At the time I was attracted to pure science — physics — where you could speculate and be creative. It’s equivalent to being an artist. If you get the chance, and the cards fall right, there’s no difference. The intellectual play and spirit are the same.
— David Byrne (interviewed by Timothy Leary), 2000
I’ve commented more than once here on the myth of the Mad Scientist: contrary to popular belief, there are no easy shortcuts to scientific greatness. It’s true that some of our creative processes are subconscious, that we sometimes come up with ideas on vacation or after a good night’s sleep. No one, however, becomes a great scientist by just sleeping a lot. Our subconscious faculties only become engaged after we’ve studied the problem and thought about it extensively, often to the point of exhaustion. They don’t kick in every time, and when they do, their input is not even always useful. (I’ve woken up many times with shiny new ideas that did not hold up on inspection.) Excitement, inspiration and quality vacationing can make it easier to put in the sustained, disciplined work of constructing correct and complete mathematical arguments, but does not replace it. As for the relation to actual mental illness, I’ve linked before to a relevant interview with John Nash.
I didn’t get any disagreement on that from math and science types. We understand well enough how the creative process works. We know that being all fired up to prove the Riemann hypothesis is different from actually doing it. Imagine my surprise, then, when I attended a discussion on art and science in the “Philosopher’s Cafe” series a few weeks ago. Scientists and mathematicians came in good numbers, and many of them professed exactly the same kind of misconceptions about art that they would dismiss outright with regard to science. Art, if you believe them, is all about feelings. When a work of art evokes strong emotions, we assume that the artist was overwhelmed accordingly at the moment of creation, leaving no room for intellectual mediation or for calculated, deliberate activity. In other words, the artist experiences an intense feeling, whips himself into a state of rapture, and bang, a painting or whatever materializes in front of him in a puff of magic dust. The Mad Artist swipes his cape and takes a bow, in all his fictional glory.
I’m a research mathematician of some renown. (The regulars here know that, but I’ll say it explicitly anyway, for those who might find this post via links and google searches.) I’m also an amateur photographer (see my Google+ page for samples), and I’ve been attracted to visual arts all my life in some way or other. I’m finding in my own practice that the creative processes in art and in mathematics are often more similar than it might first appear, and I’ve had plenty of confirmation of that from both sides of the aisle. This post is about that, with emphasis on the mathy and sciencey side of art. (Time permitting, there will also be a follow-up post in the converse direction.)
This is not a post about “mathematical art.” Honestly, I have little interest in most of it. I write research papers about fractals, but I find neither mathematical insight nor artistic value in the rainbow-coloured pictures of fractals usually found at math art exhibitions. Don’t even think about sending me links to math rap songs, either. I don’t need art to talk to me about mathematics. I want it to speak to me as art, on its own merits, with no special bonus points for math themes or content.
I’m interested in the less obvious but more organic similarities on the level of the creative process. I’m hardly the first to observe them. Just last year, I attended an artist talk where a painter spoke of his work in terms of “solving the mathematical equation.” Yet, it was plainly in evidence in that discussion a few weeks ago that too many scientists think of art as a softer, lower grade kind of creative endeavour where the concepts of logical thinking and problem solving are pretty much unknown. In that regard, here are a few points to consider.
I’ll be talking mostly about photography, and to some extent about painting, because that’s what I know best. If you think it’s different in other arts, I’ll refer you to Ursula Le Guin’s excellent description of a physicist’s creative process in The Disposessed; I can’t find a link now, but I recall reading somewhere that it was based on her own experiences with writing. If you think that it’s just me thinking that way, that’s very easy to check. There are many artists out there who have blogs, public Facebook or Google+ pages. They might post pictures of work in progress, talk about their influences, recount how a particular piece came about. They might be using different, less “scientific” language, but you will still find a good deal of premeditation, problem solving and analytic thought in what many of them do. And if you tell me that not all art is that great… well, yeah. Not every math paper is a towering pinnacle of intellectual achievement, either. We all do what we can.
Knowing your subject. This is pretty basic, but should be mentioned anyway: an artist needs to know his or her subject. How well? Ancient Greek sculptors spent centuries studying human anatomy and learning to depict it in statues. Both Leonardo da Vinci and Michelangelo studied human anatomy by dissecting corpses. Leonardo in particular produced hundreds of pages of notes and drawings, including studies of the human skeleton, muscles, internal organs, the cardiovascular system. He documented facial expressions, effects of age and illness. He was not just both an artist and a scientist; his art and science blended into each other, both being manifestations of the same curiosity about all things human.
Here’s a less known but perhaps even better example. Take a look at this gallery of paintings of horses by the Polish painter Juliusz Kossak, a horse buff who spent a great deal of time hunting, visiting stables, even hanging around abattoirs in order to study equine anatomy. Note in particular the third image, from 1868. Why is the date important? Because at the time, it would have been disputed whether this pose was even possible. There was no agreement on the exact details of a horse’s trot or gallop, the motion being too quick for the human eye to break down. Most artists painted a horse in trot with at least one foot on the ground, and in gallop with both front legs extended forward and both hind legs extended to the rear. Eventually, however, the arguments were settled by a series of photographic studies starting in 1872. Kossak was totally vindicated. The artist beat the scientists to figuring it out.
Seeing. When you paint or take photographs, what do you want to depict? If “what I see” is what you’re thinking, here’s why this is not really an answer.
The human eye does not work like a camera lens. It is, in fact, not an especially good optical instrument. The image formed on the retina is at best fragmented and incomplete. The detailed, continuous picture we think we see is believed to be a product of our brain instead, stacking and editing multiple images, combining optical input with prior knowledge and educated guesswork. This process is known as “unconscious inference,” a term that goes back to Hermann von Helmholtz in the 19th century.
Exactly how this happens is not yet fully understood, but several partial theories have been proposed. Gestalt psychology suggests that we organize visual data according to patterns such as similarity, closure or continuity. We group similar objects together. We perceive geometric shapes (circles, triangles) as complete even when parts of them are missing. We choose to see them as continuous rather than broken, symmetrical rather than arbitrary. Well known optical illusions corroborate the theory. M.~C.~Escher’s paradoxical drawings invite the viewer to indulge her temptation to attach a familiar meaning to the visual stimuli, then turn the tables on her when an impossible object is revealed.
We classify visual input according to broad templates and interpret it based on probabilities. Often, we only take a quick glance at an object before we allow the part of our brain that handles sensory perception to go on autopilot. We expect a house to have a door, windows, roof, so we take them for granted. Afterwards, we will be sure that we have seen them, but we will have no recollection of their shape or colour. On the other hand, unlikely or incongruent events tend to go unnoticed even when we believe that we are paying close attention. The Simons Lab at the University of Illinois at Urbana-Champaign has a series of mind-blowing videos demonstrating just how much visual information we can miss. People walk in and out of the room, objects get rearranged, one actor gets substituted for another in mid-action, and we remain oblivious to it all, never realizing that anything has ever been out of order.
Researchers have examined and confirmed the notion that our perception of the world depends strongly on our relationship to it. In experiments, hills appear steeper when we are carrying a heavy backpack, and distances seem larger when we are cheated into perceiving our bodies as smaller.
There are lessons here for the photographer. For one thing, it explains many cases where my photos did not turn out the way I’d expected: I had made lazy assumptions and did not take the time to really look at what I was shooting. “Look with your eyes,” Syrio Forel tells Arya Stark in Game of Thrones, and indeed when I do that, I see more. Fortunately, it is a skill that can be learned and practiced. But this is also where we get to art as problem-solving.
Solving the puzzle. What should a photograph capture? In light of the last few paragraphs, there is no reason for the answer to be obvious. Should we aim for “objective” truth, whatever that might be? Should we maximize the amount of information included, try to capture the mood, account for our own feelings? Should we try to make the hill look steeper depending on the weight of our backpack?
Would an unedited image shot on automatic settings be “natural”? Consider for instance what happens when some parts of the scene are much brighter than others. Our eyes compensate for this, adjusting the aperture constantly by dilating and shrinking the pupils, so that we see the details of both the bright and the dark objects. In photography, though, the aperture is fixed. To mimic the way we process the scene in real life, the photographer might edit the image or blend several exposures. She might also, however, leave the photo unedited, with parts of the scene underexposed. That could still be a fine image, but it would not necessarily look more “natural” to us than the edited one.
We don’t even look at the photograph the same way that we look at its subject. Take landscapes. In real life, a landscape surrounds us, and we explore it gradually as the eye wanders from one object to the next one, changing focus, taking the time to cross the spaces between them. All of this can be lost easily in a small two-dimensional image. Good photography restores the order and the empty spaces that would otherwise collapse in the loss of a dimension, establishes and explores visual relations between objects. It does so through the tools of art: framing, composition, light, focus, colours, contrast.
Artistic creativity and vision are forced on us when we try to simply depict the world as it is and discover that this, in fact, is generally not possible. Any rendering of the view before us as a photograph is a format conversion that calls for artistic choices.
Here’s a very small example to illustrate what I have in mind. The photo at the top of this post shows the view from Reef Bay, Mayne Island, with Mount Baker in the background. Mount Baker is about 110 km from Reef Bay in a straight line. It looks like a mirage, as it probably should (I’ve made a back-of-the-envelope calculation to the effect that, while in principle it could be visible from that far, it should not be as prominent as it is). It’s just barely visible in most photographs (trust me, I’ve tried), a vague outline blending into the background; nonetheless, when you’re there in person, it attracts attention to the point of dominating the landscape. How can I get that on camera?
The first point is that the photographs are probably true to the actual optical reality. Our own in-person perception is far more subjective; if our attention centres on Mount Baker, that’s likely because we’re so eager to place it there. The second point is, how to induce a similar effect in a photograph? This is where the knowledge of Gestalt theories, optical illusions and other such can come in handy. In this particular case, when I saw the bleached drift logs by the entrance to the beach, I figured I had what I needed. With the two whitish objects near the bottom of the image, the viewer would look for a third one higher up as a counterpoint, and sure enough he would find it.
I’m mentioning all this because it’s not necessarily obvious, to a bystander, that this aspect of art even exists. I’m not sure that I understood it myself before I took up photography. But there it is, presenting you with a puzzle, demanding your problem-solving skills.
Once we realize this and accept that there is, in fact, a puzzle to solve, it’s easy to draw broad parallels with other endeavours requiring similar intellectual work, from mathematics research to occupations usually not thought of as creative. There’s inquiry and curiosity. There’s constructing a solution, reconciling competing demands, making the pieces fit together. There’s the same process of gradual improvement over time, setbacks, learning from mistakes, getting better with practice.
It also becomes apparent that, contra the Mad Artist, art is work that often involves planning and strategizing. I’ve learned a good deal from both reading up on science related to vision and spending time around art and paintings. My knowledge of the subject, mostly the Pacific Northwest and its nature, informs the process, starting with where and when I choose to go shooting. I have, for example, driven to specific locations at specific times of day and year because of prior knowledge of what the light would be like. There’s room for happy surprises and unexpected coincidences, but to quote Louis Pasteur, “chance favours the prepared mind,” in photography as it does in science.
On the shoulders of giants. Here’s another misconception: unlike researchers, who always build on the work of others and contribute to a greater body of knowledge, artists create each piece as a self-contained object, pretty much independently of everything else in the universe. That, at least, is how I understood what some of the folks at Philosopher’s Cafe were saying.
I suppose an argument could be made that we don’t need to study the history or theory of art to enjoy a particular piece, but even that might be a bit flimsy. Concepts such as beauty and harmony are very much culture-dependent (our modern culture has one ideal of female beauty, but others have been known to see it differently), and additionally, each art form has its own conventions that can be impermeable for those outside the culture. I don’t, for example, understand kabuki theatre. Along the same lines, someone else might be looking at the Mona Lisa and not seeing the big deal.
On the artist’s side, though, the argument breaks down altogether. If we view art, as I do, as (among other things) the study of human perception of reality, then it only makes sense to rely on the accumulated body of knowledge. Today’s painters don’t spend their time reinventing the wheel linear perspective. They can draw on what’s available. It doesn’t have to be a matter of deciding “I want to paint like X.” More likely, it’s “I want to paint seascapes. Now, X, Y and Z also painted seascapes, and I really like the way X and Y did it, so I’ll look carefully at what they did just so I have a starting point.” This is much like what we do in math research: when we start working on a problem, the first thing we often do is find out who has worked on similar questions, what they proved and how they did it.
Photography did not even exist until the 19th century, either as art or as technology, so one might think it was totally invented out of whole cloth at the time. Except that’s not true. It is all but forgotten today that early photography aimed explicitly to imitate paintings. This was done through composing images carefully, adjusting focus and exposure, manipulation of colour and contrast in developing the negatives. Long before the invention of Photoshop, photographs were being airbrushed and had details sketched into them. Multiple negatives were combined into one photograph, often through literal cutting and pasting. Edward Steichen, often considered to be the first fashion photographer in history, wrote in 1903:
In the very beginning, when the operator controls and regulates his time of exposure, when in dark-room the developer is mixed for detail, breadth, flatness or contrast, faking has been resorted to. In fact, every photograph is a fake from start to finish, a purely impersonal, unmanipulated photograph being practically impossible. When all is said, it still remains entirely a matter of degree and ability.
Both painting and photography have evolved a great deal since then. And yet, the rules of composition in photography are basically identical to those in painting. (Abstract paintings possibly excluded, so that we’re comparing apples to apples.) Incidentally, if you do click through or look for other similar links, you’ll find that many of the rules are reminiscent of the Gestalt theories I mentioned earlier: using leading lines, geometric shapes, repetitions and patterns. You’ll also find that painters did all this and more long before the term “Gestalt psychology” was even coined. I learned a good deal of it just by having been exposed to the European art canon since I was a little kid.
The “rules” are only a toolkit. They’re not absolute, and they’re pretty much worthless when not backed up by genuine intellectual and emotional engagement with the subject. Cheap prescriptivism doesn’t work. This applies especially to the more “mathematical” rules, such as the rule of thirds or any similar numerical recipes. I’ve seen arguments to the effect that such recipes should make the resulting work more interesting to me as a mathematician, because I should like numbers and equations, shouldn’t I? In reality, they’re just as likely to do the exact opposite. But I’ve said that already.
How do we know, then, which rules to follow and which ones to ignore? If you substitute “method” for “rule,” that’s the same question that calculus students ask us every year in class. The answer is the same, too. It’s analyzing the problem until one truly understands it. It’s practice, experience, judgement, trial and error. It’s something that I, as a mathematician, can actually connect with.
(Revised on September 3 because I was not happy with the middle part.)
The Accidental Mathematician 
