Mathematical Movement: Gesture
Mathematics is infected by the ancient idea of being a science of quantity – which is doubly false, being neither a science nor more necessarily concerned with quantity than anything else. It is a practice and comparable to dance. Paul Valery[i]
In this sense [about a blindfold, solo dance piece], the thought of movement coincides with a movement of thought. Rather than being abstractly separated and antecedent to the moving body, thought coincides with movement in the critical moment of its appearance. Stamatia Portanova[ii]
How might movement and in particular gesture, a basic physiological capacity of all animate bodies, be connected to the in-animate — invisible, impalpable — objects of mathematics? What is the link between the movement of abstract mathematical thought and physical movements of the human body? Between virtual and actual entities? Certainly, gesture is deeply embedded in human culture, prior to, along with, and inside spoken language, as a means, a constitutive principle, of thought; and, as we shall see, the same is true for mathematical language.
This should not be surprising: before all else, there is gesture’s role in the production of ‘the human’. Gesture is the ur-medium of hominization, a crucial route and operative means of it: the proffered breast, the smile, facial ‘expressions of emotion’, tones of voice, the innumerable gesturo-haptic movements of nurture, control, and communion (cradling, pointing, eye contact, turn-taking, and so on) that scaffold the production of a psyche. Gesture, in short, serves to induct the infant into speech and initiate the psychic mirroring of the other that allows a human subject to come into being. As a result, it is a source of pre-linguistic psychic affect, employed with powerful effect in rituals, ceremonies of social cohesion and collective enunciations, not least the many kinds of disciplined body practices – gestures of submission and piety — prescribed for their adherents by religions. And, of course, art forms such as dance, music and song deploy figured visuo-kinetic and sonic movements of the body – precise gestures – as their semiotic and ontological resource; a single move of a dancing body, for example, is an assemblage of choreographed disciplined movements – gestures — at many levels: from neuro-somatic and proprioceptive actions to those which relate the individual body to an ensemble of gesturing bodies. Theatrical art is predicated on the invention and staging of gesture: classical rules of pose, posture, delivery, gait for actors; Bertolt Brecht’s post-Aristotelean concept of theatrical gestus — gesture signifying at the level of the entire play; Antonin Artaud’s theatre of cruelty where “gesture … instead of serving as a decoration, an accompaniment of thought, instead causes its movement” (1958, 39); contemporary forms of dance- and movement-theatre in which gesture is a means of narration.
Less directly, for Gilles Deleuze the cinema is before all else a ‘cinema of bodies’, an art form within which what he calls the ‘movement-image’, a spatio-temporal entity that is to be understood corporeally, precisely as a gesture, is the irreducible element of filmic thought. According to Giorgio Agamben ‘to gesture’ initiates the ethical, “What characterizes gesture”, he insists, “is nothing is being produced or acted, but rather something is being endured. The gesture, in other words, opens the sphere of ethos.” (2000:57) For Ludwig Wittgenstein, buildings, as soon as they exceed the purely instrumental, cannot but express ideas, display attitudes, make gestures in relation to their inhabitants and surroundings. “Architecture”, he writes, “is a gesture, [it] expresses a thought, it makes one want to respond with a gesture.” (1980,22e) From a phenomenological perspective, Maurice Merleau-Ponty insists, one is “led to recognize a gestural or existential significance to speech” which is the “the subject’s taking up of a position in the world of his meanings.” (1962:193) Similarly: “Wittgenstein’s finest insight”, Antonio Negri tells us, “is precisely that language is a gestural form: to speak is to gesture [gestire], that is, to manage the body [gestire il corpo]; he taught us that language is all there within the body.” (Negri 2004:164)
Can mathematics be understood in terms such as these? In what sense could an abstract symbolic language be ‘all there within the body’? How might gestures be an irreducible element of mathematical thought?
Mathematics is an operational practice. Mathematicians speak constantly of performing or executing or constructing or carrying out something such as a calculation, a geometrical construction, a computation, an algebraic operation, a mapping, an algorithm. They engage in adding, multiplying, differentiating, integrating, verifying, averaging, triangulating, along with numerous other operative practices, including the key actions of defining object-concepts and proving (showing, demonstrating, validating) a theorem. The objects of all this activity, the mathematical entities being acted upon, as well as the procedures and narratives organizing them, are immaterial; the actions occur not in actual space but in the virtual space of thought, instantaneously, at ‘infinite speed’ as it is said. Where are the bodies of the mathematicians doing the thinking here? What’s their connection, if any, to the carrying out of these virtual actions? Or, before that, how were bodies involved in inventing or creating the entities in the first place? For the most part, these bodies sit, scribble symbols and diagrams, stare into the distance, go walk about, and sketch demonstrations on blackboards. It is difficult to make any meaningful link between the nature of these bodily activities and the conceptual activity or origin of their thoughts. It would seem, as many would have it, that mathematical activity is in fact incorporeal; creating and practicing it, and certainly its content, has nothing to do with the body. Seen thus, mathematics is entirely the work of res cogitans, a species of pure, disembodied thought, a vast thinking machine that discovers absolute truths about invisible un-embodied objects.
But this long, widely held view of mathematics, though natural when grappling with these invisibles – when thinking and imagining them somehow ‘out there’ — is mistaken. On the contrary, mathematics’ links to physical movement, though mediated, are undeniable: each of the primary object-concepts of the mathematical universe – number, relation, set, function, operation, variable, line, point, space, equation – can be seen to emerge from a small set of pre-conceptual physical actions, disciplined bodily movements or gestures, that, with their subsequent transformations, make up the corporeal wherewithal of mathematical thought. In other words, contemporary mathematics, though habitually understood in terms of static disembodied object-concepts, is constructed in/by a language whose basic conceptual vocabulary is rooted in gestural movement–schemata of the body. Chief among these are: the gestures of pairing two things together; combining two things to make a third; replacing one thing by another; pointing at a thing; showing, exhibiting or manifesting a thing; displacing or extending the body in its space; making/altering a mark; and the meta-gesture of repetition, of doing the gesture again. So that, for example, the object-concept ‘number’ can be seen as rooted in the gesture of making a stroke, an undifferentiated mark, and then repeating it; likewise the object-concepts ‘equation’, and ‘relation’, are different conceptualizations of the gesture of pairing; and a ‘binary operation’ the gesture of combining two things to make a third; the object-concept ‘function’ is constructed out of two collections of things by repeating the gesture of pairing them; a ‘line’ or ‘curve in space’ realizes the gestural displacement of a body, of a thing; and so on.
These primary objects form the conceptual matrix from which mathematics develops, augmented by concepts invented to solve external problems of science and transformed through an internal dynamic of self-appropriation and recursive colonization that mathematics performs on its own products. In this, object-concepts can pair with each other and fuse to create new genres of objects and relations between them, as in a topological group or an entity in algebraic geometry; relations between objects can become objects in their own right, as in the action of counting becoming an infinite sequence, or the action of ordering things is objectified as a token of an order-type, giving rise to an ordering of types. In fact, any mathematical procedure or activity operating on objects can, in principle, become an object able to be operated upon; either by another object or by the same object recursively, for example, as in function of a function, or a category of categories. This self-enfolding dynamic of mathematics is without any apparent limit, guaranteeing a potentially infinite expansion of mathematical problems and thought from within itself.
The primary objects, and their transformations, mask any evidence of their embodied origins: once it has emerged, its movement folded into a formal mathematical object, a gesture disappears from view; its movement has become the mathematical attributes and agency the object possesses, its potential for reacting with and operating on other objects. To say more about the process behind the enfolding, whereby a gesture might give rise to and inhabit an abstract mathematical object, one has to consider the gesture before it becomes implicit in the object. This entails examining the response to problems outside mathematics; finding sites in the development of the subject at which a gesture is introduced into the language of mathematics as a solution to a problem; and examining its passage from the body to its entry and disappearance inside a new mathematical object-concept. Precisely this is the project behind the mathematician-philosopher Gilles Châtelet’s historical account of mathematical thought.
His essay Figuring Space playfully introduces itself as drawing the “moral of a vaudeville play with three protagonists whose disputes have animated various debates about ideas for two centuries”. (1) It tracks the creation of mathematico-geometric space through the triple optic of mathematics, physics, and philosophy in the form of Gottfried Leibniz’s concept of a flexible and elastic space and Gilles Deleuze’s ontology of the virtual. Châtelet opens up the operative meaning, embodied origin, and intellectual potential – the virtuality — of such fundamental concepts as ‘point’, ‘space’, ‘dimension’, ‘curve’, and ‘line’ through a series of historical case studies drawn from the work of Einstein, Maxwell, Grassman, Ampere, and Hamilton. The project is to re-think the geometric/scientific construction of space through gesture and diagrams, particularly the “patrimony of those [gestures] that set it alight and multiply it.” (14) His aim for mathematics, understood as the creation of solutions to problems from sources external to it, is to reveal this patrimony.
Mathematicians think about disembodied, abstract entities. Châtelet articulates two prostheses they employ to overcome the difficulty of engaging with and thinking about invisible, immaterial ‘things’. One — semiotic, public and formal — is the symbolic apparatus, “the official crutch of the literal text, which accounts for the carrying out of the operations and which guarantees the transmission of knowledge”. The other — sensual, virtual and ruminative — is “a more subtle crutch, reserved for initiates, who are able to sense a whole network of allusions interlaced with the literal text and continually overflowing it.” The latter prosthesis is the one that interests him, namely the nature of this overflow; its virtual resources and their role in the creation/discovery of new symbolized operations and functions.
Refusing the Aristotelean division between movable matter and immovable mathematics, Châtelet insists throughout that mathematics cannot be divorced from “sensible matter”, from the movement and material agency of unconscious as well as conscious bodies. Mathematics is an “embodied rumination”, inseparable from the contemplative, a-logical and intuitive operations of thought.
Emphasizing the creative importance of what is before thought, what is sensed, he cites mathematician Andre Weil’s observation that “Nothing is more fertile, all mathematicians know, than these obscure analogies, these murky reflections of one theory in another, these furtive caresses, these inexplicable tiffs; also nothing gives as much pleasure to the researcher […] as when the illusion vanishes and presentiment turns into certainty.” (7) Certainly, many a-logical, barely grasped mathematical ‘sensings’ – allusions, hunches, intuitions, suspicions, feelings, convictions, premonitions and pre-knowledge — dance in and out of everyday mathematical thought.
Châtelet’s project tracks the movement of thought, the transition from “figures tracing contemplation” to “formulae actualizing operations”; a process whereby “metaphors shed their skin”, metaphysics catalysizes thought, and the “passage from premonition to certainty is cemented”. Pulsing through this movement, folded and implicit in it but invisible within the symbolized operations that enshrine it, is gesture. Citing Jean Cavailles’ insistence that one should rediscover the “central intuition […] graspable in action of a theory”, he takes from this a notion of gesture that is “crucial in our approach to the amplifying abstraction of mathematics.” A notion, Châtelet insists, that “eludes rationalizing paraphrase, […] metaphors and their confused fascinations and, above all, formal systems that like to buckle shut a grammar of gesture.” Moreover, not only is gesture not formalizable, but it does not operate as a sign: it is neither referential “it doesn’t throw out bridges between us and things”, nor intentional, the grasp that it exerts is not pre-determined, “no algorithm controls its strategy.” It would be better, he says, to speak of a “propulsion, which gathers itself up again as an impulse, of a single gesture that strips a structure bare and awakens in us other gestures.” He emphasizes that gesture refers to a “disciplined distribution of mobility before any transfer takes place: one is infused with the gesture before knowing it.” Gestures are not in other words, intentional acts; their effects cannot be understood or explained in terms prior to their action; their mathematical consequences are created “before one knows it”. (9-10)
The propensity for a gesture to “awaken other gestures” enables it to “store up all an allusion’s provocative virtualities without”, as he puts it, “debasing it into abbreviations” (10) Virtuality concerns what is actualizable in an event, its potential, all the futures it could/might give rise to. Unlike the possible, which refers to determinations that are fixed but lack the conditions to realize them, the virtual is inseparable from tensions, problems, and open questions. And, since for Châtelet mathematical creation is primarily the business of finding determinate solutions to problems, virtuality is the source for the novel abstractions solutions demand: “Virtuality invents and decides on a mode of elasticity: it prepares, cuts out and propels new plastic units. Virtuality awakens gestures: it solicits determination, it does not snatch it.” (20)
To elaborate, Châtelet introduces a notion crucial to his account of the formation of mathematical abstraction, that of a diagram. But, in contrast to its usual meaning as a visual representation, he uses it in a radically different sense. He takes this from Gilles Deleuze, for whom it is related to the capacity of matter to become an active agent in the creation of form.[iii] As such, it is intimately linked to the potential of ‘sensible matter’ and the body’s gestures to engage mathematical thought and produce novel mathematical objects. Specifically, he understands diagrams as arrested gestures, gestures caught mid-flight in the creation of a formal concept. “A diagram can transfix a gesture, bring it to rest, long before it curls up into a sign.” Diagrams distill action and experience, they function as a “technique of allusions” that relay gestures and he speaks of dynasties of them, “families of diagrams of increasingly precise and ambitious allusions”, playfully adding “For those capable of attention, they are the moments where being is glimpsed with a smile.” (10) Diagrams are distinct from metaphors, though in practice, in the moment of innovation, they occur together as the common fruit of a gesture. Discussing “the screw as bold metaphor” in Maxwell’s work on electromagnetism, he describes how it “endows the length with perforation power through a flick of the wrist [as] part of a whole set of diagrams and metaphors.” (177). True, like metaphors, diagrams can “leap out in order to create spaces and reduce gaps”, but unlike metaphors, they can “prolong themselves into an operation which keeps them from being worn out.” Diagrams, then, are always unfinished business vis a vis the presence and action of gestures: “when a diagram immobilizes a gesture in order to set down an operation, it does so by sketching a gesture that then cuts out another.” (10)[iv]
Evidently, Châtelet is indifferent to diagrams in the everyday sense of visual representations of concepts or procedures; they form no part of his scheme of mathematical ontogenesis. Likewise, with the entire (visual) apparatus of mathematical symbols, which he diminishes as no more than a “crutch” necessary for the carrying out of operations and transmitting knowledge. This means that his understanding of mathematics has no purchase on its modes of internal growth, since the mechanisms of self-appropriation fueling it are inextricable from definitions, notation systems, and the formal, symbolic and linguistic dimensions of mathematical thought.[v] In relation to the question of mathematical movement, then, I shall change tack and approach diagrams from a direction quite outside Châtelet’s approach. Specifically, one that understands them as semiotic/linguistic devices, representations, or rather, presentations of mathematical object-concepts built into them.
To this end I shall shift attention to an algebraic language that conceptualizes mathematical entities as diagrams of arrows between objects. The language is that of categories. Introduced some 60 years ago, category theory has since rewritten the presentation of algebraic objects and colonized large areas of topology, mathematical logic and theoretical computer science.[vi] As we shall see, the diagrams, far from freezing motion mid-flight on its way to curling up into a sign, is an essential element of the signifying apparatus itself.
Simply put, a category is a bunch of objects with arrows between them. (The formal name for arrow is morphism as in iso- and homomorphism). The objects can be any kind of mathematical entity – numbers, functions, knots, groups, relations, topological spaces, graphs, and (importantly) arrows and categories themselves – whilst the arrows must satisfy the following three simple axioms, each of which can be expressed syntactically or equivalently as a commuting diagram; meaning any two paths in the diagram with the same source and target objects are equal.
Composition. If f: A -> B and g: B -> C are arrows from A to B and B to C respectively, then there exists an arrow g o f: A -> C from A to C which is their composition. In other words, a category is closed with respect to concatenation of arrows.
Identity. Every object A has an identity arrow idA to itself which is inert in the sense that for any arrows f and g going in and out of it, the equations idA o f = f and g o idA = g hold. Observe, idA is the analogue for the operation of composition what zero is to the operation of addition and one is to multiplication.
Associativity. The operation of composition is associative. If f: A -> B, g: B -> C and h: C -> D then f o (g o h) = (f o g) o h. In other words, the two possible paths from A to D are the same.
(We follow standard practice here and omit the identity arrows from diagrams.)
Our aim is to externalize the movement inside a category diagram. There are several features of such diagrams that commend them for such a project. In addition to presenting concepts, such diagrams are integral to expositing and deploying them: they allow one to think diagrammatically: one can construct a concept or prove a theorem by tracing paths of connections through a diagram and seeing that it commutes. More importantly here, they encapsulate movement. What is immediately striking about category language, in contrast to that of sets, is the suggestion of motion evoked by an arrow. Arrows are of course used universally as signifiers outside mathematics, as a semiotic for a movement, a transformation, a change of state from source to target, an itinerary, a causal chain of consequences, a flow of information, a path to be followed, and so on. In C. S. Peirce’s vocabulary, arrows are at once icons, symbols, and indices of a directed dynamic.
Given a diagram of category theory, made up of objects, arrows, and equality such as for example one of the axioms just presented defining the formalism, how might we actualize the movement inhabiting it? We need to interpret the objects as fixed entities, the arrows as movements in some sort of space, and the equality relation between arrows as a relation actualized in this space. There are several possibilities. The space can be that of physical locomotion, the objects as fixed locations within it, the arrows actualized as possible paths, to be instantiated by particular routes from source to target locations. Or the space can be acoustic-aural, with objects as fixed rhythms or riffs, and an arrow indicating a sonic movement that effects a transition from source to a target rhythm. Or the space can be purely gestural, with objects as fixed gestural configurations, arrows understood as performers morphing from source to target gesture. Or the space could be chromatic, objects as fixed colours, arrows as modulations between them. For all the possibilities, the interpretation of the equality relation can be synchrony: namely, the equality between two arrows is interpreted as the simultaneous arrival of their actualizations at their target.
I’ll consider here only the first of these possibilities, which interprets space as familiar physical space and arrows as locomotion within it. Further, we shall concretize the scheme by having locomotion be an embodied performance, that is, a movement scheme enacted by dancers. Thus, objects will be designated locations in a performance space, arrows interpreted as paths traversed by the dancers, and composition of arrows realized as one path followed by another. Equality between arrows will be realized as synchrony: if two routes from object A to object B are equal then the dancers materializing them arrive at B simultaneously.
Observe that the scheme — an arrow from A to B is realized by a dancer moving from location A to B — allows each of the axiom-diagrams to be performed. For the identity diagram the arrow idA is interpreted as a dancer moving from A to A in a loop back to herself. For the composition diagram, dancer 1 moves from A to B to C, arriving at C simultaneously with dancer 2 moving from A to C. The pattern is repeated for the associativity diagram with four dancers: dancer 1 moving from A to B, dancer 4 moving from A to D, with dancers 2 traversing path A to C to D and dancer 3 the path from B to D, all arriving simultaneously at D.
Diagrams are themselves mathematical object-concepts and in fact the axiom-diagrams themselves are categories, that is, they satisfy themselves. As objects, they are, as one would expect, elemental: they are precisely how category theory defines/presents the ordinal numbers. The identity diagram is 1; the diagram of an isolated arrow is ordinal 2; the composition diagram is ordinal 3; and the associativity axiom is ordinal 4. Being able to perform the axioms suggests the possibility of performing other diagrams of the same kind. The axioms count to 4. The simplest diagram to perform beyond them will be the next number: the diagram of ordinal number 5, which will again be a category[vii]. Together a choreographer and I produced a dance piece, Ordinal 5, which performed the diagram of the ordinal number 5 as part of an event on Topology organized by the Tate Modern in London.[viii] The diagram annotated with the paths traversed by performers 1 to 6 is given here; all of them are assumed to issue from A and converge, at the same moment, at E.
Observe, there’s nothing special about either the choice of the dance medium to realize the movement inhabiting the diagram or the interpretation of equality of arrows as simultaneity. As indicated, the movement could just as well be realized in a sonic or gestural or chromatic space. In the first, the overall effect would resemble the structure of certain piano pieces composed by Steve Reich, and in the second that of the video installation Quintet of the Astonished by Bill Viola.
Finally there is the advent of digital gesture. Digital motion, technologies of motion capture, enable gesture to be recorded, electronically coded and digitized. And not merely recorded, but transmitted, studied, altered, processed, simulated, invented, and programmed; to be deployed/implemented across digital environments from touch screen/haptic interfaces to remote surgery and the control of military drones. Several millennia after writing enabled speech to be captured, motion capture has provided gesture with its electronic medium. The latter is larger in technical scope than its predecessor: recording gesture encompasses recording speech, since speech, after all is a particular (coded, auditory) form of gesture. What writing did to/for spoken language, digital capture is beginning to do for gesture. With the medium come new modes of visual representation, new kinds of diagrams. And how a movement inhabits or is captured by a diagram is embedded in the larger question: how are the contemporary practices of digital visualization producing, capturing and inflecting thought. But this is outside the focus here (see Rotman 2008).
To conclude: one can understand mathematical motion with Châtelet, as the movement of/within mathematical thought from the body’s disciplined mobilities by way of their frozen forms to formalized abstractions. Or one can, through the kind of project I’ve just outlined, arrive at mathematical movement from the differently conceived visual diagrams of category theory. The two conceptions link mathematical and corporeal motion from opposite directions: the first from the material body through the Deleuzian diagram to abstract, immaterial object, the second from diagrammatically presented abstract objects to bodies performing in space. Each reveals, folded deep inside every mathematical object, the gestures of bodies; a fact that, among other things, is central to the ability of these very bodies to deploy mathematics to successfully map out and predict the world they physically experience.
Agamben, Giorgio. Means without Ends, Translated V. Binetti and C. Casarino. Minneapolis: U. of Minnesota Press, 2000
Artaud, Antonin. The Theatre and its Double. Trans. Mary Caroline Richards. New York: Grove Press, 1958
Châtelet, Gilles. Figuring Space: Philosophy, Mathematics, and Physics. Translated Shaw, Robert and Zagha, Muriel. Dordrecht, Kluwer Academic Publishers, 2000
De Freitas, Elizabeth and Sinclair, Nathalie. Mathematics and the Body: Entanglements in the Classroom. Cambridge U. Press, to appear, 2014
Krauthausen, Karin. “Paul Valery and Geometry: Instrument, Writing Model, Practice”, Configurations, 18, (3), pp. 231-49, 2010
DeLanda, Manuel. “Deleuze, Diagrams, and the Genesis of Form”, Amerikastudien, 45, (1), pp. 33-41, 2000
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Portanova, Stamatia. “Infinity in a Step: On the compression and complexity of a movement of thought” Inflexions, 1.1, consulted January 2014 www.inflexions.org.
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Rotman, Brian. Mathematics as Sign: Writing, Imagining, Counting. Stanford U. Press, 2000
Rotman, Brian. Becoming Beside Ourselves: the Alphabet, Ghosts, and Distributed Human Being. Duke U. Press, 2008
Rotman, Brian. “Topology, Algebra, Diagrams”. Theory, Culture & Society, 29 (4/5), pp. 1-14, 2012
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[i] Quoted in Krauthausen 2010, p. 245
[ii] Portanova 2014
[iii] More specifically, it has to do with the refusal of a hylomorphic understanding of material morphogenesis, that is, the imposition on supposedly inert matter of an external idea or pattern, and, an embrace of the contrary idea of the internal production of form from within matter itself. See De Landa 2000.
[iv] Concrete examples of this ‘cutting out’ process have been shown to arise in the mathematics classroom. In fact, inspired by and extending his essay, Elizabeth de Freitas and Nathalie Sinclair (2014) have theorized ‘the body’ and its gestures to include its material environment, allowing an amplification of Châtelet’s gesture-diagram account of mathematics to emerge in practical work with students learning mathematics.
[v] The issue here is the manner in which the creation of mathematical ‘content’ is or is not extricable from its symbolic apparatus. I claim that it is not. That, while appearing to describe (refer to, organize, name) a universe of entities external and prior to itself, the symbolic language of mathematics, on the contrary, has a causal or constitutive relation to their character and existential status. Thus the ordinals are not, as commonly understood, names of ‘numbers’ that exist prior to them, but are co-existent with them, objects imagined to be named by them. See Rotman 1987 for a semiotic elaboration of this capacity in relation to the zero symbol, and Rotman 2000 for a wider discussion. For comment on a much earlier exploration of the crucial role of writing in the operation of mathematical thought and imagination by Paul Valery, see Krauthausen 2010.
[vi] More specifically, category theory conceives a mathematical object not as an isolated set but as an element within a collectivity – a category – of objects that are kin to it, objects linked to it by arrows. For the definitional and epistemological difference between set-theoretical and category-theoretical formulations of mathematics, see Rotman 2012, https://brianrotman.wordpress.com/articles/topology-algebra-diagrams/
[vii] By comparison to the diagrammatic language of categories, in set-theoretical language the ordinal number n is perforce defined as a set: specifically the set of all previous ordinal numbers starting from 0. Writing e for the empty set, and [x, y, …] for the set whose members are x, y, …, the following set s spells out the number 5:
s = [0, [e], [e, [e]], [e, [e], [e, [e]]], [e, [e], [e, [e], [e, [e], [e, [e]]]]]
[viii] My collaborator, who was primarily responsible for the aesthetic (individual gestures, movement style, affect) of the dance, was choreographer Jeanine Thompson (Theatre department, Ohio State University). The piece, Ordinal 5, was performed, with music by Dan Scott, by students from the departments of theatre and dance at the University, at the Tate Modern, London in November 2011. A film of the dance produced by Janet Parrott can be viewed at https://vimeo.com/80409215. An earlier, proof of concept piece, based on ordinal 4, was constructed by Nick Salarzar Sutil and performed at Goldsmith’s College, University of London.