Matter and two concepts of continuity in Leibniz

Philosophical Studies

Volumn 94, Issue 1-2, 1999, Pages 81-118

  Levey, Samuel ^a  

^a Lamont Doherty Earth Observatory   (United States)

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EID: 53149154852     PISSN: 00318116     EISSN: 15730883     Source Type: Journal    
DOI: 10.1023/a:1004455121551     Document Type: Article

References (44)

1. The open territory here remains very large. In this paper I shall leave aside (among the many related topics in Leibniz) many of the nuances of Leibniz's later thought concerning continuity and continuous orderings, his views about the construction of points, lines, and planes, his account of a link between continuity and possibility, etc. For a good discussion of some aspects of Leibniz's later accounts of continuity not taken up in the present paper, see Timothy Crockett, "Continuity in Leibniz's Mature Metaphysics," also in this issue.
2. I am responsible throughout for translations of Leibniz and Descartes, but in translating Descartes I have consulted Cottingham, Stoothoff, and Murdoch, eds., The Philosophical Writings of Descartes Vol. 1 (Cambridge: Cambridge University Press, 1985), and in translating
3. Leibniz, Leroy Loemker, ed., Philosophical Papers and Letters (Dordrecht: Kluwer Academic Publishers, 1969) and
4. R. Ariew and D. Garber, eds., Philosophical Essays (Indianapolis: Hackett Publishing Co., 1989).
5. Also, for Leibniz's writings in A VI,3, I have consulted manuscripts of Richard Arthur's forthcoming translation volume, Leibniz and the Labyrinth of the Continuum: Leibniz's Writings on the Continuum 1672-1686,
6. to be published in the Yale Leibniz Series. I abbreviate the primary texts thus: A = Berlin Academy Edition, Samtliche Schriften und Briefe: Philosophische Schriften, Series VI, Vols. 1-3, 6 (Berlin: Akademie-Verlag. 1923-80);
7. AT = Adam and Tannery, eds., Oeuvres de Descartes, revised edition (Paris: Vrin/C.N.R.S., 1964-76);
8. C = Couturat, ed., Opuscules et Fragments inédits de Leibniz (Paris: Alcan, 1903);
9. G = Gerhardt, ed., Die Philosophischen Schriften, Vols. 1-7 (Berlin: Weidmannsche Buchhandlung, 1875-90);
10. GM = Gerhardt, ed., Mathematische Schriften von Gottfried Wilhelm Leibniz, Vols. 1-7 (Berlin: A. Asher; Halle: H.W. Schmidt, 1849-63). References to AT, G and GM are to volume and page numbers; references to C and VE are to page numbers; references to A are to series, volume and page numbers. Also, I follow Arthur in adopting a notational convention (initiated by Edwin Curley) for translating the Latin term seu, which (like sive) corresponds to the English term 'or' where it implies equivalence: seu = or in other words, that is to say, etc. In the translations of the Latin, I mark the occurrences of this " 'or' of equivalence" in Leibniz's text with a circumflex, thus: or.
11. The earliest passage I know of where Leibniz explicitly defends the plenum hypothesis is A, VI,525 dated to March 1676. And there he takes pains to point out that the hypothesis is not merely presupposed; rather, he argues for it. But the view of nature as a plenum seems clearly to be a backdrop for many of his discussions from 1671 to 1672 onward.
12. In fact in this letter Leibniz presents two possible ways in which discontinuity might arise: "first, in such a way that contiguity is at the same time destroyed, when the parts are so pulled apart from each other that a vacuum is left; or in such a way that contiguity remains" (A VI,2,435). His ensuing discussion makes it clear that it is the latter possibility which keenly interests him, and that it is the one he holds to be how discontinuity is actually introduced into matter.
13. Aristotle's definition occurs at Physics 231a21, 227a10-15, and
14. Metaphysics 1069a5-8.
15. Perhaps we should say it is a mere contiguum: a contiguum that is not also a continuum. For on the usual Aristotelian account that Leibniz observes here, the continuous is a special case (the limiting case) of the contiguous: the continuous are both "those whose boundaries are together" and "those whose boundaries are one." See, for example, Aristotle, Physics 227a10f., and Leibniz's remarks about the continuous and the contiguous at A VI,3,94, and 537.
16. It's hard to capture the subtle imagery at work here in a close translation. Leibniz's materia turbans evokes a picture of a body of matter as a disorderly crowd: a multitude of agitating individuals, each with its own distinctive actions, yet nonetheless collectively displaying a sort of unity by participating in a greater common motion.
17. In a piece whose date is not clearly established (the Akademie editors place it anywhere between 1677-1695), Leibniz writes in much the same vein: "The principle of cohesion is harmonizing motion [motus conspirans], and that of fluidity is varying motion" (VE 495).
18. This is to remain Leibniz's view, in the slightly evolved form of the thesis that nothing is absolutely fluid or solid, but rather that all matter is to some degree or other pliant, and that motion accounts for this. In a piece dated to 1683-6, "On the Existing World," he writes: "Therefore it must be said that no point can be assigned in the world which is not set in motion somewhat differently from any other point however near to it, but, on the other hand, that no point can be assigned which does not have some motion in common with some other given point in the world; under the former head, all bodies are fluid; under the latter, all are cohering. But to the extent that a common or proper motion is more or less observable, a body is called on solid, or a separate body, or perhaps even a fluid" (VE 420).
19. There's a lot of text here. For an overview, see Hartz, G. and Cover, J. "Space and Time in the Leibnizian Metaphysic," Noûs 22 (1988): 493-519.
20. See also, J.E. McGuire, "'Labyrinthus Continui': Leibniz on Substance, Activity and Matter," in Machamer and Turnbull (eds.), Motion and Time, Space and Matter (Columbus: Ohio State University Press, 1976) 290-326;
21. Glenn Hartz, "Leibniz's Phenomenalisms," The Philosophical Review 101 (1992): 511-49;
22. Richard Arthur, "Russell's Conundrum: On the Relation of Leibniz's Monads to the Continuum." In Brown and Mittelstrass (eds.), An Intimate Relation (Dordrecht: Reidel, 1989) 171-201;
23. and Chapter III of my "Matter, Unity and Infinity in Early Leibniz" (Ph.D. dissertation, Syracuse University, 1997). Although the period of Leibniz's writings that I have in mind here is a vaguely bounded one, spanning more than a decade, I believe the purest expressions of the views hat he holds during this time emerge near its end, in the years closely surrounding 1705. For that reason, and for brevity, I shall refer to the accounts of matter and continuity that Leibniz defends in this period as belonging to "the 1705 metaphysics."
24. Elsewhere is "Leibniz's Constructivism and Infinitely Folded Matter," forthcoming in Genarro and Huenemann (eds.), New Essays on the Rationalists (Oxford University Press, 1999). Also, in section 1 of that paper the modalized spin on continuity is discussed briefly.
25. More precisely, it's something that is not a feature of the Cartesian view as Leibniz would have seen it. For in Descartes' Optics one can find an ideas of an "action or tendency to move" (which "it is necessary to distinguish from" movement itself) that could naturally be read as a cousin concept to Leibniz's conatus (cf. AT VI,88). But while Leibniz certainly read Principles 1-2 closely, there is no clear evidence to suggest that he is (in 1670-2, at any rate) aware of that particular element of Descartes' views in the Optics. My thanks to Alison Simmons for bringing that element of the Optics to my attention.
26. In one important sense, however, the conatus physics does stay alive. The picture of the material world as invested throughout with centers of motion - the picture of the conatus world with its principles of action that are in each of the infinitely many parts of matter and ground their physical properties - this will quietly keep its grip on Leibniz's thought for some time, and rearise explicitly in his later efforts to found physics upon a metaphysics of immaterial and "active" first principles: another role that is to be ascribed to the monads. In this way one might say that the ontology of conati "transforms" into the ontology of monads in the later years. But as a basic account of physical phenomena - of motion, cohesion, impact, and so on - the early theory of endeavors has run its course by late spring of 1676. For some (albeit brief) discussion of Leibniz's rejection of infinitesimals see fn. 8 of my "Leibniz on Mathematics and the Actually Infinite Division of Matter," The Philosophical Review, Vol. 107, No. 1 (January 1998).
27. Michael White is the first to make this point that the concept of a seamless or natural whole finds expression in the contemporary topological concept of connectedness. For a fine discussion of some related issues in Aristotle's philosophy (including Physics 227a6 and 227a10-15 that I quote below on pages 24-5) and the philosophy of topology,
28. see his "On Continuity: Aristotle Versus Topology?" History and Philosophy of Logic, Vol. 9, No. 1 (1988), 1-12.
29. I appeal to point-set topology in order to motivate in a clear way the concept of connectedness; I am not proposing here to model Leibniz's account of matter and continuity in that framework. The latter project would not be a straightforward one, for (inter alia) the idea of "indistant" contiguous closed regions would not be consistent with the usual metric topology. That is, the usual way of defining a metric on a topological space won't allow for two distinct surfaces to be at "no distance" from one another (though a non-standard approach might still do it). More importantly, however, the ontology of point sets would simply be anathema to Leibniz. The right approach to capturing Leibniz's account would, I suspect, require a model in combinatorial topology. I discuss this approach in a work in progress, tentatively entitled "Discontinuity and the Structure of Motion in Leibniz's Metaphysics."
30. Cf. James Munkres, Topology (Englewood Cliffs, NJ: Prentice Hall, 1975);
31. Michael Henle, A Combinatorial Introduction to Topology (New York: Dover, 1979);
32. Paul Alexandroff, Elementary Concepts of Topology (New York: Dover, 1961).
33. is a set of open subsets of space S. (1) The whole space S and the empty set are elements of S. (2) The union of any number (finite, countable or uncountable) of sets of from is in. (3) The intersection of any finite number of sets from is in.
34. We shall assume throughout that all the point spaces discussed have the "usual topology" for Euclidean spaces, namely, the order topology and its self-products.
35. In fact, this definition of a limit point is broad enough to include all the interior points of a set as limit points, as well as those that fall on the boundary. For our purposes, we shall be considering only those limit points that are also boundary points (i.e. those limit points that are not also interior points).
36. A separation of a topological space S is more usually defined as a pair of nonempty, disjoint, open sets S and V such that the union (U ∪ V) = S. Our definition in terms of limit points is equivalent, but brings out more clearly the issues of our immediate concern.
37. Leibniz's fullest expression of the argument for this conclusion, which I won't rehearse here, occurs near the outset of Pacidius Philalethei (cf. A, VI,3,534-41), though when he writes the Pacidius in autumn of 1676 he has been warming up to that line of argument for some time.
38. It should be noted that in a passage near the end of the Pacidius's main discussion of middle states there is a remark that poses a prima facie challenge to my suggestion here that the argument against middle states might be seen as ruling out the existence of common boundaries and with it the related account of cohesion. The primary interlocutor, Charinus, while offering an example of a perfectly round sphere resting on a perfectly flat table, says in passing: "It is clear that the sphere does not cohere with the plane, and that they have no extrema in common, otherwise one would not be able to move without the other" (A VI,3,537). This certainly suggests that the account of cohesion as boundary-sharing is in the air (even if it's not exactly asserted), and the interlocutors do not go on later to point out that the argument against middle states defeats that account of cohesion. But in my view, also to be defended in "Discontinuity and the Structure of Motion," this account of cohesion, like so many of the ideas mentioned early in the Pacidius, is rehearsed as a feature of Leibniz's previous thought, and it is not an element of the most considered views that are ultimately endorsed in the dialogue; moreover, Leibniz's final commitments in the Pacidius, and the overall gist of his remarks about middle states and continuity that occur as the dialogue progresses, do seem to militate against the account of cohesion as boundary-sharing.
39. This is not to say that the texts are always unwaveringly clear on this point. As Robert Adams has pointed out, Leibniz can be "quoted on both sides of the question whether bodies are continuous or whether they only appear to be continuous" (Adams, Leibniz: Determinist, Theist, Idealist. (New York: Oxford University Press, 1994), p. 233). Yet I do not believe there is a very serious worry about Leibniz's views even in most of the passages (after 1676) in which he can be quoted apparently on the other side of the question. In the years shortly following 1676 Leibniz will sometimes call matter 'continuous', but I think it is fairly clear in those contexts that he only means that matter is not interrupted by void spaces (which accords with a use of 'continuous' that he mentions occasionally in 1676; cf. A VI,3,542). Adams cites a troubling passage at G IV,394, from 1702, where Leibniz seems to say outright that body [corpus] is continuous. Though continuity itself does seem to be at stake there, it is less than clear to me that in the crucial sentence Leibniz is characterizing the nature of actual material bodies and not, say, an abstract concept of "body" that simply does not weigh-in on the question whether actual bodies are ultimately discrete or continuous. Still, even if Leibniz wavers in places, his considered view about material bodies is abundantly clear and a deeply committed feature of his metaphysics.
40. The Latin term cointegrantes derives from integro = "to make a whole," so Leibniz's subsequent gloss on "co-integrating parts" is in fact nicely captured already in the base meaning of his technical term, though that does not come through in my translation.
41. See my "Leibniz on Mathematics and the Actually Infinite Division of Matter," op. cit.
42. The relevant principles are: (1) No infinite aggregate can form a true whole, or be truly one; and (2) what is not truly one does not truly exist (cf. G 11,97,251). The first of those is especially motivated by the link between the concept of an infinite whole and Galileo's paradox. I take this up in detail in "Leibniz on Mathematics," op. cit.
43. Leibniz's own statement of the regress problem - and the argument it yields for the existence of immaterial substantial unities - can be found at, for example, G 11,261, 267, 296. I discuss the regress problem together with some others (including the "unity problem" mentioned in the previous footnote) in "Leibniz's Constructivism and Infinitely Folded Matter," op. cit.
44. Earlier versions of this paper were presented at Dartmouth College and at the March 1998 Pacific Division meetings of the American Philosophical Association. My thanks to both audiences for helpful input. Special thanks also to Christie Thomas and Walter Sinnott-Armstrong for providing comments on earlier written drafts, and to John O'Leary-Hawthorne, Jan Cover and Timothy Crockett for helpful discussion.