matter аhiМh admitted the elastiМitв required for oЛserving the prinМiple of Мontinuitв. He elaЛorated a sвstem of point partiМles eбerting forМes on eaМh other and making up material Лodies. The main differenМe Лetаeen the sвstems of Kant and BosМoviМh is that BosМoviМh postulated one unified forМe аith Мhanging influenМe instead of an interplaв of tаo distinМt forМes. AММording to BosМoviМh’s forМe laа, the forМe projeМted Лв a point partiМle repels at verв short distanМes and inМreases infinitelв as the distanМe diminishes infinitelв. At someаhat farther (Лut still rather short) distanМes, the forМe varies Лetаeen repulsion and attraМtion in a аave-like manner. At greater distanМes, the forМe attraМts and аeakens аith the distanМe in aММordanМe аith σeаton’s inverse square laа of attraМtion (Roger J. BosМoviМhμ Theory of Natural Philosophy, transl. Лв James εark Child, CamЛridge 19θθ, §§ 7–1η.). 29 IЛid., § 1ζ2. 30 IЛid., § 197. 27 8ζ Idan Shimonв eбistenМe of material elements, and sinМe, further, matter in BosМoviМh’s theorв is Мomposed of indivisiЛle points and does not Мontinuouslв eбtend, real time and spaМe are also disМrete rather than Мontinuous. The Мontinuitв аe asМriЛe to spaМe is merelв ideal. When аe aЛstraМt from the aМtual points of matter and МonМeive the possibility of interposing as manв points as аe like Лetаeen anв tаo given points, аe form for ourselves the idea of an imaginary Мontinuous spaМe.31 Thus BosМoviМh МonМludesμ “I reМognize no Мoeбisting Мontinuum… for, in mв opinion, spaМe is not anв real Мontinuum, Лut onlв an imaginarв one.”32 τne deМisive feature of BosМoviМh’s theorв is the prioritв of matter over spaМe. DisМrete points of matter are the fundamental elements of his aММount. Everвthing else depends on them and the forМe laа governing their aМtivitв. Thus, even though BosМoviМh аishes to staв aаaв from the dispute regarding the nature of spaМe, his theorв implies a version of a derivative vieа of spaМe, one that, moreover, entails the disМrete nature of real spaМe from the non-Мontinuous МharaМter of material points. 4. Conclusion I disМussed in this paper δeiЛniz’s aММount of relational spaМe and its elaЛorations Лв Kant and BosМoviМh. All three thinkers regarded matter as logiМallв prior to spaМe and advanМed relational and derivative vieаs of spaМe. δeiЛniz argued for relational and ideal spaМe in order to resolve the proЛlem of the Мonstitution of matter. Kant had the same motivation for suggesting a relational aММount of spaМe. Вet he suggested that to resolve this proЛlem it is suffiМient to postulate a relational and real spaМe and monads аhiМh are phвsiМal rather than mental. For BosМoviМh, the vieа that real spaМe is derivative and disМrete аas the outМome of a theorв of matter designed to preserve δeiЛniz’s prinМiple of Мontinuitв. Thus, δeiЛniz’s notion of relational spaМe proved to Лe a fruitful philosophiМal idea. It вielded Лold and intriguing attempts to deМipher the nature of spaМe. For all three thinkers, the relational aММount of spaМe аas МomЛined аith a dвnamiМal vieа of matter. Thus it аas integrated in theories of nature that introduМed novel sМientifiМ ideasμ δeiЛniz’s theorв of nature suggested, for eбample, a rudimentarв formulation of the prinМiple of Мonservation of energв, аhile the Kant-BosМoviМh model of matter is of historiМal 31 32 See iЛid., § 198. IЛid., § 1ζ2. δeiЛniz, the Вoung Kant, and BosМoviМh on the Relationalitв of SpaМe 8η importanМe as the forerunner of 19th Мenturв field theories and Мertain dвnamiМal МonМeptions of matter in modern phвsiМs.33 For an analвsis of δeiЛniz’s disМussion of Мonservation prinМiples, seeμ Idan Shimonвμ “δeiЛniz and the Vis Viva Controversв,” in εarМelo DasМal (ed.)μ The Practice of Reason. Leibniz and His Controversies, Philadelphia 2010, pp. η1–73. For a sвstematiМ surveв of the KantBosМoviМh model of matter, see Thomas Holdenμ The Architecture of Matter. Galileo to Kant, τбford 200ζ, pp. 23θ–272. See also εaб Jammerμ Concepts of Force. A Study in the Foundations of Dynamics, σeа Вork 19θ2, pp. 1η8–187ν Thomas Hankinsμ “Eighteenth-Centurв Attempts to Resolve the Vis viva Controversв,” inμ Isis ηθ (19θη), pp. 281–297, here pp. 291– 297ν Jeffreв Edаardsμ Substance, Force, and the Possibility of Knowledge. On Kant’s Philosophy of Material Nature, Berkeleв 2000, pp. 103–10η. 33