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!
Kurt Gödel's Open World
1 day ago in The Curious Wavefunction