A Note on Gaussian Spaces and Multiplicities

I confess that I've often felt disgruntled by the secondary literature on Deleuze, and have often experienced a desire to distance myself from it. Somehow the constant talk of rhizomes, becomings, deterritorializations, and such rings hollow to me; like it is a new normative vocabulary designed to appeal more to the narcissistic image of the ego than to engage in tough critical work. In this regard, my interests in Deleuze have tended to gravitate more towards his earlier, more sober works such as Difference and Repetition and The Logic of Sense. There, in Deleuze's account of different/ciation, I found the resources to conceptualize the immanent organization of situations, the relationship between genesis and structure, the primacy of relations over elements, and, most importantly, the critical tools for diagnosing false problems and discerning the manner in which these false problems are solutions of a poorly formed sort.

Very roughly, Deleuze's account of different/ciation is designed to account for the conditions of actualization moving from the virtual to the actual. On the one hand, the process of differenTiation refers to the formation of virtual multiplicities organized around differential relations and the singularities that constitute them. In speaking of multiplicities-- or what Deleuze, following and subverting Plato and alluding to Kant in the transcendental dialectic of the Critique of Pure Reason refers to as "Ideas" --Deleuze is speaking of neither the one nor the many, but of an organization belonging to the many as such (DR, 182). Unlike the old ouisiological or identity based ontology, multiplicities need refer to no enduring substance or essence, but is a structure or system that comes to be without referring to any prior identities. In this regard, Deleuze attributes three properties to Ideas or virtual multiplicities: 1) they are undetermined in relation to themselves (dy is literally nothing in relation to y, just as dx is literally nothing in relation to x), 2) they are determinable in relation to one another (dy/dx generates a value), 3) and they bear an ideal of complete determination with regard to their potentialities.

From the standpoint of differential ontology, the importance of the differential and multiplicities is clear. What needs to be avoided is treating differences as identities. That is, we must avoid the temptation to think of difference as diversity (DR, 222), for a diversity is just a collection of self-identical atoms, which thereby returns us to the ontology of identity. Insofar as the differential is literally nothing in itself, it avoids this problem. However, insofar as differentials take on value in their reciprocal determination, we are given a principle describing the chance-governed emergence of order through various syntheses. If I've often felt disgruntled by talk of deterritorialization, then this is precisely because all too often it seems to ignore the question of how order emerges. Deterritorialization to what end? In response to what problem? In describing these multiplicities, Deleuze remarks that "'Multiplicity', which replaces the one no less than the multiple, is the true substantive, substance itself" (DR, 182). Multiplicity has come to replace the old substance that endured beneath accidental changes. It is substance as an effect of difference... An ironic substance when viewed through an Aristotlean or Cartesian lense.

In my previous entry I claimed that the death of God has two consequences: On the one hand, it follows from the death of God that identity must come second, that it must be a product or an effect. On the other hand, it follows from the death of God that all situations are local, that there is no global whole or totality; or, as Deleuze puts it, that the whole is open. The concept of multiplicity responds to both of these problems. In developing the concept of multiplicity, Deleuze characterizes multiplicities as "problems". Here Deleuze follows the Kant of the first critique, where the Ideas of reason are conceived as problems pertaining to how the disparate experiences generated by the synthesis of intuition and understanding are organized into a whole. We might think of the difference between reason and understanding in Kant as the difference between a scientific treatise (such as Newton's Principia or Hjelmslev's Prolegomena to a Theory of Language) where the aim is to discover the principles that govern a disparate set of phenomena and an encyclopedia that gives us disparate "facts" about the world without showing how they are interconnected.

As Hjelmslev describes the project of science, "A priori it would seem to be a generally valid thesis that for every process there is a corresponding system, by which the process can be analyzed and described by means of a limited number of premises. It must be assumed that any process can be analyzed into a limited number of elements recurring in various combinations. Then, on the basis of this analysis, it should be possible to order these elements into classes according to their possibilities of combination. And it should be further possible to set up a general and exhaustive calculus of the possible combinations... in the theory of which all events (possible combinations of elements) are foreseen and the conditions for their realization established" (Hjelmslev 1961, 9). Of course, the idea of system that Hjelmslev is here proposing is not that which we find in autopoietic theory, complexity theory, or systems theory. Rather, Hjelmslev's ideal of theory is closer to that of Newton's, where on the basis of a few principles of motion, three in all, he claims he is able to exhaustively account for all varieties of motion we see in the physical world around us (this, of course, would progressively be undermined with the emergence of thermodynamics and chemistry, where such processes are no longer reversible in the order of time). In this regard, the reason that the Ideas of reason in Kant are problematic is that they refer to the continuous problematic of thought in integrating or subsuming the disparate cases of experience under general principles in a total system that articulates how they all hold together. Since experience is infinite, the task of reason is therefore infinite or endlessly open (Kant refers to this as the "unrest of reason") and the problems of reasons are not problems solved once and for all, but continuous problems that condition the way mind relates to world (reason perpetually searches for the manner in which parts systematically belong to the whole, unlike the encyclopedia that is content simply to collect "facts" without seeking the more basic principles that underlie them).

For Kant the problematic Ideas of reason are an epistemological affair pertaining to how mind organizes its experience into a system; whereas for Deleuze Ideas or multiplicities aren't mental entities at all (though there are psychological multiplicities), but rather dimensions of being presiding over the genesis of individuals or beings. Deleuze retains the notion that Ideas or multiplicities are problematic, while wresting them from their epistemological interpretation in Kant. Thus Deleuze's account of problematic multiplicities is something of a synthesis between Kant, Plato, and Leibniz. From Kant Deleuze takes the notion that Ideas are problems. From Plato Deleuze takes the notion that Ideas are (that they aren't simply mental entities). And from Leibniz Deleuze takes the notion that Ideas are not abstract forms, but coincide with the individual or the actual like the inflection of a point on a curve (or a tangent to a curve).

If Ideas or multiplicities are problems, then this is because they pose the question of how to synthesize a differential field of relations and singularities in emerging an actuality or phenomenon. Thus, in a beautifully poetic passage from chapter 2 of Difference and Repetition, Deleuze writes, "What we call wheat is a contraction of the earth and humidity, and this contraction is both a contemplation and the auto-satisfaction of that contemplation. By its existence along, the lilly of the field sings the glory of the heavens, the goddesses and god-- in other words, the elments that it contemplates in contracting. What organism is not made of elements and cases of repetitition, of contemplated and contracted water, nitrogen, carbon, chlorides and sulphates, thereby intertwining all the habits of which it is composed" (DR, 75)? Drawing on Deleuze's language from differential calculus, the wheat and the lily are "integrations" or solutions of a differential field defined in terms of water, nitrogen, carbon, chlorides, sulphates, proteins, etc. and the singular points that emerge when these elements are brought together. The world of actuality, the world of phenomena, consists of solutions to problems posed when these various series are brought together. A multiplicity defines the problematic field through which these integrations or solutions occur, and are thus the real conditions for the phenomena of the world.

In actualizing or integrating this differential field-- what Deleuze refers to as the process of differenCiation, presiding over the genesis of qualities, species/kinds, and parts --identity is produced as an effect or product of these differences, which Deleuze refers to as "simulacra" by virtue of their status as effect and repetition. From the example of the lily and the wheat above, it is clear that problems or multiplicities are not forms of negativity that disappear in the solution, but rather insist in the actuality as virtual conditions with regard to which the actuality remains in communication. The lily remains in communication with sunshine, nitrogen, water, earth, etc. in a state of perpetual feedback, continuously actualizing itself over time. That is, identity is a continuous process that finds its limit in exhaustion. As Deleuze will write, "...the logical relation of causality is inseparable from a physical process of signalling, without which it would not be translated into action. By 'signal' we mean a system with orders of disparate size, endowed with elements of dissymmetry; by 'sign' we mean what happens within such a system, what flashes across the intervals when a communiation takes place between disaparates. The sign is indeed an effect, but an effect with two aspects: in one of these it express, qua sign, the productive dissymmetry; in the other it tends to cancel it" (DR, 20). An actualized entity just is such a signal-sign system, remaining in perpetual communication with the virtual multiplicity it actualizes, constituting its own elements and identity. Such is Deleuze's account of what Badiou refers to as the operation of the "count-for-one"... An operation that is immanent to being itself and energetic in character.

Yet how are we to discern the local nature of ontological situations, the manner in which there is no global, overarching global situation in which all local situations are embedded as if parts of a whole? In part the above passage already responds to this question. If series must be brought into relation to communicate in order for causality to occur, and if the resonance of series is always a matter of chance, of a throw of the dice, like Lacan's objet a where no series enjoys primacy of model over copy, then all we have are local situations without a total situation. Unlike Heidegger's analysis of Dasein where Dasein is always being-in-THE-world, Deleuze's simulacra or actualizations are always divergent, enjoying only local relations. Heidegger argues that in order to understand the being of Dasein we must understand the manner in which Dasein is a being-in-THE-world. By contrast, Deleuze's concept of multiplicity provides us with the principle of a new structuralism, of a new local ontology, that allows us to understand the immanent organization of a multiplicity without referring it to an embedding global space. As Delanda so beautifully puts it in relation to the mathematician Gauss, "...when Gauss began to tap into these differential resources, a curved two-dimension surface was studied using the old Cartesian method: the surface was embedded in three-dimensional space complete with its own fixed set of axes; then, using those axes, coordinates would be assigned to every point of the surface; finally, the geometric links between points determining the form of the surface would be expressed as algebraic relations between the numbers. But Gauss realized that the calculus, focusing as it does on infinitesimal points on the surface itself (that is, operating entirely with local information), allowed the study of the surface without any reference to a global embedding space. Basically, Gauss developed a method to implant the coordinate axes on the surface itself (that is, a method to implant the coordinate axes on the surface itself (that is, a method of 'coordinatizing' the surface) and, once points had been so translated into numbers, to use differential (not algebraic) equations to characterize their relations. As the mathematician and historian Morris Kline observes, by getting rid of the global embedding space and dealing with the surface through its own local properties, 'Gauss advanced the totally new concept that a surface is a space in itself'" (Intensive Science and Virtual Philosophy, 11-12).

According to the old Cartesian method, we can only outline the properties of a space by relativizing it to a global space in terms of which it is then mapped. By contrast, Gauss is able to explore a space in terms of its intrinsic metric and organization as a local space, without referencing it to a whole of which it is conceptualized as a part. Deleuze sometimes spoke of getting up behind an author and "creating a monster", as if his reading methodology somehow consisted of a distortion of the philosopher's or author's thought. Certainly there is a case to be made for this with regard to the process of deterritorialization, where something is wrested from a territory and reterritorialized upon a new territory, like the animal paw that is deterritorialized from the earth and reterritorialized on the branch. However, it seems to me that the more interesting aspect of Deleuze's approach to other philosophers and art is not so much his "monstrous creations" (I seldom find them particularly monstrous), but rather their Gaussian or Riemannian style, where he explores them in terms of their own internal organization and metric, without reducing them to something alien such as history, society, biology (reductivism), or the signifier. What we find in Deleuze's approach to phenomena is a Gaussian technique. For instance, take Deleuze's books on Cinema, his book on Francis Bacon, or his study of Sacher-Masoch. In the first instance, Deleuze's carefully separates cinema from narrative and the signifier, studying it in terms of its specific organization pertaining to the production of images. In the case of Bacon, Deleuze doesn't look for an underlying narrative or "meaning", but instead studies the manner in which Bacon composes and organizes his images and lines, both in terms of their production and actuality, so as to liberate a logic of sensation. Finally, in approaching Sacher-Masoch, Deleuze allows Sacher-Masoch's novels to speak for themselves in terms of their desire and relation to pain, stalwartly refusing to reduce Masochism to the complement of Sadism. In each case, we have a local exploration of a "space" of multiplicities that is extremely precise.

Along these lines, a Deleuzian spirit can be found in many domains of contemporary inquiry. Systems theory, autopoietic theory, cybernetics, structuralism, contemporary quantum mechanics, and post-Darwinian evolutionary theory can all be seen as studying systems that both constitute their own elements or produce identity as an effect, and which study the local organizations of multiplicities without referring to a global space of resemblances. Thus, for instance, the Lacanian clinic engages with the immanent organization of an analysand's unconscious, without referring it to a global set of diagnostic categories, seeking to determine how this analysand's symptom here is organized, rather than seeing the symptom as an instance of a kind such as we find in Anglo-American approaches to psychotherapy where the patient's symptom is referred to a set of diagnostic categories (thereby remaining in the heritage of Linnaeus' biology). Deleuze provides the ontology proper to these "structuralisms", unfolding both the ontological status of these "ideal" systems of relation, but also the manner in which these systems come to be actualized in the phenomena we see about us.

Rather than talk of deterritorialization and becoming-animal, it is this rigorous and careful analysis of multiplicities and their becomings that I would like to see more of. As Deleuze writes in Difference and Repetition, "Ideas are varieties which include in themselves sub-varieties. We can distinguish three dimensions of variety. In the first, vertical dimension we can distinguish ordinal varieties according to the nature of the elements and the differential relations: for example, mathematical, mathematico-physical, chemical, biological, psychical, sociological and linguistic Ideas... Each level implies differentials of a different dialectical 'order', but the elements of one order can pass over into those of another under new relations, either by being dissolved in the larger superior order or by being reflected in the inferior order. In the second, horizontal dimension we can distinguish characteristic varieties corresponding to the degrees of a differential relation within a given order, and to the distribution of singular points for each degree (such as the equation for conic sections which gives according to the case an ellipse, a hyperbola, a parabola or a straight line; or the varieties of animal ordered from the point of view of unity of composition; or the varieties of language ordered from the point of view of their phonological system). Finally, in depth we can distinguish axiomatic varieties which determine a common axiom for differential relations of a different order, on condition that this axiom itself coincides with a third-order differential relation (for example, the addition of real numbers and the composition of displacements; or, in an altogether different domain, the weaving speech practised by the Graiule Dogons)" (DR, 187). It is in the analysis of these multiplicities and their relations that real advances in learning are to be made and the real possibility of social and political change occurs. Yet this requires rigor and precision, not the haphazard formation of "rhizomes" like we see in so many celebrations of Deleuze and Guattari. Of particular interest here are the relations among different orders of Ideas or multiplicities, such as the relation between psychological and sociological ideas described by Niklas Luhmann in Social Systems, where the psychic system is structurally coupled with the sociological system to form systems of communication such that individuals cannot be said to belong to society but occur in the environoment of society. Here we have an instance of the social system constituting its own elements by using the elements of another system.

However, it is perhaps above all of importance to analyze the genesis of false problems and the poor solutions that accompany them. Too little of this has been done, as there has been a tendency to adopt Deleuze's ontological theses about affirmation tout court, without loooking at the negative as an effect that plagues us at the level of the actual. As Deleuze writes in relation to Marx, "The famous phrase of the Contribution to the Critique of Political Economy, 'mankind always sets itself only such tasks as it can solve', does not mean that the problems are only apparent or that they are already solved, but, on the contrary, that the economic conditions of a problem determine or give rise to the manner in which it finds a solution within the framework of real relations of the society. Not that the observer can draw the least optimism from this, for these 'solutions' may involve stupidity or cruelty, the horror of war or 'the solution of the Jewish problem'. More precisely, the solution is always that which a society deserves or gives rise to as a consequence of the manner in which, given its real relations, it is able to pose the problems set within it and to it by the differential relations it incarnates" (DR, 186). Seldom do we see any discussion of these types of "solutions", yet one of the ancient tasks of philosophy inaugurated by Epicurus, Lucretius, Epictetus, Machiavelli, and Spinoza was the diagnosis of humanity's superstititions and false passions, so as to rescue us from sad passions and poor solutions. In the same space as this passage, Deleuze references Althusser who discovers a properly "static" reading of Marx, rescuing it from its historicism, and analyzing the multiplicities that underlie actual social relations. Should not the name Althusser lead us to evoke the analysis of ideology as a necessary moment in the formulation of genuine problems? And does this not lead us to look at the relationship between Zizek, Lacan, and Deleuze (despite Zizek's Hegelianism which is questionable) in a different light as part of the ongoing critical project of diagnosing and overcoming superstitition? Need we necessarily see a strict opposition between the approaches of Althusser and Zizek, and those of Foucault and Bourdieu? Or rather, are the tools these thinkers give us all tools that allow us to examine the organization of Ideas or multiplicities at different levels of granularity, at different telescopings of Gaussian space?