Online project: Conversations
Below are transcripts of the informal conversations that took place in the early spring 2018. The first one was about the notion of space in mathematics. 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. Then I wondered if the way the mathematicians think about the notion of space could somehow resonate with the artist’s practice. This was a starting point for the second conversation, an attempt to solve the unsolvable hen-and-egg problem: does a painter start with a notion of space or an object?— 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: We started with the objects, they came first, then wondered how they were functioning in a space, and now we are talking about different 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 …
AV: Usually what’s important is not that you put 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 compare 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.
A Conversation in the Łazienki Park
WS: We were talking about the notion of space in mathematics and as a starting point we had objects, and then by putting them together, considering the relations between them, we somehow arrived at the notion of space. Let’s say we are now in the field of making representations—the medium of painting, in which we also deal with objects (representations of mental objects), but it seems that one needs to have a notion of a space first, that you can’t start painting not knowing what your space is…
AV: What about paintings in a cave, Lascaux cave paintings? There was no frame, I do not know if there was a space, there were just animals or people chasing the animals, or in the primitive paintings where there is a fish here or there …
WS: I think that in order to paint on the wall of the cave, you have to have some kind of an idea that this is a space for your objects.
AV: The sense of space seems to be inner in the early paintings, they don’t question it, it’s there to tell the story. What is this space? It’s like air or it’s like fish in the water, they don’t know they are in water, unless they are taken out of water. Normally we are not aware that we live in the air, unless the air becomes so bad, that we have to worry about it. I think that a canvas representing a space becomes an object of reflection once a problem, some limit arises with the original, implicit space of the canvas. I am not saying there was no space, I am just thinking that the awareness of space is heighten by dealing with limits. If we are dealing with representations of streets, we might need a perspective, some form of perspective, but it comes later.
WS: Before we start talking about different representations of space, we have to start with the concept of space that serves as a screen for the painter’s creation. There are two different stages… A wall in a cave or any kind of wall, or a canvas, whatever it is, seems to be just a surface. It needs to be transformed into a screen on which mental objects are then projected. In that sense, I am wondering, if one creates space first.
AV: There is a stage of creating one space and there is a later stage of compering different spaces. This comes much later, this kind of awareness, it’s much more advanced.