Geometry and Abjection
by Victor Burgin
abstracted from "Geometry and Abjection", published in the AA Files # 15
Space has a history. In the cosmology of classical Greece, as F. M. Cornford writes, ‘the universe of being was finite and spherical, with no endless stretch of emptiness beyond. Space had the form of . . . a sphere with centre and circumference.’ This classical-space essentially survived the biblically derived ‘flat earth’ of early Christian doctrine, to re-emerge in the late Middle Ages. In medieval cosmology, supercelestial and celestial spheres encompassed, but did not touch, a terrestrial sphere - the space of human action - in which every being, and each thing, had a place preordained by God and was subject to His omnivoyant gaze. Foucault has termed this medieval space the ‘space of emplacement’; this space, he observes, was effectively destroyed by Galileo: ‘For the real scandal of Galileo’s work lay not so much in his discovery, or rediscovery, that the earth revolved around the sun, but in his constitution of an infinite, and infinitely open space. In such a space the place of the Middle Ages turned out to be dissolved ... starting with Galileo and the seventeenth century, extension was substituted for localisation.
The vehicle of this changed cosmology was Euclidean geometry. Euclid wrote the Elements of Geometry around 300 BC. Husserl, in The Origin of Geometry, supposes that this system arose out of practical activities, such as building. However, the classical conception of space seems to have been based upon visual evidence rather than technique the horizon appears to encircle us, and the heavens appear to be vaulted above us. In the Renaissance this conflict between observation and intellection, between hyperbolic and Euclidean space, wa.4 played out during the early stages of the invention of perspective. (The absence of a necessary connection between knowledge of Euclidean geometry and the development of perspective is evident from the example of the Islamic world.) In the West, the primacy of geometry over perception was stressed by St Augustine, who wrote: ‘reason advanced to the province of the eyes ... It found ... that nothing which the eyes beheld, could in any way be compared with what the mind discerned. These distinct and separate realities it also reduced to a branch of learning, and called it geometry.’
Although dependent upon Euclid’s Elements, Renaissance perspective took its most fundamental concept from Euclid’s Optics. The concept is that of the ‘cone of vision’. Some two thousand years after Euclid, Brunelleschi conceives of this same cone as intersected by a plane surface - the picture-plane. By means of this model, something of the pre-modern world view passes into the Copernican universe - a universe which is no longer geocentric, but which is nevertheless homocentric and egocentric. A basic principle of Euclidean geometry is that space extends infinitely in three dimensions. The effect of monocular perspective, however, is to maintain (he idea that this space does nevertheless have a centre - the observer. By degrees the sovereign gaze is transferred from God to Man. With the ‘emplacement’ of the medieval world now dissolved, this ocular subject of perspective, and of mercantile capitalism, is free to pursue its entrepreneurial ambitions wherever trade winds blow.
Entrepreneurial humanism first took liberties with, then eventually replaced, theocentric determinism, according to a model which is implicitly Aristotelian. and in a manner which exemplifies the way in which spatial conceptions are projected into the representation of political relationships. In Aristotle’s cosmological physics it was assumed that the preponderance of one or other of the four elements first posited by Empedocles (earth, water, air and fire) would determine the place of that body within a continuum from the centre to the periphery of the universe. This continuum of actual and potential ‘places’ constituted space. Analogously, the idea that a human being will find his or her natural place within the social space of differential privileges according to his or her ‘inherent’ qualities has remained a corner-stone of humanist-derived political philosophies. Newton disengaged space per se from Aristotelian ‘place’, and Newtonian physics was in turn overtaken by the physics of Einstein, in which, in the words of Minkowski, ‘space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality’. More recently, the precepts of general relativity have themselves come into question in ‘quantum theory’. The cosmology of modern physics has nevertheless had little impact on the commonly held world view in the West, which is still predominantly an amalgam of Newton and Aristotelianism - ‘places in space’, a system of centres of human affairs (homes, work-places, cities) deployed within a uniformly regular and vaguely endless ‘space in itself’.
In the modernist avant-garde in art, references to a mutation in the apprehension of space and time brought about by modern physics and mathematics are not unusual. Thus, for example, in 1925 El Lissitsky wrote: ‘Perspective bounded and enclosed space, but science has since brought about a fundamental revision. The rigidity of Euclidean space has been annihilated by Lobachevsky, Gauss, and Riemann. Nevertheless, modernists more commonly ascribed the changed apprehension of space not to scientific concepts per se, but rather to technology. Thus Vertov wrote: ‘I am the cinema-eye. I am a mechanical eye. I, a machine, can show you the world as only I can see it . . . I ascend with aeroplanes, I fall and rise together with failing and rising bodies.’ Constrained by mechanical metaphors, Russian futurism, like cubism, ultimately failed - notwithstanding El Lissitsky’s pronouncement - to abandon Euclidean geometry. The mirror of perspectival representation was broken only in order that its fragments, each representing a distinct point of view, be reassembled according to classical geometric principles - to be returned, finally, to the frame and the proscenium arch.
In the modern period, space was predominantly space traversed (by this token we judge that the prisoner has little of it). In the ‘postmodern’ period, the speed with which space is traversed is no longer governed by the mechanical speed of machines such as aeroplanes, but rather by the electronic speed of machines such as computers and video links, which operate at nearly the speed of light. A mutation in technology therefore has, arguably, brought the technologism inherited from the spatial perceptions of modernist aesthetics into line with the perceptions of modern physics. Thus, for example, Paul Virilio writes that ‘technological space ... is not a geographical space, but a space of time’. In this space/time of electronic communications, operating at the speed of light, we see things, he observes, ‘in a different light’- the ‘light of speed’. Moreover, this space seems to be moving, once again, towards self-enclosure. For example, David Bolter, a classics professor writing about computer programming, concludes, ‘In sum, electronic space has the feel of ancient geometric space.’ One of the phenomenological effects of the public applications of new electronic technologies is to cause space to be apprehended as ‘folding back’ upon itself. Spaces once conceived of as separated, segregated, now overlap: live pictures from Voyager II, as it passes through the rings of Saturn, may appear on television sandwiched between equally ‘live’ pictures of internal organs, transmitted by surgical probes, and footage from Soweto. A counterpart, in the political sphere, of the fold-over spaces of information technologies is terrorism. In the economic sphere it is the tendency of multinational capitalism to produce First World irruptions in Third World countries, while creating Second World pockets in the developed nations. To contemplate such phenomena is no longer to inhabit an imaginary space ordered by the subject-object ‘stand-off’ of Euclidean perspective. The analogies which fit best are now to be found in non-Euclidean geometries - the topologist’s Mobius strip, for example, where the apparently opposing sides prove to be formed from a single, continuous. surface.
Space, then, has a history. It is not, as Kant would have it, a product of a priori, inherently Euclidean. categories. It is a product of representations. Pre-modern space is bounded; things within it are assigned a place along a predominantly vertical axis - ‘heaven-earth-hell’, or the ‘chain of being’, extending from God down to stones. Modern space (inaugurated in the Renaissance) is Euclidean, horizontal, infinitely extensible, and therefore, in principle, boundless. In the early modern period it is the space of the humanist subject in its mercantile entrepreneurial incarnation. In the late modern period it is the space of industrial capitalism, the space of an exponentially increased pace of dispersal, displacement and dissemination of people and things. In the ‘post-modern’ period it is the space of financial capitalism - the former space in the process of imploding or ‘unfolding’; to appropriate a Derridean term, it is space in the process of ‘intravagination’. Twenty years ago Guy Debord wrote about the unified space of capitalist production, ‘which is no longer bounded by external societies’, the abstract space of the market which ‘had to destroy the autonomy and quality of places’, and he commented: ‘The society which eliminates geographical distance reproduces distance internally as spectacular separation’. Such ‘internal distance’ is that of Psychical space. Nevertheless, as I have already remarked, psychoanalytically inspired theories of representation have tended in recent years to remain faithful to the Euclidean geometrical-optical metaphors of the modern period.