2012 Topology, Algebra, Diagrams

Theory, Culture & Society, 29 (4/5), 1-14

Abstract

Starting from Poincare´’s assignment of an algebraic object to a topological manifold, namely the fundamental group, this article introduces the concept of categories and their language of arrows that has, since their mid-20th-century inception, altered how large areas of mathematics, from algebra to abstract logic and computer programming, are conceptualized. The assignment of the fundamental group is an exam- ple of a functor, an arrow construction central to the notion of a category. The exposition of category theory’s arrows, which operate at three distinct but deeply interconnected levels, is framed by a comparison with the language and outlook of set theory founded on the concept of membership; sets and their theorization having provided, famously through the Bourbaki initiative, the basic ontological and epis- temological vocabulary for defining and handling all mathematical entities. The com- parison with sets emphasizes how categories offer a form of diagrammatic argument and thought against set theory’s fidelity to syntax-based proofs; how categories invert set theory’s priority of objects and their attributes over relations by making the relations of an object to others of its kin primary; and how categories replace identity, that is, equality, between objects, by the weaker notion of isomorphism, restricting equality to identity between arrows. The article concludes with a return to topology and some remarks about the question of its possible use in articulating/ characterizing cultural dynamics.

Diagrams .. . for those capable of attention are the moments where being is glimpsed smiling. (Chaˆtelet, 2000: 10)

What is a topological space? Two responses: one, palpable and familiar examples, the other abstract and alien.

The first response visualizes movement in material space, bodies, con- crete contours of shapes and figures: discs, circles, a Mo¨bius strip,  the surface of a sphere, a cylinder, a torus, a sphere with holes; a  braid, a spiral, a knot threading through Euclidean space, perhaps a Klein Bottle. Topology studies the properties of spaces left unchanged by continuous deformations – stretching, twisting, folding, bending  and so on. Two spaces are topologically  identical  –  homeomorphic  – if  each  can  be deformed into the other. For example, the surfaces  of a sphere, cube, pyramid are homeomorphic; a coffee cup is homeomorphic to a donut, a coffee pot to a donut with a tunnel (equivalently a sphere with two tunnels). Topology offers  mathematical models of  continuous  analogue transformation in  contrast to the discrete changes and discontinuities of digital models. Deforming one space into another exemplifies a fundamental fact:  mathematical  entities  in  general,  not  just  topological spaces, are  never isolated individuals: they belong to types or species or families  of related objects to  which they  are structurally akin (of which more later) and their study involves the transformations between them that preserve their species kinship.1

The second response is the offcial, axiomatic definition. A topological space is a set X together with a family of its subsets – open sets – defined by the property that any finite intersection and infinite union of them is open; and a continuous deformation of spaces is a function f(x) from X to Y such that if f(S) is an open set in Y, then its inverse image f-1(S) is an open subset of X. This point-set formulation of an abstract or general topological space is the universally accepted mathematical definition of the concept. The definition is of maximum generality. It assumes nothing about the elements of X – they could literally be points in material space, or algebraic structures, or vectors, and so on – and nothing about the nature of the open sets beyond their definition. Moreover the ‘space’ is simply a set, a naked or featureless multiplicity: one cannot visualize it, nor does it make any reference to bodies or material space or physical movement. The idea of ‘space’ it offers, then, is unconnected to our palpable relation to curves and surfaces. It encompasses spaces with n dimensions and strange properties, spaces with an infinite number of dimensions, abstract spaces of functions such as Hilbert space, and so on.

Abstract Sets

The mainstream picture we have (have been given by the mathematical community) of mathematics since the early decades of the 20th century is couched in the language of abstract sets. By thelate 19th century, Cantor’s theory of a hierarchy of infinite sets with different infinite mag- nitudes had been accepted as legitimate mathematics, but not without the presence of  paradoxes,  such  as  ‘Is  the  set  of  all  sets  which  are  not members of themselves, a member of itself?’, which demanded an answer to the question: What exactly – rigorously – is a set?

In response, mathematicians axiomatized the concept by constructing a system of axioms whose intended objects were sets and whose only primitive, undefined relation was ‘is a member (element) of’. The axioms posited the existence of certain sets – the empty set, an infinite set – together with ways of producing new sets from existing ones (power set, choice set) and conceived equality between sets extensionally: regard- less of any intensive differences, sets are equal if every member of one is a member of the other. The axioms freed mathematics from the taint of paradox associated with the idea of ‘set’ and allowed mathematicians to extend the foundational project (already initiated by Weierstrass’s arithmetization of a limit in terms of sets of points and Dedekind’s definition of the continuum of real numbers as subsets of the rational numbers) to the whole of mathematics (Gowers, 2008: 771–2, 776).

A   collective   of   mathematicians   writing   under   the   pseudonym Bourbaki initiated the project in the 1930s to do just that, producing over the following decades thousands of pages of rigorously re-written mathematics in which every mathematical object and relation is a set and every mathematical argument, construction and definition is translated into  the language  of sets  subject ultimately  to  Boolean apparatus of logical quantifiers ranging over a universe of sets.

Ontologically, the enterprise successfully realizes a late 19th-century foundational desire, parallel to the contemporary atomism in physics, to

identify the fundamental ‘Dinge’ of mathematics. But its authors enclose it within an extreme, puritanical interpretation of mathematical rigour, according to which nothing – no notation, definition, construction, conjecture, concept, theorem or proof – is allowed to refer to or invoke or rely on any attribute, body or process of the physical world, not least any reference to the mathematician’s corporeality. An interdict that – significantly, for reasons to emerge – applies to drawing diagrams but does not how – could it? – extend to writing, to the physical inscribing of material symbols  by  mathematicians.  As a  result,  contrary  to  normal  (naı¨ ve, unformalized) mathematical practice,  which  is  rarely  free  of  figures, not one of their thousands of pages, Bourbaki proudly declare, contains a single diagram. Certain  features  of  this  set-based  characterization  of  mathematics stand out. First, objects are primary, relations between them secondary.

Although ontologically every mathematical entity whether an object (a number, group, topological space, vector, ordered set, matrix and so on) or a relation (a connection between objects, a function) is translatable into the language of sets, the ontology is not flat: the two are not imagined to be on the same level. Conceptually – epistemologically, definitionally – objects have a prior status: one defines a structure (a group, a space) as a set together with an operation on its elements or subsets, then one considers how, as an entity, it might be related to entities other than itself. The idea of a primacy of objects resonates with a recent initiative in contemporary philosophy, associated with Graham Harman (2005: 187– 9), which he dubs ‘a weird realism’. This is the project of ‘object-oriented ontology’ that ‘features a world packed full of ghostly real objects signaling to each other from inscrutable depths, unable to touch one another fully’. Rejecting the doctrine that being and thought are the same, adhered to by many from Parmenides to Badiou, Harman is obliged to revive the problem of causation and ‘reawaken the metaphysical question of what relation means’. The result is a post-Heideggerian phenomenology having little to do with set-theoretical mathematics.

Second, set-theoretic onto-epistemology is entirely intrinsic: objects are self-contained, isolated entities ultimately specifiable as structure of sets, their specific content an interior knowable without reference to that of any other object; a Platonic universe of ideal abstract multiplicities without histories or any relation to bodies. Translating the entire corpus of mathematics into the language of sets is an impressive and highly influential meta-theoretical achievement of 20th-century mathem-

atics. It has dominated theoretical discussion of mathematics as well as the norms for ‘correct’, ‘rigorous’ presentations of the subject for a better part of the century. It is reproduced within contemporary philosophy beyond its purpose of securing a rigorous ontology for mathematics. For Alain Badiou it settles the question of ontology as such, ontology in general, by virtue of ‘the equation that ‘‘ontology = axiomatic set theory’’, since mathematics alone thinks being, and it is only in axiomatic set theory that mathematics adequately thinks itself and constitutes a condition of philosophy’.2

The claim is contentious. Elaborating it would take us too far afield. Instead, we might note, by way of contrast, various examples of extrinsic relational approaches to the onto-epistemology of objects. Outside mathematics, in structural linguistics, the rejection of intrinsic content is precisely Saussure’s turn from a referential understanding of language with ‘positive’ terms to one in which the value of an item consists of its differential relations with other items. On a different terrain, the shift from internal psychological structure to external social relations lies behind the varied formulations of the individual ‘I’ by Vygotsky, Voloshinov and Mead. In mathematics, before set theory’s instauration, external relations (movements) govern Klein’s Erlanger Programm of 1872, which classifies geometries not in terms of the intrinsic properties of the objects, the figures they study, but through external movements, the symmetry groups the figures conform to. In Poincare´ ’s understanding of science it is the excision of the Kantian Ding an Sich: ‘The things themselves are not what science can reach .. . but only the relations between things. Outside of these relations there is no knowable reality’ (1905: 2). I’ll describe below a final, far-reaching example of  an  extrinsic  epistemology  of objects (one that arose in the wake of Poincare´ ’s study of  topological manifolds) provided by categories (see the section on   ‘Functorial Thought and Algebraic Topology’).

Third, set-theory’s foundational remit, its task of securing mathemat- ics’ ontology, is inherently formalistic. The signifier-driven Bourbaki pro- gramme of ensuring that ‘naive’ mathematics be translated into a first- order logic and vocabulary of sets and membership assures its fidelity to a severely abstract, linear, logico-syntactical language and style of exposition, a language that obviously excludes diagrams. If one understands diagrams pictorially, as visible icons of ideas, their exclusion reveals an essentially iconoclastic dimension of Bourbakist ‘rigour’ that lies deep in Platonist suspicion of images. But, though  visually  apprehended,  diagrams  operate  in  a  gesturo-haptic  register, which points to another aspect of their absence from the Bourbaki pages. As we’ll see later (see the section on ‘Topology and Embodied Space’) diagrams, according to Gilles Chaˆtelet, play a pivotal role in mathematical ontogenesis, operating in the space between the body and the written – in the present case set-theoretically framed – symbol. From this perspective, the exclusion of diagrams both protects the purity of mathematical objects from any kind of physical or corporeal contamination and cannot but be silent regarding mathematics’ becoming.

N-dimensional Space

Set theory’s formal – a-spatial – definition of a topological space cited earlier proves to be powerfully suggestive. It allows one to abstract the spatial nature – extension and orthogonality – of the three axes of Euclidean space and treat them simply as independent variables, replacing three by n to produce an n-dimensional topological space that, in its natural formulation, is a manifold, meaning it is locally Euclidean – any small enough region of it can be continuously mapped onto a region of Euclidean space. An important subspecies of manifolds, differentiable manifolds (introduced earlier in all but name by Riemann),  occurs when the mapping functions are not merely continuous but infinitely smooth, when they are differentiable in the sense of calculus, and so allow its techniques to be deployed in modelling the behaviour of material systems in time measured along the continuum (a one-dimensional differentiable manifold) of real numbers. Thus, given a dynamical system, it will have a number of degrees of freedom, independent ways it can vary, whose values are taken to constitute a full description of its state. For example, a bicycle has a number, say six, degrees of freedom, so its behaviour in time, the smooth changes in its state, can be understood as the path of a single point in a six-dimensional differentiable manifold whose topological structure is given by the differential equa- tions and vector field which the degrees of freedom satisfy. In this way, the dynamic behaviour of any physical process can be modelled as the path of a point in an n-dimensional space. According to Manuel DeLanda (2002), such a topological account of material change, coupled with symmetry-breaking discontinuities, offers a precise working out of the dynamic underlying Gilles Deleuze’s (1994: 214ff.) account of becoming, of the ontogenesis of the actual, the material world of bodies and physical processes, from the virtual.

The topological vocabulary of n-dimensional differentiable manifolds –      orbits, attractors, basins of attraction – bundled together with the (quite different) theory of chaos and fractals is widely evoked. But one can ask two questions about the appropriateness and cultural utility of this approach. First, how suitable is the concept of independent ‘degrees of freedom’, derived from modelling the dynamics of purely mechanical systems, for theorizing the modes of variation of cultural apparatuses? Second, does the approach go beyond a physics of culture, a science of its material forms? Even if one assumes its total success in capturing physical dynamics, why should the topological vocabulary on offer, based as it is entirely on differential equations and the techniques of calculus – the science of extensive, material change – have more than a limited relevance for the intensive phenomena and their dynamics operating in the socio-cultural universe? Certainly, one doesn’t, except within quantitative forms of sociology (statistics, sociometrics), physical geography and soon, find much use for models based on numerical or metric concepts, let alone calculus, in the literature of social and cultural theory. What, in any event, do infinitely smooth functions and the discrete infinity of numbered points on the one-dimensional continuum have to do with social time and cultural temporality? And why should time be modelled as a set of points along a line?3

Functorial Thought and Algebraic Topology

But topology, the mathematics of continuous transformations, is inher- ently indifferent to questions of measurement (differentiable or other- wise). Its interest is  with shape, with the spatial characteristics  of a topological space. Presented with a surface, for example, it asks: Does it have one or more holes? Does it have an edge, a boundary? What curves if any can be drawn on its surface? What are its formal properties? For surfaces of dimension higher than two – because they can’t be visualized as objects in familiar three-dimensional sensory space – answering such questions requires new methods, methods pioneered by Poincare´ for the case of manifolds. Manifolds are a fundamental species of topological space. All the familiar curves, knots and surfaces we began with are one- or two-dimensional manifolds. Poincare´, investigating the properties of three- and higher-dimensional manifolds, introduced in 1895 a radically new concept, the fundamental group of a manifold, initiating what came to be called algebraic topology, a field joining algebra and topology at the centre of contemporary understanding of the concept; and the field, as we’ll see, wherein the set-theoretical understanding of mathematical objects is revealed as conceptually inadequate.

Because they are locally Euclidean, one is able to define ‘paths’ in manifolds.  Poincare´’s  invention  was  to  construct  an  algebraic  object, namely a group, from classes of closed paths – loops – in a manifold. One picks an arbitrary point p in a manifold M and then considers all loops starting and ending at p. If a loop can be continuously deformed into another loop the two are identified as members of a single class. This leaves a set S of classes of loops through p not deformable into each other. A binary operation, composition of loops, is defined on S making it into a group – the fundamental group G(M) of M.

The assignment of the fundamental group to a topological manifold (akin to the 17th-century assignation of an algebraic equation to a geometric curve) allows manifolds to be conceptualized in terms of group properties. This is because G does more than assign a group to a manifold; it assigns group homomorphisms to continuous functions and it preserves the latter’s connections: if two functions compose, then so do their corresponding homomorphisms.4 This means that topological prop- erties and relations can be systematically translated into those of groups, which was Poincare´ ’s intention, namely, thinking topology algebraically. Poincare´ ’s approach – finding invariant aspects of a space alive to algebraic formalization – is surely suggestive. Invariants are as important to transformations of social and culture space as they are within mathematics and their dynamics will likewise exhibit structural features. Thinking algebraically, then, might be a fruitful route into the topology of sociocultural phenomena. Whether this is so is an open question since algebra, at least so far, is a little used resource in the toolbox of the social sciences.

The question arises as to whether the systematicity of this translation, a higher-level phenomenon that emerges from the interaction between algebra and topology, can be articulated within the set-theoretical language which defines and circumscribes them. A negative answer to this question – the recognition that a new language was needed – crystallized in the 1940s when Samuel Eilenberg, a topologist, observed to Saunders Maclane, an algebraist, that the latter’s calculation of a certain algebraic structure looked identical to a calculation in topology of a well-known homology group. Out of their joint attempt to say why this might be and to understand what mathematical moves were common to the two calculations, they formulated a language of what they dubbed ‘categories’ – a diagrammatic language of arrows and configurations of arrows that in the 60 or so years since its formulation has more than accomplished what they had in mind, leading not only to a re-writing of algebra and topology and their relation but to a radical impact on the theory of programming as well as a geometrical reformulation of mathematical logic.

What is a category? A category is a collection of objects A, B, .. . and arrows or morphisms A ! B from a source object A to a target object B, obeying a few simple axioms, namely, arrows A -> B and B -> C can be composed to form a new arrow A -> C, the operation of composition is associative, and every object has an identity arrow which leaves any arrow with which it can be composed unchanged.5 Each axiom can be expressed equivalently as an equation or as a commuting diagram of arrows. The components of a category are objects, arrows, composition of arrows and equality between arrows. It is widely believed that these four concepts are sufficient to encompass all mathematical structures in the following sense: ‘To each species of mathematical structure, there corresponds a category whose objects have that structure and whose morphisms preserve it’ (Goguen, 1991: 3). Some examples are: the category SET has sets for its objects and functions on their elements as arrows; the category GP of groups has objects, groups and arrows as homo- morphisms; the objects of the category TOP are topological spaces with arrows continuous transformations. Any kind of mathematical entity – a function, relation, graph, geometrical object, a family of sets, and so on – can, with appropriate arrows, serve as the objects of a category.

In particular, and importantly so, the objects of a category can themselves be categories. In this case, the arrows of the larger category of categories are called functors. Functors, arrows between categories, are functions  of  two  variables,  sending  objects  to  objects  and  arrows  to arrows in such a way that composition relations between arrows are preserved.6 The originating concept of algebraic topology, the assign- ment of a fundamental group of a manifold, is a functor from the category MAN of topological manifolds to the category GP of groups. Functors are the nuts and bolts of category-theoretic thinking, and relations between them are important, prompting them to be considered as objects in a category of functors. Relations between them, the arrows in this new category, are called natural transformations.7

Categories contrast to set-theoretical framing of mathematics in two respects. First, prioritizing arrows over objects mandates an exterior epistemology in contrast to the interior version built into set theory. Unlike the set-based version, an object in a category is understood rela- tionally, through external difference, not as an autonomous, internally structured entity; it is known and constituted entirely in terms of the arrows entering and exiting it from other objects. In other words, cate- gories think a mathematical object from the outside, in a ‘bio-social’ register from species to individual, and not as an isolated, self-contained entity whose relations to others proceed from the inside out. In this categories echo the external, sociologized epistemologies  of  Saussure and others, mentioned earlier.

Second, in contrast to the fixity of sets and the membership relation, arrows   and   composition   connote   movement   or   transformation. Categories deliver a dynamic logic through schemes of arrows that allow mathematics to be understood and practised as diagrammatic thought. Diagrams, interdicted by the Bourbaki enterprise as un-rigorous and extraneous to mathematical content, are not only a legitimate constituent of categorical language but are the means of definitions and proof. Categories also constitute, in their operation as an algebraic formalism, a species of structuralism. Not in the dominant mid-20th-century sense associated with Le´vi-Strauss who derived it from the linguist Roman Jakobson. For Jakobson  speech  is  constituted  from pairs of binary oppositions (vowel/consonant, acute/grave) that operate together to form triangular phonological structures. Le´ vi-Strauss  transferred  this schematic  from  phonology  to  anthropology  producing, for example, from the  binary  oppositions  culture/nature  and  normal/transformed applied  to  food  the  triangle  of  the  raw,  the cooked and the rotten. Not, then, the structuralism founded on oppositions between yes/no properties compatible/complicit with the  set-theoretic thinking of the period, with  its  ontology  of  identity   and pure unchanging ‘being’ wedded to a binary logic of excluded middle. Rather, a mobile structuralism of n-ary relations and transformations, a mathematical universe of things-in-formation.

The language of categories has successfully colonized large areas of contemporary mathematics, including the study of topology. The two contrasts with set theory indicate why categorical language might be a more relevant and interesting way of thinking for social and cultural theory in relation to topological spaces than that offered by the set- theoretical model of point-set topology. One can add a programmatic point. A category is a multiplicity, a family or species of kindred objects understood relationally and not as isolated individuals. It would be surprising if the theory’s success as a transparent language of mathematical structure had no transfer or application to questions of structure and relationality  within  social  and  cultural  multiplicities  One  possibility: adapt Poincare´ ’s approach and formulate functors assigning  algebraic structures – invariants – to the topological transformation of multiplicities. But again, given the dearth of algebraic thinking within socio-cultural theory (a reaction perhaps against the crudity of set-theoretic structuralism), it is difficult to evaluate the feasibility of the suggestion.

Topology and Embodied Space

That categories deploy diagrams in a substantive way differentiates them from set theory, but conveys no hint of a deeper sense of diagrams, namely, the pivotal role they play in mathematical ontogenesis. According to Deleuze, there are two poles of mathematical activity: what he terms the axiomatic, articulated here as the translation of math- ematics into axiomatically based structures of sets; and the problematic pole, according to which mathematics is produced in response to prob- lems (inside and outside mathematics) whose solutions account for the ontogenesis and character of these very structures. For Chaˆtelet,  dia- grams coupled with gestures are the very means of ontogenesis, a principal strand in the becoming of mathematical ideas, objects and relations. Refusing the Aristotelean division between movable matter and immov- able  mathematics,  Chaˆtelet  insists  that  mathematics  can  neither  be divorced  from  ‘sensible  matter’,  from  the   movement  and  material agency of bodies, nor from the  contemplative, a-logical and intuitive operations of thought: it combines them as ‘embodied rumination’. He offers a  material/corporeal account of mathematics, wherein gestures – which arise from ‘disciplined distributions of mobility’ of the body – are the physical vectors of mathematical thought. A gesture is not referential, it doesn’t ‘throw out bridges between us and things’ and it doesn’t operate along predetermined routes – ‘no algorithm controls its staging’. Gestures are not conscious, intentional acts: ‘One is infused with a gesture before knowing it.’ Gestures are not communicative acts by an individual ‘mind’: they are outside – before – the domain of signs, not subject to a pre-given signifier/signified code of interpretation. The gesture’s mode of meaning is enactive, it performs: it is a material event that engenders mathematical substance by virtue of occurring. It expresses thought, as Deleuze would say.

Diagrams arise in the wake of gestures, and they facilitate other gestures. A gesture arrested in mid-flight creates a diagram, a movement captured ahead of itself: ‘A diagram can transfix a gesture, bring it to rest, long before it curls up into a sign.’ And it is the source of – ‘it cuts out’ or ‘alludes to’ – another gesture. Gestures and diagrams are inseparable. Diagrams/gestures operate in a pivotal space: they are embodied acts that bridge the gulf between thought and the sign. On one side: intuition, a-articulated images, and rumination; on the other: syntax, representation,  symbols.  Observation  in  the  mathematics  classroom reveals the impulse to gesture and to draw diagrams as ever-present in learning  and  solving  problems.  This  being  so,  one  would  expect  a Chaˆ telet/Deleuze  account  of  embodied  ontogenesis  to  provide  useful insights for pedagogical research. Such proves to be the case; a contemporary study concludes that students’ diagrams are ‘precisely what Chaˆtelet  found  in  historical  developments  in  mathematics:  inventive ‘‘cutting out’’ gestures that interfere and trouble assumed spatial principles .. . and [which show] the emergence of new perspectival dis-symme- tries within the given surface’.8

Gestures of the body, disciplined mobility in space, figures and their deformations, lie at the origin of topological thought. Chatelet’s  work suggests the possibility of accessing the intrinsic corporeality of  topo- logical ideas through a kind of reverse engineering: retrieve the gestures that have been operationalized/internalized  into  symbols,  make  the problems they respond to and the intuitions guiding them physically explicit within a material context. Of course, many kinds of artwork incorporate topological and geometrical ideas in  material  form,  but they do so implicitly in the service of aesthesis and not as an explicit retrieval of an abstract, operationalized gesture. There are, however, conceptually motivated rather than artistic examples that might be noted. Here are two.

The hyperbolic plane is an abstractly defined mathematical object, a non-Euclidean surface whose curvature, opposite to the everywhere positive curvature of a sphere, is everywhere negative. A recent project shows how negative curvature, ‘the geometrical equivalent of negative numbers’, occurs naturally within the process of crocheting: if one increases the number of stitches in successive rows the resulting crocheted object curls inward and exhibits a negative curvature; one materializes in crocheted form the topology of a hyperbolic plane.9 Alternatively, rather than manually produced material objects, one can materialize a mathematical concept through physical performance. For example, as we have seen, the commuting diagrams of category theory are patterns of arrows that capture mathematical concepts. Interpreting arrows topologically, as movements of bodies in space, together with suitable physical correl- ates of composition and equality of arrows, delivers a movement scheme for a dance. Even simple concepts – counting and the ordinal numbers it produces – have interesting enough diagrams of arrows for the idea to work, as was recently demonstrated by the performance of a movement piece, Ordinal 5, based on the commuting diagram for the number five conceived in category theoretic terms.10

Observe in conclusion that locating topology in relation to the sensory modalities and activity of bodies is where we came in, with a picture of topological spaces as palpable spatial entities – Mobius strip, sphere, Klein Bottle, knots and so on. All known and studied in the period before set-theoretical thinking rigorously excluded the body. If, with Chaˆtelet, we accept the corporeal dimension of mathematical concepts, we might return to these palpable – gesturally accessible – origins and seek to extract topological concepts from them, hopefully ones of interest to social and cultural theory.11 For example, knots (and one is reminded of crocheting), which are an object of much contemporary research, offer a rich and heterogeneous site of topological complexity.12

It would seem that simple one- and two-dimensional surfaces are surprisingly  useful  sources  of  topological,  culturally  interesting  insights. Thus, as Sloterdijk has demonstrated in his study of sphericity, even the sphere, as banally familiar and simple a surface as one can imagine, can give rise to a topologically framed, philosophical anthropology, an existential account of spatial being, in terms of enclosures, containers, bubbles, globes and foam.

Notes

  1. In algebra, homomorphisms. For example, if A and B are groups with an operation of +, a homomorphism is a function f from A to B that preserves the operation, that is, f maps x + y onto f(x) + f(y). For vector spaces, func- tions from one space to another which preserve addition of vectors and scalar multiplication are linear transformations; and so on.
  2. See Smith (2004: 78), where Badiou’s ‘meta-ontological’ claim is exposited in relation to Deleuze’s opposing view of multiplicities.
  3. The question is outside the topic here, so a brief response by way of two way of challenging the one-dimensional continuum model of time. Far from infinite smoothness, the time-line might be discontinuous, infinitely un-smooth, composed of ever-smaller bits, its dimension a fractal, so that ‘linear’, successive time could have a fractal dimension that hovered (perhaps indiscernibly) around 1, or at a perceptible difference from 1. Or, abandoning a line entirely, time could be conceptualized as a surface, a fully two-dimensional temporality yielding non-linear space-like possibilities of ‘succession’. For example, time might be imagined, Michel Serres suggests, as a ‘crumpled handkerchief’,  thus  opening  the  possibility  of  distant  moments  being  in close temporal proximity to each other. There are two philosophical concepts of time: metric time, the one-dimensional irreversible medium in which all material and immaterial processes take place. Virtual time, expounded by Deleuze,  as  that  from  which  time  is  expressed,  a  becoming  inseparable from  these  very  processes.  Fractal  time  and  two-dimensional  models  of time differently problematize the dimensionality of the former.
  1. In other words, G is a particular kind of function from topological spaces to groups which preserves composition of continuous functions, meaning that the image G(f * g) of a composite of continuous functions f and g is the composite G(f) * G(g) of their images G(f) and G(g).
  1. Axiom 1: Composition. If f: A -> B and g: B -> C then there exists a composite arrow g * f : A -> C. In other words, categories are closed with respect to concatenation of arrows.

Axiom 2: Identity. Every object A has an identity arrow idA such that idA * f  = f * idA  =  f for any arrow f. In other words, idA is to the operation * what 0 is to addition and 1 is to multiplication.

Axiom 3: Associativity. The operation of composition is associative. If f: A -> B, g: B -> C and h: C -> D then f * (g * h)  = (f * g) * h. In other words, the two possible paths from A to D are the same.

  1. The double action of a functor G from a category C to category D goes as follows. G maps objects X and Y of C to objects G(X) and G(Y) in D. G maps arrows f: X -> Y and h: Y -> Z in C to arrows G(f): G(X) -> G(Y) and G(h): G(Y) -> G(Z) in D such that composition of arrows is preserved. That is, the G-transform of a composite G(h * f) is equal to the composite G(h) * G(f) of G-transforms (Figure 1).
  2. As an arrow between functors, a natural transformation must satisfy two objectives. If G and H are functors from C to D one needs to compare their images, that is, relate G(X) with H(X) and G(Y) with H(Y) as X and Y range over C. And one needs to compare the action on the arrows between f                                h                           G (f)                              G (h)
  3. Figure 1. A functor G from C to D.

the objects of C. We can only do this via a family of arrows in D – an arrow µX from G(X) to H(X) for each X in C. But G and H as functors also assign arrows in D to those in C. The naturality condition ensures that the two assignments of arrows are compatible. Functors can go in opposite directions. G might assign a group to a manifold and H a functor in the reverse direction, assigning a manifold to a group. In general such functors will not be opposites or inverses of each other but various conceptual versions of these; the concept of adjointness captures such an idea. A basic example: if G is a forgetful functor then F, a functor which produces the freest struc- tures possible that could have been forgotten by G, is adjoint to G.

  1. See de Freitas and Sinclair (2011: 18). This is an empirically based study of Chaˆtelet’s ideas framed within Deleuze’s  conception  of   mathematical thought. It is elaborated in the context of communication  in de Freitas (to appear).
  2. There are now large-scale, communally produced crocheted artifacts, fash- ioned after naturally occurring surfaces such as coral reefs (Wertheim et al., 2004/5). Conceived by the author. Performed at the Tate Modern, 19–20 November 2011. See Julian Henriques (2012) for a full report of the event and the topological sound art that accompanied it.
  1. The impulse here to return topology to a founding material and spatial intuition is pursued very differently in Richeson (2008), who presents the subject as growing out of and responding to Euler’s formula connecting the vertices, faces and edges of polyhedra.
  1. Lacan, for example, who incorporates topological concepts throughout his work,  advocates  a  knot-based  model  of  psychic  structure  based  on  the Borromeo knot. See Ragland and Milovanovic (2004).

Refernces

Chaˆ telet, Gilles (2000) Figuring Space: Philosophy, Mathematics, and  Physics, trans. Robert Shaw and Muriel Zagha. Dordrecht: Kluwer.

De Freitas, Elizabeth (forthcoming) ‘What Were You Thinking? A Deleuzian/ Guattarian Analysis of Communication in the Mathematics Classroom’.

De Freitas, Elizabeth and Nathalie Sinclair (2011) ‘Diagram, Gesture, Agency: Theorizing Embodiment in the Mathematics Classroom’, Educational Studies in Mathematics 8 December: 1–20.

DeLanda,  Manuel  (2002)  Intensive  Science  and  Virtual  Philosophy.  London: Continuum Press.

Deleuze, Gilles (1994) Difference and Repetition, trans. Paul Patton. New York: Columbia University Press.

Goguen, Joseph A (1991) ‘A Categorical Manifesto’, Mathematical Structures in Computer Science 1(1): 49–67.

Gowers, Timothy (2008) The Princeton Companion to Mathematics. Princeton, NJ: Princeton University Press.

Harman, Graham (2005) ‘On Vicarious Causation’, URL (consulted May 2012): www.scribd.com/doc/27861684/Harman-vicarious-causation.

Henriques, Julian (2012) ‘Hearing Things and Dancing Numbers: Embodying Transformation’, Theory, Culture & Society 29(4/5).

Poincare´ , Henri (1905) Science and Hypothesis. New York: The Science Press. Ragland, Ellie and Dragan Milovanovic (2004) Lacan: Topologically Speaking. New York: Other Press.

Richeson, David S (2008) Euler’s Gem: The Polyhedron Formula and the Birth of Topology. Princeton, NJ: Princeton University Press.

Smith, Daniel W. (2004) ‘Badiou and Deleuze on the Ontology of Mathematics’, in  Peter  Hallward  (ed.)  Think  Again:  Alain  Badiou  and  the  Future  of Philosophy. London: Continuum.

Wertheim,   Margaret,   David   Henderson   and   Daina   Taimina   (2004/5) ‘Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina’, Cabinet 16.

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