logic convers abab

Online Project: Conversations

Below are transcripts of informal conversations that took place in Warsaw in early spring of 2018. The first conversation was about the notion of space in mathematics. What is it? Is there “a space” in general or 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. We talked about mathematical objects and their role in constructing mathematical spaces and getting to know them. The way in which mathematicians talked about the space made me wonder if a similar approach could be possible in the artistic practice. Could the relations between elements of a future artwork be considered beforehand, without a preconfigured idea of the space which they will inhabit? This hen-and-egg problem was the subject of the second conversation.

— W. S.

 

What Comes First: A Conversation at the Kitchen Table

Roman Kossak: 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.

Wanda Siedlecka: So, you start from the object.

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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 the 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: whether 4-dimensional time space which curves this way or that way, or, if you go to quantum physics, Hilbert’s space, so you are faced with a variety, a diversity of spaces, it’s not “the space” where things happen but different spaces, and then there is a next step when you put all those spaces together and study them together, how they differ.

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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 objects, they came first, then we 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?

kitchen-3b

Andrés Villaveces: 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 together 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 comparing 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.

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AV: You start seeing a lot of things by putting a lot of triangles together, and also maybe by putting together lines, parabolas, and circles. You put them all together and start making them interact, not focusing on one triangle, or seven, or seventeen, 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, actually appears only from the interaction of infinitely many lines.

ogniskowa

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

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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 there was a fish here or there.

WS: I think that in order to paint on the wall of the cave, you need to come up with an idea that this is the space for your objects.

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AV: The sense of space seems to be inner in the early paintings, they don’t question it, it’s there for telling the story. What is this space? It’s like water for the fish, 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 heightened 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 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 a 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 comparing different spaces. This comes much later, this kind of awareness, it’s much more advanced.

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