Since its discovery more than thirty years ago, interest in the “pathological” mathematical object known as the Mandelbrot set (M-set) has steadily grown. This can be attributed both to its compelling graphic qualities,1 and because it allows researchers to observe the self-similar, or “fractal” nature of problems as encountered in a wide range of disciplines. The M-set (and fractal geometry, of which it is the paradigmatic example) has made the quality of roughness quantifiable, and has helped us to re-conceptualize the nature of a ‘boundary,’ bringing into relation objects and processes as disparate as galaxies and Brownian motion, bronchia and market fluctuations. Perhaps for this reason alone it deserves to be recognized as one of the most significant discoveries of the 20th century, but reinforcing this view is the fact that the M-set, avowedly the most complex mathematical object ever discovered, also provides us insights into how we might structure our posthuman ontology. For as the computational revolution continues to reveal ever finer-grained details of phenomena at all scales of organization it becomes possible and even imperative that we begin to imagine a reality defined not by a succession of ever-smaller and simpler objects, as is true of the reigning positivist ontology, but by a continuum of complexity in which the latter is understood not to diminish with scale and in which the human no longer occupies the privileged position by which such determinations are made. In a fractal ontology, scale and complexity take on a relative rather than absolute cast, foregrounding in the process the agentic capacities of other entities. The M-set, then, should be numbered among the most important discoveries of the last century, yes, because it allows us to understand at least one facet of the relationship between quality and quantity, the affective and the effective, but also because it makes legible the paradox the human now confronts: that its survival may well depend on its ability to remove itself from the center of its own ontological constructions and to integrate itself within an indefinitely complex continuum.
Even with the M-set as a guide, this will be no easy feat. Indeed, so foreign is the above concept to Western rationality that it is one which we might imagine could only be conceived outside of its orienting frame; a logic so improbable that it could only be the product of an alien intelligence. But in reality, evidence for the possible cogency of a fractal ontology has arisen, paradoxically, from an intelligence which happens to represent the epitome of analytical thought, that of the digital computer. Once the latter’s Boolean outputs were coupled with those of the cathode ray tube and high-powered graphics software, the human was provided a window onto an entirely novel ontological terrain, one to which Benoit Mandelbrot, an IBM researcher for 35 years, was among the first to have access. The digital tools provided him by his job allowed him to recognize early on the power of employing computer-generated graphics to solve seemingly intractable mathematical problems. Being familiar with the work of Pierre Fatou and Gaston Julia on the iteration of rational functions, Mandelbrot was also primed to recognize statistical resonances between problems in such disparate disciplines as information theory, linguistics, economics, mathematics and physics. But without the ability to model these resonant patterns on the computer screen even he, by his own admission, would have been unable to identify their common geometry.
In the last 10 years, continued improvements in digital technology have pushed philosophers to begin to engage with some of the more radical implications of the fractal geometry Mandelbrot popularized, including the hypothesis of an infinitely complex, and expanding, universe. Self-described “object-oriented ontologists” have recently espoused what they call a “flat ontology,” or the radically non-anthropocentric concept that all entities/objects participate equally in the space of relations regardless of the scale at which they operate. For philosopher Manuel Delanda this means that “institutional organizations, urban centres or nation states are…not abstract totalities but concrete social individuals, with the same ontological status as individual human beings but operating at larger spatio-temporal scales.”2 All entities in his scheme comprise “component parts of species, much as individual cells are parts of the organisms themselves, so that cells, organisms and species form a nested set of individuals at different spatial scales."3 The difference between a flat ontology and the reigning positivist model should be clear: where the latter assigns agentic capacities to entities such as nations, institutions, hurricanes, etc. only metaphorically, as a rhetorical trope, the former assigns them literally such that a cell and a nation-state are able to enter into relations with other objects on an equal footing. Unfortunately, though this ontology appears to present a fractal model of object/entity relations, it still embraces an absolute concept of complexity such that the latter reduces with scale. As with the positivist model it understands complexity and specificity to inhere in an object according to the position it holds within a scalar hierarchy. Though seemingly opposed to each other at a fundamental level, positivist and flat ontologies are more alike than not, agreeing, additionally, on such critical philosophical tenets as the possibility of the observer being isolated from that which he observes and on the status of the bounded object as the primary ontological unit.
Characteristics of a Fractal Ontology
By contrast, I believe the Mandelbrot set implies an ontology radically different from either of the above. It evokes one characterized by a self-similar, scale-free and, thus, truly fractal architecture, one in which: complexity and specificity are deemed to be relational rather than essential qualities of the object and to have no lower (or upper) bound; objects are understood to be at once unities and multiplicities, diffractionally enfolded in one another and thus always linked, however tenuously;4 and objects are at once entities, processes, events, or non-existent relative to the other objects which perturb, constitute, or pass through them, the determination being dependent on the speed/scale at which each operates. Thus an acorn is at once non-existent from the perspective of a fast-moving neutrino, an event for the ant on whose leg it lands, a process for the morphogenetically-linked molecules which participate in its unfolding and an object for the squirrel who buries it in anticipation of the coming winter.
Reality as revealed by the computational revolution, then, has much in common with the concept of “nature” as defined by philosopher and ecocritic Timothy Morton; it is more than simply one’s “environment,” more than something one can point to “over there.” Instead, it is that which “…flickers between things – it is both/and or neither/nor…it is both the set and the contents of the set. It is the world and entities in that world. It appears like a ghost at the never-arriving end of an infinite series....”5, And, I would add, it is a reality in which parts and wholes can contain each other, where scale-invariance is the norm and where going forward means returning to the spot (or one nearly identical to) where one began.6 It is the world that begins where analytical reason ends or begins to blur into the metaphysical as the digital economy makes legible ever finer-grained details of reality’s constituent parts. What non-linearity and Morton both point us to, then, is a fractal ontology derived from combining two seemingly paradoxical stances with respect to the status of objects/entities; a radical perspectivism in which the qualities of an object are as manifold as the objects which make it up and/or prehend it (such that it might be thought of as more pattern than substance); and in architecture theorist Sanford Kwinter’s terms, an “immanent transcendental” such that “‘things,’ phenomena, though sundered from the metaphysical structure that grounds them in ‘meaning,’ now find their principle of being nowhere else but within themselves.”7 The first is evocative of Leibniz’s Monadology, though without the teleological pull of the “unmoved mover” and attendant perfection hierarchy and with diffracted rather than reflected monads,8 while the second resonates with object-oriented ontologist Graham Harman’s withdrawn object9 – with the exception that the object’s withdrawn interior, like that of the Mandelbrot set (described below), never achieves closure and describes not a true object but at best an invagination or fold in the line that circumscribes its boundary.10 Thus we are presented with an ontology in which objects are both insubstantial quasi-objects, defined not by their contents or essential qualities, but, as with a text, by how other quasi-objects interpret them; yet they are also quasi-closed systems, worlds unto themselves in as much as any object, including the universe, can be said to exist independently of all others.
This makes it an obscure ontology in that, unlike the reigning positivist model, its elements, consisting as they do of the relationships between the real and the virtual, or what can and cannot be quantitatively assessed, largely escape attempts to model or otherwise visualize them. This puts those who would popularize such a fractal ontology at a distinct disadvantage, one only reinforced by the fact that the geometric optics of vision, or that faculty most trusted as an arbiter of truth, tends to reify the ideal, closed forms of Euclidian geometry which ground the positivist ontology. Collectively, vision, ideal geometric forms and the positivist ontology form a closed, self-reinforcing system, one which has, over time, become naturalized within the material-discursive environment. Challenging the positivist, self-reinforcing loop that results from this reification then requires we articulate the ways in which how we see plays a vital, if largely overlooked, role in determining what we see, an articulation scholars such as Alva Noë and Andy Clark have been pursuing to great effect.
Complementing the work of the above scientists has been that of mathematicians such as Mandelbrot who have employed the computer to aid them in their explorations of formal relationships. Arising from their efforts have been images that have guided those who would understand the dynamic between the mediators of sight (curved lens, stereopsis, orthogonal framing) and the material-discursive forms (atomized individual, social hierarchy, ordering grid) which are regularly reified by the latter’s operation. Fractal geometry, for example, has provided imagery which appears to occupy the neutral ground between the Real and the Ideal, making possible the contemplation of alternatives to the above naturalized, discursive structures even as it opens up new, fractional dimensions11 between seemingly bounded entities. With its ability to capture the finest, formal details of everything from clouds, mountains, ferns and lightning bolts to the behaviors of heart muscles, neurons, intestinal filtering systems and protein growth,12 fractal geometry unites exterior and interior topologies. Perhaps even more surprising, however, is that the ratios common to all of the above “natural” objects/entities are also to be found in such cultural phenomena as market fluctuations and relative word usage in texts thus making a continuum of these once formerly opposed taxonomic categories. It begins to make it possible to conceive of the human existing of nature rather than simply in nature. The most graphic communication of this surprising result is to be found, of course, in the Mandelbrot set, itself. Being the paradigmatic fractal, it best conveys the complexities of our otherwise all-but-impossible-to-schematize material-discursive environment.
Properties of the Mandelbrot Set
Many readers are, no doubt, already familiar with the image of the Mandelbrot set – a fuzzy, (typically) black, insect-like shape outlined by brightly-colored, contoured bands. While originally presented as a static image, ever-more detailed and multi-dimensional renderings of the M-set are regularly produced in conjunction with improvements in computing technologies.13 The dark interior of the set represents all numbers in the complex plane that remain within a circle of radius 2 and do not grow exponentially when iterated in the function Z= z² + c, while the colorful contours which constitute the set’s exterior indicate the rate at which the output of successive iterations moves towards infinity. The result is a rather peculiar object – a finite area paradoxically enclosed by an infinite perimeter. The ever-receding, filigreed vanishing point familiar to anyone who has experienced a deep zoom into the M-set is then a trip down (or up?) into the infinite boundary separating the set’s interior from its exterior.14 However, because this boundary (being infinite) never attains closure, one can never ascertain that such a division truly exists, all apparent indications to the contrary; that is, unlike with classical geometric forms such as a circle or square, one can never extricate the set from its surrounding topography because the co-constitutive entwinings between inside and outside extend indefinitely into the infinite reaches of the complex plane constituting an unimaginably intricate example of a plane-filling, one-dimensional line (as the line never meets itself). In this sense the boundary of the M-set resembles other ambiguous divisions: “The taut skin of the drum, even of the eardrum, separates inside from outside like a margin, and gives rise to resonant sound when struck...Is this drum, this margin, part of the inside or the outside?”15
It should be clear that the implications for our ontological models would be profound were it to be demonstrated that this condition of non-closure applied not just to the M-set but to objects more generally; that is, that the closure of objects as presented in deterministic models of relations (and to the eye) is partially or wholly illusory – a harmless heuristic, at best, a gross distortion of reality at worst. And, in fact, the computational revolution has repeatedly brought to light evidence demonstrating precisely this, that material reality is permeated by fractal structures and processes such that fractals have been described as “the ‘signature’ of dynamical processes at work,”16 essential to descriptions of everything from the random movement of particles suspended in a fluid17 to the structure of interstellar gas.18 This then should give those who employ idealized, atomistic objects in their models and theories reason to doubt the efficacy of the latter, as, indeed, increasingly appears to be the case in disciplines as varied as economics19 and medicine. Health care researchers, for example, have discovered that physiological “dis-orders” are associated not with a lack of order but rather with too much:
The antithesis of a scale-free (fractal) system (i.e., one with multiple scales) is one that is dominated by a single frequency or scale. A system that has only one dominant scale becomes especially easy to recognize and characterize because such a system is by definition periodic - it repeats its behavior in a highly predictable (regular) pattern. The theory underlying this prediction may account for a clinical paradox: namely, that a wide range of illnesses are associated with markedly periodic (regular) behavior even though the disease states themselves are commonly termed "dis-orders"…Findings from nonlinear dynamics have also challenged conventional mechanisms of physiological control based on classical homeostasis, which presumes that healthy systems seek to attain a constant steady state. In contrast, nonlinear systems with fractal dynamics, such as the neuroautonomic mechanisms regulating heart rate variability, behave as if they were driven far from equilibrium under basal conditions. This kind of complex variability, rather than a regular homeostatic steady state, appears to define the free-running function of many biological systems [italics in original].20
Scholars in the humanities and social sciences who explore issues of recursivity and affect implicitly recognize the limitations of the atomistic model as do scientists who specialize in non-linear dynamical systems and/or quantum mechanics. Cognitive scientist Andy Clark, sociologist Hannah Landecker, cultural theorist Sarah Ahmed, theoretical physicist Karen Barad and economist Yanis Varoufakis are just a few of the scholars who are trying to understand and model the interface between inside and outside with the understanding that the two are co-constitutive and thus reflexively linked. This aspect of the M-set reinforces the belief held by many who work in these and related disciplines that an absolute division between observer and observed, subject and object can, at best, be approached only asymptotically rather than established absolutely, a point at variance with the beliefs of positivists and object-oriented philosophers alike (though the latter, at least, have posited the novel concept of an inexhaustibly creative, withdrawn interior). While this is not a new concept, with, among others, Henri Bergson arguing over a century ago for recognition of the porosity of bodies,21 it is one which is only now gaining real traction as a result of the computational revolution. The non-closure aspect of the M-set, as revealed by the new technologies, thus has the potential to help us think, or, rather, feel our way through some of the paradoxes that confront anyone who would try to model either objects or their inter-relations.
But the M-set, in addition to this property of non-closure, has other distinguishing features which help us better understand the peculiarities of the posthuman, material-discursive topology. Though derived from a simple equation requiring nothing more than multiplication and addition, the M-set extends to infinity without ever duplicating any part of itself. As such, it is a graphic example of Deleuze’s “self-differentiating difference” or that which “unfolds itself and thereby creates a universe.”22 Like the whole in which it is contained, each “object” within the set is unique and acts as a potential portal into unexplored realms. At the same time, each is a multitude made up of patterns which repeat at all different scales such that there are patterns, and patterns of patterns, and so on, including periodic repetitions of the insect-like “interior” of the set (though never are these repetitions identical to the original).23 Self-similar at all scales, every part can then be said to contain the whole even as it is contained by a whole which can never be fully encompassed. This quality of the set, like that of its non-closure, also has implications for our ontological models, as it allows us to see how it might be possible for parts to contain wholes as readily as wholes contain parts. This is what I refer to as the “diffractional”24 aspect of the M-set, which I liken to Karen Barad’s call for a move away from “common-sense” representationalism and towards performativity.25 We have already seen how the M-set challenges what Barad refers to as “one of representationalism’s fundamental metaphysical assumptions: the view that the world is composed of individual entities with separately determinate properties.”26 It is precisely this “assumption of thingness”27 and its association with the real that must be destabilized if we are to move beyond jejune ontological debates such as those which take place between, for example, scientific realists and social constructivists. Both camps adhere to an ontological model premised on the idea that representations of scientific knowledge “mediate our access to the material world.”28 However, contrary to what Barad refers to as Descartes’ “asymmetrical faith in word over world,”29 there is no reason to assume that we have greater access to our thoughts about things, and the images we use to represent them, than we have to the things themselves. One does not precede the other but indicates the on-going process of subjectification, a process aptly conveyed by Whitehead in his theory of concrescence, where the “throb of feeling is not perceived by a subject as such but rather constitutes the actual occasion out of which the distinction between subject and object emerges.”30
Another key feature of the M-set and the last I will review here is the insight it provides into the inter-relationships among scale, complexity and locatedness of the viewer. Scale (in the sense of organizational level) and complexity are not absolute but relative distinctions made by an observer located at a certain scalar level. Leibniz was perhaps the first to make this point explicit:
Each portion of matter may be conceived as like a garden full of plants and like a pond full of fishes. But each branch of every plant, each member of every animal, each drop of its liquid parts is also some such garden or pond…Thus there is nothing fallow, nothing sterile, nothing dead in the universe, no chaos, no confusion save in appearance, somewhat as it might appear to be in a pond at a distance, in which one would see a confused movement and, as it were, a swarming of fish in the pond, without separately distinguishing the fish themselves.31
Ruyer, Simondon,32 Serres and Tarde33 also suggest, to one degree or another, that it is not complexity which diminishes with scale but rather our ability to perceive the inner-workings of entities/events operating at speeds and timeframes radically removed from our own. Biologist Stanley Salthe, arguing along the same lines, constructs a thought experiment:
…imagine how a human being might appear to a higher-level entity like a city. Since the cogent moment of the higher-level individual is considerably longer than that of the lower-level entity, the latter would be splayed out over a larger level moment so that a human being might appear like an amoeba with its center at, e.g., its bed and with its pseudopods being habitual trajectories.34
Where Salthe likens the relationship existing between the individual and the city to that which exists between the amoeba and the individual, a scale-invariant model such as the M-set (or, to some degree, Leibniz’s Monadology) suggests this relationship be understood to play out ad infinitum. Classical notions of complexity, time/scale and perspective work against this understanding, however, collectively creating the impression of a hierarchically organized natural world in which the small and quick have fewer parts and are thus less complex than the large and slow. But technologies such as those which make possible the experience of virtual reality and the M-set allow us to think, visualize and feel reality as a continuum rather than (or in addition to) a hierarchy, one in which parts and wholes are understood to extend indefinitely without diminishment.35
Visualizing Paradox
The M-set is an odd sort of map, then, in that it can help determine neither origin nor direction, neither progress nor goal. It can’t even provide the sense of control typically imparted by maps, as the bird’s-eye view it presents demarcates not a static terrain but an environment that might be said to move through the observer even as she moves through it. It is a map of interpenetrating parts and wholes, making legible in Bergson’s terms “life’s own domain…reciprocal interpenetration, endlessly continued creation.”36 As such, it cannot help her distinguish inside from outside because, as it indicates, the patterns of which these domains consist are self-enfolding and self-similar “all the way down.” It provides her instead with an image of a “topology of multiplicities,” one which marks “the end of dialectics”37 as its symmetry is radial rather than bilateral, always moving out from the center towards the observer as the latter moves towards it. In this respect, it is a deeply disorienting map, and, hence, one I believe perfectly suited to helping us navigate the turbulence of an era defined by the collapse of meta-narratives wedded to static frames of reference.
With the outputs from its algorithm constituting a hybrid of both real and imaginary numbers,38 the M-set is the anti-meta-narrative, incapable of helping viewers/readers determine even the actual from the virtual. Instead, it invites readers/viewers to explore a realm somewhere in-between, a realm in which reductionism and ever-greater powers of magnification lead not to simplification but only to an ever-receding vanishing point of complexity and novelty. The more one chases after this vanishing point the more destabilized one’s own inertial frame becomes. In the terms philosopher Keith Ansell-Pearson articulates, the M-set might be said to enact a map of the “non-chronological time that constitutes our interiority.”39 Viewers of a deep zoom into its boundary can find it difficult to decide at times whether it is they who are moving or the surrounding environment. Exploring the territory described by this map, in other words, is profoundly disorienting, not only because, being self-similar at all scales, it is devoid of absolute reference points, but because one becomes implicated within its infinite folds. To arrive at one’s goal, perhaps one of the “alluring”40 quasi-objects that appear off in the “distance,” is to encounter not one’s telos but only the interplay of a figure and ground traced from a single line, one which includes the viewer as well. This evokes the sense that it is not only the set’s boundary which is infinite but one’s own, a feeling which leads science fiction and science writer Arthur C. Clarke to speculate that “there is some structure…deep inside the human mind [and, I would add, body] that resonates to the patterns of the M-set.”41 That neuroscientists, among others, have recently confirmed this to be the case should then come as no surprise to those with personal experience of this “map.”42
So then, one might ask, of what value is this “map”? From the perspective of someone who might be setting out on a journey with all ports-of-call pre-determined, perhaps none at all. Navigating such an itinerary, or one laid out by a traveler more interested in what anthropologist Tim Ingold refers to as “transport” rather than “wayfaring,”43 requires little more than a point of origin and a small-scale representation of the terrain to be traversed. Always anticipating his next move, the transported traveler is content to skim the spatiotemporal surface, never becoming fully present to the places through which he passes. Transforming space into place would require he allow the spaces through which he moves to move through and transform him, a double movement that might inconveniently challenge the illusion, reinforced by his map, that he is a bounded entity navigating a fixed geometry of absolute time and space. The M-set as map, by contrast, illuminates and even enacts, when set in motion, the geometry of this double movement. It illustrates how one’s entanglements at the macro level are not only mirrored by, but are enfolded within, those at the micro level and vice versa, with only the temporal/scalar dimension intervening to separate and make them legible.
…consequently every body feels the effect of all that takes place in the universe, so that he who sees all might read in each what is happening everywhere, and even what has happened or shall happen, observing in the present that which is far off as well in time as in place: sympnoia panta, as Hippocrates said. But a soul can read in itself only that which is there represented distinctly; it cannot all at once unroll everything that is enfolded in it, for its complexity is infinite.44
The M-set, of course, obviates the need for Leibniz’s concept of “he who sees all,” as its infinite recursivity and scale-invariance illuminate how it might be possible for parts and wholes to relate to each other non-hierarchically. The downside, however, is that it has little to say about points of arrival and departure, finding the shortest distance between two points, or judging progress along a pre-planned route, making it comparatively useless to the transported traveler.
On the other hand, it is potentially invaluable to Ingold’s “wayfarer,” as its meanderings help her make sense of her own, the course of her journey and that of the M-set being similarly determined by (what are perceived to be) “chance” encounters. The wayfarer recognizes that the serpentine contours of their twinned paths derive from the accretion of unpredictable, iterative outputs from a feedback-looped, eventive “anarchitecture.”45 What the next stop in the itinerary of either will be can never be known in advance, only discovered by playing out events in real time. Both exhibit in Barad’s terms a performative aspect akin to Judith Butler’s concept of “gendering… an iterated doing through which subjects come into being.”46 While in the short term the incremental movements of both the M-set and the wayfarer seem disconnected and haphazard, they take on meaning over time as patterns linking individual events or iterations begin to coalesce. The meanderings of both the wayfarer and the M-set then confirm Leibniz’s earlier suggestion that determinations of chaos and order (and hence meaning) are not inherent qualities of the object or system but shift in relation to the distance – temporal, spatial, scalar – between observer and observed.
The value of the M-set as a map, then, is manifold. While it might not have much to tell us about moving from point A to point B, it graphically illustrates some of the more paradoxical aspects of the fractal ontology that technology is making legible. It allows us to visualize how it is possible for the closure of objects to be not absolute but scale- (and hence speed-) dependent; for a simple formula to generate both an infinity of novel instances and self-similar patterns; for parts to contain wholes as readily as wholes contain parts; and for binaries, such as inside/outside, order/chaos, figure/ground, to indicate not an essential quality of the object or system but rather to be descriptive of the relationship between observer and observed, neither of which can be said to exist independently of the other. As Morton observes, “There is a Buddhist saying that reality is ‘not one and not two.’ Dualistic interpretations are highly dubious. But so are monist ones. There is no (single, independent, lasting) ‘thing’ underneath the dualist concept.”47 The M-set then, like the technology of the mandala, has the potential to help us explore and even experience, under controlled conditions, the topology of binaries in tension, or the topology of “not one and not two.” More than a visual guide to a stable topography, it might be compared to a flight or surfing simulator in which one learns to synchronize one’s proprioceptive and haptic sensibilities to the surrounding topology. But because the user’s body is already familiar with the topology of the M-set, it being a map of the body’s own, the loop between outside and inside flows in the opposite direction. The M-set, then, encourages the user to actualize Bergson’s concept of “life” as “endlessly continued creation” by transforming her external environment to align more fluidly with her interior affect. As such, it constitutes a technology for exploring and realizing a fractal ontology as it both lessens the distance between map (representation, Ideal) and territory (object, Real) and reveals the two to constitute not separate domains but a navigable, if performative, diffractional continuum. Clarke is then right to suggest that the implications of the M-set for our understanding of reality are profound: “We’ve all read stories of maps that revealed the location of some hidden treasure. In this case the map is the treasure!”48
In his memoir, The Fractalist, Mandlebrot speaks of his admiration for his hero Johannes Kepler. By “breaking the circle,” or perceived perfection of the orbits of the planets as understood by astronomers, Kepler was able to bring order to a small piece of the universe. By employing mathematics as a research tool rather than accept what was “universally” understood to be an a priori truth, he was able to uncover not only new physical laws but also the rewards to be gained by rejecting conventional wisdom. Mandelbrot, wishing to make similar contributions to science, learned this important lesson and arguably achieved this goal. Much as Kepler’s iconoclastic efforts later would result in the human learning that it did not occupy the center of the universe, so Mandelbrot’s discoveries invite us to question our a priori understanding of the relationship between scale and complexity, or that which places the human at the central, scalar position with respect to other of the universe’s organizational levels. And just as Kepler’s rejection of the ideal circular orbit prompted a wholesale reconfiguration of humans’ understanding of, and relationship to, material-discursive forces, so does Mandelbrot’s revelation of the relative nature of scale/complexity invite us to reevaluate our relationships to other of the universe’s material-discursive phenomena. No longer being the pivot point around which the universe revolves the human might be free to recognize and invent new configurations and alliances between itself and other objects/entities. Critical to this endeavor, of course, has been and will be the ability of technological innovations such as computer simulation and mathematical modeling to make accessible entities/worlds once undiscoverable either through theory or experimentation.49 While the new digital technologies make it possible to conceive of a fractally-structured ontology, they do not guarantee that it will be taken up as a viable alternative to the reigning positivist model. But given where the latter appears to be taking us, one can only hope that such an alternative might be given a fair hearing.
1 The graphic qualities of the Mandelbrot set and fractal geometry, more generally, have broad appeal, in part because their unusual yet familiar repetitions of form make geometry intelligible even to non-mathematicians. For a recent example of this phenomenon see : (http://www.nytimes.com/2013/01/22/science/usc-exhibit-shows-fractals-built-from-paper.html?_r=0&adxnnl= 1&adxnnlx=1358964334-VcPUDmqErL9vcNoMajrpkQ). Accessed 1/23/13.
2 Manuel Delanda. 2005. Intensive Science and Virtual Philosophy. Continuum: New York. p. 63.
3 Ibid., p. 85.
4 Leibniz anticipated much of what technology is now making legible with respect to a fractal/flat ontology: “And as the same town, looked at from various sides, appears quite different and becomes as it were numerous in aspects [perspectivement]; even so, as a result of the infinite number of simple substances, it is as if there were so many different universes, which, nevertheless are nothing but aspects [perspectives] of a single universe, according to the special point of view of each Monad.” Gottfried F. Leibniz. 1998. “Monadology.” Philosophical Texts. Trans. and ed. by R.S. Woolhouse and R. Francks. New York: Oxford University Press. Paragraph 57.
5 Tim Morton. 2007. Ecology Without Nature: Rethinking Environmental Aesthetics. Harvard University Press: Cambridge, MA. p. 18.
6 Philip Pilkington interviews Greek economist Yanis Varoufakis on the collapse of neo-classical and neo-liberal economics. “What can be salvaged from the theoretical wreck that is economics? My answer is: The process of discovering the limits of analytical reason. By studying critically all models, we end up none the wiser about quantitative outcomes but much, much smarter about the complexities of really existing capitalism. Our exploration of economics may take us, in the end, right back to the point we started: Not having a clue about when the next crisis will hit, what sectors will dominate, whether the stock exchange will pick up soon or not. But, while we shall not have a determinate model of prices and quantities, we shall be much more appreciative of capitalism’s motivated irrationality, its penchant for surprising even the powers that be, its capacity to create incredible wealth and untold suffering by means of precisely the same process.” (http://www.nakedcapitalism.com /2012/03/the-new-priesthood-an-interview-with-yanis-varoufakis-part-ii.html). Accessesd 4/2/12.
7 Sanford Kwinter. 2002. Architectures of Time: Towards a Theory of the Event in Modernist Culture. MIT Press: Cambridge, Massachusetts. p. 40.
8 Gottfried Leibniz. Ibid., “Monadology.” Paragraphs 82-84. “…rational animals have this peculiarity, that their little spermatic animals, as long as they remain such, have only ordinary or sensuous souls, but those of them which are, so to speak, elected, attain by actual conception to human nature, and their sensuous souls are raised to the rank of reason and to the prerogative of spirit. Among the differences that there are between ordinary souls and spirits, some of which I have already instanced, there is also this, that while souls in general are living mirrors or images of the universe of created things, spirits are also images of the Deity himself or of the author of nature. They are capable of knowing the system of the universe, and of imitating some features of it by means of artificial models, each spirit being like a small divinity in its own sphere. Therefore, spirits are able to enter into a sort of social relationship with God, and with respect to them he is not only what an inventor is to his machine (as in his relation to the other created things), but he is also what a prince is to his subjects, and even what a father is to his children.”
9 Graham Harman: “The world not only withdraws from human access, but objects withdraw from each other as well. There is no such thing as direct contact between any two entities: causal relations must be indirect, vicarious.” (http://figureground.ca/interviews/graham-harman/). Accessed 10/ 10/12.
10 Much as Niklas Luhmann’s observer is forced to rely on the observations of others to complete him. From Luhmann’s perspective, the self-reflexive system springs from a constitutive paradox in that its ability to maintain its corporeal integrity is dependent on the unity/duality paradox of its origins remaining invisible to the system itself. Rather than this creating a world of closed, solipsistic systems, however, he views this blind spot at the center of every autopoietic system as creating the possibility of sociality, as “only an [other] observer is able to realize what systems themselves are unable to realize” (1990. Essays on Self-reference. Columbia University Press: New York. p.127.) because “reality is what one does not perceive when one perceives it.” 1990. “The Cognitive Program of Constructivism and a Reality That Remains Unknown.” In Wolfgang Krohn, Gunter Kuppers and Helga Nowotny eds. Self-organization: Portrait of a Scientific Revolution. Dordrecht: Kluwer Academic Publishers. p. 76. The significance of this insight for social theory is aptly described by Cary Wolfe: It is only in the mutual observations of different observers that a critical view of any observed system can be formulated. If we are stuck with distinctions that are paradoxical and must live with blind spots at the heart of our observations, Luhmann writes, “Perhaps, then, the problem can be distributed among a plurality of interlocked observers” who are of necessity joined to the world and to each other by their constitutive but different blind spots. The work of social theory would then consist in developing “thoughtful procedures for observing observation, with a special emphasis on that which, for the other, is a paradox and, therefore, cannot be observed by him.” William Rasch, Cary Wolfe. Eds. 2000. Observing Complexity: Systems Theory and Postmodernity. University of Minnesota Press: Minneapolis. pp. 260-261.
11 The self-similar folds in a crumpled sheet of paper, for example, typically give such an object a dimension of 2.5 because its dimensionality lies midway between a plane (a flat sheet) and a volume (a compressed ball).
12http://katachriston.wordpress.com/2010/12/27/fibonacci-fractals-and-inorganic-teleology/). Accessed 4/6/12.
13 4D rendering of the Mandelbrot set. (http://www.youtube.com/watch?v=5ej3dj4x64k). Accessed 9/3/12.
14 Deep zoom into the Mandelbrot set. (http://www.youtube.com/watch?v=G_GBwuYuOOs). Accessed 9/3/12.
15 Morton is here referring to Derrida’s discussion of the ambiguity of “margins”: “Derrida has shown how this suggestive term evokes the difficulty of distinguishing between inside and outside.” Tim Morton. Ibid., p. 40.
16 S. N. K. Watt. 1993. “Fractal behaviour analysis.” In Prospects for Artificial Intelligence. Eds. A. Sloman, D. Hogg, G. Humphreys, A. Ramsay, & D. Partridge. IOS Press: Amsterdam. p. 112.
17 (http://www.doc.ic.ac.uk/~nd/surprise_95/journal/vol4/ykl/report.html). Accessed 5/6/12.
18 (http://iopscience.iop.org/0067-0049/178/1/1/pdf/74198.web.pdf). Accessed 5/6/12.
19 Author and financial analyst Yves Smith (of “Naked Capitalism” blog) discusses how most economists fail to effectively model the economic landscape because of their aversion to non-linear/fractal models: “And this problem is made worse by the fact that economists have long been allergic to the sort of mathematics and modeling approaches best suited to this type of analysis, namely systems dynamics and chaos theory. I discussed both these aesthetic biases at length in ECONNED, but the very short version is that following Paul Samuelson, economists have wanted to put the discipline on a “scientific” footing, and that meant embracing the “ergodic” axiom. Warning: a lot of natural systems aren’t ergodic. The ergodic assumption means no path dependence and no tendencies to instability. If you get a good enough sample of past behavior, you can predict future behavior. If you think these are good foundation for modeling financial markets, I have a bridge I’d like to sell you.” (http://www.nakedcapitalism.com /2012/03/gillian-tett-exhibits-undue-faith-in-data-and-models.html). Accessed 3/9/12.
20 Ary L. Goldberger. 1999. “Nonlinear dynamics, fractals, and chaos theory: implications for neuroautonomic heart rate control in health and disease.” Eds. C. L. Bolis, J. Licinio. The Autonomic Nervous System. World Health Organization: Geneva. pp. 135 –152.
21 Keith Ansell-Pearson. 1999. Germinal Life: The Difference and Repetition of Deleuze. Routledge: London. p. 25. “Throughout his work Bergson performs an important critique of natural perception, which is a model of perception that reduces the activity and becoming of life (movement) to a centred subject of perception. Hence he writes: ‘to perceive consists in condensing enormous periods of an infinitely diluted existence into a few more differentiated moments of an intenser life, and in this summing up a very long history, to perceive means to immobilize’ (233; 208). Not only is the border between an organism and its environment never clear-cut, being always porous and sympathetic, but so are the boundaries which separate and divide bodies: ‘the close solidarity which binds all the objects of the material universe, the perpetuality of their reciprocal actions and reactions, is sufficient to prove that they have not the precise limits which we attribute to them’ (209). ‘True evolutionism’, therefore, must assume the form of a study of ‘becoming’ (Bergson 1962:369; 1983:370). But this requires that we do not follow the path of perception which would reduce an ‘infinite multiplicity of becomings’ to the single representation of a ‘becoming in general’.” (pp. 303-4).
22 Ronald Bogue. 2003. Deleuze on Literature. Routledge: New York. p. 4.
23 To give some idea of the frequency with which this occurs, one can imagine that while the image of the original set takes up six inches on a computer screen, magnifying this so as to arrive at the fourth generation of the insect-like core (which is similar to the original but slightly different) creates an image half a mile wide.
24 Whitehead’s “extensive continuum,” as interpreted by Steve Goodman, presents another sort of ‘diffractional’ reading of reality such that: “… there is a resonance of actual occasions, which are able to enter into one another by selecting potentials or eternal objects. It is in such a potential coalescence of one region with another that an affective encounter between distinct actual entities occurs. The vibratory resonance between actual occasions in their own regions of space-time occurs through the rhythmic potential of eternal objects, which enables the participation of one entity in another.” Steve Goodman. 2009. Sonic Warfare: Sound, Affect and the Ecology of Fear. MIT Press: Cambridge, MA.
25 Karen Barad. 2007. Meeting the Universe Halfway: Quantum Physics and the Entanglement of Matter and Meaning. Duke University Press: Durham. pp. 48-49.
26 Ibid., p. 55.
27 Ibid., p. 56.
28 Ibid., p. 48.
29 Ibid
30 Steve Goodman. Ibid., p. 97.
31 Gottfried F. Leibniz. Ibid., “Monadology.” Paragraphs 67 and 69.
32 Gilbert Simondon . 2009. The Position of the Problem of Ontogenesis. Trans. by Gregory Flanders. Parrhesia. Number 7. pp. 4–16. “In this way, it becomes possible to think of the relation that is interior and exterior to the individual as participation, without referring to new substances. The psychic and the collective are constituted by individuations that occur after the vital individuation. The psychic is the continuation of the vital individuation in a being that, in order to resolve its own problematic, must itself intervene as an element of the problem by its action, as a subject. The subject can be conceived of as the unity of being as an individuated living being, and as a being that represents its actions through the world to itself as an element and as a dimension of the world. The vital problems are not closed upon themselves; their open axiomatic can only be saturated by an undefined series of successive individuations that engage ever more of the preindividual reality and that incorporate it into the relation to the environment.” p. 14.
33 Gabriel Tarde. 2012 (1893). Monadology and Sociology. Trans. Theo Lorenc. Re.press: Melbourne. “In truth, one might justifiably wonder…whether it is really certain that our own intelligence and will, those great egos disposing of the vast resources of a gigantic cerebral state, are superior to those of the tiny egos confined in the miniscule city of animal or even plant cell. Surely, if we were not blinded by the prejudice of always considering ourselves superior to everything, such comparisons would not be to our advantage. At root, it is this prejudice which prevents us from believing in monads.” p. 22.
34 Stanley Salthe. 1985. Evolving Hierarchical Systems. Columbia University Press: New York. p. 284. Salthe defines the cogent moment as “the duration of the smallest natural time unit for some system. Systems of different scale will have cogent moments of relatively different durations.”
35 This, of course, runs counter to the Standard Model of Physics which states that nothing can exist below the Planck length though, as Arthur C. Clarke points out, “no one know s for sure.” Meanwhile, theories speculating on the possibility that we live not in a universe but a “multiverse” are proliferating as they seem to offer the best solution to the problem of accounting for the recent discovery that seventy percent of our universe’s mass consists of dark energy. These concepts are not necessarily related other than that they both point to the fact that some of our most basic understandings of matter and cosmology are still and will probably always remain open to debate.
36 Keith Ansell-Pearson. Ibid., p. 24. “The ‘ethical’ character of this method of philosophy resides, therefore, in the cultivation of a ‘sympathetic communication’ that it seeks to establish between the human and the rest of living matter (Merleau-Ponty is one of the few commentators to note that an ‘ethics’ informs the entirety of Bergson’s thinking, 1988:31-2)…By expanding consciousness ‘it introduces us into life’s own domain, which is reciprocal interpenetration, endlessly continued creation’(Bergson 1962:179; 1983:178).
37 Ibid.,p. 103. Citing Deleuze and Guattari (A Thousand Plateaus. 1980:603-6; 1988:483-5).
38 Leibniz described the imaginary numbers as being “a wonderful flight of God's spirit… [because] they are almost an amphibian between being and not being.” Clifford A. Pickover. 2009. The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Sterling Publishing Co. : New York. p. 124.
39 Keith Ansell-Pearson. Ibid. p. 43. Ansell-Pearson characterizes Bergson’s concept of duration as understood by Deleuze: “Subjectivity is never ours, it is time, that is, the soul or the spirit, the virtual.” (Deleuze 1985:110-11; 1989:82-3): the separation between things, objects, and environments is neither absolute nor clear-cut, for ‘the close solidarity which binds all the objects of the material universe, the perpetuality of their reciprocal actions and reactions, is sufficient to prove that they have not the precise limits which we attribute to them’ (235; 209). p. 24.
40 This is Graham Harman’s term for the special quality of an object that draws others to it: “Allure is a special and intermittent experience in which the intimate bond between a thing’s unity and its plurality of notes somehow partially disintegrates.” 2005. Guerrilla Metaphysics: Phenomenology and the Carpentry of Things. Open Court: Chicago.
41 Arthur C. Clarke. “The Colors of Infinity.” Video. (http://www.youtube.com/watch?feature=player_ embedded &v=qB8m85p7GsU). Accessed 3/26/12.
42 Steven B. Lowe, Larry S. Liebovitch, John A. White. May 1999. “Fractal ion-channel behavior generates fractal firing patterns in neuronal models.” Article in Physical Review. Vol. 59. No. 5. (http://www.ccs.fau. edu/~liebovitch/pre59b.pdf). Accessed 3/8/12.
43 Tim Ingold. Lines: A Brief History. Taylor and Francis, Ltd.: London pp. 75-6.
44 Leibniz. Ibid., “Monadology.” Paragraph 61.
45 “Based in New York in the 1970s, Anarchitecture was an artists' group whose members included artists Laurie Anderson, Tina Girouard, Carol Goodden, Suzanne Harris, Jene Highstein, Bernard Kirschenbaun, Richard Landry and Richard Nonas, as well as the architecturally trained artist Gordon Matta Clark (1943-1978). The group’s name, a mixture of 'anarchy' and 'architecture', was conceived in informal conversation, one of the main ways through which the group collaborated. In 1974, they produced an exhibition of the same name, which encapsulated their critique of the modernist impulses of contemporary culture within which architecture was conceived as a symbol for that culture's worst excesses and drawbacks. Anarchitecture was very critical of the stasis in cultural attitudes and what Richard Nonas called the 'hard shell', or resistance to change, that architecture epitomized. All contributions to the show were anonymous and followed an agreed format to emphasize their collective approach. The central role accorded to architecture was perhaps a reaction to Matta Clark's own experience of architectural education at Cornell University, from where he graduated in 1968. The group as a whole tackled architecture's complicity in capitalist modes of production, using wordplay and found photographs to explore issues related to cities, ways of inhabiting buildings and the role of property.” (http://www.spatialagency.net/database/the. anarchitecture.group). Accessed 3/19/12.
46 Karen Barad. Ibid., p. 57.
47 Tim Morton. Ibid., p. 48.
48 Arthur C. Clarke. Ibid.
49 Timothy Lenoir (2002) "Makeover: Writing the Body into the Posthuman Technoscape. Part One: Embracing the Posthuman," in Timothy Lenoir, ed., Makeover: Writing the Body into the Posthuman Technoscape, Baltimore: Johns Hopkins University Press. Configurations, Vol. 10, Number 2, Spring, pp. 203-220.