Spatial Questions: Cultural Topologies and Social Spatialisations

Books

Rob Shields

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  • Chapters
  • Front Matter
  • Back Matter
  • Subject Index
  • Dedication

    To Sophie and Bohdana

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    List of Illustrations

    • 1.1 Illustration from the Shan Hai Jing4
    • 1.2 Mappa Mundi from Al Idrisi, Kitab Nuzhat, c. 1154 6
    • 2.1 Panoramic view. Historical poster of the Chateau Frontenac, c. 1926 37
    • 3.1 Table of philosophers of space 54
    • 3.2 Map and reverse chronology of key thinkers on space 58
    • 3.3 Raphael, School of Athens, 1505 60
    • 5.1 Engraving of Königsburg, 1613 108
    • 5.2 Euler's illustration of the bridges of Königsberg and Kneiphof Island, c. 1736 111
    • 5.3 Reverse chronology and map of key figures in topology 116
    • 5.4 René Thom's topological archetypes (after Thom, 1975 and Demers, 2010) 121

    About the Author

    Dr Rob Shields is Henry Marshall Tory Chair and Director of the Faculty of Extension's City-Region Studies Centre and teaches in the Departments of Sociology and of Art and Design at the University of Alberta. Before being awarded the Tory Chair, he was Professor of Sociology and past Director of the Institute of Interdisciplinary Studies at Carleton University.

    His focus has been urban cultural studies, particularly the social use and meanings of Places on the Margin, urban spaces and regions, including tourist destinations and place identities. In co-edited collections such as Rereading Jean-François Lyotard, and Demystifying Deleuze and in texts such as Lefebvre, Love and Struggle, Shields considers changing spatialisations. This intellectual project has been extended through a peer-reviewed journal, Space and Culture, Curb planning magazine and many talks on the city, the Strip-Appeal of malls and other Ecologies of Affect, and the importance of intangibles and The Virtual to everyday life.

    Preface and Acknowledgements

    This book is a work of thresholds, edges and folds: a topology in many senses – it explores spatial distinctions and boundaries established by markers such as a doorway, as well as less tangible distinctions such as near and far or cultural divisions between social fields. It marks a threshold of new research that shifts from space as a constitutive dimension and spatialisation of social life and personal identities to understanding competing and contested time-space milieux which are elaborated by social interactions, texts and institutional relationships. This shift from questions of social spatialisation to time-space topologies structures the sequence of chapters in this book.

    We begin by shifting from the discourse of ‘space’ to social spatialisation from the beginning. For a detailed discussion of social spatialisation, the reader is refered to Places on the Margin (1991), with its case histories of how social action and place-images plays out as a nuanced geography of interlinked places and regions. Here, ‘social spatialisation’ or just ‘spatialisation’ is used to capture the sense in which places are ‘cast’ as ‘places-for-this’ or ‘places-for-that’. ‘Spatialisation’ in French and English popular usage has meant ‘making spatial’. In Chapter 2, ‘social spatialisation’ redefines ‘space’ as a problematic term by locating its partiality and identifying the cultural role that it plays by constructing a crucible and arena for the play of capital, art and technology. It seeks to not only translate but move beyond Lefebvre's Marxist-Hegelian analysis by stressing its Leibnizean, Nietzschean and even structural qualities and drawing on Foucault and Deleuze's equally Nietzschean engagements with cultural and psychological structures of power. Their debt to twentieth-century French thought – Durkheim, Cassirer, Bachelard and Canguilhem – is well established despite their critiques and denials. However, spatialisation takes on a distinct meaning in my usage given that Foucault and Deleuze tend to use the term ‘spatialisation’ in its French sense as localisation, ‘placing’.

    A ‘Glossary’ is appended after the last chapter to provide a ready reference to the terminology used in this book. Terms such as the ‘virtual’, ‘material’, ‘actual’, ‘real’, ‘ideal’, ‘possible’, ‘abstract’ and ‘probably’ are ontological aspects of events and objects drawn from The Virtual (Shields, 2003, 2006b). They stand in the relation to each other described by Proust's definition of memory, developed in part by Deleuze, as virtual: ‘real but not actual, ideal but not abstract’.1 Any aspect of spatialisation could profitably be developed on its own or its relationships explored, for example the process of the realising of fictional place-images as the character of a place or the actualisation of that quality in a built environment such as a tourist attraction. Although none occur purely, for example a virtuality in isolation from material object or vice versa, the terms can be diagrammed as a four-part ontology:

    RealPossible
    IdealVirtual (memory, intangibles, spatialisations)Abstract (representations, maps, place-images, borders)
    ActualConcrete or material (tangible objects, constructions, material boundaries, fences)Probable (statistical %, e.g. risks, political economics of values)

    Rather than ‘either-or’, this table is a synopsis of a set of modalities of existence that co-occur and provide the stations of a continuously changing reality, where objects appear of use at hand in their capacity to occupy all of these positions, that is, to be simultaneously ‘actually’ one thing but ‘as if’ something else, ‘possibly’ effective for other uses and ‘metaphoric’ of others. Table 1.1 is thus marked by continuous movement between its categories: realisation of the possible, actualisation of the ideal, virtualisation of the actual, and so on, and also illegitimate transformations such as from the abstract to the concrete, described as the ‘miraculous’ in everyday speech (see Shields, 2003). ‘Syncretic’ phenomena that are undecidable, hovering on the boundary between categories are also of great interest.

    Since its presentation in Places on the Margin, spatialisation seems to have provided a tool-kit for geographical analyses of a much broader set of phenomena (Shields, 2006a; see for example, the usage in Hall, 2003: 151). The term has been disseminated as a keyword across human geography in particular. Contributors to a recent reference book use the term over a hundred times in 350 pages to profile Key Thinkers on Space and Place (Hubbard et al., 2004). Historically, one might ask whether or not it constitutes a paradigm shift in geographical thought? The argument pursued in this book goes further. Do questions of spatialisation in turn lead to questions of time, history and memory that demand a less genealogical, more topological approach to space-time? The affective quality of hoped-for futures and nostalgic pasts has been the topic of previous collective research by the Space and Culture Research Group at the University of Alberta, culminating in the edited book-work Ecologies of Affect (Davidson et al., 2011). This research group of students, colleagues and so many students who have become colleagues, continues in the writing groups, theory retreats, visiting speakers and conferences on a wide range of topics. These recollected and anticipated spatialisations set the stage for the introduction of topology.

    The survey of the development of non-Euclidean understandings of topology will inevitably strike mathematicians as crude and social theorists as incomplete, but to my knowledge it represents the only non-algebraic introduction to topology in plain English to outline both the history and the implications of this approach. Other efforts at this time include work on social networks and special issues of Space and Culture and of Theory Culture & Society. My efforts to go beyond the metaphorical usage of topological terms by connecting-up cultural studies with the rigour of topological concepts also benefitted from comments at symposia organised at Cultural Studies centres at the University of Western Sydney and at Monash University in Melbourne Australia in 2010, and at Shanghai International Studies University in 2011 and the University of Toronto's Semiotic Circle in 2012. I deeply appreciate the efforts of organisers and funders, including the Henry Marshall Tory Chair fund that I have benefitted from abroad, at the University of Alberta, and the support of the Social Sciences and Humanities Research Council of Canada.

    Many of the chapters in this book are the result of almost three decades of work. Some have appeared in earlier versions that anticipated the presentation here. However, all have been revised and updated and include orienting commentary that shows how they form the building blocks of a larger edifice that itself has matured over time. In particular they have benefited from discussion at the Space and Culture Research Group at the University of Alberta over the last four years and in sessions organised at the American Association of Geographers Conference in 2011, and the Bristol Symposium on Event Space sponsored by the World University Network in late 2011. Many individual conversations over coffee have provided food for thought. In Chapter 1, while I alone am liable for problems resulting from the too brief treatment of the ancient Chinese text Shan Hai Jing, this has benefited from the critique and commentary of scholars including at the Chinese Academy of Social Sciences in Beijing in September 2011. Professor Jin Huimin and Professor Xu Dejin provided valuable context. I am most grateful for this and many others’ academic and personal hospitality.

    In as much as this book draws on a long period of research, students and colleagues in Sociology, Cultural Studies and other Departments at Lancaster, Carleton, and University of Alberta; a sabbatical at the Faculty of Communications, Universidad Federal da Bahia, Salvador Brazil; professional networks of the City Region Studies Centre in the Faculty of Extension and its research staff including Dr Kevin Jones, and visiting academics such as Professor Hamid Abdollayan, Professor Sergio Benicio di Melo, and Professor André Lemos; networks through the American Association of Geographers, the Association for Cultural Studies, the editorial group of Theory Culture & Society and the all-important virtual network of the journal, Space and Culture have all added to the shape of this work. I am especially grateful to all those who hosted these symposia or took the time to talk with me. It is impossible to thank everyone personally or properly. My family have given me the most benefit of patience and tolerated travels to the events mentioned above, or family holidays that turned into proofreading days or detours to archives. I am especially grateful for Bohdana and Sophie's willingness to take on the challenges of travel and dislocation. Thank you.

    Edmonton, AlbertaCanada, 19 May 2012
    Note

    1 In Shanghai, Beijing Professor Huang Zhuo Yue reminded me of the critique of Storch's defence of the bourgeoisie and nobility as engaged in immaterial production (such as service, or knowledge work) as discussed in Marx's Capital Vol. 4 (online Ch. 4 Section 16). However, the virtual is not intended as an activity in and of itself but an aspect of all historically located activities and objects, including material products. Immaterial labour has received much critical attention recently, notably in the work of Maurizio Lazzaratto and in relation to the accumulation and potlatch of excess in the work of Georges Bataille.

  • Glossary of Terms

    • abstract Ideal and possible entities such as ideas and concepts.
    • Actual Actively and tangibly existing as a phenomenon in the present moment, not just potentially, conceptually or latently. See Table 1.1.
    • Analysis Situs Literally, analysis of situation. According to Poincaré, it describes topology as a branch of Geometry that describes the relative situation of points, lines and surfaces without consideration of their size.
    • Area Quantity that expresses an extent of a surface, typically of two-dimensions.
    • Atlas A collection of maps that covers the earth, the universe or a manifold.
    • Boundary The edge of a manifold or the closure of a set minus its interior. If a manifold M has a boundary, then it is finite. The boundary is a manifold of one less dimension than the dimension of M.
    • Chart A chart is a local coordinate system of a topological neighbourhood. A number of charts can be collected into an atlas.
    • Clinamen Lucretius’ name for the indeterminate and unpredictable swerve of atoms that founds the origin of freedom from predictability propounded by Epicurus.
    • Closed Path A path (that is, a curve) on a manifold that begins and ends at the same point.
    • Compact A manifold is compact if it has an atlas with a finite number of charts.
    • Concrete (also Material): Real and actual entities with extension and effects, Descartes’ res extensa. See Table 1.1.
    • Connected Sum The manifold that results from cutting a solid ball out of each of two manifolds and identifying the points on the two spheres that bound the complements.
    • Continuous A function from one space to another is continuous if the preimage or inverse of every open set is also open.
    • Complex Numbers Set of numbers obtained by augmenting the real numbers with the square roots of negative numbers.
    • Corollary A proposition that follows easily from a theorem or another proposition.
    • Curvature A mathematical object that measures the deviation of the sum of angles of triangles from 180 degrees. In a two-dimensional manifold, the curvature at each point is a number.
    • Diagram An icon, projection or visualisation of an entity, space or process using geometric and symbolic representation techniques to show relationships between elements within some sort of matrix or mechanism.
    • Differential Equation An equation in which one specifies how rates of change (the ‘differential’) in one mathematical object (variable) is related to other objects. An example would be process affected by changing forces that determine the actual outcome in ways that are difficult to foresee by simply extrapolating from the current moment. The general solution is a ‘function’ whose various ‘derivatives’ satisfy the differential equation over a lengthy span of time.
    • Dimension The number of independent degrees of freedom in a set. Or, the minimum number of real numbers (that is, the number of coordinates) that it takes to specify a proposition near a given point in a set.
    • Durée The subjective experience of time as duration with a characteristic tempo proposed by Bergson as a critique of time measured as a series of instants, which loses the mobility of time.
    • Entourage Set of elements or neighbourhoods.
    • Euclidean Space For each positive integer n, Euclidean space of dimension n is n-space with distance defined by the Pythagorean theorem.
    • Field An organised expanse or corpus of some integral quality that contains relations that either have a causative power (e.g. a magnetic field), or between causes and effects or other relations of dependency (social field, algebraic field).
    • Fifth Postulate The fifth, and most complex, of the five postulates in Euclid's Elements: If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles (that is, whose sum is less than 180 degrees), the two straight lines, if extended indefinitely, meet on that side.
    • Flat A space is flat (a surface) if the sum of the angles of every triangle in the space is 180 degrees.
    • Geometrisation Conjecture The conjecture that every three-dimensional manifold can be cut up along spheres and torn into pieces that have one of eight geometries.
    • Geodesic A curve that traces the shortest distance between any two of its points.
    • Geometric Property A property that depends for definition on a distance or symmetry (for example: straight, angle measure, circle).
    • Geometry The structure that results from having a distance defined on a manifold.
    • Great Circle A circle on the (two-dimensional, round) sphere cut out by intersecting the sphere with a plane through its centre.
    • Group (of Transformations) A set of transformations with the properties that (1) the result of performing one transformation in the set, then another, is a transformation that again is in the set, and (2) the transformation that undoes a given transformation in the set is in the set. (That is, a group of a transformations is a set that is closed under the operations of taking products and forming inverses.)
    • Homeomorphism A one-to-one correspondence between two manifolds in which nearby points correspond to nearby points. Homeomorphic objects can be remolded or deformed into each other without tearing being required. The most common example is of the transformation of a doughnut shape (a torus) into a mug where the hole in the doughnut becomes the hole in the handle while part of the body of the doughnut is hollowed out to form the cup.
    • Homogeneous A manifold is homogeneous if the space looks the same at every point.
    • Ideal Not actually present but considered as present when limits of infinity are included or when the intangible or pure form is included as part of existence. The Ideal includes both abstractions such as ideas and real but virtual entities. See Table 1.1.
    • Lemma A mathematical result whose main purpose and interest is as a stepping-stone to proving another result.
    • Loop A path that begins and ends at the same point. More technically, a continuous mapping of an interval into a one-dimensional manifold in which both endpoints get mapped to the same point. That is, both ends ‘loop back’ to the same point.
    • Manifold A mathematical set that looks like Euclidean space at each point. (More formally, regions sufficiently near any point are homeomorphic to n-space.)
    • Map A representation of an area or field that normally has some one-to-one translation between the field and the representation itself, which may be a diagram. Maps may depict a real, imagined or virtual space.
    • Material See Concrete.
    • Metric A rule for specifying the distance between any two points of a set. In a manifold, a metric can be given by specifying a rule for measuring speed along curves.
    • Milieu Serre's notion of a medium that is between a communication channel and an environment. The milieu is an in-between transmission of space and time. It is thus a topology and is characterised by noise, mediation and intersections.
    • N-Space The set of all ordered n-tuples of real numbers.
    • Negative Curvature A region in a manifold has negative curvature if the sums of all angles of all triangles in that region are less than 180 degrees.
    • Neighbourhood The neighbourhood to point or set x is an open set that contains the x.
    • Non-Euclidean A space or mathematics in which Euclid's laws do not hold.
    • Open Set A set is open if any member or point in it can be varied or moved slightly and still remain within the set. Thus in topology an open set is the set of points near each other or a given point.
    • Partial Differential Equation A type of differential equation in which one specifies rates of change at different points in different directions. The solutions of partial differential equations are objects that have the desired rates change at all points in all directions. Many equations of mathematical physics are partial differential equations.
    • Path A continuous mapping within a manifold.
    • Phase Space Phase spaces are logically-defined spaces similar to those which allow us to visualise data in graphs; they have x-y-z or more dimensions. Each point corresponds to a unique state of a given system so that the phase space includes all possible states of the system.
    • Place a portion of space, a set or a field that has an assigned identity whether coordinates, location, or a name. Those defined in indefinite (unbounded) areas can be identified as places.
    • Place-image A specific image or representation associated with a place, either metaphorically, through action or history, or directly as in a photograph of a site.
    • Place-myth A set of place-images that constitute an overall representation of a place or location.
    • Poincaré Conjecture The conjecture, not proved, that every simply connected, compact three-dimensional manifold without boundary is homeomorphic to the three-dimensional sphere.
    • Positive Curvature A region in a manifold has positive curvature if the sums of angles of all triangles in that region are greater than 180 degrees.
    • Possible Entities that exist but without direct effect, such as abstractions (ideas, concepts that are ideal possibilities) and probabilities (mathematical objects such as percentages that indicate actual possibility). See Table 1.1.
    • Postulate An assertion accepted without proof. Synonymous with axiom.
    • Probable Actual and possible entities as in mathematical probabilities such as risks. See Table 1.1.
    • Proof A complete argument in which each assertion is an axiom or previously proved proposition, or else follows by formal rules of logic. It begins with axioms and known propositions and ends with the statement that was to be proved.
    • Proposition A statement that is derived from postulates and previously proved propositions using mathematical reasoning.
    • Pythagorean Theorem The statement that the square of the length of the hypotenuse of a right-angle triangle is equal to the sum of the squares of the two sides meeting at the right angle. (The hypotenuse is the side across from the right angle.)
    • Real Entities that exist as realised, not as merely possible or thought. Real entities are defined by having tangible, material effects.
    • Res Extensa Literally, extended thing (res). According to Descartes, ‘corporeal substance’ is one of three substances alongside ideas and God. Objects are combinations of these substance attributes, thus one can recognise a melted piece of dripping wax as another form of solid wax through knowledge of the idea, even as its corporeal substance changes in nature.
    • Riemann Curvature Tensor A mathematical object that assigns a value to every planar direction through the point (which reflects how much angle sums of tiny geodesic triangles in that direction tend to deviate from 180 degrees).
    • Round Sphere The word round refers to a sphere that has the same curvature at every point and is used when one wants to make a distinction between such spheres and topologically equivalent spheres that may have bumps. The surface of our earth is not a round sphere because it is flattened at the poles. A round sphere is isometric to the set of points at some fixed distance from a point in Euclidean space.
    • Simply Connected A manifold is simply connected if any loop can be shrunk to a point. This is equivalent to the statement that the fundamental group consists of a single element (necessarily, the identity).
    • Social Spatialisation (Spatialisation) The ongoing spatial organisation of all facets and scales of life, in the context of previously created spatial organisations to create a cultural formation that includes dispositions, representations and framings of the environment as a space of action. A ‘spacing’ of activities and ideas into ‘places for this’ and ‘places for that’. Some archetypes of spatialisation include division into centre and periphery, binary divisions (here versus there, near versus far), division by scale (global versus local), and mosaic patterns.
    • Space An extent or context defined by a social spatialisation. Spaces are manifolds characterised by a specific dimensionality.
    • Sphere Without additional qualification, refers to a two-dimensional sphere which is any object homeomorphic to the set of points in three-dimensional space at a fixed distance from a given point. The surface of a ball is a sphere. There are, however, spheres of every dimension. The simplest definition of a sphere of a given dimension is as the set of points of fixed distance from a single point in the Euclidean space of one dimension higher. So, for example, the set of points at distance from a fixed point in six-dimensional Euclidean space is a five-dimensional sphere.
    • Surface A two-dimensional manifold.
    • Syncresis The interface of the ideal and the actual. Interaction across this interface or mobility between the ideal and actual as continuous actualisation and virtualisation. Syncresis is a porous, incomplete merging of different elements which preserves enough of the originals’ identities to not fully subsume them into a new synthesis. This produces a flickering or equivocal states of real entities such as a ‘flame’.
    • Tensor A mathematical object that assigns real numbers to a prescribed number of vectors (that is, velocities) at each point of a manifold.
    • Theorem An especially important proposition.
    • Three-dimensional Manifold (or Three-Manifold or Three-Space) An idealised mathematical volume that models the shapes that three-dimensional spaces, like our everyday universe, might have. The region around every point can be mapped onto the inside of a solid aquarium. Put differently, the region near every point looks like three-space.
    • Three-Sphere A manifold constructed by taking two solid balls and matching up the points on the (spherical) boundary of each. The set of points at a fixed distance from a point in four-dimensional space is a three-sphere.
    • Three-Torus The manifold constructed by connecting the opposite faces of a solid rectangular box.
    • Time A linear dimension ordering events sequentially within any given system or context. For modern cultures, time is normally progressive; that is, unidimensional and directional towards the future, but is also understood to be reversible, cyclical, divisible in epochs or phases, and to have varying velocity, or tempo, depending on the experience of durée. Its measurability is the subject of debate.
    • Topological Property (Topological Invariant) A property that is invariant under continuous homeomorphisms. Examples are connectivity, simple connectedness, dimension.
    • Topologically Equivalent Two manifolds are topologically equivalent if they are homeomorphic.
    • Topology The study of shapes and their transformation. The area of mathematics concerned with relationships between entities, points and lines, such that their properties are preserved under continuous deformations without tearing or suturing. It emerged from ‘analysis situs’ to become a study of manifolds and connectivity.
    • Torus The surface of a doughnut.
    • Two-Dimensional Manifold (or Two-Manifold) An idealised mathematical shape that models the surfaces of possible worlds. The region around every point can be mapped onto a sheet of paper (that is, the inside of a rectangle in the plane).
    • Two-Space The plane envisaged by Euclid that goes on forever in two independent directions. As a set, it is just the set of pairs of real numbers.
    • Umwelt The environment or world as detected, often partially, by the senses of a biological organism.
    • Virtual Real but ideal, or in Proust's definition, ‘Real but not actual, ideal but not abstract’. Virtualities are the qualities or latent capacities of an entity, and of intangible but real entities such as ‘community’ or ‘goodwill’. Virtualities are known through their actual effects, but do not have extension, therefore are not captured in the Cartesian understanding of reality. See Table 1.1.

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