The ‘Properties’ of Leibnizian Space: Whither Relationism?

Abstract

The metaphysical foundation of Leibniz's theory of space is examined against the backdrop of the subtantivalism/relationism debate, while largely confining the analysis to the ontological level of material bodies and properties. It emerges that the details of Leibniz' theory defy a straightforward categorization which employs the standard relationism that is often attributed to his views. Rather, a more careful analysis of his metaphysical doctrines on bodies and space reveals the importance of a host of concepts, such as the foundational role of God, the holism of both geometry and the material world's interconnections, and the viability and adequacy of a property theory in characterizing his natural philosophy of space.

Notes

¹ C.D. Broad, ‘Leibniz's Last Controversy with the Newtonians’ in Leibniz: Metaphysics and Philosophy of Science, edited by R.S. Woolhouse (Oxford: Oxford University Press, 1981 [1946]), 157–174 (171–173); L. Sklar, Space, Time, and Spacetime (Berkeley, CA: University of California Press, 1974), 169; M. Friedman, Foundations of Space-Time Theories (Princeton, NJ: Princeton University Press, 1983), 219; see also S. Auyang, How is Quantum Field Theory Possible? (New York, NY: Oxford University Press, 1995), 247.

² E. Khamara, ‘Leibniz's Theory of Space: A Reconstruction’, Philosophical Quarterly, 43 (1993), 472–488 (478); G. Belot, Geometric Possibility (Oxford: Oxford University Press, 2011), 173–185.

³ See, for example, M.J. Futch, Leibniz's Metaphysics of Time and Space (Berlin: Springer-Verlag, 2008), 48; and generally for relationists, C. Hooker, ‘The Relational Doctrines of Space and Time’, British Journal for the Philosophy of Science, 22 (1971), 97–130 (111).

⁴ Citations to original works will list the original source on its first appearance but not after, followed by an English language translation, when available. [C]: Opuscules et fragments inédits de Leibniz, edited by L. Couturat (Paris, 1903); [A]: Sämtliche Schriften und Briefe, edited by Akademie der Wissenschaften der DDR (Darmstadt and Berlin: Akademie-Verlag, 1923), cited with series, volume, and page; [GM]: Leibnizens mathematische schriften, edited by C.I. Gerhardt (Hildesheim: Olms, 1962), cited with volume and page; [G]: Die philosophischen schriften von Leibniz, edited by C.I. Gerhardt (C.I. Hildesheim: Olms, 1965), cited with volume and page; [L]: Leibniz: philosophical letters and papers, edited and translated by L.E. Loemker, second edition (Dordrecht: Kluwer, 1969); [AT]: R. Descartes, Oeuvres de Descartes, edited by C. Adams and P. Tannery (Paris: J. Vrin, 1976), cited with volume and page; [AG]: Leibniz: philosophical essays, edited and translated by R. Ariew and D. Garber (Indianapolis, IN: Hackett, 1989); [MP]: Leibniz: Philosophical Writings, edited and translated by M. Morris and G.H.R. Parkinson (Rutland, VT: C. Tuttle, 1995); [NE]: New Essay on Human Understanding, edited and translated by P. Remnant and J. Bennett (Cambridge: Cambridge University Press, 1996), cited with book, chapter, and section; [LC]: Leibniz and Clarke Correspondence, edited and translated by R. Ariew (Indianapolis, IN: Hackett, 2000), cited with author, [L] or [C], letter, and section; [LoC]: The Labyrinth of the Continuum: Writings of 1672 to 1686, edited and translated by R. Arthur (New Haven, CT: Yale University Press, 2001).

⁵ G.VII.345–440; LC: L.V.62.

⁶ For example, Descartes, Principles of Philosophy, II.16; AT VIIIA 49.

⁷ To J. Bernoulli, January 1699, GM.III.565; AG 170.

⁸ LC: L.IV.17.

⁹ LC: L.V.30.

¹⁰ This definition is paraphrased, for our purposes, from J. Earman, World Enough and Space-Time (Cambridge, MA: MIT Press, 1989), 12.

¹¹ G.V.39–509; NE: II.xiii.8.

¹² LC: L.V.47.

¹³ NE: II.xiii.8.

¹⁴ I. Newton, Philosophical Writings, translated and edited by A. Janiak and C. Johnson (Cambridge: Cambridge University Press, 2004), 66.

¹⁷ LC: L.V.46.

¹⁵ LC: L.V.47.

¹⁶ The following characterization of ‘same place’ by R. Arthur, on the other hand, seems entirely appropriate, and does not bring into play Galilean transformations: ‘Thus the hypothesis of fixed existents allows us to define place in terms of an equivalence: it is the equivalence class of all things that bear the same situation to our (fictitious) fixed existents. And when we take all possible situations relative to these fixed existents, we have a manifold of places, or abstract space’, R.T.W. Arthur, ‘Space and Relativity in Newton and Leibniz’, British Journal for the Philosophy of Science, 45 (1994), 219–240 (237). On a similar note, it should be mentioned that the findings in above present obstacles to line of thought present in several contemporary assessments of Leibniz's theory, i.e. that Leibniz's space is relational, or quasi-relational, but that his treatment of motion is absolute; see, e.g., J.W. Cook, ‘A Reappraisal of Leibniz's Views on Space, Time, and Motion’, Philosophical Investigations, 2 (1979), 22–63; J.T. Roberts, ‘Leibniz on force and absolute motion’, Philosophy of Science, 70 (2003), 553–573. A full assessment of relational motion as regards Leibniz is beyond the bounds of this essay, but, for a careful analysis, see P. Lodge, ‘Leibniz on Relativity and the Motion of Bodies’, Philosophical Topics, 31 (2003), 277–308. On the whole, De Risi and Arthur have contributed greatly to the cause of disassociating Leibniz from the traditional, reductive, and external relationism that most modern philosophers of space and time have tended to read into his philosophy.

¹⁸ LC: L.V.49.

¹⁹ For example, LC: L.V.47.

²⁰ Earman, World Enough and Space-Time, 13.

²¹ Earman, World Enough and Space-Time, 14.

²² This minimalist interpretation, (R2*), is presumed in and all earlier works by the author, but included within the general (R2) category. For instance, E.Slowik, ‘The “Dynamics” of Leibnizian Relationism: Reference Frames and Force in Leibniz's Plenum’, Studies in History and Philosophy of Modern Physics, 37 (2006), 617–634.

²³ Earman, World Enough and Space-Time, 14–15.

²⁴ LC: L.III.5.

²⁵ Leibniz, Discourse on Metaphysics, 18; G.IV.427–63; AG 51.

²⁷ LC: L.V.53; also AG 51.

²⁶ For example, Leibniz, Specimen Dynamicum; GM.VI.234-254; AG 135.

²⁸ To Huygens, June 1694, GM.II.185; AG 308.

²⁹ AG 51.

³⁰ For example, Roberts, ‘Leibniz on force and absolute motion’.

³¹ C 590–593; AG 92.

³³ A.VI.iii.519; LoC 119–121.

³⁴ A.VI.iv.1641; LoC 335.

³² A.VI.iii.391; LoC 55.

³⁶ NE: II.xv.2.

³⁵ NE: II.xiii.17.

³⁷ NE: II.xvii.3.

³⁸ NE: II.xxiii.21.

³⁹ NE: II.xxiii.21.

⁴⁰ See E. Slowik, ‘Newton's Neo-Platonic Ontology of Space’ (2008), philsci-archive.pitt.edu/id/eprint/4184.

⁴¹ W. Charleton, Physiologia Epicuro-Gassendo-Charletoniana (London, 1654), 70.

⁴² LC: L.III.12.

⁴³ H. More, Henry More's Manual of Metaphysics: A Translation of the Enchiridium Metaphysicum (1679), Parts I and II [Enchiridium], translated by A. Jacob (Hildesheim: Olms, 1995), 98–148.

⁴⁴ LC: L.III.12.

⁴⁵ LC: L.V.106.

⁴⁶ LC: L.IV.15.

⁴⁷ LC: L.V.79, emphasis added.

⁴⁸ For example LC: L.V.104.

⁴⁹ I. Barrow, The Mathematical Works of Isaac Barrow D.D. [Works], edited by W. Whewell (Cambridge: Cambridge University Press, 1860), 154.

⁵⁰ I. Barrow, Lectiones Geometricae, translated by M. Capek, in The Concepts of Space and Time: Their Structure and Development [Lectiones], edited by M. Cˇapek (Dordrecht: Reidel, 1976), 203.

⁵¹ See E. Grant, Much Ado About Nothing: Theories of Space and Vacuum from the Middle Ages to the Scientific Revolution (Cambridge: Cambridge University Press, 1981), chapter 6.

⁵² Barrow, Lectiones, 204.

⁵³ Barrow, Works, 158.

⁵⁴ V. De Risi, Geometry and Monadology: Leibniz's Analysis Situs and Philosophy of Space (Basel: Birkhäuser, 2007), 564.

⁵⁵ A.R. Hall, Henry More and the Scientific Revolution (Cambridge: Cambridge University Press, 1990), 210. Yet, as Futch explains (Futch, Leibniz's Metaphysics of Time and Space, chapter 2), many aspects of Barrow's treatment of time are closer to the absolutists; in particular, he separates time from bodily change, much like Newton, whereas Leibniz remains somewhat wedded to the older Scholastic tradition. Nevertheless, on the issue of space, Barrow and Leibniz are nearly identical, save for Leibniz's possibly having attributed quantity to empty spaces that are bounded by matter or at least measurable (NE: II.xiii.22), but, since God's essence grounds the possibility of a body occupying that space, it is thus not true to say that it constitutes an ontologically unsupported spatial extension (see ‘Substance, Accident, and Relations’). Likewise, Barrow claims that ‘space’, i.e. as a capacity, exists prior to bodies, but, as argued above, Leibniz holds the very same view, although he does not refer to this sheer possibility using the term ‘space’ (rather, space only co-exists with bodies). As regards our later analysis in ‘Holism: Physical and Geometrical’, it is interesting to note that Barrow claims that numbers, like points, have position, since position requires a multiplicity (Barrow, Works, 62), which is, undoubtedly, very reminiscent of Leibniz's approach.

⁵⁶ Notes on Foucher's Objections, G.IV.491–2; AG 146.

⁵⁷ NE II.xiii.17; Newton, Philosophical Writings, 21.

⁵⁸ LC: L.V.39.

⁵⁹ LC: L.IV.9.

⁶⁰ LC: L.IV.9.

⁶¹ LC: L.V.47.

⁶² See, for example, B. Mates, The Philosophy of Leibniz: Metaphysics and Language (New York, NY: Oxford University Press, 1986), 58–69, for more details.

⁶³ See, for example, Leibniz, Discourse, 9; AG 42.

⁶⁵ MP 133.

⁶⁶ LC: L.V.47.

⁶⁴ MP 133–134.

⁶⁷ Auyang, How is Quantum Field Theory Possible?, 247–251.

⁶⁸ On Leibniz on relations, see M. Mugnai, Leibniz' Theory Of Relations (Stuttgart: Verlag, 1992).

⁷² NE: II.iv.5.

⁷³ LC: L.V.104.

⁶⁹ L. Sklar, Philosophy and Spacetime Physics (Berkeley, CA: University of California Press, 1985), 234–248.

⁷⁰ On the many ways that the world could be filled with matter, Leibniz comments that ‘there would be as much as there possibly can be, given the capacity of time and space (that is, the capacity of the order of possible existence); in a word, it is just like tiles laid down so as to contain as many as possible in a given area’. (G.VII.304, On the Ultimate Origination of Things, 1697; AG 151). This suggests, in keeping with the analysis above, that spatial structure is not determined by matter, contra modern reductive relationism (instead, the spatial structure determines the possible material configurations). Furthermore, Euclidean geometry appears to be that determinate structure. After defining ‘distance’ as ‘the size of the shortest possible line that can be drawn from one [point or extended object] to another’, he comments that ‘[t]his distance can be taken either absolutely or relative to some figure which contains the two distant things’, but adds that ‘a straight line is absolutely the distance between two points’ (as opposed to the arc of a great circle on a spherical surface; NE: II.xii.3). By defining a straight line as ‘absolute distance’, in contrast to ‘relative distance’ – i.e. relative to the various figures or surfaces that can be delineated within Euclidean (three dimensional) geometry – this implies that the overall determinative structure of space is Euclidean. Interestingly, it would seem to follow that a limited material world with, say, a spherical shape would have a non-Euclidean metric on that surface, given his notion of relative distance. On the larger issue of ‘compossibility’, see D. Rutherford, Leibniz and the Rational Order of Nature (Cambridge: Cambridge University Press, 1995); and Futch, Leibniz' Metaphysics of Time and Space. Finally, Leibniz's complex analysis of infinity is also beyond the bounds of this essay, but see, e.g., R. Arthur, ‘Leibniz's Theory of Space’, Foundations of Science (2011, forthcoming); and S. Levey, ‘Leibniz on Mathematics and the Actually Infinite Division of Matter’, The Philosophical Review, 107 (1998), 49–96.

⁷¹ NE: II.iv.5.

⁷⁴ LC: L.V.54.

⁷⁵ In the work previously quoted from 1676, Leibniz comments that ‘Space is only a consequence of [the Immensum], as a property is of an essence’. (LoC 55). Compare with: ‘Thus, by making space a property, the author falls in with my opinion, which makes it an order of things and not anything absolute’ (LC: L.IV.9). Of course, Leibniz does not sanction the ontology of a straightforward property theory of space in the way that, say, More's late Enchiridium seems to espouse (see More, Enchiridium, 56–57, where space becomes God's internal property). Yet, as argued above, ∼ (R3) is viewed as a key requirement for a property view, and thus it is imperative that we investigate the manner by which Leibniz's theory resembles a property theory on this point, indeed, as argued in this essay, Leibniz's theory of space more accurately reflects a property theory orientation than modern relationism.

⁷⁶ See LC: L.V.47.

⁷⁷ M. Furth, ‘Monadology’, Philosophical Review, 76 (1967), 169–200.

⁷⁸ MP 133, emphasis added.

⁷⁹ ‘Metaphysical Consequences of the Principle of Reason’, c.1712, MP 176; also, The Monadology, 61; AG 221.

⁸⁰ What is an Idea?, G.VII.263–264; L 207.

⁸¹ Leibniz, Discourse, 14; AG 47.

⁸² L 207.

⁸³ MP 133.

⁸⁶ To Des Bosses, July 31, 1709, G.II.379; translated by De Risi, Geometry and Monadology, 567–568.

⁸⁴ See, De Risi, Geometry and Monadology, 176, for the metrical basis of Leibniz's theory.

⁸⁵ To Masson, 1716, G.VI.624–629; AG 228.

⁸⁷ NE: II.xiv.27.

⁸⁸ NE: II.xvii.1.

⁸⁹ NE: II.xvii.5.

⁹⁰ NE: II.iv.5.

⁹¹ LC: L.V.104.

⁹² NE: II.xiv.27.

⁹³ LC: L.III.5.

⁹⁴ LC: L.V.47.

⁹⁵ NE: II.xiv.26.

⁹⁶ NE: II.xvii.5.

⁹⁷ See, especially, the quotes from MP 133. The use of such terms as, ‘instantiate’, ‘exemplify’, etc., is drawn from structuralist conceptions of mathematics: see, S. Shapiro, Philosophy of Mathematics: Structure and Ontology (Oxford: Oxford University Press, 1997). Given God's role as the basis of these eternal (geometric) truths, the non-reductive, in re (nominalist) version of mathematical structuralism would seem to best fit Leibniz's conception, although more traditional (non-reductionist, non-fictionalist) strands of nominalism would also be applicable (and hence our analysis does not rely on a structuralist reading). For more on these issues, see E. Slowik, ‘Spacetime, Ontology, and Structural Realism’, International Studies in the Philosophy of Science, 19 (2005), 147–166.

⁹⁸ LC: L.V.62.

⁹⁹ LC: L.V.41.

¹⁰⁰ Newton, Philosophical Writings, 22.

¹⁰¹ LC: L.III.5.

¹⁰² The static and kinematic shift arguments also explain why Leibniz's theory does not exactly correlate with a strict or thorough property theory, ∼ (R3). On a strict property theory, a static or kinematic shift of the world would allow new positions of the world's shifted bodies to be obtained, just as they do as regards passage (a), where the individual dynamic changes of bodies relative to one another brings about new spatial positions in universal place. However, since Leibniz sees space as arising from the dynamic interconnections of the whole world, this fact limits the spatial locations, and thus the changes of spatial location, to changes in location among individual bodies. Hence, Leibniz's dynamic holism provides a conclusion that is quite similar to traditional relationism on this issue alone. But, as argued above, this aspect of his natural philosophy stems from a commitment to his blend of nominalism and dynamic holism, which is quite different from a body-centered (R2) relationism (but not the sophisticated (R2*) variety).

¹⁰³ Leibniz, Principles of Nature and Grace, 1714, G.VI.598–606; AG 207.

¹⁰⁵ G.II.438–439; AG 199.

¹⁰⁴ To Des Bosses, 5 February 1712, G.II.435–436; AG 199.

¹⁰⁶ Compare with Newton, Philosophical Writings, 22–30.

¹⁰⁷ To Des Bosses, 26 May 1712, G.II.444; AG 201.

¹⁰⁸ To De Volder, 30 June 1704; G.II.248–253; AG 178.

¹⁰⁹ See, e.g., J. Cover and G. Hartz, ‘Are Leibnizian Monads Spatial?’, History of Philosophy Quarterly, 11 (1994), 295–316.

¹¹⁰ AG 179; see, e.g., Rutherford, Leibniz and the Rational Order of Nature, for an extended analysis.

¹¹¹ See D. Garber, Leibniz: Body, Substance, Monad (Oxford: Oxford University Press, 2009), 383–384, who briefly suggests a similar interpretation of the goals of Leibniz's monadic metaphysics.