**Dancing Ordinal 5**

** **This project arose from my interest in gesture and the body’s role in the business of thinking; in particular, mathematical thinking.

As one knows, mathematical thought is abstract, its ideas and entities are baffling and alien to many. Perhaps one reason for this is because mathematical entities – numbers, equations, relations, groups, spaces, functions, and operations – are disembodied. They are real but not actual; they are virtual: invisible, intangible, ideational; things of potentiality, imagination and thought. Their existence, their meanings, how they behave, are controlled by a vast language of written signifiers — symbols, ideograms, and diagrams manipulated according to rules and schemes whose variety and complexity far exceed the familiar syntactical resources of natural languages. Might it be possible, however, to transduce mathematical ideas from this purely symbolic realm into a material medium?

What would it mean to give material form, to actualize a mathematical idea? Could one capture a mathematical abstraction in a concrete state of affairs? Could one realize it dynamically as a physical process in time; for example, as an embodied performance, such as a dance in physical space or a musical composition in sonic space? What follows is a response to these questions in the shape of a dance-piece that embodies the mathematical idea enshrined in the diagram of a basic mathematical object.

The diagram is drawn from a language of diagrams, Category theory, mathematicians have developed over the past sixty years. The concept of a category is extremely simple and mathematically generic: all of contemporary algebra. much of topology as well as stretches of mathematical logic are now described in its terms. A category is a collection of objects A, B, … with arrows f, g, … between them, where the arrows obey the following three axioms. Equations between arrows are always expressible as commuting diagrams.

**Axiom** Contiguous arrows f and g compose to form a new arrow g o f.

**Axiom** Every object A has an **identity arrow** id_{A} to itself which is inert, in the sense that for any arrows f and g going in and out of it, the equations id_{A} _{o} f = f and g _{o} id_{A} = g hold. Observe, id_{A} is the analogue for the operation of composition what zero is to the operation of addition and one is to multiplication.

**Axiom** The composition operation is **associative: **the two possible paths h o (g o f) and (h o g) o f from object A to a fourth object D are equal.

The idea is to invent a movement scheme that would enable these two diagrams to be danced. Since they are the axioms, any diagram of objects and arrows obeying them. In other words any category could — in principle — likewise be danced. The scheme is as follows: objects A, B, … are interpreted as fixed locations in space; an arrow f from source A to a target location B is interpreted as a path of a dancer; equality f = g between arrows is interpreted as the dancers performing f and g must arrive simultaneously at their common target.

The particular diagram chosen to enact this scheme is the one representing the ordinal number 5 in the language of categories:

Ordinal 5

The scheme, illustrated below, has dancers 1 through 6 all starting simultaneously at A, obeying the simultaneity requirements at C, D and ending all at the same time at E.

Ordinal 5 with performers

The scheme of paths and synchronies is only the abstract form of a dance. To create a dance proper, it needs to be inserted into a choreographic space: particular routes on the dance floor realizing the arrows, as well as gestures and styles of movement need to be constructed and combined into an ensemble. This was the work of Jeanine Thompson, movement professor in OSU Theatre department. The dance she created, *Ordinal 5*, was performed, with music by Dan Scott, by students from the departments of theatre and dance at the Ohio State University, at the Tate Modern, London in November 2011.

A film of the project, with the same title, directed and edited by Janet Parrott is available at https://vimeo.com/80409215