Leibniz's Mathematical Argument against a Soul of the World

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Notes

A Gottfried Wilhelm Leibniz: Sämtliche Schriften und Briefe, edited by the Deutsche Akademie der Wissenscaften (Darmstadt and Berlin: Akademie-Verlag, 1923–). Cited by series, volume and page, e.g. A.VI.iv.587.AG Leibniz: Philosophical Essays, ed. and trans. by R. Ariew and D.Garber (Indianapolis: Hackett 1989).C Opuscles et fragments inédits de Leibniz, ed. by L. Couturat (Paris: Alcan 1903; reprint Hildesheim: Olms 1961).DM Discourse on Metaphysics, cited by section.DMP De mundo praesenti (March 1684–Spring 1686): A.VI.iv.1505–13 = RA.282–96.DNMA Deum non esse mundi animam (1683–Winter 1685/86(?)): A.VI.iv.1492.G Die Philosophische Schriften von Gottfried Wilhelm Leibniz, 7 vols., ed. by C. I. Gerhardt (Berlin: Weidmann 1875–1890; reprint Hildesheim: Olms 1960). Cited by volume and page, e.g. G.III.235.GM Leibnizens Mathematische Schriften, ed. by C. I. Gerhardt, 7 vols. (Berlin and Halled: Asher and Schmidt 1849–1863; reprint Hildesheim: Olms 1971. Cited by volume and page, e.g. GM.III.58.HGA The Leibniz Clarke Correspondence, ed. with introduction and notes by H. G. Alexander (Manchester: Manchester University Press, 1956). Leibniz's letters are cited by number and section, e.g. Lz.V.47.L Gottfried Wilhelm Leibniz: Philosophical Papers and Letters, ed. and trans. by Leroy Loemker (Dordrecht: D. Reidel, 1969).LA The Leibniz–Arnauld Correspondence, ed. and trans. by H. T. Mason with an introduction by G. H. R. Parkinson (Manchester University Press 1967).LDB Letter to Des Bosses, 11 March 1706. G.II.304–8.LH Leibniz-Handschriften: Niedersächsische Landesbibliothek, Hanoever, as listed in Eduard Bodemann, [IT]Die Leibniz-Handschriften der königlichen öfftenlichen Bibliothek zu Hannover (Hanover and Leipzig: Hann'sche Buchhandlung, 1895; reprint Hildesheim: Olms 1966).MP Gottfried Wilhelm Leibniz: Philosophical Writings, ed. and trans. by M. Morris and G. H. R. Parkinson (London: J. M. Dent & Sons, 1973).NE New Essays on Human Understanding, ed. and trans. by P. Remnant and J. Bennett (Cambridge: Cambridge University Press, 1996).P G. W. Leibniz, De Summa Rerum: Metaphysical Papers and Letters, 1675–1676, ed. and trans. by G. H. R. Parkinson (New Haven: Yale University Press, 1992).RA G. W. Leibniz: The Labyrinth of the Continuum, ed. and transl. with an intro. by R. Arthur (New Haven: Yale University Press, 2001).T G. W. Leibniz: Theodicy, ed. with an intro. by A. Farrer and trans. by E. M. Huggard (La Salle, Illinois: Open Court, 1985). Cited by section, e.g. Theodicy §195, and by page, e.g. T.87.

¹Wittgenstein 1956: I, Appendix II, § 3; as quoted in Benardete 1964: 44–5.

²In this passage, Russell does not explicitly claim that for Leibniz, if something is a substantial unity, then it is a whole. What he does explicitly claim is that Leibniz held the converse: if something is a whole, then it is a substantial unity (‘the notion of a whole can only be applied to what is substantially indivisible’). But it is the former proposition that is necessary to establish that the world is not a substantial unity on the grounds that it is not a whole. The latter proposition provides no avenue for establishing that the world is not a substantial unity. Moreover, as will become clear later, it is not even something that Leibniz held. For Leibniz thought that things could be wholes without being substantial unities.

³The reason that Carlin says that the argument he thinks is suggested by Russell's interpretation ‘requires the premise that only bodies which are one per se can have a soul’ is presumably because in the passage that he quotes, Russell does not attribute to Leibniz the claim that if something is a substantial unity, then it is a whole, which is required to establish what Russell apparently wants to claim, namely, that for Leibniz the world cannot be a substantial unity because it is not a whole. So if we add to what Russell actually attributes to Leibniz that which is required, namely, that if something is a whole, then it is a substantial unity, we end up with the equivalence of being a whole and being a substantial unity, or being one per se. As I suggested earlier, the claim that ‘one whole must be one substance’, which Russell attributes to Leibniz, may have been intended by Russell to express this equivalence.

⁴I previously argued that such a distinction was not relevant to the argument. See Brown 1998: 115–16.

⁵It should be noted that DMP was not generally available at the time Carlin published his article in 1997. It appeared subsequently (1999) in A.VI.iv.1505–13 and (2001) in RA.283–97.

⁶The passage I have translated above from DNMA constitutes the whole of that piece.

⁷Arthur notes that ‘above the Ens of both Ens per se and Ens per accidens Leibniz had written the word unum (a unity); I have instead set it in parentheses afterwards’ (RA.421n3).

⁸That is, in his letter of 8 December 1686. See G.II.76 = LA.94 = AG.79.

⁹Recall that in the passage from DMP quoted at the beginning of this section, Leibniz declares that the principle of unity ‘in us is called a soul, and in every body a substantial form, provided that the body is a unity in itself’. I mention this to illustrate the point that Leibniz often uses the term ‘body’ to refer to a substantial unity or ‘a unity in itself’, that is, a corporeal substance. But the context, as in the present passage, generally makes it clear that a ‘body’ so understood possesses a substantial form. A body that lacks a substantial form is a mere aggregate that lacks substantial unity; and so a body that actually possesses a soul, but is being considered apart from soul, must be considered as an aggregate, even though it is in fact and independently of how we are considering it a substantial unity in virtue of actually possessing a soul.

¹⁰As the preceding passage makes clear, Leibniz uses the term ‘notion’ in a broader sense than ‘idea’. Notions include thoughts of both consistent and inconsistent concepts, but there can be ideas only of consistent concepts.

¹¹Although Leibniz himself does not use the expressions ‘finite in magnitude’ and ‘infinite in magnitude’, using rather ‘finite’ and ‘infinite’ simpliciter, I do not believe the terms he uses in this context can bear any other reading.

¹²In fact, considered abstractly, apart from their subjects, I take it that for Leibniz the extension of space and the extension of body are the same. In his Fifth Letter to Clarke, Leibniz does say that ‘finite space is not the extension of bodies: as time is not their duration’. Rather, ‘things keep their extension; but they do not always keep their space’. So ‘every thing has its own extension, its own duration; but it has not its own time, and does not keep its own space’ (Lz.V.46, G.VII.399 = HGA.69). But I assume that the distinction between finite space and extension of body is made because, as Leibniz says in the Entretien de Philarete et d'Ariste, the extension of the one, namely, space, relates to the ‘diffusion or continuation of situation or locality’, whereas the extension of the other, namely, body, relates to the ‘diffusion of antitypy or materiality’. But considered abstractly, I take it that for Leibniz the extension of space and of body are the same, which is why, in the New Essays, he denies that there is a double extension, one of body and one of space:

Although it is true that in conceiving body one conceives something in addition to space, it does not follow that there are two extensions, that of space and that of body. Similarly, in conceiving several things at once one conceives something in addition to the number, namely the things numbered; and yet there are not two pluralities, one of them abstract (for the number) and the other concrete (for the things numbered). In the same way, there is no need to postulate two extensions, one abstract (for space) and the other concrete (for body). For the concrete one is as it is only by virtue of the abstract one: just as bodies pass from one position in space to another, i.e. change how they are ordered in relation to one another, so things pass also from one position to another within an ordering or enumeration – as when the first becomes the second, the second becomes the third, etc. In fact, time and place are only kinds of order.

(A.VI.vi.127 = NE.127)

¹³It is true that the argument as it is presented in DMP begins with the assumption that ‘the aggregate of all bodies is called the world’ (A.VI.iv.1509 = RA.287), and that two paragraphs earlier, Leibniz had argued that ‘bodies that lack a substantial form … are merely aggregates of bodies, like a woodpile or a heap of stones’ (A.VI.iv.1508 = RA.287). But since the assumption that the world is an aggregate plays no essential role in the argument presented there against the possibility of a world soul—at least as I interpret that argument—I think that the term ‘aggregate’ should, in the context of the argument, be interpreted in some neutral sense that does not rule out the possibility that the world might be one in multitude.

¹⁴It is easy to get the impression that Carlin takes Leibniz's argument to be that the world cannot become a substance because it is not a whole. But this, I think, would be a confusion. For Leibniz often points out that corporeal substances do not come to be, that no body that lacks a soul ever comes to possess one, although a body may come to be incorporated within the body of an existing corporeal substance and hence come to be a part of a body that possesses a soul. Thus, for example, at Monadology §§72–6, Leibniz wrote:

The soul changes body only little by little and by degrees, so that it is never stripped at once of all its organs … That is why there is never total generation nor, strictly speaking, perfect death, death consisting in separation of the soul. And what we call generations are developments and growths, as what we call deaths are enfoldings and diminutions… . Today, when exact inquiries on plants, insects, and animals have shown us that organic bodies in nature are never produced from chaos or putrefaction, but always through seeds in which there is, no doubt, some preformation, it has been judged that, not only the organic body was already there before conception, but there was also a soul in this body; in brief, the animal itself was there, and through conception this animal was merely prepared for a great transformation, in order to become an animal of another kind… . I have, therefore, held that if the animal never begins naturally, it does not end naturally, either; and not only will there be no generation, but also no complete destruction, nor any death, strictly speaking.

(G.VI.619–20 = AG.222–23; cf. G.VI.601–2 = AG.209, A.VI.iv.1466 = RA.264–5; A.VI.iv.1649 = AG.34 = MP.92, G.IV.480–81 = AG.140–1 = MP.118–19)

Thus whatever corporeal substances there are have, according to Leibniz, existed from creation and, barring miraculous annihilation, will never cease to be. Consequently, it would make little sense on Leibniz's view to suggest, as Carlin seems to, that a body cannot come to possess a soul in virtue of its lacking some particular property or other; for on Leibniz's view, no body whatsoever can come to possess a soul. The issue for Leibniz is not whether a body can come to possess a soul but rather, as I have argued, whether it actually does possess a soul; and according to Leibniz the criterion for determining whether a body does possess a soul is whether the body is an unum per se.

¹⁵In response to questions that were raised about how he was conceiving the relationship between wholes and substances, Arthur notes that it ‘was quite correct to challenge any hint in my previous paper that ‘true wholes’ could be identified with substances, which have no parts’ (Arthur 2001: 115n20). But there were more than simply hints to that effect. For example, at one place Arthur argued as follows:

There is no bound to an infinite division, such as occurs in any body, so that the infinite parts of a body do not constitute it as an infinite collection or true whole. Consequently, a Cartesian body, regarded as pure extension, is not something complete, and cannot be a substance.

(Arthur 1999: 111)

But if, as is asserted here, it follows from the fact that ‘the infinite parts of a body do not constitute it as an infinite collection or true whole’ that a Cartesian body ‘is not something complete, and cannot be a substance’, then a substance must, in fact, be a ‘true whole’. In the same paper again, Arthur wrote that if Leibniz had ‘embraced infinite numbers and wholes … (b)odies would be real wholes’ and ‘matter would … be real, and would not need immaterial principles to complete it’. Furthermore, ‘human bodies would form real unities without the need of immaterial souls’ (ibid.). But if bodies would be ‘real wholes’ and ‘matter would be real … and not need immaterial principles to complete it’ under the assumption that infinite number and wholes are possible, then it is hard to see how this does not involve the assumption that ‘real wholes’ are substances. And it seems that the only way to understand the claim that if infinite number and wholes were possible, then ‘human bodies would form real unities’ is by assuming that wholes are ‘real unities’. Finally, in the introduction to his recently published book, Arthur still writes that ‘in order for the various parts to be said truly to belong to the same whole, to the same substance, it is necessary that there should be a substantial form, a principle that underlies the changing phenomena’ (RA.lxx). But this surely seems to imply that substances are wholes with parts.

In any event, one reason it is difficult to keep the notions of whole and substance distinct in Leibniz is, I think, because he does hold that substances must be wholes in magnitude; but, as I have argued, this does not violate the assumption that una per se lack actual parts, since the parts involved in measuring magnitude are not actual parts of a body or of a corporeal substance. I must confess, however, that I myself had previously thought that it was Leibniz's position that substances could not be wholes on the grounds that he held that substances, unlike wholes, cannot have parts.

¹⁶Arthur 2001: 113. It should be noted that Carlin himself does not say that ‘it seems to be a tacit premise for Leibniz that only such a phenomenal body is a candidate for being the body of a real substance’.

¹⁷Leibniz had arrived at the argument that he presented to Malebranche several years earlier. In a paper from December 1675, he had argued that ‘the number of all numbers is a contradiction, i.e. there is no idea of it; for otherwise it would follow that the whole is equal to the part, or that there are as many numbers as there are square numbers’ (A.IV.iii.463 = P.7).

¹⁸One of these is a piece that the editors of the Akademie edition entitle De minimo et maximo and date between the fall of 1672 and the winter of 1672/73, and the statement of the paradox may be found at A.VI.iii.97–8. The other is Pacidius Philalethi (1676), and Leibniz's statement of the paradox in this work may be found at A.VI.iii.549–50 = RA.175–7.

¹⁹See Brown 2000: 22.

²⁰Levey 1998: 62; cf. also Benardete 1964: 47–8.

²¹Leibniz had in fact formulated his ‘demonstration’ of the part–whole axiom many years before this, in a paper that the editors of the Akademie edition entitle Demonstratio propositionum primarum and date between fall 1671 and the beginning of 1672 (see A.VI.ii.482–3).

²²See Arthur 2001: 113.

²³I am very grateful to Mark Kulstad for the many invigorating conversations we have had concerning the matters discussed in this paper, and for the many helpful comments and suggestions that he has made along the way. I should also like to thank Laurence Carlin and Richard Arthur who, each in his own way, have forced me to think much harder than I had thought was necessary to understand Leibniz's odd little argument against the possibility of a world soul.