![Sample our Humanities journals, sign in here to start your FREE access for 14 days]({"type":"image" , "src" : "/sda/5939/TOC-Subject-Sample-2021-Humanities.jpg"})
ABSTRACT
How did the traditional doctrine of parts and wholes evolve into contemporary formal mereology? This paper argues that a crucial missing link may lie in the early modern and especially Wolffian transformation of mereology into a systematic sub-discipline of ontology devoted to quantity. After some remarks on the traditional scholastic approach to parts and wholes (Sect. 1), Wolff's mature mereology is reconstructed as an attempt to provide an ontological foundation for mathematics (Sects. 2–3). On the basis of Wolff's earlier mereologies (Sect. 4), the origin of this foundational project is traced back to one of Wolff's private conversations with Leibniz (Sect. 5) and especially to the former's appropriation of the latter's notion of similarity as a means to define quantity (Sect. 6). Despite some hesitancy concerning the ultimate characterization of quantity (Sect. 7), Wolff's contribution was historically significant and influential. By developing a quantitative, extensional account of mereological relations, Wolff departed from the received doctrine and paved the way for the later revival of mereology at the intersection of ontology and mathematics.
KEYWORDS:
Acknowledgements
I am grateful to Giorgio Lando for a number of insightful comments on an earlier draft. I owe many thanks to two anonymous BJHP referees, especially Referee 2, for valuable suggestions and observations.
Notes
1 See Barnes, ‘Bits and Pieces’; Henry, Medieval Mereology; and Arlig, ‘Medieval Mereology’.
2 See Smith and Mulligan, ‘Pieces of a Theory’; Smith, ‘Annotated Bibliography’; and Simons, ‘Part/Whole II’.
3 See Kaulbach, Oeing-Hanhoff and Beck, ‘Ganzes/Teil’; Burkhardt and Dufour, ‘Part/Whole I’; Pasnau, Metaphysical Themes 1274–1671; and Normore and Brown, ‘On Bits and Pieces’. Leibniz's mereology has received some more sustained attention: see Boehm, Le ‘vinculum substantiale’; Schmidt, ‘Ganzes und Teil bei Leibniz’; Burkhardt, ‘The Part-Whole Relation’; Burkhardt and Degen, ‘Mereology in Leibniz's Logic’; Cook, ‘Monads and Mathematics’; and Mugnai ‘Leibniz's Mereology’.
4 See Gruszczyński and Varzi, ‘Mereology Then and Now’. As is well known, a second, independent mereological trend originated from Whitehead's work and Leonard and Goodman's calculus of individuals.
5 Leibniz's essay ‘De primae philosophiae emendatione, et de notione substantiae’, published in the Acta Eruditorum of March 1694, had an early and powerful influence on Wolff. See Carboncini, ‘L’ontologia di Wolff’.
6 See, e.g. Alsted, Metaphysica, Book I, Ch. 17; Scharf, Exemplaris metaphysica, Book II, Ch. 17; and Calov, Metaphysica divina, Ch. 17. Some authors even treated the doctrine of whole and part in both logic and metaphysics: see Freedman, European Academic Philosophy, vol. 2, 722n.
7 Wolff's application of mereology to natural sciences cannot be pursued in this paper. See the Conclusions for some hints.
8 On the historical significance of this idea of ontology, see de Boer, ‘Transformations of Transcendental Philosophy’, 62.
9 The book is mentioned in Wolff, Mathematisches Lexikon, s.v. ‘Pars’, col. 1024: ‘Barlaam Monachus hat in seiner Logistica lib. 1. p. 4 et seqq. vieles von dem Theile demonstriret’. Some of the distinctions expounded in this entry (such as those between pars aliquanta and pars aliquota, and between even and odd part) may derive from Barlaam, Logistica, 3–4. Wolff's Kurtzer Unterricht, §7, provides a short presentation of Barlaam's contribution to mathematics.
10 ‘Pars est magnitudo magnitudinis, minor maioris, quando metitur maiorem’. Barlaam, Logistica, 3.
11 Literally, the first lines of the passage seem to rule out the possibility of a whole that is exhaustively divided into more than two parts – which amounts to the implausible claim that every partition must be a bipartition. A more charitable reading is that A need not be one single part but may in fact consist of several sub-parts.
12 The historical background of this distinction is the scholastic debate on actual and potential parts. See Pasnau, Metaphysical Themes 1274–1671, 606–32.
13 As is well known, this still appears as a theorem in Leśniewski's 1916 first system of mereology (‘Foundations’, 131).
14 For instance, (T1) is demonstrated as follows: ‘Quaelibet enim pars totius est sibimetipsi, adeoque parti totius per hypoth. aequalis. Est igitur toto minor’. Ontologia, §357. Wolff praises this demonstration as an example of ‘perfect analysis’ (Ontologia, §357n), since its only premisses are a definition and an axiom proper, i.e. an identity, in conformity with Leibniz's ideal of an analysis of axioms (see below). See Ontologia, §387n, §390n.
15 Wolff makes it clear that his proposition is equivalent to the principle usually formulated by mathematicians in terms of ratio: ‘Mathematici cum relationem, de qua hic loquimur, rationem appellent, propositionem praesentem ita latius efferunt: Idem ad aequalia eandem rationem habet’. Ontologia, §369n; see §370, §370n.
16 Vulgar numbers, explains Wolff, are the numbers used in ordinary talk: ‘A Mathematicis hodie dicuntur Numeri rationales integri’. Ontologia, §340.
17 In ‘Elementa arithmeticae’, §40, this same phrase occurs as the very definition of the integer rational number (Elementa, vol. 1: 27). By contrast, Ontologia (§340) defines the integer rational number simply as ‘multitude of units’, which was Euclid's definition of number in general. See below.
18 ‘Wenn man viele eintzelne Dinge von einer Art zusammen nimmet, entstehet daraus eine Zahl. Und daher erkläret Euclides die Zahl durch eine Menge der Einheiten’. Wolff, ‘Anfangs-Gründe der Rechen-Kunst’, §5, in Anfangs-Gründe, vol. 1: 38.
19 The first edition reads: ‘A will be assumed to be one’, but the lapsus calami was pointed out by Leibniz (see below).
20 As pointed out by Cantù (‘La matematica’, 122n), Wolff's definition of number is reminiscent of Newton's definition of it in terms of the ratio of a quantity to another (see Newton, Arithmetica universalis, 2). Indeed, Wolff reviewed Newton's book in the Acta Eruditorum of November 1708, but without mentioning the definition of number.
21 In the subsequent editions of the Elementa, Wolff corrected the lapsus calami but never considered the reformulation suggested by Leibniz.
22 Nevertheless, Wolff also retains the definition given in the Elementa and claims that the two definitions are interderivable (Ontologia, §407 and §415n).
23 The current English translation of this passage misses Leibniz's allusion to an oral exchange and wrongly identifies Leibniz's reference to the Acta Eruditorum (see Leibniz, Philosophical Essays, 231 and note).
24 This clarification concerning the two orders is taken from ‘De methodo mathematica brevis commentatio’, §55, §57 (in Elementa, vol. 1: 16–17).
25 Although this review is not featured in Wolff's Sämtliche Rezensionen, this collection includes Wolff's review of the new 1730 edition of the Elementa. The latter review begins by referring to the previous one in the first person plural: ‘Cum haec Matheseos universae Elementa primum in lucem publicam prodirent, de iis abunde diximus in Actis A. 1714 p. 249 et seqq.’ Acta Eruditorum, August 1730: 340.
26 Mathematisches Lexikon, s.v. ‘Totum’, col. 1409: ‘Totum, ein gantzes, wird in der Mathematick genennet, was als eines angesehen wird, und doch vieles in sich hält, so von einander unterschieden werden kan, dem es zusammen gleich ist. … Nemlich es wird eine Sache in Ansehung ihrer Theile ein gantzes genennet’. As an example, Wolff mentions a line divided into segments.
27 The definitions of ‘lesser’ and ‘greater’ correspond to those of Ontologia, §352 (see above).
28 We may charitably assume that parts are of a certain size, which would make it acceptable to conclude that the more parts one thing has, the greater it is.
29 Cantù, ‘Mathematik als Größenlehre’, points out that Grösse is characterized as discrete algebraic quantity, whereas the characterization of mathematics as the science of quantities refers to Grösse as continuous quantity: see Wolff, Mathematisches Lexikon, col. 1143 and 864 respectively.
30 Ontologia, §434. This proposition is also used to argue that even the degrees of qualities have magnitude, albeit merely an imaginary one since the parts of degrees are imaginary (§§751–3).
31 The following passages actually reproduce the definition of similarity and its corollary from the Elementa arithmeticae, §24 and §26 (quoted above).
32 A further interesting application of mereology to natural science concerns the distinction between material parts and organs (i.e. functional parts) in Wolff's Cosmologia, §10.
33 See Berka, ‘Christian Wolff und Bernard Bolzano’; Cataldi Madonna, ‘Wolff, Bolzano e la probabilità’; Russ, The Mathematical Works of Bolzano, 14–19; and Rumore, ‘L'ontologia di Wolff’, 212–13.