spring-summer 2018 online project
Below is a transcript of an informal conversation that took place one morning in the early spring at the kitchen table. It was about the notion of space. What is it? Is there “a space” in general? Are there many different spaces? I was asking two mathematicians how one gets from an intuitive understanding of the term to a more formal description. Briefly, the conversation was about mathematical objects and their role in constructing and getting to know mathematical space.
I wonder if the way the mathematicians talk about the notion of space resonates with the artist’s practice. Would love to hear from you my dear artist friend. — W. S.
What Comes First: A Conversation at the Kitchen Table
RK: First you have objects, you have sets or you have geometric objects, but then you say “Wait a minute, where do all those objects live?” And they live in the space. For sets, there is a universe of sets, and for the geometric objects there is Euclidean space …
WS: So, you start from the object.
RK: Not that we start from the object, but there is this line of thought that first identifies the objects that are of interest. Space as a defined mathematical object is not unique. You don’t have to choose really, but in case of physics one has to make a decision, because if you are using mathematical space as some kind of a model of reality then you have to decide which space: weather 4-dimensional time space which curves this way or that way, or, if you go to quantum physics, then you are going to Hilbert’s space, so you are faced with a variety, diversity of spaces, it’s not “the space” where things happen but there are different spaces, and then there is a next step when you put all those spaces together and study them together, how they differ.
WS: What kind of spaces do you put together?
RK: For example, models of arithmetic or groups… or even functions, functional spaces. You can measure a distance between functions, you can ask how far is one line from another, it can be defined in different ways, because what defines space is either a concept of a distance, or a concept of a neighborhood, so you study those spaces. It gives you a global view, you look at all possible functions together, not a single object.
WS: Let me ask you a stupid question: first we started with the objects, they came first, then we wondered how they were functioning in space, and then we started talking about defining spaces …
RK: No, they are not functioning in a space, they form a space. Taken together.
WS: So, you start with objects, you put them together and they form a space. Does it depend on the kind of objects that you take…
AV: Usually what’s important is not that you bunch them together, but how they interact with one another. So, I have a bunch of parallel lines, the fact that they are parallel tells you something important, or you put another family of lines going in different directions, and you can compare globally the two directions and you form a new level, a new layer, where all the former objects start behaving almost like points, and now, as Roman said, you can start compering them. There is a general property that captures the notion of the space, the way we are placing these functions, they are not thrown into a bag, but they are placed in certain ways which have certain coherent properties.
WS: I imagine that you don’t take objects at random, you select certain objects having something in mind.
RK: Look, in geometry we are interested in some properties of geometric figures, you have a right triangle and want to know how one side is related to another, or you have lines which cut through the midpoint of a triangle. But you aren’t interested in any old triangle, you are interested in some special triangles, because they are perhaps used in architecture.
AV: You start seeing a lot of things by putting a lot of triangles together, and also may be by putting lines, parabolas, and circles, you put them all together and start making them to interact not focusing on one triangle, or seven, or seventeen triangles, but many of them, all possible triangles, and then you start seeing things that you couldn’t see while looking at one particular triangle, you start seeing regularities, you also start seeing singularities.
WS: Not only that you take many different triangles, but you throw in other geometric objects in order to see certain properties…
AV: Yes, you have a line and the line moves and when the line moves it creates a parabola, but it moves in a certain way, there are some proportional distances, you can only see the property when you see the whole family of lines, the parabola, the new shape, appears only from the interaction of actually infinitely many lines …
So that’s a space, this emergence of something new out of the family of objects that have some regularity. That’s how people start being interested in looking not at one object but at collections of objects that are somehow smoothly organized along some property.
WS: In other words, when you analyze the objects, compare them, see the relations, through that process a notion of space emerges.