In their famous text A Thousand Plateaus1, Deleuze and Guattari make the distinction between what they call smooth and striated space. The latter is a space that is segmented and divided into various territorialities, while the former is a space that is unstratified. In smooth space various lines can freely explore and perhaps realize their creative potentialities. This distinction between spaces is derived from the distinction between the nomadic and the sedentary. The goal of the nomad is to occupy a smooth space, a space in which there are no clear cut restrictions on exploring one area or another. There are forces sure, but these forces are not conceptualized forms. The smooth space forms a sum of flows, similar to Deleuze’s theory of forces in Nietzsche and Philosophy2. A force here can explore its own potentialities, it has no “slave”-esque restriction upon forces. The striated space by contrast constantly runs into categories, into territorialities, which segment the space. Lines, forces, affirmations, what have you, are always constrained through their representation. Deleuze and Guattari compare the difference between the two forms of space to the difference between a Riemann space and a Euclidean Space. Euclidean space was designed to model physical reality through our experience of three-dimensional space. As such the space is constrained by a wide variety of properties that make it practical for basic geometry and modeling. A Riemann space by contrast is based on a manifold that is defined by the smoothness of its tangent space, defined by the continuous presence of derivatives. Regardless of the mathematical jargon used, this means that a Riemann space has an infinite potentiality of connections to be made between points, it has the possibility of producing a rhizome. Riemann geometry is the mathematical representation of smooth space, of rhizomatics. Einstein famously used Riemann’s manifolds in his theory of general relativity, which distorts the classical Newtonian account of forces. Forces in Einstein can be distorted, warped, by various properties of spacetime. Riemann and Einstein in a sense “smoothed out” their fields, producing potentialities once thought impossible.
Deleuze frequently referred to extra-philosophical concepts within mathematics, what he and Guattari would call functors. These concepts are functors because the concepts are inherently tied to the discourses of math and science, they are used for their function rather than for creativity. They are productivist, pragmatic concepts, rather than transcendental ones. Through his use of these functors on philosophical grounds they become concepts and something resembling conceptual mathematics might become possible, a mathematics not constrained to pure rigor but allowed to freely explore and create. One mathematical functor Deleuze makes great use of are differentials. He particularly uses them in Difference and Repetition3 to ground his philosophy of difference. The differential to Deleuze is a mathematical development that mirrors his own philosophy of pre-conceptual difference. In classical mathematics difference is conceived through the relationship between two entities: a y/x. Yet through the differential these stable differences relying upon identities and reparation become displaced, as the moment of difference becomes conceivable through the process of the limit. A derivative, the relationship dy/dx, creates the grounding of the conceptual difference of identities, as it determines the rate of change of the entire relationship. The classical idea of difference is thus reversed, the differential mirroring the potentiality of difference prior to its repetition. Deleuze uses this smoothing out of difference, outside of its striated space of conceptual difference, to ground a Bergsonian account of time. In Bergson there is no such thing as the moment, the event, instead there is only flow. The flow of time through a succession of representations, each relying on other “moments” or “events” to ground themselves. To Deleuze this Bergsonian account of time is grounded by his philosophy of the differential, as the time defined by the succession of transcendent events is determined by their smooth repetition of difference. His view of time is transcendental as it is defined through repetition, reflecting his metaphysical goal of filling in the gaps between affects. Bergsonian time is thus a transcendental empiricist account of time, a smooth time. Land uses this theory of time to ground his theory of accelerationism. Accelerationism is the smoothing out of historical time, not merely the subjective time of experience and affect. Time is perpetually haunted by capital, which manifests its becoming-real in all moments and replaces humanity and modernist will as the object of history. There is only one way out to the accelerationist, through going through capital’s inherent deterritorial tendency. Depending on the accelerationism either capital’s alienation itself or a new subjective agency is from there found and flattened to a flatline. Accelerationism and its associated Bergsonian smoothness is the basis for the cybernetic control found in our current age. Despite the radical potentialities that figures such as Plant4 and the early Fisher5 find in transcendental time, an inter-subjective smoothness can only be found in the singularity itself, outside the smooth plane that constitutes the basis of cybernetics and acceleration.
An outside to smoothness that does not venture back into a striated space can perhaps be found through the mathematical object of the fractal. The fractal operates in direct opposition to the smoothness of Deleuze’s metaphysics. In his references to mathematics and physics, Deleuze assumes that the base ontology behind repetition and change is fundamentally smooth. Each affect can pass from one to the next in a differentiable and continuous fashion. Only through this smoothness can a transcendent time be grounded, as only through it can affects and moments clearly relate to each other and ground a coherent experience. Of course this smoothness allows a coherent creativity to emerge, but a creativity of what? Is it a personal creativity or is it the inherent schizophrenic creativity of capital? Fractal space contests this smoothness and instead posits a space formed by non-virtual singularities. These singularities are non-virtual because they are not smoothed out into a plane, instead they are particular expressions in themselves. The fractal space of singularities mirrors the mathematical functor of the fractal as it is itself infinitely jagged and complex, it is comprised of singularities that cannot be reduced to a smooth relationship. It is neither continuous nor differentiable if one is to speak in the terms of calculus, instead it is porous and jagged. The fractal space offers a third space to the dichotomy of smooth and striated, as it displaces both virtual smoothness and the conceptual actuality of striation. Instead it offers an account of pure immanence, offering a basis in the life of singularities.
Fractal space offers a before to the supposedly pre-conceptual space of smoothness and the plane of immanence. Of course the plane of immanence offers a preconceptual space, as the intensities of the plane form concepts and constitute a plane of transcendence. Yet the plane of immanence has a prior conception of form, through the metaphysics of smoothness and flattening. Fractal space has no such prior form, it merely has the singularities that occupy the space. The labeling of the space as “fractal” is somewhat misleading, as the singularities do not have to resemble a fractal, in fact it can resemble planes of immanence, transcendence, or any other ontological form. Instead it operates only upon a lack of assumption of smoothness or striation at the basis of ontology, instead establishing a pure immanence of non-virtual singularities. Fractal space thus is directly related to Deleuze’s idea of pure immanence as a life, established in his aptly titled Pure Immanence: A Life6. Pure immanence to Deleuze is prior to its own virtuality through the plane, instead self-deconstructing his own Spinozism through placing immanence immanent to itself. Thus there can be immanence and monism without substance, forming what Deleuze calls a life. A life is not grounded on the transcendent subject of the phenomenologists, but is instead the totality of what is between affects. A life is the object of transcendental empiricism, what it seeks to discover and explore through its creative processes. Fractal space is thus the exploration of life itself, of being-as-itself and its connotation. The object of fractal space is the pure immanence of a life, yet it does not assume this pure immanence will form a virtual plane. The object of liberatory philosophy and politics is to form a liberated plane, yet fractal space does not operate upon this assumption of becoming-smooth. Instead it allows singularities to be and to become only as ends in themselves, rather than constituting flows. The movement of becoming between the three spaces goes as follows. The singularities of fractal space become virtual and constitute a plane. From there this plane actualizes singularities into concrete striated concepts. Here striated space begins through the formation of a plane of transcendence.
Deleuze and Guattari once famously declared “Never believe that a smooth space will suffice to save us.”7 Smoothness, despite its potential for radicality in the form of the nomadic war-machine, is also the basis for cybernetic control. This is not at the fault of Deleuze’s metaphysics, as many claim, as Deleuze is throughout his work aware of how apparatuses capture nomadic forces. Regardless it has become clear that a smooth nomadism can now only be expressed in a Baudrillardian reversibility, which assumes spectacle’s becoming-real and can only navigate capital through an accelerationist form of exit. Instead a proper nomadism, a proper war-machine, can only be founded upon a fractal politics. The fractal through its displacement of rhizomatic smoothness creates the potentiality for situations and outsides by which a proper overcoming of the current state of things can be grounded. Fractal politics allows for the possibility of presence through the collapse of its transcendent representation. Forms-of-life has the possibility to be and become outside of its own categorization and striation. The singularity here is the inverse of Badiou’s political event, which is expressed within transcendent political discourse. Instead the singularity is a being-as-itself, a being with the potential to be outside the dominant discourse. Here situations can be found, riots and insurrections can be given concrete basis, and the free exploration of life can occur. Through fractal politics a becoming-virtual can occur, where these singularities and situations form planes of communication that can form new futures. An inter-subjective plane can result in which desires, forms-of-life, and connections can become ends in themselves, outside of their own representation and reduction. Here subjectivity is able to explore its own potentialities, to establish a nomadism that is fundamentally its own. Fractal politics allows one to separate subjectivity and singularity from its control through cybernetics, to posit that one is in control of subjectivities production. Through it perhaps the creative nothing, insurrection, ownness, and true unmediated connection can be rediscovered.
Gilles Deleuze, Félix Guattari 2014. A Thousand Plateaus : Capitalism and Schizophrenia. Minneapolis; London: University Of Minnesota Press.
Gilles Deleuze. 2013. Nietzsche and Philosophy. London ; New York: Bloomsbury.
Deleuze, Gilles. 2014. Difference and Repetition. Translated by Paul Patton. London: Bloomsbury.
Plant, Sadie. 1998. Zeros and Ones: Digital Women and the New Technoculture. London Fourth Estate
Fisher, Mark. 2018. Flatline Constructs : Gothic Materialism and Cybernetic Theory-Fiction. New York, New York: Exmilitary Press.
Gilles Deleuze, John Rajchman, and Anne Boyman. 2012. Pure Immanence: Essays on a Life. New York: Zone Books.
Gilles Deleuze, Félix Guattari 2014. A Thousand Plateaus : Capitalism and Schizophrenia. Minneapolis; London: University Of Minnesota Press.