On math and magic

I've been on a kick lately of re-reading my old favorite fantasy novels. I started with some of Lloyd Alexander's Prydain Chronicles, and am now going back through Ursula K. LeGuin's Earthsea Trilogy. I haven't touched this books—or anything in the fantasy genre—since my early teens, and its been interesting to see how differently I relate to them now.

...from another former obsession

One moment in particular struck me. In LeGuin's A Wizard of Earthsea, there's a scene in which a young apprentice-mage sneaks a look at his master's dusty old spellbooks and becomes transfixed by the ancient runes inside. I realized that the visceral feeling evoked by this passage (and others like it throughout the fantasy genre) is exactly what I felt as a college freshman exploring the math section of my undergraduate science library. I would spend hours at a time browsing dusty old math books, the more arcane the better, trying to decipher their internal logic. Yes, I wanted to learn new math, but I was also hooked on the feeling of being lost in these mysterious tomes. Like the mage's spellbooks, these math books contained strange symbols describing deep and powerful truths, which could only be understood through long, deep study.

A sample from a recent article of mine. Doesn't math look cool?

Reflecting back on these moments highlights how my relationship to mathematics has changed.  I was initially drawn to math because of its beauty, elegance, mystery, and because it contained a kind of absolute truth.  But after teaching for three years and studying differential geometry for one, I found that abstract beauty and truth were no longer enough to sustain my excitement.  I wanted to discover and describe important patterns in the world, not just relationships between abstract constructs.  Metaphorically speaking, I wanted to work my magic in the world, not just study it for its own sake.  This lead me to study study of complex systems and eventually evolutionary dynamics.  Mathematics has lost none of its beauty or mystery for me, but my focus now is on its connection to the world rather than its absolute, self-contained truths.

This parallels, in some ways, the differences I've noticed in the way I approach these fantasy novels now.  As a hyper-imaginative pre-teen, I wanted to lose myself in these fantasy worlds, to blur the lines in my mind between these worlds and my own.  Re-reading them now, I have no desire to escape into these worlds.  Rather I look for metaphors and themes connecting these worlds to mine. These books (and the genre as a whole) seem obsessed with the idea of power: discovering one's own power, learning about different sources of power, coming to grips with the dangers and limitations of power, avoiding the temptation to use power for evil.  As a researcher, a future professor, and simply an adult actor in this world, I have a certain measure of real-world power now that I lacked as a bookish pre-teen. In these books, I'm finding an opportunity to reflect on how to wield that power, and the responsibility that comes with it.

Perhaps the larger theme is this: I used to think I needed to escape from the world in order to be myself.  Now my goal is to connect to the world, as much as possible, while still being deeply, authentically, myself.

Inside the Mathematician's Studio

One of the aims of this blog is to give the general public a sense of what we applied mathematicians and other non-laboratory scientists actually do with our time. An earlier post addressed the content of what we do: the development and analysis of models. This post, on the other hand, will focus on process. Specifically, my process: how I actually do math.  This post is a joint project with my partner Anna, whose beautiful sequence of illustrated text about the nature of the creative process appears on her site drawmedy.

Of course, there are many aspects of what I do. Activities such as reading through the literature, meeting with collaborators, and writing up results, don't require much explanation. I focus here on the parts of my job that makes me feel most like a mathematician: coming up with new ideas and developing them into mathematical arguments.

It starts with a problem. Most often I'm trying to prove some result of the form "In this model, under these conditions, this kind of behavior can arise". Sometimes these questions can be addressed using textbook-style sequences of steps, or even using programs like Mathematica. But such straightforward solutions don't interest me as a mathematician, and I like to leave this kind of work to other people. What really makes me come alive are the questions for which new mathematical approaches must be conceived.

This is an inherently creative process. There is no way of knowing at the outset what the solution may look like, or even whether a solution will be found. All you start with is your toolbox of mathematical techniques, and some hunches about which tools might work if applied correctly.

From this starting point, it's a process of trying approaches, failing, trying other approaches, asking questions, re-framing the problem, working out simple examples, and trying to make connections between different areas of my knowledge. This process plays out in pencil scratchings on my bound notebooks, two pages of which I've reproduced here:

These two (non-consecutive) pages show some of my musings on Prisoner's Dilemma games played on networks.   On the first page I'm mainly working through some visual examples.  You can also see some of the general questions these examples inspired. ("Maybe this is all about...")

The first half of the second page shows me asking questions (indicated by the Q:) and formulating hypotheses about how different models might be connected.  I typically jot down my thoughts in real time as they occur to me, so that it almost feels like journalling.  I tend to write in complete sentences, but sometimes a thought will end mid-sentence as something else occurs to me.  I'll also go back and write in the margins (e.g. the circled questions at the top right of the second page) if I have an idea that connects to something I wrote earlier. 

The second half of the second page shows some calculations as I test one of the hypotheses generated above.  Note the circled line with the words "NOT TRUE" to the right.  Mistakes and retractions are ubiquitous in my notebooks (as they probably are in the scratchwork of most mathematicians).

My favorite position for such notebook-scribblings is reclining in a couch or comfy chair, as Anna deftly illustrates:

I tend to get antsy when sitting upright for too long. In fact, I'm a big fan of changing scene in general. If I'm stuck in one room with no good ideas, I'm liable to go searching for another room to work in.  Perhaps this helps me get a new perspective on what I'm doing, or maybe it just stops frustration from building up.

I should add that many of my best ideas actually come in the shower, or jogging, or in other situations where my brain has the time and space to chart its own course.  Other mathematicians I've spoken to share this experience.  If you've been focusing on a single problem for long enough, it can seep into your subconscious, which may continue to generate ideas even when you're doing other things.  Back in college (when I was a pure mathematician) I even got to the point of solving homework problems in my sleep, though the sleep was not exactly what you'd call "restful".

I'll end with a call to other science bloggers and writers.  The Paris Review, since the 1950's, has conducted a series of interviews with world's preeminent writers on their process: how they generate their ideas and shape them into finished pieces of writing.  Collectively, these interviews have helped shape public perception of writing as an occupation, and illustrated the variety of methods that writers employ.  In this age where science is increasingly misunderstood and distorted in the public eye, I think it would be powerful to have a similar series of documents illustrating the daily processes of scientists.  So I'd encourage any science bloggers/writers reading this to consider expressing your own personal "scientific method" to the general public, and pass the word along!

A Mathematician's Apology

First, the apology: I started a summer job last week and it's taking up a huge amount of time. So posts will be infrequent until my job ends in August.

But since it's on my mind, I'd like to share a bit about this job. I'm assisting in the PROMYS for Teachers program. The goal of the program is to give math teachers an experience similar to the way mathematicians do math.

We mathematicians approach math differently from the way anyone else does. Most people learn math by watching a teacher explain a concept and demonstrate some examples. Students then apply these concepts to some practice problems, and that's pretty much it.

Mathematicians are in the business of discovering new mathematics, not reproducing what is known. To do this, we do what all scientists do: we experiment. Except that for us, experimentation involves only a pencil and paper (computers are sometimes used, but not as much as you might think.) We take numbers or shapes or other mathematical objects and play around with them. This can be a frustrating and fruitless process, but eventually we hope to discover something interesting about the way these objects work.

The next step after discovery is to describe our discovery. This is harder than it sounds, because mathematical language is very precise. One's first few attempts to describe a discovery are often wrong in some way; perhaps an important qualifier has been left out. Sometimes we have to invent new language to describe a discovery.

Finally, we try to justify our discovery by proving it from first principles or from other established theorems. This process can range from easy (a few minutes of thinking) to moderately hard (a few weeks) to epic (a few centuries.) The most famous mathematical theorems come from discoveries that are simple to describe but surprisingly difficult to prove.

In the PROMYS for Teachers program, we try to give teachers a taste of this experience. We give them numerical problems that hint at deep mathematical patterns. We then ask them to describe these patterns precisely and prove them if possible. This is often frustrating for them, since they haven't been shown how to do the problems or proofs beforehand. But by making their own discoveries, they take ownership of the mathematics, and when the process works it is tremendously exciting.

This program is modelled on the Ross program at Ohio State University, which I attended as a high school student. That program basically made me into a mathematician, so it is very rewarding to me to be able to share this experience with teachers.