Research Focus Badiou Category Theory and Integrated Studies

Research Focus: Badiou, Category Theory and Integrated Studies • AU MAIS graduate - interdisciplinary focus • Reader of Alain Badiou - living French philosopher • Logics of Worlds With this work Badiou deploys Category Theory to develop a phenomenological account of the appearing of entities within a given world as a network of relations.

Category Theory Background • Developed from attempts to solve problems at the intersection of algebra and geometry. • Framework for all of mathematical theory Category Theory itself has been used to study the relations between branches of mathematics and to help solve problems across branches. Category Theory’s power comes from it’s ability to map and preserve abstract structures between domains.

Proposal for Integrated Studies • Category Theory: a tool for interdisciplinary research • Model how it operates between branches in mathematics for doing research between disciplines Category Theory provides the argument for the autonomous discipline of interdisciplinary research. It would be deployed as a method to clarify underlying abstract structures within disciplines and for mapping the relation of these structures between disciplines leading to the discovery of new disciplines beyond just hybridized disciplines. Worlds = Categories = Disciplines

Diagrammatic Orientation • Diagrams play integral role

Basic Category Theory Operators • Object represented by a dot • Relation represented with an arrow • Combined you have a directed arrow between two objects The arrow can be conceptualized as a morphism, relation, affect, action, touches, or sees.

Requirements for a category Require an identity arrow to each object. Composition of arrows to commute. Associativity maintained in composition: i ( g f ) = (i g) f º º Minimal operators and requirements results in maximal expressive power at the level of abstraction.

Usage 1: Internal to discipline (category) • Relations within a category can be mapped • Structure formations within a category can be identified, e. g. limit

Plato’s Four Virtues of State Chapter four of Plato’s Republic identifies four virtues of state with justice being the highest virtue:

Usage 2: Socrates Relates State to Individual “Let the discovery which we made be now applied to the individual - if they agree, we shall be satisfied” State Functors Individual

Usage 3: Categories of Functors - Emergence of IS State Individual

Summary • Category Theory: a tool for interdisciplinary research • Model behaviour inside a discipline • Model behaviour between disciplines • Model new discoveries in the relational space between disciplines.

Challenges • • Category Theory is still rigorous We still have the problem of initially mapping a discipline’s objects and relations.

Further Reading • Diagrammatic Immanence, Rocco Gangle • Mathematics of the Transcendental, Alain Badiou