10

LEIBNIZ’S HARLEQUINADE

Nature, Infinity, and the Limits of Mathematization

JUSTIN E. H. SMITH

DESCARTES: “IN THE SAME STYLE AS THE REST”

René Descartes writes to Marin Mersenne in a letter of 1639 that if his account of the circulation of blood, among other things, “turns out to be false, then the rest of my philosophy is entirely worthless” (1964–76, AT 2 501). He does not say that if his account is wrong then his circulation theory, or his medicine, or his physiology will be worthless. He says that his philosophy depends on the correctness of his explanation of cardiac motion. This is, indeed, putting quite a lot of weight on a question that appears to be of only the remotest concern to philosophers today—there is, for example, no specialization in the philosophy of cardiology—and at the very least it should motivate any historian of early modern philosophy to reconsider the way he or she conceptualizes the philosophical project, and to strive to study this project in its early modern expression in a way that is adequate to the self-conception of its leading exponents. For Descartes, the medical philosophy aimed to prolong the human body’s life, and at the base of any human or animal body’s life is the continuation of certain vital processes, most particularly respiration, digestion, and circulation. These, in turn, must be understood if they are to be influenced by human art and intervention. The project of understanding, in turn, for Descartes, means first and foremost, as he put it in the 1637 Discourse on Method, explaining a given domain of nature “in the same style as the rest”1 (1964–76, AT 6 45). But what style is this?

Many commentators would be tempted to call it ‘the mechanical style,’ and to see this as synonymous with ‘the mathematical,’ but these categories are rightly being subjected to a certain degree of scholarly revision in recent scholarship, including the present volume. Roger Ariew, in chapter 4, emphasizes the polysemy of the notion of mathematization in the early modern period: different thinkers meant different things when they asserted of their own projects that these were mathematical, or that they involved mathematization of domains of nature that previously had not been given such a treatment. According to Ariew, the historiographical approach to early modern natural philosophy that prevailed in the twentieth century, in particular in the work of Burtt (1925), Dijksterhuis (1961), and Koyré (1957) (hereafter, collectively, BDK), is fundamentally misguided. On the new, revisionist approach, there has been a basic conflation of mechanization and mathematization. Certainly, if we understand the latter notion as involving the substitution of “mathematical idealities for the concrete things of the intuitively given surrounding world,”2 then there is indeed little primary-textual evidence that Descartes and his successors were intent on mathematizing the natural world in this way. The twentieth-century consensus on this point is now rightly in the course of being displaced. But we may also understand by ‘mathematization’ another sort of endeavor: that of accounting quantitatively for natural processes that had previously been dealt with in strictly qualitative terms. Typically, such a quantitative approach would be elaborated in terms of the mass, figure, and motion of the particles involved in the process: particles that could be described in terms of their size, weight, shape, and speed. This description would then, it was hoped, render large-scale natural processes comprehensible.

It is plain that such a description was part of what Descartes had in mind when he expressed the desire to explain physiological processes “in the same manner as the rest.” And while it may be the case that Descartes himself did not think of this sort of explanation as part of a mathematization of nature, there can be no doubt that subsequent physiologists, many of them working under the banner of Cartesianism, did so conceive it. Lorenzo Bellini and Niels Stensen, certainly, to cite two prominent examples, thought of their own iatromechanical projects as a sort of mathematization of medicine, and moreover the model of such mathematizing was geometry. This much is clear from the very title of a work such as Steno’s Elementorum myologiae specimen seu Musculi descriptio geometrica [Specimen of the Elements of Myology, or, A Geometrical Description of the Muscle] of 1667 (see Kardel 1994; also, e.g., Bellini 1662). And Leibniz, for his part, who early in his career praised both Bellini and Stensen for “mathematizing in medicine” (Smith 2011, 84) evidently takes ‘mechanical’ and ‘mathematical’ as synonyms, at least in their application to the study of physiological processes. Thus he writes to Arnauld: “One must always explain nature mathematically and mechanically, provided that one understands that the very principles or laws of mechanics of force do not depend on mathematics alone, but on certain metaphysical reasons (Leibniz 1875–90, G 2 58).3 In sum, Ariew is certainly right to insist that we must not conflate mathematization and mechanization in Descartes himself, and it is also certainly wrong to suppose that mathematizing iatromechanists would think of mathematics as an idealizing abstraction away from bodies. And yet, the supposition that the approach to mechanical nature as something mathematically tractable, the approach more geometrico, lay at the heart of the mechanical project is by no means an invention of mid-twentieth-century historiography. It goes back to the very self-understanding of early modern mechanical philosophers themselves. This is, at least, the conclusion we must draw if we limit our consideration of early modern mechanical philosophy to iatromechanism, or to the medical-scientific and physiological study of living bodies. In this respect, as in no doubt many others, it is crucial for the new, post-BDK historiography to consider the life sciences alongside the other domains of natural philosophy, in order to arrive at a sufficiently clear picture of just how much from the BDK thesis needs to be thrown out, and how much deserves to be retained.

Stensen, Bellini, and others were hoping to extend what they saw as the recent successes of ‘mathematizing’ in the study of the nonliving world. But it would be a mistake to suppose that this order of operations, this attempt at an extension of mathematical methods (again, not necessarily in the BDK sense, but simply in the sense of quantitative tractability) from celestial mechanics and ballistics to medicine, amounted to a movement from the most pressing to the less pressing of matters, or that the foundational sciences were taken care of first, and then the less important sciences were followed up by the arrière-garde. It would be more correct to say, in fact, that from the perspective of early modern thinkers themselves, physics was mathematized first only because physics is relatively easy compared to physiology and medicine. The fact that its objects more readily submit to a mathematical treatment than the parts of living bodies may have meant that lessons could be extended from physics to the life sciences, but this did not entail that physics was seen as somehow more fundamental. Quite the contrary, its mathematization appears to have been seen as a sort of preliminary skirmish in the build-up to the great battle, whose victory would have been the crowning achievement of the new philosophy, namely, the mathematization of medicine. It is not that the mathematization of medicine would provide a peripheral confirmation of the robustness of an explanatory approach already thoroughly established in a more foundational science, but rather that the failure to mathematize medicine, the inability to account for it “in the same style as the rest,” would amount to a falsification of the entire endeavor, indeed of ‘philosophy’ as Descartes understands it.

But alas, by the end of the century, physics will have its Principia mathematica, while the philosophical treatment of living bodies, by contrast, will have fragmented into a sort of small-‘e’ empiricist agnosticism, on the one hand, and a fairly reactionary vitalism on the other, which took the phenomena of life as in principle unsusceptible to a treatment “in the same style as the rest.” By the end of the following century, we would find Kant declaring that in principle there could never be, as he put it, “a Newton of the blade of grass” (G 5 400, 18ff). What happened? Why did this program fail? And what can its failure show us about the project of modern natural philosophy as a whole? In this chapter, I would like to argue that the philosophy of G. W. Leibniz provides significant insight into the fate of the early modern project of mathematizing living nature. While early on he had hoped to see mathematical methods extended to the analysis of the composition of organic bodies, and while in his mature analysis of these bodies the idea of an actual infinity plays a central role, nonetheless, for reasons I will proceed to spell out, Leibniz’s mature theory of living bodies, according to which they consist in infinitely many bodies in hierarchical relations to one another ad infinitum, cannot be considered a victory for mathematization. Leibniz was no Newton for the blade of grass.

LEIBNIZ: “C’EST TOUT COMME ICI”

There is an interesting, if at first not obvious, connection between this last desideratum of Descartes’ philosophical program, on the one hand, and, on the other, the so-called Harlequin principle, as stated a few decades later by Leibniz: c’est tout comme ici, “it’s all as it is here.” For both Descartes and Leibniz, it is a core conviction that different domains of nature must not be seen as requiring different sorts of explanation. Concretely, this means both a collapse of the Aristotelian separation between the superlunar and the sublunar, as well as of that between the living and the nonliving. Planets, projectiles, and muscles must all be subjected to the same sort of treatment as the others. The implications of the Harlequin principle for Leibniz are however rather different from those of Descartes’ desire to explain everything “in the same style as the rest.” Leibniz, in fact, appears to be drawing this principle from a few, likely unexpected, sources, and it is worthwhile to trace his version of the principle back to them, in order to gain a clearer picture of what he himself intends.

Harlequin—‘Arlequin’ or ‘Arlecchino’—is a stock character of the Italian and French commedia dell’arte traditions. He is known, in some of his multiple iterations, for wearing a many-layered costume that makes it impossible to disrobe him. Thus Leibniz describes his conception of the infinite folds of organic bodies in the New Essays of 1704 as follows: “It is … like Harlequin, whom they wanted to disrobe on stage, but could never arrive at the end, because he had I don’t know how many clothes the ones on top of the others: although these replications of organic bodies to infinity, which are in animals, are neither so similar to one another nor as layered upon one another as the clothes, the artifice of nature being of a completely different subtility” (G 5 309).4

In a letter to Damaris Masham of the same year, Leibniz makes oblique reference to Anne Mauduit Nolant de Fatouville’s Arlequin, Empereur dans la lune, a comedy that first appeared in 1683 or 1684 (G 3 343).5 He recalls the expression of a view according to which “everywhere and all the time, everything’s the same as here.” In fact, in this play, the phrase, “c’est tout comme ici” is repeated several times by different characters (the Doctor, Colombine, Isabelle), listening to Arlequin’s description of life on the moon and affirming that this is “just like” life on Earth. The conclusion of the play consists in a resounding repetition of this phrase by the entire cast, yet this is in response to Arlequin’s description of the daily habits of lunar women—they wake up past noon, take three hours to get dressed, travel to the opera in carriages—and it has nothing at all to do with the structure of the matter making up the lunar world or anything of the sort.6

What, then, is the connection between the two references to Harlequin in Leibniz—the reference to the character’s onionlike costume, on the one hand, and the reference to the “tout comme ici” principle on the other? In order to answer this question, we need to take stock of the full range of Leibniz’s interest in his era’s science fiction. In the New Essays, in addition to referencing Nolant de Fatouville’s fantasy about the emperor of the moon, Leibniz also cites another work describing travel through the solar system, and indeed one that shares many familiar themes from Nolant de Fatouville’s work: Cyrano de Bergerac’s Histoire comique des États et Empires du Soleil, first published in 1662 as a sequel to his Histoire comique des États et Empires de la Lune, published posthumously in 1655.7 “I am also of the opinion,” Leibniz writes, “that genii apperceive things in a way that has some relationship with ours, even if they had that curious gift that the imaginative Cyrano attributes to some animated natures in the Sun, composed of an infinity of small birds that, moving according to the command of the dominant soul, form bodies of every kind” (G 5 204).8 In the tale of the voyage to the sun, Cyrano’s narrator describes several encounters with composite beings. For example, he describes an encounter with a miniature man who emerges out of a pomegranate that has fallen from a tree. This man identifies himself as “the king of all the people who constitute that tree” from which the fruit has just fallen (de Bergerac 1858, 195). Soon after this, “all the fruits, all the flowers, all the leaves, all the branches, and finally the entire tree, fragmented into little men: seeing, sensing, and walking” (196). And soon the men begin to dance: “As the dance grew tighter, the dancers blurred into a much more rapid and indiscriminate stampede: it seemed that the purpose of the Ballet was to represent an enormous Giant; for by dint of coming together and augmenting the speed of their movements, they mixed so closely that I now perceived only one great Colossus … This human mass, previously boundless, reduced itself little by little so as to form a young Man, of an average size” (198–99).

The little king proceeds to jump into the composite man’s mouth, and, in the role of ‘dominant monad,’ proceeds to give the composite golem its principle of unity: “All this mass of little men did not, before now, give any sign of life; but as soon as it had swallowed its little King, it no longer felt itself to be anything but one” (ibid.).9

In his earlier work, describing a no less delirious lunar voyage, Cyrano’s narrator defends heliocentrism, but does so on the grounds that “all bodies that are in Nature need this radical fire,” and therefore it must be “at the heart of this Kingdom, so as to be able to promptly satisfy the needs of each part” (de Bergerac 1858, 35). He compares this placement to the location of an animal’s genital organs at the center of the body, or to “the pits at the center of their fruit; and just as the onion conserves, under the protection of a hundred skins that surround it, the precious germ, where ten million others will draw their essence; for this apple is a little universe of its own” (ibid.).

Assuming that Leibniz in fact read and was influenced by both Cyrano de Bergerac and Nolant de Fatouville, as he reports in the New Essays, we are now in a position to see the connection between the two versions of the Harlequin principle: everything is as it is here, which is to say, first of all, that there is no need to invoke a different sort of explanation for superlunar bodies as for sublunar bodies. Second, the sort of explanations proffered for macroscopic bodies will be the same as that for microscopic ones; scale is of no relevance in determining the appropriate explanation for the structure and motion of a given portion of the natural world. Finally, the structure that characterizes every body in the natural world is one of bodily individuals that are in turn capable of constituting greater composite bodily individuals. Individual bodily beings are simultaneously worlds apart, but are also, at the same time and no less, implicated in the constitution of greater bodily beings. A given bodily being can ‘fragment,’ but no being can pass from a bodily state into a nonbodily one, nor indeed from bodily existence to nonexistence tout court, just as the king of the people formerly constituting the tree falls away in the form of a fruit and then emerges from the fruit as a little man.

For Leibniz, the need to account for everything “in the same manner as the rest” is articulated in terms of the tout comme ici principle. In turn, the way the world is structured ici, in the sublunar sphere, and more particularly in what we would call the ‘biosphere’—and indeed even more particularly in the plant or animal body—is unlike anything Descartes was prepared to imagine. For Leibniz, the world is conceptualized on the model of the animal body, which is in turn conceptualized as an infinitely structured assemblage of infinitely many constituents—corporeal substances, all of which stand in hierarchical relations of domination and subordination relative to all others. Leibniz does not look at the animal body and hope that it might be explained in the same manner as the planets and projectiles; rather, he looks at the planets and projectiles and asserts that they are to be explained in the same manner as the animal body. What remains constant from Descartes to Leibniz, then, is the desire to explain the entire world in a unified way, and to do so, in some broad sense, ‘mathematically’; what changes, though, are both the domain of nature from which the explanation is to be drawn, as well as the particular branch of mathematics that is looked to as a potential source of answers. The evolution is from the explanation of everything on the model of inanimate bodies, and by appeal to geometry, to the explanation of everything on the model of animate bodies, and by appeal to infinity.

MATHEMATICAL AND BODILY INFINITY

On Leibniz’s mature view, already clearly in evidence by the 1704 New Essays, everything is as it is here, and moreover here it consists in infinitely structured or ‘folded’ body. For Leibniz, this infinite structure is synonymous with ‘organism,’ understood not as a count noun (there is no talk of ‘organisms’), but as an abstract noun characterizing all natural bodies. To be characterized by organism is to be infinitely structured, which, to follow out Leibniz’s trail of synonyms still further, is to be a ‘divine machine.’ Thus infinity lies at the very heart of Leibniz’s mature account of the natural world, and in this respect he could not differ more sharply from Descartes. What it is for each philosopher to offer an account of any given segment of nature “in the same style as the rest,” then, will be very different in each case.

As Ohad Nachtomy rightly notes, “Descartes delineated an irreconcilable gap between the infinite creator and its finite creatures, suggesting that it would be not only cognitively impossible but also morally and theologically wrong for us to investigate the infinite” (Nachtomy 2014, 9–28). In this connection, Descartes is hewing far more closely to the traditional understanding of infinity, among the vast majority of philosophers up until his time. Aristotle effectively curtailed serious commitment to an actual infinity with his well-known observation that “nature avoids what is infinite, because the infinity lacks completion and finality, whereas this is what nature always seeks” (Aristotle 1943, 7; 1.1.715b15). Leibniz was emboldened in his embrace of an actual infinity in large part by the innovative, imaginative, and fairly radical philosophical speculation of predecessors such as Henry More and Giordano Bruno. But, more important still, he was able to incorporate infinity into his philosophy in a very concrete, and not merely speculative way, as a result of important attainments of his in pure mathematics. As Nachtomy well explains: “Leibniz discovered a rational method to treat infinity in mathematics. By translating infinitesimal quantities into finite ones, arguing that they can be regarded as variables, smaller or larger than any assignable quantity, he showed that infinitesimals could in fact be used in calculations. Leibniz’s sophisticated approach (evident in his early work in mathematics) certainly contributed to his applying infinity in other domains of his philosophy as well. For, given this approach, one could feel free using infinity without falling into paradox” (Nachtomy 2014, 12). Liberated from fear of paradox, Leibniz is also freed up to develop what might be considered a highly counterintuitive account of the structure of the natural world, as consisting in infinitely structured bodies, which result from the conspiracy of infinitely many subordinate bodies, yet cannot be said to be made up out of these subordinate bodies, since these subordinate bodies have exactly the same sort of structure as the bodies they in turn serve to constitute, and so on ad infinitum. Indeed in the end there simply is no rock-bottom level of the physical world that serves to make up composite entities in the way that bricks make up a house. Divine machines—and in the end such machines are all there is—are divine precisely to the extent that they cannot be analyzed into fundamental constituent parts; which is in the end just another way of saying that they cannot be analyzed away, and are therefore immortal.

Leibniz repeats this account of the structure of natural bodies numerous times, along with its various corollaries such as that of the immortality of corporeal substance. Thus in a letter to Malebranche of 1679, Leibniz writes: “There is even room to fear that there are no elements at all, everything being effectively divided to infinity in organic bodies. For if these microscopic animals are in turn composed of animals or plants or other heterogeneous bodies, and so on to infinity, it is apparent, that there would not be any elements” (Smith 2011, 235).10 In a note on a letter of Michelangelo Fardella from 1690, amply discussed in a recent study by Dan Garber (2009), we find a similar expression of the commitment to the infinite structure of matter, though now expressed by explicit analogy to geometry: “There are substances everywhere in matter, just as points are everywhere in a line … Just as there is no portion of a line in which there is not an infinite number of points, there is no portion of matter which does not contain an infinite number of substances” (AG 1989, 105). Consider, finally, this passage from a text entitled “On Body and Force, against the Cartesians,” written just two years before the New Essays, in which Leibniz nicely brings together the infinite structure of the organic body with the divinity and immortality of the corporeal substance: “Moreover, a natural machine has the great advantage over an artificial machine, that, displaying the mark of an infinite creator, it is made up of an infinity of entangled organs. And thus, a natural machine can never be absolutely destroyed just as it can never absolutely begin, but it only decreases or increases, enfolds or unfolds, always preserving in itself some degree of life or, if you prefer, some degree of primitive activity” (AG 253). This is Leibniz’s mature account of the structure and activity of natural bodies in general, and it is at the same time his account of living bodies, since again, for Leibniz, there is no body that is not living: to be alive just is to have organic structure, which is to say to consist in individual corporeal substances standing in relations toward one another of nestedness ad infinitum, and there simply are no bodies that are not like this. This means that in the final analysis it is the study of the living body, and in actual practice the study of the human body, that will serve as the model and guide for the study of nature in general. Sometimes, the discipline that sees to this study is described as ‘physiology,’ sometimes as ‘animal economy.’ But most often it is called, simply, ‘medicine.’

“MATHEMATIZING IN MEDICINE”

We have seen that Leibniz, like Descartes, wishes to explain everything in nature in the same way, and we have seen that for Leibniz this way will come to involve the application of infinity to the account of the structure of natural bodies.11 For Leibniz, moreover, on the final analysis all natural bodies are living bodies, in the sense that there is nothing that is not a divine machine, nothing that is not characterized by organism. We have also seen that Leibniz and many of his contemporaries suppose—notwithstanding any revisions that need to be made to the BDK thesis as applied to Descartes—that it is a desirable thing to ‘mathematize nature,’ that is, to render all natural bodies tractable by subjecting them to quantitative methods of analysis, and to suppose that what is causing the qualitative features of living bodies are in the end the mass, shape, and motion of those bodies’ microanatomical constituents. The question now arises, though, whether Leibniz would himself consider his analysis of living bodies in terms of their infinite structure a case of successful “mathematizing in medicine.”

The young Leibniz is very optimistic about the project of mathematizing medicine, and, as we have seen, he cites Stensen and Bellini as his models for this undertaking. In the Directiones ad rem medicam pertinentes of 1671, he anticipates that medicine will eventually be exhaustively explicable in terms of the mass, figure, and motion of bodily particles. Yet later on Leibniz will come to see the project as only partially realizable. As he writes to Michelotti in a letter on animal secretion of 1715: “There may be many mechanical causes that explain secretion. I suspect however that one should sooner explain the thing in terms of physical causes. Even if in the final analysis all physical causes lead back to mechanical causes, nonetheless I am in the habit of calling ‘physical’ those causes of which the mechanism is hidden” (Leibniz 1768). Here, then, ‘physical’ contrasts with ‘mechanical’ to the extent that the latter lends itself to immediate mathematization, given the state of our knowledge and our capacity for observation, whereas physical explanation remains avowedly hypothetical.

Early on, for Leibniz, it had been the work of Steno that served as a model for the possible mathematization of medicine and related fields of investigation. Steno had argued, most importantly in his Elementorum myologiae specimen seu Musculi descriptio geometrica of 1667, that the nerves and muscles alike contract and expand without the influx or efflux of any new material. For the Danish physician, this argument was of particular importance for the broader argument against the animal spirits playing a role in animal motion. Steno is effectively attempting to demonstrate the mechanism of contraction more geometrico, namely, by showing how the shortening of the fibers that constitute the muscles is alone sufficient to account for muscular contraction.

Ultimately, Leibniz will find the Stenonian account inadequate, and will come to believe that the sort of mechanical explanation that should be hypothesized in accounting for the motion of the muscles will be one that attributes an important role to elasticity. The elasticity of the inner parts is conceived as a sort of force (vis elastica) that keeps the body in motion through countless imperceptible vibrations in a manner analogous to the “vibrations” of perceptions that endure in the soul as memories. As Leibniz writes in a letter to Bernoulli of May 6, 1712: “In organic beings many things seem to consist in perpetual, imperceptible vibrations, which, when we perceive them to be at rest, are in fact being held back by contrary vibrations. Thus in truth we are led back to an elastic force. I suspect that memory itself consists in the endurance of vibrations. Thus there does not appear to be any use for a fluid that goes by the name of animal spirits, unless it is traced back to the reason itself of the elastic force” (G 3.2 884–85). This strategy of explaining the dilation and contraction of the parts of the body in terms of an effervescence that brings about a sort of vibration far antedates Leibniz’s correspondence with Bernoulli and seems to be traceable most directly to the influence of Boyle’s New Experiments Physico-Mechanicall, touching the Spring of the Air, and its Effects of 1660. Leibniz writes as early as the Corpus hominis of the mid-1680s:

While it is granted that the seat of effervescence is in the heart, it nonetheless is easily communicated to the whole body by the blood vessels, just as [when] we attempt to heat an enormous cask of wine with a small fire, if the fire be applied through a small copper utensil, connected with the vessel through a tube. Seeing moreover that in any ebullition there is an excessive dilatation, the vapor is nevertheless not expelled, but rather it is necessary that it in turn be pushed along, whence arises respiration, indeed in all exceedingly great efficient [causes] there is a certain reciprocation of restitutions such as we note in oscillating pendula, or in vibrating chords. (Smith 2011, 295)

What will be new by the time of the 1712 letter to Bernoulli is Leibniz’s interest in describing a mental process such as memory as parallel to the bodily vibration that is brought about by the elastic force and that keeps the body in perpetual motion. Bernoulli would hold that “in the whole machine of the human body, every smallest particle involved in a movement is moved either directly by an order of the soul or by muscles. All these muscles follow strictly and steadily the laws of mechanics.” Leibniz was very impressed with Bernoulli’s work, yet there is no way, given Leibniz’s conception of body and soul as parallel automata, that he could have agreed with his physician friend as to the dual sources of motion in the body. Leibniz would certainly agree that the muscles follow the laws of mechanics, and that the origin of motion in the muscles is a mechanically explicable pyrotechnical event, yet for him no “order of the soul” could make a difference in the succession of a corporeal substance’s states. The reason for this is spelled out at length in Leibniz’s arguments against Stahl’s account of how the soul moves the body.

Leibniz agrees that the body contains the principles of its own motion, and this will be the major point of contention around which his debate with Stahl circles. The debate, at least as both of its participants understood it, was not about ‘vitalism,’ a notion that would not even come to be meaningful until well after the deaths of both Leibniz and his opponent. Yet if we must categorize Leibniz anachronistically in terms of this doctrine, we may say with firm conviction that he is an antivitalist: for him, the growth, motion, and preservation of a living body can be exhaustively accounted for without appeal to the soul. The soul is not responsible for life.

In many respects, Leibniz’s development with respect to the question of mechanism reflects the development of his philosophy as a whole: we see an early, fervent commitment to the explanatory promise of the mechanical philosophy, followed by a gradual reintroduction of a role for teleological explanation, one that sees it as coexisting with mechanical explanation, and that sees each of the two types of explanation as accounting for one and the same world at different metaphysical levels and in view of different epistemological exigencies. Leibniz’s rediscovery of teleology, however, and his mature view that natural beings are not ‘mere’ machines but rather corporeal substances, should not be seen entirely as an abandonment of the project of mathematizing nature. Rather, the conception of both the structure and complexity of nature on the one hand, and on the other the sort of mathematical model that could be useful in the study of nature, changed in tandem. For Leibniz, in both mathematics and in nature, the key concept would be one that, as we have already seen, had been utterly excluded in first-wave mechanical philosophy such as that of Descartes: infinity.

DIVINE MACHINES

Leibniz’s mature antivitalism is far from a further development of approaching anatomy and physiology more geometrico. If there is any extent to which mathematics continues to provide a model for the study of living bodies, then the domain of mathematics that suggests itself is not geometry, but rather the infinitesimal calculus. As we have already seen, Nachtomy perceives a close relationship in Leibniz between the mathematics of infinity on the one hand and his analysis of the structure of organic bodies on the other. However, there are some grounds for caution in perceiving Leibniz’s engagement with the mathematical problem of infinitesimals as shining any sort of light on his metaphysics of body. There are no smoking-gun texts that lead us directly from the one to the other, though Nachtomy does make a rather compelling case that the infinite nestedness of parts within parts that defines Leibniz’s mature conception of the organic body could not have taken shape in the way it did if Leibniz had not been working through the problem in many other areas of his philosophical reflection. This includes his reflection on the composition of the continuum and his contributions to the development of the infinitesimal calculus, as well as his reflections on the nature of freedom and the idea of infinite analysis of complete concepts. In a very broad sense, then, we may cautiously say that for Leibniz infinity is a central concern in many aspects of his thought, and that this concern is reflected equally, with interesting parallels across both domains, in Leibniz’s mathematics and in his account of the structure and motion of so-called living bodies. Leibniz’s ultimate account of these bodies, we might say, differs from the account of respiration, circulation, generation, and so on, that had been sought after by first-wave mechanists in roughly the same way the infinitesimal calculus differs from geometry.

As Nachtomy has well noted, there is no domain of the natural world that does not involve infinity. Rather than rejecting infinity, as both Aristotle and Descartes had recommended, Leibniz is insistent that any adequate explanation of nature must involve the notion of infinity. As he writes to Foucher: “I am so much in favor of actual infinity that, instead of admitting that nature rejects it, as it is vulgarly said, I hold that it affects it everywhere, for better marking the perfections of its author” (G 1 416). Nachtomy rightly stresses however that Leibniz employs different notions of infinity in different contexts, and that he is particularly careful to distinguish between infinity in a mathematical context, “which concerns abstract and ideal entities,” and in a metaphysical context, “which concerns concrete and real beings” (Nachtomy 2014).

The principal difference between the two contexts, as Nachtomy’s distinction suggests, is that mathematics concerns itself with ideal entities, which thus have no real divisions in them but rather are literally continuous, that is, are such that they are not constituted out of real parts, but rather any part or section can be taken out of any given part or section at will. Thus ideal continua are actually infinitely divisible, for Leibniz, whereas concrete and real beings are actually infinitely divided. Recall, in this connection, Leibniz’s notes on a letter to Fardella, from 1690. There, he drew an analogy between the composition of the organic body on the one hand and the relationship between points and lines on the other, arguing that bodies are no more built up out of fundamental parts than lines are built up out of points. But there is a crucial difference, namely, that there is no conceivable need to account for how lines are built up at all, since on the final analysis there simply are no such things as real lines. Bodies, however, demand to be accounted for; they are real, and therefore really constituted in some way or other, even if they are not constituted out of physical atoms. The answer, again, is that bodies are constituted as divine machines, which is to say that they result from the conspiracy of infinitely many other organic bodies, and so on without end, ad infinitum, no matter how far down you may wish to go in your analysis of the organic bodies constituting other organic bodies.

From the mid-1690s until his death, Leibniz gives several explicit accounts of what he means by this technical term, all of which amount to variations on the same core idea.12 He says to Stahl in 1709, for example, that “organic machines are nothing other than machines in which divine invention and intention are expressed to a greater extent” (Leibniz 1720, 135). And five years earlier, in a letter to Damaris Masham, he writes: “I define organism or a natural machine, as a machine each of whose parts is a machine, and consequently the subtlety of its artifice extends to infinity, nothing being so small as to be neglected, whereas the parts of our artificial machines are not machines. This is the essential difference between nature and art, which our moderns have not considered sufficiently” (G 3 356).

“Organic,” as an adjective, in the seventeenth century had no particular biological connotation (and of course “biological” had no connotation whatsoever). Rather, it was first and foremost a description of anything that has interrelated, working parts, whether physical or conceptual; anything, that is, that the Greeks would have recognized as an organon, a term any serviceable Greek-English lexicon would translate as “instrument” or “tool.” Working with this minimal definition, we arrive already at the surprising conclusion that if we wish to avoid anachronism we must stop reading early modern occurrences of the term “organic” as antonyms of “mechanical,” and instead interpret them as synonyms.

Anne Conway illustrates this original synonymy of “organic” and “mechanical” very clearly in her Principles of the Most Ancient and Modern Philosophy, published posthumously in 1690, when she writes that an animal is not “a mere Organical body like a Clock, wherein there is not a vital Principle of Motion” (Conway 1996). Similarly, in his Lexicon Philosophicum of 1662, Johannes Micraelius (1996) defines “organic parts” as “composite heterogeneous parts … They are members of the body, which nature exploits for uses that are necessary for life.” The “inorganic” in turn is the “intellect, for it does not have its own organ of the body of which it makes use” (Micraelius 1662). The organic is whatever has working parts—machines and animals alike—and the inorganic is that which lacks parts, which is to say, that which is mental or intellectual. But Micraelius gives no possibility for distinguishing among different kinds of organicity, just as some efforts to describe animals as simple machines offered no criteria for distinguishing among different varieties of mechanicity. This is what Leibniz would provide: for him, organicity is a special variety of mechanicity; for Leibniz, in contrast with Conway, the horse’s body is “organical.” A body is organic, Leibniz explains, “when it forms a kind of automaton or natural machine.” For Leibniz, unlike Conway, the horse’s body, even though it is organic, is not simply like a watch, since to be an organic body is not to be a “mere” organic body. Leibniz defends the organicity of the horse by denying it to the watch. This distinction might seem obvious today, but until Leibniz made it, it went against the very meanings of the words involved.

Leibniz would agree with Aristotle’s general line of reasoning, according to which it is the function, and not the material constitution, of an organ or an animal that makes it the sort of organ or animal that it is. For him it is impious to argue that eyes see simply because they are so structured as to be able to see, rather than that they were structured in order to see. Leibniz will favor the function of the organ by tracing its existence to a divine creator and to its, so to speak, “intelligent design.” Such a consideration certainly could not have interested Aristotle, yet in his as in Leibniz’s case the organ exists for the execution of a function, rather than that it happens to fulfill that function simply because it exists. Leibniz would certainly also agree with Aristotle that just as a blind eye is an eye in name only, so, too, a cadaver is a man only by convention. What makes the blind eye merely a nonfunctioning organ, rather than a dead animal, is that the seeing eye that it once was, was what it was only insofar as it contributed to the telos of the creature as a whole.

While Leibniz’s understanding of “organic” does mark a new turn in the history of the concept, it is still not the antonym of “mechanical” that many commentators have taken it to be. In Leibniz’s view, an organic body is distinct from a clock with respect to the complexity of its constitution, but Leibniz continues to agree with Conway that an organic body, considered in itself, lacks a single, dominant, vital principle. For Leibniz, an organic body is distinct from a mere mechanical body in that it is infinitely complex, but this does not mean that the organic body per se is something the explanation of which requires the introduction of an immaterial vital principle. It is true that metaphysically speaking an organic body is always dominated by the soul or form of the animal or corporeal substance to which it belongs, but physically speaking, the difference between an organic body and an inorganic body is found in the complexity of the organic body: it and all of its parts and the parts of the parts, ad infinitum, are machines of nature.

The organic body of the fish, then, insofar as it is the body of the fish, will in fact never be without a dominant monad or unifying entelechy. Yet the block of marble, at least as a whole, is always without one, even though every part of the organic matter making up the block of marble is part of some corporeal substance. An organic body can at most be conceptually distinct from a corporeal substance, while in fact there is never an organic body that is not the organic body of a corporeal substance.

Any arbitrarily chosen parcel of matter is extremely unlikely to constitute in itself one organic body, even if there is no part of it that is not so constituted. As Leibniz writes in 1702, the organic body, taken separately, is just a special kind of aggregate, while the union of this organic body with an entelechy is one per se, and not a mere aggregate of many substances, for there is a great difference between an animal, for example, and a flock. And further, this entelechy is either a soul or something analogous to a soul, and always naturally activates some organic body. Which, taken separately, indeed, set apart or removed from the soul, is not one substance but an aggregate of many, in a word, a machine of nature.

The organic body, then, is a machine of nature, even if, taken together with the soul rather than separately, the whole thing is not a machine at all, but a corporeal substance. Insofar as we are considering the organic body of the fish, as distinct from its soul, we are considering something on an ontological par with a pile of sawdust, even though the fish, which consists in this organic body and an ichthyoid soul, is of an ontologically higher rank than the pile. The block of marble is made up entirely of organic matter, but is only an aggregate, insofar as it is not, as a whole, unified by a dominant monad or entelechy. The fish’s body is also made up entirely of organic matter, but the fish itself is a corporeal substance and not an aggregate, insofar as there is a dominant monad, the fish’s soul, uniting the organic body. While it is true that souls and bodies are not really separable, their conceptual separation is of central importance.

We find the same point also in the New Essays. Animated bodies, Leibniz says there, can be picked out by their interior structures. Body and soul can each be taken separately, and each suffices for the determination of the identity of the thing in question. Neither influences the other, but each expresses the other perfectly, the one being the concentration in a unity of what the other disperses throughout a multitude. Leibniz emphasizes that the organic body may be taken separately (pris à part), which is to say that organic bodies just are the machines of nature, or that which remains mechanical in its least parts, and which does not require the introduction of the capacity for perception that would be required in the exhaustive account of a corporeal substance.

In Divine Machines, I have developed at greater length this distinction between the organic body and the corporeal substance. There is no need to dwell on it any further here. It is enough to be clear that, for Leibniz, the corporeal substance is to be understood in relation to its ends, which it has in virtue of the domination relation of its soul or entelechy to the infinitely many other monads implicated in it; the natural or divine machine, by contrast, which is to say the organic body, is to be understood without regard for its ends or for its unification under the domination of an entelechy, but only with regard to its infinite structure.

It is precisely this infinite difference between the natural and the artificial machine that, for the mature Leibniz, will come to be coextensive with, and also come to replace, the more familiar distinction between the living and the nonliving. As Leibniz makes particularly clear already in the Protogaea of the early 1690s, the formation of crystals, by contrast with that of animals and plants, can be exhaustively analyzed in terms of “external contiguity,” that is to say in terms of the regular repetition of radial and polygonal shapes. This is important, because it sharply delineates anything formed geologically, including crystals, from the realm of the organic. Crystals and organic bodies are in fundamentally different ontological categories, yet this difference cannot be accounted for by the fact that the former are ‘nonliving’ while the latter are ‘living,’ since, strictly speaking, for Leibniz organic bodies are not living. Rather, organic bodies can be explained exhaustively in terms of their vegetative structure, while life, in turn, is simply a capacity of immaterial perceiving monads. As Leibniz writes to Stahl, in response to the Halle physician’s account of supposedly irreducibly vital processes: “I do not wish to quarrel over words. It is the author’s wish to call ‘life’ what others call ‘vegetation’ ” (Leibniz 1720, 11). Animal bodies vegetate but are not alive, for the mature Leibniz, and vegetative structure consists precisely in this: that whereas crystals, for example, are generated out of the finite repetition of regular geometric forms, vegetative bodies are ungenerable, to the extent that they consist in an infinite structure with no lower limit to its composition, and thus no possibility of ever being decomposed into its elementary constituents, or of ever having been built up in time from such constituents.

At this point, we have considered just about every aspect of the structure and nature of organic bodies, or divine machines, and of corporeal substance that Leibniz was willing or able to impart to us. We have seen that they are real, infinitely structured entities, whose composition Leibniz on occasion wishes to describe on analogy to the relationship between points and lines, even though he knows that this analogy cannot be terribly helpful, to the extent that lines and bodies belong to entirely different ontological categories, and lines, as ideal entities, do not really need to be constituted at all. But how, if bodies are real, composite entities, can they fail to be built up out of fundamental parts, as houses are from bricks, rather than simply resulting from requisites? What is it that yields a big body if not several smaller bodies? But if there is no lower limit to the analysis of smaller bodies into smaller bodies still, then how can a body of any size ever be yielded by composition?

Nachtomy, as we have seen, believes that Leibniz is emboldened in accounting for the composition of real bodies by appeal to infinity as a result of his parallel success in the mathematical treatment of infinite quantities. Having banished paradox from the mathematics of the infinite, Leibniz was now ready to explain the bodily world as well by appeal to infinity. On Nachtomy’s view, this was a great coup de grâce of Leibniz’s philosophy: to give us a novel, postmechanist philosophy of nature by extending his successes in the mathematics of the infinite to the study of the natural world. Yet as I have been arguing here, the not-merely-mechanical, end-directed corporeal substance is something quite distinct from the infinitely structured organic body. To the extent that Leibniz invokes infinity in his analysis of natural bodies, in other words, he does so precisely in order to preserve a variety of mechanism, even if this amounts to a mechanism with a very considerable twist. This is precisely the point of Leibniz’s insistence that, as he writes to Stahl, “organism is in truth mechanism, but more exquisite” (Leibniz 1720, 9). ‘Exquisiteness,’ for Leibniz, does not reach back to a pre-Cartesian understanding of nature, but rather radically modifies the mechanical philosophy in order to give an account of nature that is adequate to its complexity, and that is also almost wholly original in Leibniz (with, obviously, a complex prehistory in figures such as Giordano Bruno, Nicholas of Cusa, Henry More, and many others).

A final problem with the suggestion that Leibniz is offering us a postmechanical nature by extending his successes in the study of the mathematics of the infinite to his account of the natural world is precisely that we may doubt that his invocation of ‘infinity’ in his account of the structure of bodies has much to do with mathematical infinity at all. It is certainly true, as Nachtomy suggests, that Leibniz was able to some extent to banish paradox from the mathematical treatment of infinity by treating infinitesimal quantities as variables. But again, the freedom to treat them in this way in the end rested for Leibniz on the fact that mathematics is only concerned with ideal entities, yet the very challenge that the natural world poses for Leibniz is that it requires us to account for something real, actually existing, and resistant to fictions. Leibniz determines that the matter making up this world is actually infinitely divided, and that the organic bodies, in which all existent matter is wrapped up, consist in bodies nested within one another ad infinitum. It is not clear that in elaborating this remarkable account of the natural world, which is to say of the living world (to say that the two are coextensive is in the end exactly the same as to say that all matter is wrapped up in organic bodies or divine machines), Leibniz successfully steers clear of paradox.

CONCLUSION

Leibniz’s theory of divine machines may, in sum, be seen as a continuation of the project of the mathematization of nature, even if it amounts to a continuation in such different terms as to be almost unrecognizable. From the treatment of nature more geometrico that inspired early mechanical philosophy, we witness a shift to an approach to nature inspired more by the mathematics of the infinite and of infinitesimals. But this shift is based, ultimately, on an untenable analogy between two different realms, the ideal and the real, and in the end it does not seem that Leibniz’s success in advancing a paradox-free treatment of the mathematical infinite enabled him to provide a fully compelling account of the infinite structure of natural bodies.

We may ask whether an early mechanical philosopher such as Descartes would have seen the introduction of the concept of the organic, which is to say, again, the concept of the infinite structure of bodies, as a failure of the mechanical program, or rather as a final perfection of it. But one thing that is certain is that Descartes and Leibniz both saw the need to account for the structure and origins of what are commonly called ‘living beings’ as likely the most pressing task for the new natural philosophy to fulfill. Leibniz’s account of living bodies would come to serve for him as the explanation of body in general, and in this respect we may say, with Garber (1985), that for Leibniz it is what we would call ‘biology,’ rather than physics, that is the foundational science of nature. Leibniz hoped early on to ‘mathematize’ biology, or what he called ‘medicine,’ but seems to have grown skeptical of the possibility of doing so as his mature philosophy developed. At the same time as his mature philosophy developed, however, he became increasingly enthusiastic about an account of living nature that rested on the notion of infinity. As I have attempted to argue, however, against Nachtomy’s provocative and in many respects compelling account, this employment of the notion of infinity in the mature account of nature does not amount to a successful instance of the mathematization of nature, nor even a conscious attempt to mathematize it.

There are, of course, many sources feeding into Leibniz’s use of infinity in his philosophy, and in particular into his account of the infinitely nested structure of natural bodies. His work in the mathematics of infinity no doubt played a role. Although the full case has yet to be made, however, a no less significant source for Leibniz’s conception of worlds within worlds, and of the constitution of composite beings out of countless other such beings, appears to have come from the science fiction of which he was an avid reader: the imaginative flights of fancy from the likes of Cyrano de Bergerac, for example, who dreamed up a trip to the sun, and an encounter there with deliriously strange beings. Leibniz, the genius eclectic, did not dismiss Cyrano’s pomegranate men as the products of a mere rêverie, but instead saw in them a reflection of the very earnest account of the natural world he spent much of his intellectual energy, drawing on sundry and often surprising sources, to elaborate.13

ABBREVIATIONS

NOTES

1. “De la description des cors inanimez & des plantes, ie passay a celle des animaux & particulierement a celle des hommes. Mais, pourceque ie n’en auois pas encore assez de connoissance, pour en parler du mesme style que du reste, c’est a dire, en demonstrant les effets par les causes, & faisant voir de quelles semences, & en quelle faôn, la Nature les doit produire, ie me contentay de supposer que Dieu formast le cors d’un homme, entierement semblable a l’un des nostres, tant en la figure exterieure de ses membres qu’en la conformation interieure de ses organes.”

2. Citing Ariew’s citation of Sophie Roux summarizing Husserl on mathematization (see Ariew, chapter 4, this volume).

3. “Il faut tousjours expliquer la nature mathematiquement et mecaniquement, pourveu qu’on sçache que les principes mêmes ou loix de mecanique ou de la force ne dependent pas de la seule étendue mathematique, mais de quelques raisons metaphysiques.”

4. Nouveaux essais 2, ch. 7, §42. “[C]’est … comme Arlequin qu’on voulait dépouiller sur le théâtre, mais on n’en put venir à bout, parce qu’il avait je ne sais combien d’habits les uns sur les autres: quoique ces réplications des corps organiques à l’infini, qui sont dans un animal, ne soient pas si semblables ni si appliqués les unes aux autres, comme des habits, l’artifice de la nature étant d’une tout autre subtilité.”

5. Leibniz to Masham, May 8, 1704.

6. Anne Mauduit Nolant de Fatouville (credited anonymously as ‘Monsieur D***’), Arlequin Empereur dans la Lune (de Fatouville between 1765 and 1814).

7. Antonio Nunziante has offered an excellent analysis of the relevance of this part of Cyrano’s work for our understanding of Leibniz’s theory of corporeal substance, and it is Nunziante who first brought this connection to my attention (see Nunziante 2011).

8. Nouveaux essais 2, ch. 23, §43. “Au reste je suis aussi d’avis que les Genies appercoivent les choses d’une maniere qui ait quelque rapport à la nostre, quand même ils auroient le plaisant avantage, que l’imaginatif Cyrano attribue à quelques Natures animées dans le Soleil, composées d’une infinité de petits volatiles, qui en se transportant selon le commendement de l’ame dominante forment toutes sortes de corps.”

9. “Tout cet amas de petits hommes n’avoit point encore, avant cela, donné aucune marque de vie; mais, sitot qu’il eut avalé son petit Roi, il ne se sentit plus être qu’un.”

10. Aristotle 1, 2:719.

11. Portions of this section were developed previously in Smith (2011) though in the course of making a very different argument about the place of medicine and physiology in Leibniz’s philosophy.

12. Portions of the present treatment of the distinction between “organism,” “organic body,” “mechanism,” and “corporeal substance” were previously developed in Smith (2011) though, again, in the course of making a very different argument about the significance of these concepts in Leibniz’s philosophy.

13. Elsewhere, I have argued for the important role of the empirical discoveries of microscopy in the development of Leibniz’s theory of composite substance, and indeed here I am not at all seeking to subvert that account, nor to refute Nachtomy’s, but only to recommend that here, as in so many other areas of Leibniz’s thought, no single monocausal account will do. See, in particular, Smith (2011).

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