**Joan Grubin: ****Musings on “Logic and Structure”**

Full disclosure: I know nothing about contemporary mathematics. Although I often use arithmetic in making calculations for my paper constructions, I suspect arithmetic is to mathematics as a 12” ruler is to a supercomputer. So the invitation from the Painters Gallery to participate in an exhibition of work made with the intention of exploring the relationship between the language of art and the language of mathematics intrigued me.

The catalyst for the show was the mathematician Roman Kossak’s “Logic and Structure”, a short text laying out some heady ideas about mathematical language, and proposing an art project. Reading it made my brain ache to the point of near-explosion with the effort at comprehension. But I was fascinated by the idea that there is a language of math capable of describing any physical object with a formal system of signs and symbols. This idea seemed to share a surprising kinship with art, which also has a language and vocabulary of its own.

I soon found that ignorance coupled with curiosity can stimulate the imagination!

Plus, I just like saying the words “logic and structure”. They feel good in the mouth, and imply something toothsome and substantial, like *al dente* pasta.

The curators challenged the artists selected for the exhibition, of which I was one, to make something that could be subjected to mathematical analysis. The simple guidelines required only that the image or object be comprised of elements (“…points, lines, regions, particular shapes or objects”), and some relationships between them. That seemed doable: although to me a work of art is more than the relationship between its parts, this project was an effort to explore ways in which a work of art could be analyzed in mathematical terms. The deal was – the artists provide the work, the mathematicians would see what could be “logically expressed about the structure” of such singular “visualizations”.

Like many artists, I have a hard time following the rules. I devised a series of circles constructed out of paper, each with the same circumference. My first circle had nested concentric elements; the second was a circular single plane of woven paper, and the third became an irregularly woven curved half-sphere. What I was making seemed to have a logic of its own that veered away from the rules. But if my three circles were considered as elements of a work in three parts, then it might be possible to describe the relations between them in mathematical language.

In the end I came away thinking about the meaning for art of the concepts of “logic and structure”, and how an art work is convincing to the degree it develops an internal logic and structure that drives its coming into being. Neither artist nor viewer needs to quantify, or even be fully conscious of it, but I know that in the process of making something, when an underlying logic begins to emerge, it feels like you’re on the right track! It’s as if an art object has an intelligence of its own. It starts to tell you what it wants to do. It’s not a manifestation of logic in the sense of reason as the basis for decision-making, but more an intuitive apprehension of an intrinsic organizing principle that is governing the thing you are making.

Perhaps the notion of an “intuitive logic” is related to what artists mean when we speak of “voice” – that ineffable but indisputable unique aggregate of qualities, signs, marks, images, and tone that emanates from the work – what Alex Katz calls a work’s “inside energy”.

So maybe the logic governing mathematical thinking is rational, while the nature of another kind of logic underlying artistic thought bends toward the intuitive. But maybe not.

It makes me wonder how intuition plays a role in mathematical thinking. I’ll await a mathematician’s response.

Joan Grubin

**Roman Kossak’s response**

Intuition plays enormous role in mathematics. Mathematicians analyze structures. Those structures are rigorously defined and we know some of their basic properties, but by far not all. The goal is to describe them as fully as we can and to classify them. We are in a sense “looking at them” waiting for some inspiration that will allow us to notice patterns and regularities. There are no predesigned strategies that would help us. One has to find one’s own way. If a structure we interested in is infinite, there is no way to visualize it in its totality, but we can imagine and draw finite parts of it. Then we try to mentally put all separate pieces together. This is not easy, but mathematical logic helps. It does not offer recipes for thinking, but it provides a language that guides us in attempts to see structures: What are the elements? What are the most important relations between them? Can more complex relations be reduced to simpler ones in an intelligible way?

We are not computing much, we are not applying rules, formulas, and patterns. We are constructing and searching. Are we artists of sorts?