Category theory is a mathematical formalism that can be used to represent many aspects of mathematics within a coherent framework that uses 'mappings' (i.e. arrows) between objects (i.e. points). It was formulated by Saunders Mac Lane in the 1940s - that is, at the same time as other remarkable mathematical and technological developments were taking place which led to the development of Cybernetics (from the Macy conferences in 1948) and the succeeding developments that span out of Von Neumann, Wiener, McCulloch, Shannon, Mead, Bateson and Ashby in economics, computer science, telecommunications, psychology, psychiatry, anthropology, philosophy, education, art, management science, and so on. The metaphor of 'mapping' would have found particular resonance with the functional transformations and feedback loops of computers, and indeed, Category Theory has its strongest foothold in computer science departments as an important formalism for thinking about algorithms.
My first encounter with it was when trying to understand some fascinating work by Louis Kauffman on "Time" at the American Society for Cybernetics. Kauffman presented a way in which time could be seen to be immanent in processes of perception, producing an elegant description that showed how the square root of minus one could be seen to be a "clock". Powerful stuff but quite challenging in the depth of its implications and importance (I am convinced that 'time' is one of the biggest problems in cybernetics - no mechanism works outside time! - and I'm not entirely convinced by Kauffman's explanation, but no doubt my understanding will deepen). My reticence in accepting Kauffman's description of time partly revolved around my suspicion that cybernetics has not been entirely successful in dealing with 'absence'. Bateson wrote well about absence, but I still felt that the whole business of cybernetic 'difference' is too 'singular'. Differences to me are never 'points' - they are always surrounded by a context, much of which we can't see. Differences have dark matter.
But then a curious coincidence was encountering Alain Badiou's work. There's much in Badiou which is consistent with Critical Realism, which I have been interested in for a long time (but which also has some problems). Most importantly for me, Badiou too has dealt with absence. Equally important, he has taken on the problem of 'difference' - or what he prefers to call the 'event'. But the really cool thing about Badiou is he knows his maths - and in recent years, the branch of maths he has done most of his work is Category Theory. It's always when I get the same thing from two different places that I start to take a serious interest.
Since then I've been reading Badiou's recently published "Mathematics of the Transcendental", whilst glancing at Lawvere's "Sets for Mathematics" and other texts. Understanding emerges only slowly with this stuff. But the DIAGRAMS are fascinating. Here we have networks of lines and algebraic interpretations which say things about absence, and being. In fact, Lawvere has even written recently about Category Theory being able to characterise "becoming": phenomenology has always suffered from a lack of formalism (although Husserl's wrote some fascinating things about geometry).
Why are the diagrams fascinating? Because they look like social networks! In the early days of category theory and computer science, I guess the network that mattered was the wiring between components and the transformations that occurred. It seems to be that the archetypal image of computing now is the diagram of electronic utterances between people connected by a network, or the diagram of document terms in a corpus: the diagram of the relation of ideas. On the whole we are poor at thinking about the meaning of these topologies. Gordon Pask complained years ago that the digital computer fascinated us like a 'magic lantern' and actually stopped us thinking, rather than being a tool for thought. Von Foerster made a similar comment (in 1971!) that
The question is how to think about our existent problems. The problem is that technology is part of our existent problems. Social network diagrams and our poor understanding of what they mean are part of our existent problems. We need better ways of reasoning about the world, and by extension, reasoning about technology. That means using our analytical and critical capacities to dig into the underlying logic of what is happening to us. Maybe Category Theory can provide a way of doing that.
My first encounter with it was when trying to understand some fascinating work by Louis Kauffman on "Time" at the American Society for Cybernetics. Kauffman presented a way in which time could be seen to be immanent in processes of perception, producing an elegant description that showed how the square root of minus one could be seen to be a "clock". Powerful stuff but quite challenging in the depth of its implications and importance (I am convinced that 'time' is one of the biggest problems in cybernetics - no mechanism works outside time! - and I'm not entirely convinced by Kauffman's explanation, but no doubt my understanding will deepen). My reticence in accepting Kauffman's description of time partly revolved around my suspicion that cybernetics has not been entirely successful in dealing with 'absence'. Bateson wrote well about absence, but I still felt that the whole business of cybernetic 'difference' is too 'singular'. Differences to me are never 'points' - they are always surrounded by a context, much of which we can't see. Differences have dark matter.
But then a curious coincidence was encountering Alain Badiou's work. There's much in Badiou which is consistent with Critical Realism, which I have been interested in for a long time (but which also has some problems). Most importantly for me, Badiou too has dealt with absence. Equally important, he has taken on the problem of 'difference' - or what he prefers to call the 'event'. But the really cool thing about Badiou is he knows his maths - and in recent years, the branch of maths he has done most of his work is Category Theory. It's always when I get the same thing from two different places that I start to take a serious interest.
Since then I've been reading Badiou's recently published "Mathematics of the Transcendental", whilst glancing at Lawvere's "Sets for Mathematics" and other texts. Understanding emerges only slowly with this stuff. But the DIAGRAMS are fascinating. Here we have networks of lines and algebraic interpretations which say things about absence, and being. In fact, Lawvere has even written recently about Category Theory being able to characterise "becoming": phenomenology has always suffered from a lack of formalism (although Husserl's wrote some fascinating things about geometry).
Why are the diagrams fascinating? Because they look like social networks! In the early days of category theory and computer science, I guess the network that mattered was the wiring between components and the transformations that occurred. It seems to be that the archetypal image of computing now is the diagram of electronic utterances between people connected by a network, or the diagram of document terms in a corpus: the diagram of the relation of ideas. On the whole we are poor at thinking about the meaning of these topologies. Gordon Pask complained years ago that the digital computer fascinated us like a 'magic lantern' and actually stopped us thinking, rather than being a tool for thought. Von Foerster made a similar comment (in 1971!) that
"we have, hopefully only temporarily, relinquished our responsibility to ask for a technology that will solve existent problems. Instead we have allowed existent technology to create problems it can solve."That was before the internet. Things are worse now.
The question is how to think about our existent problems. The problem is that technology is part of our existent problems. Social network diagrams and our poor understanding of what they mean are part of our existent problems. We need better ways of reasoning about the world, and by extension, reasoning about technology. That means using our analytical and critical capacities to dig into the underlying logic of what is happening to us. Maybe Category Theory can provide a way of doing that.
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