functions

Today we discussed a couple of issues. We are considering what topology allows us to consider as a logic of organization difficult if not impossible to conceive with just geometrical concepts and tools. One way of looking at this is ontologically -- and that only means the way in which something is said to exist. If i asked you to describe this water bottle, this seems non-controversial: it has such and such geometrical properties, such and such measurements, and so on. But now i throw the water bottle and ask you to describe it mathematically. All of sudden geometry seems insufficient and we need another mathematical tool to describe the arc of its movement. As Eric pointed out, that would imply the introduction of time, the change of position, etc. It would imply calculus. In order to mathematically describe the bottle being thrown, we’d have to introduce the mathematics of calculus. But note: there are also two different ontologies here. The thing qua thing -- that is the object as such as static. Then there is the thing qua event -- the bottle moving through the air. Renee: Good point, once we draw the trajectory we reintroduce geometry, we reintroduce a geometrical artifact, a curve – but also note that the curve would not have the curvature without calculus. This also plays into our discussion of diagrams. The curve is a description of the behavior, it is a diagram. The curve doesn’t necessarily describe the arc of movement, though it could – what it describes the rate of change, as long as it is continous. Here’s an interesting question: if I draw the bottle in two dimensions using geometry am I making a diagram? Ok, so, what I am asking you to consider is the nature of a diagram and how it can be used to describe the behavior of events. If we consider Plato’s ontology in relation to the Meno and the Phaedo, the highest form of existence is really ideas and forms: Eidos/Form. The discussion of Equality in the Phaedo, and Beauty and the rest in Meno are discussion about Universal Principals which we know only as pure souls but which we have the capacity to recollect. As humans we acquire knowledge through experience, but this is contingent – not universal. The mathematical demonstration in Plato then belongs to a kind of Ideality – things like geometry are Universal facts, they are not contingent on our experience: they transcend it. Mathematics as always been thought of in these terms and architecture has continuously purchased Uinversal principals of meaning through using such models. This is partially what I mean by a model of meaning. Le Corbusier’s use of the golden ratio and the modular are such examples. Ok, now consider the example from Bentham’s Panopticon and Foucault’s discussion of it as a function of a function. It is a diagram of relations of force. The incredible thing is that it doesn’t have any particular formation as, say geometrical object or space – it is a way of networking and creating space. Deleuze thought that this was important: a diagram is not a thing, but a series of relations through which things come into being. They are more event-related. And this is also why he is critical of the Platonist tradition and offers an important reading of ontology through Stoic ontology and the way in which it privilages events over things or ideas. So, one way to consider the problem of what is topology is to consider it as having a different form of existence than objects. This does not define it, but helps clarify and distinguish it from geometry. When you eat and apple there is an apple and then there are a series of processes that convert that apple into something else. We can consider these functions – could we say the functions map the apple on to other functions, nourishment, energy, etc? Well, at least we could say the entire digestive process of which there a numerous different events is not really a visible process – its visibility is not the same as seeing the apple. But we could diagram those functions and consider their topological properties – that is their forms of continuity. And this might now help – we can’t really see as diagram as a thing, but rather as something that relates, that networks a series of functions. So it is not a picture of a thing so much as a state of affairs and in that sense has to be abstract. When Choisy discovered the principle of asymmetry in the Acropolis and introduced into architectural notation for the first time a vector describing the arc and movement of the spectator he was introducing a new ontological concept in architecture – the experience of walking through the site, not just seeing the building as such, and this is what accounted for the odd juxtaposition of the buildings. Le Corbusier took this diagram and turned it into the architectural promenade of which there are now countless variations of which Koolhaas’s work is just one. Here is a case in which a notion of organization had to take on features of topology (that is continuity of functions movement, building, space, time, position, etc.) through the logic of the diagram. What this introcued to architecture was the concept of the event. Tschumi’s work is principally based on this. It is principally diagrammatic.