ARTICLE
FIVE FIGURES OF FOLDING: DELEUZE ON LEIBNIZ'S MONADOLOGICAL METAPHYSICS
This article is about Gilles Deleuze's book Le Pli. Leibniz et le Baroque from 1988. It shows how Deleuze's notion of folding captures some basic intuitions in Leibniz and how they relate to each other. To this purpose, I propose five figures, all referring to the same basic fold, all illustrating how the consideration of such figures allows developing central elements of Leibniz's monadology. These figures can help, I hope, alleviate some of the fundamental difficulties in understanding Deleuze's approach to the Monadology from the non-Deleuzian perspective of contemporary Leibniz scholarship and give a sense of the synthetic, explanatory force that Deleuze's notion of folding has in relation to Leibniz's monadological metaphysics.
1. INTRODUCTION
Gilles Deleuze's book Le Pli. Leibniz et le Baroque from 1988 has been used for a great many purposes.1 It is much appreciated by literary scholars, art academy students and architects who have adopted, or rather adapted, many of Deleuze's ideas about ‘the fold to their own practices, often widening the spectrum of applications but equally often losing sight of the systematic, conceptual background of Deleuze's concept and how exactly it relates to Leibniz's monadological metaphysics. While early Deleuze scholars who studied Le Pli often produced mediocre commentary,2 more recent commentators, including Simon Duffy, Daniel W. Smith and others, have contributed substantially to our understanding of how Le Pli relates to other work by Deleuze.3 Deleuze scholars are however rarely sufficiently equipped with plain historical knowledge about Leibniz's philosophy to assess the merits of Deleuze's book as a piece of Leibniz scholarship. As for Leibniz scholars, the situation is the opposite. In most cases unequipped with broader knowledge of Deleuze's philosophy, they struggle to make good sense of Le Pli’s densely written analyses which draw on Deleuze's other work in multiple ways.4 This said, use of Le Pli among non-Deleuzian Leibniz scholars is not infrequent. We can, for example, find friendly references in Christia Mercer's Leibniz's Metaphysics from 2001 and active use of important aspects of the Deleuzian perspective in Richard Arthur's recent book Leibniz from 2014 (see Mercer, Leibniz's Metaphysics, 8; Arthur, Leibniz, chapters 3, 5, and 9). In none of these cases, however, does the use of Deleuze's book presuppose any deeper understanding of the notion of folding and how it is related to the basic structure of Leibniz's monadological metaphysics.
Although I do think Deleuze's analysis of Leibniz in Le Pli has considerable appeal, the following is not a plea in favour of his reading, but only an attempt to show how the notion of folding can capture, in an extraordinarily synthetic way, some basic intuitions in Leibniz and how they relate to each other. To this purpose, I provide five simple graphic figures that I hope can help in understanding what Deleuze had in mind. These figures have their limitations. They do not capture all the intuitions behind Deleuze's fold-concept, neither do they comprehend the full scope of the intuitions that they do capture. Most importantly, in order not to complicate matters unduly, I everywhere represent the fold in two dimensions as a folded line. There is, however, a good argument to be made for rather representing it as a folded three-dimensional surface. Moreover, the figures below do not relate the dynamic aspect of the fold, that is, the fact that it moves, folds in and folds out. This is an aspect that becomes relevant, for example, in Leibniz's account of birth and death, according to which ‘what we call generation are developments and growths, as what we call deaths are enfoldings and dimunitions' (Leibniz, La Monadologie, § 73, GP VI, 619, trans. AG, 222). A perfectly adequate description of the folding paradigm would thus not be in terms of a curve, or folded line, but rather in terms of a moving, folded surface, maybe something like the surface of the sea or a bed sheet on a washing line. I have nonetheless opted for considering the properties of a simple curve in order not to get entangled in mathematical considerations too complicated for my main purpose, which is more pedagogical than philosophical (let alone mathematical). Despite such limitations, however, I hope and believe that the five figures of folding I develop below can help provide a better grasp of Deleuze's interpretation.
2. MONADOLOGICAL METAPHYSICS
Before delving into the details of Deleuze's reading, some preliminary remarks should be made about how he approaches Leibniz's metaphysics in Le Pli and how this reading relates to the Monadology.
Let me begin by forestalling a possible misunderstanding. Le Pli is not a book about Leibniz. It is first of all a work in the history of aesthetics that should be classified along with Deleuze's other work on aesthetics from the 1980es, notably the short book on Francis Bacon and the two volumes on cinema.5 Indeed, Le Pli is no more a work about Leibniz than L'Image-temps and L'Image-movement are works about Henri Bergson or Charles Sanders Peirce. Leibniz's philosophy is, in Le Pli, used to give conceptual consistency to the otherwise very elusive epochal and aesthetic category of the Baroque. Like other commentators, most importantly Walter Benjamin, Herbert Knecht and Peter Fenves,6 Deleuze believes that baroque aesthetics finds a philosophical equivalent in Leibniz's monadology. Hence, for Deleuze, Leibniz ‘gives [the Baroque] the philosophy it was lacking'.7 However, Le Pli also provides the basic elements of a global interpretation of Leibniz's metaphysics and it is the way that those elements are organized around the notion of folding that is the topic of the following. But it is important to keep in mind that this is not the primary purpose of Deleuze's book.
It should equally be kept in mind that there are other books where Deleuze engages with Leibniz's philosophy.8 Throughout Spinoza et le problème de l'expression, particularly in the conclusion, Deleuze proposes very dense comparative analyses of Spinoza and Leibniz centred around their common anti-Cartesianism and different concepts of expression (see Deleuze, Spinoza et le problème de l'expression, 299–311). In Différence et répétition, when criticizing the so-called ‘regimes of representation', he describes Leibniz as the philosopher who, like Hegel albeit in a different way, made representation infinite, making him one of the major villains in Deleuze's catalogue of philosophers, occupying the spot right after the German master of negativity (see Deleuze, Différence et répétition, 62–3, 71, 119, 338–40). In a more affirmative mode, Deleuze's conceptualization of the ‘alogical incompatibility' between ideal series that he terms ‘vicediction', and that constitutes a very important element in his account of how individual things come to be, is basically modelled on Leibniz's notion of ‘compossibility’ (see Deleuze, Différence et répétition, 68–9, 338–9; see also Le Pli, 79). This also comes through in Logique du sens, sixteenth series, where Deleuze appeals explicitly to the notion of ‘compossibility' in order to conceptualize what he calls the ‘static ontological genesis’ (Deleuze, Logique du sens, 133–41, 198–203). In these earlier texts, however, Deleuze's use of Leibniz is surgical. His aim is not so much systematic exposition as it is to highlight specific features of Leibniz's philosophy, sometimes only for comparative purposes. We find a more comprehensive reading of Leibniz's philosophy if we turn to Deleuze's lectures at the University of Paris VIII in 1980 and in 1986–7, most of which are available online in French transcription and English translation.9 Those lectures, eminently pedagogical and fairly straightforward in their approach to Leibniz's texts, are immensely helpful for grasping Deleuze's reading in the Le Pli, otherwise a difficult work to penetrate for the uninitiated.
None of Deleuze's works, including Le Pli, provide a full exposition of Leibniz's philosophy. If, however, we combine the early readings and the Paris VIII lectures with the insights developed in Le Pli, it is possible to extract a near-complete reading of Leibniz's monadological metaphysics from Deleuze's texts. I here speak very consciously of a ‘monadological metaphysics', although it is an expression that Leibniz never uses, that Deleuze does not explicitly use, and that I would myself use only with the most extreme caution. Anyone who has witnessed the ‘genetic' turn in Leibniz studies over the last decades – spearheaded by Michel Fichant and Daniel Garber, and followed by a great many other Leibniz scholars (see Fichant, ‘L'invention métaphysique’, 7–140; Garber, Leibniz. Body, Substance, Monad) – would immediately object: What is ‘monadological metaphysics'? When and where exactly does Leibniz develop it? Is it even possible to determine exactly when it is complete? Does it not constantly evolve? Multiple problems regarding the real, historical elaboration of Leibniz's doctrine will arise.
Deleuze is perfectly indifferent to such concerns. Whether he speaks of the 1673 Confessio philosophi or the 1710 Essais de théodicée, the 1676 Pacidius philalethi or the 1714 Principes de la nature et de la grâce – it is all for Deleuze the same doctrine in different formulations. Unsurprisingly, the commentator for whom he declares most unequivocally his admiration is Michel Serres.10 In his 1968 thesis Le Système de Leibniz et ses modèles mathématiques, Serres develops a structuralistic reading of Leibniz according to which everything Leibniz ever wrote, from the De arte combinatoria to the Monadology, represents different aspects of one and the same system, ‘scenographic' variations on the same virtual ‘ichnography' as Serres paraphrases a distinction we find in some late Leibniz texts.11 Hence, for Serres, as also for Deleuze who follows him, the 1714 Monadology is just a highly compressed summary of a ‘monadological metaphysics' that permeates all of Leibniz's philosophical writings.
This stubbornly synchronic view of Leibniz's philosophical enterprise is, from the historical point of view, an obvious flaw in Serres's approach and it is so in Deleuze's as well. Fortunately, in Deleuze's case, this flaw is in part counterbalanced by another flaw, namely the fact that Deleuze, in contrast to Serres, mostly ignores the parts of Leibniz's philosophical work that fall outside the standard corpus contained in the editions of Gerhardt and Couturat. In particular, he shuns the philosophical work of the young Leibniz, most of which is published in the series VI of the Academy edition, an edition that Deleuze simply does not use.12 Consequently – with a few exceptions such as the Confessio philosophi and the Pacidius philalethi –, he rarely cites texts that do not belong to Leibniz's ‘mature' philosophical work, that is to say, from around or after 1686, making it considerably less shocking, although still not quite pardonable, to talk about Leibniz's philosophy as a single ‘system' or ‘monadological metaphysics'.
3. THE FOLD IN LEIBNIZ'S TEXTS
What does this ‘monadological metaphysics' look like for Deleuze? As is often the case in Deleuze's highly synthetic interpretations of other philosophers, his reading of Leibniz is governed by a single interpretive move, expressed by a single operative notion. That notion is, unsurprisingly, that of the fold. As Deleuze concludes his book, ‘we remain Leibnizian because it is all still about folding, unfolding, refolding' (Deleuze, Le Pli, 189). Let me in this section say a few words about the textual basis for that general interpretive strategy.
Deleuze's references to texts in Leibniz where there is explicitly a question of folding are very few. The most important one is to a passage of a 1676 paper, the Pacidius philalethi, where Leibniz explains how ‘the division of the continuum must not be considered to be like the division of sand in grains, but like that of a sheet of paper or tunic into folds. And so, although there occurs some folds smaller than others infinite in number, a body is never thereby dissolved into points or minima' (A VI, iii, 555, trans. Arthur, 185).13 This early passage is Leibniz's most developed development of the paradigm of folding. The second text that Deleuze references (but does not quote) is the Protogaea, Leibniz's account of the origin of the earth, written around 1690–1, and in particular chapter VIII which is concerned with ‘deposits of metals in the earth and a description of veins' (see Leibniz, Protogaea, chap. VIII, 20–5). Finally, Deleuze references a letter to Des Billettes from 1696, where, in a context similar to that of the Protogaea, Leibniz is trying to explain the origin of magnetism in the following hermetic passage, describing the geological formation of veins of magnetic material:
[ … ] it seems that the notion that particles are transformed into folds is not necessary; it suffices that after a while, the passages accommodate themselves to what passes through them that going backwards is somehow prevented and, so to speak, like brushing against the sense of the hair (GP VII, 453).
On these textual grounds alone, Deleuze's approach is clearly a philological non-starter. The first text, the Pacidius philalethi, supports Deleuze's use of the image of the fold, but this is a very early text that can hardly be qualified as part of his monadological metaphysics – in fact, in this text Leibniz embraces a metaphysical position closer to occasionalism.14 As for the other, later, passages, they hardly advance Deleuze's case, since the Protogaea only suggests folds and the letter to Billettes declares them unnecessary rather than the contrary. In the end, Deleuze's most promising references are some suggestive passages in the Monadology, § 61, where Leibniz writes that ‘a soul can read in itself only what is distinctly represented; it cannot unfold all its folds at once, because they go to infinity' (GP VI, 617, trans. AG, 221), and in the Principes de la nature et de la grâce. § 13, where Leibniz states that ‘one could know the beauty of the universe in each soul, if one could unfold all its folds, which only open perceptibly with time' (GP VI, 604, trans. AG, 211).
If, however, we turn to texts more recently made available in the Academy edition, and in particular to the massive volume VI, iv, published in 1999, we find further material bolstering Deleuze's case in texts that most likely were perfectly unknown to Deleuze himself. Thus, in a fragment entitled Definitiones cogitationesque metaphysicae, written around 1678–81, Leibniz notes that ‘the parts of one body constitute one continuum. For a unity always lasts as long as it can without destroying multiplicity, and this happens if bodies are understood to be folded rather than divided' (A VI, iv, 1401, trans. Arthur, 249–51). And again, in the Conspectus libelli elementorum physicae from around 1678–9: ‘It is seen that in truth everything is fluid, and is only folded in different ways, without the continuity ever being broken' (A VI, iv, 1900). And finally, in an undated text on Saint Augustin, Leibniz notes: ‘The whole universe is one continuous body. Nowhere divided, but transfigured like wax, or folded in different ways like a garment' (A VI, iv, 1687). The image of folding thus shows up fairly frequently in texts from the late 1670s and early 1680s, but remains a component in later texts as well. For the strict historian of philosophy, Deleuze's recourse to the image or notion of folding thus presents itself as textually somewhat better grounded than one would suspect at first sight, indeed better than what Deleuze probably himself suspected.
4. THE FOLD OF THE WORLD
Let me now turn to the graphic figures. According to Deleuze, the world is for Leibniz comparable to an infinitely folded curve that extends to infinity.15 Figure 1 represents a segment of that infinite fold or curve, with two stationary points, P and Q, and an inflexion point Z.
Figure 1. The fold of the world.
Such a curve can be expressed mathematically by means of a function. Now, for Leibniz, there is a specific function for any possible curve passing through any possible configuration of points, and, conversely, any possible configuration of points can be generated by a specific function. Leibniz writes:
Thus, let us assume, for example, that someone jots down a number of points at random on a piece of paper, as do those who practice the ridiculous art of geomancy. I maintain that it is possible to find a geometric line whose notion is constant and uniform, following a certain rule, such that this line passes through all the points in the same order in which the hand jotted them down. And if someone traced a continuous line which is sometimes straight, sometimes circular, and sometimes of another nature, it is possible to find a notion, or rule, or equation common to all the points on this line, in virtue of which these changes must occur (A VI, iv, 1538, trans. AG, 39).
In this passage, drawn from the Section 6 of the Discours de métaphysique, Leibniz explains that any compossible set of things or events – that is, things or events that can exist together without contradiction – can constitute a world, that is to say, be brought under a given function governing that world as its universal law. The underlying idea here is that any possible world unfolds following a pre-established plan in the way that a given mathematical curve is drawn in accordance with a given function, but that some worlds develop according to a simpler function than others.16 God will choose to bring into existence the world that balances the simplicity of the function to the richness in variations of the resulting curve. God, Leibniz continues, will create the world ‘which is at the same time the simplest in hypotheses and richest in phenomena, as might be a geometric line whose construction would be easy but whose properties and effects would be very remarkable and of a wide reach' (A VI, iv, 1538, trans. AG, 39).
Moreover, according to Leibniz, ‘each possible world depends on certain principal designs or purposes of God which are distinctive of it'.17 Hence, each possible world is characterized by a certain number of ‘remarkable' events, supported by infinitely many ‘ordinary' events leading up to each of them and connecting them to each other (before Caesar crosses the Rubicon, he first approaches the Rubicon, gets off his horse, wades a few steps into the water, etc. … ). We can observe that on the curve: P and Q, the two ‘remarkable' points, are related by any number of ‘ordinary' points in between: p1, p2, p3, and q1, q2, q3, etc.18 Elaborating on an example that Deleuze occasionally uses, one could, for example, argue that our world is built up around the two remarkable or singular events in world history that are Adam's fall and the expiation of our inherited sin through the death and resurrection of Christ.19 Everything that happens in between these ‘principal designs', P and Q, happens in order to establish continuity between them. The inflexion point Z is where the one is prolonged into the other, the place where the ordinary events begin to relate to and support the second rather than to the first remarkable event. In this case, the inflexion point Z could, for example, mark the birth of Christ or the place in the complete biblical story where we pass from the Old to the New Testament. In sum, the world we live in is the best, brought into existence by God because it could be created according to simple laws giving rise to maximal variety, and because it included a number of particularly ‘remarkable' events or ‘principal designs' that God felt it was best to include. God then created the world in such a way that those features were connected to each other in a world history that is maximally rich in phenomena while still following regular, simple laws.
5. THE FOLDS OF MATTER
Concretely, in the existing world, that is to say, the phenomenal physical world, the fold of the world is expressed as a ‘fold of matter'. So, a rock, a plant, or a human body, are all constituted by folded matter. As we have already seen, there are numerous texts where Leibniz uses the notion of folding in order to account for the apparently paradoxical double affirmation that matter is, at the same time, continuous and actually divided to infinity. That matter is actually infinitely divided is a very characteristic feature of Leibniz's conception of the physical world. It runs counter to standard Aristotelian wisdom according to which actual infinity implies contradiction.20 Hence, according to the Monadology, § 65, ‘each portion of matter is not only divisible to infinity, as the ancients have recognized, but is actually subdivided without end, each part divided into parts having some motion of their own' (GP VI, 618, trans. AG, 221; see also GP VI, 599, trans. AG, 207). At the same time, however, such division cannot be division into an infinite number of discrete parts, since Leibniz is also committed to a principle of continuity, that is, the principle according to which nature is ‘full' (GP VI, 617, trans. AG, 221) and ‘makes no leaps'.21 This brings us to the second figure of the fold, a kind of microscopy of the basic fold, illustrating how it can be subdivided into infinitely many other folds (Figure 2).
The image of a ‘fold going to infinity' or infinitely folded curve very effectively expresses a conception of the extended world that honours those two criteria at the same time: infinity and continuity. Moreover, it allows grasping an important point in Leibniz's anti-atomistic conception of the physical world, namely the fact that no physical body as such allows for a precise limitation, but that the border between the inside and the outside of the body is blurred and fluffy, exactly because any closer look at a given body will reveal that its surface is infinitely folded. Leibniz writes to Arnauld:
[ … ] the very shape which is of the essence of a delimited extended mass is never really exact and determined in nature, because of the actual infinite division of the parts of matter. There is never a globe that is not uneven and no straight line without curves mingled into it, neither is there any curve of a certain finite nature which is not mixed with some other [curve], and this in the small parts as much as in the large ones, something which implies that shape, far from being constitutive of bodies, is not just an entirely real and determined quality outside the mind, and one can never assign to a body a certain precise surface as one could if there were atoms.22
In relation to these aspects of Leibniz's theory – the continuity and infinite division of matter, and the fluffy shape of extended things – the idea of the ‘folds of matter' has both intuitive appeal and good support in the texts. Moreover, the ‘mingling' of curves into curves that the letter to Arnauld speaks of is quite nicely illustrated by Figure 2. So far, the folding paradigm captures in a very straightforward way Leibniz's basic conceptions.
Figure 2. The infinite division of matter.
6. EVENTS
From here on it gets a little more complicated. For the folds of matter cannot be folds of passive matter. Leibniz was one of the foremost critics of the Cartesian physics, in particular of the Cartesian conception of extension as essentially inert mass. In his ‘reformed physics', or ‘dynamics', Leibniz thus formulated a physics according to which physical objects are not determined in the standard mechanist way by extension and movement, but by action and force. As Leibniz put it in the Discourse of Metaphysics, ‘the nature of body does not consist merely in extension, that is, in size, shape and motion' but ‘force [ … ] is something more real' (A VI, iv, 1545, trans. AG, 44, and A VI, iv, 1559, trans. AG, 51). Hence, for Leibniz, the physical world of bodies is animated by derivative forces that are metaphysically grounded in the primitive forces of individual substances, or monads. Each physical body must be considered a specific configuration of such derivative forces. Correlatively, on the metaphysical level, a body is an aggregate of substances endowed with primitive active and passive force.23
But how are we to reconcile these two apparently different conceptions of extended bodies and the physical world, that is, as folded matter and as complexes of derivative forces? Deleuze clearly tries to reconcile them, affirming, for example, that ‘matter is a force that refolds itself incessantly'.24 But how exactly is this supposed to work? We need here to make some very basic mathematical considerations. Let us look again at the fold. A fold is an inflexion on a curve, that is, a curve folds insofar as it curves. Now, as everyone knows, Leibniz is the inventor of the differential calculus. We all also know that the calculus is a kind mathematics that is concerned with the infinitely small and, in some interpretations, with limit values. But what sort of operation is it exactly that the differential calculus allows us to perform when it comes to curves? Installing our fold in installing in a coordinate system coordinate system can help elucidating that question (Figure 3).
Figure 3. The differential calculus.
One of the things that the differential calculus allows doing is, in each point of a curve, to determine a particular value that is habitually called the ‘derivative' or the ‘differential quotient', expressed by the formula dy/dx. The differential quotient dy/dx is the limit of the ratio between Δx and Δy when Δx is vanishing, or infinitely close to 0. We express that as follows:
What does this value dy/dx express in relation to the curve? It does not express the position of the curve, which is given by the simple co-ordinates (x, y). Neither does it express the relation between two points on the curve, for example, the relation between (x, y) and (x+Δx, y+Δy). This relation is expressed by the secant passing through those two points. Instead, since Δx is vanishing, and Δy along with it, dy/dx expresses the inclination of the curve, not between two points, but at a point, that is to say, the inclination of the tangent passing through the point (x, y). This tangent indicates exactly what happens in (x, y), that is, not where or what the curve is, but what the curve does at this particular point. Moreover, it indicates how the curve in any given point is already moving on to something else, how it is already in the process of taking up a relation to another point. Hence, the calculus allows grasping the nature of the curve, not as a composition of points (x, y), but as continuous succession of differential relations, each expressing the curving itself, that is, the way in which the curve varies, or the fact that something happens in each point of the curve (see Deleuze, Le Pli, 24; see also Pourparlers, 214). In short, the differential calculus allows the mathematician to express a curve as a continuous series of events. This is the reason why, already in Logique du sens, Deleuze affirms that ‘the first theoretician of the event is Leibniz’ (Deleuze, Logique du sens, 200; see also Le Pli, 25, and Pourparlers, 216).
Considering how the differential calculus describes curves thus serves to make mathematical sense of the notion that the existing world is not a set of objects, but rather a series of events.25 According to this paradigm, all things are fundamentally conceived as something that occurs, eliminating the difference between things and events: ‘Sinning', ‘Crossing the Rubicon', or ‘the Pyramids' are all happenings, as it were, complex spatiotemporal organizations of events, each expressing their particular segment of the world fold.26
7. INCLUSION IN THE MONADS
As we have seen, according to Deleuze, the physical world is like an infinitely long sequence of inflexions on the world fold, a long series of events. This world of matter consists, as Leibniz puts it, of ‘real phenomena'.27 They have reality insofar as they are in fact enacted. Before those events are realized in the existing world, they are however first constituted as what Deleuze calls ‘ideal events'. As he writes in Logique du sens: ‘What is an ideal event? It is a singularity. Or rather it is a set of singularities, singular points which characterize a mathematical curve [ … ]. They are points of turning back, of inflexion, etc.' (Deleuze, Logique du sens, 67). Why does Deleuze speak here of inflexions in terms of ‘ideal' events and how do such ideal events come into existence as ‘real phenomena'?
According to Leibniz, the world is pre-established. It is conceived by God before any of the individual substances that inhabit it are created. Hence, before they are realized, all events of the world have an ideal or merely possible existence in God's mind.28 The same applies to all conceivable worlds: they exist qua possible in God's mind. In Deleuze's reading of Leibniz, such possible worlds are, in a sense, worlds without subjects, in any case not constituted by individual substances.29 These worlds and the individuals that inhabit them exist only in God's mind, as sequences or bundles of ideal events. For example, there is in God a concept of Adam's sinning, indeed concepts of every property or event that pertains to Adam, together constituting the complete concept of Adam. But Adam himself, an individual substance expressing those properties or events, is as yet nowhere in view: ‘The world, as the common term expressed by all the monads, is pre-existent in relation to its expressions [ … ]. God has not created Adam the sinner, but first the world in which Adam sins' (Deleuze, Différence et répétition, 68). Individual substances, or monads, only become relevant when God decides to create one of the possible worlds, namely the one he considers to be the best. For, in order for ideal events to pass from a state of mere ideality or possibility into reality, they must be enacted by substances. Events cannot realize themselves. In this context, Leibniz often appeals to the scholastic axiom according to which actions are actions of subjects (actiones sunt supppositorum).30 For example, in order for the ideal event of sinning to become real, it must be enacted by Adam; or in order for the ideal event of crossing the Rubicon to become real, it must be enacted by Julius Caesar. The way God brings a sequence of ideal events, that is, a possible world, into existence is thus by creating the subjects or monads that incarnate that sequence of events: God creates Adam so that he can sin and Caesar so that he can cross the Rubicon. Strictly speaking, then, God does not create the best possible world. Rather, he creates the substances appropriate for realizing or incarnating that world. One text where Leibniz intimates such a reading of his doctrine is the section 14 of the Discours de métaphysique:
For God, so to speak, turns on all sides and in all ways the general system of phenomena which he finds it good to produce in order to manifest his glory, and he views all the faces of the world in all ways possible, since there is no relation that escapes his omniscience. The result of each view of the universe, as seen in a certain position, is a substance which expresses the universe in conformity with this view, should God see fit to render his thought actual and produce this substance.
(A VI, iv, 1549–50, trans. AG, 46–7)
Deleuze formulates this relation between the ideal events of the world and the active powers of created substances by means of the Leibnizian notion of inclusion. Inclusion is closely related to the logical relation of inherence, associated with Leibniz's famous principle of inesse, that is, the principle developed in the Discours de métaphysique according to which, in all true propositions, the predicate is included in the subject (A VI, iv, 1539–42, trans. AG, 40–2). Logically speaking, monads are like subjects.31 The world, or the events of the world, are like properties or predicates that inhere in those subjects.32 Correlatively, the substances enact ideal events or actualize the world by including them, that is to say, by expressing them internally.33 Hence, the events of the chosen possible world become real phenomena in virtue of being expressed as properties of created substances. Now, monads are soul-like substances, and their activity is an inner one that can be modelled on our mental life.34 Their internal activity – basing ourselves on the analogy with the soul – can be formulated in terms of perception and appetition (i.e. the tendency to move from one perception to the next) (Leibniz, La Monadologie, § 14–15, GP, 608–9, trans. AG, 214–5). Hence, the way in which monads realize or enact events is by perceiving them or representing them.35 Ideal events, in short, become real phenomena by being projected into the perceptive universe of individual substances.
We can illustrate this intricate relation between world and monads by means of the following fold figure (Figure 4).
Figure 4. Inclusion.
The world-fold is included in m(P) from the perspective of P and in m(Q) from the perspective of Q in such a way that m(P) and m(Q) express the fold from the viewpoint of P and Q, respectively. This expressive activity of monads consists in perception or mental representation. What God does, then, when creating the world is that he ‘lodges' a monad in each and every fold of the world, and the way in which they actualize the world is by expressing or perceiving that world from the perspective of the particular fold they occupy. The inflexions of the fold are, as it were, projected into – or included in – the monads m(P) and m(Q). We find these monads even in the minutest folds of the world. All the way down, we will find yet more folds with yet more monads lodged in them, each perceiving the world from their particular place. This is why Leibniz affirms that ‘there is world of creatures, of living beings, of animals, of entelechies, of souls in the least part of matter' (Leibniz, La Monadologie, § 66, GP VI, 618, trans. AG, 222).
8. PERSPECTIVISM
It is important to realize how each individual substance, on account of the ‘connection of things' as Leibniz sometimes puts it (Leibniz, Discours de métaphysique, sect. 8, A VI, iv, 1541, trans. AG, 41), represents not only a part of the world, but all of it:
[ … ] since everything is connected because of the plenitude of the world, and since each body acts on every other body, more or less, in proportion to its distance, and is itself affected by the other through reaction, it follows that each monad is a living mirror [ … ], which represents the universe from its own point of view [ … ].
(Leibniz, Principes de la nature et de la grâce, § 3, GP VI, 599,trans. AG, 207)
Hence, our monads m(P) and m(Q) do not only express the events related to P and Q, respectively, but both monads express or perceive the same complete world of events. They do however not express it from the same viewpoint, since m(P) expresses it from the position P and m(Q) from the position Q. Hence, according to the Système nouveau, ‘every substance represents the whole universe exactly and in its own way, from a certain point of view, and makes the perceptions or expressions of external things occur in the soul at a given time [ … ]' (Leibniz, Système nouveau, 1695, GP IV, 484, trans. AG, 143.). In the Monadology, § 57, Leibniz puts the point as follows:
Just as the same city viewed from different directions appears entirely different and, as it were, multiplied perspectively, in just the same way it happens that, because of the infinite multitude of simple substances, there are, as it were, just as many different universes, which are, nevertheless, only perspectives on a single one, corresponding to different points of view of each monad.
(GP VI, 616, trans. AG, 221)
We here touch upon what Deleuze designates as Leibniz's ‘perspectivism'. Such perspectivism, Deleuze insists, involves no relativism or subject-dependence of the perception of the world, but quite to the contrary a dependence of the subject itself on the viewpoint from where it is determined to perceive the world:
We call it [i.e. the subject] a viewpoint to the extent that it represents a variation or an inflection. Thus is the foundation of perspectivism. It does not mean some dependence in relation to some given defined subject. On the contrary, the subject is that which comes to the viewpoint, or rather which inhabits the viewpoint.36
The subject does not ground a viewpoint on the world, but ‘the subject is that which comes to a viewpoint',37 exactly because the fold of the world, that which determines the order of viewpoints, is pre-established. Hence, in order to become real, the world must be included, or projected into the monads’ respective perceptive universes, but the way in which the world is perceived by the monads is not determined by the monads themselves, but by the particular position they occupy in relation to the world they perceive, or express. The formula that sums up this expressive relation between the world and the subjects that inhabit it is thus the following: The world exists in the monads, but the monads exist for the world (Deleuze, Le Pli, 141–2). Deleuze often repeats this formula in order to stress the non-relativist point that the world is logically prior to the subjects that come to inhabit it: ‘There is antecedence [of the world] in relation to the monads, even though a world does not exist outside the monads which express it [ … ]' (Deleuze, Le Pli, 81).
9. CLEAR ZONES OF EXPRESSION
Each monad perceives and expresses the entire world. However, due to their different situations in relation to the fold, that is, the fact that they are lodged in different folds, each monad does not express the universe in the same way: ‘[ … ] every substance expresses the entire universe, and [ … ] its individual essence consists in nothing but this expression of the universe taken in a certain sense'.38 The individuality of each monad is thus determined, not by what it represents – because all monads represent the same world –, but by the manner or way in which it represents it. It is for this reason that Deleuze affirms that Leibniz's philosophy is a mannerism, as opposed to an essentialism (Deleuze, Le Pli, 27, 55, 70–2, 76).
This brings me to my fifth and final figure, which concerns Leibniz's theory of clear and obscure perception. The fact that each monad perceives the world from its particular point of view entails that there are differences in clarity of their perception of the world, some segments being expressed clearly by some monads and only obscurely by others. Hence, Leibniz writes in the Monadology, ‘a soul can read in itself only what is distinctly represented; it cannot unfold all its folds at once, because they go to infinity' (GP VI, 617, trans. AG, 221). Each monad has its specific ‘clear zone of expression': ‘[ … ] in every created monad there is only a small part that is distinctly expressed [ … ] and all the rest, which is infinite, is only expressed confusedly' (Leibniz, Eclaircissement des difficultés que Monsieur Bayle a trouvées, GP VI, 553). This is illustrated by Figure 5.
Figure 5. Individuation and clear zones of expression.
In the Principes de la nature et de la grâce, § 2, Leibniz writes that each monad includes the world as multiplicity in a unity, just like ‘in a center or point, though being entirely simple, we find an infinity of angles formed by the lines that meet there' (GP VI, 599, trans. AG, 207). On the figure, then, in order to determine the clear zone of expression corresponding to a given monad, we should, in each point on the curve, draw the lines perpendicular to the tangents of the curve, the so-called ‘normal' lines (e.g. the normal lines in p1, p2, p3, and q1, q2, and q3). Those normal lines that converge and meet in the position, or situs, of a given monad determine the segment of the world fold that that monad will express clearly; those normal lines that do not converge towards the situs of the monad, on the contrary, will determine the part of the world fold that it will express only obscurely. In our case, the normal lines for the segment pertaining to the stationary point P, that is, the lines in p1, p2 and p3, all converge towards m(P). After the inflexion-point Z, in q1, q2, and q3, on the contrary, they diverge with regard to m(P), but converge with regard to m(Q). This illustrates the fact that m(P) is in charge of expressing clearly the fold constructed around the point P up until the inflexion point Z, where monad m(Q) takes over, being in charge of expressing clearly the fold around Q.
This difference between clear and obscure zones of expression can also be expressed in terms of Leibniz's famous theory of minute perceptions, mostly clearly laid out in the preface to the Nouveaux essais sur l'entendement humain. Elaborating on Locke's analysis of the feeling of ‘uneasiness' in the Essay concerning human understanding, Leibniz here stresses the importance of this feeling as something that marks out psychologically our confused perception of the entire universe (A VI, vi, 38–68). Leibniz alludes to the theory in the Principes de la nature et de la grâce in the following terms: ‘Each soul knows the infinite – knows all – but confusedly [ … ]. Confused perceptions are the result of impressions that the whole universe makes upon us' (GP VI, 604, trans. AG, 211). According to this theory, each clear perception is a macro-perceptual integration of infinitely many minute perceptions:
Since each distinct perception of the soul includes an infinity of confused perceptions which embrace the whole universe, the soul itself knows the things I perceived only so far as it has distinct and heightened [relévées] perceptions; and it has perfection the extent that it has distinct perceptions. Each soul knows the infinite – knows all – but confusedly. It is like walking on the seashore and hearing the great noise of the sea: I hear the particular noises of each wave, of which the whole noise is composed, but without distinguishing them. It is the same when it comes to each monad (Leibniz, Principes de la nature de de la grâce, § 13, GP VI, 604, trans. AG, 211 (modified)).
Everything that is within the clear zone of perception of a given monad, will be perceived as an integrated distinct macro-perception, whereas all the perceptions that fall outside that clear zone of perception – that is, the monad's perception of the rest of the world – remain unintegrated and confused perceptions, like a perceptive murmur in the back of our head that relates the rest of the world to us confusedly, a ‘differential unconscious' as Deleuze himself calls it.39
This is what is aptly illustrated by the convergence and divergence of normal lines in relation to the position of the monads m(P) and m(Q) in relation to the curve: where the normal lines converge or meet in the situs of a given monad, perception of the corresponding events on the curve are integrated into clear macro-perception in that monad, for example, the events p1, p2, and p3 in relation to m(P); where the normal lines diverge from that situs, on the contrary, perception in the monad is differentiated into obscure minute perception, for example, the events q1, q2, and q3 in relation to m(P). We should finally note how this notion of clear zones of expression is yet another way of formulating a principle of individuation for monads. For since two monads cannot inhabit the same segment of the fold, each monad necessarily has a clear zone of expression that is different from that of any other: ‘Monads all go confusedly to infinity, to the whole; but they are limited and differentiated by the degrees of their distinct perceptions' (Leibniz, La Monadologie, § 60, GP VI, 617, trans. AG, 221).
10. CONCLUSION
I have attempted to describe in an accessible way how Deleuze's notion of folding captures the basic construction of Leibniz's monadological metaphysics. I have done so by means of five figures, all referring to the same basic figure of a fold. I do not pretend to have given all the fold figures one might imagine for explaining Deleuze's particular take on Leibniz's monadology. There is, I suspect, more mileage in the exercise. I have, for example, not touched upon the problem of how Leibniz's conception of monadic domination in organic bodies or his difficult theory of substantial bonds may be illustrated by elaborating further the fold-figure, although I am convinced that might very well be possible. Conversely, I do not pretend either that, simply by multiplying the figures further, all aspects of Leibniz's metaphysics could be accounted for, or even that the figures I have provided are fully adequate to illustrate entirely the aspects they are intended to capture. Nonetheless, these figures can help, I hope, alleviate some of the difficulties in understanding Deleuze's approach to the Monadology from the non-Deleuzian perspective of contemporary Leibniz scholarship and give a sense of the synthetic, explanatory force that Deleuze's notion of folding has in relation to Leibniz's monadological metaphysics.40
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NOTES
1 I will use the following abbreviations for Leibniz's work: A = Sämtliche Schriften und Briefe; AG = Philosophical Essays; Arthur = The Labyrinth of the Continuum; GM = Leibnizens mathematische Schriften; GP = Die philosophischen Schriften; L = Philosophical Papers and Letters.
2 For example, the translator of Le Pli, Tom Conley, writes in his introduction regarding Leibniz's logic: ‘By means of Leibniz's innovation [ … ] the subject is enveloped in the predicate, just as Proust's intension is folded into its effect' (Conley, ‘Translator's Foreword', xiv). In Leibniz, of course, it is the other way round, i.e. the predicate that is included in the subject, and Deleuze never says otherwise. Comments like this display deep ignorance of basic features of Leibniz's metaphysics, a curious comfortableness with nonsensical statements, and warrant great caution when using the translation. Throughout this article, I refer only to the original text and provide my own translations.
3 Among other contributions, many of which are referenced below, see in particular the volume edited by S. Van Tuinen and N. McDonnell: Deleuze and The Fold: A Critical Reader.
4 I should here make an exception of the splendid article by Bouquiaux, ‘Plis et enveloppements chez Leibniz', 39–56.
5 See Deleuze, Logique de la sensation; L'Image-mouvement; L'Image-temps. On this point, see also Peden, Spinoza Contra Phenomenology, 305, n. 10.
6 See Benjamin, The Origin of German Tragic Drama; Knecht, La Logique chez Leibniz; Fenves, ‘Autonomasia', 432–52. On Leibniz and Benjamin in Deleuze, see Lærke, ‘Four Things Deleuze Learned from Leibniz', 25–45.
7 Deleuze, Le Pli, 173.
8 For discussion of the development of Deleuze's reading of Leibniz, see Robinson, ‘Events of Difference', 141–64; and Smith, ‘Deleuze on Leibniz', 127–8.
9 See http://www.webdeleuze.com. The site includes lectures from 15/04/1980, 22/04/1980, 29/04/1980, 06/05/1980, 20/05/1980, 16/12/1986, 201/01/1987, 27/01/1987, 24/02/1987, 10/03/1987, 17/03/1987, 18/03/1987, 07/04/1987, 12/05/1987, 19/05/1987/, 20/05/1987, 25/05/1987, and one undated lecture from early 1987. Deleuze's lectures from 28/10/1986, 4/11/1986, 18/11/1986, 06/01/1987, 13/01/1987, 27/01/1987, 03/03/1987, 17/03/1987, 28/04/1987, 05/05/1987, 19/05/1987, 02/06/1987 are also available as sound-files on Gallica, the digital library of the Bibliothèque Nationale de France (http://gallica.bnf.fr).
10 Deleuze repeatedly endorses Serres's readings (see Le Pli, 12, n. 18; 25, n. 9; 30, n. 16; 40, n. 5; 102, n. 38; 104, n. 2; 120, n. 13; 176, n. 17). He does, however, disagree with Serres on a particular point concerning the position of a mathematical law in relation to the series it grounds: Is the law outside or inside the series? Deleuze, contrary to Serres, argues that it is outside, but stresses that ‘this is one of the only points where I cannot follow Serres' (Le pli, 68, n. 20).
11 See, for example, Leibniz to Des Bosses, 15 February 1712, in The Leibniz-Des Bosses Correspondence, 232–3.
12 In his defense, it should be mentioned that volume A VI, iv (covering the years 1677–1690) only appeared in 1999. The other three volumes were available to him: A VI, i (1663–1672) appeared in 1930, A VI, ii (1663–1672) in 1966, and A VI, iii (1673–1676) in 1980.
13 Deleuze cites the text in Le Pli, 9
14 See the exchange on ‘transcreation' in Pacidius philalethi, October 1676, A VI, iii, 567–9, trans. Arthur, 21–7. Leibniz here holds that things do not pass from one state to another in virtue of an inherent power of change, but that change should rather be explained by the fact that ‘a permanent substance [ … ] has both destroyed the first state and produced the new one'. On this text, see also Levey, ‘The Interval of Motion', 371–416.
15 Deleuze, Lecture on Leibniz, 06/05/1980: ‘But you can make this abstraction, you consider the world. How do you consider it? You consider it as a complex curve [ … ]. For Leibniz, that is what the world is'. See also Le Pli, 81; and Pourparlers, 217.
16 On this passage, see also Lærke, ‘Compossibility, Compatibility, Congruity'.
17 Leibniz to Arnauld, 14 July 1686, A II, ii, 73.
18 See Deleuze, Le Pli, 81, 121. Deleuze also distinguishes between the ‘remarkable' and the ‘ordinary' in terms of the ‘regular' and the ‘singular'. Hence, ‘a singularity is surrounded by a cloud of ordinaries or singulars' (Le Pli, 81).
19 See Deleuze, Lecture on Leibniz, 22/04/1980: ‘Why is Adam's sin included in the world that has the maximum of continuity? We have to believe that Adam's sin is a formidable connection, that it is a connection that assures continuities of series. There is a direct connection between Adam's sin and the Incarnation and the Redemption by Christ. There is continuity.'
20 For Aristotle, see Metaphysics, II, 2, 994a, in The Complete Works, vol. II, 1570.
21 See, for example, Leibniz, Nouveaux essais sur l'entendement humain, IV, xvi, § 12, A VI, vi, 473. For a somewhat dated, but very insightful paper on the principle of continuity, see Arthur, ‘Leibniz in Continuity', 107–15.
22 Leibniz to Arnauld, 9 October 1687, A II, ii, 250. See also Leibniz to Arnauld, 16 April 1687, A II, ii, 171.
23 For synthetic introduction to Leibniz's dynamics, see, for example, the Specimen dynamicum, 1695, GM, 234–54, trans. AG, 117–38.
24 Deleuze, Lecture on Leibniz, 16/12/1986.
25 To be sure, there is an element of anachronism in my explanation to the extent that Leibniz himself did not interpret the derivative dy/dx in terms of a limiting process. It is, however, a graphically clear way of illustrating how Deleuze understood the value dy/dx as a kind of pure relation, a relation with no relata, and how this fits with his conception of the Leibnizian predicate as pure event. It is worth noting that, in his interpretation of the calculus, Deleuze owed a great deal to the structuralist reading proposed by the Nicolas Bourbaki group. On this, see Deleuze, ‘A Quoi reconnaît-on le structuralisme?', 299–335. For the notion of structure according to the Bourbaki mathematicians, see, for example, ‘L'architechture des mathématiques’, 35–47. On Deleuze's interpretation of Leibniz's calculus, see also Duffy, ‘Deleuze, Leibniz and Projective Geometry in The Fold', 129–47.
26 Deleuze, Le Pli, 103. By this should not be understood that such organizations are constituted within a given spatiotemporal framework, but rather that they are constitutive of the spatiotemporal framework, to the extent that, for Leibniz, space and time are nothing by orders of simultaneity and succession derived from the relative order among phenomena (or here: events). See, for example, Leibniz, Third paper to Clarke, 25 February 1716, GP VII, 363–4, trans. AG, 325.
27 For the notion of real phenomena, see Leibniz, De modo distinguendi phaenomena realia ab imaginariis, approx. 1683–85, in A VI, iv, 1498–504, trans. L, 363–6.
28 I here follow Deleuze's (in my own view problematic) reconstruction which describes the ‘existentification' of things as a realization of the possible that is distinct from the actualization of the virtual. Deleuze thus argues that there is ‘actuality that remains possible and which is not necessarily real' (see Le Pli, 140). There is however in Leibniz's own texts some evidence in favour of the contrary notion that, by virtue of being conceived in God's mind, the merely possible rather remains real while not being actual. The problem may simply be one of terminology.
29 This is one crucial point where I think Deleuze's reading of Leibniz's modal ontology diverges importantly from current standard readings according to which the complete concepts of possible individuals conceived in God's mind can, in some important sense, also be understood as possible substances. On Deleuze's reading, and rigorously speaking, there is no such thing as a possible substance. Complete concepts of things are, insofar as they are conceived in the divine mind, more like bundles of predicates. It would take us too far afield to assess the merits of either reading here.
30 See for exemple Leibniz, Discours de métaphysique, sect. 8, A VI, iv, 1539–40. For a commentary, see Fichant, ‘Actiones sunt suppositorum’, 135–48.
31 Deleuze implicitly relies on Louis Couturat's classic conception of the monad as ‘nothing but the logical subject elevated to substancehood' (see Couturat, ‘Sur la métaphysique de Leibniz', 9).
32 This implies that all properties expressed by predicates are, when properly analysed, reducible to events or actions. Thus, according to Deleuze, the fundamental propositional form in Leibniz's logic is not the traditional attributive proposition S is P (such as ‘Adam is a sinner’), but rather a verbal proposition of the form subject–verb (‘Adam sins’) (see Le Pli, 69–72; Pourparlers, 218; Logique du sens, 200–1; Lecture on Leibniz, 20/01/1987). Leibniz's favourite examples do indeed suggest something like that: ‘Adam sins', ‘Caesar crosses the Rubicon', ‘Alexander combats Porus and Darius.' The point is that the predicates ascribed to subjects in propositional logic correspond to actions ascribed to subjects or, more precisely, to events enacted by subjects. The reading is far from being unwarranted (for a similar reading of the Leibnizian predicate, see Fichant, ‘L'invention philosophique', 57).
33 Leibniz mainly develops his notion of expression in the Discours de métaphysique and the correspondence with Arnauld. The literature on the topic is extensive. See, for example, Kulstad, ‘Leibniz's Conception of Expression', 56–76; and Swoyer, ‘Leibnizian Expression', 65–99.
34 See, for example, Leibniz to Arnauld, 28 November (8 December) 1686, A II, ii, 121.
35 A monad is ‘by its very nature representative' (La Monadologie, § 60, GP VI, 617, trans. AG, 220).
36 Deleuze, Le Pli, 27.
37 Lecture on Leibniz, 16/12/1986.
38 Leibniz to Arnauld, 30 April 1687, A II, ii, 167.
39 This will prove salutary for Leibniz's philosophy in the context of Deleuze's own Nietzschean position, and explains in some measure why Deleuze occasionally is willing to label himself a ‘Leibnizian': the ‘Dionysian' rumbles in the depths of the differential unconsciousness of minute perceptions. On Deleuze's conception of the differential unconscious, see Le Pli, 114–7, and Différence et répétition, 214, 275–6.
40 I am grateful to Paul Lodge and to the anonymous reviewers of the BJHP for numerous corrections and suggestions that much helped me improve this paper.