the mathematical science of the infinite

For reasons that I cannot yet fully articulate, I am trying to understand set theory. Trying and failing. How is it that I simply cannot comprehend even the most basic abstract mathematical concepts? I read a sentence or two. Read them again. And back to the beginning. I miss the logic from one statement to the next. 

Frustrated, my mind wanders. I find myself thinking about Good Will Hunting and then that Russell Crowe film where he plays a mathematician. I think about what it must feel like to grasp some difficult abstraction intellectually for the first time. I have no recent relevant experience to compare this to, but I think perhaps it must be like the flush of erotic excitement? An overwhelming hunger satiated fleetingly: a bright hot flash and then a lingering pleasantness as the knowledge, this new, deeper level of understanding seeps into the cells. 

Despite my limited understanding, I find the language of mathematics beautiful. Consider a few of the topics in the articles I find on set theory:

The Continuum Hypothesis; The Axiom of Choice; Independence Proofs; Natural Numbers; Cardinalities of Sets. 

“Set theory is the mathematical science of the infinite,” one post in the Stanford Encyclopedia of Philosophy begins. Oh yes, I think. I want to study the science of the infinite. But when I return to the page, I am flummoxed.

As any object of our study, the ordered pair has to be a set. It should be defined in such a way that two ordered pairs are equal if and only if their first coordinates are equal and their second coordinates are equal. This guarantees in particular that (ab) ≠ (b,a) if a ≠ b.

Definition. (ab) = {{a}, {ab}}.

I followed until the definition. Why is the second coordinate now represented {a,b} and not simply {b}?

“In any case, it seems obvious…” the article goes on, but sadly, it is not obvious to me and I question my ability to think well and deeply about anything at all. How might I have this understanding inserted into my brain? How have I gone on this long without the ability to grasp these concepts? What re-wiring would it take for this to make sense to me?

Why is this important? It is important to me now because the language is important.

In the new project I am working on, I envision three sections, the last of which is Set Theory. I encountered the term in my research and recognized that it followed a pattern which could include the other two section titles (Problem Sets, Data Sets, Set Theory). The three titles together seemed to suggest a kind of linguistic progression and, as far as content, could be interrelated. Each section (or “element” in set theory) could in fact be convincing as a “member” (set theory, again) of the same set.

It is a way to understand the project, too, I think, the scope of it.

Theresa Hak Kyung Cha (1951-1982) was a Korean-born American visual artist, filmmaker and writer best known for her book Dictee, published in 1982 in which she weaves together biography, history, poetry, found texts, and images to tell a story about alienation, dislocation, memory, and about the limitations of language itself. She spent most of her life in the San Francisco Bay area, then moved to New York in 1980 to further her career, which had begun developing momentum. In November of 1982, weeks before Dictee was released, she was attacked, raped, and murdered by the security guard who worked in the building where she was meeting her husband, photographer Richard Barnes. They had been married earlier that year.

I was introduced to Dictee in 1998 by Carole Maso, with whom I was studying. I was struggling with writing about my mother’s recent death and about my own coming of age as a Korean-born adoptee raised by first-generation immigrant parents in a working-class suburb of New York City. Cha’s concerns – the limitations of language, separation from one’s own history, the representation of women’s lives and the role of religion and of myth – had powerful resonance.

Like many, I read about the short life and work of photographer Francesca Woodman (1958-1981) before I had seen many of her photographs. Woodman was raised by artists, who imprinted her with a seriousness of purpose about the making of art. She studied at the Rhode Island School of Design in Providence where I have lived for the past two decades. She moved to New York in 1979 to take up her art career in earnest. In January of 1981, she took her life by jumping from the window of her apartment building.

I was drawn to Woodman at first by the mythology that had developed around her, but soon found her photographs to have resonance with formal artistic concerns of my own – representations of self, representations of women’s bodies, stillness and motion, light and shadow, disappearance and erasure.

At some point, I realized that Cha and Woodman were living in New York at the same time. This is not in itself remarkable in a city of millions, but I began to imagine them, walking the streets of their city, working, dreaming, their lives unfolding. I do not know that they ever met, but I hold in my mind the image of a heat map of the city, with the spaces around each of them glowing red, pulsing.

Common themes can be found in their work and they seem to have shared some influences – French philosophers and writers, filmmakers. Both women worked almost exclusively in black and white, and used their bodies as vehicles to creative expression.

I do not mean to compare and contrast their work, or to draw some grand conclusion about them as artists or as women. I am not particularly compelled by Cha’s status as victim or in Woodman’s psychological state. My primary interest is that their lives were cut short at a time when their creative work was blossoming, at a time of great possibility, so that we are left to puzzle over what is left behind – these fragments, these incomplete stories – and to imagine what might have been.

What I am working on is an attempt to take these pieces of what is left and hold them near each other, to explore what the spaces between and around them might yield. I think that as someone who lives with incomplete information about my own story, my own history, I am particularly compelled by working with fragments – with rupture, disappearance, erasure – and by attempts to invoke some imagined possibility that the reality of life (and death) renders impossible.

As to set theory:

The language of set theory is based on a single fundamental relation, called membership. We say that A is a member of B (in symbols A ∈ B), or that the set B contains A as its element. The understanding is that a set is determined by its elements; in other words, two sets are deemed equal if they have exactly the same elements. In practice, one considers sets of numbers, sets of points, sets of functions, sets of some other sets and so on. In theory, it is not necessary to distinguish between objects that are members and objects that contain members — the only objects one needs for the theory are sets. 

These women’s lives. The work they made, the influences they had, the artistic concerns; their vision and ambitions. The cities they lived in. The languages they spoke. The city in which they died. I think there must be a way to represent these as interrelated sets and that there is something to be learned from articulating the properties of the sets they inhabited. If the  ultimate goal of set theory is “to describe the structure of the mathematical universe,” there must be room, beneath this mantle to evoke and observe the work of Cha and Woodman, to build an armature to support close examination of their lives.