Monads, Composition, and Force
Ariadnean Threads through Leibniz’s Labyrinth
Richard T. W. Arthur
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For Gabriella
Preface
This book has undergone a profound transformation from the proposal that I first submitted to Peter Momtchiloff of Oxford University Press in the summer of 2008. Back then it consisted of eight chapters based largely on previously published essays, each chapter concerning itself with one of the issues that Leibniz had identified (in 1676) as making up the labyrinth of the continuum: “on the composition of the continuum, on time, place, motion, atoms, the indivisible and the infinite” (A VI iii 77/ DSR 88–91). But as I worked these chapters into a continuous narrative, it became clear not only that it was getting too big and unwieldy, but that I was trying to address two different audiences with divergent interests and competencies. The more technical material on the infinite and infinitesimals, mathematical continuity, and the theory of space and time, would appeal more to an audience of philosophers of science and mathematics, whereas the material on the theory of substance would be of more interest to historians of philosophy and metaphysicians. So I clove off the more technical material, saving it for a projected companion volume, and concentrated on providing an account of how monads, Leibniz’s atoms of substance, were introduced by him in order to solve the problem of the composition of matter and motion, both of which appear to the senses as continuous phenomena. That is not to say, of course, that Leibniz’s views on the infinite, on time and space as relational, and even on the differential calculus, are of no relevance to what I discuss here—just that I do not discuss them here in any technical detail.
Even shorn of the technical material, this is already quite a large book. Partly this is a consequence of my genetic approach, trying to follow the development of Leibniz’s ideas from his youth to his maturity. But also, in order to explain otherwise puzzling features of Leibniz’s thought, I found it necessary to include a substantial amount of exposition of the views to which he was reacting. This is particularly the case in Chapters 3 and 4, where I treat the writings of early atomists and scholastic thinkers who comprise that context, and whose views are less likely to be familiar to those scholars of early modern philosophy who are used to beginning with Descartes. On the suggestion of one of the OUP readers, I have appended a glossary of these thinkers as an aid to understanding that historical context.
I have freely availed myself of material from previous publications. Most obviously the first two chapters owe much to my “Russell’s Conundrum” (1989), the third to “The Enigma of Leibniz’s Atomism” (2003), the fourth to “Animal Generation” (2006), although there are sprinklings of text from other papers too. In some ways this progression duplicates the course of my own thought on Leibniz’s solution to the continuum problem over those years, although all of it has been subjected to revision and reinterpretation in response to the reception of those papers.
There are very many scholars and friends who have helped in the formulation of the ideas of this book. I immediately found common ground with Dan Garber when I met him in the late 1980s in Chicago in our shared opposition to the traditional branding of Leibniz as an idealist tout court—although my heterodoxy was born of my reading of the Leibniz–Clarke correspondence, whereas Dan has always been more reticent to extend a non-idealist interpretation into the eighteenth century, when he believes Leibniz eventually opted for a “monadological metaphysics”. Our conversations on this have continued from the early days of the Midwest Seminar in Early Modern Philosophy until now, and I am greatly indebted to him for his inclusiveness and generosity of spirit. My interactions with Don Rutherford extend back even farther, to Robert Butts’ seminar at Western in 1984, when I was teaching Applied Mathematics and he was there on a post-doc. Although our interpretations of Leibniz diverge subtly on the status of corporeal substance in Leibniz’s mature thought, I have profited immensely from his scholarship (as will be evident below), and Don has been exceptionally generous in giving me feedback in writing on early drafts of this book. Sam Levey was a doctoral student of Jonathan Bennett’s at Syracuse when he first contacted me (in 1995), and I shared with him drafts of the translations I had made for the Yale University Press volume that was eventually published in 2001. His penetrating questions and suggestions for alternative interpretations were of huge benefit to me then, and he has been keeping me on my toes ever since. I have known Massimo Mugnai since we met at the 5th Internationaler Leibniz-Kongreß in Hanover in 1988, and got to know him well during my time as visiting professor at the Università di Bologna in Italy in 1998, an appointment he had helped engineer. Massimo has been an enduring source of encouragement and fruitful dialogue on the interplay between Leibniz’s logic and his metaphysics ever since, and I am particularly grateful to him for invitations to workshops in Florence and Pisa, and to give a series of lectures at the Scuola Normale Superiore in Pisa. Pauline Phemister and I were pleasantly surprised when we met in 1997 to find that we had been thinking along similar lines, and we have been exchanging ideas on Leibniz ever since. Our agreement crystallized around the shared conviction that, for Leibniz, once God had created simple substances, he had ipso facto created corporeal substances too. Peter Loptson and I had already worked together as department chairs in the Tri-U Ph.D. programme (Guelph, McMaster, and Wilfrid Laurier) for a couple of years before we began engaging on Leibniz interpretation. An amicable collaboration led to a joint talk to the Department of Philosophy at McMaster in 2006 and a co-authored publication in the Leibniz Review of that year, outlining our agreements and differences. Peter’s stroke in 2012 unfortunately prevented us from collaborating further. It is also a pleasure to acknowledge the input of my former MA student, Adam Harmer, who persuaded me that my arguments implied that even the body of a corporeal substance, for Leibniz, is still a plurality and not a unity, the unity being the form adapted to it. I have a special debt to Hiro Hirai and Kuni Sakamoto for deepening my understanding of Scaliger and Sennert (evident in Chapter 4 and the Glossary), and also to Shinji Ikeda for helping arrange my visit to Tokyo and the workshop
in Toyama in 2016 so we could discuss all this in person; and also to Andreas Blank for stimulating exchanges of view on the same topics. Chapter 4 owes its origin, however, to the stimulus of Justin Smith, who, in the course of many profitable exchanges of views, had encouraged me to develop what I had written about Scaligerian and Sennertian themes in Leibniz into an article (Arthur 2006), on which that chapter is based. My appreciation of the importance of matters biological and medical to Leibniz’s thinking is largely due to Justin’s work. Lately that appreciation has been deepened by my increased familiarity with the work of François Duchesneau, and I have been much encouraged by the closeness of our readings of Leibniz’s metaphysics. This is equally true of Ohad Nachtomy, with whom I have had many enthusiastic interchanges, both on his idea of nested individuals in Leibniz, and about the infinite. To Maria Rosa Antognazza, too, I am indebted for stimulating challenges to my reading of Leibniz’s syncategorematic understanding of the infinite, as well as for all I learned from her wonderful biography of Leibniz. Everything to do with my understanding of Leibniz’s mathematics, even if it is in the background in this book, is informed by the wise and benign counsel of David Rabouin, to whom I am grateful for invitations to Paris, long discussions, and many subtle insights. It is also informed by the extraordinary erudition of Vincenzo De Risi, whose invitations to Berlin and Leipzig have given me many opportunities for profitable discussions both with him and with other colleagues in Europe. I am also indebted to many other fine Leibniz scholars for our interchanges down the years: Bob Sleigh, Glenn Hartz, Christia Mercer, Jan Cover, Philip Beeley, Nico Bertoloni Meli, Catherine Wilson, Nicholas Rescher, Ursula Goldenbaum, Hidé Ishiguro, Mark Kulstad, Gregory Brown, Brad Bassler, Doug Jesseph, Herbert Breger, Enrico Pasini, Robert Adams, Brandon Look, Stefano Di Bella, Jeff McDonough, Martha Bolton, Paul Lodge, Jean-Pascal Anfray, Stephen Puryear, Michael Futch, and John Whipple. No doubt there are other scholars whose names are not appearing distinctly to me in my confused concept of contributions to my reading of Leibniz, for which I duly apologize: you are in there somewhere!
I owe no small debt also to students of my seminars on Leibniz, on whom I have foisted drafts of the manuscript of this book. Particular thanks for their feedback, ranging from the substantial to the finding of typos, go to Sean Dudley, Paul O’Hagan, Joshua Maurin, Katie Boudreau, Ryan Mosoff, Marten Kaas, John Roman, Matthew Gardhouse, David Corman, Michael Bennett, James Sikkema, Kathi Gardhouse, and Jason Coutts.
It remains to acknowledge the Oxford University Press readers for their detailed feedback on the draft manuscript I submitted in May 2016, and for their excellent constructive suggestions on how to improve it. That obliged me to do a lot of rewriting, but if the book is the better for it, it is in no small part due to their contribution.
And finally, a particular debt of thanks is due to my family for their love and encouragement, and toleration of all the many hours I had to sequester myself away to complete this project.
5.1
List of Figures
3.1 Leibniz’s diagram for the multiplication of souls in vertices
5.1 Zeno’s dichotomy
5.2 Indivisibles in the continuum
5.3 Infinitely small things in the continuum
Introduction
Leibniz frequently stressed that his theory of substance was intended as a solution to the problem of the “labyrinth of the continuum”, the problem of whether or how continuous entities, such as matter or motion, could be composed of indivisible elements. Thus, in the course of a concerted attempt to explain his position to Antoine Arnauld in October 1687, he wrote:
Also, philosophers have recognized that it is form that gives determinate being to matter, and those who have not taken this into consideration will never escape from the labyrinth of the composition of the continuum once they enter into it. It is only indivisible substances and their states that are absolutely real. (A II ii 250/L 343)
Similar claims abound in his writings. “No one will arrive at a truly solid metaphysics who has not passed through that labyrinth” (A VI 3, 449), he wrote in 1676, and the same concern is reiterated as late as 1710 in the preface of his Theodicy. There (as elsewhere) he depicts the labyrinth of the continuum as the second of the two famous labyrinths in which the human mind is wont to lose its way.1 The first labyrinth, of course, is the subject of the Theodicy itself, a thorough discussion of determinism, fatalism, the nature of freedom and the origin of evil, and related theological controversies. But in 1710 Leibniz is still expressing the desire to give a similar treatment of the second labyrinth, consisting in “the discussion of continuity and the indivisibles which appear to be elements in it, which must involve a consideration of the infinite”, and to point out how, “for lack of a true conception of the nature of substance and matter, people have taken up false positions leading to insurmountable difficulties” (GP VI 29/H 53).2
On most interpretations of Leibniz’s philosophy of substance, though, these are puzzling pronouncements. Leibniz is famous for his doctrine of monads: these are the
1 Cf. On Freedom (c.1689): “There are two labyrinths of the human mind: one concerns the composition of the continuum, and the other the nature of freedom, and both spring from the same source: the infinite” (A VI iv 1654/AG 95); and this from an unpublished manuscript of 1686: “There are two famous labyrinths of errors, one of which has principally exercised theologians, the other philosophers: the former, that of freedom, the latter, that of the composition of the continuum; for the former touches on the inner nature of the mind, the latter, that of body” (A VI iv 1528).
2 “I shall have perchance at another time an opportunity to declare myself on the second, and to point out that, for lack of a true conception of the nature of substance and matter, people have taken up false positions leading to insurmountable difficulties, difficulties which should properly be applied to the overthrow of these very positions” (GP VI 29/H 53).
indivisible substances that he took to be the basic constituents of the world and the foundation of all the phenomena we experience. He conceived each of these substances as endowed with perception (whereby it represents all the others, each such representation being a state), and as undergoing continuous changes of state through appetition, its inbuilt tendency to pass from present to future states.
This much is clear. But exactly how to interpret this doctrine has been a source of controversy ever since. One tendency, beginning with some of Leibniz’s own contemporaries, has been to see his philosophy as a species of idealism. Because monads are described as essentially immaterial and analogous to souls in having perception and appetition, it has been widely supposed that Leibniz’s aim was therefore to reduce all created reality to an infinite plurality of purely mental entities, with the whole physical world being simply the content of the mental representations of these beings. Such a view had particular appeal to later idealists rebelling against Hegel’s monism, for whom Leibniz’s essay, the Monadology, came to be seen as a manifesto for the refutation of materialism through the reassertion of the primacy and plurality of individual souls.3 A recent example of such a reading of Leibniz as an idealist is Robert Adams’ carefully argued Leibniz: Determinist, Theist, Idealist (Adams 1994). For Adams, assertions such as Leibniz’s insistence to Burchard de Volder in 1704 that “there is nothing in things except simple substances, and in them, perception and appetite” (GP II 270/ L 537), exemplify “a broadly idealist approach to metaphysics” (Adams 1994, 217), in which bodies and other wellfounded phenomena are simply the harmoniously interrelated intentional contents of an infinite multitude of perceivers.
On this reading, monads are the only actual existents, and the solution to the continuum problem is simply to see that as such they could not be constituents of a mathematical continuum, which is an ideal object. All else—bodies, light, motion, animals, and the rest of the material world—are mere phenomena resulting from monads’ perceptions, some of them well founded in the mutual agreement of perceptions, and some of them not.
This phenomenalist reading does not fit very well, however, with the characteristic way in which Leibniz describes his solution. A classic instance of this is what he wrote to Nicolas Remond in July 1714:
In the ideal or the continuum, the whole is prior to the parts, as the arithmetical unity is prior to the fractions which divide it, and which can be assigned arbitrarily; its parts are merely potential. In the real, however, the simple is prior to the aggregates, parts are actual, and are prior to the whole. (GP III 622)
Here “the simple” uncontroversially refers to Leibniz’s “simple substances”, another term he used for monads. But what about the “actual parts” that are supposed to be
3 This includes Hermann Lotze (1817–81), Eduard Dillman (1891), Dietrich Mahnke (1848–1939), and Ernst Cassirer (1874–1945) in Germany, and those in England opposing the neoHegelian Absolute Idealism of F. H. Bradley and T. H. Green, for whom reality consisted in a single unified conscious spirit. Cf. Rutherford’s remarks in his (1990b, 13, n. 4).
prior to the whole? These can’t be monads composing to form a whole, since Leibniz denies that monads are parts of anything. And what are the “aggregates” to which they are alleged to be prior in real things? Well, it is clear from numerous passages that it is bodies that are aggregates of monads. As Leibniz told Michel Angelo Fardella in 1690: “A body is not a substance, but an aggregate of substances” (A VI iv 1668/AG 103).4 This is a real difficulty for the phenomenalist interpretation, since it is hard to see how immaterial substances can be “aggregated” to produce a material body if the latter is simply the content of veridical perceptions.5 Moreover, Leibniz’s inclusion of aggregates of simple substances among “the real” appears to imply that bodies, in the dichotomy between the real and the ideal, fall on the side of the real.
There are also numerous passages in Leibniz’s writings where he refers to matter as containing monads, as in this letter Leibniz wrote to Des Bosses (14 February 1706):
Since monads or principles of substantial unity are everywhere in matter, it follows from this that there is also an actual infinity, for there is no part, or part of a part, that does not contain monads. (LDB 24–5)
Some commentators acknowledge that bodies must be more than intentional contents of perceptions if they are to be aggregated, but want the word ‘contain’ not to be taken as a literal containment, otherwise something immaterial would be spatially located. An example of this kind of interpretation is that of Donald Rutherford (1990a, 2008). Acknowledging that bodies have a certain reality independent of perceivers, Rutherford interprets this as a reality of essence, rather than of existence: bodies may be said to be aggregates of monads external to the perceiver in the sense that their reality essentially presupposes that of the monads that constitute them; but the monads do not exist in space, since space is simply ideal.
But Leibniz seems to intend his talk of bodies containing monads more literally than this. For example, when asked by Des Bosses for clarification of his meaning in the abovequoted remark, Leibniz replied as follows:
When I say there is no part of matter that does not contain monads, I illustrate this with the example of the human body or that of some other animal, any of whose solid and fluid parts again contain in themselves other animals and plants. And this, I believe, must be reiterated of any part of these living things, and so on to infinity. (11 March 1706; LDB 35)
4 Similarly, Leibniz writes to Arnauld, also in 1690: “bodies are only aggregates, and not substances properly speaking; it must therefore be the case that there are substances everywhere in bodies, substances that are indivisible, ungenerable and incorruptible, having something corresponding to souls” (A IIb 311–12/ WFT 136).
5 Following Rutherford, I take a phenomenalist interpretation of Leibniz’s metaphysics as a species of idealism that consists minimally in the claim that “the content of a monad’s perceptions as though of bodies reflects no external reality; the being or reality of material things consists solely in the fact that they are perceived” (1990b, 15). As he observes, various differing phenomenalist interpretations have been given in the literature. Adams (1994, 219–24) distinguishes the phenomenalism he ascribes to Leibniz from a Berkeleyan view where bodies are identified with perceptions; but he still takes them to have existence only in the modifications of perceiving substances (223).
This passage suggests a notion of containment that is as literal as the idea that my stomach contains microorganisms that aid in my digestion. Likewise, in Considerations on the Principles of Life, and on Plastic Natures published in 1695, Leibniz is quite explicit that matter is infinitely divided into parts, each containing organic bodies equipped with a soul (or soulanalogue):
I have already said . . . that there is no part of matter which is not actually divided, and which does not contain organic bodies; that there are also souls everywhere, just as there are bodies everywhere; that the souls, and even the animals, always subsist; that the organic bodies are never without souls, and the souls are never separated from all organic body—although it is true, however, that there is no portion of matter of which you could say that it is always affected by the same soul. (GP VI 545)
In these passages Leibniz’s monads appear to be living things, soullike substances with their own organic bodies. If indeed a soul is never separated from an organic body, perhaps this is the intended meaning of monad: a substance with a body, a corporeal substance. Pauline Phemister has given an interpretation along these lines in her Leibniz and the Natural World (2005). Leibniz presented his simple substances as purely immaterial, she argues, so as to make his views more acceptable to the Cartesian audiences he was addressing. His official view, though, was a variety of hylomorphism: every substance, every monad, has both an immaterial and a material aspect, form, and matter. A monad is simply a corporeal substance. Insofar as it is actual, its matter is “secondary matter”, an aggregate of further monads made actual by its dominant form.
Such an interpretation is consistent with Leibniz’s remark to Arnauld, quoted above, “that it is form that gives determinate being to matter”. When Leibniz “rehabilitated” the substantial forms of the scholastics in the late 1670s, he did so in order to distinguish true unities from mere aggregates: a true unity (or ens per se) is distinguished from a mere aggregate (or ens per aggregationem) by the possession of a substantial form. But interpreting corporeal substances as these true unities engenders some troubling difficulties about composition. For Leibniz distinguishes corporeal substances from simple substances like souls or the ego, and there are numerous occasions on which he clearly states a kind of reduction to simple substances, for instance, in his last letter to De Volder: “there can be nothing real in nature but simple substances and the aggregates resulting from them” (19 January 1706; GP ii 282/AG 185).
Scholars are split on how to interpret this quandary. Some, like Adams, doubt that Leibniz’s commitment to corporeal substances was ever sincere (in spite of evidence that he continued to uphold them till the end), whereas others attribute to him different views in different periods. Thus Daniel Garber has been influential in arguing that Leibniz did indeed adopt a hylomorphic ontology in his “middle period”, where a corporeal substance is an aggregate of other corporeal substances to infinity, each one being a composite of organic body and form (Garber 1985, 2009). But this corporeal substance ontology is unstable, since nothing composite should properly be regarded
as substance; certainly, it cannot be regarded as “simple”.6 Consequently, Garber contends, Leibniz abandoned corporeal substances in the early 1700s (even if reluctantly, and not without reversions) for a monadological idealism.
None of these readings sits well, however, with Leibniz’s claims that monads, his indivisible substances, are the key to resolving the problem of the composition of the continuum. If the idealist reading is correct, and bodies are merely intentional objects of the perceptual states of monads, it is hard to see why their composition would be a problem that would exercise Leibniz at all. If they exist only in the representations of perceiving monads, their extension might as well be continuous, without this implying anything more about the monads than would the continuous spectrum of colour (since for Leibniz extension and colour alike are secondary qualities). Again, if bodies are aggregates of monads, but their aggregation is interpreted to mean only that their reality presupposes the reality of monads, we lose the connection that Leibniz stresses between bodies as the actual parts of matter and the discreteness of aggregates of reals. Why couldn’t a really continuous matter, divisible into all possible parts, still presuppose monads for its existence?
There are also difficulties, though, for the nonidealistic readings. If it is “only indivisible substances and their states that are absolutely real”, as Leibniz was already claiming to Arnauld in 1687, this would seem to rule out a corporeal substance interpretation: for corporeal substances are not indivisible if they have bodies that are actually infinitely divided. If, on the other hand, one insists that “corporeal substance is an indivisible whole without parts . . . even though its organic body is divided to infinity” (Phemister 2005, 117), it is hard to escape the suspicion that no escape from paradox has been found. Aggregates are contrasted with true wholes for Leibniz: so how could the possession of a dominant form weld the substances constituting a body into a true whole?7 There is also the difficulty pointed out by Garber of how to square the infinite complexity of corporeal substance with the simplicity of simple substances. Yet if Leibniz had converted to a monadological idealism in the early 1700s as a result of this and other difficulties with the notion of corporeal substance, as Garber proposes, then it is odd that in his explanations to Des Bosses in 1706 he should have chosen realistic talk about animals to illustrate what he meant by monads being contained in bodies.
A different picture emerges, I submit, if instead of beginning with interpretations of Leibniz’s theory of substance and then considering how to make sense of his remarks about the problem of the composition of the continuum, we begin by taking seriously
6 Garber (2009, 88): “Even though corporeal substances are indivisible and immortal, they are not simple. Indeed, they are infinitely complex, bugs in bugs that go to infinity.”
7 Phemister acknowledges difficulties with this account in her chapter on the composition of the continuum. She argues that “the presence of souls and substantial forms has not served to introduce determinate divisions throughout the material continuum since, although the souls and substantial forms unify the corporeal substance itself, they do not unify the organic bodies themselves”, and their introduction into aggregates to “mark the boundaries of indivisible creatures seems unable to alter the ideality of the extended in the slightest degree” (2005, 121).
his claim that this is a central problem of metaphysics for which he introduced indivisible substances as the solution, and examine how his theory of substance crystallizes around this as his thought progresses. When we adopt this perspective, Leibniz’s introduction of monads appears closely interwoven with foundational problems in natural philosophy, and far less like a fanciful exercise in dogmatic metaphysics. We find ourselves obliged to examine exactly what the connection is between the actually infinite division of matter and the necessity of introducing true unities, what hangs on the contrast between the discreteness of division and the continuity of bodies and motions considered abstractly, and what is involved in monadic aggregation. To understand why Leibniz thought it necessary to revive forms or entelechies, we need to investigate what function they performed in late scholastic philosophy, and his response to the criticisms of that philosophy offered by leading mechanists like Robert Boyle. We need to explore not only how Leibniz’s reinterpretation of forms as forces is supposed to provide a foundation for the continuity of motion, but also how it is supposed to solve the problem of the relativity of motion, as he claimed. And this will require us to reexamine his new science of dynamics without prejudging it as being in competition with his theory of forms.
This is the approach I take in this book. Each chapter treats an issue that Leibniz himself described as being implicated in the labyrinth of the composition of the continuum: composition, aggregation, atoms, forms, motion, substance, and continuation in existence. Also embroiled in the labyrinth are other issues of a more mathematical nature concerning time, space, the infinite, and indivisibles;8 I will consider those here only insofar as they are relevant to his theory of substance, deferring a fuller treatment to another work. The first two chapters of this book deal with the problem of composition of matter and how monads are supposed to function as the solution; the third and fourth with Leibniz’s atomism and his rehabilitation of substantial forms, showing the relevance to his views of seventeenthcentury theories of chemical composition and animal generation, as well as the Latin pluralists’ views on forms; the fifth examines Leibniz’s argument that motion is a phenomenon whose reality requires a foundation in force; the sixth treats spatial continuity in relation to Leibniz’s mature views on substance, especially in relation to his analysis of extension and his flirtation with substantial bonds; and finally the seventh deals with continuity of existence through time, including a detailed examination of Leibniz’s position on divine concurrence and continuous creation.
Some foretaste of the argument can be given by returning to the issues raised above, and examining them from the perspective of the continuum problem. A prominent theme is the notion of the indivisibility of substance. As we have seen in his exchanges
8 Cf. Leibniz’s description of his second labyrinth as concerning “the composition of the continuum, time, place, motion, atoms, the indivisible and the infinite” (Guillielmi Pacidi de Rerum Arcanis of 1676; A VI iii 527).
with Des Bosses quoted above, Leibniz insists that the bodies of corporeal substances are actually infinitely divided. But this does not prevent him from giving animals as an illustration of entities possessing the kind of real unity that he saw as required for a “truly solid metaphysics” that escapes the labyrinth. The real unity possessed by living beings is contrasted with kind of unity possessed by bodies, which are mere beings by aggregation. The idea is that a living being, as a being, cannot be divided, even if its body can. A living being has a body, according to Leibniz, but dividing this (as in amputation) does not divide the animal itself, which has a metaphysical unity. As he explained to Des Bosses in his letter of 11 March 1706:
A material unity . . . , according to the mathematicians, may be composed of two halves, when their subject allows, as ½ + ½ = 1, for example, as the value of a groschen is an aggregate of the values of two half groschens. But I was speaking of substances. A fraction of an animal, or a halfanimal, therefore, is not a unity per se, since this can only be understood of the body of an animal, which is not a unity per se, but an aggregate, and has an arithmetical unity, not a metaphysical one. (LDB 30–1)
Leibniz went further, holding that no division would divide the animal, as opposed to its body. From this it follows that each animal is coeval with the created universe: “an animal never has a beginning naturally, neither does it end naturally, so that there will be not only no generation, but also, rigorously speaking, no total destruction or death”, Leibniz wrote in the Monadology (§76/L 650); “What we call generation is a development and an increase, just as what we call death is an envelopment and diminution” (§73/L 650). Leibniz held this view from the late 1670s until the last year of his life, even through the vicissitudes of his epistolary exchange with Des Bosses: “as I have often said,” he wrote in a letter of 29 May 1716, “it is not only the soul that endures, but the animal as well” (LDB 369).
It is quite otherwise with bodies taken in themselves. In the continuation of the above passage from his letter to Des Bosses, Leibniz wrote: “just as matter itself, if it lacks an adequate entelechy, does not make one being, neither does a part of it. . . . Matter (that is, secondary matter), or a part of matter, exists in the same manner as does a herd or house, that is, as a being by aggregation” (LDB 31). This is the same position that Leibniz had proposed in rehabilitating substantial forms in the 1670s and 1680s. Without a form or entelechy adequate to it, a body is merely an aggregate of beings, and not itself a unity. It has a merely arithmetical unity, not a substantial unity.
So we have the following picture. Each monad is the principle of unity of a living being. It is a principle of unity in the sense that it remains the same one thing through all the changes undergone by its body. His favourite example of this is the transformation of a caterpillar into a butterfly: it remains the same animal with the same genetic information encoded within it despite the drastic change of its body, most of which is sloughed off in the process. The difference is that Leibniz conceives such metamorphoses as continual in each animal: no living thing retains precisely the same
body, in the sense of possessing the same material parts, for longer than a moment. Every animal has a body at each moment: in this sense it is a corporeal substance.9 But what is substantial is the principle of unity itself, that which explains how it is the same individual (its principle of individuation); and that which, so Leibniz holds, is the source of all its actions. This principle is not material, else it would be divisible. It is what Leibniz means by the indivisible substance.
This gives the theory of substance in its broadest outlines. But it does not answer any of the questions raised above in relation to the problem of the continuum. Why did Leibniz think a body could not be a substantial unity? What does Leibniz mean by describing bodies and motion as phenomena, and in what sense are bodies aggregates of monads? Why does he rehabilitate substantial forms to explain what endures through time instead of adopting atoms like his contemporaries? And what relation could there be between this revised Aristotelianism and his new science of dynamics?
These are the questions I shall try to answer in the chapters that follow. In keeping with the theme of the labyrinth, I proceed by following certain clues that function as Ariadnean threads suggesting a path through the labyrinth towards an exit. The thread I pursue in the first chapter is the question of what Leibniz means by “actual parts”. The most naïve interpretation is to suppose that monads are the indivisibles one arrives at after a process of infinite division: they would be the “actual parts” that are prior to the whole in real things, as opposed to the merely potential parts into which an ideal mathematical continuum might be divided. Although most scholars have recognized that this runs counter to Leibniz’s insistence that monads are not parts of matter, many have nevertheless found themselves forced to equate his indivisible substances with the “actual parts” of phenomenal bodies, even if they put the expression in scare quotes to show that this should not be taken literally. What moves them to do so is the conviction that only monads are actual, while phenomena are not; so, even if monads are not properly parts, they must be what Leibniz is referring to. Pursuing this thread, however, we find that, despite their being classed as “actual”, the actual parts Leibniz refers to are in fact parts of bodies: they are the determinate parts into which phenomenal bodies are divided by the motions internal to them. This has profound consequences for understanding Leibniz on the continuum. For it means that the contrast he makes between the ideal and the actual is not between the ideal realm and that of monads:
9 Cf. Adams: “The monad can be called a corporeal substance inasmuch as it has a body” (1994, 269). Adams dubs this reading of Leibniz’s corporeal substances the “qualified monad conception”, attributing it to Ernst Cassirer, who claimed that it “denotes the substance ‘insofar as it is endowed with a determinate organic body, according to which it represents and desires’”, citing Erdmann (E 678, n. 5) for the quotation (Cassirer 1902, 408). Cassirer qualifies this by adding: “This by no means records a new and alien moment in the monad, which seemingly treads close to the function of consciousness, but concerns only a determination in the content of consciousness itself.” This makes his reading a wholly idealistic one, in contrast to the reading I give it, where the organic body, although derivative, is not reducible to a “determination of consciousness”—not only because most monadic representations are not conscious, but also because what is represented (namely, the substances constituting the body) exist independently of their being represented, and thus underwrite the reality of the body.
phenomena themselves are actual when they have determinate parts, as opposed to the indeterminate parts into which they could be divided when regarded abstractly.
Moreover, we see that Leibniz holds not just that some bodies are actually divided by motions internal to them, but that every body, and every part of every body, is so divided by the differing motions of its parts. He reiterates this on numerous occasions in both published and unpublished papers, early and late. Thus in the Monadology of 1714 we find: “Every portion of matter is not only divisible to infinity, as the ancients realized, but is again actually subdivided without end, each part into further parts, each one of which has its own motion” (§65; GP VI 618). This idea that each actual part of matter is individuated by its own motion is the key to the argument for monads from the actual division of matter, for each such motion must have its own principle, and this is one of the things that monads provide, as we shall see. That matter is divided by its internal motions into comoving portions constituting actual parts is Cartesian in origin, although Leibniz insists that such a division is not merely indefinite, as Descartes held, but actually infinite.
Implicit in this criticism of Descartes is an understanding of the actual infinite in multitude that Leibniz inherits from certain scholastic philosophers. According to this “syncategorematic” interpretation, advocated by such thinkers as William of Ockham, there is no infinite number. To say that there are infinitely many parts in a division is simply to say that whatever number may be assigned to them, there are more. This conception of the infinite (arising especially out of Leibniz’s work in mathematics) is of profound significance for his philosophy. For it is part and parcel of his argument that there is no “bottom level” of division. If the division of matter resulted in actual indivisibles, then these could indeed be regarded as the elements of matter: simple substances would be the primary elements out of which all bodies are aggregated and therefore composed. As I argue in Chapter 1, such an interpretation would make for a physical monadology somewhat like Wolff’s. On Leibniz’s interpretation, by contrast, matter is actually divided in such a way that every part of it is further divided, but there is no least part. It follows that none of its parts can be indivisible. Since substance is by definition indivisible, it follows that no part of matter can therefore be a substance.
But there is a further conclusion to be drawn from this argument from infinite division. This is that since each actual part of a body is individuated by its motion, and these motions necessarily differ at each different moment, the aggregate of parts making up a body are not the same from one moment to another. That is, body as an aggregate does not remain precisely the same thing through time. Leibniz habitually alludes to Plato in this connection. As he writes in the Conversation of Philarète and Ariste of c. 1711, “Thus it seems that in philosophical rigour bodies do not deserve the name substances; this seems already to have been the view of Plato, who remarked that they are transitory beings that never subsist longer than a moment” (GP VI 586/L 623). They are not substances, enduring beings, but phenomena, always changing. This is one sense in which bodies are phenomena for Leibniz: they are phenomena because, unlike substances, they do not remain the same being from one moment to another.
