Abstract
Mereology, the doctrine of the relations of part to whole and of parts to parts, has so far awoken the interest of only a small number of Leibniz’s scholars. Since the publication of the pioneering paper of Hans Burkhardt and Wolfgang Degen (Topoi 9(1):3–13, 1990), entirely devoted to Leibniz’s mereology, very few works have been published on the same topic. Moreover, these works tend to consider mereology in the general setting of Leibniz’s metaphysics and do not pay due attention to those essays where Leibniz systematically develops a mereological calculus. In the years 1686–1690, indeed, Leibniz wrote a series of essays, concerning the so-called ‘plus-minus calculus’, where a very interesting mereological doctrine is developed.
While several scholars have investigated the ancient and medieval attempts to develop a more or less embryonic mereology, modern theories of parthood are less explored. In this paper my aim is to fill at least partially this gap, focusing on Leibniz’s mereological ideas. Leibniz, indeed, just in the Dissertation on Combinatorial Art (1666) elaborated the project of constructing a very general mereological doctrine, which evolved later in a series of papers centered on the logical operation of ‘real addition’. Leibniz considered the real addition as a kind of non restricted sum, capable of being applied to any sort of things and satisfying the conditions of ‘idempotence’, ‘reflexivity’ and ‘transitivity’. With the real addition, the relation of containment (reflexive, transitive and anti-symmetric), and the notion of proper parthood, Leibniz elaborates a quite interesting mereology. My main purpose is to offer an exhaustive analysis of the essays (written around 1690) in which Leibniz proposes his mereological calculi based on ‘real addition’.
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Notes
- 1.
- 2.
A VI, 1, p. 187. All passages from the Dissertation on Combinatorial Art are quoted from Martin Wilson’s forthcoming translation for Oxford University Press.
- 3.
A VI, 1, p. 187.
- 4.
A VI, 1, pp. 187–88.
- 5.
- 6.
Antognazza (2009), pp. 57–83.
- 7.
Cf. Mugnai (1973).
- 8.
On the influence of Neoplatonic ideas on Leibniz’s philosophy cf. Mercer (2001).
- 9.
Leibniz (1998a), p. 268.
- 10.
Leibniz (1998a), p. 145.
- 11.
Specimen Calculi Coincidentium (A VI, 4A, pp. 816–22); De casibus in quibus componendo nihil novi fieri potest (A VI, 4A, pp. 823–28); Specimen Calculi coincidentium et inexistentium (A VI, 4A, pp. 830–45); Non inelegans Specimen demonstrandi in abstractis (A VI, 4A, pp. 845–55); De Calculo irrepetibilium (A VI, 4A, pp. 855–58).
- 12.
A VI, 4A, p. 837.
- 13.
Lewis (1960), p. 299.
- 14.
LP, p. 135.
- 15.
Leibniz (1998b), p. 416.
- 16.
Cf., for instance Leibniz (1992), p. 157.
- 17.
A VI, 4 A, p. 823.
- 18.
Cf. for instance A VI, 4 A, p. 823: “Et generaliter ex quotcunque rebus […] nihil fieri potest novi” [“And in general, from any number whatsoever of things […] nothing new can be made”]
- 19.
A VI, 4A, p. 1001.
- 20.
Lenzen (2000), pp. 79–82.
- 21.
LP, pp. 18–19.
- 22.
LP, p. 20.
- 23.
LP, p. 122 (here and in the following quotations from this edition the translation has been slightly modified according to the remarks made at the beginning of the present paper); A VI, 4A, p. 846.
- 24.
LP, p. 132.
- 25.
LP, p. 133; A VI, 4A, p. 835.
- 26.
LP, p. 126; A VI, 4A, p. 850.
- 27.
LP, p. 136; A VI, 4A, p. 839.
- 28.
LP, p. 141; A VI, 4A, pp. 832–33.
- 29.
Cf. NE, p. 487: “For when I say Every man is an animal I mean that all the men are included amongst all the animals; but at the same time I mean that the idea of animal is included in the idea of man. ‘Animal’ comprises more individuals than ‘man’ does but ‘man’ comprises more ideas or more attributes: one has more instances, the other more degrees of reality; one has the greater extension, the other the greater intension.”
- 30.
LP, p. 136; A VI, 4A, pp. 838–39. In the Gerhardt’s edition this essay has no title and Parkinson, who translates from this edition, entitles it A Study in the Calculus of Real Addition. The edition of the Academy has Specimen calculi coincidentium et inexistentium (A VI, 4A, p. 830).
- 31.
Adjectio realis, cf. A VI 4A, p. 834.
- 32.
A VI, 4A, p. 627.
- 33.
- 34.
A VI, 4A, p. 858.
- 35.
On the differences between real addition and the arithmetical operation of sum in Leibniz, cf. Lenzen (1989).
- 36.
LP, p. 142; A VI, 4A, p. 834
- 37.
LP, pp. 142–43; A VI, 4A, p. 834.
- 38.
LP, p. 132; A VI, 4A, p. 834.
- 39.
LP, p. 139; A VI, 4A, p. 842.
- 40.
LP, p. 126; A VI, 4A, p. 850.
- 41.
LP, p. 126; A VI, 4A, p. 851.
- 42.
GM 7, p. 261.
- 43.
GM 7, p. 261.
- 44.
Cf. A, VI, 4A, pp. 823–28.
- 45.
LP, pp. 122–23; A VI, 4A, pp. 846–47.
- 46.
GM 7, p. 274.
- 47.
The circle and the square, indeed, are homogeneous: they have two dimensions, whereas the side of the square has only one dimension, and therefore it is not homogeneous with the circle.
- 48.
A VI, 4A, p. 821
- 49.
GM 7, p. 274.
- 50.
C, p. 476.
- 51.
GM 7, p. 30.
- 52.
GM 7, p. 274.
- 53.
GM 7, p. 274.
- 54.
A VI, 4A, p. 872.
- 55.
GM 7, p. 30.
- 56.
A VI, 4A, p. 418.
- 57.
A VI, 4A, p. 392.
- 58.
A VI, 4A, p. 418.
- 59.
GM 7, p. 282.
- 60.
A VI, 3, p. 483.
- 61.
Ibidem.
- 62.
A VI, 6, p. 63.
- 63.
A VI, 4A, p. 418.
- 64.
A VI, 4A, p. 311.
- 65.
A VI, 4A, p. 278.
- 66.
A VI, 4A, p. 393.
- 67.
NE, p. 486.
- 68.
A VI, 4A, p. 846.
- 69.
Ibidem.
- 70.
Cf. Kauppi (1960), pp. 71–76.
- 71.
LP, p. 136; A VI, 4A, p. 839.
- 72.
LP, p. 123; A VI, 4A, p. 847.
- 73.
Cf. Varzi (2016), p. 15. Usually the relation of overlapping is defined as Oxy = def. ∃z(Pzx ∧ Pzy).
- 74.
LP, p. 123; A VI, 4A, p. 847.
- 75.
For an exhaustive presentation of the supplementation principles, cf. Varzi (2016), pp. 19–36.
- 76.
A VI, 4A, p. 821.
- 77.
LP, p. 124; A VI, 4A, p. 848.
- 78.
A VI, 4A, p. 819.
- 79.
Ibidem.
- 80.
LP, p. 124; A VI, 4A, p. 848.
- 81.
A VI, 4A, p. 819.
- 82.
LP, p. 126; A VI, 4A, 851.
- 83.
- 84.
Cf. Varzi (2016), p. 14.
- 85.
A VI, 4A, p. 1064.
- 86.
Rescher (1991), pp. 227–228.
- 87.
AG, p. 167.
- 88.
A VI, 4B, p. 1669.
- 89.
Cf. Leibniz’s letter to De Volder (AG, p. 177): “Therefore I distinguish: (1) the primitive entelechy or soul; (2) the matter, namely, the primary matter or primitive passive power; (3) the monad made up of these two things; (4) the mass [massa] or secondary matter, or the organic machine in which innumerable subordinate monads come together; and (5) the animal, that is, the corporeal substance, which the dominating monad in the machine makes one.”
- 90.
AG, pp. 289–90.
- 91.
Rescher (1991), pp. 227–228.
- 92.
- 93.
A, followed by number of Series and volume = G. W. Leibniz. 1923. Sämtliche Schriften und Briefe. Darmstadt/Berlin.
AG = G. W. Leibniz. 1989. Philosophical Essays, ed. and trans. Roger Ariew and Daniel Garber. Indianapolis/Cambridge: Hackett Publishing Company.
C = L. Couturat (ed.). 1903. Opuscules et fragments inédits de Leibniz. Paris: Alcan.
GM = C. I. Gerhardt (ed.). 1849–63. G. W. Leibniz: Mathematische Schriften, 7 vols., Berlin/Halle: A. Asher/W. H. Schmidt.
LP = G. W. Leibniz. 1966. Logical Papers. A Selection, trans. and ed. with an introd. G. H. R. Parkinson. Oxford.
NE = G. W. Leibniz. 1981. New Essais on Human Understanding, trans. and ed. P. Remnant and J. Bennett. Cambridge: Cambridge University Press.
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Mugnai, M. (2019). Leibniz’s Mereology in the Essays on Logical Calculus of 1686–1690. In: De Risi, V. (eds) Leibniz and the Structure of Sciences. Boston Studies in the Philosophy and History of Science, vol 337. Springer, Cham. https://doi.org/10.1007/978-3-030-25572-5_2
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