A Demonstrator's Themes

Imagination

At what point does imagination come into play? There is first the conscious effort of mind, which is the process by which we attempt to understand the complex world we live in. There are a myriad of ways by which this is achieved. The particular history, talents, and temperament of the given individual are the primary factors that determine the manner of this conscious exploration. After a certain period and depth of this effort to understand, a state of unconscious activity becomes established. This activity may continue even if the individual has ceased working. And often it is rather unexpectedly either upon taking up the activity again or simply out-of-the-blue while doing something else that an original sequence of movements comes into being or a particular insight (usually complete) thrusts itself before the mind. It is this sudden new development on the foundation of what is already known that constitutes the process of imagination, which culminates in the creative act. What characterizes the imaginative step is the appearance of an unanticipated gesture, something that is unexpected, bold, and surprising establishing a connection that was not heretofore known. The greater the degree of surprise and the deeper the connection, the greater the imaginative leap.

Inspiration

Each of us is inundated with an endless stream of sensations at every moment of our lives. On any given day, we are conscious of only a few events that stand out from the rest. The creative mind builds, over the course of many years, an idiosyncratic aesthetic filter that allows it to quickly and accurately evaluate external sensory data. This is achieved by a dedication to understand as much as possible of one's particular discipline and its position in a larger context. Inspiration is the perception of an unusual sequence of sensory data that resonates with the creative individual's aesthetic. This induces a state of mind that leads to a period of experimentation that may culminate in the creation of something original. The sources of inspiration are often carefully guarded so that they are kept pure and are not polluted by the arrival of too many eager visitors.

Process (including local, global aspects)

Unlike many artists, my process generally starts with a very strong idea, which usually takes the form of a compelling mathematical statement. The particular theorem is something that inspires my imagination to the extent that I am driven to try to understand it. A substantial bit of effort is required to appreciate the basis upon which the theorem is built and the statement of the theorem itself. But having done this preliminary work, I am ready to attempt to create a drawing from this mathematical idea. Initially, I restrict myself to the rigid limits of the mathematics thereby establishing the framework of the artwork. This translates to mathematical notation and the first layer or two of drawing that serves as the structural basis. As the work progresses, the hand and eye, guided by its aesthetic filter and the instinct born of years of practice, goes forth unconscious of the requirements of mathematical rigor. It is at this stage that creativity comes in. And it is during these moments of egoless aesthetic awareness, induced by the mechanical procedure of drawing, that the essence of the art comes into being. One can say that the whole of the process of an artist's life is to access this state of mind wherein she is more of a medium through which the art is created rather than the author of the art. For me, the entire process of creativity is one of intense initial effort followed by an effortless coming together. It is an incredibly joyful and liberating process.

Rules (separate from process though they may govern it)

An artist establishes over the years an unconscious set of rules by which his creative process operates. For me they are like the rules of math, which govern the way things are put together. At first restrictive, they become gradually familiar, natural, and flexible. With the years, these rules are wholly internalized to the point where they can be bent and sometimes outright broken. A given rule takes form when a certain procedure results in an effective means of expression. The mind of the artist takes note of this and stores it in its storehouse of fundamental processes. Likewise, if a certain rule is no longer effective, it is gradually phased out.

Though many of these rules come to being in the early period of an artist's development, the set of rules by no means stops growing. As an artist reaches maturity, the set of rules tends to stabilize with only slight tweaks that suffice to further streamline the rules ultimately resulting in the minimum set of highly effective rules. This is the trajectory of the majority of artists except for a few truly courageous visionaries. These are people who are never satisfied with any given stable set of rules, which results in an established mature style. Instead they abandon entire sets of effective rules that may have taken the effort of years to build and turn their minds to making new sets of rules to create new worlds of art. Never satisfied to rest on the laurels of their creative genius, their maverick protean nature is ever spawning new frontiers until their minds can no longer sustain the strain of such demanding creativity. These are the strong artists who are responsible for the paradigm shifts in each age.

Theory

The very nature of real art is its originality. In essence, the art of any period is the exploration of that which is not known of the contemporary human experience of that time. Once a theory has been developed about a kind of art, that art is no longer seminal and hence no longer art. A theory of art is its death knell.

Notation

In the second semester of my freshman year, my Chinese literature class was right after what must have been a graduate economics class (as it was in Braker Hall). The blackboard was filled will matrices and variables that seemed filled with mystery. I remember one day the phrase “the life of x” written on the board amongst a forest of mathematical notation. And I wondered out loud how x could have a life! A couple of my classmates turned around and laughed at me. But that little phrase planted a seed in my mind. From that day on, I wanted to understand this strange language that seemed so mysterious and impenetrable to me and perhaps more difficult than Chinese. Not long after, I began for the first time in my life to apply myself seriously to my studies. Instead of skipping classes and wandering the streets of Boston at night wondering about the meaning of existence, I immersed myself in third semester calculus. I gradually abandoned all my liberal arts classes (except drawing and painting), which seemed so random and arbitrary, for the austere rigor of theoretical math.

In Calculus III, I had to really work as I had goofed off in Calc I and II. In that one course, I learned all three semesters of calculus and got an A. Montserrat Teixidor was so patient with me in her office hours explaining to me things that probably most high school students know in Spain and probably the rest of the world. I really appreciated her help and her support.

It was during that time that I began to understand the notation of mathematics. How incredibly succinct it was. Leibniz's integral sign was the distillation of so many concepts.

Sketch (artistic v. mathematical meaning)

A sketch for an artist is a useful tool to clarify one's ideas. It is one that I use often when exploring possibilities for new work. This is probably an extension of my experience with figurative painting. I used to always carry with me a sketchbook in which I recorded scenes from my day-to-day life. During my years living in China, I filled dozens of such books from which I would glean material for my oil paintings. Nowadays, though my work is largely abstract, I continue to adhere to the sketch as a first step to a larger work. Of course, these “sketches” are not the same as sketches from life. In fact, they often don't look like sketches at all, but are instead scribbles of mathematical notation that are remnants of the working out of the details to a proof or an example. Yet the role that these “mathematical sketches” plays is the same: it serves to further develop the ideas of the work.

Curiously, a “sketch” for a mathematician is not an initial step towards a theorem or finished piece of mathematics. Instead, it is the brief synopsis of an already accomplish work of math. Thus, there is often the presentation of a “proof sketch” wherein the justification for a theorem is presented with only the broad strokes of its main ideas that serve as a plausible argument for the veracity of the proposition. What's missing are the actual details that support those main ideas. Of course, to really understand the mathematics it is imperative that the details be scrutinized, but for non-experts this is not necessary. The proof sketch is accepted and we have a vague idea of the importance of the theorem.

This difference in the definition of a “sketch” brings to light an interesting point. Whereas an artistic work is the interpretation or expression of a certain reality experienced by the artist, a mathematical work is a piece of mathematical reality. In this way, a sketch whether artistic or mathematical is derived from reality. The difference seems to be the role that a sketch plays in the two disciplines. In art, a sketch points to the work while in math, it originates from the work. And in my art, a sketch seems to incorporate both meanings—it is at once the broad summation of a mathematical fact and a step in the process of developing an artistic expression.

A deeper and more important point that this discussion of the word “sketch” unearths is the difference between artistic and mathematical creativity. While a sketch is clearly a step in the process of creating art, what plays an analogous role in the process of mathematical discovery? What are the tools by which a mathematician makes her theorems?

Communication

The impetus for both artistic and mathematical endeavors is the discovery of new patterns, representations, interpretations, and connections in reality. For the artist this either takes the form of the expression of these new revelations or is the very expression itself. By and large, the mathematician seeks also to communicate her new vision. But often the world is not ready. Both artists and mathematicians are not understood by society at large, because in their initial effort to communicate they tend to use the contemporary language when what is needed is a new idiom/vocabulary to describe what only they have seen. What ensues is perhaps not as hopeless as Eugene Ionesco's play “The Bald Soprano,” but has undeniably similar features.

Certainly, there is quite a difference in the manner in which artistic and mathematical visions fail to be grasped by the society of the time. Mathematical insight takes place in a realm so abstract and rarefied that the language used to describe it cannot possibly be understood by a non-expert. Not only is its vocabulary very different, but the absolute precision of the language, which reflects the rigor of mathematical thought, can only be mastered after a long period of intense devoted study. It is generally accepted by all that mathematical discovery cannot possibly be understood properly without sufficient training. As a result society's ability to judge the value of a piece of mathematics is virtually nonexistent. Thus, while it is usually clear to a group of experts what the great mathematics of the day are, it is understood by no one else.

On the other hand, because art springs from life, it is commonly thought to be accessible to everyone. As a result the vast majority of people who purport an interest in the arts believe with conviction that they know what it is they are seeing. This is wonderful except that almost everything is then considered art and nobody really knows what's what. Anyone who wants to can with some effort be seen or heard now and then. The situation is exactly the opposite that of mathematics. In art, everyone thinks they understand the great art of today when the truth is that no one really knows.

The futility of communication may well be reason for one to adopt the position of a solipsist following the mathematician and philosopher L.E.J. Brouwer. But in spite of his solipsism, Brouwer was driven to write, lecture, and publish his ideas, which seems to suggest that even if you are the only person in the world you need to somehow try to make a record of your having existed even if just for yourself. Curiously, with time artists and mathematicians are accepted and thought to be understood. This happens perhaps because of the effectiveness of their particular idiom or vocabulary, which then becomes adopted by the collective language. But whether or not they really are understood is probably irrelevant. In the end, art and mathematics may simply be the futile hope of authentic communication.

Beauty

Beauty is not the objective of seminal art though it is definitely a characteristic that is associated with such art when it becomes accepted into the collective oeuvre. So there is the aesthetic of the artist and then the general collective aesthetic. When a strong artist is able to lay down her aesthetic as the defining one of her era, then it becomes part of the cultural heritage. In general, the aesthetic of the creative artist cannot be called beautiful according to society's standards. Work which is deemed beautiful according to contemporary standards may often be derivative as it is not the outgrowth of a new process based on a new set of rules. Instead it is simply the recombination of what has already been unanimously accepted as good art. One prominent marker of derivative art is superlative technical skill substituting for original and radical ideas.

It is precisely the surprising radical new connections that are made stemming from the known that underlies the new aesthetic. The re-presentation of reality in this radical light, which is often not immediately apparent to most people, is what the artist may call beautiful for want of a better word.

Visualization

I began to visualize consciously when I was fourteen, which was right after I got my first oil painting set. The reason for this was that I was trying to make abstract paintings that were generalized color fields, which were probably inspired by the palettes of Rothko and Klee; I found myself continually changing patches of colors wiping them out with a cloth steeped in turpentine and then inserting another color. This process was exciting, challenging, difficult, and, when things went wrong, very frustrating. This was especially true because of the difficulty I had with the boundaries of each color that I replaced. Each color modification no matter how subtle caused a tremendous shift in the entire painting and established a new set of relationships between all colors. Adjacent colors would suddenly pop up or sink down depending on the relative differences in value and hue [needs work] resulting in a new visual deformation of the color patch.

This perpetual balancing act of color consumed me and around that time I started to visualize color fields. This happened so that I could anticipate what a given substitution might do to the painting before actually going through the painting process. I think it began by my having dreams of my paintings and upon waking I would visualize the particular painting I was working on and start making my color substitutions and imagining the changes that would take place. In this manner painting taught me to visualize.

Although I was not aware of it at the time, it was an exercise that had very mathematical features—a set of things, the relations between these things, the boundaries of these things, the sum of these things, the arrangement of these things, notions of adjacency, substitution, etc. It was only later that I realized how comfortable I was in analysis and topology because I could visualize almost all of the things we covered. It was this need to come up with an image of the math that I was studying that is in the kernel of my art today.

Representation

For artists, representation usually suggests figurative work, the traditional means by which reality is recreated in a work of art. I have been trained through years of drawing and sketching the physical world to accurately represent the reality that I see with my eyes. Although, I am currently making what looks like abstract work, I often think that it is representational in the sense that I am representing mathematical reality, which of course happens to be very abstract.

An interesting point that Prof. Sandor Kovacs brought up in his talk is that there are no originals in mathematics. In other words, there is no actual object from which something is made, i.e., there is no original basket of fruits from which a still-life is generated. So when I create a drawing from a mathematical theorem, from where does the work come from? There is a given theorem, which could be the work of a mathematician that I know or the work of some celebrated mathematician from the past. I try to understand it, which is difficult and if it is someone's current research, nearly impossible. But in the process of coming to terms with the theorem, an image begins to form in my mind containing certain salient features of the proposition. Gradually this visualization stabilizes and I begin my drawing.

Parameters/parameterization

As was clear in one of our dialogs, the word “parameter” in the vernacular has come to mean a limit. This results from its being confused with perimeter. But this is not what it means in mathematics. Thus, in our discussion, though we were all talking about parameters, there was a misunderstanding about what was being said since the non-mathematicians were thinking of one thing while the mathematicians were talking about another. This was a very interesting example of how easy it is to assume that we are speaking the same language when in fact we have very different definitions and how that can quickly lead to confusion.

In mathematics, a parameter is a variable. So when we say something depends on several parameters, we mean that a variation in any of the parameters results in a change in that something. For example, in general, a person's mood is a function of several parameters such as digestion, weather, day of the week, etc.; if say the weather were to change from sunny to rainy (while the other parameters remain fixed), there is a resultant change in mood from happy to not so happy, or if we change the day from Monday to Friday (keeping other parameters fixed), the mood may go from resigned to hopeful.

We can generalize this notion and think of each parameter as generating a particular thing. In this way, we get a “listing” of sorts. Depending on the structure of what we are observing, we can consider the manner in which a particular thing behaves as we vary a given parameter. Can we predict what kinds of things that can happen within a certain parametrization? Shafarevich's Conjecture, which Sandor Kovacs shared with us, gives us deep and interesting answers to what we can say about such things. For details, I defer to Sandor.

Communities (artistic & mathematical)

The creative spirit is perched precariously at the cusp of community and individuality. She finds root in a complex network of often intersecting communities with which she is associated to varying degrees (more often weakly associated than strongly) and derives inspiration from these connections. At the same time she tends to remain aloof preferring not to develop strong bonds so that she maintains her freedom to wander on her own. In this way, she modulates between these worlds continuously seeking for the perfect balance that will optimize her creativity.

Communities whether artistic, mathematical, or otherwise provide a foundation for its members and a place for them to get sustenance in their work. The camaraderie not only provides energy and inspiration, but also creates channels for the development of the individual and the group as a whole. In artistic circles, this often manifests in the joining of similar minds to create a collective identity that is often a powerful means of projecting a certain kind of expression. This unity in variety when it works can be a very effective force in the overall development of the field. The inherent danger of such movements is that the individual artist may become overwhelmed and subsumed by the group vision and inadvertently forfeit her true artistic nature. The need to identify with a community is best balanced by the need to differentiate. When such a balance is struck, the artist is in the perfect position to create.