Topology,Algebra,Diagrams

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 the late 19th century, Cantor’s theory of a hierarchy of infinite sets with different infinite magnitudes 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: regardless 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 (naïve, 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 structures 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 mathematics. 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 Poincaré’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 Poincaré’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 mathematics’ ontology, is inherently formalistic. The signifier-driven Bourbaki programme of ensuring that ‘naïve’ 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 Châtelet, 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 Spaces

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 equations 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 so on, 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 inherently indifferent to questions of measurement (differentiable or otherwise). 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 Poincaré 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. Poincaré, 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. Poincaré’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 properties and relations can be systematically translated into those of groups, which was Poincaré’s intention, namely, thinking topology algebraically.

Poincaré’s approach – finding invariant aspects of a space alive to algebraic formalization – is surely suggestive. Invariants are as important to transformations of social and cultural 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 socio-cultural 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 homomorphisms; 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 assignment 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 relationally, 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, categories 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 Lévi-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. Lévi-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 Poincaré’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 mathematics into axiomatically based structures of sets; and the problematic pole, according to which mathematics is produced in response to problems (inside and outside mathematics) whose solutions account for the ontogenesis and character of these very structures. For Châtelet, diagrams 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 immovable mathematics, Châtelet 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 Châtelet/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 Châtelet 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-symmetries within the given surface’. ⁸

Gestures of the body, disciplined mobility in space, figures and their deformations, lie at the origin of topological thought. Châtelet’s work suggests the possibility of accessing the intrinsic corporeality of topological 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 correlates 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 Châtelet, 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.