Draft entry for a Michel Serres Dictionary: Le Système de Leibniz et ses modèles mathématiques (1968)

Le Système de Leibniz was published during the heady anni mirabiles of late 1960s French thought. It appeared in 1968, the same year as Roland Barthes’s short essay ‘The Death of the Author’, one year after Derrida’s Of Grammatology and Deleuze’s Difference and Repetition, and two years after Foucault’s The Order of Things. Like Derrida’s and Deleuze’s volumes Le Système was written as a major doctoral thesis (French doctoral candidates submit a major and a minor thesis), the fruit of research under the supervision of Jean Hyppolite at the Ecole Normale Suprérieure, Rue d’Ulm. Like those other works it stands both as a rich and sinuous study in its own right and also as a radical declaration of philosophical intent from a philosopher elaborating the shapes of thought that will accompany him through his subsequent writings.

The importance of the work can be summarised under three headings. First, it argues for a new reading of Leibniz’s philosophy which highlights the parochialism of previous commentators, offering a system for coming to terms not only Leibniz’s spectacularly diverse œuvre but with the whole of human knowledge. Secondly, it challenges the dominant narrative of modernity that traces a “philosophy of consciousness” from Descartes through Kant to Husserl, replacing it with a more adequate account of the genesis of modern thought. Thirdly, it offers a paradigm for understanding contemporary society which, though still prophetic in 1968, has been progressively vindicated from that time until now. As Serres was writes on the cover blurb, Leibniz is “of our time, he is our predecessor”, and he was later to add that “we are all neo-Leibnizians now” (H4 275).

A new Leibniz

In the late 1960s the world of Leibniz scholarship was dominated by Louis Couturat and Bertrand Russell and Bertrand Russell, for whom Leibniz was to be understood first and foremost as a logician working from a determinable series of axioms. Serres does not argue that this approach is incorrect in its own terms, but that it is insufficient and Procrustean. In fact, his own decisive contribution to Leibniz studies in Le Système is to argue that any attempt whatsoever to gather Leibniz’s thought under the banner of a particular paradigm, discipline or model is reductive and fails to follow the indications and signposts found in that thought itself. Each commentator on Leibniz adopts a limited perspective on his thought, Serres argues, because he or she chooses to enter Leibniz’s thought through only one of its many doors (SdL 26). If we read Leibniz through Leibniz’s work on logic, we should not be surprised that Leibniz emerges as a logician: given the starting point, the destination is inevitable. Serres’s approach seeks to reverse this trend in two ways. First, he sets out to do justice to the diversity of Leibniz’s thought and to resist using any particular disciplinary approach is the “key” to the whole. Secondly, rather than reading Leibniz in terms of the intellectual trends of the day, he attempts to read the modern and contemporary world through and in the light of Leibniz’s system.

In terms of the first of these two aims, it is hard to see how a broad reading of Leibniz’s many different contributions could be anything but reductive. In mathematics alone the philosopher of Leipzig was decisive in the elaboration of differential calculus, binary arithmetic, topology, symbolic logic, and more; he produced enduring contributions to epistemology, theories of theodicy and possible worlds, the philosophy of physics, the philosophy of mind, jurisprudence, geology and history, as well as making pioneering discoveries in cryptography and inventing a calculating machine. Surely any all-encompassing account of such diversity would inevitably to privilege some parts over others and airbrush away any apparent contradictions and incommensurabilities?

Serres’s solution to this considerable problem is to see in Leibniz’s work neither a simple unity nor an unrelated diversity, but as a complex harmony of domains none of which stands over and above the others, each with its own integrity, and with the possibility of local translations from one to the other. Borrowing the language of Vitruvian architectural theory, Serres argues that each commentator provides a scenography (a view from a particular perspective) of Leibniz’s work, whereas in Le Système he is elaborating an ichnography, a geometrical ground plan of the system, projected as from above but not beholden to a single perspective (there is no vanishing point in an ichnographic plan). In the same way that a building cannot be grasped as a whole without moving from perspectival to geometrical representation, so also Leibniz’s system cannot be appreciated as a system until it is abstracted and formalised. The tripartite structure of Le Système reflects this approach. Part I, “Stars”, explores the nature of the points of intersection in the Leibnizian system where various paths and passages between different domains converge to form a star-like nodes. Part II, “schemas”, considers the system as a whole, and the nature of the connections between different stars. Part III, “Points”, is a Leibnizian critique of the modern age’s search for a fixed point of reference.

In seeking to translate from one domain of Leibniz’s thought to another, Serres takes inspiration from Leibniz himself. In fact, the great interest of Leibniz’s thought, he argues, is not in his many discoveries and ideas in isolation but in the way in which he seeks to link them together (SdL 289-90). These links in the Leibnizian system do not create a simple reductive unity but a complex harmony which Serres sums up on more than one occasion with the comment that the model of Leibniz’s system is the system of Leibniz’s models (SdL 37, 52, 321). By “model” (or “paradigm”) Serres means here the content of Leibniz’s thought in a particular domain or discipline (SdL 309-10) such as logic or geometry: the previous commentators elevated one such model to an explanation of Leibniz’s system as a whole. Each of these models contains elements which are structurally analogous to elements in other models, but these analogies only emerge after a work of formalisation and abstraction, and it is the formalised structure shared by each of the models that constitutes Leibniz’s “system”. It would be a mistake to understand this system as a totalising whole. It is not a unity but a complex web of translations from one model to another, inspired by Leibniz’s own work on harmonic tables and understood as a vast multilingual dictionary or encyclopaedia, the entries in which are neither subordinated to one primary entry (Serres calls such a chimerical chief entry an “umbilical discipline”, SdL 250-1), nor unrelated to each other. There is no single hegemonic discourse or domain, and each discipline remains expressive in its own area (SdL 383-4).  As in Leibniz’s theory of monads, each point in the Leibnizian system can be an origin, a link, or an end; each reflects the whole, and each can act as a guide to the system as a whole (SdL 3).

The relation between models is not direct but analogical (SdL 4-5) or isomorphic (SdL 41-3), a principle of variation which reduces neither to absolute identity nor to absolute difference (SdL 59). In fact, for Serres generic diversity walks hand in hand with formal purity in Leibniz’s thought (SdL 41), a theme that will recur later in his own writing and which he expresses in terms of a harmonious relation between the white Pierrot (purity) and the multi-coloured Harlequin (complexity). In the course of Le Système, one example he gives of such an analogical or isomorphic harmony is the notion of the “primitive”, a “‘structure’ which has its faithful models in every region of the encyclopaedia” (SdL 137) including arithmetic, mechanics, language, and colour theory.

As in the case of Vitruvian ichnography, in order to understand the Leibnizian system without reducing it to one of its models it is necessary to adopt the correct formalised point of view, a lesson which Serres learns from Leibniz’s work on conic sections. The elements discovered and formalised by ancient geometry—such as the point, the angle, the circle, the parabola, the ellipse and the hyperbole—each exhibit different properties and seem to obey different laws, until it is discovered that they can be expressed as sections of a cone (SdL 690). The tip of the cone is the point of view from which all the seemingly unrelated shapes show themselves to obey a higher order (SdL 692). In the same way, in Leibniz’s philosophy all the disorder and suffering of the world reveals itself to be part of a greater system when considered from the (to human minds inaccessible) point of view of God.  Serres turns this Leibnizian commitment to the harmonising point of view back on Leibniz’s philosophy itself, and understands his contributions to different areas of human knowledge as so many seemingly unrelated and irreconcilable geometrical elements that can only be seen as part of a system from a particular, formalised point of view. However, in the same way that Leibniz’s God is like a circle whose centre is everywhere in circumference nowhere, the point of view from which everything makes sense is, precisely, everywhere (SdL 251): the whole of Leibniz’s system is contained in each of its models, in each “total part” (SdL 279-80), and there is no need to adopt a transcendent divine standpoint. On a number of occasions throughout the book, Serres sums up this insistence upon harmony in Leibniz’s system with a quotation from his Philosophische Schriften, paraphrasing the inquisition of Harlequin who, having returned from a trip to the moon, is quizzed about the lunar landscape: “we could say, as in the Emperor of the Moon, that in all places and at all times it is exactly as it is here, though it varies in degrees of size and perfection” (SdL 1, 38, 357).

A more adequate modernity

In the third part of Le Système Serres offers a critique of the conventional narrative of modernity, a narrative which traces a philosophy of consciousness from Descartes through Kant to Husserl and beyond. Serres sets up Descartes and Leibniz as the titular heads of two rival accounts of modernity, such that when it comes to Leibniz’s system, “it is necessary, in order to understand its elementary articulations, to adopt point four point to adopt a language opposed to […] Cartesian organisation” (SdL 23-4). Both Descartes and Leibniz are concerned with the question of the fixed point, Serres argues, but in very different ways. For Descartes, the originary cogito and the zero point of Cartesian coordinates serve as such fixed points, providing a foundation for a whole system of knowledge. In contrast to Cartesian thought, Leibniz elaborates a philosophy not of consciousness but of harmony, and not of being but of relation, in which the very idea of the fixed point is inflected through mathematical accounts of the infinite. In this early Leibnizian critique of modernity we can already begin to see Serres’s importance for the rejection of Cartesian modernity in contemporary new materialisms and object oriented thought.

The cogito

The classical account of modern thought begins with the zero point of the cogito, a point which Serres characterises the pitiful scrap of knowledge to which the modern is happy to cling, assured as she is of its foundational solidity (SdL 215). The problem comes when I seek to build upon this zero point by employing the methodical rules of the Cartesian system, for I find that in simply applying a pre-established method I discover nothing at all that was not already contained in it (SdL 215-6). Here we see a striking similarity between Serres’s critique of Descartes and his dismissal of Bertrand Russell’s book on Leibniz: to start with a fixed set of assumptions and axioms is to condemn oneself never to move beyond them, and never to invent: Descartes and those who follow him are prisoners in a labyrinth of their own making (SdL 216) and gods in their own (exceedingly small) kingdom. In a further critique of the Cartesian method Serres argues that, in seeking to demonstrate the validity of the cogito, Descartes goes beyond the geometrical paradigm that he professes to follow. As Serres points out geometry itself admits varying levels of assent beyond the true/false dyad (SdL 132), and is far from being in a position to demonstrate its own axioms. If we were to demand that geometry demonstrate its first principles, as Descartes seems to demand that his own method demonstrate the validity of its starting point, then we would have no geometry at all today (SdL 132, 550).

There is yet a further flaw in the Cartesian method, evident both in the cogito and also in the system of Cartesian coordinates: it relies in the final analysis on the veiled dogmatism of arbitrarily imposing a centre or zero point of all knowledge (in the case of the cogito) and all measurement (in the case of Cartesian coordinates). Why count from here and not there? Why my own thoughts and not some other foundation of knowledge? Descartes cannot escape the fact, Serres insists, that the decision for the cogito itself is situated, the result of a methodological fiat: there is no point of view without a point of view (SdL 669-70).

Whereas Descartes begins with the individual “I”, Leibniz’s monadology is a philosophy of the irreducible relation. In a move which fore-echoes Jean-Luc Nancy’s insistence upon the primordiality of being-with (Mitsein) in Being Singular Plural, Serres argues that the Leibnizian monad is no “granular atom” (SdL 313) from which a multiplicity can subsequently be constructed, but it is both self-sufficient and a reflection of the whole: the monad is a chiasmus between the one and the multiple (SdL 298). Serres’s Leibniz offers us a philosophy of the originary and irreducible “we” and of composition, not of the primacy of the one or the multiple (SdL 295). Unlike Nancy’s singular plural, however, it does so in a way that is not beholden to a modernity the genealogy of which runs back from Heidegger through Kant to Descartes.

Similarly, the monad cannot be reduced to the dichotomy of res cogitans and res existensa; it is a “subject-object” (SdL 789) which does not allow consciousness to be pitted over against the world and which must be understood in terms of pre-established harmony between thought and matter (SdL 506n2). In this exposition of the monad we see the germ of much of Serres’s later ecological thought and his influential notion of the quasi-object and quasi-subject. We also hear a for-echo of Bruno Latour’s critique of the Cartesian subject in We Have Never Been Modern and elsewhere, though Serres’s account in Le Système is broader, more detailed, of greater rigour and more philosophically informed.

Geometry and the infinite

If Serres’s Leibnizian account of modernity cuts the Gordian knot of the one and the many in this way, it also avoids the problems attendant on the modern (and post-modern) preoccupation with the dichotomy of the same and the other, a dichotomy that relies both on the Cartesian methodological preference for the individual res cogitans and also on Descartes’s two-dimensional geometry. Descartes thinks that with the cogito he has found the fixed point upon which to build the whole edifice of knowledge, but his system is inadequate because he builds it according to the narrow paradigm of algebraic geometry (SdL 219). Leibniz does not refute this geometrical paradigm but expands it to include other branches of mathematics. One of the most important problems issuing from Descartes’s decision to base his epistemology on a geometrical paradigm is that it affords him no robust understanding of the infinite. Leibniz’s embrace of the Desarguesian principle that parallel lines meet at an infinitely distant point (SdL 154) allows for a rigorously mathematical approach to understanding how, given only the correct point of view, the seemingly opposite or unrelated can be harmonised. Leibnizian analogy and harmony allow us to think “the pluralism of Sames [des Mêmes] essentially Other [Autres], or of Others essentially the Same” (SdL 152). Leibniz’s mathematical system of harmonies pluralises the Same into Others, and unifies Others into the Same (SdL 152-3) in a way that Descartes’s narrowly geometrical paradigm could never countenance.

The infinite is also crucial in Serres’s Leibnizian argument against Descartes’s conception of the fixed point. Any pair of scales with arms of finite length will have a pivot point at which the scales balance. However, if the arms are infinitely long there is no such balancing point. Or rather, every point would then be the balancing point (SdL 648). Similarly, an infinitely large solid would have both no centre and its centre would be everywhere. We see in this duality a further fore-echo of the Harlequin/Pierrot doublet that will punctuate Serres’s later writing: the ubiquity of the balancing point and the absence of a single balancing point emerge together and are not in opposition to each other. In this way, the concept of the fixed point is utterly transformed when it is thought in terms of the infinite. Pascal well understands the implications of infinity for the thinking of the fixed point in the modern age, Serres argues, and therefore he insists that the human being has no centre. Descartes, on the other hand, fails to discuss it. This is a grave omission on Descartes’s part and one that sets him at the periphery of his era, for the chief question with which the modern age must wrestle is the question of the existence—not the location—of any fixed point (SdL 663).

Doubt, truth, and certainty

Leibniz and Descartes also offer conflicting accounts of truth, doubt and certainty. For his part, Descartes is committed to doubt all things were he finds the least suspicion of uncertainty, and the slightest impurity provokes a hyperbole of rejection (SdL 117-8). One bad apple ruins the whole barrel of truth and so, to avoid the spread of falsehood, the entire contents must be thrown away. Leibniz’s approach to truth is quite the opposite: if there is even the slightest truth at all in an impure mixture of truth and falsehood then it is to be purified and preserved (SdL 119). As Serres points out, there are veins of gold to be found in the sterile rocks of scholasticism, and yes, there is truth even in poetry and the novel. In the insistence that “Leibniz’s is not an analytic mind but a mind that makes combinations” (SdL 66) we can see the inspiration for his own insistence that philosophy is not about analysing but about federating (Mort 53).

The Leibnizian approach to truth applies a series of filters to an initial obscurity, whereas Descartes’s analytic method makes a series of excisions in an original impurity, cutting it back to leave a series of “clear and distinct” ideas. The difference is that, for Descartes, with each new excision the empire and scope of truth becomes smaller, but each new Leibnizian filter maintains a view of the whole but sees it differently (SdL 122-3). Whereas for Descartes truth is single, eternal and universal, for Leibniz it is plural (SdL 120), progressive and local or regional (SdL 123). Whereas Descartes cuts away the obscure to leave a skeletal truth, Leibniz harmonises disorder to find the point of view from which it reveals itself as a higher order. For Descartes, the suppression of error is total and primary; for Leibniz it is progressive and continuous (SdL 130). In other words, Leibniz’s notion of truth rolls forward with the gradualism of evolution, whereas Descartes apes the miracle of ex nihilo creation.

Leibniz crowns and critiques Descartes

Furthermore, Descartes’s understanding of doubt is adversarial and agonistic, for he fancifully imagines an evil genius both cunning and interested enough to seek to deceive him (SdL 221n1). For Leibniz, by contrast, there is no imaginary adversary truth is found not through negation, opposition and antagonism, but through correspondence, complementarities and dualities (SdL 338). For Leibniz “the whole truth is not here or over there, it is in the passage, in the translation, in substitution” (SdL 634-5). Given that the adversarial is itself at issue between Descartes and Leibniz, it is important to grasp that, for Serres, there is no straightforward, adversarial opposition between Descartes and Leibniz themselves (for that would fall back into a Cartesian agonistic conception of truth, and would be quite foreign to Leibniz’s own thought). Rather, Leibniz is figured by Serres both as the greatest culmination, and most piercing critique, of the Cartesian method.

Serres’s complaint is not that Descartes is wrong in what he says, but that he does not follow through sufficiently on his own convictions. When Leibniz broadens Descartes’s geometrical paradigm to embrace other mathematical disciplines, the lessons of methodology gleaned from these additional areas “formalise and crown the Cartesian method as they critique it”, transforming “an impoverished and strict discipline into a region overflowing with riches and novelties” (SdL 281) in a Leibnizian system which Serres does not hesitate to call “baroque”. The contention that Leibniz both completes and critiques the Cartesian method recurs a number of times in Le Système (SdL 215, 232, 281), and this idea of crowning-and-critiquing is in itself an instance of the Leibnizian search for truth through composition and correspondence: insight is gained by broadening and harmonising, not by narrowing and excising, by embracing complexity rather than by seeking a zero point of knowledge shorn of all obscurity. Crowning-and-critiquing is also one of the distinctive moves of Serres’s own thought. It is no reductio ad absurdum (if it were, then it would not be appropriate to describe Leibniz as crowning Descartes, as well as critiquing him), nor necessarily a transcendental critique of the conditions of possibility of a given philosophy. Nor again would it be accurate to describe this move is deconstructive, although it provides a compelling and fruitful alternative to a deconstructive approach. Perhaps the best way to describe it would be as “going the extra mile”, pursuing an insight or a commitment in its own terms beyond the point at which its originator was content to let it rest.

The chain and the web

In a further contrast, truth for Descartes is established in a linear, unidirectional chain of reasoning, beginning always with the cogito. The image appropriate to Leibnizian truth, however, is not the chain but the web (SdL 14), a “tabular space with an infinity of entries” (SdL 1) which can be entered at any point, within which any node (or “star”) can be reached from any other node. Furthermore, there are multiple paths from one node to another, and there is always a route from any point to any other point in the web (SdL 440). For Descartes, knowledge grows like a lengthening line, but for Leibniz and Serres it grows like an expanding circle (SdL 391). Within this context, Descartes’s linear reasoning can be understood as one path within a wider web, and we can see that Leibniz’s system is not opposed to Descartes’s method but contains and exceeds it. The figure of the Leibnizian web regularly recurs through Serres’s subsequent work, and can serve as a shorthand to sum up his entire reading of Leibniz (GB 90). It also has profound implications for Serres’s understanding of space (it is topological), time (it is plural and local) and progress (it is spatial, not temporal, and it is plural), upon which he will elaborate many times in later works.

The Copernican Revolution

As well as elaborating at length upon Leibniz’s difference from Descartes, Serres also touches upon a Leibnizian critique of Kant. Just as Leibniz’s system both crowns and critiques Descartes’s more geometrico with his own more mathematico, he also crowns and critiques Kant. Focusing on the Copernican Revolution of the first Critique, Serres argues that the question of the location of the fixed point (whether the earth or the sun is to be considered to be unmoving) is secondary and derivative compared to the question of whether there is a fixed point at all. Compared to this prior question, the Copernican Revolution is trivial, though to say so is still today considered a “philosophical blasphemy” (SdL 650). For Leibniz himself, every monad is a fixed point, and no monad is the fixed point. There is no single Copernican Revolution (any more than there is a single fixed point), but many local revolutions (SdL 250): our sun will be the earth for another sun (SdL 663), and so on ad infinitum. This means that the Leibnizian web is sufficiently described neither as Ptolemaic nor as Copernican; both Ptolemaic and Copernican moments are necessary to his “centred-decentred” system within which “everything is a centre in its way” (SdL 634). “Copernican Revolution” is a term to describe a moment in a continuing series, not an event that cleaves history in two (SdL 639), and today we need to do to Kant what Kant and Copernicus did to Ptolemy, for if Ptolemy is the Copernicus of the earth, then Copernicus is the Ptolemy of the sun (SdL 633-4).

A foundational work

Le Système de Leibniz is foundational for much of Serres’s later thought and writing. To begin with, the Leibnizian web provides the model for his understanding of contemporary society. Leibniz, Serres claims, is our predecessor: he foresaw not only our mathematics and physical sciences but also our networks of communication and the importance of data in our society (SdL blurb). More than one commentator has noted that Serres’s account of the Leibnizian web constitutes a remarkable adumbration of the World Wide Web, and in a 2011 interview Serres makes the same point (LSD). Like the World Wide Web (as opposed to the “internet”: the material infrastructure supporting it), Leibniz’s system is topological: any point can be adjacent to any other point, and physical distance becomes an irrelevance. Both webs have no centre, no necessary entry point, and no telos. In Serres’s own summary, Leibniz announces our decentred modernity, and shows it to us before it arrives (SdL 810).

Secondly, the Leibnizian insistence upon harmony and translation that sets the stage for Serres’s contention that, despite the claims of Marxism, our society is not best understood as one which fundamentally functions in terms of production and exchange, but in terms of communication and translation (Mort 51).

Thirdly, to recognise Leibniz’s commitment to truth as harmony and systematisation is a lodestar in understanding Serres’s own subsequent thought, for we see in his later writing a marked absence of the adversarial confrontation and self-defence that characterise some academic philosophical prose. Through his argument in Le Système we can see that this rejection of the adversarial on Serres’s part is not a weakness or an unwillingness to engage in political debate, as it has sometimes mistakenly been characterised, but a rigorous consequence of his rejection of a particularly threadbare and geometrical account of modernity and a of passionate commitment to a Leibnizian paradigm.

Fourthly, reading Le Système allows us to lift the bonnet on Serres’s deep commitment to cross-disciplinarity and understand how it, too, is born of rigorous philosophical commitments. Serres’s writings that bring together the arts and the sciences (in books such as Feux et signaux de brume: Zola or La Naissance de la physique dans le texte de Lucrèce), are too often quickly dismissed as vague and unsystematic gestures at crude synthesis by those who understand neither the imposing Leibnizian edifice that stands behind them nor its rejection of the easy equivalences of numerical equality and its insistence upon the arduous work of ichnographical formalisation. As one witty reviewer in the Nouvel Observateur put it, “for those who reproach Michel Serres for mixing apples with oranges [the rather more vivid French phrase is “marrying the carp and the rabbit”], algebra with music, this majestic book will vindicate the propriety of a thought that seeks to conceive simultaneously of the most diverse multiplicities and the unity of a single point.”

Finally, the importance of the Leibniz book becomes even greater when we consider its idiom. Unlike Serres’s later writing Le Système is written in conventional academic prose, extensively footnoted and maintaining a close engagement with the writing of Leibniz and others. It therefore provides the underlying and assumed philosophical armature for many of the ideas which, in his later writing, he elaborates and explores in different ways.

See also: Descartes, René; Harlequin and Pierrot; Quasi-object; Leibniz, Gottfried Wilhelm; Modernity; Structure.