Abstract
Ohad Nachtomy’s paper considers a fundamental metaphysical problem in Leibniz: the relation between infinity and being. Nachtomy argues that, for Leibniz, both a nonactive law prescribing an infinite program of action, and also a source of action or primitive force, are required in order to account for an actual being.
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Notes
- 1.
The abbreviations used are listed at the end of this article.
- 2.
In a letter to Conring (1677) Leibniz writes: “At qui subtiliores sunt adversarii ajunt Ens perfectissumum tam implicare contraditionem quam numerum maximum” (A 325).
- 3.
“There comes to mind a similar line of reasoning conspicuous in Galileo’s writings. The number of all squares is less than the number of all numbers, since there are some numbers which are non square. On the other hand, the number of all squares is equal to the number of all numbers, which I show as follows: there is no number which does not have its own corresponding square, therefore the number of all numbers is not greater than the number of all squares; on the other hand, every square number has a number as its side: therefore, the number of squares is not greater than the number of all numbers. Therefore, the number of all numbers (square and non-square) will be neither greater than nor less than, but equal to the number of all squares: the whole will be equal to the part, which is absurd” (A 550–51; Arthur, 177). See also A 6.3. 11; Arthur 5.
- 4.
In a letter to Conring (1677) Leibniz writes: “At qui subtiliores sunt adversarii ajunt Ens perfectissumum tam implicare contraditionem quam numerum maximum” (A 325). Leibniz’s possibility proof is given in the following passage: “Demonstrationem reperisse videor, quod Ens perfectissimum, seu quod omnem Essentiam contineat, seu quod omnes habeat Qualitates, seu omnia attributa affirmativa, sit possibile, seu non implicet contradictionem. Hoc patebit si ostendero omnia attributa (positiva) esse inter se compatibilia. Sunt autem attributa aut resolubilia, aut irresolubilia, si resolubilia sunt erunt aggregatum eorum in quae resolvuntur; suffecerit ergo ostendisse compatibilitatem omnium primorum, sive irresolubilium attributorum, sive quae per se concipiuntur, ita enim si singula compatibilia erunt, etiam plura erunt, adeoque et composita. Tantum ergo suffecerit ostendere Ens intelligi posse, quod omnia attributa prima contineat, seu duo quaelibet attributa prima esse inter se compatibilia” (A 6.3 572; DSR, p. 91–93).
- 5.
- 6.
In this period (as well as later ones, cf. 1984; Discours de métaphysique, New Essays), it is clear that Leibniz is investigating these notions by comparing and contrasting them. It is arguable that Leibniz’s concern regarding the possibility of the infinite being (and perhaps of possibility proofs in general) is driven by his concerns about the contradiction he discerns in the notion of infinite number, most rapid motion, and its likes. See my (2005).
- 7.
“A series is a multitude with a rule of order” (A 6.4 1426).
- 8.
As Couturat nicely points out (1961, 471; my translation), “one can say that Leibniz remains a nominalist in an entirely negative sense, namely that he rejects realism and denies universals a real and substantial existence. But he does not thereby refuse to assign them objective value, like the nominalists who reduce them to names. Rather, he adapts and intermediate position, which one designate by the name conceptualism, … ” As Mugnai notes (1992, 25), “there are no ideas without the intellectual activity of someone thinking (be it God or man or some other rational being).”
- 9.
“The law of order … constitutes the individuality of each particular substance” (GP IV, 518; L 493). “For me nothing is permanent in things except the law itself … The fact that a certain law persists, which involves the future states of what we conceive to be the same – this is the very fact, I say, that constitutes that same substance” (GP II 263–64; L 534–35). See also Theodicy 291.
- 10.
- 11.
“Numbers, modes, and relations are not entities” (A 6.3 463; DSR 7).
- 12.
There are two other important distinctions that I can only mention here: (1) Leibniz’s distinction between complete and incomplete beings, and (2) his distinction between the abstract and the concrete.
- 13.
Quoted in Brown (2000, 41).
- 14.
In his mature philosophy, Leibniz defines the world as an aggregate of finite things (aggregatum rerum finitarum) (cf. GP VII, 302). Similarly, in his letter to Gabriel Wagner of March 3 1698, Leibniz defines the world as an aggregate of changeable things or things that are susceptible of imperfection (cf. Grua, 396; Adams 1994, 15).
- 15.
I believe that this is the reason why what Levey has recently called “the construction problem” rests on a misconstrued version of Leibniz’s view of being and unity (Levey 2007, 64–66).
- 16.
“The aggregate of all bodies is called the world, which, if it is infinite, is not even one entity, any more than an infinite straight line or the greatest number are. So God cannot be understood as the World Soul: not the soul of a finite world because God himself is infinite, and not of an infinite world because an infinite body cannot be understood as one entity, but that which is not an entity in itself has no substantial form, and therefore no soul. So Martianus Capella is right to call God an extramundane intelligence” (A 6.4 1509; Arthur, 287).
- 17.
A corollary to this view is Leibniz’s definition of infinite series. He does not define infinite series as a sum of numbers but as a product of its formation rule. In this connection, see Couturat’s interesting discussion (1973, 476). Couturat cites this passage from the letter to des Bosses (of 11 March, 1706): “Neque enim negari potest, omnium numerorum possibilium naturas revera dari, saltem in divina mente, adeoque numerorum multidudinem esse infinitiam.”
- 18.
See Gurwitsch (1974, 65–72), section II d., on Generative Definitions.
- 19.
I have argued for this point in my (Nachtomy, 2002, 31–58).
- 20.
For the origin and the meaning of this doctrine, see Richard Arthur’s introduction to Arthur.
- 21.
In his Principles of Philosophy article 203, Descartes seems to assimilate the artificial and the natural. For him, artificial machines serve as models to explain the natural ones. Natural machines are like artificial ones, except much more complicated. He wants to establish that they are of the same kind. He uses the notion of divine created machines to show that the subtle parts of machines are extremely complex and invisible to us. While both Descartes and Leibniz argue that machines are extremely subtle, Descartes uses this point to argue for his view that, in the final analysis, animals are nothing but subtle machines. By contrast, Leibniz uses this point to argue that there is a categorical difference between them. See also Les passions de l’ame, first part, articles 5 and 6 where he writes e.g., that the body has in it “the corporeal source of movement” (art. 6).
- 22.
See for example, Leibniz’s controversy with Stahl (Carvallo 2004, 80), where Leibniz criticizes the Moderns for pretending that “nihil aliud sit natura corporum quam Mechanismus” (there is nothing in the nature of bodies but mechanism).
- 23.
“… since I am truly a single indivisible substance, unresolvable into any others, the permanent and constant subject of my actions and passions, it is necessary that there be a persisting individual substance over and above the organic body” (Comments on Fardella, AG 104).
- 24.
See also GP II 252; GP VII 502 and C 13–14.
- 25.
Je ne compte pour substances corporelles que les machines de la nature qui ont des âmes ou quelque chose d’analogique; autrement il n’y aura point de vraie unité (A Jaquelot, 22 mars 1703, GP III, 457).
- 26.
See Fichant (2003) and Duchesneau (1998).
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Nachtomy, O. (2010). Leibniz on Infinite Beings and Non-beings. In: Fraenkel, C., Perinetti, D., Smith, J. (eds) The Rationalists: Between Tradition and Innovation. The New Synthese Historical Library, vol 65. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9385-1_11
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